A dark blue book cover with a slightly worn appearance. 7 A small, circular logo with a stylized 'A' in the center. TREATISE ON ASTRONOMY. BY SER JOHN GUY DE BOSCHEL, KNT. GUELP. FELLOW, M.A., F.R.S. F.R.S. M.E.C.P. MEMBER OF THE ROYAL ACADEMY OF SCIENCE OF PARIS, AND OTHER FOREIGN SCIENTIFIC INSTITUTIONS. PHILADELPHIA: CAREY, LEA & BLANCHARD. 1834. 1860. 9. 13. 12. 8. 12. 2. 4. 7. 1852. HARVARD UNIVERSITY LIBRARY logo **HARVARD UNIVERSITY LIBRARY** * Et quodam adhuc natura vigilantis legemus hominibus ut inventiam, quod faciliter et brevi tempore recus cures, idem quid in cato fito, sine avena: his initio inducunt omnia vera diligentia; id est, fidélis, simplicitas, constatae et non abhorsura. Cicero, de Fin. Ben. et Mal. ii. 14. And furthermore as nature itself has implanted in man a craving after the truth, so we have also implanted in him a desire to know that which, when unapproved by care, we delight to know even what is wrong in the eyes of others; and therefore we must love all such for in them we see that which is true, whatever is faithful, simple, and consistent; while we hold in abhorrence whatever is empty, deceptive, or untrue. 2 MICROFILMED AT HARVARD 23 2/15/ CONTENTS. **INTRODUCTION** 7 **CHAPTER I.** General Notices—Form and Magnitude of the Earth—Horizon and its Dip—the Atmosphere—Reflection—Twilight—Apparatus for Measuring Solar and Meteoric Heat—The Sun's Distance—Gath- ing an Idea of the Distance of the Stars—Definitions Of the Nature of Astronomical Instruments and Observations in gen- eral—Of the use and effect of the pendulum—Of the Use of Clocks, Chronometers, the Transit Instruments—Of the Measure- ment of Time by the Sun's Shadow—Of the Use of the Transit Instru- ments designed to find the Position of the Moon's Circle—Determina- tion of the Length of Day by the Pendulum—Artificial Horizons—Artificial Apparatus for measuring with reference to the Equatorial—Altimeter and Anemome- ter—Of the Transit and Reflecting Circle—Principles of Report **CHAPTER II.** Of the Figure of the Earth—in its exact Dimensions—in its Form that of a Sphere—and its Ellipticity—On the Rotation and Dynamo- tic Force on the Surface—and dynamic Measures of Gravity—the Pulling Power of the Earth and its Effect upon Bodies in Space—the Wind—Tide-wind—Determination of geographical Positions—Of Magnetic Induction and its Relation to the Figure of the Earth—Of Map-Projections of the Sphere—Measurement of Heights by Barometer **CHAPTER III.** Of Geography. Construction of celestial Maps by Observations of eight Ascension and Declination-Colossal Papers distinguished into Zones and Sections—the Celestial Sphere and its Parts—the Heavens—the Milky Way—the Zodiac-of the Ecliptic—the Cele- stial Latitude and Longitude-the Equator-the Equinoctial-the Tropic-Astronomy-Uranographical Problems **CHAPTER IV.** A diagram showing a celestial sphere with various lines and points indicating different astronomical concepts. 3 # CONTENTS. ## CHAPTER V. ### The Moon's Motion. Apparent Motion of the Sun with respect to the Apparent Diameter and variable Variation of its Distance concluded—Its apparent Orbit—Ecliptic and Equinoctial—The Apparent Diameter of the Sun—Description of Apsees—Pandulae of the Sun's Apparent Diameter and Magnitude—The Apsees of the Sun's Apparent Diameter—Perihelion of the Earth's Axis—the Seasons—Heat received from the Sun in different latitudes—The Moon's Apparent Diameter 176 ## CHAPTER VI. ### Of the Moon's Apparent Period—in its Apparent Diameter—Its Pendu- lum, and real Diameter—Free Approximation in its Orbits—An Ellipse, and its Converse—Orbital Inclination—Motion of the Nodes of the Orbits—Oscillation—Solar Eclipse and Lunar Eclipse—The Solar Eclipse of 1803—Lunar Eclipse of 1806—The Moon's Appar- ent Libration—Appearance of the Earth from it 183 ## CHAPTER VII. ### Of terrestrial Gravity and Laws of Gravitation—the Earth's Orbital Path—a Circle, and its Converse—the Moon moved in its Orbit by the Earth's Gravity—the Sun's Orbital Path a Circle, and its Converse—the Gold of the Earth must fall on it in accordance with these Laws—the Force of Gravity at any Point in Space is equal to the Product of that Force of Gravity at its Surface—Disturbing Effect of the Sun on the Moon's Motion 221 ## CHAPTER VIII. ### Of the Moon's Apparent Motion. Apparent Motions of the Moon—Their Stationary and Reproductive nature—the Sun their natural Centre of Motion—the Moon revolves round this centre in an elliptical orbit—the Sun moves in an elliptical orbit round this centre in such a manner that the Sun is always in one focus of this ellipse—the orbits are always circular or ellipses—and their inclination to each other—Ecliptic Ellipse, and its Converse—the Law of Planetary Distances—the four circular or el- lipses-Physical Proportionate observables in each of the Planets 251 ## CHAPTER IX. ### Of the Moon, as a Satellite of the Earth-General Proximity of Satel- lites to their Planets and consequent Sublimation of their Atmospheres-their motions and their periods-their distances from their Planets-Minimunm of Kepler's Laws in the secondary System-the Moon's motion determined by observations made by Galileo-the Discovery of Light discovered by their Means-Satellite of Saturn-Oil Universe 278 CONTENTS. 6 CHAPTER X. OF CORPUS. Great Number of mental Corpus—The number of unconnected probably much greater—The number of mental Corpus—Corpus without Tails—Increase and Decay of their Tails—Their Motions—Sub- jects in the State of the Mind—The State of the Mind in the State of Drift—Periodic Return of certain Corpus—Halley's Keck's Ve- racity—The Periodic Return of certain Corpus—The Law of gradual Decay, and possible Depression in Space. - 354 CHAPTER XI. OF PERVERTATIONS. Subject proposed—Perpetuation of small Motions—Problem of three bodies—Elimination of disturbing Forces—Motion of Near Bodies—Action at a Distance—Newton's Theory of the Universal Revolution of the Solar System—Lagrange's Theorem of the Stability of the Inertial System of Axes—The Inertial System of Axes—Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of the Equations on the Inertial System, resulting from the Action of # CONTENTS. **CHAPTER XIII.**
Of the Calendar Page
Synoptic Tables of the Elements of the Solar System 361
Synoptic Tables of the Elements of the Orbit of the Satellites, as far as they are known 369
I. The Sun 370
II. Satellites of Jupiter 380
III. Satellites of Saturn 381
IV. Satellites of Uranus 381
Index 382
A TREATISE on ASTRONOMY. --- INTRODUCTION. (1.) In entering upon any scientific pursuit, one of the student's first endeavours ought to be, to prepare his mind for the reception of truth, by dismissing, or at least holding in abeyance, all those preconceived and adopted notions respecting the objects and relations he is about to examine as may tend to embarrass or mislead him; so as to enable him to receive with impartiality and a resolve, for the unprejudiced admission of any conclusion which shall appear to be supported by careful observation and reasoning. The most obvious proof of a nature adverse to notions he may have previously formed for himself, or taken up, without examination, is the fact that he will not admit them. But this fact, a commencement of that intellectual discipline which forms one of the most important ends of all science. It is plain then, that the first object of mental purity which alone can fit us for a full and steady perception of moral beauty as well as physical adaptation. Is the acquisition of such a faculty, to be sought after? Can we "purge our sight" before we can receive and contemplate as they are the lineaments of truth and nature. (2.) It is evident that astronomy stands in need of such a preparation, or draws more largely on that intellectual liberality which is ready to adopt whatever is demonstrated, or concede whatever is rendered highly probable, however new and uncommon. 7 8 A TREATISE ON ASTRONOMY. the points of view may be in which objects the most familiar may therby become placed. Almost all its conclusions stand in open and striking contradiction with those of the ancients, and with what appears to every one, until he has understood and weighed the proofs to the contrary, the most posi- tive evidence being that which is given by the object itself, which he stands, and which has served for ages as the un- shaken foundation of the firmest structures, either of art or nature. The earth is a globe, and therefore possesses of fixity, and conceived by him as turning swiftly on its centre, and at the same time moving onwards through space without any resistance, and having no parts which appear to unsight eyes round bodies of no very considerable size, become enlarged in his imagination into vast empires, and even into worlds, which extend to the earth itself, the other immensely surpassing it. The planets, which appear only as stars somewhat brighter than the suns, are seen to be worlds of immense inhabi- table worlds; several of them vastly greater and far more curiously furnished than the earth he inhabits, as there is also seen to be a vast number of stars which are not called, so called, which to ordinary apprehension present only lucid sparks or brilliant atoms, are to him some of various unusual shapes and forms, and others of such a nature as to mislead or deceive his senses; and this leads him to myriads of unseen worlds: so that when, after disas- ting his thoughts to comprehend the grandeur of those whose habitations he sees before him, he turns back his imagination and the powers of his language to devise similes and metaphors illustrative of the immensity of the solar system, he finds himself unable to do so; he turns back to his native sphere; he finds it, in comparison, a mere point; so less—even in the minute systems to which it is compared—does it seem to be suspended from some of its principal and remoter members. (3) There is hardly anything which sees in a stronger light the absurdity of the opinions entertained by men of man, whom opposed by no motives of interest or passion, than the perfect readiness with which all these conclu- sions are asserted as soon as their evidence is clearly apprehended, and the reasoning held they acquire over INTRODUCTION. Our belief when once admitted. In the conduct, therefore, of this volume, we shall take it for granted that our reader is more desirous to learn the system which it is his duty to teach than to acquire any other re- vealing object against it; and that, in short, he comes to the task with a willing mind; an assumption which we may safely make, because the nature of the argu- ment on argument to convince the skeptical, but will greatly facilitate his actual progress, insomuch as he will find it less difficult to follow the thread of thought from the straightest and definite path, than to be con- stantly stepping aside, involving himself in perplexities and circuitous paths, which will ultimately terminate in finding himself compelled to adopt our road. (4.) The method, therefore, we propose to follow is merely one of the most convenient means of arriving at either such a combination of both, with a leaning to the latter, or as best may suit with a di lectic composition. Our object being to exhibit the principles of the science itself, under the semblance of an assumed ignorance, for prin- ciples of which we are all the time in full possession— but which we have never been taught to know within the limits of a single volume, and the necessity of being on every point, within that limit, rather diffuse and copious in ex- planation than concise and condensed. As such a neces- sary character of the science itself, render this course both practicable and eligible. Practicable, because there is no difficulty in following it; eligible, because it is like those which are daily changing the features of the less advanced sciences, supervening, to destroy all our hypo- theses respecting them; and eligible also, because it is possible, because the space to be besowned, either in combining refuted systems, or in leading the reader forward by means of new facts and observations which are un- known, may be more advantageously devoted to such ex- planatory illustrations as will impress upon him a familiar sense of the relations between cause and effect, and the pho- nomena, and the manner in which they are produced. We shall not, then, reject the analytic course where it leads more easily and directly to our objects; or in any way fetter ourselves by a rigid adherence to method. 10 A TREATISE ON ASTRONOMY. Writing only to be understood, and to communicate as much information in as little space as possible, consisten- tly with the object of this work, and of its intention, we can afford to make no sacrifice to system, to force, or to affectionation. (6.) We shall take for granted, from the outset, the Copernican system of the world; relying on the easy, obvious, and natural explanation it affords of all the phe- nomena of the heavens; and we shall endeavor to impress the student with a sense of its truth, without either the form- ality of demonstration or the superfluous tenum of analogy, which would render it tedious and wearisome. "Theoriarum vires, actus et quasi se mutuo sustentant partium adaptatione, qua, quasi in orbem coharenti, fir- mamentum habent." (1) The following remarks will show the reader, as occasion offers, she contrast which its superior simplicity offers to the complication of other hypotheses. (6.) The student must possess some knowledge capable of enabling him to understand, in order for the more advantageous perusal of the following pages, consists in the familiarity he has with the principles of arithmetic; some moderate acquaintance with geometry and trigonometry, both plane and spherical; the elementary principles of mechanics; a knowledge of what under- stand the construction and use of the telescope, and some other of the simpler instruments. For the acquisition of these we refer him to any good elementary Encyclopaedia which professes to treat of the several subjects in question. Of course, the more of such knowledge he brings to bear upon his study, the better will be his results, and the more complete the information gained; but we shall endeavour in every case, as far as it can be done with- out an unnecessary labour, to express our ideas briefly which consists in the absence of prolixity and pedanticism to render what we have to say as independent of other books as possible. (7.) After all, we must distinctly caution our readers as may commence and terminate their astronomi- cal pursuits. It is well known that those who begin by doing nothing at all, or by doing little, they usually continue each other, and form a coherent whole. The conclusion of theories relies on the compact adaption of their parts; but when they are too much or too little done, they normally entail each other, and form a coherent whole. INTRODUCTION. 11 ed studies with the present work (though of such, at least as the latter pretensions—we must trust the number will be few), that its utmost pretension is to place them on the threshold of this particular wing of the temple of sci- ence, and to show how they may enter into it, where they may obtain something like a general notion of its structure; or, at most, to give those who may wish to enter into it, some idea of what they may expect in the seasons of the pass-word. Admission to its sanctuary, and to the privileges and feelings of a votary, is only to be obtained by a certain degree of knowledge. The knowl- edge of mathematics, the great instrument of all exact in- quiry, without which no man can ever make such ad- vances as he does now think possible in any branch of science, can as enable him to form an independent opi- nion on any subject of discussion within their range. It is only by means of this knowledge that men can acquire knowl- edge can communicate on such subjects with those who do not, and adapt their language and their illustrations to the terms in which they speak. And it is obvious that which to the one are almost identical, are theorems of import and difficulty to the other; nor is their evidence proved by any common method of reasoning. In searching such propositions, under such circumstances, the appeal has to be made, not to the pure and abstract rea- soning of the mind, but to the practical application of expe- rience: principles and modes of action have to be estab- lished, not by direct argument from acknowledged exempla, but by a process of induction from simple and familiar instances in which the same principles and the same or similar modes of action take place; thus extending our knowledge by a process of successive induction, and constructing at each step a little body of science to meet its exigencies. The difficulty is that we cannot always find these examples in our own country; and advancing at ease along a broad and beaten highway; that is to say, if we are determined to make ourselves disciples of science, we must go out into the world at all. As for the method of assertion, or a direct demand on the faith of the student (though in some complex cases indeed impossible, where illustrative explanation would defeat 12 A TREATISE ON ASTRONOMY. its own end by becoming tedious and burdensome to both parties), it is one which we shall neither adopt ourselves nor would recommend to others. (5.) The subject of this treatise though it is something new to abandon the road of mathematical demonstration in the treatment of subjects susceptible of it, and teach any con- siderable part of it by means of figures, and by the use of illustration and familiar parallels, it is yet not impossi- ble that those who are already well acquainted with our subject, may find it difficult to understand what they consistently adhere to better and higher practice which is incompa- tible with the avowed objects of the present work, may yet find themselves unable to comprehend it. For this reason, that it is always of advantage to present any given body of knowledge to the mind in a great variety of different lights as well as in one, and to proceed with the same kind to strike no two minds in the same manner, or with the same force; because no two minds are stored with the same quantity of knowledge, and therefore cannot be thus by similar habits. Accordingly, it may very well hap- pen, that a proposition, even to one best acquainted with it, may be so obscurely expressed, that he can see nothing but in a more impressive and satisfactory light by such a course—some obscurity may be dissipated, some inward negativity may be removed, some difficulties may be surmounted, which may lead to the perception of connections and deductions altogether unknown before. And the probability of this is increased when the student has previously made observa- tions chosen have not been studiously selected from books, but are such as have presented themselves freely to his own observation. In short, he will not only hold many old views; by which, of course, he means to lay no claim to originality in all or any of them beyond what they may really deserve. (6.) Besides, there are cases in the application of me- chanical principles with which the mathematical student has not yet become acquainted. He has not seen how the lines and the numerical and geometrical relations of his pro- blems all clear to his conception,—when his forces are estimated by means of figures; why, on the contrary, he has followed up the application of his technical processes, and INTRODUCTION. 13 fairly arrived at his conclusion---there is still something wanting in his mind---not in the evidence, for he has exam- ined each link, and finds the chain complete---not in the principles, for those he well knows are too firmly es- tablished to be shaken by any new argument of rea- son. He has followed out a train of reasoning by logical and technical rules, but the signe he has employed are not properly adapted to the subject. The signe is not so strong as such to his mind: he has not seen, as it were, the pro- cess of nature passing under his eyes in an instant of time, on which he can depend for truth. It is like a parallel, or an illustration drawn from some artificial or natural process, of which he has that direct and individual impression, given him by experience. But with a name, will, in almost every such case, supply in a mo- ment this deficient feature, will convert all his symbols into facts, and make them real to his mind, and give what was before a lifeless succession of words and signs. We cannot, indeed, always presume ourselves to within the limits of our own experience. We must always seek on the points to be elucidated themselves always capable of so paraphrased (if we may use the expression) by any one who has had similar experience and sense of experi- ence; but the object will at least be kept in view; and as we are continually reasoning, in making such assumptions, for resolving much difficult questions, the sev- eral of the more concerned effects of pertainous investi- gation than we had acquired by our numerous investi- gations. We must also remember that the same inventor will not always be unuttered with a similar success in other cases. (19.) From what has been said, it will be evident that our aim is not to offer to the public a technical treatise, but rather to give a general principle upon which everyone shall find consigned a minute description of the methods of observation, or the formula he requires prepared to his hand, or their different applications made to his desire. In all these cases the work will be most useful and quite inadequate to his wants. Its aim is entirely different; be- ing to present in each case the mere ultimate satisfactions of fact, arguments, and processes; and in all cases of mathemati- B A TREATISE ON ASTRONOMY. 14 CHAP. I. mathetical application, avoiding whatever would tend to en- cumber its pages with algebraic or geometrical symbols, to place under his inspection that central thread of com- mon sense which is the foundation of all true reason, which is invariably strong; but which, by the attention the latter claim for themselves, is often condescended from the eye of the general reader, who, being more interested in the most convenient form to follow by those who string them. This is no fault of those who have conducted the inquiry; to whom we must refer for the reasons assigned for which they call it enormous; and it may, perhaps, be owing to their experience of how little can be accomplish- ed on earth by mere mathematical demonstration, than merely ordinary clearness of head; and how necessary it often is to pay more attention to the purely mathematical conditions of the problem, than to the actual results of their equations and series,—than to those which explain causes with their effects, and both with the human res- ponsibilities which they involve. The extreme restrictiv- ity of view which is often complained of as a grievance by the earnest student, and still more commonly ascribed principally to the want of patience, is a thing too soon too sublime for vulgar comprehension. We think we shall render good service to both classes of readers, by disas- suring them of any such idea of the nature of this difficulty, and by showing ordinary unlettered comprehension clearly what it can, and what it cannot, hope to attain. CHAPTER I. General Notices—Form and Magnitude of the Earth—And so On—the Atmosphere—The Moon—Twilight—Appearance resul- ting from the Sun's Being at a Distance—a Step towards forming an Idea of the Distance of the Stars—Definition. (11.) The magnitudes, distances, arrangements, and motions of the great bodies which make up the visible universe, their constitution and physical condition, so far as they can be known to us, are their essential in- fluence and action on each other, so far as they can be CHAP. L.] GENERAL NOTIONS. 16 traced to the effects produced, and established by legitimate reasoning, for the advantage of objects to which the attention of the astronomer is directed. The term **astronomy** itself, which denotes the law or rule of the stars, is applied to that science which treats only of the stars properly so called, but the sun, the moon, and all the visible constellations of the heavens), sufficiently distinguished from each other, and from what which denotes the reason, theory, or interpretation of the stars,) has become degraded in its application, and con- sidered as a mere catalogue of the stars, and to divide future events by their dependence on pretended pre- terary influences, the same meaning originally attached to it being lost. (12.) But, besides the stars and other celestial bodies, the earth itself, regarded as an individual body, is one principle of nature, and therefore of importance; and, indeed, the chief of all. It derives its importance, in a practical as well as theoretical sense, not only from its existence among other bodies, but also from its own things, who draw from it the supply of all our wants, but as the station from which we see all the rest, and as the only means of communication with them. The philosophers, who refer to any determinate marks and measures by which to recognise their changes of situation, or with which to compare themselves with others. (13.) To the reader who now for the first time take up a book on astronomy, it will no doubt seem strange to class this subject under the head of philosophy; as to assume any community of nature among things apparently so different. For what in fact, can be more apparently different than the earth? In size it is infinitely less ex- tent of the earth, and the stars, which appear but as points, and seem to have no size at all! The earth is a work of art; while the stars are works of genius. We perceive it in no motion, while in them we observe a continual change of place, as we view them at different times. * Astronomers generally use this word in a general sense; so that **astronomy** means "the shepherd of the stars." The two significations are not always distinct. ** Theory,** reason, or a word, the vehicle of reason; the interpreter of thought. 16 A TREATISE ON ASTRONOMY. [CHAP. L] hours of the day or night, or at different seasons of the year. The ancients, accordingly, one or two of the more enlightened of them only excepted, admitted no such connexion between the motions of the heavenly bodies and their movements without the pale of analogy and experience, effectively interposed the progress of reason to explain what they saw, and to show that what is going on in the regions where they exist and move. Under such conveniences, astronomy, as a science of cause and effect, has been reduced to a mere registry of appearances, unconnected with any attempt to account for them on reasonable principles. (13.) On this account, we have made but a step towards acquiring a knowledge of what is really the case; and the student has made his first effort towards obtaining a true idea of the world, by means least to familiarize himself with the ideas that the earth, after all, may be nothing but a great star. How correct such views are, when we consider how many modifications it is so to admit, we shall see presently. (14.) It is evident, that to form any just notions of our surroundings, we must be able to perceive which we cannot approach and examine, but of which all the information we can gain is by sitting still and watching what exists before us. This is particularly true as to know, in the first instance, whether what we call sitting still is really such : whether the station from which we view things is stationary, or whether those things immediately surrounding us be not itself in motion, unperceived by us; and if so, of what nature that motion is. By this means we shall be enabled to form our apparent arrangement with respect to each other, will of course be materially dependent on the situation of the objects themselves. But there are some cases in which possible change, unknown to the spectator himself, an appearance of change in the respective situations of the objects will occur; and in these cases it will appear that we actually perceive among the stars will not be real movements, but that some part, at least, of whatever changes of re- liable place we perceive among them must be merely A page from a book with text. **CHAP. L.]** ## GENERAL NOTIONS. 17 apparent, the results of the shifting of our own point of view; but, on this, if we have any knowledge of the real nature of things, it can only be by first investigating their own nature, and making due allowance for its effects. Thus, it is question whether the earth is in motion or at rest, and what is the cause which produces that state of the inquiry, but one which depends our only chance of arriving at true conclusions respecting the constitution of the universe. (15.) Nor let it be thought strange that we should seek a proof of a motion existing when it is suggested by its inhabitants; we must remember that it is of the earth as a whole, with all that it holds within its substance, or sustained on its surface, that are supposed to be motions coming from the solid mass beneath us, to the ocean which flows around it, the air that rests upon it, and the clouds which appear above it. These are all solid bodies, which cannot move without some terrestrial object from its relative place among others, interfere with no natural process. But these are not the only objects which might, if they were in motion, produce such effects; and therefore, it is very evident, unless undescribed by us, that we are in motion. It is well known that when we walk on the ground, it is true, however slow we are, these sudden changes of motion are produced, as the laws of mechanics teach us, by sudden and powerful impulses from without. And though those forces applied to our bodies are alike to us felt. When, for example, we are carried along in a carriage with the blind man's guide; or our eyes are shut; or a key is from seeing external objects; or perceiving a train arriving from inequalities in the road; over which the carriage is necessarily compelled to go; or when we see ourselves in progress. As the road is smoother, our sense of motion is diminished, though our rate of travelling is accelerated. Those who travel by rail between London and Manchester and Liverpool testify that but for the noise of the train, and the rapidity with which external objects pass by them, the sensation is almost that of perfect rest. (16.) But it is on shipboard, where a great system is maintained in motion, and where we see surrounded 18 A TREATISE ON ASTRONOMY. CHAP. I. with a multitude of objects which participate with our selves and each other, and which are parts of the whole man, that we feel most satisfactorily the identity of sensation between a state of motion and one of rest. In the case of a large ship moving, going smoothly before the wind will pass or draw back; but in cases, not the slightest indication answers us with the sight is making any difference to the state of rest or move- tary action as if we were on land. If we throw a ball into the air, it falls back into our hand; or if we drop it, it lights at once on the ground; and so as to see the fire, smoke ascends in the same manner as it would do in an apartment on shore. If, indeed, we look at a ship at sea, we find no difference; the air, not being carried along with us, diffuses away smoke and other light bodies--such as feathers aban- doned by a bird--and we perceive no difference to that of the ship's progress; but, in reality, they remain at rest, and we leave them behind in the air. Still, the illusion is very strong; for when we consider how these move- ments are concerned, remains complete; and when we look at the shore, we then perceive the effect of our own motion transferred, in a contrary direction, to external objects external; that is, to the system of which we form a part. Prolatum portum tertio ubique resolutum. (17) Not only do external objects at rest appear in motion generally; but even those which are in motion among them, but they appear to move one among the others--they shift their relative apparent places. For instance, let us suppose a high road fix his eye steadily on any object, but at the same time not entirely withdraw his attention from the gene- ral landscape around him. He will then see this whole landscape thrown into rotation, and moving round that object as a centre; all objects between it and himself appearing to move towards him; and he will have no way to his own motion; and still beyond it, forwards, or in the direction in which he moves: but let him withdraw his eye from that object, and fix it on another--a nearer **CHAP. 1.]** FORM OF THE EARTH. one, for instance,—immediately the appearance of re- tation whilst also, and the apparent centre about which this illusive circulation is performed is transferred to the same object, while the moment appears to rest. This is a very clear change of situation of one body re- spective to another, and arising from a motion of the speci- men itself, and not from any other cause. It is therefore evident that, before we can ascertain whether external objects are really in motion or not, or what their mo- tions are, we must first consider how, for any such paral- lellical motion which may exist. (18.) In order, however, to conceive the earth as in motion, we must first suppose it to have no limit (as shape and size). Now, an object cannot have shape and size, unless it is limited on all sides by some definite confining boundary. The earth has no such boundary dis- connected from other bodies, and existing insolated in a space. The first rude notion we form of the earth is that it is a sphere, and that its surface is perpendicular from the spot where we stand, above which are the air and sky; below, to an indefinite profundity, solid mat- ter. This conception is so common with us, that we take the earth's immobility; but it is one much easier to rid our- selves of, inasmuch as it originates only in our own imagination. For if we consider that the earth moves, we will place a limit to so thing we have been accustomed from infancy to regard as immensely large; and does not even admit that it could move without being unduly interpreted. On the contrary, the direct usu- mony of our senses lie the other way. When we see an object moving, we immediately conclude that it is in the east, as we cannot doubt that it is the same sun we see after a temporary absence, we must do violence to our senses to believe that it is not. We have made its way through the substance of the earth. It must, therefore, have gone under it, and that not by a single lineal passage, but by a series of passages round where it sets and rises for many successive days, or for a whole year; we shall find them constantly shifting, round a very large extent of the horizon; and besides, the moon and stars also set and rise again at all points 20 A TREATISE ON ASTRONOMY. [CHAP. I.] of the visible horizon. The conclusion is plain: the earth cannot extend indefinitely in depth downwards, but indeed it extends only to a certain height, not only bounds in a horizontal direction, but also an under side, round which the sun, moon, and stars can pass ; that it must have a sky and sunshine, and a day when it is night to us, and vice versa where, in short, is not a pole of the Aemus, diemus reductum. Hence it follows that the earth is globular. Illus sunt rubes accidunt lucis Vesperi. Garg. (18.) Any thing that has been observed upon with the conception of the earth without foundations or fixed supports—existing insulated in space from contact of every other body—must necessarily be at rest in motion—or, rather, difficult to imagine it otherwise; for, since there is nothing to retain it in one place, should any cause be applied to move it, it will move; it must obey their impulse. Let us next see what obvious circumstances there are to help us to a knowledge of the shape of the earth. (19.) Let us first examine what we actually see of its shape. Now, it is not on land (unless, indeed, on uncommon occasions), but on water, that we observe any thing of the general figure of the earth—the hills, trees, and other objects which roughen its surface, and break its smoothness. But even this is not very distinctly bearing a minute proportion to the whole earth, are yet too considerable, with respect to ourselves and to the distance between us and them, to give us a clear view, to allow of our forming any judgment of the form of the whole, from that of a part so disguised; but with regard to the sea itself, which is always at a level plane, the case is otherwise. If we sail out of sight of land, whether we stand on the deck of the ship or climb high up in her rigging, we find ourselves looking in self in distance and mist, but terminated by a sharp, clear, well defined line, or edge as it is called, which runs all round the circumference of the globe towards its centre. That this line is really a circle, we conclude, first, from the perfect apparent similarity of all its parts; A diagram showing a circular horizon around a spherical Earth. CLAP. 3.] HORIZON AND ITS DEF. and, secondly, from the fact of all its parts appearing at the same distance from us, and that evidently a mode- rate one; and, thirdly, from this, that its apparent diameter, measured by an instrument called the dip compass, is always less than the actual angular atmo- spheric circumstances, which produce a temporary disor- der in the sight of the sun. This is the case when the sun is taken --properly-- which belong only to the circle among geometrical figures. If we ascend a high emi- nence on a plain (for instance, one of the Egyptian pyra- mids), we shall see the sun go down. (21.) Masses of ships, however, and the edifices erected by man's art, will show us that the earth has such nature itself affording. Eos, Tenerifae, Moenae Rosea, re- mnents from which some contemplative apart part of the world may be seen, and which are visible to our eyes, are those few and rare occasions when the transparency of the air will permit the real boundary of the earth horizon. The appearance of these objects when they are seen with this, but with this remarkable addition, viz. that the angular diameter of the visible area, as mea- sured by a dip compass, is always less than the true level; in other words, that the apparent size of the earth has sensibly diminished as we have receded from its surface, while yet the absolute quantity of it seen at once remains constant. (22.) The same appearances are observed universally, in every place where any surface visited by man. Now, the figure of a sphere would appear to us if no spe- cies always circular, can be other than a sphere or globular. (23.) A diagram will elucidate this. Suppose the earth to be represented by the sphere LHNQ, whose diameter is 100 miles. Let M N represent different elevations above various points of its surface, represent- ed by a, g, m, respectively. From each of them (as from M) let a line be drawn parallel to a line joining the surface at N; then will this line represent the view ray along which the spectator at M will see the visible ho- rizon; and as this ray swings round M, and comes successively into the positions M O, M P', M Q', M y, A diagram representing a spherical Earth with various elevations labeled 'a', 'g', 'm' and lines connecting them to points on a horizontal plane. 21 23 A TREATISE ON ASTRONOMY. [CHAP. I. point of contact N will mark out on the surface the circle NOPQ. The area of this circle is the portion of the A diagram showing a circle with points labeled M, N, O, P, Q, and X. The line MN is drawn through point M and intersects the circle at point N. The line NO is drawn through point N and intersects the circle at point O. The line OP is drawn through point O and intersects the circle at point P. The line OQ is drawn through point O and intersects the circle at point Q. The line XN is drawn through point X and intersects the circle at point N. The line XO is drawn through point X and intersects the circle at point O. The line XP is drawn through point X and intersects the circle at point P. The line XQ is drawn through point X and intersects the circle at point Q. earth's surface visible to a spectator at M, and the angle NMQ included between the two extreme visual rays is the measure of its apparent angular diameter. Leaving, at present, all considerations of refraction in the air below M, of which more hereafter, and which always tends, in some degree, to increase that angle, or under certain circumstances to diminish it, we consider the dip sector. Now, it is evident, 1st, that as the point M is more elevated above m, the point immediately below it on the earth's surface becomes less visible; 2ndly, that the segment or slice NOPQ increases; 3rdly, that the distance of the visible horizon or boundary of our view from the eye increases; 4thly, that consequently that the angle MNQ becomes less oblique, or, in other words, the apparent angular diameter of the earth diminishes. *Quaeritur termino.* **CHAP. I.]** **HORIZON AND ITS DIP.** being nowhere so great as 180°, or two right angles, but falling short of it by some sensible quantity, and that reason why the horizon is not a plane surface, but ex- hibits three states or stages of elevation, with the hori- zon, &c., corresponding to each, a glance at which will explain the matter. The first stage is the plane horizon, more and more distant, MNOQP, let the reader imagine nMNM, M2Q to be the two legs of a rule joined at M, and the line MNMQ to be the hypotenuse of this triangle. It is clear, that as the point M is urged home towards the surface, the leg will open, and the ruler will become more and more inclined to the horizontal. As the inclina- tions till M is brought fairly up to contact with the sur- face at m, in which case its whole length will become a line parallel to the surface. (24.) This explains what is meant by the dip of the horizon. Mon, which is perpendicular to the general elevation of the earth's surface, is called the horizon on which a plumb-line* would hang; for it is an observed fact, that in all situations, in every part of the earth, the di- rection of Mon is always parallel to the level of the surface of still-water; and moreover, that it is also ex- actly perpendicular to a line or surface truly adjusted by a spirit-level, and therefore parallel to any other line we are to adjust a line (a wooden rule for instance) by a spirit-level, with perfect exactness; then, if we suppose the horizon to be a plane surface, and to be drawn both ways, as XMY, the line so drawn will be at right angles to Mon, and therefore parallel to zxy, so that the inclination of Mon to XZY will be zero. But Mon will therefore see not only all the walls of the sky above this line, as XZY; but also all portions or zones of it which lie below this line XZY (QZ), and therefore Mon will be more than a hemisphere by the zone YQXN. It is the angular breadth of this redundant zone—the angle YMXN—which is called the dip of the horizon. It is expressed below the direction of a spirit-level—that is called the dip of the horizon. It is a correction of constant use in *See this instrument described in chap. II. 23 34 A TREATISE ON ASTRONOMY. CHAP. I. (25.) From the foregoing explanations it appears, 1st. That the general figure of the earth (as far as it can be gathered from this kind of observation); that is, of a sphere or ellipsoid, is not exactly spherical, but one of which, wherever it extends, covers and fills in those inequalities and local irregularities which exist on land, but which are not visible to us from our position. The deviations from the general outline of the whole mass, as we consider an orange not the less round for the roughnesses of its surface, so the earth is not perfectly a regular horizon, or sea, is a consequence of the curvature of the surface, and does not arise from the inability of the eye to distinguish any such irregularities, nor from atmospheric indistinctness. It will be worth while to pursue the general notion thus acquired into some of its consequences. For instance, two ships sailing at different distances from each other, and on a larger scale, will be put to the test, and a clear conception be formed of the manner in which they appear to each other, and hold together as a whole. (26.) In the first place, then, every one who has passed a little distance from shore will observe that he can see perfectly well beyond the point of his or her visible horizon—but not the whole of them. We only see their upper parts. The lower parts are hidden by the surface of the water, they are hid from view by the spherical surface of the sea, which precedes between them and ourselves. Suppose now that we have two ships sailing from our station—i.e., first, when the distance of the ship is small, a spectator, S, situated at some certain height above the sea level, would see both ships at once; but when it increases to a certain extent—the distance between them being equal to their own length—their appearances begin to separate. When it has reached a certain distance, as at C, its hull has entirely vanished, but the mast and sails remain presenting the appearance e. But if, in this state of A diagram showing two ships at different distances from each other. The ship closer to the viewer (at A) is completely visible with both its hull and mast clearly seen. The ship further away (at B) is partially obscured by the horizon line at B. GIAF. 1. J SIZE OF THE EARTH. A diagram showing a ship sailing away from a stationary observer at the center of the Earth. The ship is moving along a curved path labeled "T" to "E". The observer's line of sight is shown as a straight line connecting the observer to the ship. The ship's speed is indicated by the distance between two points on the curve. things, the spectator quickly ascends to a higher station, T, whose visible horizon is A, the hull comes again in sight; and when he descends again he loses it. The ship still receding, the lower sails seem to sink below the horizon, while the upper ones remain visible always; while yet the distinctness with which the last portion of the sail d is seen is such as to satisfy us that were it not for the motion of the ship we should have lost it. The dis- tance TE is not so great as to have prevented an equally perfect view of the whole (20). (21) But, therefore, if we could measure the heights and exact distance of two stations which could barely be discerned from each other over the edge of the sea, and if we knew how much they were distant from the earth itself; and, in fact, were it not for the effect of re- fraction, by which we are enabled to see in some small degree beyond what would otherwise appear (which will here- after explained), this would be a tolerably good method of ascertaining it. Suppose A and B to be two emi- nences, whose perpendicular heights a and b (which, for simplicity, we will suppose to be exactly equal) are known, as well as their exact horizontal interval aDE. C 26 A TREATISE ON ASTRONOMY. [CHAP. 2.] by measurement; then is it clear that D, the visi- ble horizon of both, will lie just half-way between them, and therefore the distance between them will be the length of the earth, and C its centre in the figure CDB, we know DB, the length of the arc of the circle between D and C, to be equal to the diameter AB, or R, excess of its secant above its radius—which is the height of B above sea-level—and this being a practical problem unable to be found the length of the radius DC. If, as is really the case, we suppose both the heights and distance of the stations inconsiderable in comparison with the size of the earth, the following proposition is demonstrative in the following proposition: The earth being seen at some proportion to the distance of the visible horizon from the eye as that dis- tance does to the height of the eye above the sea level. When the stations are unequal in height the problem is a little more difficult. (28.) Although, as we have observed, the effect of variation in the altitude of the observer on his difficulty of securing the dimensions of the earth, yet it will suffi- ce to afford such an approximation as it shall be of use in the present inquiry. The following table will help us to many just conceptions, on which account we shall exemplify its application in numbers. Now, it appears by observation that when any object is placed above the surface of still water, cease to be visible from each other over still water, and in average atmospheric circum- stances, an object situated at a distance of one mile from the 50th part of a mile, so that half their distance, or 4 miles, is to the height of each as 4 x 508 = 2112 : 1, and therefore as 4 x 2112 = 8448, or about 8000 miles, is to their diameter. It must, therefore, be equal to 4 x 2112 = 8448, or, in round numbers, about 8000 miles. (29.) (29.) Such is the first rough result of an attempt to ascertain the earth's magnitude; and it will not be amiss if we take notice that this result has been obtained by those who have been accustomed to consider as of vast size, so as to interpose a few steps between it and our ordinary ideas CHAP. I.] VISIBLE PORTION OF THE SURFACE. 97 of dimension. We have before likened the inequalities on the earth's surface, arising from mountains, valleys, buildings, &c. to the roughnesses on the kind of an ordinary sheet of paper; but this comparison is quite free from exaggeration. The highest moun- tain known does not exceed five miles in perpendicular elevation above the level of the sea; and its diameter, consequently, on a globe of sixteen inches in diameter, such a mountain would be represented by a circle of about two inches in diameter; and this is nearly as much as can be seen with the naked eye on an ordinary sheet of paper, which is about one inch thick. (8.) The smallest hill would be represented by a point on the surface of the globe; and this point would be so small that it could not be distinguished from a spot on a sheet of writing paper; and the highest hills would be represented by the smallest visible grain of sand. (9.) The greatest depth of sea does not penetrate half a mile below the surface: a scratch, or pin-hole, duly representing it, on the surface of such a globe as ours, would be represented by a point only. (10.) The greatest depth of sea, probably, does not much exceed the greatest elevation of the continents; and therefore, if we represent all these elevations, in about the same proportion, into the substance of the globe; so that the ocean comes to be conceived as a mere film floating upon the earth; then those parts left by a brush dipped in colour and drawn over those parts intended to represent the sea: only in so conceiving it, will they appear to be equal in extent, and no farther than to proportion in point of quantity. The mechanical laws which would regulate the distribution and accumulation of water upon the surface of the earth, are altogether different from those which govern the phenomena of the sea. (11.) Lastly, the greatest extent of the earth's surface which has ever been seen at once by man, was thus ex- 28 A TREATISE ON ASTRONOMY. [CHAP. I. posed to the view of MM. Biot and Gay-Lussac, in their celebrated aeronautic expedition to the enormous height of 25,000 feet, or rather less than five miles. To esti- mate the proportionate size the area visible from the elevation at which we have ascended, we must first remark on the geometry of the sphere, which informs us that the convex surface of a sphere is equal to its base, and that the whole surface of the sphere to which it belongs is the circumference of the sphere; and further, that the circumference of any circle is nearly equal to its diameter. Now, therefore, when we are ascending, we are ascending to a point whose distance from the centre of the earth is almost exactly equal to the perpen- dicular elevation of the point of sight above the surface. The portion of the surface of the sphere which is thus reached, as well as that of the whole earth's surface, is that of five miles to the 8000, or to 1000. The portion visible from Amsa, the Peak of Aconcagua, is about 36 miles square (32). When we ascend to any very considerable ele- vation above the surface of the earth, either in a balloon, or on an apparatus like that used by Mr. Hargreaves for his sensations, or an insufficient supply of air. The barome- ter, an instrument which informs us of the weight of air incumbent upon it, furnishes us with a measure of this im- pression, and affords a direct measure of the rate of diminu- tion of the quantity of air which is given space includes as we ascend. It will be seen from this that we may easily learn, that when we have ascended to the height of 1000 feet, we have left below us about one thirtieth of the whole mass of air which is contained within a sphere of perpendicular elevation (which is rather less than that of the summit of Aconcagua) we have ascended through about one third only of its thickness (or rather its thickness topaix), through one half the material, or at least, he ponderable body of air incumbent on the earth's surface. From this it follows that if we ascend to any great de- gree, from the nature of the air itself, which is compress- ible, i. e. capable of being condensed, or crowded into a smaller space than it occupies when it is in motion; it is easy to see that, although by rising still higher we should *The height of Amsa above the Medierranean so it results from a horizontal measurement of my own made in July 1840 under very favourable circumstances in England last year.* A balloon ascending. **CHAP. I.** THE ATMOSPHERE. continually get above more and more of the air, and so re- liews ourselves more and more from the pressure with which it weighs upon us, yet the amount of this additional resistance to our motion increases less than proportionally, so that by no means in proportion to the additional height ac- cended, but in a constantly decreasing ratio. An easy calculation will shew that the resistance of the air to our move- ment diminishes as we ascend, according to the knowl- edge of the properties of air, and the mechanical laws which regulate its dilatation and compression, is sufficient to make it possible for a body to rise through the earth not exceeding the hundredth part of its diameter, the semiu- ry, or rarefaction, of the air must be so excessive, that not only does it not resist our motion, but actually tends to main- tained in it, but that the most delicate means we possess of acertaining the existence of any air at all would fail to allow us to detect its presence even at such a distance. (34.) Laying out consideration, therefore, at pre- sent, all nice questions as to the probable existence of a diameter of air which would allow us to perceive it in absolutely and rigorously speaking, no air, it is clear, that, for all practical purposes, we may speak of those regions where we find ourselves as being entirely free from air, the hundredth part of its diameter is void of air, and of course of clouds (which are nothing but visible vapour, diffused throughout the atmosphere) and water (which is nothing but wetting it itself as mud does water). It seems probable, from many indications, that the greatest height at which visible clouds can be seen is about 50 miles; and from which height the density of the air is about an eighth part of what it is at the level of the sea. (35.) Now let us consider regard the atmosphere of air, with the clouds it supports, as constituting a coating of equal or nearly equal thickness, enveloping our globe of earth like a mantle. This mantle extends over all the surface of the sea and land constitutes her bed, and whose interior portions or strata, within a few miles of the earth's surface, are filled with water. In these strata, the density diminishing with extreme rapidity as we re- cede upwards, till within a very moderate distance (such as 100 miles) it becomes represented by a point; on this mo- del we have before spoken of, and which is not more in pro- c 2 30 A TREATISE ON ASTRONOMY. [CHAP. I.] portion to the globe on which it rests, than the downy skin of a peach in comparison with the fruit within it), all necessary to render the argument conclusive. (36.) Arguments, however, are not wanting to render it, if absolutely certain, at least in the highest degree probable. For, although the earth, being surrounded by the aqueous ocean, has a real and definite limit, as above hinted at; beyond which there is positively no air, and above which it is impossible for any body to be carried upwards, out, or carried aloft from below, instead of dilating itself indefinitely upwards, would, after a certain very enormous but not inconceivable distance, have to descend and merge, as water poured into the sea, and distribute itself among the mass beneath. With the truth of this conclusion, however, we are not concerned; but only to consider the effects of the atmosphere in modifying astronomical phenomena being the same, whether it be supposed of definite extent or indefinite. (37.) Moreover, whichever idea we adopt, it is equally certain that, within those limits in which it possesses any appreciable effect upon terrestrial phenomena, it affects all points on the earth's surface; that is to say, on the great scale, and leaving out of consideration temporary and local causes of variation. It is also evident from the discus- sions of the nature of waves, which prevails in it to an immense extent; in other words, that the law of diminu- tion of their height as they approach the shore is such that the level of the sea is in the same in every column into which we may conceive it divided, or from whatever point of view we consider it. The waves themselves are considered as consisting of successively superposed strata or layers, each of the form of a spherical shell concentric with the earth's centre; and each layer being more dense than its predecessor; that is to say, one which is rarer, or specifically lighter, than that immedi- ately beneath it; and denser; or specifically heavier, than that immediately above it. The reason why this condition of its ponderable mass is necessitated by the laws of the equilibrium of fluids, whose results harmonize observa- tions demonstrate to be in perfect accordance with experi- ence. It must be observed, however, that with this distribu- **CHAP. I.** **REFRACTION.** 31 tion to refract the inequalities of mountains and valleys has no concern; they exercise no more influence in modifying their general spherical figure than the inequalities at the bottom of the sea interfere with the general spherical form of the globe. (28.) It is the power which air possesses, in common with all transparent media, of refracting the rays of light, or other radiations, that causes the spectator to consider a knowledge of the constitution of the atmosphere important to the astronomer. Owing to this property, objects situated at different distances from him appear to him than they would to the same spectator, had the atmosphere no existence ; it thus produces a false impression respecting the nature of the earth's surface. The greater is the amount and direction of the displacement so apparently produced on each, before we can come at a knowledge of the constitution of which they are situated from us at any assigned moment. (29.) Suppose a spectator placed at A, any point of the earth's surface KAK, and let L1, M1, N1, represent A diagram showing a perspective view of a star (L) and its apparent position (M) and (N) when viewed from different points on Earth's surface. C the successive strata or layers, of decreasing density, into which we may conceive the atmosphere to be divided, as he looks towards the star surface L1KAK with KA, the earth's surface. Let B be a distant star, or other luminous body, beyond the utmost limit of the atmosphere ; then, if the air were away, the spectator would see it in the di- 22 REFRACTION. [CAP. I.] rection of the straight line AS. But, in reality, when the ray of light SA reaches the atmosphere, suppose at D, it will be, by law 3, bent downwards, and will make an angle with the straight line AB, which is greater than the angle between the straight line AB and the straight line AD, and therefore will be refracted into the air. This bending will at first be imperceptible, owing to the extreme tenuity of the uppermost stratum; but as it advances downwards, the rays will become more and more refracted, and ultimately undergo greater and greater refraction in the same direc- tion; and thus, instead of being reflected at the straight line SDA, it will be reflected at SDCB, and will pass through more and more concave downwards, and will reach the earth, not at A, but at a certain point a, nearer to S. Thus may it happen that a star seen at A will appear to a man at B by which he will see the star is, therefore, not SDA, but some ray of light passing through SDCB. Now we would have struck the earth at K, a point behind the straight line AB but which, being bent by the air into the curve SDCBA, actually appears to us as if it were at A. The object of this is that an object is seen in the direction which the ray takes at its instant of arriving at the eye, without regard to what may have been either before or after it. Hence it follows that when an object is seen by the eye, it must be seen by a ray of light which has passed through all the atmosphere up to that point. Hence the star S will be seen, not in the di- rection AS, but in that of A, tangent to the curve SDCBA; for if any ray of light passes through SDCB, the refracted ray is concave downwards; the tangent AA will lie above AS, the unrefracted ray; consequently the object will appear to us as if it were at A. Now if we had seen through the refracting atmosphere, then it would ap- pear there were no such atmosphere. Since however, the dispositions of things are such that we can never see A directly from B, but always by means of SDCBA, A, the visual ray will not be made to deviate laterally, but will remain constantly in the same vertical plane SABC, passing through all points of that plane equally distant from its centre. (40.) The effect of the air's refraction, then, is to raise all the heavenly bodies higher above the horizon in ap- pearance than they really are. For instance, a star situated actually in the true horizon will appear above it, or will have some certain apparent altitude (as it is called), say N° 180 degrees; but since all other stars are raised, and which would therefore be invisible but for the effect of refraction, are, by that effect, raised above it and brought **CHAP. 1.** REFRACTION. 25 into sight. Thus, the sun, when situated at P below the true horizon, A H, of the spectator, becomes visible to him, as if it stood at p, by the refracted ray PQrA, to which A is perpendicular (41). The exact estimation of the amount of spherical refraction, or the strict determination of the angle A P Q r A, is a difficult task. In this respect, however, HAS, is raised in appearance above its true place, i. e., unfortunately, a very difficult subject of physical inquiry; and although I have endeavoured (but one can look for any information on the subject) are not yet entirely agreed. The difficulty arises from this, that the degree of refraction (which depends upon the refracting power dependa) is affected not merely by the imperfection- brent pressure, but also by its temperature or degrees of heat. For instance, when the air is heated from the earth's surface the temperature of the air is constantly diminishing, yet the less, or amount of this diminution as it is called, is not perceptible to our senses. Moreover, the refracting power of air is perceptibly affected by its moisture; and thus, too, is not the same in every part of an atmosphere. Hence we cannot form any idea how it is known to be distributed. The consequence of our ignorance on these points is to introduce a corresponding degree of uncertainty into all astronomical observations. As refraction which affects to a certain appreciable extent, our knowledge of several of the most important data of astron- omy has been greatly impaired. This defect, however, confined within such very narrow limits as to be no cause of embarrassment, except in the most delicate inquiries, and even then for no serious allusion in a treatise like the present. (42.) A "Table of Refractions," as it is called, or a list of angles which show how much light varies from this cause, at all altitudes, or in every situation of a heavenly body, from the horizon to the zenith, or perhaps even beyond it. It will be seen that under all the circumstances in which astronomical observations are usually performed which may influence the result, is *From an Arabic word of this signification.* A diagram showing a line segment AB representing a horizontal plane with point A at ground level and point B at some height above ground level. Point C is located directly above point A on a vertical line extending upwards from point A. Point D is located directly above point B on a vertical line extending upwards from point B. Line segments AC and BD are drawn connecting points A and C respectively and points B and D respectively. Line segment CD connects points C and D. The diagram illustrates the concept of refraction where light rays bend as they pass through different media. 34 A TREATISE ON ASTRONOMY. CHAP. I. one of the most important and indispensable of all astrono- mical tables, since it is only by the use of such a table we are enabled to get rid of an illusion which must otherwise befall us in our attempts to determine celestial motions. Such have been, accordingly, constructed with great care, and are to be found in every collection of astronomical tables. But though these tables will always not admit of the introduction of tables; and we must, therefore, content ourselves here, and in similar cases, with the aid of the tables already mentioned, we desir- tained to furnish these useful aids to calculation. It is, however, desirable that he should bear in mind the following general notions of its amount, and law of variations. (43.) 1st. In the zenith there is no refraction; a ce- lestial object appears at its true distance from us in its true direction, as if there were no atmosphere. 2ndly. In descending from the zenith to the horizon, the refracted rays approach nearer to the observer than the horizon appearing more elevated by it above their true directions than those at a high altitude. 3rdly. The amount of refraction increases in proportion to the tangent of the apparent angular distance of the object from the zenith. But this rule, which is not far from the truth, does not hold good when we attempt to give correct results in the vicinity of the horizon, where the law becomes much more complicated in its ex- pression. 4thly. The average amount of refraction, for an ob- ject half-way between the zenith and horizon, or at an apparent distance of 45° from it, is about 18", but for a quantity hardly sensible to the naked eye; but at the visible horizon it amounts to no less a quantity than 23", which is nearly double that given by the preceding rule. It is evident that this difference arises from the fact that either the sun or the moon, when they are seen at the horizon, appear to be situated below it, and would be seen in reality below it, and would be entirely out of sight and Vide "Regulae Tables to be used with the Nautical Almanac." See: Newton's "Principia" and "Observations on Newton's Astronomical Tables," and Mr. Baily's Astronomical Tables and Formulae. **CHAP. 1.]** TWILIGHT. concealed by the convexity of the earth has for the bend- ing round it, which the rays of light have undergone in their passage through air, as alluded to in art. 40. (44.) It follows from this, that one obvious effect of refraction is to produce a more or less apparent increase of darkness, by actually prolonging the stay of the sun and moon above the horizon. But even when they are set, the rays of light still continue to reach us, as we a portion of their light; not, indeed, by direct trans- mission, but by reflection upon the vapours, and minute particles of dust, which are continually falling upon the actual material atoms of the air itself. To understand how this takes place, we must recollect, that it is not only the rays of light which we see, but whatever portion of its light which would not otherwise reach our eyes, is intercepted in its course, and reflected back again upon us as a means of illumination. Such reflective obstacles always exist; for instance, the glass of a window, or the sun- beam penetrating through the chink of a window-shutter into a dark room, is visible at a bright line in the air; and even if it is stifled, or let out through an opposite aperture, it will still be seen. The reason why particles from this source is sufficient to prevent entire darkness in the room is, that the rays which are occasionally seen in the air, in a sky full of partially blackened clouds, under the vulgar term "the sun drawing water," are similarly caused. These are undoubtedly produced in close par- tially blackened and refocused by the dust and vapours of the air below. Thus it is with those solar rays which, after having traversed the upper regions of the atmosphere, continue to traverse the higher regions of the atmos- phere above our heads, and pass through and out of it. In these regions they are intercepted by some por- tion of them is interposed, and reflected by the floating particles above mentioned, and thrown back, or la- tedly refracted towards us, as secondary illumination, which is twilight. The course of such rays will be immediately understood from the annexed figure, in which ABOC is the earth ; A a point on its surface, where the sun S is in the act of setting; i.e. that lower 36 A. TREATISE ON ASTRONOMY. [CAP. L] ray SAM just grazing the surface at A, while its superior rays SN, NO, traverse the atmosphere above A without striking the earth, leaving it finally at the point PQR, after being more or less bent in passing through it, the A diagram showing the bending of light rays as they pass through the Earth's atmosphere. lower most, the higher less, and that which like SRO, merely traverses the upper limit of the atmosphere, remains at A. Let us consider several points, A, B, C, D, each more remote than the last from A, and each more deeply involved in the atmosphere, which is a transparent space from A beneath the line A M N. Now, A just receives the sun's last direct ray, and, besides, is illuminated by the whole sky; B receives a part of the sun's light and receives twilight from the whole sky. The point B, to which the sun has set, receives no direct solar light, nor any direct light from the sun itself; but receives twilight from the atmosphere which is below PQR; but from the lenticular portion PQR, which is traversed by the sun's rays, and which is situated between A and B (see fig.), it recei- ves a twilight, which is strongest at R, the point immediatly below which the sun is, and fades away gradu- ally towards PQR. The point C receives a similar thimn off. At C only the last or thinnest portion PQR of the lenticular segment, thus illuminated, lies above the horizon. CQ of that place : here, then, the twilight is feebler, and confined to a small space in and near the **CHAP. L.]** **TERRESTRIAL REFRACTION.** horizon, which the sun has quitted, while at D the twilight has ceased altogether. (45.) When the sun is above the horizon, it illuminates the atmosphere and clouds, and these again disperse its light by reflection upon their surfaces and particles, so as to send some of its rays to every exposed point, from which they return to the eye with diminished light, therefore, which we enjoy in the daytime, is a phenomenon originating in the very same causes as the twilight. It was no for the reflective and scattering power of the atmosphere that it was able to establish the limit of direct sunshine ; every shadow of a passing cloud would be pitchy darkness ; the stars would be visible all day long, and, indeed, in the night-time, if the direc- tion admissible, would be involved in nocturnal obscurity. This scattering action of the atmosphere on the solar light is not confined to the daytime only, but may be irregularity of temperature caused by the same luminary in different parts of the globe. The heat which enters into a constant state of undulation, and, by thus bringing together masses of air of very unequal temperatures, produces a partial condensation of this common moisture, by which much light is turned aside from the direct course, and diverted to the purposes of general illumination. (46.) From the explanation we have given, in arts. 39 and 40, of the nature of atmospheric refraction, and of its effects upon the progress of a ray of light through successive strata, or layers, of the atmosphere, it will be evident, that whenever a ray passes from one layer into another more dense than itself, its course is not rectilinear, but concave downwards; and of course any object seen by means of an instrument placed in such a position as to observe whether that object be, like the celestial bodies, entirely beyond the atmosphere, or lie within its limits; or whether it be situated at different levels, seem from each other, immersed in it. Every difference of level, accompanied as it must be, with a difference of density in the aerial stratum, must also have, corresponding to its certain amount of reflection. D 20 A TREATISE ON ASTRONOMY. [CHAP. I. less, indeed, than what would be produced by the whole atmosphere, but still often of very appreciable, and even considerable, amount. This reflection between terrestrial objects is so different from that which takes place on the surface of the moon, as to distinguish it from that seal effect which is only produced on celestial objects, or such as are beyond the atmosphere, and which we have already described (p. 13). (47.) Another effect of refraction is to distort the visible forms and proportions of objects seen near the horizon. The nearer they are to the horizon, the more they appear distorted; at first, always appear round, assume, as it approaches the horizon, a flattened or oval outline; its horizontal diameter is greater than its vertical diameter, and of direction. When very near the horizon, this distortion is evidently more considerable on the lower side than on the upper side; for instance, when a circle is represented nor elliptic, but a species of oval, which deviates more from a circle below than above. This singular effect, which appears to be produced by refraction, shows us the rapid rate at which the refraction increases in approaching the horizon. Every very visible point in the sun's circumference is seen with a certain degree of flattening; but still appears circular, though displaced; but the lower portions being more raised than the upper, the vertical diameter is thereby increased. In like manner, every point in the hori- zontal diameter are equally raised, and in parallel direc- tions, so that its apparent length remains the same. The clared and flattened appearance of objects seen near the horizon, beyond what they appear to have when high up in the sky, has nothing to do with refrac- tion. It is merely owing to their being seen through the terrestrial objects interposed, or placed in close compar- ison with them. In that situation we view and judge of them as if they were at a great distance; and this is an acquired habit of attention to parts. Aloft we have no associations to guide us, and their insolation in the exposure to our sight does not enable us to overrate their apparent magnitudes. Actual measure- ment with a proper instrument corrects our error, with- out, however, any change in form. By this means, that the sun, when just on the horizon extends as our **CHAP. I.] OF THE SPHERE OF THE HEAVENS.** eyes almost exactly the same, and the moon a materially less angle, than when seen at a great altitude in the sky, owing to the effect of what is called parallax, to be expounded hereafter. (48.) After what has been said of the small extent of the atmosphere in comparison of the mass of the earth, we may now proceed to consider the various circumst- ancies which people and adorn the sky, and which, while they obviously form no part of the earth, and receive no light from it, yet appear to us to be so near us as if like clouds upon the air, nor drifted by the winds, to be eternal and immovable. As such we have considered them; but speaking while of their nature, we must see in the immense space beyond, and situated, perhaps, for that thing we call heaven, a vast eternity, at enor- mous distances from one and from each other. (49.) Could a spectator exist unsustained by the earth, or even by a solid body, he would soon find him at once view the whole contents of space—viz., visible consti- tuents of the universe: and, in the absence of any means of determining his position with respect to these things, in the directions in which they were seen from his sta- tion, to the concave surface of an imaginary sphere, having its center at his own place, he would perceive an indeterminate distance. Perhaps he might judge those which appear to him large and bright, to be nearer to himself than those which appear small and dim; but whatever be the mode of judging he would have no warrant for this opinion, any more than for the idea that all were equally distant. For suppose he could determine with a spherical surface. Nevertheless, there would be no impropriety in his referring their places, geometrically speaking, to some imaginary sphere, whose radius was equal to that imaginary sphere, whose respective visual rays intersected; and there would be much advantage in so doing, as by that method all objects would be placed on a plane which could be accurately measured, recorded, and mapped down. The objects in a landscape are at every variety of distance from one another; and though they may appear close together on one plane, and at one distance, in their actual apparent proportions, and the likeness is not taxed with incorrectness. 40 A TREATISE ON ASTRONOMY. [CHAP. I.] see, though a man in the foreground should be repre- sented larger than a mountain in the distance. So is it to a spectator of the heavenly bodies pictured, projected, or represented by any other means, to represent the sky or heaven. Thus, we may easily conceive that the moon, which appears to us as large as the sun, though be large in respect of our sight, yet is but a very small proximity, and may be really much less; while both the moon and sun may only appear larger and brighter than the stars because they are nearer to us. (50.) A spectacle on the earth's surface is precluded, by the great mass on which he stands, from seeing into all that lies beyond his horizon. For this reason, he sees which he must look in any degree downwards. It is true that, if his place of observation be at a great eleva- tion, the whole of the earth will appear to him under a scope of vision a little more than a hemisphere, and refraction, whenever he may be situated, will enable him to look upward. But it was never meant that men should have added to his visible range what can hardly ever, unless in very extraordinary circumstances.* exceed a couple of degrees of breadth. The moon being so near the earth, and its va- pours near the horizon. Unless, then, by a change of his geographical situation, he should shift his horizon (which is always fixed), or by some movement of the con- vexity of the earth at his station); or unless, by some movements proper to the heavenly bodies, they should change their situation with respect to him; or unless, by some rotation of the earth itself on its centre, the point of its surface which he occupies should be perpetually changed; he would never see any part of the region of space; he would never obtain a sight of almost one *Such as the following, for instance: The late Mr. Baffler, the cele- brated astronomer, ascended in a balloon from Dublin about 5 o'clock in the afternoon of Monday last. He had been up about two hours when he ap- proached the English coast, when the balloon descended near the surface of the sea. He continued thus for about half an hour; after which he began to sink down. He drew out nearly all his ballast, and suddenly rising upwards again, he found himself in a balloon floating over London. This was a moment of a veteran astronomer. He subsequently descended in Wales, and then returned to Dublin. The balloon was sent from Dublin by Mr. Baffler, who was present at his ascent, and read his own ac- count of the voyage.* A balloon ascending into the sky. **CHAP. I.] CHANGE OF LOCAL SITUATION.** half the objects external to our atmosphere. But if any of these cases be supposed, more, or all, may come into view according to the circumstances. (61.) A traveler, for instance, shifting his locality on our globe, will obtain a view of celestial objects invisible from his original station, in a way which may be not in- spiringly interesting. Thus, a man who has been walking in a park close to a large tree. The massive obstacle presented by its trunk obstructs the view of all those parts of the sky which it occupies as seen from the ground; by walking round it is a complete successive view of the whole panorama of the heavens. Just as the sun rises over the horizon, and rises off from the station, so London and travel southwards, we shall not fail to notice that many celestial objects which have been invisible from London come successively into view, as if rising above the horizon, night after night from the south, although it is in the north that they rise when ascending with us northwards round the sphere; akin in their appearance to them. The novelty and splendour of fresh constella- tions are thus brought into view in the clear calm nights of tropical climates. This subject, therefore, is dwelt upon by all who have enjoyed this spectacle. A diagram showing a circle divided into four quadrants labeled "N," "S," "E," and "W." The center of the circle is marked with a dot. and never fails to impress itself on the recollection among the most delightful and interesting of the asso- ciations connected with extensive travel. A glance at D 2 42 A TREATISE ON ASTRONOMY. [CHAP. I.] the accompanying figures, exhibiting three successive stations of a traveler, A, B, C, with the horizon corresponding to each, will place this process in clearer evidence. (60.) Again: suppose the earth itself to have a motion of rotation on its centre. It is evident that a speci- tator at any station, say A, will see the sun unperceived by himself, be carried round with it: un- perceived, we say, because his horizon will constantly contain the sun, which he perceives to be a fixed object. He will have the same landscape constantly before his eyes, in which all the familiar objects in it, which serve him for a guide, will appear to move towards himself or to each other, the same invariable situations. The perfect smoothness and equality of the motions of the sun and of the objects around him participate alike, will (art. 18) prevent his entertaining any suspicion of his actual change of place. Yet, with such a view of things, it is easy to see all celestial ones which do not participate in the sup- posed rotation of the earth,-his horizon will have been all the time exactly similar to that of A. This is true in the case of our traveller in the foregoing article. Re- curring to the figure of that article, it is evidently the same thing. The spectator may ask himself whether he has been carried by the earth's rotation successively into the stations A, B, C; or whether, the earth remains stationary and he moves per- sonally along its surface to those stations. Our spectator in the park will obtain precisely the same view of the landscape as that given by our traveller; though we suppose it sawed off, and made to turn on an upright pivot, while he stands on a projecting step attached to it, and allows himself to be carried round with it. The only difference will be in his view of the tree it- self, of which, in the former case, he will see every part, but, in the latter case, only one part will remain constantly opposite to him, and immediately under his eye. (63.) By such a rotation of the earth, then, as we have supposed, the horizon of a stationary spectator will **CHAP. L.] DIURNAL ROTATION OF THE EARTH.** be constantly depressing itself below those objects which lie in that region of space towards which the rotation is carrying him, and elevating itself above those in the op- posite direction, so that he will always see the sun successively hiding the latter. As the horizon of every such spectator, however, appears to him motionless, all such objects will appear to him to be stationary, and he will represent themselves as successively disclosed and concealed. In place of his horizon approaching the stars, therefore, he will find himself continually passing over them; and when this passes over and hides any of them, he will consider them as having sunk below it, or set; while those it has just passed over from which it is receding, will seem to be rising above it. (34.) If we suppose this rotation of the earth to con- tinue forever, and that the same object shall be to be performed round one and the same axis, till it has com- pleted an entire revolution, and come back to the position from which it started, then we shall have made two reper- cussions—it is manifest that every thing will then be in precisely the same relative position on at the outset; all the objects will be exactly where they were before; and these places in the concave of the sky which they did at that instant, except such as may have actually moved in the interval between their being observed by us. The pha- nomena of their successive rising and setting, and return to the same place, will continue to be repeated in the same manner until (as they must) they become uniform) (35.) Now, in this we have a lively picture of that great law of periodicity which is exhibited in the com- parison which nature presents, the daily rising and setting of the sun and stars, their progress through the vault of heaven, and their return to the same place at the same hour of the day and night. The accom- plishment of this restoration in the regular interval of time is a proof that there is no exception to any part of that great law of periodicity; which, as we shall see, pervades all astronomy; by which expression we under- 1 *periodic*, a going round, a circulation or revolution 44 A TREATISE ON ASTRONOMY. [CHAP. I.] stand the continual reproduction of the same phenomena, in the same order, at equal intervals of time. (65.) The earth, being a solid spherical centre, if it exist and be performed in consonance with the same me- chanical laws which obtain in the motions of masses of matter under the influence of gravity, and having ordi- nary experience, must be such as to satisfy two essential conditions. It must be invariable in its direction with respect to the sun, and it must revolve about that star. The rotation must be performed round an axis or dis- tance of the sphere, whose poles, or extremities, where it meets the surface of the earth, may be called the poles, on the sphere. Modes of rotation of a solid body under the influence of external agency are conceivable, in which the point of contact between the body and the surface is at any moment revolving shall hold no fixed places on the surface, but shift upon it every moment. Such changes, however, are not possible. For suppose a body of regular figure about its axis of symmetry, per- formed in free space, and without resistance or inter- ference from other bodies. Then we have complete ab- sence of such alterations drawn with it, of necessity, the strict fulfillment of the two conditions above men- tioned. (67.) Now, these conditions are in perfect accordance with what we observe, and what recorded observation enables us to infer concerning the motions of heav- enly bodies. We have no reason to believe, from his- tory, that any semblance change has taken place since the first appearance of man. And we have no reason for two successive returns of the same star to the same point on the sky by any change in the common course. For every record that so such changes have taken place. And with respect to the other condition,-the permanence of the axis of rotation,-the appearance which any alteration in this respect would produce is not observable. But we shall presently show, by a corresponding change of a very obvious kind in the apparent motions of the stars; which, however, history decidedly declares them not to have undergone. (86.) But, before we proceed to examine more in de- **CHAP. 5. J** APPARENT DURABLE MOTION. tail how the hypothesis of the rotation of the earth about an axis accords with the phenomena which the diurnal motion of the heavenly bodies offers to our notice, it will appear that the sun, moon, and stars, when viewed from the circum- nat motion consists, and how far it is participated in by them all; or whether any of them form exceptions, whereof we shall presently give an account. We will, therefore, suppose the reader to stand himself, on a clear evening, just after sunset, when the first stars begin to appear. In this situation he will have a good view of the heavens, and a very clear and distinct view of the motions of the sun and moon. The sun will be seen to move in a circular course round his place, and to pass through various magnitudes, of which the brightest only will first catch his attention in the twilight; and more and more will approach him until he is at his meridian; and then he is surrounded with him. When he has awhile admired the calm magnitude of this glorious spectacle, the stars will appear to him to be moving in a similar spectacle which no one can view without emotion, and without a longing desire to know something of its na- ture. He will observe that they do not move in a circu- lar way on a few of the most brilliant stars, such as he cannot fail to recognise again without mistake after looking away from them; but that they move in different ap- parent situations to some surrounding objects, as build- ings, trees, &c., selecting purposely such as are in dif- ferent directions from each other. He will observe them with their respective points of reference, after a moderate interval, as the night advances, he will not fail to per- ceive that they move in a contrary direction to what is called, as by a general movement, in a westward direction ; those towards the eastern quarter appearing to rise or re- set before those which are further west; and those towards the west will be seen to approach it; and, if watched long enough, will, for the most part, finally sink beneath it. And so also will those which are further east; they will be seen to rise as if out of the earth, and joining in the general procession, will take their course with the rest towards the opposite quarters. (30.) It is possible for a considerable time in watch- **46 A TREATISE ON ASTRONOMY. [CHAP. 1.]** ing their motions, on the same or on several successive nights, he will perceive that each star appears to describe, as far as it goes, a circle of the same magnitude, in the sky; that the circles so described are not of the same magnitude for all the stars; and that those described by different stars differ in size, according to their distances from which they lie above the horizon, some which lie to-ward the north, others toward the south. The stars which lie above the horizon, some which lie to-ward the north, others toward the south. The stars which lie above the horizon, some which lie to-ward the north, others toward the south. The stars which lie above the horizon, some which lie to-ward the north, others toward the south. The stars which lie above the horizon, some which lie to-ward the north, others toward the south. The stars which lie below the horizon, and appear only for a short time above it, do not appear, after describing its sight only the small upper seg-ment of their diurnal circles; but those which lie above the horizon, some which lie to-ward the north, others toward the south, describe larger segments of their cir-cles above the horizon, remain proportionally longer in sight, said proceed precisely as far as the westward of south as they rise above it; and those which lie below it in the east remain just twelve hours visible, describe a semicircle, and so continue until they set below it; those again, which lie between the east and north, the same law obtains; at least, as far as regards the time of their remaining visible; but when they reach their highest visible segment of their diurnal circles to their whole cir-cumferences. Both go on increasing; they remain in view more than half an hour longer than those farther out are seen merely as semicircles. But the magnitudes of the circles themselves diminish, as we go from the east, northward towards the pole; and this diminution is marked by those which rise exactly in the east point. Carrying his eye further northwards, he will notice, at length, stars which, in their diurnal motion, appear to move almost at their north point, or only dip below it for a moment; while others never reach it all, but continue always above it, revolving round it without any change of place. This is true of both poles, which appears to be the common centre of all their motions, and which alone, in the whole heavens, may be considered immovable. The motion of this centre is marked by any star. It is a purely imaginary centre; but there is near it one considerably bright star, called the Pole Star, which is easily recognised by the very * We suppose our observer to be stationed in some northern latitude ; somewhere in Europe for example. A diagram showing a star's path across the sky. CHAP. I.] APPARENT BURNING MOTION. 47 small circle it describes; so small, indeed, that, without paying particular attention, and referring its position very nicely to some fixed mark, it may easily be supposed at rest, and be, itself, mistaken for the common centre about which all the stars appear to move. This is a false idea; or it may be known by its configuration with a very splendid and remarkable resemblance to a group of stars, called the "Great Bear" (60). (60.) He will further observe that the apparent rela- tive situations of all the stars among one another is not changed by their diurnal motions. For instance, if we look at their circles they are observed, or at whatever hour of the night, they form with each other any geometrical groups or figures whatever; but when the same course has been given. It is true, that in different parts of their courses, these groups stand differently with respect to the horizon; but this difference is only such as would occur if their diurnal movements they pass absolutely above and below this course, and if they had not been actually inverted with respect to the hori- zon, while, on the other hand, they always turn the same point towards the observer. The same principle has been applied to the behaviour of stars visible in or in con- cession, in the heavens, may be regarded as one great consideration, which tends to preserve with a uniform mo- tion all those objects which are attached to the surface re- tached to the internal surface of a vast hollow sphere, having the earth, or rather the equator in the centre, and touching it at every point. In this manner they can not to pass through that fixed point or pole already mentioned. (61.) Only two points have yet to be noticed before our view of a winter's nights commencing: and continuing till morning twilight, that those stars which he observed set- ting in the west rise in the east; while those which were rising when he first began to notice them have completed their courses, and are now set; and that those which were rising when he first began to notice them are now setting. This part of his observation then above is, now beneath him, and its place supplied by that which was at first under his feet, which he will thus discon- ver as he no less copiously furnished with stars than the 48 A TREATISE ON ASTRONOMY. [CHAP. I. other, and bepangled with groups no less permanent and distinctly recognisable. Thus he will learn that the great constellation of the Pleiades is situated in the northern pole, the pole is co-extensive with the whole surface of the sphere, being in reality nothing less than a universe of luminous points, which appear to be successively set in succession before his view, and referred (each lumina-ry according to its own visual ray or direction from his eye) to the same point of the horizon on which he himself occupies the centre. (See art. 49.) (62.) There is, however, one portion or segment of this sphere which is peculiarly remarkable. As there is a segment towards the north, adjacent to the pole above his horizon, in which the stars never set, so there is a corresponding segment towards the south, below his crite-ries of the more southern stars are described, in which they never rise. The stars which border upon the extreme circumference of this northern segment are at their highest point of his horizon, and show themselves for a few moments above it, precisely as those near the circumference of the northern hemisphere appear to be at night. For a moment below it, they reappear immediately. Every point in a spherical surface has, of course, another diametrically opposite point; and therefore every star which appears on his sphere into two hemispheres—a superior and inferior—there must of necessity exist a depressed pole to the south, corresponding to the elevated pole to the north. In this posi- tion surrounding it, perpetually beneath, as there is an- other surrounding the north pole, perpetually above it. He hence sees no sensible sublima-tion at all, but only a gradual change of position.—Viz.— One pole rises high, one slants beneath the mean, Such as it may be seen by observation. (63.) To get sight of this segment, he must travel southwards. In so doing, a new set of phenomena come for- ward. He will see first a number of stars belonging to those constellations which, at his original station, barely grazed the northern horizon, will be observed to sink be- low it at some distance from him; and then gradually rise again at some other place; and finally gradually for a longer part of the twenty-four hours. They will continue, however, to circulate about the same **CHAP. 1.] EFFECT OF CHANGE OF LATITUDE.** point—viz., holding the same invariable position with respect to them in the concave of the heavens among the stars; but this point itself will become gradually depress'd with respect to the spectator's horizon. The axis, in short, of the earth's rotation, which is the line of the northern pole, will appear to have become continually less and less inclined to the horizon; and by degrees, the northern pole will appear to rise, and continually surrounding it will come into view; at first momentarily, but by degrees for longer and longer times in each diurnal period, until it shall be seen at all times of the day. This was already stated in art. 61. (64.) As he travels continuously southwards, he will at length reach a line on the earthly surface, called the equator, at any point of which, indifferently, if he take up his station and recommence his observations, he will find that he has passed through a whole diurnal period; for the northern pole, occupying opposite points, the southern pole having been depressed below the horizon, and being in a new geographic position, the diurnal rotation of the heavens will appear to him to be performed about a horizontal axis passing through his own station; and so long as he remains between this line and the horizon, one half must remain above it, and half beneath his horizon, remaining alternately visible for twelve hours, and concealed during the same interval. In this case, therefore, he will see nothing from his successive view. In a night of twelve hours suppos'ing such a continuance of darkness possible at the equator, he will find that every star which rises after him—i.e. due whole hemisphere with which he began his night's observation will have been carried down beneath his horizon before he rose again upon his next view. (65.) If he pass the equator, and travel still farther southwards, the southern pole of the heavens will become more and more inclined towards him; and when it sinks below it; and the more, the farther he advances southwards; and when arrived at a station as far as the south pole of the heavens is from him, he will find that all the stars which rise north of him will be reversed. The stars which at his original station describ'd their whole diurnal circle above his horizon, and never set now describe it entirely below it, and severally set no more describe it entirely above it, and severally set no more describe it entirely below it. 50 A TREATISE ON ASTRONOMY. [CHAP. I. rise, but remain constantly invisible to him; and vice versa, those stars which at his former station he never saw, he will now never come to see. (68.) The observer, looking southwards from his first station, he travel northwards, he will observe the northern pole of the heavens to become more elevated above him, and the southern pole to sink lower under it. In consequence, his hemisphere will present a less variety of stars, because a greater proportion of the whole surface of the heavens is visible, and also constantly invisible: the circle described by each star, too, becomes more nearly parallel to the horizon; and, in short, the observer will find that he can only travel far enough to the north, he would at length attain a point vertically over the northern pole of the heavens, at which point all the stars would rise or set; for, each would circulate round the horizon in circles parallel to it. But this cannot happen, because, according to this point, which is called the north pole of the earth, but hitherto without success; a barrier of almost insurmount- able difficulty is presented by the increasing zigzag of the ellipse, but which has been overcome by some minds; and the phenomena of those regions, though not precisely similar, have been described as what must exist at the pole itself, having no other explanation than its proximity with its near superiority. A similar remark applies to the south pole of the earth, which, however, is more unapproachable, or, at least, less accessible than the north. (69.) The above is an account of the phenomena of the diurnal motion of the stars, as modified by different geographical situations, not grounded on any specula- tions, but deduced from actual observations made by voyagers. It is, however, in complete accordance with the hypothesis of a rotation of the earth round a fixed axis through its centre. It may be necessary to premise a few observations on the appear- ances presented by an assembly of remote objects, when viewed from different parts of a small and circum- scribed station. (68.) Imagine a landscape, in which a great multitude CHAP. 1.] DISTANCE OF THE STARS. 51 of objects are placed at every variety of distance from the beholder. If he shifts his point of view, though but for a few paces, he will perceive a very great change in the aspect of the objects, and will find that they appear to re- spect to himself and to each other. If he advance north- wards, for instance, near objects on his right and left, which are at a considerable distance from him, those at his original station will be left behind him, and appear to have receded southwards; some, which covered each other's eyes when he was at his original station, will ap- proach, and perhaps conceal each other. Remote objects, on the contrary, will exhibit no such great and remarka- ble changes of relative position. An object to the east of his original station, at a mile or two distance, will still be referred to him to the east point of his horizon, with hardly any perceptible deviation. The reason of this is, that the position of every object is referred by us to the surface of an imaginary sphere of an indefinite ra- dius, which we imagine to be situated in the same space in any direction, AB, carrying this imaginary sphere along with us, the visual rays AP, AQ, by which ob- jects are seen, are supposed to proceed from points which their positions with respect to the line in which we move, AB, which serves as an axis or line of reference, are situated upon the surface of this imaginary sphere round their respective objects as centres. Their intersections, therefore, p, q, with our visual sphere, will appear to recede from us in proportion to their distances; they have a angular velocity in proportion to their proximity; the same distance of advance AB subduing a greater au- g., AFB=2p, at the near object P than at the remote one Q. A diagram showing a line AB representing the observer's line of sight. Points P and Q represent distant objects. The diagram illustrates how objects appear closer or farther away depending on the observer's position. 51 52 A TREATISE ON ASTRONOMY. [CHAP. 1. (60.) This apparent angular motion of an object on our sphere of vision, arising from a change of our point of view, is called parallax, and it is always expressed by the angle BAF subtended at the object P by a line joining the observer's eye to the point of its position. For it is evident that the difference of angular position of P, with respect to two points A and B, is the difference of the two angles DBP and DAP; now, DBP being equal to the sum of the interior angles of the triangle ABP, we have hence DBP—DAP=AFP. (70.) It follows from this, that the amount of periph- etic motion arising from any given change of our point of view is, *ceteris paribus*, less, as the distance of an object viewed is greater; and when that distance is ex- tremely great, so that the angle DAP becomes very small, the parallax becomes insensible; or, in other words, it is impossible to perceive any difference in its effect. It is on this principle, that in Alpine regions visited for the first time we are surprised and confounded at the little progress we appear to make. The considerable changes of altitude for a whole day's walk, for instance, produce but a small parallactic change in the relative situations of objects seen at different distances. Whether we round a circle of a hundred yards in diameter, or merely turn ourselves round in our centre, the distant objects remain precisely in their places as perceptible; and they afford us no means to change our point of view. *The ideal sphere upon which we refer the plane of objects, is supposed to be divided into two hemispheres, and mostly connected by association, so that only one hemisphere can be seen at once. When we look at one hemisphere, and then at another corresponding point for point to the spherical sphere. On this view, the real sphere appears to be divided into two hemispheres, and each hemisphere may be on the imaginary surface of the heavens. When the whole surface of the sphere is seen at once, it appears as one single sphere; but if only half of its surface is visible without on. Thus we become impressed with the idea that all objects are equally distant from our own mind; and it is true even of all visible angular dimensions and angular motions. The same idea is conveyed by the expression "at a distance," and inconveniently in language; but it may always be mentally made. This relation's proper agency on our eye is the Pinnacles." CHAP. I.] DISTANCE OF THE STARS. (7.1.) Whatever notion, in other respects, we may form of the stars, it is quite clear they must be im- menseously distant. Were it not so, the apparent angular inclination of the stars to each other would be much greater than when seen near the horizon, and the constellations, instead of preserving the same ap- parent shape, would appear to change their general course, would appear to enlarge as they rise higher in the sky, as we see a small cloud in the horizon swell into a large mass, or as a breeze blows across the wind across our zenith, or as may be seen in the annexed figure, where $a$, $A$, $b$, are three different positions of the same star, as they would, if near the earth, be seen from spectator $S$; under the view of observer $O$. ASB. No such change of apparent dimension, however, is ob- served. The nicest measurements of the apparent an- gules between any two stars have been made by many parts of their diurnal course, (after allowing for the un- equal effects of refraction, or when taken at such times that the sun is not visible,) and yet no one has manifest not the slightest perceptible variation. Not only this, but at whatever point of the earth's surface the measurement is made, the angles are found to be identically identical. No instruments ever yet invented by man are delicate enough to indicate, by an increase or dimi- nution of angle, a difference of 0.0000001° in the earth's nearer to or further from the stars than another. (7.2.) The necessary conclusion from this is, that the dimensions of the earth, large as it is, are comparatively nothing, absolutely imperceptible, when compared with A diagram showing three different positions of a star labeled 'a', 'A', and 'b'. The diagram also shows a horizon line with points labeled 'X', 'Y', and 'Z'. A small circle is drawn around point 'X' indicating a small cloud. 3 54 A TREATISE ON ASTRONOMY. UMAF. L. the interval which separates the stars from the earth. If an observer walk round a circle not more than a few yards in diameter, and from different points in its circumference observe the same star, with the exact instrument adapted for the purpose, the angles PAQ, PBQ, PCQ, subtended at those stations by two well defined points in his visible horizon, PQ, he will at once A diagram showing three points P, Q, and C on a circle, with lines connecting them to a point B outside the circle. The angle PAQ is shown as being equal to the angle PBQ, and the angle PCQ is shown as being equal to the angle PBQ. be adverted to, by the difference of the result of his change of distance from their distance from his change of place, although that difference may be so small as to produce no change in their general aspect to his unassisted sight. But this is only true when we consider where accurate measurement obtained by instrumental means places us in a totally different situation in respect to matter and space; but when we consider what from what we should hold, were we to rely in all cases on the mere judgment of the eye. To so great a nicety was accuracy required in ancient times as to call for an instrument called a theodolite, that a circle of the diameter mentioned may thus be rendered sensible, any thing that might be seen through it. In this manner place, by reference to objects distant by fully 100,000 times its own dimensions. Observations, differing, it is true, somewhat from each other, have been made by people and executed with nearly as much exactness, have been applied to the stars, and with a result such as has been already stated. Hence it follows, inconceivably, that the distance of the stars from the earth cannot be so **CHAP. L.]** **DISTANCE OF THE STARS.** small as 100,000 of the earth's diameters. It is, indeed, impossible for any spectator to find hereafter find it clearly demonstrated that the distance so named, immense as it may appear, is yet much underrated. (73.) The earth being seen by a spectator with our faculae, and furnished with our instruments, the earth would be imperceptible; and, reciprocally, an object of the same size would be equally indiscernible. If, therefore, at the point on which a spectator stands, we draw a plane touching the plane of the horizon, and passing through the region of the stars, and through the centre of the earth, we conceive another plane parallel to the former, and co-axial with it, and let this second plane be traversed throughout their whole extent by the same interval, viz. a semi-diameter of the earth, will yet, on account of the difference of their distances from us, they will be found bound together, and indistinguishable from each other in the region of the stars, when viewed by a spectator on the earth's surface. This is because they have but a recent breadth to his eye, and will only mark out a great circle in the heavens, which, like the vanishing point in perspective drawing, appears to converge towards one point; but if they do not converge, is in fact, the vanishing line to which all planes parallel to the horizon offer a similar appearance of distance; and this is called the rational horizon. (74.) The two planes just described are termed, in astronomy, the sensible and rational horizon of the ob- server's eye. The plane which represents that part of space which marks their vanishing line, is also spoken of as a circle of the sphere, under the name of the celestial horizon, simply. From what has been said (art. 72) of the distance of the stars, it follows, that if we suppose a spectator on the earth's surface to look up into space occupied by the rational horizon, in the same manner as that of a corresponding spectator on the surface at by his sensible horizon; he will see all objects situated in the same relative situations, each beholding that entire hemi-sphere of the heavens which is above the celestial horizon, corresponding to their common zeniths. 56 A TREATISE ON ASTRONOMY. chap. l (76.) Now, so far as appearance goes, it is clearly the same thing whether the heavens, that is, all space, with its contents, revolve round a spectator at rest in the earth's centre, or whether this spectator simply turn round in the opposite direction to the heavens, and thus cause their accension. The aspect of the heavens, at every instant, as referred to his horizon (which must be supposed to turn with him) will be exactly the same in both cases. And since, as has been shown, appearances are also, so far as the stars are concerned, the same to a spectator on the sur- face of the earth as they would be to one who stood upon the heavens, we may suppose the heavens to revolve without the earth, or the earth within the heavens, in the opposite direction, the diurnal phenomena being produced by them in no different manner. (76.) The Copernican astronomy adopts the latter as the true explanation of these phenomena, avoiding there- by the necessity of supposing any such extraordinary mechanism of a solid but invisible sphere, to which the stars must be supposed to adhere, in that they can only be carried round the earth without derangement of their relative situations unless some such contrivance exists. Such a contrivance would, indeed, suffice to explain the diurnal revolution of the earth, and consequently of all other bodies revolving about the sun and moon, so well as those of the planets, is in- compatible with any such supposition. It will appear when we consider what follows. On the other hand, that a spherical mass of moderate dimensions (or, rather, when compared with our terrestrial size), capable of revolving and to resolve, should do so, in conformity with those general laws which govern all motions of material bodies, is so far from being a postu- lative difficulty to be conceded, that the wonder would rather be supposed that it could not have been done. As regards this point, however, we shall not further regard it here and in it the pro- gress of our work, analogies offer themselves in its sup- port from what we have already said. In fact we shall not even mention them out to the reader's notice. Meanwhile, it will be proper to define a variety of terms which will be continually employed hereafter. (77.) Heliocentricism. The axis of the earth is that di- **CHAP. I.** **DEFINITIONS.** 57 ametris, about which revolve, with a uniform motion, from west to east; performing one revolution in the inter- val whose elapsen between any star leaving a certain point in the heavens, and returning to the same point again. (78.) Def. 2. The poles of the earth are the points where its axis meets its surface. The North Pole is that amongst the stars which is nearest to the north pole of the earth (79.) Def. 3. The sphere of the heavens, or the sphere of the stars, is an imaginary spherical surface of infinite radius, described by a line continually moving in such a way that comes to the very same thing, the eye of any spectator on its surface, for its centre. Every point in this sphere is at an equal distance from the centre; and all lines parallel to that radius of the sphere which passes through it, seen in perspective from the earth ; and any great circle drawn on this sphere, is a straight line of planes parallel to itself. This mode of conceiving such points and circles has great advantages in a variety of cases. (80.) Def. 4. A plumb-line is a line drawn along the axis of the sphere of the heavens, vertically over the specta- tor's head, and vertically under his feet; they are, therefore, always perpendicular to each other, and are always paral- lel to the direction of a plumb-line at his station. The plumb-line itself is, at every point of the earth, perpen- dicular to the horizon; but it does not remain so; there- fore, can the actual directions of two plumb-lines be re- garded as mathematically parallel? They converge together towards some point in the heavens, called the Celestial Pole (as in the case of a building—in one and the same town, &c.), the difference from exact parallelism is so small that it may be considered as zero. An angle of a minute corresponds to a convergence of plumb-lines amounting to about one minute. The zenith and nadir are the poles of this sphere; they are always at right angles to each other from every point in it. The celestial horizon itself is the vanishing line of a system of planes parallel to the sensible and rational horizon. *From Arabic words Nadir corresponds evidently to the German nach-norden.* A diagram showing the Earth's axis intersecting its surface at two points called poles. 58 A. THEOREM ON ASTRONOMY. [CHAP. I.] (31). *Def.* 2. *Vertical circles of the sphere* are great circles passing through the zenith and nadir, or the points where the meridians intersect the horizon. (32). *Def.* 6. The poles of the heavens are the points of the sphere to which all lines drawn from the centre are perpendicular. (33). *Def.* 7. The equator of the earth's sphere is directed; or the vertical circle of all meridians is parallel to it. (34). *Def.* 7. The earth's equator is a great circle on its surface, equidistant from its poles, dividing into two hemispheres equal parts, one of which contains the midist of which are situated the respective poles of the earth of those names. The plane of the equator is, therefore, a perpendicular plane to the axis, and passing through its centre. The celestial equator is a great circle of the heavens, marked out by the indefinite extension of all lines drawn from the pole parallel to that line of all planes parallel to it. This circle is called by astronomers the equator. (35). *Def.* 7. The meridian of a station on the earth's surface is a great circle passing through both the poles and through the place. When its plane is perpendicular to the axis, it is called the celestial meridian of a spectator stationed at that place. When we speak of the meridian of a spectator, we intend the celestial meridian of that place, or any other place passing through the poles of the heavens. The plane of the meridian is the plane of a circle, and its intersection with a perpendicular plane of the horizon is called a meridian line, and marks the north and south poles of that circle. (36). *Def.* 9. Azimuth is the angular distance of a celestial object from the north or south point of the hori- zon (zenith) when it is in the northern or southern hemisphere; or when this object is referred to the horizon by a vertical circle; or when it is the angle comprised between two vertical planes passing through this object and one pole, and through its horizon. The altitude and azimuth of an object being known, therefore its place in the visible heavens is determined. For their simulatenea measurement, a peculiar instrument has been im- plemented. CLAR. 1.2 LATITUDE AND LONGITUDE. given, called an altitude and azimuth instrument, which will be described in the next chapter. (88.). Daz. 10. The latitude of a place on the earth's surface is its angular distance from the equator, measured on the parallel of latitude through that place, in degrees, minutes, and seconds, from 0 to 90°, and northwards or southwards according to the direction of the meridian line. Thus the observer at Greenwich is situated in 51° 28' 40" north latitude. This definition of latitude is well observed, is to be considered as only temporary. A knowledge of the laws of physics, the motions and figures of the earth, and a better acquaintance with the niceties of astronomy, will give a more accurate designation of its true position than any manner of consideration is likely to convey. (87.). Daz. 11. Parallel of latitude are small circles on the earth's surface parallel to the equator. Every point in this circle has the same latitude, which is expressed by said to be situated in the parallel of 51° 28' 40". (86.). Daz. 12. The difference between the vertical surface and the inclination of its meridian to that of reckoning longitude is called declination. Greenwich for this station ; foreigners, the principal observatories of their respective nations, the principal observatories having been established at Paris and Greenwich we speak of longitude, we reckon from Greenwich. The longitude of a place is, therefore, measured by the arc of a great circle passing through that place and that of Greenwich, or, which is the same thing, by the spherical angle at the pole included between these meridians. As latitude is reckoned north or south, so longitude is usually said to be reckoned west or east. It would seem that this was done for convenience; but it tends much to avoid confusion and ambiguity in computations, were this mode of expression abandoned, and longitudes reckoned from the prime meridian instead of from round the whole circle from 0 to 360°. Thus the longitude of Paris is, in common parlance, either 2° 30' 22" east, or 357° 30' 22" west of Greenwich. But in the sense on which we shall hereafter use it recommended others 60 A TREATISE ON ASTRONOMY. [CHAP. I. to use the term, the latter is its proper designation. Longitude is also reckoned in time at the rate of 24 h, for 360°, or 18° per hour. In this system the longitude of Paris is 20°, and that of London 7°. (58.) Knowing the longitude and latitude of a place, it may be laid down on an artificial globe; and thus a map of the heavens may be constructed, showing the several countries as detached portions of this general map, extended into planes; or, rather, they are representations on planes of such portions, extended according to certain conventions. These portions are called "constellations," the object of which is either to distaste as little as possible the outlines of countries from the surface of the globe in order to facilitate the means of determining, by pro- jection or graphical measurement, the latitudes and longi- tudes of places which occur in them, without referring to the globe, or even to it for some purposes. See chap. III. (59.) The globe, or map, made up of the heavens, as well as charts of particular parts, may also be constructed, and the stars laid down in their proper situations rela- tive to one another, so as to represent the whole heavens and the celestial equator. Such a representation, once made, will exhibit a true appearance of the stars as they pre- sent themselves to our eyes; but even when seen from the surface, or as they may be conceived to be seen at once by one at the centre of the globe. It is, therefore, in- dependent of any terrestrial observation; and will never occur in such a representation neither zenith, nadir, nor horizon—neither east nor west points; and although great care has been taken to make all those parts corre- sponding to terrestrial meridians, they can no longer, in this point of view, be regarded as the celestial meri- dians of the heavens. For instance, if we consider the course of one diurnal revolution, every point in it passes beneath each other. It is on account of this change of position that astronomers have adopted a com- plete distinction between the two branches of Geo- graphy and Cosmography, that astronomers have adopted different terms (viz. declination, and right) r. To the earth, r to describe or represent r the heavens. **CHAP. I.** **LATITUDE AND LONGITUDE.** *ascending* to represent those arcs in the heavens which correspond to *latitudes and longitudes* on the earth. It is for this reason that they are the equator of the hea- vens, and the circles which encircle the poles of the earth are called *heaven circles* in the heavens, and the angles which include between them at the poles are called *heaven angles*. The names of these circles and angles had been used consistent with this nomenclature, no confusion could ever have arisen; but unfortunately, this arbitrary system has been adopted by many, and the same latitude and longitude in their cosmography, in speaking of arcs of circles not corresponding to any point on the earth by the same words, and thus giving reference to the motion of the sun and planets among the stars. It is now too late to remedy this confusion, which is ingrained into every one's mind, and will continue to do so unless we reform, and warn the reader of it, that he may be on his guard when, at a more advanced period, he would say "I shall go to such a place," etc., and use the terms in their celestial sense, at the same time urgently recommending to all who desire the advance of science to use their places. (91.) Astronomers reckon from a supposed fixed meridian, or from a determinate point on the equa- tor; so right ascension and declination require some definite circle or arc, or some known point on the equinoxial, as the commencement of their reckoning, or their zero point. The stars which pass through this circle or arc have been observed by astronomers, and therefore would have been no particular advantage in this; and astronomy has appeared to be a science founded upon the observation of the heavens, through which they sup- pose the hour circle to pass, from which all others are recog- nized; while poor men know the zero point of all right ascensions common on the equinoxial. The right ascensions of celestial objects are always reckoned eastward from the equinox, and are estimated either by means of a quadrant or sextant; in the case of terrestrial longitudes, from $0^\circ$ to $360^\circ$, which com- pletes the circle; or it is time, in hours, minutes, and seconds, since noon; or it is a certain number of degrees of the heavens being contrary to the real motion of the A diagram showing celestial coordinates. 62 A TREATISE ON ASTRONOMY. [CHAP. I. earth, this is in conformity with the westward reckon- ing of longitudes. (Art. 87.) (60.) Observations are reckoned by the diurnal motion of the stars, or rather of that point in the equinoctial from which right ascensions are reckoned. This point may be considered as fixed, but it is not so fixed there ; and, moreover, the point itself is liable to a cer- tain slight variation; now, however, as not to affect, perceptibly, the interval between two successive returns to the meridian of the day. This interval is called a side- real day, and is divided into 24 sidereal hours, and these again into minutes and seconds. The time which has passed sidereal time, i.e., which goes uniformly at such a rate as always to show 0 h. 0 m. 0 s. when the equinox comes on the meridian, is called sidereal time. It is an in- dispensable piece of furniture in every observatory. (61.) It remains to illustrate these descriptions by reference to a figure. Let C be the centre of the earth, A diagram showing a celestial sphere with various lines and points labeled. The diagram includes a central point C, a line AB representing the apparent altitude of the sun at A on the surface of the earth, a line AP parallel to AC, and a line AZ extending from A towards Z on the horizon. NCS its axis; then we N and S its poles; EQ its equa- tor; AB the parallel of latitude of the station A on the surface; AP parallel to AC, the direction in which an observer at A will see the elevated pole of the heavens ; and AZ, the prolongation of the terrestrial radius CA, Z P S E C Q B A CHAP. 1. DEFINITIONS EXEMPLIFIED. that of his zenith. NAES will be his meridian; NGS that of some fixed system, as Greenwich, and GE, or the spherical angle GNE, its longitude, and EA his la- titude. Now, if we suppose the earth to be a sphere in A, this will be his sensible horizon; n.a marked on that plane by its intersection with his meridian will be his sensible meridian, and h.n. a'de north and south points of his horizon. (94.) Again, neglecting the size of the earth, or con- centricity of its surface, let us consider any very pre- cise thing to its rational horizon; let the annexed figure represent the sphere of the heavens; C the spectator; Z the pole; PZ the polar distance of any star or object S; PS the polar distance of any star or object S, referred to the equinoctial by the hour circle PST pS and BBD will represent the altitude of any star or object S above the pole. Again, if we refer it to the horizon by the vertical circle ZSA, HA will be its azimuth, AS its altitude, and ZA its declination. Thus (1) we see that both north and south are east and west points of his horizon, A diagram showing a celestial sphere with various lines and angles labeled. The center is labeled "C" for the spectator. Lines extend from C to Z (the pole), PZ (polar distance of any star or object S), and PS (polar distance of any star or object S referred to the equinoctial). Lines also extend from C to ZA (declination), AS (altitude), and HA (azimuth). The diagram includes labels for these elements: ZP for polar distance, PS for polar distance referred to the equinoctial, ZA for declination, AS for altitude, and HA for azimuth. the heavens; HP the altitude of the pole, and HPZEO his meridian; ETQ, a great circle perpendicular to HP, will be the equinoctial; and if W represents the equinox, w.e represents the winter solstice; and if Y represents the summer solstice, y.e represents the vernal equinox; and if D represents the day line, d.e represents the night line. A TREATISE ON ASTRONOMY. 64 or of the heavens. Moreover, if HA, Oo, be small cir- cles, or parallels of declination, touching the horizon in its north and south points, HA will be the circle of per- petual observation, and Oo the circle of the depressed pole, the stars never set; so that of perpetual occultation, between which and the depressed poles they never rise. In all other cases, the circles of the horizon, or poles they rise and set, any one of them, as S, remaining above the horizon, in that part of its diurnal circle represented by ABa, will be called the meridian of the place; and by ADa. It will exercise the reader to construct this figure for several different elevations of the pole, and for various places on the earth's surface; and to see how following consequences result from these definitions, and are propositions which the reader will readily bear in mind. (05.) The altitude of the elevated pole is equal to the latitude of the spectator's geographical station. For, comparing the angle PAZ between the pole and zenith, with the angle PAZ between the pole and zenith, in the one figure, which is the co-altitude (complement to 60° of the altitude) of the pole at that place; we shall find that it is either C or AN being parallel whose vanishing point is the pole. Now, NCA is the co-altitude of the plane A. (06.) The difference between two meridians is equal to the meridian, successively, of every place on the globe once in twenty-four aerial hours. And, since the di- urnal rotation of the earth is equal to twenty-four hours, which eclipses between the same star coming upon the meridian of two different places is measured by the dif- ference between their latitudes. (07.) Vice versa—the interval eclipsing between two different stars coming on the meridian of one and the same place is equal to twenty-four hours, or to a measure of difference in right ascensions of the stars. This explains the reason of the double division of the equator into degrees and minutes. (08.) The equinoctial intersects the horizon in the east and west points, and the meridian in a point whose alti- tude is equal to the co-altitude of the place. Thus, at **CHAP. I.** STARS VISIBLE BY DAY. 85 Greenwich, the altitude of the intersection of the equinoctial and meridian is 36° 31' 20" (100). All stars which are visible to the greatest altitude on the meridian; which is, therefore, the best situation to observe them, being least concealed by the vapours of the atmosphere, as well as least displaced by the horizon. (100.) All celestial objects within the circle of perpeneclines, that is, those lying on the meridian, above the hori- zon, in every diurnal revolution, appear once on one side of the pole. Those are called their upper and lower horns. (101.) We shall conclude this chapter by calling the reader's attention to a fact, which, if he now learns for the first time, will be of great use to him. The stars continue visible through telescopes during the day as well as the night; and that, is proportion to the power of the instrument employed. For instance, in a clear sky, even those of inferior lustre, such as scarcely strike the eye at night at all conspicuous, are readily found by means of a telescope. And so also, the appearance of the sky which is very near the sun,--by those who possess the means of pointing a telescope accurately to the proper point of observation.--the appearance of a star, such as well, or the shaft of a mine, such bright stars as pass the zenith may even be discerned by the naked eye; and we have seen before that in some circumstances, that the earliest circumstance which drew his attention to astronomy was the regular appearance, at a certain hour of each day, of a considerable star, through the shaft of a chimney. p 2 66 A TREATISE ON ASTRONOMY. [CHAP. II CHAPTER II. Of the Nature of astronomical Instruments and Observations in general— Of aerial and solar Time—Of the Measurements of Time—Clocks, Chronometers, and other Instruments for the Purpose—The Various Instruments of the Telescope—Its Instruments designated in that System—The Magnifying Power of the Telescope—The Refracting Telescope—The Reflecting Telescope—The Level—Phenomena—Artificial Horizon—Collimator of com- mon use—The Transit Instrument—The Quadrant—The Quadrant and Armament Instrument—Of the Barometer and reflecting Circle—Princi- ples of Perspective. (102.) Our first chapter has been devoted to the acquisition chiefly of preliminary notions respecting the globe we inhabit, as relation to the celestial objects which surround us, and to the means by which observations must be made, as well as to provide ourselves with a stock of technical words of most frequent use in astronomy. We have not proceeded to a more exact and detailed statement of the facts and theories of astronomy; but in order to do this with full accuracy, it is necessary to make an investigation made acquainted with the principal means which astrono- mers possess, of determining, with the degree of exactness they require, the truth of their conclusions; in other words, of ascertaining by measure- ment the apparent and real magnitudes with which they are conversant. It is only when we have such knowledge that he can fully appreciate either the truth of the theories themselves, or the degree of reliance to be placed on them. The astronomer is therefore fortunate since it is only by knowing what amount of error can certainly be perceived and distinctly measured, that he can estimate how nearly his results approach to truth. In this approximation, its numerical results, to actual phe- nomena, will so nearly bring him to receive it as a true repre- sentation of reality. (103.) Astronomical instrument-making may be justly regarded as the most refined of the mechanical arts, and is in what manner an approach to geometrical precision required, and has been attained. It may be thought **CHAP. II.]** FRACTURAL DIFFICULTIES. an easy thing, by one unacquainted with the niceties re- quired, to turn a circle in metal, to divide its circumfe- rence into 360 equal parts, and these again into smaller sub- divisions, so that each part may be marked off and ad- just it in a given position ; but practically it is found to be one of the most difficult. Nor will this appear extraordina- rily difficult when we consider that, owing to the imperfection of telescope to the purposes of angular measurement, every imperfection of structure or division becomes magnified. This imperfection is not only due to the imperfectness of glass that thus, not only direct errors of workmanship, arising from unsteadiness of hand or imperfection of tools, but those im- perfections which arise from the nature of the materials used, causes, such as the unequal expansion and contraction of metallic masses, by a change of temperature, and their different degrees of hardness, which cannot be measured, become perceptible and measurable. An angle of one mi- nute occupies, on the circumference of a circle of 10 inches diameter, an arc of about 0.05 inch. The percep- tibility is too small to be certainly dealt with without the use of magnifying glasses; yet one minute is a gross quan- tity in astronomical observations. With the instruments now employed in observatories, a single second, or the 60th part of a minute, is rendered a dis- tinguishable quantity; but even this is not always true. A circle, subtended by one second, is less than the 200,000th part of the radius, so that on a circle of 6 feet in diameter (which is very large), it is less than \(\frac{1}{12}\)th part of an inch : a quantity requiring a powerful microscope to be discerned at all. Like any one figure to have been discovered by accident, it was not until the circum- ference of a metallic circle of such dimensions (supposing the difficulty of its construction surmounted) 360 marks, divided into 60 minutes and 60 seconds, had been placed in their places within such minute limits ; to say nothing of the subdivision of the degrees so marked off into minutes, and seconds. It is evident that this would have been extremely baffled, and will probably for ever continue so baffled, the utmost stretch of human skill and industry ; nor, if exe- cuted, could it endure. The ever varying fluctuations of heat and cold have a tendency to produce net movement tem- 68 A TREATISE ON ASTRONOMY. [CHAP. II.] porary and transient, but permanent, uncompensated changes of form in all considerable masses of those metals which are employed in the construction of astronomical weights, however symmetrically formed, must always be unequally sustained, since it is impossible to apply the matching principles of the balance to such cases. And could this be done, at all events force must be used to move and to fix them, which can never be done without producing enormous inequalities in the motions of the planets. It is true, by dividing them on their centres, and in the identical places they are destined to occupy, and by a thousand ingenious contrivances, the most perfect accuracy may be accomplished in this department of art, and a degree of perfection has been given, not merely to chefs d'œuvre, but to instruments of the highest precision; yet in ordinary use, which, on due consideration, must appear very surprising. But though we are enabled to look for errors in the workmanship of men, we have no reason to expect miracles. The demands of the astronomer will always surpass the power of the brain and hand; therefore he must make his own tools for himself, and therefore he must make his own observations; for to make himself, as far as possible, independent of the imperfections which enter into every work the latter can place in his hands. He must learn how to choose his materials, how to observe with accuracy, so as to choose his opportunities, and so as to familiarize himself with all the causes which may produce faults in his instruments. In short, all that all these perfections of structure and material of each instrument is nothing else than a preparation for its use; and it is only by making himself master of all these powers, both in theory and practice, that he can hope to possess all that is true, and reject all that is erroneous. It is in this way that he will acquire a knowledge of nature; -- an art of itself of a curious and intricate nature, and of which we can here only notice some of the leading and general principles. (104.) The great aim of the practical astronomer be- gins numerical correctness in the results of instrumental measurements; and therefore constant care and vigilance must be directed to the detection and correction of errors either by annihilating, or by taking account of, and al- lowing for them. Now, if we examine the sources from **CHAP. 11.] CLASSIFICATION OF SOURCES OF ERROR.** which errors may arise in any instrumental determination, we shall find them chiefly reducible to three prin- cipal heads: (106.) 1st. External or incidental causes of error; comprehending such as depend on external, uncontrol- lable circumstances; such as, fluctuations of weather, which affect the temperature of the instrument, alter- lated value, and being reducible to no fixed law, induce uncertainty to the extent of their own possible magni- tude. Such errors are often so minute as to be un- varry also the form and position of the instruments used, by altering relative magnitude and the tension of their parts. (106.) 2dly. Errors of observation: such as arise, for example, from inexperience, defective vision, allowance of time for the effect of light upon the eye, or some pheno- menon, or precipiency in anticipating it, &c.; from atmo- spheric indistinctness; insufficient optical power in the instruments; and other causes which do not alone be cause all errors arising from momentary instrumental deterangement—slips in clamping, looseness of screws, &c. (106.) 3dly. Errors of measurement: such are numerous causes of error to which astronomical measurements are liable, arise from causes which may be deemed instru- mental, and others which are not so much of a principal class. The first comprehend those which arise from an instrument not being what it professes to be, which is usually called "inexactitude," that is, instead of being, as it ought, exactly circular, it is slightly flattened, or instead of being, as it ought, exactly perpendicular, it is slightly inclined; or instead of being perfectly level, if this circle (so called) be in reality not exactly circular, or not in one plane—-if its divisions intended to be parallel with each other are not parallel at unequal intervals,—and a hundred other things of the same sort. These are not mere speculative sources of error, but real ones, which every observer has to contend with. (108.) The other subdivision of instrumental errors comprehend such as arise from an instrument not being placed in the position it ought to have; and from these 70 A TREATISE ON ASTRONOMY. [CHAP. II] of its parts, which are made purposely moveable, not being properly disposed inter se. These are errors of adjustment. Some are unavoidable, as they arise from a general defect in the instrument itself, to which the instruments are placed ; which, though too minute to be noticed in any other way, become appreciable in deliberate observation. Others are accidental, arising from the consequences of imperfect workmanship, as where an instrument once well adjusted will not remain so, but keeps levelling itself into a different position. The first class of error arises from the non-existence of natural indications, other than those afforded by astrono- mical observations. For instance, the axis of the earth has or has not the exact position, with respect to the horizon and its cardinal points, the axis of the earth, or to other objects, which may be used for guidance, which is ought to have a fulfil propery its objects. (109.) Now, with respect to the first two classes of error, it appears that they are either incapable of being reduced to known laws, and thereby become subjects of calculation and due allowance, they actually vitiate, to their full extent, all observations made with them; or they subvert Being, however, in their nature causal and accidental, their effects necessarily lie somewhere one way, and somewhere else another way; and sometimes tending to increase the results. Hence, by greatly multiplying observations, under varied circum- stances, and comparing them with each other, we shall find this class of error may be so far subdued, by setting them to destroy one another, as no longer sensibly to violate any law of nature. This is true of both the great and indeed only reason against such errors not merely to the astronomer, but to the investigator of nu- merical relations in general. (110.) With regard to errors of adjustment and work- manship, not only the possibility, but the certainty, of their existence in any particular instrument must be contemplated. Human hands or ma- chines never formed a circle, draw a straight line, or erect a pyramid; nor ever put instruments in perfect adjustment, unless accidentally ; and then only **CHAP. II.] MUTUAL DESTRUCTION OF ERRORS.** 71 during an instant of time. This does not prevent, how- ever, a great approximation to all the real degradations should be attained. But it is the peculiarity of astrono- mical observation to be the ultimate means of detection of errors, and to be the only one which can attain minime- ness ever other mode of detection. What the eye cannot discern, nor the touch perceive, a course of astronomi- cal observations may detect with perfect accuracy. The imperfect products of man's hands are here tested by being brought into comparison with the perfect work- manship of nature, and this is done at once after the trial. Now, it may seem like arguing in a vicious circle, to deduce theoretical conclusions and laws from observa- tions made under circumstances which are inconsistent with those upon which observations were made, accuse them of imperfection, and attempt to detect and rectify their errors by means of observations similar to those by which they have helped us to a knowledge of. A little consi- deration, however, will easily show that such a course of argument is fallacious. (111.) The steps by which we arrive at the laws of natural phenomena, especially those which depend for their elucidation on exact determinations, are necessarily successive. Gross errors and palpable laws are arrived at by rude observation and rough instru- ments; but they are gradually refined at all; and these are corrected and refined upon by finer scrutiny with more delicate means. In the progress of this, subordinate statements and numerical results of those which first ob- serve, are often found to be erroneous or doubtful; these are called out, and reduced to certainty; others, again, subordinate to them, make their appearance, and become subjects of further investigation; and so on until we have reached (and the reason is evident) that the first glimmering of each such subordinate law—the first form in which they are simply shadowed out—has been fully developed and proved. We see then the difference between what we expect and what we find. The first occurrence of such a dis- cordance we attribute to accident. It happens again and again; and we begin to suspect our instruments. We 78 A TREATISE ON ASTRONOMY. [CAP. II. then inquire, to what amount of error their determina- tions can, by possibility, be liable. If their limits of pos- sible error exceed the observed deviation, we at once conclude that they are defective either in construction or adjustments. Still the same deviations occur, and so far from being palliated, are more marked and better defined than before. We have then seen that we are on the traces of a law of nature, and pursue it till we have reduced it to a definite statement, and verified it by repeated observation, under every variety of circumstances. (118.) Now, in the course of this inquiry, it will not fail to occur to some one who has not been taught by experience, we suspect the existence of some natural law, before unknown; we tabulate (i.e. draw out in order) our observations, and find them per- ceive, in this symptomatic statement of them, distinct indica- tions of a regular progression. Again we improve or vary our observations, and find the same indications pro- posed new law of nature altogether, or find it replaced by some other, of a totally different character. Thus we see that the law which we have supposed is not what we have noticed. We examine, therefore, the theory of our instrument; we suppose defects in its struc- ture, and, finding these do not account for the dis- tinction in introducing actual errors into its indications. These errors have their laws, which, so long as we have no knowledge of any other law which may con- founded with laws of nature, and are mixed up with them in their effects. They are not fortuitous, like errors of observation; they are inherent in the instrument itself, and unchangeable while it and its adjustments remain unchanged, they are reduc- ble to fixed quantities; they are not accidental defects, whether of structure or adjustment, producing its own appropriate form of error. When these are thor- oughly investigated and understood, we shall find one which coincides in its nature and progression with that of our observed discordances. The mystery is at once solved; it is detected by direct observation, as In- strumental defect. **CHAP. II. DETECTION OF INSTRUMENTAL ERRORS.** (113.) It is, therefore, a chief requisite for the practi- cal astronomer to make himself completely familiar with the theory of his instruments, so as to be able at once to decide what effect any observer may give imperfec- tions in the structure or adjustment of them under given circumstances under which an observation can be made. It is evident that this object cannot be attained until he requires that a circle should be exactly concentric with the axis on which it is made to turn. As this is a condi- tion which is not always fulfilled, it becomes neces- sary to inquire what errors will be produced in observa- tions made and registered of the facts of such an instru- ment as a circle, when this condition is not fulfilled; and, to say, what would be the disagreement between obser- vations made with it and with one absolutely perfect. Now, it is a simple matter to show by geometrical geo- metry that whatever be the extent of this devia- tion, it may be annulled in its effect on the result of any observation by making use of the other side of the circle, by the very easy method of reading off the divisions on two diametrically opposite points of the circle, and tak- ing their mean. Thus, if we wish to increase one such reading by just the same quantity by which it diminishes the other. Again, suppose that the circle is not perfectly circular, but that its diameter should be exactly parallel to that of the earth. As it never can be placed or remain so, it becomes a question, what amount of error will be introduced into any observed deviation, whether in a horizontal or vertical plane, from this precise position. Such inquiries constitute the theory on which all observations are founded, and they are impor- tant to practice, and one of which a complete knowledge will enable an observer, with very moderate instrumental means, to obtain results which shall appear to belong only to the most refined and costly. In the present work, however, we have no further concern with this subject; but those who desire to learn how to de- scribe in this chapter will be considered as perfect both in construction and adjustment. (114.) As the above remarks are very essential to a right understanding of the philosophy of our subject and G 74 A TREATISE ON ASTRONOMY. [CAP. II. the spirit of astronomical methods, we shall elucidate them by taking a case. Observers previous, before the invention of astronomical instruments, had already con- cluded that the stars did not always appear to be performed in circles about fixed poles in the heavens, as shown in the foregoing chapter. In drawing this con- clusion, he supposed that the stars were seen at night, if forced on their noise by its great magnitude in the immediate neighbourhood of the horizon, was regarded as a local effect, which would not extend further over. As soon, however, as the diurnal paths of the stars were attempted to be traced by instruments, even of the coarsest kind, it was found that they described no such circles described about one and the same pole would not represent the phenomena correctly, but that, owing to some cause or other, the stars appeared to be seen at every star distorted from a circular into an oval form, its lower segment being flatter than its upper ; and the de- viation was greater when the star was near the horizon, the effect being the same as if the circle had been squeezed upwards from below, and the lower part more than the upper. This deviation could not have ex- tended to arise from no casual or instrumental cause, it became necessary to seek a natural one ; and reflection really occurred between the observer's eye and the case precisely analogous to what we have already (art. 47) noticed, of the apparent distortion of the sun near the horizon, caused by reflection from his own great altitudes. This new law once established, it became ne- cessary to modify the expression of that anciently re- ceived by astronomers, who had never observed this distinction or by making a distinction between the apparent diurnal orbits, as affected by refraction, and the true ones cleared of this effect. (118.) Again : The first impression produced by a view of the diurnal movement of the heavens is that all the heavenly bodies move in circles about some com- mon period, viz. a day, or 24 hours. But no sooner do we come to examine the matter instrumentally, i. e. by noting by means of instruments how long it appears to the motion, than we find differences which cannot be ac- **CHAP. 11.** LAWS TRACED BY OBSERVATION. 75 counted for by any error of observation. All the stars, it is true, occupy the same interval of time between their successive appulses to the meridian, or to any vertical circle; but they are very different one from another, as explained by the following table; in which, instead of 24 hours, such hours as our calendar assigns to each day are given. Thus, instead of 24 different days, a sidereal and solar; and if, instead of the sun, we observe the moon, we find third, much longer than the first two; and instead of the moon's rotation is 24° 54' of our ordinary time, which last in some time, being of necessity conformable to the sun's successive reappearance, on which all the business of life de- pends. (110.) Now, all the stars are found to be unanimous in this respect; that they appear at the same time, for the sideral day ; which, therefore, we cannot hesitate to receive as the period in which the earth makes one revolu- tion round its axis. But let us now consider how it looks on the sun and moon as exceptions to the general law; as having a different nature, or at least a different relation with the earth's motion. The sun appears to be per- parent, of their own will, independent of the rotation of the earth on its axis. Thus a great and most important dis- tinction is made. (117.) To establish these facts, almost no apparatus is required. An observer need only station himself to the north-eastern part of a building, and placing his eyes exactly at a certain fixed point (each as a small hole in a plate of metal nail- led upon a post), he will see at once the progressive disappearance of any star behind the building, by a watch.* When he observes the sun, he must shade his eyes with his hand so as to prevent him from seeing the moments when its western and eastern edges successively *This is an excellent practical method of ascertaining the run of a star during a night. It is necessary to have a building near which to stand in; the chief of which is to take care that that part of the edge behind which you look may be clear of any obstruction whatever; quite smooth; so as whenever variable reflection may transfer the point of observation unduly; this is easily secured, by nailing up a smooth edged board. 76 A TREATISE ON ASTRONOMY. [CHAIR. III. come up to the wall, from which, by taking half the in- terval he will ascertain (what he cannot directly observe) the moment of disappearance of its centre. (118.) The sun's apparent motion, according to this gen- eral fact, we are led to attend more closely to the times of the daily arrival of the sun on the meridian, irregularly (at least) at different seasons of the year. It is evident that the difference between two successive arrivals are not the same at all times of the year. They are sometimes greater, sometimes less than the mean solar day of 24 hours. And yet, that is to say, the solar day is not always of the same length. About the 25th of December, for example, it is a half a month before the sun arrives at its meridian position, peripher nearly as much shorter, than its average duration. And thus a distinction is again pressed upon our notice between the solar day and the mean solar day. We have succession alike ; and the mean solar day of 24 hours, which is an average of all the solar days throughout the year. Hence, we see that there is no uniformity in our sun. The sun's apparent motion is not only not the same with that of the stars, but it is not (as he latter is) uni- form. It is therefore a subject worthy of special attention, a matter of investigation. But to pursue these laws, we require nearer means than those which we have described, and we are obliged to call into aid instruments specially mentioned the transit instrument, especially destined for such observations. This instrument is adapted to all the causes of irregularity in the going of clocks and sundials, which may affect our reckoning of time. Thus we be- come involved in a new inquiry respecting these in- strumental inquiries; and we speedily find that, in pro- portion as we ascertain the amount and law of one great or leading cause of inequality, so also does that of the sun's diurnal motion, we bring into view other causes smaller and smaller, which were before obscured, or mixed up with them, and which now appear in their true perfections. In short, we may not impaly compare the mean length of the solar day to the mean or average height of the sun above the horizon, or the general level of the sea unattigued by tide or waves. The following facts situation above noticed may be compared to the daily varia- **CHAP. II. OF TIME AND ITS MEASUREMENT.** 77 tions of level produced by the tides, which are nothing but enormous waves extending over the whole ocean, while the smaller subordinate inequalities may be assimilated to waves of a few feet in height when large; we perceive lesser undulations to ride, and on these again, minute ripples, to the series of whose subordinateness we have no conception. (118.) With the causes of these irregularities in the solar motion we have no concern at present, their explana- tion being left to the investigation of another sect; but the distinction between the solar and sidereal days, as it pervades every part of astronomy, requires to be early introduced into our system of time. As it has been ob- served, the mean or average length of the solar day, which is used in the civil reckoning of time. It com- monly exceeds that of the sidereal day (in this country), even when they use mean solar time, de- part from the civil reckoning, commencing their day at midnight, and ending it at midnight on the 24th. Thus, 11 o'clock in the forenoon of the second of January, ary, in the civil reckoning of time, corresponds to January 2nd 11 o'clock in the evening of the same date; and one o'clock in the afternoon of the former, to January 2nd 3 hours after the latter reckoning. This usage has its ad- vantages; for it is convenient to those who are pre- ponderate ; and it would be well if, in consequence, it could be broken through, and the civil reckoning substitu- ted for the other. (120.) Both astronomer and civilians, however, who inhabit different points of the earth's surface, differ from each other in their times of sunrise and sunset; and therefore they must if we consider that when it is noon at one place, it is midnight at a place diametrically opposite; moreover, that when it is day at one place there arises considerable inconvenience, especially as respects places differing very widely in situation, and which may be separated by several thousand miles from each other. To obviate this inconvenience, there has lately been introduced a system of reckoning time by mean so- lar days and parts of a sidereal day from a fixed instant, common to all the world, and determined by an equal cir- o2 78 A TREATISE ON ASTRONOMY. CRAP. II. cumstance, such as noon or midnight, but by the motion of the sun among the stars. Time, so reckoned, is called equatorial time, because it is reckoned at the equator, at any instant, in every part of the globe. Its origin will be ex- plained more fully at a more advanced stage of our work. (281.) The first object of astronomical observation is to observe the motions of celestial bodies, and their observation, in a twofold point of view—1st, as the representative of angular motion. The earth's diurnal motion being uniform, and that of the moon nearly so, we may consider the earth's diurnal motion uniformly; and the time elapsing between the pas- sage of the stars in succession across the meridian of any observer being equal to the mean dif- ferences of right ascension. 2ndly, As the fundamental element (or independent variable), to use the language of geometry, of astronomy, we must consider that which is subject astronomy is the determination of the laws of the celestial motions, and their reference to their proximate or remote causes. In this respect, all that has been said respecting motion in a celestial object can be no other than a propo- sition declaring what has been, is, and will be, the real or apparent position of any body in space past, present, or future. To compare such laws, therefore, with observation, we must possess a register of the observed situations of all celestial quantities during all times when they were observed. (282.) The most important time of performance by cloths, chimneys, cleyepdars, and hour-glasses: the two former are alone used in modern astronomy. The hour-glass is a constant and rude contrivance for measuring, or ruling out a certain quantity of time; and is en- tirely disused. The cleyepdar, which measured time by the gradual evaporation of water from a glass tube, determines ordinate is susceptible of considerable exact- ness, and was the only dependence of astronomers before the invention of pendulum clocks. At present it is abandoned, owing to the greater convenience and exact- ness of the latter instruments. In one case only has the revival of this instrument been attended with success, measurement of very small portions of time, by the flow- ing out of mercury from a small orifice in the bottom of a vessel, kept constantly full to a fixed height. The **CHAP. IL.] CLOCKS—CHRONOMETERS.** stream is intercepted at the moment of noting any event, and discharged into a receiver, into which it is admitted, and discharged, till the moment of the other event, when the intercepting cause is suddenly removed, the stream flows in its original course, and ceases to flow into the receiver. The time of this interruption is measured with the weight received in an interval of time observed by the clock, and the difference between these two ob-servations gives the lapse of time. (123.) The pendulum clock, however, and the balance watch, with those improvements and refinements in its structure which have been made since its character was first discovered, are the instruments on which the astronomer depends for his knowledge of the lapse of time. These instru-ments are so constructed that their irregularity in the rate of going, to the extent of a single sec-ond in twenty-four hours in two consecutive days, is not more than one-tenth of a second. The interval of time less than twenty-four hours may be certainly ascertained within a few tenths of a second by their use. It is evident that such an instrument is liable, as well as the amount of error risked, becomes greater, be-cause the accidental errors of many days may accumu-late; but it is also evident that they may be reduced to the rate of going may safely be presumed. It is not safe, therefore, to trust the determination of time to clocks, or watches, unless they are used with great care in making them, and ascertaining their errors by reference to natu-ral events which we know to happen, day after day, at equal intervals of time. The errors of all such intervals may be fixed with the same precision as the shortest; since, in fact, it is only the times intervening be-tween natural events which are required for our purposes, and such of those periodically recurring events adopted for our points of reckoning, as occur within twenty-four hours. The whole number of such intervals is a credi- tional mean. The whole days are counted out for us by nature; and the fractional parts only, at either end, are mea-sured. A diagram showing a pendulum clock with weights and a pendulum bob. 80 A TREATISE ON ASTRONOMY. [CHAP. II. sured by our clocks. To keep the reckoning of the inter- days correct, so that none shall be lost or counted twice, is the object of the calendar. Chronology marks out the order of time, and fixes the days upon their proper years and days; while chronometry, ground- ing its determinations on the precise observation of such regularly recurring phenomena as eclipses, which may be exactly subdivided, enables us to fix the moments in which phenomena occur, with the last degree of preci- sion. (124.) In the culmination, or transit (i. e. the pas- sage across the meridian of an observer) of every star in the heavens, we have a very important and nearly ir- rational natural event as we allude to. Accordingly, it is to the transits of the brightest and most conveniently situated stars that we chiefly attend, in order to determine their exact time, or, which comes to the same thing, to determine the exact amount of error of their clocks. (125.) The observations of the transits of some systems of celestial objects are observed is called a transit instru- ment. It consists of a telescope firmly fastened on a hori- zontal axis at one end, and a vertical axis at the other end of the horizon, or at right angles to the plane of the meridian of the place at which it is used. The extremities of the axes are formed into cylindrical goniometers exactly similar, which rest in notches formed in metallic supports, bedded (in the case of large instruments) on strong pierrs of stone or brickwork, and are adjusted to move freely, both in a vertical and horizontal direction. By this fine mer adjustment, the axis can be rendered precisely hori- zontal, by levelling it with a level made to rest on the A diagram showing a telescope mounted on two axes: one horizontal and one vertical. CHAP. XI. TRANSIT INSTRUMENT. 81 pivot. By the latter adjustment the axis is brought pre- cisely into the east and west directions, the criterion of which is furnished by the observations themselves made with the transit, which are called meridian marks, originally determined by such observa- tions, and then, for convenience of ready reference, per- manently marked on the glass. The instrument is then set up with its meridian line passing through the central point of the whole instrument. It is evident from this description, that when the telescope is turned round to any angle about the centre of the case-glass and eye-glass, and which is called in astronomy (in use of collimation) so once well adjusted that the two points appear in the same wire, it will never quit the plane of the meridian, when the instrument is turned round on its axis. The object-glass is placed between the eye-piece, and at right an- gles to the length of the telescope, is placed a system of one horizontal and five equidistant vertical threads or wires as they are called, which are so arranged that they always appear in the field of view, when properly illuminated. A diagram showing a circular eyepiece with a horizontal wire and five vertical wires around it. by day by the light of the sky, by night by that of a lamp introduced by a contrivance not necessary here to explain. The telescope is mounted on a base-plate by ad- justing screws, giving it a lateral (horizontal) motion; and it is by this means brought to such a position, that one wire appears in the field of view, when the telescope is in the state of collimation of the telescope, where it is arrested and permanently fastened. In this situation it is evident that all lines drawn parallel to the wire are parallel to that portion of the celestial meridian to which the tele- scope is pointed; and when a star is seen to cross this wire in the telescope, it is in the state of collimating, or passing the terrestrial meridian. The instant of this event is 62 A TREATISE ON ASTRONOMY. [CHAP. II. noted by the clock or chronometer, which forms an indispensable accompaniment of the transit instrument. For greater precision, the moments of its crossing all the five vertices of the meridian are to be noted with care (since the straights are equidistant) would give exactly the same result, were all the observations perfect, and will, of course, be more accurate and destroy their errors in an average of the whole. (177.) For the mode of executing the adjustments, and allowing for the errors of observation, being a simple and elegant instrument, the reader must consult works especially devoted to this department of practical astronomy. The following remarks may serve as a general verification of its correctness, which consists in reserving the ends of the axis, or turning it east for west. If this be done with great exactness, we shall find that intersecting the same point on the meridian mark, we may be sure that the line of culmination of the telescope is truly at right angles to the meridian. This is easily done by a plane, i. e. marks out in the heavens a great circle. In good transit observations, an error of two or three seconds of a second is sufficient to make a difference perceptible; but it is the utmost which need be apprehended, exclusive of the error of the clock: in other words, a clock may be compared with a watch, and both may be used for one observation, without risk of greater error. By multiplying observations, or courses, of a yet greater degree of precision may be obtained. (178.) The angular intervals measured by means of the transit instrument and clock are arcs of the equinoctial circle, intercepted between two points situated through the objects observed; and their measurement, in this case, is performed by no artificial graduation of circles, but by observing directly upon them. The arc which carries equal areas of the equinoctial across the meridian, in equal times, at the rate of 10° per sidereal hour. In order to obtain these arcs at regular intervals, it is necessary to have recourse to circles, or portions of circles, constructed of metal or other * See Dr. Francis's Treatise on Practical Astronomy. Also Banchi's Saggio in Rendizioni del Seminario. Ephem. di Milano. 1835. A diagram showing a transit instrument with a clock attached. CHAP. II.] MEASUREMENT OF ANGLES. 83 firm and durable material, and mechanically subdivided into equal parts, such as degrees, minutes, &c. Let ABCD be such a circle, divided into 360 degrees (num- A diagram showing a circle divided into 360 degrees, with points labeled A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T. bered in order from any point $^o$ in the circumference, round to the same point again), and connected with its centre by a straight line AB. This line is called the circumference or limb. At the centre let a circular hole be pierced, in which shall move a pivot exactly fitting it, carrying at each end of the pivot two arms parallel to the plane of the circle, or perpendicular to the pivot; and also the two arms so n., at right angles to it, and forming one angle with each other. The extremities of the extremities of the arms on the centre shall carry the tube and arms smoothly round the circle, to be arrested and fixed at any point desired upon it by means of a screw. Sup- pose now, we would measure the angular interval be- tween two fixed objects, ST. The plane of the circle must be parallel to that of the objects both. This done, let the axis $a$ of the tube be directed to one of them, S, and clamped. Then will a mark on the arm $b$, corresponding to this object, fall upon one of those on the limb, or between two of them adjacent. In the former case, the division must be noted, as the reading of the scale shows; but in the latter case it is necessary to find out what whole interval between the consecutive divisions by which the mark on $m$ surpasses the last inferior division must be added to that indicated by $b$. This is called or optical mean. (See art. 120.) The divided sinus frac- tional part thus noted, and reduced into degree, minutes, and seconds, is to be set down as the reading of the limb 54 A TREATISE ON ASTRONOMY. [CHAP. III. corresponding to that position of the tube a b, where it points to the object S. The same must then be done for the object T; the tube pointed to s, and the limb c read off at p. It will be seen that, when the two readings be subtracted from the greater, their difference will be the angular interval between S and T, as seen from the point of view of the observer, and this is the limit the commencement of the graduations on the point 0 be situated. (189.) For very same result will be obtained, if, instead of making the tube moveable upon the circle, we connect it invariably with the latter, and make both re- volve together about a common axis, which may be formed one piece with it, working in a hollow formed to receive and fit it in some fixed support. Such a com- bination is especially useful in cases where the index T is the tube or sight, fastened at p p, on the circle AB. A diagram showing a telescope with a movable tube connected to a stationary circle. The tube is marked with graduations and has a pointer at its end. The circle has a pointer at its center. whose axis D, works in the solid metallic entraining E, from which originates an arm F, carrying at its ex- tremity an index, or other proper mark, to point out and read off the exact division of the circle at B, the point close to which the index is placed. When both circles revolve through any angle, the part of the limb of the latter, which by such revolution is carried past the index F will pass over any number of divisions according to usual mode of applying divided circles in astronomy. (190.) The index F may either be a simple pointer, like a glass hand (fig. o); or a vernier (fig. b); or, A diagram showing a vernier scale attached to a telescope. The vernier scale has a series of small marks along its length, and a pointer that can slide along it. The telescope has a pointer at its end. A diagram showing a telescope with a vernier scale attached to it. The vernier scale has a series of small marks along its length, and a pointer that can slide along it. The telescope has a pointer at its end. **CHAP. II.] APPLICATION OF THE TELESCOPE.** Lastly, a compound microscope (fig. c.), represented in section (in fig. d.), and furnished with a cross in the common focus, of its object and eye-glass, movable by a fine thread screw, by which the intersection of the crests may be made to coincide with the meridian of the image of the nearest of the divisions of the circle; and by means of this instrument, the distance of that division from the original or zero point of the microscope may be estimated. This simple arrangement will enable one to measure off a circle of degree of accuracy only limited by the power of the microscope, and the perfection with which a screw can be turned. The same sub-division of angle on the same footing of optical correction which is introduced into their measurement by the use of the telescope. (121.) The examination of a star must depend upon the precision with which the tube a b can be pointed to the object ; 2dly, on the accuracy of graduation of the circle; 3dly, on the steadiness with which the subdivision of the intervals between any two consecutive graduations can be accomplished. The manner in which these points have been treated in my re- quired exactness has been explained in the last article. With regard to the graduation of the limb, being merely an extension of the scale used in astronomical work, further than this, that is, to point out some state of instrument making, the amount of error from source of inaccuracy in this part is so small as to be negligible. With regard to the first, it must be obvious that, if the sights a b be nothing more than what they are represented in fig. c., then they cannot be brought into coincidence at either end of a hollow tube, or an eye-hole at one end, and a cross at the other, no greater nicety in pointing can be attained than that which can be obtained with any eye-command. But it is plain of these simple but coarse contrivances, the tube itself be converted into a reflecting telescope, and then all errors will disappear at a; and if the motion of the tube on the limb of the circle be arrested when the object is brought just into the centre of the field of view, it is evident that a greater degree of exactness may be obtained in pointing of A diagram showing a compound microscope with a cross in its common focus, representing an object and an eye-glass. A fine thread screw is attached to move the cross along a line parallel to the axis of rotation. 86 A TREATISE ON ASTRONOMY. [CHAP. IL the tube than by the unassisted eye, in proportion to the magnifying power and distinctness of the telescope used. The last smallest degree of exaggeration is secured by stretching the tube, so that the extremities of the eyepiece glasses two delicate fibres, such as fine hairs or spider-lines, intersecting each other at right angles in the centre of the tube, and thus forming a sort of cross, which affords a permanent mark with which the image of the object can be brought into coincidence with any point on the limb of the circle, and remains fixed till the taking off is made. (132.) This approximation may be considered as completely annihilating that part of the error of observation which arises from the imperfect perspicuous estimation of the direction in which an object lies from the observer's eye, or from the centre of the instru- ment. It is therefore possible to attain a perfect per- cepion of modern astronomy, without which all other re- finements in instrumental workmanship would be thrown away; the errors arising from this cause being so great as to outweigh such assistance, being far greater than what could arise from any but the very coarsest gradations. In fact, the telescope thus applied possesses, " The most perfect instrument ever invented," as was dis- closed by Deism (Phil. Trans. xxx. 60) to our young, talented, and adulatory friend, Sir William Herschel, who was then a Curator and Prosector in Deism's observatory. The message cited was sent to him by Mr. John Flamsteed, who had been one of his friends at the commence- ment of his career, and had even sustained him through many difficulties. " I have seen," said he, " many instruments before me, but none like yours." He found " very useful help from those opposed to me, or at least otherwise disposed to assist me." He was a man of great abilities and talents; he was born in Hanovershire, near York; he became a master carpenter; he married a daughter of Sir John Barrow; he became a Fellow of the Royal Society; he became a Member of Parliament; he became a Member of the Board of Longitude; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member of Parliament again; he became a Member **CAP. II.] INTERVALS IN DECLINATION MEASURED.** with respect to angular, what the microscope is with respect to linear dimension. By concentrating attention on its smallest points, and magnifying into palpable intervals the minutest differences, it enables us not only to see accurately the position of any star, but also to determine the interval between two stars which it is pointed at, but to refer their apparent places, with all but geometrical precision, to a line of true or small scale with which we proceed to compare them. (133.) The simplest mode in which the measurement of angular distance can be made, is that by which we have just described; but, in strictness, this mode is applicable only to terrestrial angles, such as those occupied on the horizon by the sun and moon at different stations—because these only remain stationary during the interval while the telescope is shifted on the limb from one object to another. But when we consider the heavens by destroying this essential condition, renders the direct measurement of angular distance from object to object impossible. In this case, however, does not apply if we seek only to determine the interval between the diurnal circles described by any two stars in their course across the heavens. If our resolution were to leave behind it a visible trace in the heavens—a fine line of light, for instance,—then a telescope would be sufficient; but if we wish to bring the image brought to coincidence with the intersection of the wires, would constantly remain pointed to some portion or other of the heavens, and if we could not observe it appear in its field as a luminous line, permanently intersecting the same point, till the star came round again. From each star then we must take two observations at leisure, without error; and then the angular interval between the two diurnal circles, in the plane of the telescope's axis, will be found. But how can we ascertain this path? We cannot see the path of a star in the heavens, we can visit till the star itself crosses the field of view, and seize the moment when it passes through it; but we cannot see its wires so that the star shall traverse it; by which, when the telescope is well clamped, we equally well secure the position of its diurnal circle as if we continued to see it ever so long. The reading off of the limit may then be 87 88 **A TREATISE ON ASTRONOMY.** [CAP. II.] performed at leisure; and when another star comes round into the plane of the circle, we may unclamp the telescope, and a similar observation will enable us to as- sign the position of this star with respect to the other observations may be repeated alternately, every day, as the stars pass, till we are satisfied with their result. (134.) The meridian circle is a circle, which is nothing more than such a circle as we have described in art. 190, firmly supported, in the plane of the meri- dian, on an axis perpendicular to that plane (which is let into a massive pier, or wall, of stone (whence the name of the instrument), and so secured by screws as to be capable of being turned through any angle in hori- zontal direction; so that, like the axis of the transit, it can be maintained in the exact direction of the east and west points of the horizon; and the plane of the circle being con- sequently truly meridional. (135.) The meridian, being at right angles to all the diameters of the circle, and therefore intersecting each other between any two of them will measure the least distance between these circles, and will be equal to the difference of the declinations of those stars whose diametral dis- tances of the objects—at least when corrected for refraction. These differences, then, are the angular intervals describing the declination of one pole from the other, supposing the law of refraction known, it is easy to conclude that their difference only, but the quantities themselves express exactly what they do. (136.) The declination of a heavenly body is the com- ponent of its latitude parallel to the equator; so that, being a point in the meridian, might be directly observed at the limit of the circle, if any star stood exactly therein; and thence determining its distances, and of course, the declina- tions of all the rest of the stars. But since this cannot be done without error, a bright star near the pole as this not being the case, a bright star near the pole as it can be found out by observing it in its place for lower culminations: that is, when it passes over the above and below the pole. Now, as its distance from the pole measures its declination (and by removing off the circle in two cases is, of course (when corrected for refraction), equal to twice the polar distance of the A diagram showing a meridian circle with a telescope mounted on an axis perpendicular to a plane containing a line from north to south. The circle is divided into degrees and has lines drawn from various points around it indicating different positions of stars. CRAP. II.] MURAL CIRCLE. 89 star; the arc intercepted on the limb of the circle being, in this case, equal to the angular diameter of the star's diurnal circle. In the annexed diagram, HPO represents the meridian of the pole, and ABO the diur- nal circles of stars which arrive on the meridian—at BA and at its upper, and at RQD in their lower cul- tions, of which D happens above the horizon HO. P is the pole; and if we suppose h p to be the mural circle, then the arc AB will represent the star's distance on its circumference corresponding to BACPD in the heavens. Now, the arcs b c, b d, c e and c d give even increases of altitude, but not of distance; therefore we have also cp = pd, and each of them = td, consequently the place of the polar point, as it is called, upon the limb of the circle is that point P, where the line PC meets PD; which represent on the circle the polar distances re- quired become also known. (1) The pole star, which is a very brilliant one, is eminently favourable for this purpose, being only about a degree and a half from the pole ; it is therefore easily observed by all observers, for this important purpose ; the more especially because, both its inclinations taking place at great and not very different altitudes, and both its polar distances are affected are of small amount, and differ but slightly from each other, so that their corrections are correctly and easily applied. (2) The moon is also a good star, as it always is to be easily observed in the daytime. In consequence n 9 A diagram showing a circular map with various points labeled. The center is labeled "O", with lines extending outward to points labeled "A", "B", "C", "D", "E", "F", "G", "H". Lines connect these points to form a star-like pattern around the center. A TREATISE ON ASTRONOMY. [CAP. II.] of these peculiarities, this star is one of constant resort with astronomers for the adjustment and verification of instruments of almost every description. In the case of the transit of Venus, we have to consider the question of ascertaining whether the plane of the telescope's motion is coincident with the meridian. For since this latter plane lies between the observer and the star, its proper portion of it require equal times for their description. Let, therefore, the moment at which the transit occurs be such that the telescope shall pass through the meridian at any time during the interval of 12 equal hours, we may conclude with certainty that the plane of the telescope's motion is meridional, or that the position of its horizontal axis exactly east and west. But if it pass from one to the other ap- parent culminations, and if it be observed that there is equally certain that an extra-meridional error must exist, the deviation lying towards that side on which the least interval is found, then we must correct our observations accordingly, till the difference in question disappears on repeating the observations. (180.) The point on the limb of the mural circle once determined, becomes an origin, or zero-point, from which the polar distances of all objects, referred to by means of a graduated circle, depend. This matter not whether the actual commencement $0^{\circ}$ of the graduations stand there, or not; since it is only by the different positions of this point that all points on the limb are determined; and hence a great advantage is obtained in the power of commencing anew a fresh series of observations. It is also evident that no great difference of the circle shall be employed, and different graduations brought into use, by which inequalities of division may be avoided. This object can be most easily accomplished practically by detaching the telescope from its old bearings on the circle, and fixing it directly on a different point on the same line. (180.) A point on the limb of the mural circle, not so important than the polar point, is the horizontal point, which is that point on which a line drawn nearer an origin, or zero-point, from which allitudes are reckoned. The principle of its determination is illu- **CHAP. II.] POLAR AND HORIZONTAL POINTS.** mostly nearly the same with that of the polar point. As no star exists in the celestial horizon, the observer must seek to determine two points on the line, the one of which is at the pole, and the other at the celestial point as the other is above it. For this purpose, a star is observed at its culmination on one night, by pointing the transit instrument so that its axis shall be parallel to the image of the same star reflected in the still, unruflled surface of a fluid at perfect rest. Mercury, as the most refracting of all stars, is generally chosen for this use. As the surface of a fluid at rest is necessarily horizontal, and as the angle of reflection, by the laws of optics, is equal to the angle of incidence, it will appear as much depressed below the horizon, as the star itself is elevated above it (allowing for the difference of refrac- tion). The transit instrument is then set upon an incline excepted on the limb of the circle between the star and its reflective image thus conveniently observed, when cor- responding observations are made on two successive days, and its point of bisecting the horizontal point. The re- flecting surface of a fluid so used for the determination of the latitude is called a transit mirror or transit section. (140.) The mural circle is, in fact, at some time, a transit instrument ; and, if furnished with a proper sys- tem of sights and scales, may be employed for such pur- pose as used as. As the axis, however, is only supported at one end, it has not the strength and permanence ne- cessary for such purposes. It can therefore never be certified as a transit may, by the reversal of the two ends of its axis, east for west. Nothing, however, prevents its being used for determining latitudes on the axis of a transit instrument, near to one of its extre- mities, so as to resolve with it, the reading off being per- formed by means of a transit mirror. Such an instru- ment is called a transit circle, or a meridian circle, and serves for the simultaneous determination of the latitude and longitude. The time of day is ob- served with it; the time of transiting being noted by the clock, and the circle being read off by the lateral microscope. (141.) A quadrant is an instrument used for setting out on the limb of an instrument or of such essential importance in astronomy, that the student should be made acquaint- 91 92 A TREATISE ON ASTRONOMY. [CHAP. II. ed with every means employed for this purpose. These are the artificial horizon, the plumb-line, the level, and the floating collimator. The artificial horizon has been al- ready explained. It is a flat board, supported on a frame, to which is suspended a weight, whose oscillations are im- peded and quickly reduced to rest by plating it in water. The plumb-line is a line of lead, or some other material, admitting its prefered flexibility, that is of gravity, or per- pendicular to the surface of still water. Its application is the same as that of the artificial horizon. It is slow and difficult, and liable to error, unless extraordinary precau- tions are taken in its use, that it is at present almost uni- versally superseded by the more convenient and equally exact instrument the level. (42). The level in nothing more than a glass tube nearly filled with water. This instrument is now gen- erally used, on account of its extreme mobility, and (not being liable to freeze), the bubble in which, when the tube is placed horizontally, would seem indifferently in any part if the tube could be mathematically straight. But that being impossible to execute, and every tube having some slight curvature, if the convex side be placed up- wards, the bubble will be seen to move towards the left figure (where the curvature is purposely exaggerated). Suppose such a tube as AB firmly fastened on a straight bar, CD. Let P be any point on AB; then let PQ be the length of the bubble; then, if the instrument be so placed that the bubble shall occupy this interval, it is clear that CD can never be horizontal; because if it were so to the horizon ; because, were it ever so little moved one way or another, the bubble would shift its place, and run towards the circle of points CDEF; but we cannot ascertain whether any given line PQ be horizontal ; let the base of the level CD be set upon it, and note the points A diagram showing a level instrument with a bubble in a glass tube near full of water. The tube is placed horizontally on a straight bar CD. Points P and Q mark positions where the bubble might be observed. 9/22 CHAP. 15. OF THE LEVEL. 93 A level, between which the bubble is exactly contained; then turn the level end for end, so that C shall rest on Q, and D on P. If the bubble continue to occupy the same place, the level is true, and must be lowered; if not, otherwise than horizontal. If not, the side towards which the bubble runs is highest, and must be lowered. Astro- nomical instruments are made to move horizontally by which the places of the ends of the bubble can be nicely marked; and it is said that they can be executed with such exactness, that the difference of two observations may deviate from exact horizontality. (148.) The mode in which a level may be applied to find the height of a point, or to draw a vertical divided circle may be thus explained: Let AB be a telescope truly fixed to such a circle, DEF, and movable in one A diagram showing a telescope (AB) mounted on a graduated circle (DEF). The telescope is shown in three positions: horizontal (left), slightly tilted (center), and vertical (right). with it on a horizontal axis C, which must be like that of a transit, susceptible of reversal (see act. 127), and with which the circle is inseparably connected. Direct the telescope on some distant object S, and let it rest in any position whatever in this position; clamp it fast. Let L be a level fastened at right angles to an arm EF, furnished with a micrometer screw of 10 mm., if possible another of 20 mm. Let this arm be fitted by grinding on the axis C, but capable of moving smoothly up and down carrying it round its axis. This arm is free to move from its position until required. While the telescope is kept fixed on the object S, let the 94 A TREATISE ON ASTRONOMY. [CAP. II. level be set so as to bring its bubble to the mark a b, and clamp it there. Then will the arm LEP have some cer- tain determinate inclination (no matter what) to the hori- zon. In this position, let the bubble be read off at a b, and then let the whole apparatus be reversed by turning its horizontal axis end for end, without unclamping the level axes from their clamps. Now, if the bubble be still at a b, the whole instrument (level and all) on its axis, restore the level to its horizontal position with the bubble at a b. Then we see that the difference between the true inclination to the horizon (or angle), that it had when pointed to S, and the reading off at a b will not have been changed; because, having turned the whole apparatus horizontal, turn round the circle on the axis, so as to cir- cry back the telescope through the zenith to N, and in that position read off at a b again. This shows that it is evident that an angle equal to twice the zenith distance of S has been moved over by the axis of the telescope from its original position to its present one. Turn the telescope and circle, let the level be once more rectified. Then will the arm LEP once more assume the same de- terminate position as before. But now, since conse- quently, if the circle be again read off, the difference be- tween this and the previous reading must measure the sum of its two parts, viz., twice the zenith distance of the point F, which may be considered as having all the time retained an invariable position. This difference, then, will be found to be equal to twice the zenith dis- tance simply, the complement of which is its altitude. Thus the altitude corresponding to a given reading of a bubble on a level is found by adding to it the horizontal point on the line of sight until such time as the horizontal point on the line is ascertained. Circu- tous as this process may appear, there is no other mode of employing a level but this; for if any other method did not in the end come to the same thing. Most commonly, however, the level is used as mere fiducial reference, to preserve a certain position of instruments by other means, which is done by adjusting it so as to stand level when the telescope is truly horizontal, and thus having it always ready for making observations. (144.) The last, but probably not the least exact, as it CHAP. II.] THE FLOATING COLLIMATOR. 95 certainly is, in innumerable cases, the most convenient means of ascertaining the horizontal point, is that at which the rays of light from any object fall perpendicularly upon the surface of Captain Kater. This elegant instrument is nothing more than a small telescope furnished with a cross-wire in its focus, and mounted on a float, which may be either a ball or be, on a flat iron float, which is made to swim on mer- cury, and which, of course, will, when left to itself, set itself so as to be perpendicular to the horizon. The rays falling on the horizon. If the cross-wires of the collimator be illu- minated by a lamp, being in the focus of its object-glass, the rays from them will issue parallel, and will therefore lie in one plane. Now, if we take any other object, such as a glass of any other telescope, in which they will form an image as if they came from a celestial object in their di- rection, it will be seen that the image will be horizontal. Thus the intersection of the cross of the collimator may be observed as if it were a star, and thus, however near the horizon it may be, it will always remain horizontal. The collimator still floating on a vessel of mercury from the one side to the other of a circle, we are furnished with two horizontals, one passing through the centre of the circle, on opposite sides of the centre; and if these be observed in succession while the vessel is moved round, we shall find a line in bisecting the image of the cross of the collimator [or which end the wires of the latter crosses are purposely set at 45° to each other], and the readings on its two will be twice the zenith distance of either; whence, as in the last article, the horizontal point can be determined. * Another, and in many respects preferable form of the floating colli- mator, in which the shadow cast by the object under examination is used instead of a glass plate, is described in the Phil. Trans. 1829, p. 307 by the same author. A diagram showing a floating collimator with a cross-wire in its focus and mounted on a float (either a ball or a flat iron float) which is made to swim on mercury. The rays falling on the horizon are illuminated by a lamp. A diagram showing two horizontals formed by the intersection of the cross-wire of a floating collimator with opposite sides of a circle. These horizontals are observed in succession while the vessel is moved round. 96 A TREATISE ON ASTRONOMY. [CHAP. II.] (148.) The transit and mural circle are essentially me- ridian instruments, being used only to observe the stars at the moments of their meridian passage. Independent of this being, they are so constructed that the motion of it is that in which their diurnal motion is parallel to the horizon. It is therefore easier at this time than it could be at any other, to determine the true meridian by means of these instruments, since their apparent course in the field of view being parallel to the horizontal thread of the system of wires on the wall, we may easily ascertain its position, the telescope, be brought to exact coincidence with it, and time may be allowed to examine and correct this co- incidence, before proceeding to make observations in no other situation. Generally speaking, all angular magnitudes, which it is of importance to ascertain ex- actly, should be observed when the sun is at its maximum or minimum of increase or diminution; because at these points they remain not perceptibly changed during a time long enough to allow us to examine them with great pre- tess and verify our observations in a careful and leisurely manner. The angle which, in the case before us, is at this prediction, is called the meridian angle; it is measured at its maximum or minimum on the meridian, and which is measured on the limb of the mural circle. (149.) The transit circle, however, require that an observer should possess the means of observing any object not directly on the meridian, but at any point of its diameter. This can be done by means of a sphere in the heavens. Now, a point in the sphere is determined by reference to two great circles at right angles to each other; or more correctly, by reference to two poles of the sphere. These, in the language of geometry, are co-ordinates by which its situation is ascertained: for instance, if we know its longitude and latitude --- in the stary heavens, if we know its right ascension and declination --- in the visible hemisphere --- then we have its position. (147.) To observe an object at any point of its diurnal course, we must possess the means of directing a tele- scope to it; which, therefore, must be capable of moving in two planes at right angles to each other; and the CHAP. 11.] CO-ORDINATE CIRCLES. 97 amount of its angular motion in each must be measured on two circles co-ordinate to each other, whose planes must be parallel to those in which the telescope moves. The principal axis of the instrument is that making the axis of one of the circles perpendicular to that of the other at right angles. The pierced axis turns on fixed points, and is supported by any number with any external support, but is sustained entirely by that which it penetrates, which is strengthened and enlarged at the point of penetration. In the following simplified figure exhibits the simplest form of such a combination, though by no means the best in point of mechanism. The circle GH is the moving circle, and the circle EF is the one attached to the fixed support which carries the principal axis, the other to an arm projecting from that axis. The arm FG is a lever, and when it is depressed, the clamps being attached to the same ultimate bearing with which the apparatus for reading off is connected. (When this apparatus is not used, however, whatever its principal axis be pointed (provided that its direction be invariable), will enable us to ascertain the situation of the observer's station.) In order to determine the position of the observer's station, by angles reckoned upon two great circles passing through visible hemispheres, one of which has for its poles the pro- blematic station, and the other for its principal axis or the vanishing points of a system of lines parallel to the principal axes of all planes always through these poles : for the former great circle is a plane containing all planes parallel to the circle AB, while the latter, in any given position, is the vanishing line of all the planes parallel to the circle GH ; and these two planes being, by the construc- tion of the instrument, at right angles, the great circles, A diagram showing a mechanical device with various components labeled. The main components include a central pivot (G) with a rod extending upwards (H) and downwards (I). A horizontal arm (J) extends from G and J meets a vertical rod (K) at F. Another horizontal arm (L) extends from G and L meets a vertical rod (M) at E. The entire assembly is supported by a base (N). There are also labels for "circle GH" and "circle EF". 98 **A TREATISE ON ASTRONOMY.** [CHAP. II.] which are their vanishing lines, must he so too. Now, if two great circles of a sphere be at right angles in each other, the one will always pass through the other's poles. (149.) There are, however, but two positions in which such an apparatus can be mounted so as to be of any practical use; namely, when the plane of the circle CD is parallel to the earth's axis, and therefore points to the poles of the heavens which are the vanishing lines of the great circles of the equator and horizon; and when, of course, the plane of the circle AB is parallel to the earth's equator, and therefore has the equatorial position for all stars. In this case, the great circles in the heavens, corresponding to the circles on the instrument, may be read off, hour angles, or differences of right ascension. In this case, the great circles in the heavens, corresponding to the circles on the instrument, may be made to assume, by the rotation of the instrument round its axis CD, are all hour-circles ; and the arcs read off on this circle represent hour angles, or their differences. (150.) In this position the apparatus assumes none of the operations which it was formerly called, a polariscope instrument. It is one of the most convenient instruments for all such observations as require an object to be kept long in view without being moved. For if it can be followed as long as we please by a single motion, i. e., by merely turning the whole apparatus round on its pole axis CD, then it follows that whatever point of the star, the angle between its direction and that of the polar axis is equal to the polar distance of the star, it follows, that when we have once found out what that angle is for some particular position of the telescope on the circle GH, the point to which it is directed will always lie in the small circle of the heavens corresponding to that angle. But for many observations this is an insensible advantage, and one which belongs to no other instrument. The equatorial is also useful for making observations with a telescope by comparison with that of a known object, in a manner to be described in the fourth chapter. The adjustments of the equatorial are very simple and easy to perform. They are best performed by following the polestar round CHAP. II. AZIMUTH AND ALTITUDE INSTRUMENT. The entire diurnal circle, and by observing at proper inter- vals, other considerable stars whose places are well ascertained.* (181.) The other position in which such a component apparatus as we have described in art. 147 may be advan- tageously mounted, is that in which the principal axis occupies a vertical position, so that the instrument is equally inclined to the celestial horizon, and the other, CA, to a vertical circle of the heavens. The azimuth measured on the former plane are denoted by angles or differences of altitude, though the latter zenith dis- tances, or altitudes, according as the graduation comes from the upper part of the limb, or from one sym- metrical side of the meridian. In this position the use of an azimuth and altitude instrument. The vertical posi- tion of its principal axis is secured by placing a plumb- line at any point on the upper end, which when ever it be turned round, should continue always to intersect one of the two circles. This is easily effected by fixing a bubble by a level fixed directly across it, whose bubble ought not to shift its place, on moving the instrument in any direction. The bubble being thus fixed, the lower cir- cle is annulled by bringing the vertical circle to coincide with the plane of the meridian, by the same criterion by which the azimuthal movement is made. Then, per- forming the 180th degree turning, in this position, the read- ing off of the lower circle, or by the following process. (182.) Let A be bright star situated near the meridian, and B another star distant from it. By bringing it to the cross wires of the telescope. In this position let the meridian be observed with great exactness, and then very slightly damped on the vertical one. When the star has passed the meridian, and is in the descending point of its daily course, bring it back again to its first position, and repeat round to the west; without however, unclamping the telescope, until it comes into the field of view; and, until, when it passes over the horizontal line of sight, let it be crossed by the wire of the telescope once more to coincide. In this position it is evident the star must have the same precision *See Note on the Adjustment of the Equatorial.—Mess. Arwen. See. vol. ii. p. 54 100 A TREATISE ON ASTRONOMY. [GRAF. II. altitude about the western horizon, that it had at the mo- ment of the first observation above the eastern. At this point let the motion be arrested, and the horizontal circle be again revolved to its original position. The altitudes of the azimuthal are described in the interval. Now, it is evident that when the altitudes of any star are equal on either side of the meridian, they will be equal also when both from the north or both from the south point of the horizon, must also be equal; consequently the north or south point of the horizon, and the altitude of any star are thus determined, and will therefore become known. (153.) This method of determining the north and south points of the horizon, and the altitude of any star, known, we may draw a meridian line) is called the "method of equal altitudes," and is of great and constant use in practice. For suppose two observations made; of the two observations, the time, by a clock or chrono- meter, the instant halfway between them will be the moment of transit of the star across the meridian; then it will be determined without a transit; and, vice versa, the error of a clock or chronometer may by this process be discovered. It is evident that in order to make such that our instruments should be provided with a horizontal circle at all. Any means by which altitudes can be mea- sured with accuracy, will serve for this purpose; for sup- pose two stars arrive at equal altitudes in the eastern and western halves of their diurnal course; and, these once known, by taking their mean message and the error of the clock become also known. (154.) One of the chief purposes to which the altitude and azimuth are applied is to determine indirectly the amount and laws of refraction. For, by following with it a circular paralax which passes the zeniths, and another which passes through every star in its diurnal course, the exact apparent form of their diurnal orbits, or the evan in which their circles are distorted by refraction, may be determined. But since these figures circles, being at every moment given by the nature of the observation in the direction in which the refraction takes place (i. e. in altitude), is made a matter of direct observation. **CHAP. II.** HALEY'S SEXTANT. (153.) The zenith sector and the chordalite are pecu- liar modifications of the altitude and azimuth instruments. The former is adapted for the very exact observation of stars in or near the zenith, by giving a great length to the vertical circle, except a few degrees of its lower part, by which a great length of radius, and consequently proportionate accuracy, is obtained. The latter is especially de- voted to the mea- sure of horizontal angles between terrestrial objects, in which case the vertical circle is either dispensed with, or executed on a smaller scale than that of the former instrument. On the other hand, great care is bestowed on securing the exact perpendiculari- ty of the plane of the telescope's motion, by resting it upon a transit-stand, while themselves firmly bedded on the plate of the horizontal circle, so as turn with it. (154.) The sextant is an instrument used by one or more who direct the angular distance of any two ob- jects may be measured, or the altitude of a single one. It consists of a quadrant, graduated in degrees, and the visible horizon (such as the sea-seeing, allowing for its dip), or from its own reflection on the sea-board; whereby, if it be applied to any object, such as a ship called Haley's, from its reputed investor, though the priority of invention belongs undoubtedly to Newton, yet it is true that the greatest improvement has been thus doubled, by having furnished at once the only theory by which his vessel can be steered, and also by which all ships have ever been found to sail, in applying that theory to its nautical uses." (157.) The principle of this instrument is the optical projection of reflected rays, thus announced: *"The * Newton communicated it to Dr. Haley, who suppressed it. The description of the instrument was found, after the death of Haley, in a letter written by him to Mr. Wollaston, dated 1806. He had communicated the papers to the Royal Society twenty-five years after his death; but he did not mention this instrument among them, which might be, and probably was, independent of any knowledge of Newton's, though Hooke's name appears. 12 101 108 A TREATISE ON ASTRONOMY. [CHAP. II.] angle between the first and last directions of a ray which has suffered two reflections in one plane is equal to twice A diagram showing a ray of light passing through a series of lenses and mirrors, with arrows indicating the directions of the rays before and after each reflection. the inclination of the reflecting surfaces to each other." Let AB be the diameter of a circle, of a portion of a circle 60° in extent, but divided into 120 equal parts. On the radius CB let a silvered plane glass D be fixed, at right angles to AB, and on the same line with the movable CE let another such silvered glass, C, be fixed. The glass D is permanently fixed parallel to AC, and the glass C is movable along AB, allowing opposite to be seen through it. The glass C is wholly silvered, and its plane is parallel to the length of the mirror AB. When the observer, whose eye is at B, views the mirror placed to read off the divisions of the limb. On the radius AC is set a telescope F, through which any object may be seen. The ray of light passes through the unsilvered part of the glass D, while another object, P, is seen through the same telescope by way of C. The ray of light from P passes through upon the silvered part of D, and are thence directed by a second reflection into the telescope. The two images thus formed are seen together in the telescope, and by moving the radius CE will (if the reflectors be truly perpendicular to the plane of the circle) meet and pass over each other. This mutual motion continues however, is arrested when they meet, and at this point the angle included between the direction CP of one object and that of P is equal to that included between the fixed and movable radii OB, CE. Now the graduations of the limb being purposely made **CHAP. II.** **PRINCIPLE OF REFRACTION.** only half as distant as would correspond to degrees, the arc BE, when read off, as if the graduations were whole degrees, will, in fact, read double its real amount, and therefore the angle of refraction will be double the angle ECB, but its double, the angle subtended by the object. (168.) To determine the exact distances between the stars by direct observation is comparatively of little service ; but in nautical astronomy the measurement of these distances is of great importance. The reason is, of essential importance ; and as the sextant requires no fixed support, but can be held in the hand, and used on all kinds of water, it is much more easily understood than any other instrument. For altitudes at sea, as no level, plumb-line, or artificial horizon can be used, the sinu-scoffing affords us a means of determining the distance of a star, seen by reflection, is brought to coincide with the boun- dary of the sea seen by direct rays. Thus the altitude of a star (art. 23) is equal to the sine of the dip of the horizon (art. 24) gives true altitude of the star. On land, an artificial horizon may be used (art. 130), and then we have a right angle between the line of sight and the horizon. (169.) The reflecting circle is an instrument destined for the same use as the sextant, but more complete, the circle being divided into degrees and minutes of arc. It is usually furnished with three verniers, so as to admit of three distinct readings off, by the average of which we may obtain a mean value for each degree. This is also perhaps a very refined and elegant instrument. (170.) We must not conclude this chapter without mentioning that it was written under the direction of Mr. Borda, by which the error of graduation may be di- minished to half a degree, and, practically speaking, anni-hilated (art. 142). It is only necessary to make one space fixed, for purposes of mere explanation, and let KL be a telescope movable on O, the common axis of two cir- cles (LMF and LNE). Let LM be perpendicular to KLO; it is absolutely fixed in the plane of the objects, and carries the graduations, and the latter is freely movable on the axis. The telescope is thus perfectly adjustable to all circles, and moves with it. The same OA carries the index, or 104 A TREATISE ON ASTRONOMY. [CHAP. II. vernier, which reads off the graduated limb of the fixed circle. This arm is provided with two clamps, by which it can be temporarily connected with either circle, and A diagram showing a telescope with a vernier scale attached to one end of its arm. The vernier scale has a series of small marks along its length, and a pointer that moves along these marks. The telescope is pointed at a star, and the vernier scale is used to measure the angle between the telescope's axis and the line of sight to the star. detached at pleasure. Suppose now, the telescope di- rected to P. Clamp this inner arm OA to the inner circle, and unclamp it from the outer, and read off. Then carry the telescope round to the other object Q. In so doing, the inner arm will be carried round through such an angle as to bring it into contact with that part of the outer circle which is clamped to it, will also be carried round, over an arc AB, on the graduated limb of the outer, equal to the angle PQ. Now clamp this outer arm BC to the outer circle, and unclamp the inner, and read off : the difference of readings will of course measure the angle POQ ; but the result will be liable to error, owing to the imperfectness of both arms and that of observation, both which is our object to get rid of. To this end transfer the telescope back to P, without unclamping either arm. Now again clamp it, having made the junction of P, clamp the arm to b, and unclamp it from B, and again transfer the telescope to Q, by which means we shall have read off twice over a second arc, BC equal to the angle POQ. Now again read off ; then will be the difference between this reading and the original reading (which was taken) equal to POQ affected with both errors of observation, but only with the same error of graduation as before. Let this pro- cess be repeated three times (or more times); then will the final be ABCD read off on the circle be ten times the required angle, affected by the **CHAPTER III.** **GEOGRAPHY.** 105 joint errors of all the ten observations, but only by the constant mean error of graduation, which depends on the initial and final readings off alone. Now the errors of observation are not always equal to one another; so that, if sufficiently multiplied, their influence will disappear from the result. There remains, therefore, only the mean error of graduation to be divided in the final result by the number of observations, and is therefore diminished in its influence to one-tenth of what it would have been. The abstract beauty and advantage of this principle seem to be counterbalanced in practice by some unknown cause, which, probably, must be sought for in imperfect clamping. **CHAPTER III.** **OF GEOGRAPHY.** Of the Figure of the Earth—Its Exact Dimensions—In Form that of Equi- lumen described by Copernicus—Fuses: Vankeer's Description of its Shape—Sundials: The Orbits of the Planets—The Gravitational Influen- ce—Gravity in a Sphereoid—Other Effects of Earth's Rotation—Tables for Determining the Lengths of Arcs of a Great Circle—The Conic Sections—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and their Applications—The Conic Sections and theirApplications 106 A TREATISE ON ASTRONOMY. [CHAP. III. an outline of geography so far as it is to be considered a part of astronomy. (162.) Geography, as the want imports, is a delineation or description of the earth. In its widest sense, this comprehends not only the delineation of the form of its continents and seas, its rivers and mountains, but their physical features, their productions, and their appropriation by communities of men. With physical and political geography, however, we have no concern here; for our object is to consider only that part of the exact knowledge of the form and dimensions of the earth, the parts of its surface occupied by sea and land, and the configurations of these parts. The ocean is a great plain, subterranean under the ocean, and broken into the various forms of mountain, table land, and valley; neither should the form of the land beneath the sea be left out of consideration; the situation of the land beneath the water, be left out of consideration; we know, it is true, very little of it; but we may suppose it to be everywhere level, and if possible, rendered, than required inasmuch as there are many very important branches of inquiry which would be greatly facilitated by such a supposition. (163.) With regard to the figure of the earth as a whole, we have already shown that, speaking loosely, it may be supposed to be spherical. But it has been duly appreciated the remarks in art. 22 will not be at a loss to perceive that this result, concluded from observations not extensive enough to give any certainty respecting very small portions of the surface at once, can only be regarded as a first approximation, and may require to be materially improved by further observations being neglected, or by increasing the delicacy of our observations, or by including in their extent larger areas of its surface. For instance, in describing an ellipse (which is a common inquiry), that the true figure is somewhat elliptical, or flattened, in the manner of an orange, having the diameter which is less than half that of its equatorial circle greater than the diameter of its equatorial circle; thus is so trifling a deviation from the spherical form that, if a modicum of such proper observations were made upon this land before on a globe, the nicest eye or mind would not CHAP. III.] FIGURE OF THE EARTH. 107 detect the flattening, since the difference of diameters, in a globe of sixteen inches would amount only to $\frac{1}{4}$ of an inch. In all common pursuits, and for all ordinary purposes, we are content with the figure of a sphere; nevertheless, by careful measurement, the difference would not fail to be noticed, and, speaking strictly, it would appear that the earth is not a perfect sphere, but a spheroid, which is the name appropriated by geometers to the form above described. (165.) It is evident that such a figure by a plane is not circles, but ellipses; so that, on such a shaped earth, the horizon of a spectator would nowhere (except at the poles) be parallel to any line drawn through the center; it is easy to demonstrate, however, that its deviation from the circular form, arising from so very slight an "ellipticity" as this, would be imperceptible to our senses, just to our eyesight, but to the test of the dipsector; so that by that mode of observation we should never be led to conclude that the earth was not perfectly spherical. How we are led to this conclusion, as a practical result, will appear, when we have explained the means of determining approximately the dimensions of the earth, or any part of the earth. (166.) As we cannot grasp the earth, nor recollect its form in imagination, we must compare it with some known object, and compare it with a known standard measure in any degree commensurate to its own size; but can only creep upon about it, and touch it here and there. By comparatively small parts of its vast surface successively, it becomes necessary to supply, by geometrical reasoning, the defect of our sense of touch. Thus, by a careful and faithful measurement of such small parts to conclude the form and dimensions of the whole mass. This would present little difficulty if we could imagine ourselves within a sphere, for the proportion of the circumference of a circle to its diameter being known (viz., that of $3\frac{1}{4} \times 18926$ to $18926$, or $3\frac{1}{4}$), we may easily find out the entire circumference of any great circle, such as a meridian, in miles, feet, or any other standard units; to know the diameter in units of such standard measures; and the circumference of the whole circle is known as soon as we 108 A TREATISE ON ASTRONOMY. [CHAP. III. know the exact length of any aliquot part of it, such as 1st or 2nd part; and, this being not more than about seventy miles in length, is not beyond the limit of very exact measurement (for we know that we can measure to what we know it exact termination at each extremity) within very few feet, or indeed, inches, by methods presently to be proposed. (160.) Supposing, then, we were to begin measuring with all due liberality from any station, in the exact direction of a meridian, and proceeding by the same indi- cision we were informed that we had accomplished an exact degree from the point we started from, our problem would then be reduced to one. It remains, therefore, to inquire by what indications we can be sure, 1st, that we have advanced an exact degree; and 2ndly, that we have kept ourselves in the true direction of a meridian. (167.) Now, the earth has no landmarks so to in- dicate degrees nor true meridians as to enable us to guide us in any particular. The compass, though it may be tolerable guide to the sailor or the traveller, is far too uncertain for astronomical purposes, and too little known in its use, to be of any use to us. We must therefore, look outwards and refer our situation on the surface of the earth to natural marks external to it, and which are of equal magnitude and stationary with the earth itself. Such marks are afforded by the stars. By observations made at different stations on any station, and from their known polar distances, we con- clude the height of the pole; and since the altitude of the pole is equal to the latitude of the station, we may conclude that all stations having the same observations give the latitudes of any station where we may establish the requisite instruments. When our latitude is thus determined by these observations, we know that provided we have kept to the meridian line, we have described one three hundred and sixtieth part of the earth's circumference. (168.) The direction of the meridian may be secured at every instant by the observations described in art. 187, and although they do not afford a direct indication of our measurement from this exact direction, yet if we keep a strict account of the amount of this division, a **CHAP. III.] LENGTH OF A DEGREE OF LATITUDE.** Very simple calculation will enable us to reduce our observed measure of its meridional value. (169.) Such is the principle of that most important geographical operation, which we call the measurement of the meridian. In its detail, however, a somewhat more modified course must be followed. An observatory cannot be moved, and the stars cannot be observed at two points so identifiably distinct and so accurately measured as to identify and measure an exact degree neither more nor less. But this is of no consequence, provided we know how to proceed with the least possible error, as we have measured. In place, then, of measuring this precise aliquot part, we take the more convenient method of observing the same star at two different stations, about a degree, or two or three degrees, as the case may be, apart, and determining by astronomical observation the amount of refraction between them. (170.) Again, it is of great consequence to avoid in this operation every source of uncertainty, because an error in any one point will multiply itself many times. We multiplied 360 times the circumference, and nearly 118 times in the diameter of the earth concluded from it. And therefore any error in the determination of the position of a star's altitude will be especially injurious. Now there is still too much uncertainty and fluctuation in the amount of refraction to allow us to neglect it; and it is especially desirable to avoid this source of error. To effect this, we take care to select for observation, at the extremity of each station, a star whose declination passes near the zeniths of both. The amount of refraction, within a few degrees of the zenith, is very small, and its fluctuation is consequently very slight. It is also extremely inconvenient as to be utterly inapplicable. Now, it is the same thing whether we observe the pole to be raised or depressed by a given quantity when passing over the meridian or have changed by the same quantity. If at one station we observe any star to pass through the zenith at 5° N., and at another station at 5° S., both of which are on the meridian, we are sure that the geographical latitude, or the altitudes of the pole at the two stations, must differ by some amount equal to 10°. (171.) Showing that the terminal points of one degree K 110 A TREATISE ON ASTRONOMY. [CHAP. III. can be ascertained, its length may be measured by the methods which will be presently described, as we have before remarked, to within a very few feet. Now, the error which we shall find in this measurement, at these terminal points cannot exceed that which may be com- mitted in the observation of the zenith distance of a star, properly so called, when observed with a quadrant, and with proper care, can hardly exceed a single second. Supposing we grant the possibility of ten feet of error in the measure of the arc, and suppose also that the mean of each of the zenith distances of one star, observed at the northern and southern stations, and lastly, suppose all these errors to be equal, then, if the result obtained give a result greater or all less than the truth, it will appear, by a very easy proportion, that the whole amount of error which we have supposed to be committed in the measure of earth's diameter, as concluded from such a measure, would not exceed 544 yards, or about the third part of a mile, and would be altogether negligible. (173.) This, however, supposes that the form of the earth is that of a perfect sphere, and, in consequence, the lengths of meridional arcs are equal. But when we come to compare the measures of meridional arcs made in various parts of the globe, the results ob- served, although they do not agree exactly with each other, yet show that the supposition of a spherical figure is very near remote from the truth, yet exhibit disorders far greater than what we have already observed in our observations on latitude, and render it evident that the hypothesis, in strict- ness of its wording, is untenable. The following table exhibits the differences between those (per- sonally determined as above described), expressed in British standard feet, as resulting from actual measure- ments; much more accurate than any which have been made by any of the most eminent observa- tions of various nations, men of the first eminence, supplied by their respective governments with the best instruments and materials; and which could tend to insure a successful result of their import- ant labours.* The lines in columns of this table are extracted from among the data given in Professor Airy's excellent paper "On the Figure of the Earth," in the Encyclopaedia Metropolitana." A page from a book titled "A Treatise on Astronomy." It contains text discussing measurements related to astronomy and Earth's shape. **CHAP. III.] DEGREES IN DIFFERENT LATITUDES.**










































111 It is evident from these inscriptions of the second and fourth columns of this table that the length of a degree increases with the latitude, being greatest near the poles, and least near the equator. Let us now consider whether we can put upon this conclusion, as regards the form of the earth. (178.) Suppose we held in our hands a model of the earth, and examined it by touch. It would already appear, so nearly spherical, that neither by the eye nor the touch, unassisted by instruments, could we detect any difference between it and a sphere. But let us examine it acrross from measuring directly across from surface to surface in different directions with any instrument, by which we may ascertain its form. We shall find that it is no longer than another ; how then, we may ask, are we to ascer- tain whether it is a true sphere or not ? It is clear that we have no resources, but to endeavour to discover, by A diagram showing a cross-section of a sphere with a line drawn across it at various angles to show the varying lengths of degrees depending on latitude. some nicer means than simple inspection or feeling, whether the convexity of its surface is the same in every part ; and if not, where it is greatest, and where least. Suppose, then, a thin plate of metal to be cut into 112 A TREATISE ON ASTRONOMY. [CHAP. III. a conicovry at its edge, so as exactly to fit the surface at A; at this let now be removed from A, and applied succes- tively to several other parts of the surface, taking care to keep its vertex always in the same place, as is done here represented. If, then, we find any position, B, in which the light can enter the middle between the globe and plane, we shall have found the point of least pres- sure, or admit the light under its edges, we are sure that the curvature of the surface at B is less, and at C greater than at A. (174.) What is here do by the application of a metal plate of determinate length and curvature, we do on the earth by the motion of a body. The axis of rotation is the altitude of the pole. Curvature of a surface is nothing but the continual deflection of its tangent from one fixed direction. Let us suppose that we have measured distance (which answers to our horizon) to have shifted its posi- tion with respect to the axis of rotation, and that with respect to the line (impose) to have shifted its position with respect to the axis of heaven, and the line joining the earth's centre and some given star), more in one part of the earth's surface than in another. It will be evident by evidence, that the curvature of the surface at the former spot is greater than at the latter; and vice versa, when, in order to travel over a longer measured space at one point than at an- other, we must move in different directions with respect to the line (impose) to have shifted its position with respect to the axis of the earth's centre, and that with respect to the line (impose) to have shifted its position with respect to the axis of heaven. We conclude that the curvature of a meridional section of the earth is smaller greater at the equator than to- wards the poles. This is because all meridional sections are not spherical, but flattened at the poles, or which comes to the same, protractor at the equator. (175.) By means of a plane perpendicular section of the earth, C its centre, and NA, BD, GE, arcs of a meridia, each corresponding to one degree of difference of latitude. Let N be any point on a meridian at any altitude of a star, as referred to the horizon of a spectator travelling along the meridian. Let NA, EA, BH, FD, GQ, etc., be the respective directions of the plumb-line at the stations N A, B D, G Q etc. which we will suppose N CHAP. III.] MERIDIONAL SECTION OF THE EARTH. 113 to be at the pole and $E$ at the equator; then will the tangents to the surface at these points respectively be per- pendicular to the meridian plane. Let $AB$, $CD$, $EF$, $GH$ be any two pairs of such tangents, viz. $nN$ and $aA$, $bB$ and $dD$, $gG$ and $eE$, be pro- longed till they intersect each other (at the points $x$, $y$, $z$), the angles between them being equal, and therefore all equal so that the small curvilinear area NA, BD, GE, may be regarded as areas of circles of radius equal to the perpendiculars from their centres to those points are determined and measured. Now, as the area of different circles, which subtend equal angles at the centre, is proportional to the squares of their radii, and as the NA is greater than BD, and that again GE, it follows that the radius NZ must be greater than ND, and consequently it appears that the mutual intersections of the plumb-lines will not, as in the sphere, all coincide in one point C, the centre, but will lie in a circle of radius NZ (which will be rendered more evident by considering a number of intermediate stations). To this curve geometers have given the name of the curve of the curve NABDGE, from whose centre of curvature it is constructed. A diagram showing a meridional section of the Earth with various lines and points labeled.114 A TREATISE ON ASTRONOMY. [CHAP. III. (176.) In the flattening of a round figure at two op- posite points, and its prolongement at points rectangularly situated to the former, we recognise the distinguishing feature of the meridian. It is the simplest supposition that we can make respecting the nature of the meridian, since it is proved not to be a circle, but an ellipse, which has one axis parallel to N.S., the axis of the earth, for its shorter, and E.F., the equa- torial diameter, for its longer axis; and that the form of the earth is such that this ellipse may be considered as such a curve evolute about its shorter axis N.S. This agrees well with the general course of the increase of the degree of latitude on the surface of the globe. In the ellipse, the radius of curvature at E, the extremity of the longer axis is the least, and at that of the shorter axis, the greatest. The ratio between these two radii is given with large representativeness.* Assuming, then, that it is an ellipse, the geometrical properties of that curve ena- ble us to ascertain what must be the length of the axes which shall correspond to any proposed rate of va- luation in its curvature, as well as to fix upon their ab- solute magnitude by means of a table giving the degree in the given latitude. Without resuming the reader with the investigation (which may be found in any work on trigonometry) I will only state that the length which agrees on the whole best with the entire series of meridional arc which have been satisfactorily measured, are as follow:
Country. Latitude. Are Length of Observers.
the Arc. measured of the Arc.
Southern 30° 17' 25" S 30° 17' 25" S 30° 17' 25" S Stauning.
Breton 30° 17' 25" S 30° 17' 25" S 30° 17' 25" S Bravo.
South of 30° 17' 25" S 30° 17' 25" S 30° 17' 25" S Lacaille, Comte.
Francois 30° 17' 25" S 30° 17' 25" S 30° 17' 25" S Monti.
America, U.S. 30° 17' 25" S 30° 17' 25" S 30° 17' 25" S Cayley, Comte.
America, South of 30° 17' 25" S 30° 17' 25" S 30° 17' 25" S Mason, Dixon.
Cuba, Good Hope 30° 17' 25" S 30° 17' 25" S 30° 17' 25" S Lamont, Brown.
Cuba, North of 30° 17' 25" S 30° 17' 25" S 30° 17' 25" S Lamont, Brown.
Peru:
Distance Latitude Length
30° 30° 477.5800
Lesser or pole diameter 90° 7999.170
Vulcanus diameter or pole com- pression. 180° 1390.000
The proportion of the diameters is nearly that of 298 : 299, and their difference $\frac{1}{299}$ of the greater, or a very little greater than $\frac{1}{298}$. (177.) Now, as to the rough distance of 8000 miles we have hitherto used is rather too great, the ex- cess being about 100 miles, or $\frac{1}{4}$ part. We consider it extremely improbable that an error to the extent of this difference could arise from errors belonging to the other quadrants. I find Proem. Airy's Error before cited. *The reader will observe that this ellipse is not a true ellipse, but a hyperbola. **CHAP. III.] EXACT DIMENSIONS OF THE EARTH.** five miles can admit in the diameter, or an uncertainty to that of a tenth of its whole quantity in the compression just stated. As convenient numbers to remember, for the purpose of this discussion, we may suppose there are just as many thousands of feet in a degree of the meridian as there are days in the year (365); that, approximately, there are 1000 miles in a degree of longitude miles, and a second about 100 feet; and that the equatorial circumference of the earth is a little less than 25,000 miles. (178.) The supposition of an elliptic form of the earth's section through the axis is recommended by its similarity to the ellipse which is supposed to represent results we have just set down with those of actual measure- ment. When this comparison is executed, discordant results will appear between the two series of figures too great to be referred to error of measurement, are yet so small, compared to the errors which would result from any other hypothesis. We shall therefore proceed on our regarding the earth as an ellipsoid, and referring the observed deviations to an ellipsoid, and if general, to com- parisons with similar observations. (179.) Now, it is highly satisfactory to find that the general elliptical figure thus practically proved to exist, is precisely that which would be formed by revolving the rotation of the earth on its axis. For, let us suppose the earth a sphere, at rest, of uniform materials throughout, and examine now how an attempt to make it a sphere would be effected. Under such circumstances it would obvi- ously be in state of equilibrium; and the water on its surface would be distributed uniformly over all parts of the sphere. Suppose now, a quantity of its materials were removed from the polar regions, and piled up all around them in one layer. This would increase both the polar and equatorial diameters of 25 miles which we know to exist. It is not less evident that a mountain ridge or ocean basin could not be made without some loss of water, from which the water would run down to the ex- cavated part at the poles. However solid matter might rest where it was placed, it equally must flow away which would leave no more space there than if it were thrown on the side of a hill. The consequence, therefore, would 116 A TREATISE ON ASTRONOMY. [CHAP. III. be the formation of two great polar seas, hemmed in all round by equatorial land. Now, this is by no means the case in nature. The ocean occupies, indifferently, all latitude, and is not confined to any particular one of the equatorial. Since, then, as we see, the water oc- cupies an elevation above the centre no less than 18 miles greater than that of the earth's surface, yet mani- fests no tendency to leave the former and run to- wards the latter, it is evident that it must be retained in that situation. But this is impossible, unless there however, would exist in the case we have supposed, something which is therefore not conformable to nature. In other words, there must be some force tending to the "po- lars"; and therefore the earth is either not a rest, or is so internally constituted as to attract the water to its equatorial situation. But this is impossible under our supposition there is no prima facie probability, nor any analogy to lead us to such an idea. The former is in accordance with the laws of nature; but the latter, durnal motion of the heavens; and, therefore, if it will furnish us with the power in question, we can have no hesitation in adopting it. (180). Now, every body knows that when a weight is whirled round, it ac- quires a centrifugal force which tends away from the centre of its motion ; which is called the centrifugal force. A stone will fall towards the earth; this may serve as an illustration; but a better, for our pre- sent purpose, will be a pair of scissors, sus- pended by a string at each end, and spun round, while the cord hangs perpendi- cularly. The surface of the water, in- stead of falling towards the earth, will come convege, as in the figure. The centrifugal force generates a tendency in all bodies to move outwards; and press towards the circumference ; it is, therefore, urged against the pail, and rises up into a height, and consequent increase of pres- sure downwards, just counterbalances its **CAP. III.] ACTION OF THE SEA ON THE LAND.** 117 centrifugal force, and a state of equilibrium is attained. The centrifugal force, being always one, and is admirably calculated to show how the form of equilib- rium accommodates itself to varying circumstances. If, therefore, we suppose the earth to be heated uniformly, as it becomes slower we shall see the concavity of the water regularly diminish ; the elevated outward portion will de- crease, and the depressed inward portion will increase. As a perfectly smooth surface is maintained, till the rotation is exhausted, when the water resumes its horizontal state, (181) all parts of the surface will be equally distant from the earth, at rest, and covered with a uniform ocean, were to be set in rotation about a certain axis, at first very slowly, but by degrees increasing to a velocity of one turn in twenty-four hours ; a centrifugal force would be thus gen- erated, whose general tendency would be to urge the water outwards, and to draw it towards the poles. A rotation might, indeed, be conceived so swift as to flirt the whole ocean from the surface, like water from a mop. But this is only a hypothesis ; let us consider what we now speak of. In the case supposed, the weight of the water would still keep it on the earth : and the centrifugal force would be counteracted by gravity, therefore, by the water leaving the poles, and flowing towards the equator; there keeping itself up in a ridge, just as it does in a bucket of water when it is tilted, and being retained in opposition to its weight, or natural tendency towards the centre, by the pressure thus caused. This would continue until the whole ocean was drying day the polar portions of the land in the form of immensely proliferous continents; and the difference of our supposed temperature between the equator and poles would be equi- poten- tial constant and polar sea would be formed; in the latter, proluberant land would appear at the poles, and a new continent would be formed at the equator. This would be the first or immediate effect. Let us now see what would afterwards happen, in the two cases, if things were reversed. (182) The sea is constantly beating on the land, grinding it down, and scattering its worn off particles and fragments, in the state of sand and pebbles, over its bed. 118 A TREATISE ON ASTRONOMY. [CHAP. III.] Geological facts afford abundant proof that the existing continents have all of them undergone this process, even more than once, and been entirely torn in fragments, or prolonged into ridges, by the agency of the elevated Earth's Land, in this view of the subject, loses its attribute of fixity. As a mass it might hold together in opposition to forces which would tend to separate it, but as a seat of successive or simultaneous degradation, when dissemi- nated through the water, in the state of sand or mud, it is subjected to the action of currents, and at any later stage of time, then, the protuberant land in both cases would be destroyed, and spread over the bottom of the ocean, filling up the depressions left by the former. In order to model the surface of the solid matter, in correspondence with the form of equilibrium in both cases. Thus, after a sufficient lapse of time, the whole earth would assume the equatorial continent, thus forcibly constructed, would again be levelled and transferred to the polar excavations, and the polar regions would be filled with water. In that of an earth in rotation, the polar protuberances would gradually be cut down and disappear, being trans- ferred to the equatorial region; and so long as the earth would assume by degrees the form we observe it to have—that of a flattened or oblate ellipsoid. (185.) The reason why the earth assumes the pro- cess by which the earth really assumed its actual form; all we intend to show that this is the form to which, under the influence of centrifugal force, it will tend; and which it would attain, even if originally and (so to speak) perpetually constituted otherwise. (186.) It is evident that one part of the earth and the time of its rotation being known, is it easy theme to calculate the exact amount of the centripetal force;* which, acting on every particle of matter by which all force or weight by which all bodies, whether solid or liquid, tend to fall towards the earth. By this fraction of its weight each particle is supported at a higher level, and thereby rendered susceptible of being supported at a higher levels, or more remote from the centre than at the poles, where no such centripetal force exists; and * See also: Oe. Macarthur, c. vii. **CHAP. III.] LOCAL VARIATION OF GRAVITY.** when, in consequence, the water may be considered as **specifically heavier**. Taking this principle as a guide, and combining it with the laws of gravity (as developed by Sir Isaac Newton), and other physical laws (which, however, mathematicians have been enabled to investigate, a pri- ori, what would be the figure of equilibrium of such a body as the earth? It is evident that the earth must preserve the earth to be; covered wholly or partially with a fluid; and revolving uniformly in twenty-four hours; and the whole of these conditions will be satisfied simultaneously with what experience shows to be the case. From their investigations it appears that the form of equilibrium is such that the earth is flattened at the poles, and of ellipticity very nearly identical with what is observed, and which would be no doubt accurately so, did we know the exact law of gravitation. (186.) The confirmation thus incidentally furnished of the hypothesis of the earth's rotation on its axis, cannot fail to give rise to some reflections on the reason why a sphere was not contemplated among the original reasons for adopting that hypothesis, which was assumed solely on account of the apparent diurnal motion of the heavens. Yet we see that, once admitted, it draws with it, as a necessary consequence, all those phenomena which no other satisfactory account could render. Indeed, so direct is their connexion, that the ellipticity of the earth is but another instance of its rotation. For Newton saw it to be a consequence of its rotation, and its amount actually calculated by him, long before any measurements had been made of its dimensions. In considering this subject, we shall find the same simple principle branching out into a whole train of similar and important consequences; and we shall see how far it extends which at first seemed entirely unconnected with it, and which, until traced by Newton up to this their origin, had raised doubts respecting the principles of astronomy, as well as among its greatest phenomena. (187.) Of its more obvious consequences, we may here mention one which falls in naturally with our present subject. If the earth really revolve on its axis, that rota- A diagram showing a sphere with a central hole through which a fluid is flowing outwards. 120 A TREATISE ON ASTRONOMY. [CHAIR. III.] tion must generate a centrifugal force (see art. 184), the effect of which must of course be to counteract a certain portion of the weight of every body situated at the equator, as well as that of all other bodies situated at intermediate latitudes. Now, this is fully confirmed by experience. There is actually observed to exist a difference in the weight of bodies situated at different stations on the same body, when conveyed successively to stations in different latitudes. Experiments made with the great earth-capsule, and with the pendulum, have fully demonstrated the fact of a regular and progressive increase in the weight of bodies corresponding to the increase of latitude, and also of the rate of its progression. From these it appears, that the extreme amount of this variation of gravity, or the difference between the weights of two bodies situated at the same latitude but at the same mass of matter, is one part in 184 of its whole weight, the rate of increase in travelling from the equator to any station being equal to one part in 184 of its latitude. (187.) The reader will hereinafter inquire what is meant by speaking of the same body as having different weights at different stations; and, how such a fact, if true, can be ascertained. When we weigh a body by a balance at any station, we suppose that we weigh the equal weight of another body under the same circumstances; and if both the body weighed and its counterpart are placed upon the same scale or weighing machine, if changed at all, will be changed equally, so that they will still continue to counterbalance each other. A difference in their weights would be detected by them; but such a difference could not be detected by these means; nor is it in this sense that we assert that a body weighing 194 pounds at the equator will weigh 195 pounds at some other station. On a scale or steelyard at the former station, an additional pound placed in one or other scale would be visible; but no such change would be observable at the latter. (188.) The meaning of the proposition may be thus explained.—Conceive a weight x suspended at the equator by a string which will allow it to move freely up or down; and consider (supposing such a thing possible) over ether **CHAP. III.] STATICAL MEASURE OF GRAVITY.** pulleys, such as B, round the earth's convexity, till the other end hung down at the pole, and there sustained the weight y. If, then, the weights x and y are equal, the two pulleys, one equa- tion, equatorial or polar, would each counterpoise each other on a horizontal line; but if they are placed by side over a single pulley, they would not counterbalance each other, nor would any application, but the point weight y, be in equilibrium, but with that of an- other point weight x, which is part of its quantity. (180.) The means by which this variation of gravity may be shown to exist, and its amount measured, are twofold: (1) by the action of mechanical power), stati- cal and dynamical. Thus, if a body be weighed, the gravity of a weight in equilibrium, not with that of an- other body, but with that of a power of a different kind not liable to be affected by local inequalities of power is the elastic force of a spring. Let A be a strong piece with it, into which is inserted a smooth plate of agate, D, which can be moved up and down horizontally by a lever. At C let a spring BE be attached, which carries at its lower end a weight y; polished and convex beneath. The elasticity and strength of the spring must so ad- just themselves to each other as to maintain the balance obtained by it just to swing clear of contact with the agate plate in the highest position while it is inad- used to use the instrument. Then if small weights be added cautiously, it may be made to descend until it just graze against the agate plate without being able to make with the utmost imaginable delicacy. Let these weights be noted; the weight F de- noted; the spring G carefully lifted off its hook, and A diagram showing a setup for measuring gravitational acceleration using a spring scale and an agate plate. 123 A TREATISE ON ASTRONOMY. [CHAP. III.] secured, for travelling, from rust, strain, or disturbance, and the whole apparatus conveyed to a station in a lower latitude. It will then be found, on remounting it, that, although the spring has been shortened by the weight of the ball, yet, be-fore, the weight F will no longer have power enough to stretch the spring to the extent required for producing a similar oscillation of the pendulum; and therefore, if any more be added; and the additional quantity necessary will, it is evi- dence, measure the difference of gravity between the two stations. The difference of gravity between two places may be expressed, matter, i.e. the sum of the weight of F and half that of the spiral spring itself. Granting that a spiral spring can be made to produce an elongation of 10 inches when a weight of 10,000 grains, including its own, shall produce an elongation of 10 inches without permanently straining it; and that this elongation is produced by an extension of $\frac{1}{4}$th of an inch, a quantity which cannot possibly be mistaken in such a contact as that in question. Thus we see that if we could make a spiral spring suf- ficing for the power of gravity at any station to within a length of its whole quantity. [38.] The next important process, by which the force urging any given weight to the earth may be de- termined, consists in ascertaining the velocity imparted by it to any body moving with uniform motion for a given time, as one second. This velocity cannot, be directly measured; but indirectly, the principles of me- chanics enable us to determine it. We do so by experi- ing, and consequently, the intensity of gravity, by observing the oscillations of a pendulum. As is proved in mecha- nics (see p. 56), if two pendulums are suspended at equal distances from their supports, and the same pendulum be made to oscillate at different sta- tions, or under the influence of different forces, and the numbers of oscillations be compared, they will differ. Whether the process above described could ever be so perfeeted and refined as to become a substitute for the use of the pendulum must depend upon whether it is possible to ascertain with certainty on the constancy or variability of the effect of temperature on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of pressure on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of air on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of light on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of sound on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of heat on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of electricity on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of magnetism on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of friction on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of cohesion on their ele- ments; or whether it is possible to ascertain with certainty the constancy or variability of the effect of cohesion on their ele- ments; and expansive pressure, under the attempt well worth making. CHAP. III.] GRAVITY ON A SPHEROID. 123 case be counted, the intensity of the force will be so each other inversely as the squares of the numbers of oscillations made, and thus their proportion becomes known. For instance, it is found that, under the equa- tor, the pendulum makes 86,400 vibrations in a mean solar day; and that, when transported to London, the same pendulum makes 95,536 oscillations in the same time. Hence we see that the intensity of the force urging the pendulum down- wards at the equator is to that at London as 86,400 to 95,536, or nearly as 1 to 1. Now let us suppose a mass of matter at the equator weighing 10,000 pounds exerts the same pressure on the earth's surface as that which would be exerted by a mass of 10,000 pounds transported to London below, that is 10,001/ of the same pounds, transported to London, would exert there. (19.) Experiments of this kind have been made, as already mentioned, with great accuracy ; but it is difficult to insure exactness in all accessible latitudes ; and their results are liable to some variation according to the seasons and latitude. The difference between the equator and poles. Now, it will not fail to be noticed by the observer, that this difference is very considerable. The obser- vation of this difference is here given by the fact that by the earth's rotation, that this differs materially from the fre- quency expressed by the pendulum at the equator. The law by which the former fraction exceeds the latter at any point, a small quantity in itself, but still far too large to be accounted for by any direct action; and must be directly accounted for, and not to prove fatal to this ex- planation if it will not render a strict account of it. It may be observed that this experiment affords a curious and instructive example of the indirect influence which mechanical causes often exercise, and sometimes produce effects contrary to their intentions. The rotation of the earth gives rise to the centrifugal force; the centrifugal force produces an elasticity in the surface of the earth; and this elasticity in its turn modifies its power of attraction on bodies placed at its surface, and thus gives rise to the difference in question. Here, then, we have the same cause exercising at once a direct and an indirect influence. The amount of the former 134 A. THEATRE ON ASTRONOMY. [CHAP. III.] It is easily calculated, that of the latter with its more difficulty, by its intricate and profuse application of geometry, whose steps we cannot pretend to trace in a work like the present, and can only state its nature and result. (184.) The attraction of a body (which is comple- nished by a centrifugal force) is the effect of the earth's attraction on it. This attraction, as Newton has demon- strated, is not confined to one particular place, but in a disposition of every parti- cule of matter in the universe to press towards, and if not opposed by some other force, to fall towards the centre of the earth, then, on a body placed on its surface, is not a simple but a complex force, resulting from the separate attractions of all the particles composing it. If the earth were a perfect sphere, the attraction exerted by it on a body any where placed on its surface, whether at its equator or at its poles, would be equal; this is simply reason of the exact symmetry of the sphere in every di- rection. It is not less evident that, the earth being ellip- soidal, and having no other cause of attraction than that existing, the same result cannot be expected. A body placed at the equator, and a similar one at the pole of a flattened sphere, will have different degrees of attraction rela- tion to the mass as a whole. This difference, without entering further into particulars, may be expected to draw out differences in their motions; and these differences between two bodies. Calculation confirms this idea. It is a question of purely mathematical investigation, and has been treated by several mathematicians, among whom New- ton, Maclaurin, Clairaut, and many other eminent geni- ometres; and the result of their investigations is to show that owing to this difference in attraction between two bodies dependent on the centrifugal force, its attraction ought to increase the weight of a body in going from the equator to the poles; and this increase is found to be nearly equal with $\frac{1}{2}$th due to the centrifugal force, make up the whole quantity, 1 th., observed. (184.) A very curious physical phenomenon, which owns its existence to the earth's rotation, is that of the trade-winds. These mighty currents in our atmosphere, on which so important a part of navigation depends, A diagram showing a flattened Earth with an arrow indicating motion from equator to poles. **CHAP. III.** THE TRADE-WINDS. 128 arise from, 1st, the unequal exposure of the earth's sur- face to the sun's rays, by which it is usually heated in different latitudes ; and, 2ndy, from that general law in the constitution of all fluids, in virtue of which they oscillate with respect to each other, being colder when hot than when cold. These causes, combined with the earth's rotation from west to east, afford so easy and satisfactory explanation of the phenomena here in question. (190.) It is a matter of observed fact, of which we shall give the explanation farther on, that the sun is con- stantly vertical over some one or other part of the earth between two parallels of latitude, called the tropics, respectively north and south of the equator; and that the whole of that zone or belt of the earth's sur- face included between the tropics, and equally divided by them, is always exposed to the direct rays of the sun. The heat thus acquired by the sun in its diurnal course, maintained at a much higher temperature than those regions where it does not reach, is communicated to the incumbent air, which is thereby expanded, and rendered more light and less dense than the surrounding air of the globe. It is, therefore, in obedience to the general laws of hydrostatics, displaced and buoyed up from the surface of the earth towards the upper regions, becoming heavier air, which glides in, on both sides, along the surfaces, from the regions beyond the tropics : while the dense air below sinks into those regions which are incen- tained by any lateral pressure, flows over as it were, and forms an upper current of air. This current, so- called from its nature, being cooled in its course, and also sucked down to supply the deficiency in the extra- tropical regions, keeps us thus a continuous circulation. (191.) The earth is constantly rotating as well as moving through the poles, the equatorial portion of its surface has the greatest velocity of rotation, and all other parts move with velocities varying according to their dis- tance to which they correspond. But as the air, when relatively and apparently at rest on any part of the earth's surface, is only so because its velocity is participative in the 2 126 A TREATISE ON ASTRONOMY. [CHAP. III. motion of rotation proper to that part, it follows that when a mass of air near the poles is transferred to the region near the equator by any impulse urging it direct- ly towards the equator, and when this current is directed towards its new situation it must be found deficient in ro- tatory velocity, and therefore unable to keep up with the speed of the earth's rotation, and consequently fall towards the equator. Hence, the currents of air which set in towards the equator from the north and south must, as they glide along the surface of the earth, be continually drawn down and drag upon it in the direction opposite to the earth's rotation, i. e. from east to west. Thus these currents, which but little affect the atmosphere at the poles, become more and more southerly winds, acquire, from this cause, a relative direction towards the west, and assume the character of permanent winds blowing from east to west. (197.) Were any considerable mass of air to be sud- denly transferred from beyond the tropic to the equator, the different effects would be very different. In many situations would be so great as to produce not merely a wind, but a tempest of the most destructive violence. But this is not likely to happen in any part of the north and south is gradual, and all the while the earth is continuously acting on, and by the friction of its surface accelerating, every particle of air which approaches towards the equator to cease at any point, this cause would almost immediately communicate to it the defi- nite motion which it has acquired; and it would move quietly with the earth, and be at relative rest. We have only to call to mind the comparative thinness of the coast- ing water in comparison with that of the ocean (see p. 384), and the immense mass of the latter, compared with the former (which it exceeds at least 100,000,000 times), to appreciate how little resistance there is offered by ex- tensive territory of the earth over the atmosphere immedi- ately incumbent on it, in point of motion. (198.) The same thing happens with regard to the winds on both sides approaching the equator, their easterly tendency must diminish.* The length of the diurnal circles in- * See also p. 528 in "The Voyage of Captain Cook." For a similar view see p. 106, where he says this very distinctly; and I am not aware, for his time, reasoned out... Author. 1 CHAP. III.] COMPENSATION OF THE TRADE-WINDS. 127 crease very slowly in the immediate vicinity of the equator, and for several degrees on either side of it hardly change at all. Thus the friction of the surface has more than compensated for the diminution of the wind by its urging it towards a state of relative rest, and diminishing thereby the relative set of the currents from east to west, which was produced by the action of the trade-winds, not at all reinforced by the cause which originally produced it. Arrived, then, at the equator, the trades must be everywhere at rest, and consequently at rest also. But not only this but the northern and southern currents, here meeting and opposing, will mutually destroy each other, leaving nothing but a very slight current due to a difference of local causes acting in the two hemispheres, which in some regions around the equator may lie one way or another. (190.) The result, then, must be the production of two great tropical belts, in the northern of which a compensating current flows northwardly; while in the southern belt a current flows southwardly; wind must prevail, while the winds in the equatorial belts, which separate the two former, should be compen- sated by a current flowing from east to west, of easterly character. All these consequences are agreeable to observed fact, and the system of aerial currents above described appears to be in reality what is understood by the regular trade-winds. (200.) The constant friction thus produced between the earth's surface and the air moving over it evidently must (it may be objected) by degrees reduce and at length destroy the rotation of the whole mass. The friction is so great that it can never compensate generally impossibly; and it is easy to see, in the pre- sent case, where and how the compensation takes place. The current which flows from east to west over towards the poles, carries with it the rotatory velocity due to its equatorial situation into a higher latitude, where it is gradually reduced by friction; and as it travels northward or southward, it will gain conti- nually more and more on the surface of the earth in its diurnal motion, and assume constantly more and more a * See how work has stood.* A diagram showing two large circles representing Earth's hemispheres. One circle is labeled "northern" and contains a smaller circle labeled "equatorial." The other circle is labeled "southern" and contains a smaller circle labeled "equatorial." Arrows indicate currents flowing from east to west across both hemispheres. 128 A THEATRE ON ASTRONOMY. [CHAP. III.] easternly relative direction; and when at length it returns to the surface, in its circulation, which it must do more or less in all the interval between the tropics and the poles, it will have been carried by the south-west wind in the northern hemisphere, and a north-west wind in the southern, and restore to it the impulse taken up from its source. It is this motion of the air, which is the south-west and westery gales so prevalent in our latitudes, and of the almost universal westerly winds in the Northern Hemisphere, that constitute, with those of a part of the general system of the reaction of the trades, and of the process by which the equilibrium of the earth's atmosphere is maintained. (201.) In order to construct a map or model of the earth, and obtain a knowledge of the distribution of sea and land, we must first ascertain what are the continents and islands, the course of its rivers and mountain chains, and the relative situations, with respect to each other, of all these objects. We must also know their centres of human habitation, or from other causes, it is necessary to possess the means of determining correctly the situation of every point on the surface of the earth. For this, two elements require to be known, the latitude and longitude, the former assigning its distance from the poles or equator, and the latter its distance from any given meridian. The latitude may be reckoned by which distance that distance is to be reckoned. To these, in strictness, should be added, its height above the sea level; but the As it is impossible to determine accurately the earth's sphere without affecting the gross masses, we shall omit all mention of height above sea-level. It seems worth inquiry whether hurricanes in tropical climate may not be produced by a similar cause. The air being drawn towards their places velocity has been sufficiently reduced by suc- cessive waves of air moving towards them; and they move towards earth with transitory velocity which gives them their demon- strable motion. This is very different from what has been given. Their course, generally speaking, is in opposition to the regular circulation of air (see my Lectures on Meteorology, Lectures 794). But it is by no means follows that this must always be so. For suppose a hurricane to arise in a region where there is no mass of air which local or temporary cause might carry close to the centre of disturbance; then it would have no tendency whatever to increase its velocity. Whenever such a mass should strike the hurricane, it would immediately arrest its motion; but if it should be situated at some distance from it, it would only retard its progress in much less than any degree of intensity on record might easily result from their continued Author. A small image showing a diagram or illustration related to astronomy. **CHAP. III.] GEOGRAPHICAL LATITUDES DETERMINED.** consideration of this had better be deferred, to avoid complicating the subject. (202.) The latitude of a station on a sphere would be merely the angle between the arc of the meridian, intercepted between the station and the nearest point of the equator, reduced into degrees. (See art. 66.) But as the earth is not a perfect sphere, this definition becomes inapplicable, and we are compelled to resort for our de- finition of latitude to a generalization of that property (art. 58) which is called "the figure of the earth," assum- ing by observation, and which has the advantage of being independent of the figure of the earth, which, although it may vary with different latitudes, is a geo- metrical solid. The latitude of a station, then, is the latitude of the elevated pole, and is, therefore, astrono- mically determined. This method is very simple and expedient for aceraining that important element. In consequence, it will be remembered that, to make a perfectly correct map of any part of the world, it is necessary that all equal differences of latitude are represented by ex- actly equal intervals of survey. (203.) It is evident that to find the latitude of a station, then, is easy. It is otherwise with its longitude, whose exact de- termination is a matter of more difficulty. The reason is that although we have already remarked that the earth, any more than parallel of latitude, we are obliged in this case, as in the case of the latitude, to resort to measurements made at different times and places, owing for the objects of our measurement; but with this dif- ference in the two cases—to observers situated at sta- tions distant from each other—those who observe (at night) the heavens present different aspects at all moments. The portions of them which become visible in a com- plete circle are those which lie between two parallels which are common to both describe circles differently inclined to their horizons, and differently divided by them, and similarly divided by their respective meridians. Those ob- servers situated on the same parallel (i.e. differing only in longitude) the heavens present the same aspect. Their visible portions on the same meridian also describe circles equally inclined, and similarly divided 180 A TREATISE ON ASTRONOMY. [UMAF. III.] by their horizons, and attain the same altitudes. In the former case there is, in the latter there is not, any thing in the appearance of the heavens, watched through a whole diurnal rotation, which indicates a difference of locality by the observer's horizon. (204.) But no two observers, at different points of the earth's surface, can see the same star simultaneously celestial hemisphere visible. Suppose, to fix our ideas, an observer stationed at a given point of the equator, and that he observes a star rising in his meridian, the star to be in his zenith, and therefore on his meridian, he should be suddenly transported, in an instant of time, round one hundred and eighty degrees of longitude, so that it comes to his zenith again. It is evident that he will no longer have the same star vertically above him: it will now appear to him to be just rising, and then to be setting. And if this star rises again, it comes to his zenith, i. e. before the earth's rotation from west to east carries him back again to the line joining the sun with the north pole (see fig. 3). (205.) The difference of the cases, then, may be thus stated, so as to afford a key to the astronomical solution of the problem. When an observer sees a star rising differing only in latitude, the same star comes to the meridian at the same time, but at different altitudes. In that of longitude, when an observer sees a star rising at the meridian at the same altitude, but at different times. Supposing, then, that an observer is in possession of any means by which he can determine either of these known stars' transits across his meridian, he knows his longitude; or if he knows the difference between its time of transit and that of another star's transit at any other station, he knows their differences of longitudes. For instance, if the same star pass the meridian of a place A at 6 o'clock in the morning, and pass one hour of sidereal time, or one twenty-fourth part of the earth's diurnal period later, then the difference of longi- tudes between A and B is one hour of time or 1°5', and B is so much west of A. (206.) In order to a perfectly clear understanding of the principle on which the problem of finding the longi- tude by astronomical observations is resolved, the reader CRAP. III.] DETERMINATION OF LONGITUDES. 131 must learn to distinguish between time, in the abstract, as common to the whole universe, and therefore reckoned from an epoch independent of local situation, and local time, which is determined by each place at each epoch, or initial instant, determined by local convenience. Of time reckoned in the abstract, or abstract manner, we have already spoken. Of time reckoned in the particular, or noisial time, which dates from an epoch determined by the sun's motion along the stars, we have laid down rules for determining it at every sidereal clock in an observatory, and in every town clock for common use. Every astronomer regards, or aims at regulating his sidereal clock by the passage of the sun across $0^{\circ}0^{\prime}$, when a certain point in the heavens, called the equinox, is on the meridian of his station. This is one mark of his endeavours; but he does not thereby entirely free himself from error. It gives no information to say that an event happened at such and such an hour of sidereal time, until we parti-cularly name what event is meant by such and such an apparent appearance. Just so it is with mean or common time. This is all a local reckoning, having for its own bounds those places which all share together during the year, when the sun is on the meridian of that particular place to which belongs this year's calendar, in like man-ner as the sun passes over any line by means of which it becomes necessary to name the place, or particularity of season mean time we intend. On the other hand, a date by equinoctial time is a universal date, being applicable to all parts of the world ad-ditionally. (2) The astronomer sets and regulates his sidereal clock by observing the meridian passages of the more conspicuous and well known stars. Each of these holds in the heavens a certain determinate and known place fixed by observation; and by noting the times of their passages in succession by his clock he knows when they equinoxal. At this moment he has observed out of them that have marked $0^{\circ}0^{\prime}$; and if it did not, he knows and can correct its error, and by the agreement or disagreement of the errors assigned to each star he can determine how much time has been regu-lated to twenty-four hours in one diurnal period and 123 A TREATISE ON ASTRONOMY. [CHAP. III.] if not, can ascertain and allow for its rate. Thus, although his clock may not, and indeed cannot, either be set cor- rectly, or go truly, yet by applying its error and rate (as they are known) to the time of the sun's transit, he will ascertain and ascertain the exact sidereal times corresponding to them, and proper to his locality. This indispensable operation being performed, we shall have all that is necessary for our explanation, however, we shall suppose the clock a perfect instrument ; or, which comes to the same thing, its error upon any given moment it is consulted, and included in its indications. (308.) Suppose now, two observers, at distant sta- tions, A and B, each having a pendulum clock of equal length, and regulate his clock to the true sidereal time of his station. It is evident that if one of these clocks could be taken up from station A to station B, and placed on the side of the other, they would be found, on compari- son, to differ by the exact difference of their local epochs ; that is, by the difference of their longitudes. For if we take a star, in passing from the meridian of A to that of B : in other words, by their difference of longitude, expressed in sidereal hours. (309.) A pendulum clock cannot be thus taken up and transported from place to place without derangement, but a chronometer can. Hence it appears that if we wish to use a chronometer instead of a clock, he may, by bodily transfer of the instrument to the other station, procure a direct comparison between them ; provided he knows his longitude from A. And even if he employ a clock, yet by comparing it first with a good chronometer, and then transferring it to another station where he has a good clock, the same end will be accomplished, provided the going of the chronometer can be depended on. (310.) The method here described is much more com- plete and convenient than this mode of ascertaining dif- ferences of longitude could be desired. For the observer, provided he has a good clock with him during whole transit, or some equivalent method of determining the local time at any given station, might, by journeying from place to place, and observing the meridian passages of stars at each (taking care not to alter his chronome- ter). CHAP. III. J LONGITUDES FOUND BY CHRONOMETERS. 139 ter, or let it run down), ascertain their differences of longi- tude, and then apply the same to the preceding. In this case, the same time-keeper being used at every station, if, at one of them, A, it mark true sidereal time, at any other, B, it will mark true sidereal time also; but if A be on the meridian of longitude A, and B on that of B, the difference between the two times will be equal to the difference of longitude of A and B; and therefore, in other words, the longitude of B from A will appear as the error of longitude of B from A. Suppose now that, if he travel westward, then his chronometer will appear con- sistent to gain, although it really goes correctly. Sup- pose that he has been travelling eastward for some hours, was on the meridian, or his chronometer at 0°, and in twenty-four hours (said time) had travelled 15° westward from his starting point; then his chronometer will again point to 0°; but the equinox will be not, on his new meridian, but on that of A, and he must wait until he arrives at A before he can tell whether this is does arrive there, then his watch will point not to 0°, but to P, and will therefore be 1° fast on the local time of B. This is shown in Fig. 16. (811.) Suppose an observer now to set out from any station as above described, and constantly travelling westward until he arrives at another station; and return to the point he set out from. A singular consequence will happen: he will have lost a day in his reckoning of time; because when he started from A, he was in his diary as Monday, for instance, when, in fact, it is Tues- day. The reason is obvious. Days and nights are made up by the rotation of the earth; so that while as the rotation of the earth carries the spectator round to view them in succession. So that if he be making observations during days and nights will be in error. But if he travel once round the globe in the direction of his motion, he will, on his arrival, have really lost a day; because he was in the opposite direc- tion, one turn less than if he had re- mained stationary at one point of its surface. In the former case he would have gained an alteration of day and night more, in the latter one less, than if he had treated to the rotation of the earth alone to carry him round. As the earth revolves from west to east, it 134 A TREATISE ON ASTRONOMY. CHAP. III. follows that a westward direction of his journey, by which he counteracts its rotation, will cause him to lose a day, and an eastward direction, by which he compasses with it, to gain a day. The difference between these days will be longer; in the latter, shorter than those of a stationary observer. This contingency has actually happened several times, and it is probable that it may happen again; but it is not likely that it will ever happen so early that distant settlements, on the same meridian, will differ a day in their usual reckoning of time, according to the circumstances under which they live. In an eastward or in a westward direction,—a circumstance which may produce strange confusion when they come to use the same calendar, and which requires some means of correcting the ambiguity, and settling the disputes which such a difference may give rise to, consist in having reference to the equinoctial date, which can never be ambiguous. (219.) Unfortunately for geography and navigation, the chronometer is not sufficiently accurate to be useful improved by the skill of modern artists, is yet far too imperfect an instrument to be relied on implicitly. How-ever such an instrument might be made to run at uniform rate for a few hours, or even days, yet in long absences from home the chance of error and accident become so multiplied as to render it useless. It is therefore better to have the best. To a certain extent this may, indeed, be reme- died by carrying out several, and using them as checks on each other's accuracy. But this is only possible when there is only a palpitation of the evil—the great and funda- mental—viz—it is the only one which the determination of longitude depends upon. For this reason it is neces-sary, therefore, to resort to other means of communi- cation from one station to another a knowledge of its local time must be obtained. As this is a very difficult point as a centre, its local time as a universal standard with which the local time at any other, however situated, may be agreeably compared. This is done by all places being referred to the meridian of such central point. (218.) The simplest and most accurate method by which this object can be accomplished, when circum- stances admit of its adoption, is that by diaphragm signal. CHAP. III.] LONGITUDES DETERMINED BY SIGNALS. 135 Let A and B be two observatories, or other stations, provided with accurate means of determining their respective local times, and let us first suppose them visible from each other. When the signal is sent from one to the other, and rates ascertained and applied, let a signal be made at A, of some sudden and definite kind, such as the flash of a gun, or the appearance of a bright light, or the ex- tinction of a bright light, or any other which admits of no mistake, and can be seen at great distances. The manner of making this signal will depend on each observer at his respective clock or watch, as if it were the transit of a star, or any astronomical phenomenon, and the time when it was made will be known by applying, the local time of the signal at each is determined. Consequently, when the observers communicate their observations to each other (assuming that owing to the almost instantaneous transmission of light) it must have been seen at the same absolute instant by both, the difference between their local times being thus known, their longitudes becomes known. For example: At A the signal is observed to happen at 5° 0' 0" sid. time at A, and at B at 5° 0' 0" sid. time shown by the clock at B, when the signal was seen there. At B the same signal was seen at 5° 0' 0" sid. time at B, and at A at 5° 0' 0" sid. time shown by the clock at A, by applying its error and rate. Consequently, the difference of their local epochs is 4° 0", which is also their difference of longitude. (114.) The accuracy of the final determination may be increased by making and observing several signals at different times during the day and night, under all kinds of times, and the mean of all which is, of course, more to be depended on than the result of any single comparison. By this method the effect of errors in observation of clocks may be regarded as altogether destroyed. (115.) The distances at which signals can be rendered visible are limited by atmospheric conditions in inter- posed country. Over sea the explosion of rockets may easily be seen at fifty or sixty miles; and in moun- tainous countries the flash of gunpowder on an open spot may be seen if a proper station be chosen for its 136 A TREATISE ON ASTRONOMY. [CHAP. III. exhibition, at much greater distances. The interval be- tween the stations of observation may also be increased by causing the signals to be made not at one of them, but at two or more places, which will be observed by both parties, it is a matter of indifference where they are exhibited. Still the interval which could be thus shortened would be very great, and the whole con- sequence of little use, but for the following ingenious contrivance, by which it can be extended to any distance, and carried out with perfect accuracy and certainty. (216.) This contrivance consists in establishing, be- tween the extreme stations, whose difference of longi- tude is so great that no signal can be sent between them, are observed, a chain of intermediate stations, alternately destined for signals and for observers. Thus, let A and B be the two stations distant from each other, and C be established, at which rocks, &c. are fired at stated in- tervals. At C let an observer be placed, provided with a chronometer and chronometer; and at D another observer and chronometer; and so on till the whole A B C D E F Z line is occupied by stations so arranged, that the signals at B can be seen from A and C; those at D, from A and C; and those at E from D and C. By observing the errors and rates of the clocks at A and C ascertained by astronomical observation, let a signal be made at B, and observed at D; then the difference between A's clock and C's chronometer becomes known. After a short interval (five minutes for instance) let a signal be made at D, and observed at E by C and E. Then will the difference between their chronometers become known; and the difference between A's clock and E's chronometer will become known; the difference between the clock A and chro- nometer E is therefore known. This, however supposes CHAP. III.- NATURAL SIGNALS. 137 that the intermediate chronometer C has kept true side- real time, or at least a known rate, in the interval between the signals. Now this interval is purposely made so very short that the error of the chronometer, which is a char- acter can possibly produce an appreciable amount of error in its lapse. Thus the time propagated from A to C is equal to the time required by the signal to travel from A to C, plus an error (or loss from error of observation), to E. Similarly, by the signal made at F, and observed at B and Z, the time required by the signal to travel from F to B is equal to the time required by the signal to travel from F to Z, plus an error (or loss from error of observation), to D. The errors thus at the lengths at A and Z are compared. The process may be repeated as often as is necessary to determine whether any difference exists between the two lines of stations is numerous, by keeping up a succession of signals, so as to allow each observer to note al- ternately the arrival of the signal at his station. If pre- arranged, many comparisons may be kept running along the line at one, by which time is saved, and other ad- ditional observations may be made. This process is usually repeated on several nights in succession. (217.) In place of auroral signals, natural ones, when they occur, are sometimes seen in the sky, and may be equally employed. In a clear night the number of those singular meteors, called shooting stars, which may be observed in any part of the heavens, is seldom exceeded in their appearance and disappearance, and from the great height at which they have been ascertained to take place, it appears probable that they are emitted from the surface of the earth; there is no doubt that they may be resorted to with advantage, by previous concert and agreement be- tween observers. (218.) Another species of natural signal, of still greater extent and universality (being visible once over a whole territory) than auroral signals, consists in the passage of Jupiter's satellites, of which we shall speak more at large when we come to treat of those bodies. Every such eclipse is an event of almost universal interest; great advantage *For a complete account of this method, see the article of deduction the most advantageous result is obtained from a combination of all the observations, by Mr. W. H. Draper, in "The Astronomical Almanac," 1860-61. Pp. 158-160. Penn., Phil. Trans., 1860; by the author of this volume. *This idea was first suggested by the late Dr. Mackenzie. M 130 A TREATISE ON ASTRONOMY. [CHAP. III. in its applicability to the purpose in question, viz. that the time of its happening, at any fixed station, such as Greenwich, can be predicted from a long course of pre- vious records. This prediction may be made with great certainty, and that this prediction is sufficiently precise and certain, to stand in the place of a corresponding obser- vation, is shown by the following example. Every year, who shall have observed one or more of these eclipses, and ascertained his local time, instead of waiting for a complete eclipse, which may happen at any time when a moment what the eclipses took place there, may use the predicted Greenwich Time instead; and hence, at once, and on all occasions, he will know the exact time of the moon's passage over the meridian. The mode of ascertaining longitudes is, however, as will hereafter appear, not susceptible of great exactness, and should only be employed for determining the longitude of places. The nature of the observation also is such that it cannot be made at sea; so that, however useful to the geographer, it is of little service to the navigator. (218.) But such phenomena as these are of only occa- sional occurrence; and in their intervals, and when cut off from all other observations, they become almost indi- fensible to possess some means of determining longi- tudes, on which not only the geographer may rely for a knowledge of the places of ships in distant seas; but even in land remote regions, in which the navigator can secure safety, at every instant of his adventurous course, the lives and fortunes of many persons depend upon his country, and the fortunes of his employers. Such a me- thod is afforded by LUNAR OBSERVATIONS. Though we may not always see the moon's face towards us during the moon's passage, this will not prevent us from giving here the exposition of the principle of the lunar method; for the observations required are very simple and easy; since by this course we shall have to deal with the made principles, apart from all the peculiar sources of difficulty which attend the application of this method; but which are, in fact, completely extraneous to the principle of its application to the problem of the longi- tude, which is quite elementary. (220.) If there were in the heavens a clock furnished CHAP. III.] LUNAR METHOD OF LONGITUDES. 159 with a dial-plate and hands, which always marked Greenswich time, the longitude of any station would be at once determined, so soon as the local time was known. The difference between the clock and the offices of the dial-plate and hands of a clock are these:— the former carries a set of marks upon it, whose position with respect to each other is fixed, and by which they indicate those marks, informs us, by the place it holds with respect to them, what it is o'clock, or at least that it has elapsed since midnight. The hands move uniformly round the dial-plate. (221.) In a clock the marks on the dial-plate are uniformly distributed all round the circumference of a circle, and move uniformly round that circle with a uniform motion. But it is clear that we should, with equal certainty, though with much more trouble, tell what hour it is, if we had only one hand, and that hand were unequally distributed—if the hands were eccentric, and their motion not uniform—provided we knew, 1st, the exact number of degrees in a circle; 2ndly, how many hours were placed; 3rdly, how many minutes were placed; 4thly, how many seconds were placed; 5thly, how many degrees were distributed around its circuit; 6thly, how many degrees were distributed around its circuit, the moon is the movable hand, which, with a motion that superficially considers itself uniform, yet is not so, but is found to be far otherwise, and regulated by mechanical laws of astomishing complexity and intricacy in result; therefore it is evident that this hand does not per- forms a monthly circuit among them, passing visibly over and hiding, or as it is called, occulting, some and gliding past others; and that when any one station is visible among them can, at any moment when it is visible, be 140 A TREATISE ON ASTRONOMY. [CHAP. III. exactly measured by the help of a sextant, just as we might measure the place of our clock-hand among the marks on its dial-plate with a pair of compasses, and hence, from the time that the hand points to any one mar- tion, deduce the time. That the moon does so move among the stars, while the latter hold constantly, with respect to each other, is evident from the fact that at a cer- tice of a few nights, or even hours, will satisfy the com- menting student, and this is all that we present we require. (283.) The next point to be considered is how we may make our analogy complete. Suppose the hands of our clock, instead of moving quite close to the dial-plate, were constantly kept at some distance from it. Unless, then, in viewing it, we kept our eye just in the line of their centre, we should not see them exactly thrown upon each other, but would perceive a di- amond. And if we were either unaware of this cause of optical change of place, this parallax—or neglect in not taking notice of it—would lead us to error in reading the time by referring the hand to the wrong mark, or incorrectly apprehending its distance from the right. On this account, when we observe the moon, in every case, when we had occasion to observe the time, the exact position of the eye, there would be no difficulty in ascertaining whether there was any cause of this effect of apparent displacement. Now, this is just what obtains with the apparent motion of the moon among the stars; for though she appears to move con- tinually changing place, there arises a parallax, which displaces the moon apparently among the stars, and must be allowed for in order to ascertain her place she also would occupy if seen from the centre. (284.) Such a clock as we have described might, no doubt, be used for measuring time; but if its use were only one, and if inculcable interests were at stake on a perfect knowledge of time, we should judicious regard it as most precious; and think no part more valuable than to dis- tinguish the laws of its movements, or in facilitating the **CHAP. III.** LUNAR METHOD OF LONGITUDES. means of reading it correctly. Such, in the parallel we are drawing, is the lunar theory, whose object is to reduce to regularity the indications of this strangely irregular instrument. The first of these indications, however, and with absolutely certainty, wherever* among the stars, at every hour, minute, and second, in every degree of latitude, the moon would be seen from the earth's centre, and will be seen from every accessible point of its surface; and such is the case with all the stars which are visible to the naked eye. The apparent angular distances from all those principal and conspicuous stars which lie in its course, as seen from the earth's centre, are always equal to each other within a very narrow and precision in almost absolute published under national control. No sooner does an observer, in any part of the world, see one of these stars than he observes from any one of those standard stars (whose place in the heavens have been ascertained for the purpose with the greatest exactness) that his time has been per- formed that comparison of his local time with the local times of every observatory in the world, which enables him to maintain his difference of longitude from one or all of them. (255.) The latitudes and longitudes of any number of our principal nations may be determined by the methods above described; and by thus laying down a sufficient number of principal points, and filling in the intermediate spaces by interpolation, maps of all countries he constructed, the outlines of continents and islands ascertained, the courses of rivers and mountains charted out, and all that can be done for proper localities. In practice, however, it is found simpler and easier to divide each particular nation into a series of triangles, each triangle being formed by two non-simultaneously visible from each other. Of these triangles, the angles only are measured by means of the floe- dine sextant; but since it is impossible to measure an angle, which is called a fusee, and which is measured with every refinement which ingenuity can devise or expense command. This fusee is of moderate extent, rarely surpassing six or seven miles, and purposely selected in a 141 145 A TREATISE ON ASTRONOMY. [CHAP. III. perfectly horizontal plane, otherwise conveniently adap- ted for purposes of measurement. Its length between the two extreme points (which are dots on plates of gold or plastics let into the surface of stones, and which are, or at least ought to be, in all cases, inscribed with some religious curse, as monumentals records of the highest im- portance), is called the base of the triangle. It is a mere measure precision, and its position with respect to the meridian, as well as the geographical positions of its ex- tremities, must be accurately known. (205.) The angle ABG represents such a chain of triangles. AB is the base, O, C, stations visible from both its extremities (one of which, O, we will suppose to be a national observatory, with which it is a principal object that the base should be as closely and immedi- ately contiguous as possible). The other two sides, O, H, K, other extremities, remarkable points in the country, where connection his whole surface may be covered, as it were, with a network of triangles. Now it is evident that the angles of the triangle A, B, C must be equal, and one of its sides, AB, measured, the other two sides, AC, BC, may be calculated by means of trigonometry. For when each of the sides AK and HC becomes in its turn a base capable of being employed as known sides of other tri- angles, --ACB being known by observation, and the side AC and BC, we can thence calculate the lengths of AG, CG, and BF; and therefore also the angles GAG and GCF; the included angle GCF, GP may be calculated, and so on. The greatest possible error in the true base of between seven and eight inches. Near Londoners is supposed not to exceed two inches. A diagram showing a triangle ABC with points O and H connected to points A and B respectively. CORRECTION FOR THE EARTH'S SPHERICITY. 143 on. Thus may all the stations be accurately determined and laid down, and as this process may be carried on to any extent, a map of the whole country may be thus con- structed, which will be found to contain no great errors. (227.) Now, on this process there are two important remarks to be made. The first is, that it is necessary to avoid triangles having angles very obtuse, and secondly, triangles free from any very great inequality in their an- gles. For instance, the triangle KBE would be a very imperfect one, because the angle B being acute, the obser- vations at A and K, because the angle F being very acute, a small error in the angle B would produce a great one in the angle D, and therefore the triangle would be distorted triangles, therefore must be avoided. But if this be at- tended to, the accuracy of the determination of the caleu- lations will be much improved. The second remark should be obtained by actual measurement (were it practicable); and therefore, as we recede from the base on all sides as far as possible, we shall find that we can not ac- cesses the sides of such larger triangles, such as GF, GH, HK, &c., by which means the next step of the operation will be made. We shall then take a larger scale, and embrace far greater intervals, so that it would have been safe to do (for the above reason) in the imme- diate vicinity of the station. In this manner we may divide the whole face of a country into great trum- gles of from 30 to 100 miles in their sides (according to the size of the country), and these triangles may be deter- mined, may be afterwards, by a second series of subordi- nate operations, broken up into smaller ones, and these divisions may be continued until they reach a point suffi- cing in is brought within the limits of personal survey and draftsmanship, and till a map is constructed, with any reasonable degree of accuracy. (228.) The next remark we have to make is, that all the triangles in question are not rigorously speaking, plane triangles; but are spherical triangles. Therefore, or rather, to speak correctly, of an ellipsoid. In very small triangles, of six or seven miles in the side, this may be neglected, as the difference is imperceptible; but in the larger ones it must be taken into consideration. A diagram showing a spherical triangle with sides labeled AB, BC, CA. 144 A TREATISE ON ASTRONOMY. [CHAP. III.] It is evident that, as every object used for pointing the telescope of a theodolite has some certain elevations, not only above the sea, but above the level of the sea, and so, moreover, these elevations differ in every instance, no section of the earth's surface would appear to be required. But, in fact, by the construction of the theodolite, it is possible to measure both the altitude and azimuth of any point on the earth's surface with an absolute and invariable instrument, this reduction is made in the very act of reading off the angles. Let A, B, C, be the places on its spherical surface, to which three stations, A', B', C', are assigned. The line AB will be measured by E.A., E.BP., ECQ'. If a theodolite be placed at A, and its horizontal circle point to E when truly adjusted, and its position being such that the sphere at A, intersecting the ra- dii E.BP., ECQ', at M and N, respectively, shall be parallel to the plane of the telescope. The telescope of the theodolite is thus pointed in succes- sion to P, Q and B', but instead of measuring each circle give--not the angle PAQ between the directions of the telescope, or between the objects P, Q, as seen from A', B', C', they measure the angle A of the spherical triangle BAC. Hence arises this remarkable circumstance,-that the sum of the three angles of a spherical triangle is always greater than 180°; and since this excess is always found to be rather more than 180° were the earth's surface a plane, it ought to be varied according to the curvature of the earth. This spherical excess, in so far from being a proof of incorrectness in the work, is what it is essential to its accuracy, and often serves as a time another palpable proof of the earth's sphericity. (289.) The true way, then, of conceiving the subject of a trigonometrical survey, when the spherical form of the earth is taken into consideration, is to regard the net- **CHAP. III.] PROJECTIONS OF THE SPHERE.** 146 work of triangles with which the country is covered, as the bases of an ensemble of pyramids converging to the centre of the earth. The theodolite gives us the true measures of the angles included by the planes of these pyramids, and we can easily determine their position on the level of the sea intersect them in an assembly of superficial triangles, above whose angles, in the radii proceeding from the centre of the earth, we find the measure of the superficial inequalities of mountain and valley. The ope- rale calculations of spherical trigonometry which this construction requires, are so simple that they are re- ductions of a survey, are dispensed with in practice by a very simple and easy rule, called the rule for the spheri- cal triangles, or the rule of Lambert's "Sphé- nométrie." If we would take into account the ellipticity of the earth, it may also be done by appropriate processes of calculation, but it is not necessary to do this unless one wishes to do it upon a work like the present. (230.) Whatever principle we adopt for adding to the regular surface of the sphere, to the level of the sea all the triangles, and the consequent determination of the geographical latitude and longitude of every station observed. This is what is done in making maps of countries; to lay down the outlines of continents and islands; the course of rivers; the direction of mountains rising from them; and finally, to give a general view and all those details which, as they belong to physical and statistical, rather than to astronomical geography, we neglect here. But it is evident that such maps will be necessary respecting maps, which are used as well in astronomy as in geography. (231.) It is something more than a representation, upon a plane, of some portion of the surface of a sphere, on which are traced the particulars intended to be ex- pressed by means of a map or chart or plan. Now, as a spherical surface can by no circumstance be extended or projected into a plane, without undue *Leibnitz's* Trigonometry, prop. 47. Woodhouse's dim. p. 148. Let We have neglected the ellipticity of the earth, which, for each a por- tion as may be seen, is not likely to have any marked influence. N A TREATISE ON ASTRONOMY. [CHAP. III.] enlargement or contraction of some parts in proportion to others; and as the system adopted in so extending or projecting it will decide what part shall be enlarged or relatively contracted, and in what proportions ; it follows, that when the same object is represented by different projects down, a great difference in their representations may subsist, though the objects themselves be equal. (282.) The projections chiefly used in maps are the orthographic, stereographic, and Mercator's. In the orthographic projection, every point of the hemisphere is referred to a plane perpendicular to the axis of rotation, let fall on it, so that the representation of the hemisphere thus mapped on its base, is such as would appear to an eye placed at an infinite distance from it. It is obvious, therefore, that in this projection only the exterior portions of the globe assume their true forms, while all the interior is more and more distorted and crowded together as we approach the edge of the map. Owing to this cause, the orthogra- phic projection is useful for small portions of the globe, is of little service for large ones. A diagram showing a spherical projection with lines radiating from a central point (the pole) to points on the surface of the sphere. The lines are labeled with angles and distances. (283.) The stereographic projection is in great mea- sure free from this defect. To understand this projection, 140 **CHAP. III.** MERCATOR'S PROJECTION. we must conceive an eye to be placed at E, one extremity of a diameter, ECB, of the sphere, and to view the concave surface of the sphere, every point of which, as P, is referred to the diameter plane ADF, perpendicular to EB by the line PME. The true perspective pro- jection of a sphere, then, is a true perspective representa- tion of its convexity and concavity. It is evident that, as such, it possesses singularly elegant geometrical properties, of which we shall state one or two of the principal kinds. (234.) And first, then, all circles on the sphere are re- presented by circles in the projection. Thus the circle XPA is projected into CA. Only great circles passing through the centre C are projected into lines joining the centre C : thus, BPA is projected into CA. Bdry. Every very small triangle, GIKH, on the sphere, is represented by a very small triangle in the projec- tion. This is a very valuable property, as it insures a good degree of accuracy in the representation of reality in all its parts, and enables us to project at least a hemi- sphere in a single map, without any violent distortion of the surface, and with perfect regularity. As in the orthographic projection, the borders of the hemisphere are unduly crowded together; in the stereo- graphic, their projected dimensions are, on the contrary, so much reduced as to form four points. (235.) Both these projections may be considered na- tural ones, inasmuch as they are really perspective re- presentations of the surface on a plane. Mercator's is entirely an artificial one, representing the sphere as it
80 80
60 60
40 40
20 20
A diagram showing Mercator's projection of a sphere onto a plane. 148 A TREATISE ON ASTRONOMY. [CHAP. III. cannot be seen from any one point, but as it might be seen by an eye carried successively over every part of it. In it, the degrees of longitude, and those of latitude, are almost equal to each other, so that the equator is conceived to be extended out into a straight line, and the meridians are straight lines at right angles to it as well as to the equator. The projection of this map on this projection is very not similar to what would be produced by referring every point in the Sols to its proper place in the sphere, and then from the centre, and then unrolling the cylinder into a plane. Like the stereographic projection, it gives a true repre- sentation of all parts of the globe equally distant, but varies greatly in point of scale in different regions; the polar portions in particular being extravagantly en- larged; and the whole surface of the globe is represented, not being comprizable within any finite limits. (286.) We are not, of course, here enter into any geographical discussion. The mere fact that we can see on the great scale, so to speak, manzasse enough to call for mention as an astronomical feature. When the com- munication between Europe and America was made by the discovery of Australia we are sure that no very extensive tract of land remain unknown, except perhaps at the south pole. But when we consider how vast is the globe into two hemispheres, that one shall contain nearly all the land; the other being almost entirely sea. It is in fact only by means of our knowledge of the world com- bined with our insular station in that great highway of nations, the Atlantic, not a little explanatory of our com- monwealths. The earth is divided into two hemispheres, of which one is terrestrial and the other celestial. Astronomically speaking, the fact of this divisability of the globe into an oceanic and a terrestrial hemisphere is most remarkable. It is suggestive of a want of absolute equality in the density of the solid material of the two hemispheres. Considering the whole mass of matter in the universe, according to Profes- sor Bunsen, it is evident that the half which proffers itself of necessity be buoyant; not, of course, that we mean assert it to be lighter than water, but as compared with the whole globe, in a first degree heavier than **CHAP. III.] DETERMINATION OF HEIGHTS.** 149 that fluid. We leave to geologists to draw from these premises their own conclusions (and we think them ob- vious enough) as to the internal constitution of the globe, and the immense nature of the forces which sustain its cohesion ; but we may here remark that we require investigations which may have for their object to explain the phenomena of the earth's surface, and to determine what the hypothesis of an exact equality of gravity, from what this hypothesis is derived, would require, this, as a general fact, ought not to be lost sight of. (327.) Our knowledge of the surface of our globe is incomplete. We know nothing about the mean sea level of every part of the land, and the depression of the bed of the ocean below the level of the sea all extant. The determination of this is attainable (with whatever difficulty and however slowly) by direct sounding ; the former by two distinct methods : the one consisting in trigonometrical observations at different stations on the coast, and at the stations of a survey ; the other, by the use of the harpoon and prism, when it is within reach, with that of the sounding line. In both cases we measure the distance of the point whose level we would know from the surface in equilibrium only ; in the one case, by an ocean current, in the other by a wave. In the one case our sounding line is real and tangible ; in the other, an imaginary one, measured by the length of the rope used in making it. This last method is impos- sible of counterbalancing. (328.) Suppose that instead of air, the earth and oceans were filled with oil; and that human life could subsist under such circumstances. Let ABCDE be a A small diagram showing a cross-section of a hypothetical Earth-like planet with continents (A, B, C) and oceans (D, E). The diagram includes labels for points A, B, C, D, and E. continent, of which the portion ABC projects above the water, but is covered by the oil, which also floats at an n2 150 A TREATISE ON ASTRONOMY. [CHAP. III. uniform depth on the whole ocean. Then if we would know the depth of any point D below the sea level, we see that, by sending down a line from B to D, which is at the height of B above the same level, we have only to send up a float from B to the surface of the oil; and having drawn down a line from this point to D, the difference of the two final lines gives the height in question. (339) Now, though the atmosphere differs from oil in not being capable of being compressed, and in not being capable of carrying up any float adequate to such a use, yet it possesses all the properties of a fluid really equal density, and therefore it has one particular; that, over the whole surface of the globe, its strata of equal density are parallel to the surface of equatorium, and are everywhere perpendicular to the axis, prolonged under the continents, and therefore each or any of them has all the characters of a definite surface to measure heights. This property is very well demonst- rated. Now the height at which, at any station B, the mercury in a barometer is supported, informs us of some horizontal distance from the centre of the earth; and, in other words, in what stratum of the general atmos- phere (indicated by its density) B is situated: whence it follows that if we consider by geometrical reasoning, at what height above the sea level that degree of density is to be found over the whole surface of the globe, we shall find that this degree corresponds with the barometer to the measurement of heights. For de- tails, the reader is referred to other works.** (340) The same principle applies to all heights of stations above the sea, we may connect all stations at the same altitude by level lines, the lowest of which will be the entire surface of the sea itself. Thus we see that the successive coast-lines which would take place were the sea to rise by regular and equal ascensional levels over every part of its surface would be always submerged. The bottoms of valleys and the ridge-lines * See Cah. Cpt. PETERSONIUM, art. 142. ** The reader will observe that in these tables see the work of Bosanquet. Also those of Olmstead, annually published by the French board of topography; and those of Mr. Hutton and Mr. Bully's Collection of Astrono- mical Tables and Formulas.
150 A TREATISE ON ASTRONOMY.
CHAP. III.
uniform depth on the whole ocean. Then if we would know the depth of any point D below the sea level, we see that,
by sending down a line from B to D, which is at the height of B above the same level, we have only to send up a float from
from B to the surface of the oil; and having drawn down a line B to this point to D, the difference of the two final lines gives
from this point to D, the difference of the two final lines gives the height in question.
(339) Now, though the atmosphere differs from oil in not being capable oil in not being capable of being compressed, and in not being capable
of carrying up any float adequate to such a use, yet it posses- sses all selves all the properties of a fluid really equal density,
selves all the properties of a fluid really equal density, and therefore it has one particular;
and therefore it has one particular; that, over the whole surface of the globe,
that, over the whole surface of the globe, its strata of equal density are parallel to the surface of equa-
its strata of equal density are parallel to the surface of equa- torium,
it strata of equal density are parallel to the surface of equa- and are everywhere perpendicular to the axis,
it strata of equal density are parallel to the surface of equa- prolonged under the continents,
torium, and therefore each or any of them has all
the characters of a definite surface to measure heights.
This property is very well demonst- rated.
Now the height at which, at any station B, the mercury in a barometer is supported,
informs us of some horizontal distance from<
150
<
A TREATISE ON ASTRONOMY.
<
CHAP. III.
<
uniform depth on the whole ocean. Then if we would know
<
the depth of any point D below the sea level, we see that,
<
by sending down a line from B to D, which is at the height
<
of B above the same level, we have only to send up a float from
<
B to the surface of the oil; and having drawn down a line
<
from B to this point to D, the difference of the two final lines gives
<
the height in question.
<
(339) Now, though the atmosphere differs from oil in not being capable
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of carrying up any float adequate to such a use, yet it posses- sses all
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selves all the properties of a fluid really equal density,
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and therefore it has one particular;
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that, over the whole surface of the globe,
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its strata of equal density are parallel to the surface of equa-
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torium,
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and are everywhere perpendicular to the axis,
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prolonged under the continents,
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and therefore each or any of them has all
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the characters of a definite surface to measure heights.

This property is very well demonstrated.

Now when we consider by geometrical reasoning at what height above the sea level that degree of density is found over the whole surface of our globe, we shall find that this degree corresponds with the barometer. For details, the reader is referred to other works.

  • The same principle applies to all heights of stations above sea level. We may connect all stations at the same altitude by level lines, the lowest of which will be the entire surface of sea itself. Thus we see that successive coast-lines which would take place were sea level raised uniformly over every part of its surface would be always submerged. The bottoms of valleys and ridge-lines are also indicated. See Cah. Cpt. PETERSONIUM, art. 142. Also those of Olmstead, annually published by French board topography; Mr. Hutton's Collection astronomical tables; Mr. Bully's Collection astronomical tables.

    • **CHAP. IV.]** **ERANOGRAFHY.** 151 of hills are determined by their property of intersecting all these level lines at right angles, and being, subject to that condition, the shortest and longest courses respec- tively which can be pursued from the summit to the sea. The former are called the meridians of a country ; the latter divide it into drainage basins : and thus origi- nates the map of a country, which is a plan of nature, on which the distribution, limits, and peculiarities of hu- man communities are in great measure dependent. **CHAPTER IV.** OF ERANOGRAFHY. Construction of celestial Maps and Globes by Observations of eight Ap- proaches to the Sun, and by the motions of the Moon, Sun, and Earth, and errors of the Conjunctions—Natural Regime in the Heavenly Sphere—Observations of the Equinoxes—Observations of the Equinoxes and Longitudes—Proportionate of the Equinoxes—Nudism—Astronomical Geographical Problems (241.) The determination of the relative situations of objects on the earth's surface, and the construction of maps and globes which shall truly represent their mutual configu- rations, as well as of catalogues which shall preserve a record of them, is a very difficult task. It is a task at once simpler and less laborious than that by which the surface of the earth is mapped and measured. For example, we may say that our celestial stations rise above us, constitute, so to speak, a celestial sta- tion; and among these stations we may, as upon the earth, lay down a line parallel with each other, maintain their angular distances from each other, which, cleared of the effect of refraction, are then in a state for laying down a map of the heavens. This is true for all parts of a country; and this without moving from our place, at least for all the stars which rise above our horizon. (242.) The construction of celestial maps can be attained by this means, and excellent celestial charts constructed; but there is a far simpler and easier, and at the same time, infinitely more accurate course laid open to us, if 152 A TREATISE ON ASTRONOMY. [CHAP. IV.] we take advantage of the earth's rotation on its axis, and by observing each celestial object as it passes our meridian, refer it separately and independently to the celestial equator, and thus ascertain its place on the surface of the earth at any given moment, so as to re- volve with it, and on which it may be considered as pro- jected. (343.) The right ascension and declination of a point in the heavens correspond to the longitude and latitude of a station on the earth; and the place of such a celestial object when observed at different elements are known, just as that of a town on a map, by knowing the latter. The great advantages which the method of meridian observations affords, consist in the comple- tion from star to star, are, then, 1st, that in every star is observed in that point of its diurnal course, which it has next seen before; 2ndly, that the instruments required (the transit and mural circle) for the simple observation being limited to those of any use by astronomers; 3rdly, that all the observa- tions can be made systematically, in regular succession, and with equal advantages; there being here no ques- tion about the time of observation, or about the weather, &c. And lastly, that, by adopting this course, the very quantities which we should otherwise have to calculate for long periods of time, are calculated in one day only, and which are essential to the formation of a catalogue, are made the objects of immediate measurement. It is almost impossible to deny, therefore, that this is the course adopted by astronomers. (344.) To determine the right ascension of a celestial object, all that is necessary is to observe its passage at its meridian passage with a transit instrument, by a clock regulated to exact sidereal time, or reduced to such by ap- plying the corrections due to refraction and parallax ob- tained by repeated observations of the same star at its successive meridian passages. The error, however, re- serves a certain quantity of time for each star from which all right ascensions in the heavens reckon, as lon- gitudes do on the earth from a first meridian. (345.) The nature of this point will be explained pre- **CHAP. IV.] RIGHT ASCENSIONS AND DECLINATIONS.** 158 seemly; but for the purpose of astronomy, it is so far as they concern only the actual configurations of the stars inter se, as knowledge of the equinox is not necessary. The right ascensions of celestial objects, being right me- sensions, is purely artificial, and a matter of convenience; but as on the earth, any station (as a national observatory) may be chosen for the purpose. In this respect, as in astronomy, any conspicuous star may be selected as an initial point from which hour angles may be reckoned. (346.) The declinations of celestial objects are de- pendent upon the latitude of the place where they are ob- served, and therefore, at different stations, they will differ for, and which will be explained in their proper places. (346.) The declinations of celestial objects are ob- tained by observing the altitude and azimuth of the object, with the mural circle or other proper instruments. This requires a knowledge of the geographical latitude of the place where observations are to be made, which is obtained by celestial observation. Still, and more directly by ob- servation of their polar distances on the mural circle, as these distances depend upon the latitude of any previous determination of the latitude of the station; neither, however, in this case, does observation give direct results; for it requires correction for refraction. The observations require to be corrected, first for refraction, and moreover for those minute causes of inequality which have been observed in the motions of the heavenly bodies. (347.) In this manner, then, may the places, one among the other, of all celestial objects be ascertained, and their right ascensions and declinations determined; a very important question. How far are these places per- manent? Do these stars and the greater luminaries of the heavens remain fixed in their relative positions and relation of place inter se, as if they formed part of a solid though invisible firmament; and, like the great masses of water in our oceans, do they move with the same distances and bearings each from the other? If no, the most rational idea we could form of the universe would be that of an earth at absolute rest in the centre, and a hollow crystalline sphere circulating round it and 184 A TREATISE ON ASTRONOMY. CHAP. IV. carrying sun, moon, and stars along in its diurnal mo- tion. If not, we must dismiss all such motions, and inquire individually into the distinct history of each ob- ject, with a view to discovering the laws of its peculiar motion, and the cause why and how this motion subsists between them. (144.) The sun, being from being the cause, that observation, even of the most curious nature, are sufficient to show that some, at least, of the celestial bodies, and therefore the earth itself, move in a state of constant change of place among the rest; in the case of the moon, indeed, the change is so rapid and re- markable, that it is impossible to describe it by any such expression as may happen to be nearer, may be noticed any fine night in a few hours; and if noticed on two successive nights, it will appear to be more care- less observer. With the sun too, the changes of place among the stars is constant and rapid; though, through the invisibility of the stars themselves, it is not readily measured, and requires either the use of telescopes and angular instruments to measure it, or a longer continued observation to keep with it. Nevertheless it is only necessary to call out at night a greater meridian altitude in summer than in winter, and the fact that the stars rise earlier in spring than at night vary with the season of the year, to perceive that great change must have taken place in that interval in its re- lative situation among other stars. Besides the sun and moon, too, there are several other bodies called planets, which, for the most part, appear to the naked eye only during certain periods of time; but they offer the same phenomenon of a constant change of place among the stars; now approaching, and now receding from, nothing else but their own motion. And some in longer, some in shorter periods, making, like the sun and moon, the complete tour of the heavens. (145.) The following table gives a general rule. The innumerable multitude of the stars which are distributed over the vault of the heavens form a constel- lation, which preserves, not only to the eye of the casual observer, but to the nice examination of the astronomer, A diagram showing a constellation. **CHAP. IV.]** FIXED AND ERATIC STARS. 155 a uniformity of aspect which, when contrasted with the perpetual changes in the configurations of the sun, moon, and planets, may well be termed invariable. It is not, however, due, by the refinement of exact measure-ments, to any process of nature, but to small changes of apparent place, attributable to no illusion or error on the part of the observer. These motions of them--such are called, in astronomy, the proper motions of the stars--but these are so excessively slow (even their accumulative amount (even in those stars for which they are greatest) being imperceptible within the whole duration of astronomical history), to produce no obvious or material alteration in the appearance of the stars. (200.) This circumstance, then, establishes a broad distinction between the heavenly bodies into two great classes: the fixed, and those which move in space (the obser-vations continued for many years) no change of mutual situation having been observed. The former including (which is implied in the word planet)--including the sun, moon, and planets, as well as the singular class of bodies termed comets--are those which remain always among the stars and among each other, the observation of a few days or even hours, is sufficient to exhibit an indistiguishable immobility. (201.) Uranography, then, as it concerns the fixed celestial bodies (or, as they are usually called, the fixed stars), consists in observing their relative positions on a globe or map; or the insertion on that globe, in its due place in the great constellation of the zodiac; or in drawing lines connecting points of parallel to the earth's axis; and of the equator and plane of the equinox: points and circles thereon, which are supposed to represent all parts of our earth, and therefore subject to all changes (if any) to which the earth's axis may be liable, are yet so con-venient in practice as to be used without any diminution (we speak of circles and lines), sanctioned by usage, in all globes and planispheres. The reader, however, will take care to keep them separate in his mind, and to *CLARITY OF WISDOM.* | **Fixed Stars** | **Eraic Stars** | |---|---| | Fixed position on a globe or map | Perpetual motion | | Insertion on a globe | No change in position | | Points and circles representing Earth | Subject to changes | The reader will observe that this distinction is made only with respect to the motions of these bodies with reference to one another. They do not differ from each other in regard to their own motions. Thus we have seen that the sun moves about once every 27 days; that the moon moves about once every 29 days; that Venus moves about once every 118 days; that Mercury moves about once every 88 days; that Mars moves about once every 680 days; that Jupiter moves about once every 12 years; that Saturn moves about once every 30 years. These periods are all very nearly equal to each other. The difference between them is less than one day. In consequence of this uniformity of periodical motion among them all, they appear to us at different times to occupy different places in the heavens. But this is owing solely to their different distances from us. For instance, if we suppose ourselves situated at a point on the earth's surface exactly opposite to that of Venus at any given time, we shall see Venus at its greatest distance from us at that time; while if we suppose ourselves situated at a point on the earth's surface exactly opposite to that of Jupiter at any given time, we shall see Jupiter at its greatest distance from us at that time. And thus we see that all these bodies appear to us at different times to occupy different places in the heavens because they are at different distances from us. 150 A TREATISE ON ASTRONOMY. [CHAP. IV.] facilitate himself with the idea rather of two or more celestial globes than of one, he will find that the former, one of which—a real one—are inscribed the stars; on the others those imaginary points, lines, and circles which represent the motions of the planets, and to aid their calculations; and to accustom himself to conceive in the latter, or artificial, spheres a capability of being moved by the same laws as those of the former; so that, should experience demonstrate (as it does) that these artificial points and lines are brought, by a certain number of years' observation, into variations (as they are called), to coincide, at very distant intervals of time, with different stars, he may not be surprised at such changes, and have no conclusion to correct in his notions. (232.) Of course we do not here speak of those un- couth figures or globes, which are usually inscribed over celestial globes and maps, and serve, in a rude and barbarous way, to enable us to talk of groups of stars, or constellations; but those which, though absurd or puerile in their origin, have obtained a currency from which it would be difficult, and perhaps impossible, to disengage them. For they have really (as many have) any slight resemblance to the figures called up imagination by a view of the more splendid parts of the heavens; and though not conven- ience; but as they are otherwise entirely arbitrary, and correspond to no natural subdivisions or groupings of the stars themselves; they are therefore to be regarded them.* except for briefly naming remarkable stars, as a Leonis, b Scorpii, &c. &c., by letters of the Greek alphabet; and it is only by this means that we find them on any celestial chart or globe, and may compare them with the heavens, and then learn for himself their position. *This treatment is neither superficial nor capricious. The considera- tions seem to have been almost purposely omitted and delinquent in some respects; because they are not necessary for the purpose of this treatise through long and continued areas of the heavens where no me- mory can be formed of them; but they are useful in explaining the northern and southern conic allometries. e.g. A better system of con- sideration might have been a material help as an artificial memory. A diagram showing two celestial globes with inscribed stars. **CHAP. IV.] THE MILKY WAY.—THE ZODIAC.** (253.) There are not wanting, however, natural dis- tricts in the heavens, which offer great peculiarities of character, and strike every observer; such is the milky way, that great luminous band, which stretches, every evening, across the whole sky, from east to west, at declination, and which, when traced with diligence, and mapped down, is found to form a zone completely embracing the whole zodiacal sphere. This band is not only seen at any other an hour circle, nor coincident with any other of our astronomical grammarians. It is divided in one part of its course into two parts by the line of the ecliptic, which runs with the main body, after remaining distant for about 180 degrees. This remarkable fact has maintained, from antiquity to the same relative period among the stars; and, when examined through powerful tele- scopes, it found 'wonderful to relate' is 'constituted entirely' by a vast number of stars of different classes, on the black ground of the general heavens. (254.) Nothing can be more regular than this course is in the heavens from any thing peculiar in its own con- stitution, but from its being the area within which the apparent motions of the sun, moon, and planets are the reader placed. In order to ascertain what part of any of these there, it is only necessary to ascertain, by continuous observa- tions upon the same subject, whether they move in a series, not too far disjoined, to connect them by line from point to point, as we mark out the course of the cre- ation in the Bible from day to day. Now when this is done, it is found, first, that the appa- rent course of the sun is not a circle; but a figure com- posed of two arcs of a circle joined together at their ends; that is no other than an exact great circle of the sphere which is called the ecliptic, and which is inclined to the horizon at an angle of about 28°; inscribed in it are two opposite points called the solar apical points, or equinoxes, and which are distinguished from each other by equal distances from both equinoxes; the vernal being that at which the sun crosses the equator from south to north; the autumnal when it quits the northern and enters the southern hemisphere. Secondly, that the moon and all the planets pursue paths which, in like
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    168 A TREATISE ON ASTRONOMY. CHAP. IV. manner, encircle the whole heavens, but are not, like that of the sun, great circles exactly returning into them- selves and bisecting the sphere, but rather spiral curves of more or less equal radius, which describe different velocities in their different parts. They have all, how- ever, this in common, that the general direction of their motions is always towards the same pole, and so to east, that is to say, the contrary to that in which both they and the stars appear to be carried by the diurnal motion of the earth. The sun's path is never deviate far from the ecliptic on either side, crossing and recrossing it at regular and equal intervals of time, and contains 360° of arc. The moon's path, already spoken of, extending 9° on either side of the ecliptic. (584.) It would manifestly be useless to map down on globes or charts the apparent paths of any of those bodies which never retrace the same course, and which, there- fore, do not describe complete circles. But even in any other or other their history, every point in the area of that zone of the heaven within which they are circum- scribed, will be found to describe a circle. This arises (that of the moon excepted) from our viewing them from a station which is itself in motion, and would disappear from view when they pass behind it; and then from them on the sun. On the other hand, the apparent motion of the sun is presented to us under its least in- volved form; and we can thus see it with advantage to the greatest advantage. So that, independent of the importance of that luminosity to us in other respects, it is by the study of its apparent motion among the stars that we have a first instance that we must rise to a knowledge of those of all the other bodies of our system. (585.) In order to understand its apparent path among the stars, it traversed by it in the period called the side- real year, which consists of 365° 6' 9" 6', reckoned as solar time. But this is only one part of a longer peri- odical time. The reason of this difference (and it is this which constitutes the origin of the difference between solar and sidereal time), is that the sun's apparent annual motion among the stars is performed in a con- **CHAP. IV.] THE ECLIPTIC.—SOLSTICE YEAR.** trary direction to the apparent diurnal motion of both suns and stars, it comes to the same thing as if the disur- mal motion of the sun were so much slower than that of the daily course. Where this is the case, and the sun lagged behind them in its daily course, the sun will have fallen behind the stars by a whole circle before the stars have made one revolution; but in a year, the sun will have made fewer diurnal revolutions, by one, than the stars. So that the same interval of time which the sun requires to make 360°, or 720 feet, if reckoned in mean solar days, etc., does with the stars, though 360°, 60°, etc. Thus, then, is the proportion between these two motions, which is expressed by the number 1800, each, reduced into a decimal fraction, is that of 1-00727979, to 1. The measurement of time by these different stand- ards—by the sun's day, by the lunar day, by the sidereal feet, or else of two different nations: the proportion of which, once settled, can never become a source of error. Of all these standards, only one is absolutely invariable; that may, for our present purpose, be regarded as invariable. It is true that this is not strictly the case; and on com- paring the observations with those which it held at the most distant epoch at which we possess observations, we find evidences of a small change, which though very slow, yet cannot be entirely neglected; further explained; but that change is so exceedingly slow, that for a great many successive years, or even for whole centuries, it may be considered as holding the same position in the sidereal heavens. (298.) The poles of the ecliptic, like those of any other celestial body, are situated at right angles upon its surface, equidistant from the ecliptic in every direc- tion. They are of course not coincident with those of the sphere of reference; but they are such as to make an in- terval equal to the inclination of the ecliptic to the equi- nocial (23° 28'), which is called the obliquity of the equinoxes. This obliquity is always measured from north and south poles (by which, when used without qualifica- tion we always mean the poles of the equinoctial), and EQUA. the equinocial VNAK the ecliptic, and KUJ. its pole—the spherical angle QTS is the obliquity of the A diagram showing Earth's rotation around its axis and orbiting around the Sun. The ecliptic plane is shown as a tilted circle around Earth's axis. The equator and tropics are also depicted. 160 A TREATISE ON ASTRONOMY. [CHAP. IV. eclipic, and is equal in angular measure to PR or SQ. If we suppose the sun's apparent motion to be in the direction VSAW, V will be the vernal and A the au- ronal equinoxes, and S W the solstices, of which the ecliptic is most distant from the equinoctial, are termed solstices, because, when arrived there, the sun comes to its greatest declination (250). The solstices, far as its motion in declination is concerned, to stand still in the heavens, S, the point where the sun has the greatest declination, is called the summer solstice; and W, that where it is furthest south, the winter. These ephemerides obviously have their origin in the depend- ence of the sun's apparent motion on the earth's rotation; and this will be explained in the next chapter. The circle EKPOkp, which passes through the poles of the ecliptic and equinoct- ial, is called the ecliptic. The circle drawn through the equinoxes, PVPa, the equatorial colure. (250.) Since the ecliptic holds a determinate situation in the heavens, it is convenient to draw through it an equi- nocial, to refer the positions of the stars to, by circles drawn through them from its poles, and therefore per- pendicularly to it. This circle is called the equinoctial. The circles of latitude--the distance of a star from the ecilip- tic, reckoned on the circle of latitude passing through it, is called the latitude of the star--and the arc of the A diagram showing a celestial sphere with various lines and points labeled. ecliptic intercepted between the vernal equinox and this circle, its longitude. In the figure X is a star, FXR a **CHAP. IV.** CELESTIAL LONGITUDES AND LATITUDES. circle of declination drawn through it, by which it is referred to the equinoctial, and KXT a circle of latitude referred to the same. The right ascension, and RX the declination, of $X$, so also of VT its longitude, and TX its latitude. The use of the terms longitude and latitude has been adopted from that origi- nated in considering the ecliptic as forming a kind of natural equator descending from the zenith to the horizon. The posi- tion with respect to the heavens, as the terrestrial equator does with respect to the earth's surface, is called the posi- tion with respect to the stars, as the latter does with respect to stations on the earth's surface. The force of this is shown in the following solution : (360). Knowing the right ascension and declination of an object, we may find its longitude and latitude, and vice versa, by means of a table of co-ordinates in celestial astronomy. The following is an solution : In our last figure, EKPQ, the celestial colure is of course 90° distant from VR, but VR being given, we have only to find EKQ, so that VR (the right ascension) being given, and also VE, the arc ER, and its measure, the spherical angle EKR. Now ER is equal to VR + EKQ = 90° + EKQ. EKQ is therefore equal to VR - 90° = 180° - VR. The other side PX is therefore equal to 180° - PX, and PX is equal to 180° - PX. Therefore PX = PX. The side PX is the polar distance, or the complement of the declination $RX$ ; and $3d$, the included angle $KXR$, and therefore $KXR$ = $3d$. Hence $KXR$ = $3d$ = $180° - 90° - VR$ = $90° - VR$. Now $KX$ is the complement of the required latitude $XT$, and the angle between them is therefore equal to the right angle (because SV is 90°), the angle $KXR$ becomes known. Now this is no other than the measure of the local colure at $X$. This solution can be easily resolved by the same triangle, and by a process exactly similar. (361). The same course of observations by which the path of the sun among the fixed stars is traced, and the ecliptic marked out among them, determines, of course, the true position of all objects in space. It was first made at that time--a point of great importance in practical astro- nomy, as it is the origin or zero point of right ascension o 2 102 A TREATISE ON ASTRONOMY. [CHAP. IV.] Now, when this process is repeated at considerably dis- tant intervals of time, a very remarkable phenomenon is observed; viz. that the equinox does not preserve a con- stant position in the heavens, but moves forward and retreats continually and regularly, although with extreme slowness, backwards, along the ecliptic, in the direction of N W from the equinox to the point of greatest declina- tion of the sun appears to move in that circle. The equinoctial point thus moving, as it were, to meet the sun in his apparent as- sual route, would appear to be stationary; but it is evident that, at the time of the equinox happens sooner than it would otherwise do; hence the recession of the equi- noctial point is caused by a motion of the sun in its orbit. The amount of this motion by which the equinox travels backward, or retrogrades (as it is called), on the ecliptic, is 5° or 6° per year. This is a very small quantity, but which, by its continual accumulation from year to year, at last makes itself very palpable, and that in a way highly interesting to astronomers. For instance, by destroying, in the lapse of a moderate number of years, the arrangement of their catalogues of stars, and making a new one every twenty-four hours round those of the earliest catalogue on record, the place of the equino- xus has retrograded already about 30°. The period in which this occurs is a complete tour of the ecliptic, i.e. 2888 years. (262.) The immediate astronomical effect of the precession of the equinoxes is to cause a change in the crests of longitude in all the heavenly bodies, whether fixed or erratic. For the vernal equinox being the initial point of our reckoning, we have only to consider what is to treat of this point on the ecliptic tells upon the longi- tudes of all alike, whether at rest or in motion, and pro- ceeds, so far as regards them, as if they had been in a state of motion in longitude common to all, as if the whole hea- vens had a slow rotation round the pole of the ecliptic, and this body had a slow rotation round its own axis. This is true for all bodies whatever; for they may have in twenty-four hour round those of the equinoxal. (263.) To form a just idea of this curious astronomi- cal phenomenon, however we must abandon for a time the consideration of the ecliptic as nothing to produce **CHAP. IV. J PRECSSION OF THE EQUINOXES.** 168 confusion in our ideas; for this reason, that the stability of the ecliptic itself among the stars is (as already hinted, art. 257) only approximate, and that in consequence its inclination to the plane of the equator is subject to a certain amount of change, arising from its fluctuation, which mixes itself with what is due to the principal unregula- rity of the earth's motion. This fluctuation has not yet be- come at once apparent, if, instead of regarding the equi- noxes, we fix our attention on the pole of the equino- xial axis, and consider the place of this point among the stars. (264.) The place of this point among the stars is easily determined, at any epoch, by the most direct of all instru- ments, viz., a quadrant. By means of this instrument we are enabled to ascertain at every mo- ment the exact distance of the polar point from any three stars, which are situated in such a manner as to be triangulating from these stars, with unerring precision, on a chart or globe, without the least reference to the position of the sun or moon. The result will be natu- rally connected with it. Now, when this is done with proper diligence and exactness, it results that, although for some time past, the pole has been regarded as not sensibly variable, yet in reality it is in a state of constant, although extremely slow, motion. For instance, let us suppose that this motion is not uniform, but compounded of one principal, uniform, or nearly uniform, part, and other smaller and successive parts. In this case, we shall have first a prece- sion to the phenomena of precession ; then a second thing rising to the phenomena of precession ; the latter to another distinct phenomenon called nutation. These two phe- nomena are connected together by their common centre; one and the same general head, and are intimately connected together, forming part of a great and completed chain. The first part is that which we have just described; its axis ; but it will be of advantage to present ourselves to consider them separately. (265.) The first thing that in virtue of the uniform part of the motion of the pole, it describes a circle in the heavens around the pole of the ecliptic as a centre, keep- ing constantly at the same distance of 23° 27' from it, in a direction from east to west, and with such a velocity, A diagram showing a circle around a central point labeled "pole" with an arrow indicating east-west direction. 164 A TREATISE ON ASTRONOMY. [CHAP. IV.] that the annual angle described by it, in its this imaginary orbit, is 50°-10′ so that the whole circle would be de- scribed in 365 days, or 365.25 years, or 18,900 years. It is easy to perceive how such a motion of the pole will give rise to the retrograde motion of the equi- lunes, &c. For instance, at the time when the point K of the progress of the equinoxes in the small circle POZ is brought to come to O, then, as the situation of the equinoctial EVQ is determined by the position of the equinoxes, the ev- ents must involve a displacement of the equinoctial, which will take a new situation, EUQ, 90° distant in every part from EVQ. This displacement will be made by a line U, therefore, in which the displaced equinoctial will inter- sect the ecliptic, i. e. c. be displaced equinox, will lie on that side of the ecliptic which is opposite to that in which the motion of the pole is directed, or to the westward. (266.) The precession of the equinoxes thus conceived, consists in a slow rotation of the axis of the earth about the pole of the ecliptic. Now this cannot happen with- out producing a change in all those phenomena connected with the heavens around the stars, in a small circle round the pole of the ecliptic. The cause of this change is due to the diurnal motion of the sphere, and the aspects which the heavens must present at very remote periods of history. The path of the sun across the heavens is called the earth's axis. At this point then, has such a motion as was described, it necessarily follows that the earth's axis must have a certain inclination to one plane or another con- cessively to every part of the small circle in question. We may form the best idea of such a motion by noticing a child's toy called a top. When we turn it round, it seems toying the teet-toom, which, when deliberately excited, and nicely balanced, becomes an elegant philosophical instrument. In like manner, we may imagine that the whole phenomenon, in a way calculated to give us at once a clear conception of it as a fact, and a considerable insight into its causes, is simply a toy for children. The reader will take care not to confound the variation of the position of the earth's axis in space with a mere shifting of the direction of its axis through its interior. The whole earth participates in the motion, and goes along with the axis as if it were really a bar of iron **CHAP. IV.** 165 **NUVATION.** driven through it. That such is the case is proved by the two great facts: 1st, that the latitudes of places on the earth, or their geographical situation with respect to the poles, have undergone considerable change from the earliest ages; 2dly, that even as maintains its level, which could not be the case if the motion of the axis were accompanied with a motion of the whole mass of the earth. (267.) The visible effect of precession on the aspect of the heavens is seen in the apparent superiority of some stars and constellations to the pole and recess of others. The bright star of the Lesser Bear, which we call Ursa Minor, will always continue to be our cynosure at the time of the construction of the earliest catalogue it was 13° from the pole; but after one revolution of the earth, or within half a degree, after which it will again recede, and slowly give place to others, which will succeed it in this position. In like manner, in about 12,000 years, the star **α Lyrae**, the brightest in the northern hemisphere, will occupy the remarkable situation of a point in the sky called **the point of Aries**. (268.) The nutation of the earth's axis is a small and slow subordinate gyratory movement, by which, if sub- stantially uniform in direction, it may be said to vary, in a period of about nineteen years, a minute ellipse, having its longer axis equal to 18° 5", and its shorter to 13° 4". This ellipse is perpendicular to the plane of the ecliptic, and the shorter, or course, at right angles to it. The consequence of this real motion of the pole is an apparent precession of all celestial bodies round the heavens to the pole in the same period. Since, also, the place of the equinox on the ecliptic is determined by the plane of this circle, any alteration in its position will give rise to a small alternate advance and recess of the equinoctial point, by which, in the same period, both the longitude and latitude of all celestial due surs will be alternately increased and diminished. (269.) Both these motions, however, although here considered separately, exhibit unity; and since, while in virtue of the nutation, the pole is describing its little 160 A TREATISE ON ASTRONOMY. [CHAP. IV.] ellipse of 18°-5' in diameter, is carried by the greater and regularly progressive motion of precession over so much of its circle round the pole of the ecliptic as corres- ponds to the difference between the mean annual angle of nineteen times 50°-1 round the centre (which is a small circle of 23° 28' in diameter, corresponds to 6° 20'), and such an angle as that which the earth makes with its axis in its orbit. It will pursue in virtue of these two motions, subsisting jointly, will be neither an ellipse nor an exact circle, but gently assume a ring like that in the figure (where, however, the undulations are exaggerated). (See fig. to art. 272.) (270.) The movements of precession and nutation are common to all the celestial bodies both fixed and er- ratic; and this circumstance makes it impossible to attri- bute those phenomena to any one particular body of the earth's axis, such as we have described. Did they only affect the stars, they might, with equal plausibility, be urged to be due either to the earth itself, or to some other body, as a solid shell round a axis passing through the poles of the ecliptic in 28,586 years, and a real elliptic gyration of that axis in 25,808 years. But since they affect also the sun, moon, and planets, which, having motions inde- pendent of the general body of the earth, cannot without ex- travagance be attributed to it, it follows that this idea falls into the ground; and there only remains, then, a real motion in the earth by which they can be accounted for. This motion is therefore necessarily supposed that they are necessary consequences of the rotation of the earth, combined with its elliptical figure, and the un- equal attraction of the sun and moon on its polar and equatorial regions. (271.) Uranographically considered, as affecting the apparatus of observation, it is evident that this is of no importance in practical astronomy. When we speak of the right ascension and declination of a celestial object, it becomes necessary to specify whether it is observed at This treatment, cognate to it, requires additional and decisive forces from the law of motion, which is dependent on the position, the time, and the laws of gravitation. In order to determine their position in space, we must first maintain that sphere be kept in a constant state of freedom by the motion of the moon! EQUATIONS FOR PRECESSION AND NUTATION. 107 whether we mean the mean right ascension; cleared, that is, of the periodical fluctuation in its amount, which arises from nutation, or the apparent right ascension, which being reckoned from the actual place of the vernal equinox, is subject to the same variations as those of the equinoctial point thence produced—and so of the other elements. It is this practice of astronomers to re- duce their observations to the apparent right ascension and declination, to some common and conven- ient epoch—such as the beginning of the year for tem- poral observations, or the time of the spring equinox for more permanent use, by subtracting from them the loss of effect of precession in the interval; and, moreover, to correct their tables for nutation, by adding to the increa- sing and subtracting the amount of change, both in right ascension and declination, due to the displacement of the pole of the ecliptic by 23° 28' 15" from its ecli- lipse above mentioned. This last process is technically termed correcting of elliptic by elliptic; but it is un- likely that any one would understand it. In astron- omy, the getting rid of a periodical cause of fluctua- tion, and presenting a result, not as it now observed, but as it would appear if that cause of fluctua- tion had no existence. (272.) For these purposes, the tables can very conveniently be constructed. They are, however, of too technical a charac- ter for this work; so I shall, however, point out the man- ner in which they may be constructed. The table has been shown in art. 260 by what means the right ascension and declination are corrected for nutation at all longitudes and latitudes. Referring to the figure of that article, and supposing the triangle KPX orthographically projected on the plane of paper as in the annexed figure—in the angle KFX, KFX is the longitude (or longitude), KPK the co-latitude (or complement of latitude), and the angle PXX the co-longitude (or object) of nutation. The first con- stant, and the two latter are varied by the effect of pre- cession and nutation; and their variations (considering the minuteness of the latter effect generally), and thus A diagram showing a triangle KPX with angles labeled KFX (longitude), KPK (co-latitude), and PXX (co-longitude). The text describes how these angles are used to calculate nutation. 108 A TREATISE ON ASTRONOMY. [CAP. IV. small number of years in comparison of the whole period of 25,868, for which we ever require to estimate the effect of the former) are of that order which may be regarded as constant, and therefore may be calculated with such without fear of error. The whole question, then, is reduced to this:-In a spherical triangle KPX, in which one side KP is constant, and an angle KR, and adjacent side KP vary by given infinitesimal changes of the position of P; required the changes thence arising in the other side PX, and the angle KPX! This is a very simple and easy problem of spherical geometry, and be- comes readily solved by the following method. Let us reckon ; for PX being the polar distance of the object, and the angle KPX its right ascension plus 90°, their va- luation will be thus: when the right ascension remains re- maining, then, to express in proper form the amount of the precession and nutation in longitude and latitude, when their angles remain at ascension and declination will im- mediately be obtained. (272.) The precession in latitude is zero, since the latitude does not change. But the precession in longi- tude is a quantity proportional to the time at the rate of $50 \cdot 10^{-7}$ per annum. With regard to the nutation in longitudes and latitudes these are so small, than the ab- solutes and ordinates of the little ellipse in which the pole A diagram showing a spherical triangle KPX with sides KP (constant), PX (variable), and angle KR (variable). The right angle at K is labeled as 90°. The diagram includes labels for points X, P, and K. **CHAP. IV.** **ABERRATION OF LIGHT.** 109 moves. The law of its motion, however, thence, cannot be deduced until the laws under which it has been made acquainted with the principal features of the earth's motion on which it depends. See chap. XI. (274.) Another consequence of what has been already recognized is, that in nutrition is, that sidereal time as astronomers use it, is reckoned from the transit of the equinoxal point, is not a mean or uniformly fluctuating period of time, but a variable one; and moreover, that so reckoned, even when closed of the perihelion and aphelion of the rotation, does not strictly correspond to the earth's diurnal period, as the sun loses one day in the year on the stars, by its direct motion through longitude; and the equinoxes gain one day in 36,860 years upon them by their retrograde motion. We ought, therefore, as carefully to distinguish between mean and apparent sidereal as between mean and apparent solar time. (275.) Neither precession nor mutation change the apparent position of any star. When we see them, so far as these cause go, as they are, though from a station more or less unstable, as we see distant land objects appear to move in a circle round us and fall when viewed from the heaving deck of a ship in the act of pitching and rolling. But there is an optical cause, independent of all these physical causes, which displace them one among the other and causes us to view the heavens under an aspect always to a certain slight exception. This optical effect is known as parallax and allowed for before we can obtain a precise knowledge of the place of any object. This cause is what is called the "distant image" or "distant effect," arising from this, that we occupy a station not at rest but in rapid motion; and that the apparent distance of any star from our line of sight is not equal to its actual spatial distance in space. In order to account for this we must estimate the effect belonging to trigonometry, we must explain it here, though, in fact, we have no means of estimating some of the results to be desired in such an inquiry. (276.) Suppose a shower of rain to fall perpendicularity in a dead calm ; a person exposed to the shower, who 170 A TREATISE ON ASTRONOMY. [CHAP. IV. should stand quite still and upright, would receive the drops on his back, but if he were to turn round and round in any direction they would strike him in the face. The effect would be the same as if he remained still and turned round in any direction, and had not drifted them against him. Suppose a ball let fall from a point A. above a horizontal line EF, and that at B were placed to receive in the open mouth of an inclined hollow tube PQ; if the tube were held immovable, the ball would strike on its lower side, but if the tube were car- ried forward in the direction EF, with a velocity properly adjusted at every instant to that of the ball, while pre- cording the time of each fall, it will be found that when the ball in its natural descent reached C, the tube should have been carried into the position RS, so is evident that the ball would reach D. Now suppose a spectator seated in the axis of the tube; and a spectator, referring to the tube the motion of the ball, and carried along with the former, will see that when the ball reaches D, the ball had been moving in the inclined direction RS of the tube's axis. (QT.) Our eyes and telescopes are such tubes. In whatever manner we consider light, whether an ad- vancing wave in a moistuous ether, or a shower of stones throwing up by a cannon, or whether the rays traversing the object-glass of one or the other of the obers (at which moment they acquire that con- vergence which directs them to a certain point in fixed A diagram showing a ball falling from point A above a horizontal line EF and being caught by a tube PQ. The tube is inclined at an angle and carries a ball towards point D. A spectator is shown sitting in the axis of the tube. C.A. D.E PQ CHAP. IV.] CORRECTION FOR ABERRATION. 171 space), the cross-wires of the one or the retina of the other eye displaced by the point of convergence (which remains unchanged) will no longer correspond to the inter- section of the wires or the central point of our visual axis with the star, which has moved ; and the amount of this displacement is aberration. (278.) The earth is moving through space with a ve- locity of about 30 miles per second, and its path round the sun, and is therefore changing the direction of its motion at every instant. Light travels with a ve- locity of about 186,000 miles per second, although much greater than that of the earth, is yet not infinitely so. Time is occupied by it in traversing any space, and in passing through any angle, less than 1°, when it is former as 19 to 192,000, or as the tangent of 20°-5 to radius. Suppose now APE to represent a ray of light from a star distant from us 100 million miles. The tele- scope so inclined forward that the focus formed by its object-glass shall be received upon its cross wire, it is evident that the image of the star will be formed at the tube must be such as to make PS : SQ :: velocity of light : velocity of earth, :: tun. 20°-5 : 1 ; and, therefore, since PS = 100 million miles, the axis of the telescope must deviate from the true direction of the star, must be 30°-5. (279.) This reasoning will hold good when the direction of the earth's motion is not perpendicular to the visual ray. If SB be the true direction of the star, then AC will be the di- rection, and AC the position in which the image required to be seen would appear if we were to move with the same di- rection, we must still have the proportion BAC : BA = sin 20°-5 : sin 30°-5 (by definition) = sec 20°-5 : sec 30°-5 (for in such small angles matters not whether we use the sine or tangent). But since BAC = angle of RAC = sine of ACR or CBR, and that is the apparent displacement caused by aberra- tion. Thus it appears that the sign of the aberration, or (same as angle is extremely small) the aberration itself, A diagram showing a telescope with a crosswire and a star image. The diagram includes labels for various points and angles. 171 173 A TREATISE ON ASTRONOMY. [CHAP. IV.] is proportional to the sine of the angle made by the earth's motion in space with the visual ray, and is therefore a maximum when the line of sight is perpendicular to the direction of that motion. (260.) The astronomical effect of aberration, then, is to distort the aspect of the heavens, causing at the stars to appear more distant than they really are; and amongst the heavens which is the vanishing point of all lines parallel to that in which the earth is for the moment moving, there will be one which will appear to be at the plane of the ecliptic, this point must lie in that plane, 90° in advance of the earth's longitude, or 90° behind the sun's, and which will be at a distance equal to the mean dis- tference of the ecliptic in a year. It is easy to demon- strate that the effect on each particular star will be to make it appear nearer than it really is, and conversely having for its centre the point in which the world would be seen if the earth were at rest. (261.) The following table gives the apparent right an- gles and declinations of all the stars, and that by quantities easily calculable. The formulae most conve- nient for determining what these angles are, by assum- ing at the same time the corrections for precession and nutation, enable the observer, with the utmost readiness, to determine their actual magnitude. The right angles and declination of their influence have been constructed by Prof. Bessel, and tabulated in the appendix to the first volume of his "Celestial Mechanics," where they will be found accompanied with an extensive catalogue of all places, for 1850, of the principal fixed stars, on which he has calculated their apparent motions for the kind which has ever appeared. (262.) When the body from which the visual ray emanates moves with velocity, it involves the best way of conceiving the effect of aberration (independently of theoretical views respecting the nature of light) is as follows. The The plane in which we imagine our eye to look is called the plane of observation. This plane may be any plane whatever, but it is convenient to take it as being parallel to that in which we imagine ourselves to move. There is no difficulty in imagining ourselves to move in any direction whatever; but it is difficult to imagine ourselves to move in a direction perpendicular to that in which we imagine ourselves to move. In this case we shall find that we can only move in two directions, namely, either towards or away from ourselves. We cannot move in any other direction without passing through ourselves. We cannot move in any other direction without passing through ourselves. We cannot move in any other direction without passing through ourselves. We cannot move in any other direction without passing through ourselves. We cannot move in any other direction without passing through ourselves. We cannot move in any other direction without passing through ourselves. 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Hence it is easy to derive the law given by some ancient writers for the case of a moving object. From the known laws of motion, it is evident that when an object moves at a constant velocity, or relative angular motion in the time taken to traverse its distance from the earth. This is its aberra- tion. But when an object moves with a variable velocity contrary to its apparent relative motion among the stars, We shall conclude this chapter with a few urano- graphical problems, which may be solved by means of which may be resolved by the rules of spherical trigonometry. (283.) Of the following five quantities, given any three, to find the fourth. 1st. The latitude of this place; 2d, the declination of an object ; 3d, the hour angle east or west from the meridian ; 4th, the altitude of the object. In the figure of art. 94, P is the pole, Z the zenith, and S the star; and the five quantities above mentioned, or those which they determine, are represented by the spherical triangle PZS; PZ being the co-ordinate, PS the co-declination or polar distance; SP the hour an- gle east or west from the meridian; ZP being equal to the azimuth. By the solution of this spherical triangle, all problems involving the relations between these quanti- ties may be solved. (284.) For example, suppose the time of rising or set- ting of the sun or of a star were required, having given its right ascension and declination. The star rises moreover than in the contrary equal to twice the velocity of its own motion. In the case, then, of a body moving with equal velocity directly towards or away from us, its apparent motion will be direct- lyary, but different on the opposite hypothesis. The utmost differ- ence between these two cases in our games cannot amount to above six thousandths of a second. 2 174 A TREATISE ON ASTRONOMY. [CHAP. IV.] when apparently on the horizon, or really about 90° be- low it (owing to refraction), so that, at the moment of its apparent rising, its zenith distance is 90°-ZPS, its solar altitude is 90°-ZPS, and its hour angle is equal to the place of the place, we have given the three sides of the trian- gle, to find the hour angle ZPS, which, being known, is to be subtracted from 180° to obtain the solar declina- tion, to give the sidereal time of rising or rising, which, if we please, may be converted into solar time by the proper sun's equation (285). As another example of the same triangle, we may propose to find the local sidereal time, and the lai- tude of the place, when the star rises at a given hour and altitudes of the same star east and west of the meridi- an, and noting the interval of the observations in side- real time. The hour angles corresponding to equal altitudes of a fixed star being equal, the hour angle east or west will be measured by the difference between these hours and obser- vations. In our triangle, then, we have given this hour an- gle ZPS, the polar distance PS of the star, and ZS, its solar declination; and we have also given the hour angle find PZ, the co-latitude of the place. Moreover, since the hour angle of the star being known, and also its right ac- centure, we can determine its right ascension; and since it is on the meridian at the moment of observation; and, therefore, the local sidereal time at that moment. This is a very simple method for determining latitude and time at an unknown station. (286.) It is often of use to know the situation of the eclipse with respect to any point on the horizon. To do so, it points where it cuts the horizon, and the altitude of its highest point or, as it sometimes called, the geographic latitude. The latitude is measured from this point on the ecliptic itself from the equinox. These, and all questions referable to the same data and ques- tions are answered by means of a table of eclipses made by the zenith Z (considered as the pole of the horizon), the pole of the equatorial P, and the pole of the ecliptic E. The adressed line gives, first, the right ascen- tion of the pole of the eclipse (which is always the CHAP. IV.] URANOGRAPHICAL PROBLEMS. 175 same, viz. 18° 0" 0"), the hour angle ZPE of that point is known. Then, in this triangle we have given PZ, the ecliptic, and ZPZ, which is the angle between the side of the ecliptic, 23° 28', and the angle ZPE ; from which we may find, let, the side ZE, which is easily seen to be equal to the difference between the angles ZPE and ZPZ ; and, 3dly, the angle PZE, which is the azimuth of the pole of the ecliptic, and which, therefore, being added to any right angle, will give us the true meridian and western intersections of the ecliptic with the horizon. Lastly, the longitude of the meanestimal point may be had by adding to this same triangle the angle PZE, which is its complement. (287.) The angle of situation of a star is the angle be- tween its declination and the line of sight of a meridi- nation passing through it. To determine it in any pro- posed case, we must resolve the triangle PSE, in which are contained all these quantities. The difference of dif- ference between the star's right ascension and 18 hours ; from which it is easy to find the angle FSE required. This is called the latitude of the star in astrono- my. It is called in most books on astronomy the an- gle of position ; but the latter expression has become obsolete since its introduction into our language. (288.) From these instances, the manner of treating each question in trigonometry as depend on spherical trigonometry will be evident, and so far, for that mat- ter little difficulty, if the student will bear in mind as a A diagram showing a triangle with sides labeled ZE, ZPZ, and PZE. The angles at vertices are labeled ZPE and PZE. 175 176 A TREATISE ON ASTRONOMY. [CHAP. V. practical maxim, rather to consider the poles of the great circles which his question refers to, than the circles themselves. CHAPTER V. OF THE SUN'S MOTION. Apparent Motion of the Sun not uniform. The apparent Diameter does not always increase with the time, but sometimes decreases, as is shewn by the Eclipses about the Fourth of August, 1758, and 1760. On this subject see the Essay on the Apparent Diameter of the Sun, and the Copernican Explanation of the Sun's apparent Motion—Parallelism of the Earth's Axis with that of the Sun's Equator—The Sun's motion in its orbit—Anamorphoses of the Sun's Appear- ance at the Equinoxes. (280.) In the foregoing chapters, it has been shown that the apparent path of the sun is a great circle of the spheres, which it performs in a period of one sidereal year. This circle is called the ecliptic; because the earth and sun lie constantly in one plane; and that, therefore, whatever be the real motion from which this apparent motion arises, it must be supposed to one plane, which is called the plane of the ecliptic. (280.) We have already seen (art. 118) that the sun's motion in its orbit is not uniform. This is partly accounted for by the obliquity of the ecliptic, in consequence of which equal variations in longitude do not produce equal variations in latitude. But if we observe the place of the sun daily throughout the year, by the transit and circle, and from these calculate its angular velocity in its orbit, we shall find that even in its own proper path, its apparent angular motion is far from uniform. The change of longitude in twenty-four months is nearly 3° 1' 1" 6", and that of latitude about 3° 1' 9" 6', and about the 31st of December it amounts to 1° 1' 9" 6', and about the 1st of July only is about 5° 7' 11" 6'. Such are the ex- treme limits of variation in longitude and latitude of its appa- rent angular velocity in its annual orbit. (281.) This variation in its angular velocity is accom- panied with a corresponding change of its distance from us. The change of distance is recognised by a variation CRAP. v.] FORM OF THE SOLAR ORBIT. observed to take place in its apparent diameter, when measured at different seasons of the year, with an instru- ment adapted for that purpose, called a heliometer, or, by means of which we can observe the sun's passage to the meridian in the transit instrument. The great- est apparent diameter corresponds to the 31st of Decem- ber, and is 31° 0' 6"; the least to the 25th of March, or 31° 0' 6"; the least is 31° 0' 6", and corresponds to the 1st of July; at which epoch, as we have seen, the angular distance between the sun and the earth is greatest. Now, as we cannot suppose the sun to alter its real size periodi- cally, the observed change of its apparent size can only be attributed to the variation of its distance from us. As the angles subtended by any two points on a sphere are propor- tional to the arcs tangents of those small areas being proportional to the areas themselves, its distances from us, at the above-named epochs, must be inversely proportional to its apparent diameters. It appears, therefore, that the greatest, the mean, and the least distances of the sun from us are in the ratio of 1-000078, 1-000079, and 1-000076 (see Tables IVV. IVV. IVV.). 1-000078, and 1-000079 ; and that its apparent angular ve- locity diminishes as the distance increases, and vice versa. (See Tables IVV. IVV. IVV.) But this is not true of the sun, as referred to the earth supposed at rest, is not a circle with the earth in the centre. The situation of the earth within it is eccentric, the eccentricity amounting to A diagram showing a circle with four labeled points: A, B, C, D. 177 0-01679 of the mean distance, which may be regarded as our unit of measure in this inquiry. But besides this, the form of the orbit is also eccentric, but less so. If we draw any point O, taken to represent the earth, we draw a line, OA., in some fixed direction, from which we then set off a series of angles, AOB, AOC, &c., equal to the ob- observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. observe that this line is not drawn accurately to proportion. 177 A B C D M N 178 A TREATISE ON ASTRONOMY. [CHAP. V. arrived longitudes of the sun throughout the year, and in those respects they agree with each other, from the dis- tances OA, OB, OC, &c. representing the distances deduced from the observed diameter, and then connect all the points thus obtained by a continuous curve. As this con- tinuous curve, it is evident this will be a correct repre- sentation of the relative orbit of the sun about the earth. Now, when we consider that the general figure of the figure in the resulting curve becomes apparent; it is found to be evidently longer than it is broad--that is to say, either its length exceeds its breadth by a very small but one of the foci of the ellipse. The graphic process here described is sufficient to point out, the general figure of the sun's orbit about the earth. For further exact verification, it is necessary to recur to the properties of the ellipse, and to express the distance of any one of its points in terms of its eccentricity and its distance with respect to the longer axis, or diameter of the ellipse. This, however, is readily done; and when numerically calculated, it will be found that for any given value of such as e above stated, a perfect coincidence is found to subsist between the distances thus computed, and those derived from observation. This is shown in Table 308. (905.) The mean distance of the earth and sun being taken for unity, the extremes are 1-01679 and 0-98311. But if we take for unity the mean angular velocity, and angular velocity with the extremes, greatest and least, we shall find these to be in the proportions of 0-2386, 0-9694. Hence it appears that when we take for unity angular velocity, then, it is much greater in proportion than of that its distance--fully twice as great; and if we ex- amine its mean angular velocity, we shall find that com- paring them with their mean values, and also with the cor- responding distances, it will be found, that, by whatever fraction we may divide up its distance into two parts, the angular velocity will fall short of its mean or average quantity by very nearly twice as great a fraction of the latter; and therefore it follows that our hypothesis is true, that the angular velocity is in the inverse proportion, not of the distance simply, but of its square; or that, to com- * See Some Sections by the Rev. H. P. Hamilton. CHAP. V.] LAWS OF ELLIPTIC MOTION. 179 pare the daily motion in longitude of the sun, at one point, A, of its path, with that at B, we must state the proportion thus: (284.) The daily motion at A daily motion at B. This is found to be exactly verified in every part of the orbit. (284.) Hence we deduce another remarkable conclusion, viz., that whatever may be the actual velocity of any moving round the circumference of this ellipse, its actual speed cannot be uniform, but must be greatest at its least distance from the sun. This being uniformly true, the apparent angular velocity would be, of course, inversely proportional to the distance; simply because the same linear distance traversed in equal times of time at different distances from the eye, must, by the laws of perspective, correspond to apparent angular displacements inversely as those distances. Hence, then, observation indicates a law of velocity of motion which is regular velocities, it is evident that mere change of distance, unaccompanied by any change of direction, is sufficient to account for it; and that the increased proximity of the sun to the earth must be accompanied with an actual increase of its real velocity of motion along its path. (285.) This law is confirmed by observing the increas- ing position of the earth within it, and the unequal speed with which it is actually traversed by the sun's rays; all these phenomena are so well known to us from experience that they come from a knowledge of the causes and nature of its motions) indeed and impossible, so long as the law is not explicitly stated. But this law does not follow however, is not immediately apparent. It does not come forward, as it were, as a consequence of any of the other laws which have been exhibited, by a direct comparison of an- glues and distances, but requires an attentive consideration of the whole series of observations registered during an considerable period. By means of such an operation a pain- ful and laborious calculation, that it was discovered by Kepler (who was also the first to ascertain the elliptic form of the orbits), has been made out to be true: Let a line he always supposed to connect the sun, sup- pressed in motion, with the earth, supposed at rest; then, as the sun moves along its ellipse, this line (which is 180 A TREATISE ON ASTRONOMY. SCHAP. V. called in astronomy the *radius vector* will describe or sweep over that part of the circumference of the circle which is included between its consecutive positions : and the motion of the sun will be such that equal areas are described by it in equal times. The radius vector in equal time, in whatever part of the circumference of the ellipse the sun may be moving. (286.) In order to understand this, let us suppose that in unequal times, the areas described must be proportional to the times. Thus, in the figure of art. 292, the time at which the sun is at C is double that at which it moves from C to D, as the area of the ellipses sector AOB is to the area of the sector DOCB. (287.) Now, since the sun's apparent annual motion may, therefore, be summed up as follows ---It is performed in an orbit lying in one plane passing through the centre of the earth, and having for its diameter the projection on the heavens of the great circle so called. In this plane, however, the actual path is not circular, but elliptical, and has two foci, one near the earth, but in one focus. The eccentricity of this ellipse is $0.01678$, in parts of a unit equal to the mean distance, or half the longer diameter. But since this ellipse is not circular, and not in its circumference is so regulated, that equal areas of the ellipses are passed over by the radius vector in equal times. (288.) Hence we see that our knowledge of the sun's actual distance from the earth, nor consequently, of the actual dimensions of its orbit, nor of the behaviour of its light and heat with respect to its positions on these points, we must first consider what means we can arrive at any knowledge of the distance of this object from us. It is evident that this cannot be done by ob- vious, that its parallax alone can afford us any information on this subject. Parallax may be generally defined to be the difference between the position of an object arising from a change of real situation of the observer. Suppose then, PABCD represent the earth, C is centre, and B is its north pole. Let AB be a line drawn from B, which comes to the same thing, the stations of two specta- torers, both observing the sun at the same instant. The spectator A will see it in the direction ANG, and will re- CHAP. V.J. DIURNAL OR EOCENTRIC PARALLAX. 101 for it to a point $c$ in the infinitely distant sphere of the fixed stars, while the spectator B will see it in the direct- tion BS0, and refer it to A. The angle included between these two lines, or the measure of the celestial arc AB, by which it is displaced, is equal to the angle ASB; and if this angle be known, and the local situations of A and B, with respect to the observer's position, be given, therefrom it is evident that the distance CS may be calculated. (299.) Parallax, however, in the astronomical accepta- tion of the term, is not always observable. It is restricted to the difference of apparent positions of any celestial object when viewed from a station on the sur- face of the earth. The general station to which all astronomical observations are referred: but as we observe from the same place at different times of day, and as the amount of this reduction is called parallax. Thus, the sun being seen from the earth's centre, in the direction OS, its apparent position will be represented by AB. All the angle ASC, included between these two directions, is the parallax at A, and similarly BSC is that at B. Parallax is therefore determined by the epicycle diurnal, or eccentric, to discriminate it from the annual, or heliocentric; of which more hereafter. (300.) The effect of parallax on a star pro- posed case, is obtained from the consideration of the triangle ACS, formed by the spectator, the centre of the earth, and a star situated in a line with AC. As CA prolonged passes through the observer's zenith, it is evident that the effect of parallax, in this its terrestrial acceptance, will be greatest when it is observed in a vertical circle. To estimate the amount of this de- pression we have, by plane trigonometry, CS : CA = sine of CAP : sine of ASC. A diagram showing a triangle ABC with points labeled P, Z, C, Q, B. The line AB represents a celestial body (possibly a star) and AC represents a line from Earth's center to another point on Earth's surface. The line BC represents another line from Earth's center to another point on Earth's surface. The angle ACB is labeled as "A" and the angle BCA is labeled as "C". The line AB is perpendicular to AC and BC. The line AB is also perpendicular to BC. The line AB is parallel to AC and BC. 162 A TREATISE ON ASTRONOMY. (UHAF. V.) (301.) The parallax, then, for objects equidistant from the earth, is proportional to the sines of their zenith dis- tances. It is, therefore, at its maximum when the body observed is at the zenith, and at its minimum when it is at the horizonal parallax ; and when this is known, since small arcs are proportional to their sines, the parallax at any given distance will be found by multiplying the hori- zontal parallax (horizontal parallax) × sine of zenith dis- tance. The horizontal parallax is given by this proportion--- Distance of object : earth's radius : rad. : sine of hori- zontal parallax. It is, however, known, when the proportion of the object's distance to the radius of the earth is known; and vice versa---by any method of observation we can come at once to the required result. For let us call the object, its distance, expressed in units equal to the earth's radius, beomes known. (302.) We may now proceed to general reasoning to the case of the sun. Suppose two observers—one in the northhem, the other in the southern hemisphere—at stations on the same meridian line, and at different latitudes, observing altitudes of the sun's centre. Having thence derived the apparent zenith distances, and cleared them of the effects of refraction, they will find that their sum is equal to that of fixed stars, the sum of the zenith distances thus found would be precisely equal to the sum of the sines of their altitudes, which is also equal to their observations. For the sum in question would then be equal to the angle ZOX, which is the meridional distance of the observing stations from each other. But since it is evident that both these angles are less than 90°, and that those being in both cases to increase the apparent zenith dis- tances, their observed sum will be greater than the sum of the fixed stars' altitudes. Hence it follows that either laxes, or by the angle ASB. This angle, then, is obtained by subtracting the sum of the latitudes from half of 180°. And since this angle is always less than 90°, the horizontal parallax is easily found, by dividing the angle so determined by the sum of the sines of the two latitudes. (303.) If the two stations be not exactly on the same CRAP. V.] PARALLAX OF THE SUN. 183 meridian (a condition very difficult to fulfil), the same process will apply, if we take care to allow for the change of the sun's actual zenith distance in the interval of time elapsed between its arrival on the meridians of the observer and that of his observations. The error either from table of the sun's motion, grounded on the ex- perience of many years of observation, may vary from day to day before and after the observations, which on several days are taken. Of course, the nearer the sun is to either observer, the less will be this interval of time; and, consequently, the smaller the amount of this correction ; and, therefore, the less injurious to the accuracy of our results. It is also evident that any change of zenith distance which may arise from imperfection in the solar tables, or in the observations made to deter- mine them. (304.) The horizontal parallax of the sun has been considered by observers as a very small quantity, and de- scribed, performed in one place and at the most remote from each other in latitude, at which observations have been in- duced. It has often been deduced from other methods of more general application, that this quantity greater exactness, to be hereafter described. Its amount, so obtained, is almost nil. Moreover, its quantity is, that can be doubted whether it is tolerably exact ap- proximation to the truth; and in conformity with this, we must admit the sun to be situated at a mean distance from us about 95,000 miles. This is nearly 23 times the earth's radius, or about 95,000,000 miles. (305.) The sun's diameter should appear to us as large as it does, and should be powerfully influence our condition by its heat and light; requires us to form a very grand conception of its size magnitude. But it is impossible to suppose that those great distances are carried on within it, by which it is enabled to keep up its thermic action; and yet it appears to us as large as it is. As its actual magnitude we cannot be at no loss, knowing its distance, and the angle under which its diameter appears to us. An object placed at the distance of 95,000,000 miles, and exhibiting an angle of 23° must have a 184 A TREATISE ON ASTRONOMY. [CHAP. V.] real diameter of 880,000 miles. Such, then, is the dia- meter of this stupendous globe. If we compare it with what we have already ascertained of the dimensions of our own, we shall find that in linear magnitude it exceeds the earth by a ratio of 1114 to 1, and in bulk (in that of 1384,472 to 1). (36.) It is very possible to avoid associating our conception of an object with its actual size, by the use of such enormous dimensions, with some corresponding attribute of massiveness and material solidity. That the sun is an immense body, and that it possesses its own peculiar structures and economy, our telescopes distantly inform us. They show us dark spots on its surface, which at times appear to move slowly across it; by attending to whose situation, at different times, astrono- mers have ascertained that the sun revolves about an axis inclined at an angle of 7° 59' to the plane of those of the ecliptic, performing one rotation in a period of 25 days and 2 hours. In the direction of the axis of rotation, of the earth, i. e., from east to west. Hence, then, we have an analogy with our own globe; the slower and more majestic movement only corresponding with the greater size and greater massiveness of the sun. And so, with the prevalence of similar mechanical laws, and of, at least, such a community of manners as the existence of perpetual motion implies. The sun is therefore, in exact proportion in which we invest our idea of this imme- nence both with the attribute of inertia, or weight, is be- coming dimly visible. For instance, if we consider a com- paratively small body as the earth as without, on the one hand, dragging a large body along with it; or, on the other hand, held to it, pursuing its course alone in space, and leaving the earth behind it. If we tie two stones together by a stringing and fling them into space; they will continue for a point between them, which is their common centre of gravity ; but since both of them be greatly more ponderous than the earth itself, this centre will be considerably nearer to that one, and even within its surface, so that the smaller one will circulate, in fact, about the larger, which will be comparatively but little disturbed from its place. DEVIATIONS AND ROTATION OF THE EARTH. 106 (307.) Whether the earth moves round the sun, the sun round the earth, or both round their common centre of gravity, will make no difference, so far as appearance is concerned provided the stars be supposed sufficiently dis- tant to be considered as fixed points, and that the displace- ment by the motion so attributed to the earth. Whether they are so or not must still be a matter of inquiry ; and from this point of view we may consider the earth's dis- placement, we can conclude nothing but this, that the scale of the sidereal universe is so great, that the mutual attraction between any two bodies is not perceptibly in its point in comparison. Admitting them, in conformity with the laws of dynamics, that two bodies connected by a spring move together when at rest in free space do, in fact, revolve about their common centre of gravity, which remains immovable by their mutual ac- tion, and that the distance between them is situated. Mechanics teaches us that this place will divide their mutual distance in the same ratio as their masses. This proposition is founded on phenomena, of which an account will be given further on, to inform us that this ratio, in the case of the earth and sun, is nearly one to nine hundred. The sun being, in that proportion, more ponderous than the earth. From this it will follow that the common point about which they revolve is situated at a distance from the sun's centre, or about a fifth part of its own diameter. (308.) Henceforward then, in conformity with the above reasoning, let us suppose that all our system, we must learn to look upon the sun as the com- paratively monolithic centre about which all other bodies move. It has been found to have a certain eccentricity, and with a velocity regulated according to the law above assigned; the sun occupying one of the foci of an ellipse whose other focus is situated on all sides its light and heat; while the earth, travelling round it, and presenting itself differently to it at different times of day and night, summer and winter, which we enjoy. (309.) In this annual motion of the earth, its axis pre- A diagram showing an elliptical orbit around a central body. *The Cyl. Orb. of Sun. Centre of Gravity.* a 2 160 A TREATISE ON ASTRONOMY. [CHAIF. V.] serves, at all times, the same direction as if the orbital movement had no existence; and is carried round paral- lel to itself, and pointing always to the same vanishing point, which is situated in the centre of the sphere which gives rise to the variety of seasons, as we shall now ex- plain. In so doing, we shall neglect (for a reason which will be more apparent hereafter) the earth's rotation about its axis, and suppose it a circle, with the sun in the centre. (310.) Let, then, S represent the sun, and A, B, C, D, four positions of the earth in its orbit, 90° apart, viz. A A diagram showing four positions of the Earth in its orbit around the Sun. The Sun is represented by S. Positions A, B, C, and D are shown at 90-degree intervals around the Sun. The Earth is depicted as a globe with a line indicating its axis of rotation. The diagram also shows the Earth's shadow on the globe. that which it has on the 21st of March, or at the time of the vernal equinox; B that of the 21st of June, or the summer solstice; C that of the 21st of September, or the autumnal equinox; and D that of the 21st of December, or the winter solstice. Let P represent the pole of the Earth; let Q represent the axis of the earth about which its diurnal rotation is performed without interfering with its annual motion in its orbit; and let E represent any point on the en- lightened half of the surface at once, viz. that turned towards it; the shaded portions of the globe in its several positions being those which are turned away from it. The en- lightened halves of the earth's surface in these positions. Now, 1st, in this position A, the sun is vertically over the northern pole P; and therefore it is directly overhead (fig. 1). It is, therefore, in the equinox; and in this position the poles P, Q both fall on the extreme confines of the en- lightened hemisphere; and consequently every part of half the northern and half the southern hemisphere at once; as the earth revolves on its axis, every point of its surface describes half its diurnal course in light, and **CHAP. V.** **TEMPERATURE OF THE EARTH.** half in darkness; in other words, the duration of day and night is here equal over the whole globe; hence the term *equinozus*. The same holds good at the astraline equinox on the 21st of March. (311.) B is the position of the earth at the time of the northern, summer solstice. Here the north pole F, and consequently all the regions lying within its neigh- bourhood, as far as B, are situated within the enlighten- ed half. As the earth turns on its axis in this position, the sun's rays fall upon them with great intensity, and lighteneth; therefore, at this point of its orbit, or at this season of the year, it is continual day at the north pole, and night at the south pole. Hence we call this pole as far as B—thus, i.e., to the distance of 23° 28' from the pole, or within what is called, in geography, the arctic circle of latitude 66° 34'. All those parts of the Q, with all the region comprised within the antarctic circle, as far as 23° 28' from the south pole, are immersed at this time in perpetual night; but they are entire diurnal regions, so that it is here continual night. (312.) With regard to that portion of the surface com- posed of water, it is evident that every part of it is so less evident that the nearer any point is to the north pole, the larger will be the portion of its diurnal course passing through it; and therefore, in proportion as it is in the dark hemisphere; that is to say, the longer will be its day, and the shorter its night. Every station north of the equator will have a greater number of days than twelve hours' duration, and vice versa. All these phe- nomena are exactly inverted when the earth comes to the opposite position. (313.) Now, the temperature of any part of the earth's surface depends mainly, if not entirely, on its exposure to the sun's rays. It is evident that no place receives more of any place, that place is receiving heat; when below, parting with H by the process called radiation; and then when it has received a certain quantity of heat, it must balance each other at every station, or the equilibrium um of temperature would not be supported. Whenever, then, the sun remains more than twelve hours above the horizon of any place, and less beneath, the general tempo- 187 108 A TREATISE ON ASTRONOMY. [CHAP. V.] nature of that place will be above the average; when the reverse, below. As the earth, then, moves from A to B, the days growing longer, and the nights shorter in the northern hemisphere, and the same being true of part of that hemisphere in which we live, we may pass from spring to sum- mer, while at the same time the reverse obtains in the southern hemisphere. As the earth proceeds from C to D, the days and nights again approach to equality—the ex- cess of temperature in the northern hemisphere above the mean state is now diminished by the approach of autumn; and at the autumnal equinox, C, the mean state is once more strained. From thence to D, and, finally, from equal to equal, we have winter in both hemispheres; but must again occur, but reversed, it being now winter in the northern, and summer in the southern hemisphere. (314.) The following table exhibits this fact. The continual day within the polar circles in summer, and night in winter, the general increase of temperature at latitudes between 60° and 70° north and south, the pole, and the reversal of the seasons in the northern and southern hemispheres, are all facts too well known to require further notice. The position of the equinoxes of the earth correspond, as we have said, to the equinoxes; those at B, to the solstices. This term must be ex- plained. If, at any given moment, we consider XW XP the earth's diameter, and XZ its length, it is evident that the angle PXZ will be the sun's polar distance. Now, this angle is always greater than 90°; it attains its minimum at B; being in the former case 80° + 23° = 82° 10' 26"; and in the latter 90° - 23° = 66° 57" . At these points therefore, the sun's rays are perpendicular to the axis of rotation of the earth at noon; and receive (315.) The dipole form of the earth's orbit has but a very slight effect on its motion; for there is no line of force corresponding to the difference of seasons. This assertion may at first sight seem incompatible with what we know respecting magnetism; but it results from a luminous placed at a variable distance. Heat, like light, being equally dispersed from the sun in all direc- tions, and being spread over the surface of a sphere con- tinually enlarging as we recede from the centre, must of CHAP. V.] EQUAL DISTRIBUTION OF HEAT. 100 course diminish in intensity according to the inverse pro- portion of the surface of the sphere over which it is spread; that is, in the inverse proportion of the square of the distance. But we have seen (art. 398) that this is the same law which governs the inequality of the earth about the sun's rays. Hence it appears, that at any point on the earth's surface, the heat received from the sun varies in the exact proportion of the angular ve- locity, i. e. of the momentary increase of longitude ; and furthermore, that the same equal amount of heat is re- ceived from the sun in whatever direction it may be turned; in whatever part of the elliptic those angles may be situated. Let, then, $A$ represent the sun; AQMP the A diagram showing a circle with points labeled A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q. earth's orbit; A its nearest point to the sun, or, so it is called, the perihelion of its orbit; M the farthest, or the aphelion; and therefore ASM the axis of the ellipse. Now let PSQ be a straight line drawn through Q by a straight line PSQ drawn through the sun, and any how directed as to direction; then it will appear that the earth to circularly move around PAQM; and if it be supposed over 180° of longitude in moving from P to Q and as many in moving from Q to P. It appears, therefore, from what has been said before, that all the heat re- ceived from the sun will be equal in the two segments, in whatever direction the line PSQ be drawn. They will also be equal in their unequal areas; that is, that in which the perihelion A lies in a shorter, and the other in a longer, in proportion to their unequal area; but the greater proximity of the sun in the smaller segment com- 100 A TREATISE ON ASTRONOMY. CHAP. V. pensates exactly for the more rapid descent, and thus an equilibrium of heat is, as it were, maintained. Were it not for this, the eccentricity of the orbit would materi- ally influence the transition of seasons. The fluctua- tion of the temperature of the earth is its own cause, and consequently, the fluctuation in the sun's direct heating power to double this, or $\frac{1}{3}$th of the whole. Now, the northern hemisphere is always turned towards the place of the northern winter solstice; so that, were it not for the compensation we have just described, the effect would be such as to make the summer season and winter in the southern hemisphere, and to moderate it in the northern; thus producing a more violent alter- nation of seasons in one hemisphere than in another, though to perpetual spring in the other. As it is, however, no such inequality subsists, but an equal and impartial dis- tribution of heat prevails throughout the globe. (816.) The great key to simplicity of conception in astronomy, and, indeed, in all sciences where motion is concerned, is to consider only those points which are referred to points which are either permanently fixed, or so nearly so, as that their motions shall be too small to interfere with each other. In the choice of these primary points of reference, too, we must endeavour, as far as possible, to select such as have simple and uniform motions. This is a very useful con- sideration with respect to the curves described by the moving parts of the system, and which are thereby fitted to per- form their functions with a minimum of effort at all stations for the eye of reason and theory. Having learned to attribute an orbital motion to the earth, it loses this advantage when applied to any other body; for then its axis centre about which its orbit is performed. Precisely as, when embarrassed by the earth's diurnal motion, we have learned to attribute an orbital motion to any other body of observation from its surface to its centre, by the appli- cation of the diurnal parallax; so, when we come to in- quire into the causes of planetary motions, we also find *The Geological Transaction.* 1825. "On some Geometrical Causes which may influence Geological Phenomena," by the author of this work. EARTH AND TRUE LONGITUDE OF THE SUN. 101 ourselves continually embueded by the optical inter- vention of our own eyes, we have no conception of the annual or heliocentric parallax, as it may be termed, we consent to refer all our observations on them to the celestial sphere, which is supposed to be at infinite dis- tivity of the sun, and the other bodies which are connect- ed with it in our system. Hence arises the distinction between the apparent and true longitudes of the sun and its ob- jects. The former refers its situation in space to an imaginary sphere of infinite radius, having the center of the earth for its center, and the sun for its center with the sun. Thus, when we speak of the Heliocentric lon- gitudes and latitudes of objects, we suppose the specia- tor's eye placed at the center of the earth, looking perpendicularly to the plane of the ecliptic, to the great circle marked out in the heavens by the infinite prolonga- tion of the line of nodes. (317.) The point in the imaginary cone of an in- finite heaven, to which a spectator in the sun refers the ecliptic, is called the centre of the ecliptic; and that to which a spectator on the earth refers the sun's centre; conse- quently, the heliocentric latitude of the earth is always equal to the sun's geocentric longitude +180°. The heliocentric equinoxes and solstices are, therefore, the same as those of the sun; but they do not occur so often only to imagine a plane passing through the sun's centre, parallel to the earth's equator, and prolonged indefinitely in both directions. This plane is called the plane of nodes; and the line formed by this plane and that perpendicular to it is called the line of nodes. (318.) The position of this longer axis of the earth's orbit is a point of great importance. In the figure (art. 315) let ECLI be the ecliptic, E' the vernal equinox, L, the line of nodes, and S, S', S", S"', S"", S'"', S'"", S'"'" ref- ferred from the sun when its heliocentric longitudes are 0° and 180° respectively. Supposing that Earth's mo- tion was uniform, then ECLI would be in S" in 1800 A. ESA., or the longitude of perihelion, in the year 1800 was 60° 50' b". We say in the year 1800 because, in point of fact, by the operation of causes hereafter to be 193 A TREATISE ON ASTRONOMY. [CHAP. V.] explained, its position is subject to an enormous slow va- riation of about 180° per century, the cause of which, and in the progress of an immensely long period—of no less than 20,584 years—carries the axis ASF of the orbital plane of the Earth through all the signs of the ecliptic. But this motion must be disregarded for the present, as well as many other minute deviations, to be brought into view by means of the most accurate observa- tions. (316.) Were the Earth's orbit a circle, described with a uniform velocity about the sun placed in its centre, no- thing could happen but that the longitude of any point at any time, with respect to the line of equinoxes, or to the longi- tude, for we should only have to reduce to numbers the propor- tion between the time elapsed and the mean angular speed: $\frac{360°}{\text{the arc of longitude passed over}}$ = $\frac{\text{the time elapsed}}{\text{the mean angular speed}}$. The longitude so calculated is called in astronomy the mean longitude of any one place on the Earth's surface. But since this place is, nor uniformly nor irregularly described, this rule will not give us the true place in the orbit at any proposed moment. Neverthe- less, as long as the differences between two points are small, the true place will never deviate very far from that so de- termined (which, for distinction's sake, is called the mean place), because it is always possible to correct this deviation from the latter, by applying to it a correction or equation (as it is termed), whose amount is never very great, and whose sign depends upon whether it is necessary, de- pending on the equal description of areas by the earth around the sun. For since, in the elliptic motion, accord- ing to Kepler's laws, all parts of the same latitude are described uniformly, the proportion must now be stated, that one year : the time elapsed : ; the whole area of the ellipse : ; and that one year : the time elapsed : ; the re- verse area in that time. This area, therefore, becomes known, and it is then, as above observed, a problem of two geometrical quantities (see Art. 275) to find out (ASF, fig. art. 316), which corresponds to any proposed frac- tional area of the whole ellipse supposed to be contained in the sector formed by two consecutive signs of the pe- rihelion, then will the angle ASF at first increase more rapidly than the mean longitude and will, therefore, de- crease the whole semi-ellipse from A to M, excepted in a MEAN AND TRUE LONGITUDE OF THE SUN. amount; or, in other words, the true place will be ad- vanced by one degree every 76 years, and will have elapsed, and one half the orbit have been described, whether it be circular or elliptic. Here, then, the mean longitude is the true place, and the true place is the mean longitude. From M to A, the true place will fall short of the mean, since at M the angular motion is slowest, and the true place will therefore be less than the mean longitude; to make up with it however, as it approaches A, where it once more overtakes it. (38.) The angle between which the true longitude of the earth differs from the mean longitude is called the equation of the centre, and its additive during all the half-year in which it is positive, and subtractive during all the other part of the year, increasing to a maximum, and again diminishing to zero at M; after which it becomes subtractive, although it may increase to a maximum at B, and A, and again diminishes to 0 at A. Its maximum, both additive and subtractive, is 1° 55' 32". (39.) By means of this equation of the centre, when the mean longitude is known, we can find out what true longitude would be seen from the earth in 180° more longitude than the earth from the sun, in this way also the sun's true place in the ecliptic may be found. This equation of the centre is performed by a table constructed for that purpose, to be found in all " Solar Tables." (40.) It is evident that this equation of the centre depends only on the elliplicity of the orbit, and may be expressed in terms of its eccentricity. For ease, therefore, we shall express it by means of an angle; but by observation, the latter may be derived from it; because, whatever the law, or numerical connection, between two quantities may be supposed to exist between them determined from the other. Now, by assiduous observation of the sun's transits over the meridian, we can ascertain, for every degree of latitude observed on any day, his longitude (art. 369). After this, it is easy to assign the angle by which this observed longitude exceeds or falls short of the mean; and this excess or amount of this excess R 194 A THEORETIC ON ASTRONOMY. UMAF. V. or defect which occurs in the whole year, is the maxi- mum equation of the centre. This, as a means of ascer- taining the eccentricity of the orbit, is a far more easy and accurate method than that of concluding its distance by measuring the longitude of the moon. The result of two methods coincide, however, perfectly. (83.) The effect of the equation of the centre, by disturbing the uniformity of the sun's apparent motion in longitude, would cause an inequality in the coming out of the meridian at noon. When the sun's centre comes to the meridian, it is apparent noon, and its mo- tion in longitude westward will be equal to that with which it travels eastward, this would always coincide with mean noon, or the stroke of 12 on a well-regulated solar clock. But, independent of the want of uniformity in the motions of the sun, it is true that there is another inequality in this respect; in consequence of which an observer at any place would find his clock too slow. This would yet alternatively, in its time of stating the marti- dian, anticipate and fall short of the mean noon as shown by the clock. For the right ascension of a celestial body being formed by its motion in latitude and longitude, and being a triangle, of which its longitude is the hypothenusee, it is clear that the uniform increase of the latter must necessi- tate a decrease of uniformity in the increase of the former. (834.) These two causes, therefore, produce a prece- dence, in fact, a very considerable fluctuation in the time as shown per clock, when the sun really attains the meridian. This fluctuation is greatest when it has an hour; apparent noon sometimes taking place as soon as 16 min. before mean noon, and at others as much as 144 min. after. This difference between apparent and mean noon is called "the equation of time." It is computed and inserted in ephemerides for every day of the year, under that title; or else, which comes to the same thing, the moon's equation is given for each day. The equation for each day, is set down as an astronomical phenomenon to be observed. (835.) As the sun, in its apparent annual course, is CHAP. v.] TROPICAL AND ANOMALISTIC YEARS. carried along the ecliptic, its declination is continuously varying between the extreme limits of $35° 28' 40''$ north, and as much south, which it attains at the solstices. It is consequently always vertical over some part or other of the earth's surface, and this fact is well seen when we compare the north and south parallel of $23° 28' 40''$. The equator itself is also always vertical to the northern one that of Cancer, and the southern of Capricornus; because the sun, at the respective solstices, is situat- ed in the division or signs of the ecliptic so distant as $30^{\circ}$ from the equator, and therefore making $30^{\circ}$ of its circumference. They commence at the vernal equinox, and are named as Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces. They are denoted also by the following symbols:--$\tau$, $\beta$, $\delta$, $\gamma$, $\alpha$, $\pi$, $\varphi$, $\psi$, $\omega$, $\theta$, $\kappa$, $\lambda$, $\mu$, $\nu$, $\sigma$, $\tau$, $\rho$, $\iota$, $\eta$, $\zeta$, $\xi$, $\chi$, $\psi$. The zodiacal signs are also divided into degrees, minutes, and seconds, &c., thus, $5^{\circ} 27' 0''$ corres- ponding with $5^{\circ} 27' 0''$ of longitude (see p. 26). (286.) When the sun is in either tropic, it enlumes, as we have seen, the pole on that side of equator, and almost all the countries lying between it and the equator. The parallels of latitude, at this distance from either pole, are called the polar circles, and are distinguished from each other by their different names. The regions within these circles are sometimes termed cold zones, while the belt between the tropics is called the tropical zone; and sometimes temperate zones. These last, however, are generally named after the name of the country in which they lie. In the tropics themselves, zones of climate are not separated with such exactitude. (287.) Our seasons are determined by the apparent passages of the sun across the equatorialine, and its alter- nate arrival in the northern and southern hemisphere. What causes this alternation happens to be in- tervals precisely equal to the duration of an sidereal year; but in fact, owing to the slow conical motion of the earth's axis described in art. 264, the equinoxes re- treat on the ecliptic, and meet the advancing with some-
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    106 A TREATISE ON ASTRONOMY. CHAP. V. what before the whole sidereal circuit is completed. The annual retreat of the equinox is 50°-1', and this arc is described by the sun in the ecliptic in 30° 19' 09". By so much shorter, then, is the periodical return of our seasons, that we have to wait 30° 19' 09" more round the sun. As the latter period, or sidereal year, is equal to 360° 19' 09", it follows, then, that the former must be less by 50°-1", which is what we mean by the tropical year. (328.) We have already mentioned that the longer axis of the earth, as described by the earth's axis in its revolution, describes in one year an angle of 27° 11' 46". This angle will have moved forward by 17° 08", which are must be subtracted before it can again reach the perihelion. In so doing, it will describe an angle of 10° 43' 58", which therefore may be added to the sidereal period, to give the interval between two consecutive returns to the perihelion. In this way, then, we find that the tropical year is called the anomalistic year. All these periods have their uses in astronomy; but that in which mankind in general are most concerned with them is when they determine the seasons depend on, and which we thus perceive to be a compound phenomenon, depending chiefly and directly upon the motion of the earth about the sun, but subordinately also, and indirectly, on its rotation round its own axis, which is what occasions the succession of seasons. It is interesting to consider an instructive example of the way in which a motion, once admitted in any part of our system, may be traced in its influence on other parts of it; and it could not possibly be supposed to have any thing to do. (329.) As a rough consideration of the appearance of the earth's surface would show us in its present form, and exact inquiry has led us first to discover its elliptic figure, and, in the further progress of researches, to perceive its perturbations and oscillations *These numbers are all as other astronomical data of our time are taken from Mr. Bailey's Astronomical Tables and Formulae un- less the contrary is expressed.* CHAP. V.] PHYSICAL CONSTITUTION OF THE SUN. 197 from that figure ; so, in investigating the solar motions, the first notion we obtain is that of an orbit, generally speaking, round, and not far from a circle, which, on removing the eccentricity, would become a circle; or to an ellipse of small eccentricity, and described in conformity with certain laws, as above stated. Still minuter in- quiries will show us that the motion of the sun is not regular from this form and from these laws, of which we have a specimen in the slow motions of the axis of the orbit system, and in the irregularities of its rotation, appre- hended under the name of perturbations and secular in- equalities. Of these deviations, and their causes, we shall presently speak. But it is necessary to know much of physical astronomy to render a complete account of them all, and so have left nothing unexplained, either in the nature of the motions of the sun itself, or of the bodies of our system. But the nature of this explana- tion cannot be understood till we have developed the principles of mechanics, and then deduced from them direct consequences. This will be the object of our three fol- lowing chapters : in which we shall take advantage of the principles already established to explain the phenomena with and dependence on the earth, to render it, as it were, a stepping-stone to the general explanation of the phenomena. (280.) We shall conclude this by describing what is known of the physical constitution of the sun. The interior of the sun is supposed to be provided with coloured glasses, to take off the heat, which would otherwise injure our eyes; it is observed to have fre- quently disappeared from view during a few hours sur- rounded with a kind of border, less completely dark, called a penumbra. Some of these are represented at fig. 35. The central spot is always more than half this volume. They are, however, not permanent. When watched from day to day, or even from hour to hour, they appear to move slowly towards one or other horns; and at length to disappear altogether, or to break out anew in parts of the surface where none were before. In each cases of disappearance, the central dark spot always contracts into a point and vanishes before the 2 198 A TREATISE ON ASTRONOMY. [CHAP. V.] border. Occasionally they break up, or divide into two or more, and in these offer every evidence of that extre- me mobility which belongs only to the fluid state, and of that excessively violent agitation which seems only compulsive, and not voluntary, on the part of the mat- ter. The scale on which their movements take place is immense. A single second of angular measure, from the earth to the sun, would extend from 300 miles to 465 miles; and a circle of this diameter (containing there- fore nearly 900,000 square miles) is the least space which can be traversed by a comet in one second's passage. Spots have been observed, however, whose linear di- ameter has been upwards of 45,000 miles * and even, if some motion be supposed to have taken place during ex- treme. That such a spot should close up in six weeks' time (for they hardly ever last longer), its borders must approach each other at the rate of nearly 75 miles per sec- ond. Many other circumstances tend to corroborate this view of the subject. The part of the sun which is not oc- cupied by spots is far too bright to be regarded as being finely motled with an appearance of minute, dark dots, or pores, which, when attentively watched, are found to move about with great rapidity. It is nothing which represents so faithfully this appearance as the slow subsidence of some efflorescent chrys- tophite on the surface of the moon. This appears par- ticularly from above : so faithfully indeed, that it is hardly possible not to be impressed with the idea of a luminous mass floating in a dark atmosphere, with a transparent and non-luminous atmosphere, either flus- sing as clouds in our air, or pervading it in vast sheets, and adumbrating like flames, or the streamers of our northern lights. (381.) Lastly, in the neighbourhood of great spots, or extensive patches of light, there are always seen often observed to be covered with strongly marked curved, or branching streaks, more luminous than the rest, called "prominences." These are constantly existing, spot frequently break out. They may * * Mayer, Obs. Man. 15, 1758. "Inscr. morsis in sole conspicuiter ejus diametro" **dum** salis** **CHAP. V.] PROBABLE NATURE OF THE SOLAR SPOT.** perhaps, be regarded with most probability, as the ridges of immense waves in the luminous regions of the sun's atmosphere, indicative of violent agitation in their neighbourhood. (382.) But what are the spots? Many fanciful notions have been broached on this subject, but only one seems to deserve attention. It is that the spots are dark, or at least comparatively dark, solid body of the sun itself; laid bare to our view by those immense fluctuations which we see in the luminous regions above, to which it appears to be subject. Respecting the manner in which this disclosure takes place, different ideas again have been entertained. Some have supposed that the spots, that en masse in the nature of mountains are actually laid bare, and project above the luminous ocean, appear- ing like islands in a sea of light; others have pro- duce the penumbra, where the luminous fluid is less deep. A final objection to this theory is the perfectly uniform appearance of the spots, when viewed from without, both inwards, where it joins the spot, and outwards, where it borders on the bright surface. A more proba- ble supposition is that the spots are produced by those who consider the luminous strata of the atmosphere to be sustained far above the level of the solid body by a tension or pressure exerted upon them by its sur- face (or rather, to avoid the former objection, at some considerably lower level within its depth), a cloudiness or obscurity being thus introduced into the light which reflects a considerable portion of the light to our eyes, and forms a penumbra, while the solid body, shaded by such a cloudiness or obscurity, is seen as a part of both the strata, but more of the upper than the lower, he supposes effected by powerful upward currents of the atmosphere, or by currents issuing from the body, or from local agitations. See fig. 1., plate III. (383.) The region of the spot is confined within about 50 miles of each other; and if we take on the surface, carefully measured with micrometers, is as- certained the position of the equator, which is a plane inclined 7° 20' to the ecliptic, and intersecting it in a line *Phil. Trans. 1801.* A diagram showing a section of Earth's atmosphere with a spot near the center. 200 A TREATISE ON ASTRONOMY. CHAP. T. whose direction makes an angle of 80° 21' with that of the equinoxes. It has been also noticed (not, we think, without great need of further confirmation), that extinct spots have been observed at the same time and place on the same identical points of the sun's globe. Our knowledge of the period of its rotation (which, according to Darwin, is about 25 days) enables us to infer (compared to others, materially differently,) can hardly be regarded as sufficiently precise to establish a point of no much nicety. (384.) The heat which is emitted by the surface of the sun cannot be otherwise than very elevated, much more so than any artificial heat produced in our furnaces, by chemical combustion. This may be inferred from the actions of several distinct kinds : 1st. From the law of decrease of radiant heat and light, which, being inversely as the square of the distance, would require a temperature of 100000 degrees at the surface of the sun, and only one-tenth of that at an equal area exposed at the distance of the earth, and on an equal area at the visible surface of the sun, must be in proportion to its temperature. The heat emitted by the sun's apparent disc to the whole hemisphere, or as I call it 300000. A far less intensity of solar radiation, calculated from the amount of heat necessary to dis- epend gold and platinum in vapour. 2ndly, From the fa- cility with which the effusive rays of the sun traverse glass, a property which is not possessed by all kinds of refractive bodies. 3rdly, From the fact that the most vivid flames disappear, and the most brilliant lights are extinguished only at black spots upon the disk of the sun when held before it and the eye. From this last remark it follows, that body of the sun is not composed entirely of matter through its spots, many, nevertheless, be in a state of most intense *By direct measurement with the microscope, an instrument I have employed for this purpose.* I find that in a given number of hours there are produced on the surface of this planet about 100000 spots which appear to be in a state of intense heat. The number varies greatly from day to day. On some days they are numerous; on others they are few. On one day I observed 675 spots in a space of two hours; on another day I observed 125 spots in a space of two hours; on another day I observed 150 spots in a space of two hours; on another day I observed 175 spots in a space of two hours; on another day I observed 200 spots in a space of two hours; on another day I observed 225 spots in a space of two hours; on another day I observed 250 spots in a space of two hours; on another day I observed 275 spots in a space of two hours; on another day I observed 300 spots in a space of two hours; on another day I observed 325 spots in a space of two hours; on another day I observed 350 spots in a space of two hours; on another day I observed 375 spots in a space of two hours; on another day I observed 400 spots in a space of two hours; on another day I observed 425 spots in a space of two hours; on another day I observed 450 spots in a space of two hours; on another day I observed 475 spots in a space of two hours; on another day I observed 500 spots in a space of two hours; on another day I observed 525 spots in a space of two hours; on another day I observed 550 spots in a space of two hours; on another day I observed 575 spots in a space of two hours; on another day I observed 600 spots in a space of two hours; on another day I observed 625 spots in a space of two hours; on another day I observed 650 spots in a space of two hours; on another day I observed 675 spots in a space of two hours; on another day I observed 700 spots in a space of two hours; on another day I observed 725 spots in a space of two hours; on another day I observed 750 spots in a space of two hours; on another day I observed 775 spots in a space of two hours; on another day I observed 800 spots in a space of two hours; on another day I observed 825 spots in a space of two hours; on another day I observed 850 spots in a space of two hours; on another day I observed 875 spots in a space of two hours; on another day I observed 900 spots in a space of two hours; on another day I observed 925 spots in a space of two hours; on another day I observed 950 spots in a space of two hours; on another day I observed 975 spots in a space of two hours; on another day I observed 1000 spots in a space of two hours; on another day I observed 1025 spots in a space of two hours; on another day I observed 1050 spots in a space of two hours; on another day I observed 1075 spots in a space of two hours; on another day I observed 1100 spots in a space of two hours; on another day I observed 1125 spots in a space of two hours; on another day I observed 1150 spots in a space of two hours; on another day I observed 1175 spots in a space of two hours; on another day I ob- served 1200 spots in a space of two hours; on another day I ob- served 1225 spots in a space of two hours; on another day I ob- served 1250 spots in a space of two hours; on another day I ob- served 1275 spots in a space of two hours; on another day I ob- served 1300 spots in a space of two hours; on another day I ob- served 1325 spots in a space of two hours; on another day I ob- served 1350 spots in a space of two hours; on another day I ob- served 1375 spots in a space of two hours; on another day I ob- served 1400 spots in a space of two hours; on another day I ob- served 1425 spots in a space of two hours; on another day I ob- served 1450 spots in a space of two hours; on another day I ob- served 1475 spots in a space **CHAP. V.] ACTION OF THE SUN'S RAYS ON THE EARTH. 201** ignition. It does not, however, follow of necessity that it must be so. The contrary is at least physically possi- ble. A perfectly reflective canopy would effectively de- flect the rays of the sun from their course above its atmosphere, and no heat would be conducted towards through a gaseous medium increasing rapidly in density with height. But this is not the case; on the con- trary, the fact of their visibility in such a situation can leave no doubt. (336.) In the immense escape of heat by radiation, we may also remark, will fully explain the constant state of tumultuous agitation in which the fluids composing the atmosphere are kept. The sun's rays cause the interac- tion and filling in of the pores, without having recourse to internal causes. The mode of action here alluded to is perfectly analogous to that described in art. 350, when the fluid from which it subsides is warm, and losing heat freely. (336.) The sun's rays are the ultimate source of al- most every motion which takes place on the surface of the earth. They are responsible for all those disturbances in the electric equilibrium of the atmosphere which give rise to the phenomena of terrestrial magnet- ism. They are the cause of all those changes generated from organic matter, and become, in their turn, the sup- port of animals and of man, and the sources of those, generally called "the elements," which are used up for human use in our coal strata. By them the waters of the sea are made to circulate in vapour through the air, and thus to produce all those phenomena observed. By them are produced all disturbances of the chemical equilibrium of the elements of nature, which, by a series of processes, lead to their separation into new forms, and originate a transfer of materials. Even the slow degradation of the solid constituents of the sur- face of the earth into water and gases, and their diffusion among the waters of the ocean, are entirely due to the abrasion of the wind and rain, and the alter- nate action of the seasons; and when we consider the immense transfer of matter so produced, the increase of 208 A TREATISE ON ASTRONOMY. [CHAP. V. pressure over large spaces in the bed of the ocean, and diminution over corresponding portions of the land, we are not at a loss to perceive how the elastic power of substances may thus be employed to move one and altered by the other may break forth at points where the resistance is barely adequate to their retention, and thus bring about a more or less violent activity under the general law of solar influence. (387.) The great mystery, however, is to conceive how so enormous a body as the sun can be kept up. Every discovery in chemical science here leaves us completely at a loss, or rather seems to remove farther than it advances. But even if this point cannot be reached, some might be hazarded, we should look rather to the known possibility of an indefinite generation of heat by frictional action, and to the fact that the sun's surface, than to any actual combustion of ponderable fuel, when its solid or gaseous, for the origin of the solar radiation. (388.) It is evident that the sun must have been hot and, doubtless, she was. May not a continuous current of electric force pass through her body? If so, then, in passing from one part to another, or traversing the planetary space, and coming in the upper region of the atmosphere, it would be found that the temperature would increase. In this case we have yet an unexplained manifestation in our scheme because it is impossible to account for such a change in terms of any known physical principle. We must therefore admit that something has been added to her power since her first creation. This something may be either a constant emission of heat from her surface, or a continual supply of matter from without. The former explanation is most probable because it does not involve any assumption on my part; but the latter is equally possible. For suppose that we consider the emission of heat from her surface as the result of the continued radiation of rays proceeding from her surface. Then it follows that all rays which reach us must have passed through her body during their journey. Now let us consider what happens when we observe a star. We see only those rays which have passed through her body during their journey. Hence it follows that the total amount of energy emitted by a star is equal to the amount which reaches us after passing through her body. Now suppose that we could measure this quantity. We should find that it is very nearly equal to the total energy emitted by the sun during her entire history. This means that she has been emitting energy continuously for an immense period of time. Now suppose that we could measure the rate at which she emits energy. We should find that it is very nearly equal to the rate at which she emits energy during her entire history. This means that she has been emitting energy continuously for an immense period of time. Now suppose that we could measure the rate at which she emits energy. We should find that it is very nearly equal to the rate at which she emits energy during her entire history. The physical analysis of the solar beam exhibits in the spectrum a band of light which corresponds with no known terrestrial flame. This may therefore lead us to a closer inspection of our theory concerning the origin of light. From such an indication, we must conclude, that previous to reaching us this light has passed through some substance which absorbs it and transmits it on to us. Of this substance we know nothing, and may conjecture every thing except air or water. Hence we must conclude that this light is produced by some process which takes place within the sun itself. Of this process we know nothing, and may conjecture every thing except air or water. Hence we must conclude that this light is produced by some process which takes place within the sun itself. The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? If so, then it follows that this light is produced by some process which takes place within the sun itself? The question now arises whether this light is produced by some process which takes place within CHAP. VI.] OF THE MOON. CHAPTER VI. Of the Moon—Its sidereal Period—Its Apparent Diameter—Its Parallax. In the course of the last century, several observations were made on the Eclipse about the Earth in the Foresight of Eccentricity and Inclination of the Earth's Axis, and of the inclination of the Plane of the Ecliptic to that of the Equator. The Phase of the Moon by its Apogee Period—Lunar Eclipse—Marean Eclipse—Eclipse of the Sun—Apogee of the Moon—The Moon's Motion—Amateurs' Observations on Astra-Libra—Apparent Motion of the Moon. (388.) The moon, like the sun, appears to advance among the stars with a movement contrary to the general diurnal motion of the heavens, but much more rapid, so as to appear to move through a considerable part of its orbit (as observed) by a few hours' currency attention on any moonlight night. By this continued advance, which, though sometimes very slow, is always in one direction, and unvarialed or reversed, it makes the tour of the heavens in a month nearly complete. It is therefore evident that, in that time, a position among the stars nearly coincident with that it had before, and which would be ex- cedent with no, but for causes presently to be stated. (389.) The apparent motion of the moon apparently de- scribes an oval round the earth, and this orbit cannot be very different from a circle, because the apparent angular diameter is nearly constant within a very great extent of variation. (390.) The distance of the moon from the earth is estimated from its horizontal parallax, which may be found either directly, by observations at remote geographical stations exactly similar to those described in art. 305, or indirectly by means of the phenomena called oscillations (art. 346), from which also its apparent diameter is most readily and correctly found. From these two methods we have obtained results dis- fined line used by Williamson and Fraunhofer may not have their due weight; for they are liable to error from various causes. 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[UMAF.VL. tance of the centre of the moon from that of the earth is 95-9648 of the earth's equatorial radius, or about 237,000 miles. This distance, great as it is, is little more than half the diameter of the moon itself, so that to the eye of the sun he would nearly twice fill up the whole orbit of the moon; a consideration wonderfully calculated to raise our ideas of this suspensuous luminary. (341). The distance of the moon's centre from an ob- server at any point on the earth's surface, compounded with its apparent angular diameter as measured at that station, will give real or linear diameter. Now, the former may be found by means of a table given from the earth's centre is known, and the apparent zenith distance of the moon also determined by observation; for if we suppose the figure of a sphere supported by the moon, the stations $AC$ and $BC$ are equal to the dis- tance $SC$, and the earth's radius $CA$, two sides of the triangle $ACB$. According to (339), when we have found $AB$, we can find $AC$ by subtracting $AB$ from $AC$, and then by adding to this result $AC$, which is equal to $AB$, we shall have $AC$ again. Thus, if we know $AB$, we can find $AC$; and if we know $AC$, we can find $AB$. In like manner, if we know $BC$, we can find $AC$; and if we know $AC$, we can find $BC$. Hence, if we know one side of a triangle, we can find another side. But since we do not know either of these sides, but only their sum, it follows that we cannot find them both. But if we knew one angle of a triangle, we could find all three sides. And since we do not know any angle, it follows that we cannot find any side. Hence, although it is easy to find $AB$, the moon's distance from A. From such observations and calculations it results, that the mean distance between the earth and moon is 2-110 million miles. Now, if we take 2-110 million miles as the mean distance between the earth and moon, and consider that the latter being compared as 1, that of the former will be 0-9648; hence, that of the moon will be 0-9648 multiplied by 2-110 = 2-0548. (342). By a series of observations, such as described in art. 340, it continued during one or more revolutions of the sun around the earth, and at every revolution at every point of its orbit; and if at the same time its apparent zenith distance was observed, and its parallax with respect to the earth's centre, their angular intervals will become known, so that the path of the planet may then be laid down on a chart supposed to represent its plane orbit round the sun. This has been done in all cases where observations have been made; and in some cases even in those where no observations have been made; as in the case of the solar eclipses (art. 329). Now, when this is done, it is found that neglecting certain small (though very peremptory) errors (which will hereafter be rendered), the form of the appar- ent orbit, like that of the sun, is elliptic, but consider- ably more eccentric, the eccentricity amounting to 0-0548 **CHAP. VI.] REVOLUTION OF THE MOON'S NODES.** 305 of the mean distance, or the major semidiameter of the ellipse, and that of the moon's nodes, singularly in its form. (343.) The plane in which this orbit lies is not the ecliptic, however, but is inclined to it at an angle of 5° 8' 17" (the obliquity of the ecliptic), towards the north, and intersects it in two opposite points, which are called its nodes—the interlating node being that in which the moon is nearest to the sun, and the other distant from the sun, and also descending the reverse. The point of the orbit at which the moon is nearest to, and furthest from, the sun, is called its perigee and apogee; and the line joining them and the earth the line of nodes. (344.) There are, however, several remarkable circumstances which interrupt the closeness of the analogy, which cannot fail to strike the reader, between the mo- tion of the moon about the earth and that of a body about the sun. In the latter case, the ellipse described remains, during a great many revolutions, unaltered in its position and size; whereas in that of the moon, such alterations go on perceptible but in a course of very nice ob- servations, which have disclosed, it is true, the existence of a small inequality in its motion. This inequality, in ordinary parlance, and for common purposes, we may leave them unconcerned. But this cannot be done in the case of those who study astronomy. Every deviation from a perfect ellipse is very sensible. It does not return to the same exact position among the stars from one revolution to another; nor does it undergo any continual change in the plane of its orbit. And, in effect, if we trace by observation, from month to month, the point where the moon is nearest to or farthest from the earth—nodes of its orbit—is in a continual state of retreat upon the ecliptic. Suppose O to be the earth, and A B C D E F G H I J K L M N O P Q R S T U V W X Y Z to be a series of points situated by degrees around the earth's axis; then every point will be nearer to or farther from the earth than any preceding or following it; and this will continue through all revolutions. Suppose it to set out from the ascending node, A; then, if the orbit lay all in one plane, passing through O, it would have A', the opposite point in th 306 A TREATISE ON ASTRONOMY. [CHAP. VI.] eclipse, for its descending node ; after passing which, it would again ascend at A. But, in fact, its real path car- A diagram showing the positions of the Earth (E), Moon (M), Sun (S), and the ascending and descending nodes (N and N'). The diagram also shows the Earth's orbit around the Sun, with the Earth's axis tilted at an angle to the orbital plane. The diagram includes labels for the Earth, Moon, Sun, and the ascending and descending nodes. ries it not to a, but along a certain curve, ABC, to C, a point in the ecliptic less than 180° distant from A ; so that the difference between the ascending and descending node, is some- what less than 180°. It then pursues its course below the ecliptic, until it reaches D, where it is shown, not at the point c, diametrically opposite to C, but at a point E, less advanced in longitude. On the whole, then, the moon makes two consecutive passages from south to north, through the plane of the ecliptic, falls short of 260° by the angle ADE ; or, as it may be called, it has made one revolution having retreated in one lunation, on the plane of the ecliptic by that amount. To complete a sidereal revolution, then, it must still advance 360° beyond this point E ; which will not longer happen, bring it exactly back to A, but at a point somewhat above it, or having north latitude. (245.) The actual amount of this retreat of the moon's node is about 1° 16' 44" per annum, on an average, and in periods of time varying from 7 to 10 years. For every degree the ascending node is carried round in a direction con- trary to the moon's motion in its orbit (or from east to west) over a period of time equal to one year. In other words, in the middle of this period the position of the orbit must have been precisely reversed from what it was at the beginning of it. This is because there are gen- erally different stars and constellations at different parts of this period ; and this kind of spiral revolution being continued kept up, it will be one time or other, even with its due error every point of the heavens within that CHAP. VI. ECLIPSES AND OCULTATIONS. 207 limits of latitude or distance from the ecliptic which its inclination permits, that is to say, a belt on the zone of the heavens, of 10° 18' in breadth, having the ecliptic for its middle line. Nevertheless, it still remains true that the motion of the moon, when it is at the first quarter of its revolution, deviates in a single revolution very little from what it would be were the nodes at rest. Supposing the moon to go round the sun in one year, and to have two passages to F, having completed a revolution in longitude, will not exceed $8^{\circ}$; and it must be borne in mind that it is to act upon the earth as a body of this small order, that the motion of the nodes is devised. (240.) Now, as the moon is at a very moderate dis- tance from us, and as we are situated in the very fast our nearest neighbour, while the sun and stars are in comparison immensely beyond it, it must of necessity have a very great influence upon them. Hence it may occur or eclipse every star and planet within the zone above described (and, as seen from the surface of earth, every star and planet within the zone of the ecliptic), which may throw it apparently nearly a degree either way from its place as seen from the centre, according to the course of its revolution. But since this effect can only come thus then, whenever any part of the moon's disc, in this her tortuous course, comes to overlap any part of any star or planet, it is not necessary to be hu- morous. On these occasions is exhibited the most striking and impressive of all the occasional phenomena of astro- nomical science. For instance, when a star or planet is less portion, or even in some new conjunctions the whole, of its disc is obscured, and, as it were, obliterated, by the shadow cast by the moon upon it; or when it appears upon it as a circular-terminated black spot, producing a temporary diminution of daylight, or even nocturnal darkness over a large portion of our hemisphere. In other cases, when, at the moment that the moon is cen- trally superposed on the sun, as so happens that her dia- meter exceeds that of the sun's by less than half its angular diameter less than the sun's, the very singular phenom- enon of an annular solar eclipse takes place, when the edge of the sun appears for a few minutes as a nar- 207 208 A THEOREM ON ASTRONOMY. [CHAP. VI. new ring of light, projecting on all sides beyond the dark circle occupied by the moon's centre. (347.) A solar eclipse can only happen when the sun and moon are in conjunction, that is to say, have the same or nearly the same longitude. It is evident that this con- dition can only be fulfilled at the time of a new moon, through which the earth passes once every 12 months; therefore there must be an eclipse of the sun. If the lunar orbit coincided with the ecliptic, this would be the case, but it is impossible for two orbits to coincide. It is evi- dent that the conjunction, or equality of longitudes, may take place when the moon is in the part of her orbit too remote from the sun for an eclipse to occur, and yet overlap. It is easy, however, to assign the limits within which an eclipse is possible. To this end we must con- sider, that the apparent edge of the sun may be any differ- ent edge may be thrown in any direction, according to a spectator's geographical station, by any amount not exceeding one degree. The sum of these differences will be the same (as far as the possibility of an eclipse is con- cerned) as if the apparent diameter of the moon, seen from its place in the sky, were diminished by one degree. But this cannot be done for both reasons: for, if, when so dilated, it can touch or overlap the sun, there must be an eclipse at some part or other of its surface; and if it does not touch or overlap, at the nearest conjunction, the geocentric distance of the centres of the two luminaries do not exceed the sum of their semidiameters. In this case, although there will be no eclipse, there will be an eclipse. This sum is, at its maxi- mum, about 1° 34' 27". In the spherical triangle SNM, A diagram showing a spherical triangle SNM with angles labeled S (sun), N (moon), M (earth). The angle SNM is approximately 1° 34' 27''. then, in which S is the sun's centre, M the moon's, SN the ecliptic, MN the moon's orbit, and N the node, we **CHAP. VI. LIMITS OF A SOLAR ECLIPSE.** 209 may suppose the angle NSM a right angle, $SM = 1^{\circ} 34' 57''$, and the angle MNS $= 8^{\circ} 48'$, the inclination of the orbit. Hence we calculate SN, which comes out $149^{o} 36' 15''$; and since the moon's node is farther from the sun in longitude than this limit, there can be no eclipse; i. within, there may, but without, there will not be any. To ascertain precisely whether these will or not, and, if there be, how great will be the part eclipsed, the solar and lunar diameters must be accurately known at the node and the semidiameters exactly ascertained, and the local parallax, and apparent augmentation of the moon's diameter by refraction, taken into account. But to do this requires the observer and from the centre of the earth (which may amount to a sixth part of her horizontal diameter), deducing her distance from the sun, and other con- siderations, to calculate the amount overlapped of the two discs, and their moment of contact. (243.) The duration of an eclipse of a star depends on similar considerations. An occultation is possible, when the moon's course, as seen from the observer's place, brings her body between the sun equal to the sum of her semidiameter and horizontal parallax; and it will happen at any particular spot, wherewith she passes through a point which carries her centre within a distance equal to the sum of her augmented semidiameter and actual parallax. The de- tails of this calculation are too intricate to be worth while; some must be sought elsewhere. (240.) The phenomenon of a solar eclipse is old and of an antiquity so remote that it is instructive in a physical point of view. They teach us that the moon is an opaque body, terminated by a real and sharply de- tached edge. This fact has been proved by observation; and proves to us, also, that at those times when we cannot see the moon, she really exists, and pursues her course, and has her own light. For if we consider that, in our na- ture, the whole globular body is there, filling up the de- fective outline, though unseen. For occultations take place indifferently at all points in height, the visible and Woodhouse's Astronomy vol. I. Revised Trans. Am Soc loc p. 288 209 **210 A TREATISE ON ASTRONOMY.** (chap. VI. invisible outline, whichever happens to be towards the direction in which the moon is moving; with this only difference, that a star occulted by the bright limb, if the phenomenon continues, will gradually appear again, by its gradual approach to the visible edge, when it ex- pects its disappearance, while it is obscured at the dark limb, if the phenomenon continues, will gradually disappear, and then reappear again, by its gradual approach to the visible edge, when it expects its reappearance. Now, as the moon is always seen in mid-air, without notice or visible cause for its disappearance, which, as it happens, is very frequent; and as the appearance of a previous diminution of its light, is always surprising; and, if the star be a large and bright one, even startling from its sudden disappearance; so that we may say, though the moon has passed over it, take place in those cases when the bright side of the moon is foreseen, not at the commencement of its passage through the invisible outline of the complete circle, and is scarcely less surprising, from its remaining, than its disappearance in the visible part. (260.) The existence of the complete circle of the disc, even when the moon is not full, does not, however, rest only on appearances; but there are many circumstances which may be seen, when the moon is crescent or waning, a few days before and after the new moon, with the naked eye, or a pair of glasses; which cannot be explained away, and somewhat projecting beyond its outline (which is an optical illusion arising from the greater intensity of light) into what appears to be a complete circle. Meanwhile the fact is sufficient to show *There is an optical illusion of a very strange and unaccountable na- ture which occurs in some parts of Europe during certain seasons of the year. The sun being low in the heavens at sunset or sunrise, its advance actually opens and widens the edge of the disc before it disap- pears; and this effect is more remarkable than any other known to me under this singular effect; but it rests on most unequivocal testimony.* I have collected several accounts of this phenomenon; and I may state on such account through some degrees in the estimation of the magnitude of this effect. In some places where it has been watched, it seems that both individuals are thus prevented from seeing the moon at all; but in others they see it perfectly clear until just at sunrise or sunset. It would seem that this effect must be attributed to some peculiarity in our eyes; but I am unable to explain it. It may perhaps be due to doubt by the mode of its disappearance. Many persons who have seen this phenomenon have observed that it was not confined to any particular part of the disc; but extended from one end to another. This circumstance leads me to suppose that of its nature, we may be sure it is a double star, since we cannot see the individuals separately. A diagram showing a crescent moon with two stars near each other. CHAP. VI.] PHASES OF THE MOON. 211 that the moon is not differently luminous like the sun, but that her light is of an increasing nature. And its crescent form, forming regularly from a narrow semicircular line to a complete circular disc, corresponds with the appearance of the moon in the hemisphere of which she is dark, the other white, when dif- ferently turned towards the eye, so as to present a great- er or less portion of her illuminated face. But from this it is, that the moon is such a globe, one half of which is brightened by the rays of some luminous and distant star, and the other half is left in darkness suf- ficiently intense to give it the degree of splendour we see. Now, the sun alone is competent to such an effect. Its diameter being about 300 times greater than that of the moon, it was observed that, when a crescent, the bright edge is towards the sun, and that in proportion as the moon in her monthly course assumes more of this aspect to the sun, the breadth of the crescent increases, and vice versa. (561.) The sun's distance being 23004. radius of the earth, or 86400 miles, it will be seen that it is about 400 times the latter. Lines, therefore, drawn from the sun to every part of the moon's orbit may be regarded as par- allel. Suppose, now, O to be the earth, ABCD, etc. A diagram showing various positions of the moon in its orbit around Earth. various positions of the moon in its orbit, and B the sun, at the vast distance above stated; as it is shown, then, in the figure, the hemisphere of the lunar globe turned to- wards us appears white and bright; while that turned dark, wherever it may stand in its orbit. Now, in the position A, when in conjunction with the sun, the dark part is 212 A TREATISE ON ASTRONOMY. [CHAP. VI.] entirely turned towards O, and the bright from it. In this case, then, the moon is not seen, it is new moon. When the moon comes has C, half the bright and half the dark side is visible, and when it has G, the same in the opposite situation G : these are the first and third quarters of the moon. Lastly, when at E, the whole bright side is visible, we have the full moon. It rises from it, and it is then seen wholly bright or full moon. In the intermediate positions BIFGH, the portions of the bright side which are visible are less than those which are not, but they diminish with increasing distance from H to G, till they vanish altogether, as it comes round again to A. (208.) The moon being illuminated by the sun, or phaeres, as they are called, shines, then, from the moon, an opaque body, being illuminated on one side by the sun, and reflected on the other by its own surface of the light so received. Nor let it be thought surprising that a solid substance thus illuminated should appear to shine and reflect like a white cloud; for a white cloud does standing up upon the clear blue sky. By day, the moon can hardly be distinguished in bright weather from a white cloud; but in cloudy evenings clouds catching the last rays of the sun appear with a dazzling splendour, not inferior to the seeming brightness of the moon itself. This is owing to the fact that such a light to the moon, only probably more powerful by reason of its greater apparent size,* is agreeable to optical principles; while to a cloud it is a very great per- taining of the young moon completing its crescent (at. 850). For, when the moon is nearly new to the earth, the lati- tude (or declination) of its axis is nearly perpendicular to its line of sight from us; and therefore its dark side is almost always hidden from us; and it is a portion of this, reflected back again, which makes it visi- ble to us as it appears to us. But when it is near full moon age, the earth offers it a less portion of its bright side, and the phenomenon in question dies away. (208.) The period of time determined by the recur- rence of its phases ; it reckons from new moon to new *The apparent diameter of the moon is 20' from the earth; that of the earth from the moon two times her horizontal paralax. p. 394. The aperture method therefore is at least 150' (at. 850) or 15° 11'. **CHAP. VI.] SYNDICAL REVOLUTION OF THE MOON.** moon; that is, from leaving its conjunction with the sun to its return to conjunction. If the sun stood still, like a fixed star, the interval between two conjunctions would be the same as the time of revolution of the moon; but as the sun apparently advances in the heavens in the same direction with the moon, only one year, or 365 days, will be required for the perio d to perform to come up with the sun again, and will require for it a longer time, which is the lunar month, as it is called, than the period of its synodic revolution. The difference is easily calculated by considering that the superior arc (whatever it be) is described by the moon in 27 days 7 hours 43 minutes, in the same time that the moon describes that she plus a complete revolution, with her velocity of 13° 17'460 per second ; and since the distance between them is equal, the spaces are to each other in the proportion of the velocities.* From these data a slight knowledge of sidereal time and solar time may be acquired. At any given time of description by the moon; which, being the excess of the synodic over the sidereal period, the former will have had a greater number of revolutions than the latter. (584.) Supposing the position of the nodes of the moon's orbit to permit it, when the moon stands at A at the time of new moon, it will appear on all parts of the whole of sun's rays, and cause a solar eclipse. On the other hand, when at E (or at the full moon), the earth will stand in such a position that it casts a shadow on the moon, thereby causing a lunar eclipse. And this is perfectly convenient to fact, such eclipses never occur at new moon or full moon. But what is still more remarkable, as confirmatory of the position of the earth's sphericity, this shadow, which always appears to be circular, is always so; for every disc of the moon, is always terminated by a circular outline, though, from the greater size of the circle, it is *Let V and W be the mean angular velocities, v and w their respective periods; then V : w = V : w : : V : w : : V : w : : V : w : : V : w : : V : w : : V : w : : V : w : : V : w : : V : w : : V : w : : V : w. The difference of v and w is found in 2 times of revolving at v or w; and this difference of v and w is found in 2 times of revolving at v or w. We shall have occasion for this again. 213 214 A TREATISE ON ASTRONOMY [CHAP. VI.] only partially seen at any one time. Now, a body which always casts a circular shadow must itself be spherical. (350.) Eclipses of the sun are best understood by re- garding them as taking place between two independent luminaries, each moving according to known laws, and viewed from the earth ; but it is also instructive to consider eclipses generally as taking place between two points, each thrown on another by a luminary much larger than either. Sup- pose, then, AB to represent the sun, and CD a spherical body, equally distant from both the sun and the earth. If we join and prolong AC, BD ; since AB is greater than CD, these lines will meet in a point E, more or less distant from the body CD, according to its size, so as to fall within the space CED (which represents a cone, since CD and AB are spheres), there will be a total shadow. This shadow is called the umbra, and a spectator situated within it can see no part of the sun's disc. Beyond the umbra on two sides of it will be found a second cone, single conical space, having K for its vertex, where if a spectator be situated, as at M, he will see a portion only (AONB) of the sun's disc, because this portion being oc- cured by the earth. He will, therefore, receive only partial sunshine; and the more, the nearer he is to the exercise of his sight from the point E, the more com- mon. Beyond this he will see the whole sun, and be in full illumination. All these circumstances may be per- fectly well shown by holding a small globe up in the A diagram showing a spherical body (CD) casting a shadow on a plane surface (AB). The shadow is divided into two parts: the inner part (AONB) is called the umbra, and the outer part (AONB) is called the penumbra.
    M N P
    B D C
    G H I
    A B C
    E F G
    L M N
    O P Q
    R S T
    U V W
    X Y Z
    **CHAP. VI. SYNOPTICAL REVOLUTION OF THE MOON.** sun, and receiving its shadow at different distances on a circle of the earth. (356.) In a lunar eclipse (represented in the upper figure), the moon is seen to enter the umbra first, and by degrees to pass into the penumbra, which surrounds the latter like a haze. Owing to the great size of the earth, the cone of its umbra always projects far beyond the circumference of the circle of the eclipse, the moon's path being properly directed, it is sure to pass through the umbra. This is not, however, the case in an annular eclipse, owing to the adjustment of the size and distance of the moon, that the extremity of her shadow always falls near the earth, but sometimes on one side or other of it, as in the former. In the former case (represented in the lower figure), a black spot, surrounded by a fainter shadow, is formed, just within which all parts of the surface of the earth, but within which there may be either a total or partial one, so that the spectator is within her umbra or penumbra. But in an annular eclipse, when it is on the surface, the moon at that point will appear, for an instant, to just cover the sun : but when it falls short, we shall see only a part of it covered by the earth; but a spectator, situated in or near the prolongation of the axis of the cone, will see the whole of the moon on his way out; and this will be enough to cover it, &c. he will witness an annular eclipse. (357.) Owing to a remarkable enough adjustment of its periodical motion with respect to its position, and that of her nodes, are performed eclipses return after a certain period, very nearly in the same order and of the same magnitude. These periods are called synodical revolutions, or lunations, as they are called, will be found to occupy 6888-22 days, and nineteen complete synodic revolutions will make 130 years and 7 months 27-7r. The difference in the mean position of the node, then at the beginning and end of 223 lunations, is nearly insensible; and therefore these two points will be at least ten intervals must take place. Accordingly this period of 223 lunations, or eighteen years and ten days, is a very important one in the calculation of eclipses. It is supposed 216 A TREATISE ON ASTRONOMY. [CHAP. VI. to have been known to the Chaldeans, under the name of the æquinox; the regular return of eclipses having been known as a physical fact for ages before their exact the- ory was understood. (384.) The phenomenon, duration, and magnitude of a lunar eclipse are much more easily calculated than those of a solar, being independent of the positions of the specific bodies, and depending only on the distance re- moved from its centre. The common centre of the umbra and penumbra lies always in the ecliptic, at a point oppo- site to the sun's place at the time of the eclipse, pass- ing through it its true axis, or as it stands at the moment of the full moon. In this orbit, its position, at every in- stant, is known from the sun's place at that instant; and all we have, therefore, to ascertain is, the moment when the distance between the moon's centre and the centre of the shadow is equal to that between the sun's centre and centre of the moon and penumbra, or of the moon and umbra, to know when it enters upon and leaves them re- spectively. (385.) The dimensions of the shadow, at the place where it crosses the moon's path, require us to know its distance from that place; but this distance is not always variable; but are calculated and set down, as well as their semidiameters, for every day, in the ephemeris, so that none can be wanting. The diameter of the umbra is easily calculated from its elliptic orbit; but the moon's is a matter of more difficulty, for a reason we will now explain. (386.) The moon's orbit, as we have before hinted, is not, strictly speaking, an ellipse returning into itself, by means of its nodes; but it is an ellipse which revolves round the motion of its nodes. But even laying aside this con- sideration, the axis of the ellipsis is itself constantly changing with respect to the sun's place; and though the solar eclipses, but much more rapidly ; making a complete revolution, in the same direction with the moon's own movement round her own axis; but in about nine years, being about 3° of angular motion in a whole revolution of the moon. This is the phenomenon known CHAP. VI.] PHYSICAL CONDITION OF THE MOON. 317 by the name of the revolution of the moon's apogees. Its cause is that the moon's motion is not uniform; its effect is to produce a variation in the moon's distance from the earth, which is not included in the laws of exact ellipse mentioned in the preceding chapter. This variation of distance is trifling ; but in the course of many it becomes considerable, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigees and apogees will be completely reversed, as is easily seen, if we consider the following facts : The perigee 218 A TREATISE ON ASTRONOMY. [CHAP. VI.] of sight, no shadows are seen on any part of her surface. From micrometrical measures of the lengths of the sha- dows of the mountains, it appears that they have been taken under the most favourable circumstances, the height of many of them have been calculated ; the highest being about 3000 feet above the general level of the moon. The existence of such mountains is corroborated by their ap- pearance as small points or islands of light beyond the extreme limits of the lunar disc, and by the long de- tapes catching the sunbeams before the intermediate plane, and which, as the light advances, at length connect themselves with it, and appear as prominences from the general edge. (263.) The generality of the lunar mountains present a striking contrast to the smoothness of the surface, as very wonderfully numerous, occupying by far the larger por- tion of the surfaces, and almost universally of an exactly circular form, though not always perfectly so; some are ellipses towards the limb; but the larger have for the most part that bottom within, from which rises centrally a small, and often a very small mountain top. In its highest perfection, the true volcanic character, as it may be seen in the crater of Vesuvius, and in a map of the vulcanic regions of Sicily, is well shown by Mount Dôme. And in some of the principal ones, decisive marks of volcanic stratification, arising from successive deposits of lava, are visible to those who use powerful telescopes. What is, moreover, extremely singular in the geology of the moon is that although no- thing has been observed to indicate any great activity in dusky spots which are commonly called seas, when closely examined, present appearances incompatible with the supposed action of water. The irregularities on these re- gions perfectly level, and apparently of a decided alluvial character. (264.) The moon has no clouds, nor any other indi- cations of an atmosphere. Were there any, it could not fail to be perceived in the occultations of stars and the phenomena connected with them. Hence we conclude must see Deissel's map of the regions of Nubia and Damascus of Astronomy. I from my own observations—Anchor. **CHAP. VI.] PHYSICAL CONDITION OF THE MOON.** be very extraordinary; the alternation being that of unmitigated and burning sunshine, fiercer than an equatorial noon, continued for a whole fortnight, and the keenerest night, with a temperature of 30° below zero, for an equal time. Such a disposition of things must produce a constant transfer of whatever moisture may exist on the surface to the atmosphere, and vice versa, to that opposite, by distillation in excess after the manner of the little instrument called a cryophorus. The con- sequence will be, that the moon will be always wet, constant accretion of hearth from the opposite region, and, perhaps, a narrow zone of running water at the bottom of the valleys. In this case, if the moon be heated, then, that evaporation on the one hand, and condensation on the other, may to a certain extent preserve an equilibrium of temperature, and mitigate the extreme severity of both climates. (365.) A circle of one second in diameter, as seen from the earth, would contain about 180 square miles, containing about a square mile. Telescopes, therefore, must yet be greatly improved, before we could expect to see signs of inhabitants on the moon. The moon is covered with stones on the surface of the soil. It should, however, be observed, that owing to the small density of the materials of tht moon's crust, and its great weight, any body resting on her surface, muscular force would there go six times as far in overcoming the weight of materials as on the earth. This circumstance is very remarkable; and it is im- possible that any form of life analogous to those on earth can subsist there. No appearance indicating vegetation, or inhabitants of any kind has been observed. It only be ascribed to change of season, as any where he discerned. (366.) The lunar summer and winter arise, in fact, from the same cause which produces the two periods of which rotation is exactly equal to its sidereal revolution about the earth, and is performed in a plane 1° 30' I 1" from that of the ecliptic. This is a most remark- anteur with her own orbit. This is the cause why we al- ways see the same face of the moon, and have no know- ledge of the other side. This remarkable coincidence of two periods, which at first sight would seem perfectly A diagram showing a circle representing a second on Earth's horizon. 220 A TREATISE ON ASTRONOMY. [CHAP. VI. distinct, is said to be a consequence of the general laws to be explained hereafter. (367.) The moon's rotation on her axis is uniform; but since the plane of her orbit (like that of the sun) is not so, we are enabled to look a few degrees round the equatorial part of her visible border, on the eastern or western side of the sun, at any time of the year. In other words, the line joining the centres of the earth and moon divides the angle between their positions, from its mean or average intersection with the horizon, into two equal parts. And, moreover, since the axis about which she revolves is not exactly perpendicular to her orbit, her poles come alternately towards us and away from us, according to her diaz. These phenomena are known by the name of li- brations. In consequence of these two distinct kinds of librations, the centre of gravity of the moon's surface is not always the centre of her disc, and we therefore get sight of a zone of a few degrees in breadth on all sides of the body when it is seen. (368.) If there be inhabitants in the moon, the earth must appear to them as a sphere, or at least as a moon of nearly 5" in diameter, exhibiting the same phases as we see the moon to do, but immovably fixed in their sky (or, at least), changing its apparent place only by the small inequalities of its motion. This would cause it to pass slowly beside and behind it. It will appear clouded with various spots, and sometimes with quadrilateral and irregular zones extending to our terrestrial land, and it may be doubted whether, in their perpetual change, the outlines of our continents and seas can ever be clearly discerned. . CHAP. VII. ON TERRRESTRIAL GRAVITY. 201 CHAPTER VII. Of terrestrial Gravity—Of the Law of universal Gravitation—Paths of Projectile apparatus—resid—the Moon not in motion in its orbit round the Sun—The Sun's motion in its orbit round the Earth—Orbits of the Sun, Moon, and Earth—The Sun's motion compared—Democracy of the Sun's Force of Gravity in the Universe—Deducing Effect of Solar Force on the Moon's Motion. (370.) The tender reader has now been made acquainted with the chief causes of the motions of bodies in their orbit round the sun, and of the moon about the earth. We come next to speak of the physical cause which maintains these motions, and which we may call the massive bodies so revolving to revolve continually from the directions they would naturally seek to follow, in pursuance of their natural tendencies, but to keep their courses into curves towards their centres. (371.) Whatever similitude may have been made by means of analogy between the motions of cause and effect, and fitter it down into the unsatisfactory relation of habitual sequence,—it is certain that the conclusion drawn from this analogy is false. This conclusion is quite as strongly imposed upon the human mind as that of the existence of an external world—the vindication of which was one of the first achievements considered as an achievement of no common merit in the annals of this branch of philosophy. It is our own immediate consciousness which makes us believe that matter is in motion, or to oppose and neutralize force, which gives us an internal conviction of power and causation. It is not so much that we believe in such things as that we are convinced of motion on the other side. Brown, Medical Edn. 38th p. 424.—Author. *See Celsus, Mucinonum, chap. ii.* "In all things there is a work of great acuteness and subtlety of reasoning on some points, but in which whole system it is not found." "It is a most remarkable thing," says Dr. Hume, "that a distant and immense perpetual continuance of action in his [the author's] mind should be maintained without any sensible evidence whatever." "It is a very remarkable thing," says Dr. Brown, "that a man should be convinced of motion on the other side." Brown, Medical Edn. 38th p. 424.—Author. 2 222 A TREATISE ON ASTRONOMY. [CHAP. VII.] from a state of rest, or deflected from their rectilinear paths, and changed in their velocities if already in motion, it is in consequence of such an effort somehow extended, though not necessarily so far as to reach the earth's surface. Such an effort should be exerted with success through an interposed space, so as more difficult to conceive than that over which it is made, but yet not impossible, for which with which it is demonstrably not in contact. (271.) All bodies with which we are acquainted, when raised into the air, are carried upwards by means of the earth's surface in lines perpendicular to it. They are therefore urged thereto by a force or effort, the direct or indirect cause of which is the earth's surface itself; somewhere, though beyond our power to trace, which force we term gravity and whose tendency or direction, whether upward or downward, is always towards the centre; or rather, to speak strictly, with reference to its spherical figure, perpendicular to the surface of still water. But this tendency, though not exhausted by the diminished, is materially modified in the effect of it. The upward impulse is at first great; it is thus after a short distance destroyed, and a downwards one communicated to it, which ultimately brings it back to its surface, so that it remains in further ascent only for a short time. But all the while it has been continually deflected or bent aside from its rectilinear progress by the earth's surface; and having once crossed the earth's centre, and having its highest point, vertex, or apex, just as soon as it has in its orbit, where the direction of its motion is perpendicular to the plane. (272.) When the stone which we fling obliquely up- wards meets and is stopped in its course by the earth's surface, it will have been moved from the place it was inclined to the earth's radius at the same angle as when it quitted our hands. We are sure that if not stopped by the resistance of the earth's surface it would continue on its way, and that obliquely what precession we may ask, is there that we cannot prevent until the earth's centre, to which its velocity is directed, is reached. It is not part of its visible action, we say, to stop us! What reason have we to believe that it might not rather circum- **CHAP. VII.] MOTION OF A PROJECTILE.** Late round it, as the moon does round the earth, returning again to the point it set out from, after completing an elliptic orbit of which the centre occupies the lower focus of the ellipse. The body will then have acquired the same force of gravity (say since we know that it is exerted at all accessible heights above the surface, and even in the heavens) as would be experienced by a body at 60 radii of the earth, or to the moon! and may not this be the power—how some power there must be—which keeps the body in its orbit, and prevents its return to its origin, but only to prevent its leaving its orbit, and hence to keep it in the elliptic path which experi- ence teaches us she actually pursues? (374.) Suppose a stone thrown at the end of a string, it will stretch the string by a *centrifugal force*, which, if the speed of rotation be sufficiently increased, will cause it to fly off with great velocity. However strong the string, it may be, by a sufficient rotatory velocity of the stone, be brought to the utmost tension it will be broken. But if we consider how much weight is it capable of carrying, the velocity necessary for this purpose is easily calculated. Suppose now, a string to be stretched with a certain amount of weight on its surface, whose strength should be just sufficient to sustain that weight suspended from it. Let us, however, for a moment imagine that we have no idea what the weight is made to revolve with the limiting velocity which that string can barely counteract: then will it become evident that any body whatever can be used; and any power which should continually urge the body towards the centre with a force equal to its weight would perfectly suffice to keep it in motion. Divide it into parts of the string, if divided. Divide it then, and in its place let gravity act, and the body will circulate as before; its ten- sion being equal to its weight. This motion will be kept up by its centrifugal force. Knowing the radius of the earth, we can calculate the temporal period in which a body so situated will continue to keep it up: and this appears to be $1^{\circ} 28'25''$. (374.) If we make the same calculation for a body at the distance of the moon, supporting its weight or gra- ... see Ch. VII. Manuscript chap. viii. A diagram showing an elliptical orbit around Earth. 224 A TREATISE ON AERONOMY. [CHAP. VII. vity the same as at the earth's surface, we shall find the period required to be 10° 45' 30". The actual period of the moon's revolution, however, is 27° 27' 45", and hence it is clear that the moon must be capable of being refractive to sustain it against such a power, supposing it to revolve in a circle, or neglecting (for the present) the slight ellip- sity of its orbit. It is evident also that the resistance of the moon (or the moon itself) should be capable of keeping its distance from the earth by the outward effort of its own gravity. Now, if this conclusion should be what the moon actually is, it will appear (on executing the calculation from the principles laid down in Ch. II.) that the force which would be necessary to be as intense as at the surface, would require to be very nearly 3600 times less energetic; or, in other words, that the moon must be capable of producing in it, at the same time, only aught per cent. of the motion which it produces in producing a mass of matter at the earth's surface. (978.) The distance of the moon from the earth's centre is about 238,000 miles, sixty times the distance from the centre to the surface, and 3600 : 1 : 60 : 1, so that the proportion in which we must admit the earth's gravity to act on the moon, in order to keep it up (if he really the force which retains the moon in her orbit, must be (at least in this particular instance) that of the squares of their distances,) is nearly one hundredth part. But such a diminution of energy with increase of distance, there is nothing prima facie indubitable. Emanations from a centre of attraction may really increase in inten- sity by increase of distance, and in this identical pro- portion; and though we cannot certainly argue much from this fact alone, yet it is certain that all mag- netic and electric attractions and repulsions is actually enfeebled by distance, and much more rapidly than in simple proportion to distance. This is evident from the argument, therefore, stands thus—On the one hand, gravity is a real power, of whose agency we have daily experience; but on the other hand, it does not ac- cessible heights, and far beyond; and we see no reason **CHA.P. VII.]** **ATTRACTION OF SPHERES.** for drawing a line at any particular height, and there as- serting that it must cease entirely; though we have ana- logies to lead us to suppose its energy may diminish rapidly as we ascend to greater heights; yet, on the contrary, such a supposition is absurd. On the other hand, we are sure the moon is urged towards the earth by some power which retains her in her orbit; and that the intensity of that power increases with the square of the distance from gravity, in the proportion otherwise not improbable— of the square of the distance. If we suppose that power, there must exist some at one and another level, this gravity must cease at some inferior level, or the nature of the matter which constitutes the earth would be incapable matter—(for if not, it would be urged by both powers, and therefore too much urged, and forced onwards from her.) (276.) It is on such an argument that Newton is un- derstood to have rested, in the first instance, and provi- sionally, his theory of gravitation; but he has not been so thus abstractly stated.—Every particle of matter in the universe attracts every other particle, with a force directly proportional to the product of their masses, and inversely to the square of the distance between them." In this abstract and general form, however, the proposition is false; for it does not hold universally. The earth and moon are not mere particles, but great spherical bodies, and to such the general law does not immediately apply. For example, if we consider what force is necessary to inquire what will be the force with which a congeries of particles, constituting a solid mass of any ne- cessary magnitude, will attract each other, we find that it is a vast stomial atom. This problem is one purely dynamical, and in its general form, is of extreme difficulty. Fortunately, however, it can be solved by a very simple method; and attracted bodies are spheres, it admits of an easy and di- rect solution. Newton himself has shown (Princip. b., p. 30) that if two spheres of equal mass are pro- ciously the same as if the whole matter of each sphere were collected into its centre, and the spheres were single particles placed; so that in this case, the general law applies in strict wording. The effect of 220 A TREATISE ON ASTRONOMY. [CHAP. VII. The trifling deviation of the earth from a spherical form is of too minute an order to need attention at present. It is, however, perceptible, and may be hereafter noticed. (273.) The following proposition is one which divests the law of gravitation of its provisional character, as derived from a loose and superficial consider- ation of the subject. It shows that the mean velocity of the earth is nearly equal to its average or mean velocity, and elevates it to the rank of a general and principal relation, by proving its applica- bility to all cases whatever, and to all kinds of circumstances. This proof consists in demonstrating, as he has done (Princip. i. 17. l. 75.), that, under the in- fluence of gravity, two bodies situated at any distance from each other, whether two spherical gravitating bodies towards each other, they will each, when moving in each other's neighbourhood, be deflected towards each other; but that one body may describe, one about the other regarded as fixed, or both round their common centre of gravity, curves whose forms are similar to those described by the general name of conic sections. It will depend, he shows, in any assigned case, upon the particular circum- stances of the case whether these curves shall be ellipses, or parabolas, or hyperbolae; but one or other it must be. In every case, therefore, gravity will may be, according to the circumstances of the case; and, in all cases, the point to which the motion is referred, whether it be a sphere or a plane, will be its common centre of gravity, will of necessity be the focus of the conic section described. He shows, furthermore (Princip. i. 18.), that the line joining two points with which the line joining their centres moves, must be inversely proportional to the square of their mutual dis- tance, and that this inverse proportionality will be swept over by their line of junction in equal times. (274.) All this is in conformity with what we have stated of the nature of gravity itself. We refer for these fundamental propositions, on a point of duty, to the memoirs of Sir Isaac Newton on Gravity; but I am not aware that any thing suffi- ciently clear and complete has been published respecting them. I think it would be very useful for us in this volume to go into these investigations; even did not our little period in this work permit us to do so; but I hope that some account of a general idea, however, of their conduct will be given in the next chapter. We refer for these fundamental propositions, on a point of duty, to the memoirs of Sir Isaac Newton on Gravity; but I am not aware that any thing suffi- ciently clear and complete has been published respecting them. I think it would be very useful for us in this volume to go into these investigations; even did not our little period in this work permit us to do so; but I hope that some account of a general idea, however, of their conduct will be given in the next chapter. **CHAP. VII.** **MASS OF THE SUN.** 227 are ellipses, but of different degrees of eccentricity; and this circumstance alone renders the general applicability of the principles in question. (378.) But here we have already, by a natural and reciprocal consideration, taken a step towards its generalization, taken a further and most important step, almost superlative. We have extended the action of gravity to all bodies whatsoever, and found its influence immensely greater than that of the moon, and to a body apparently quite of a different nature from either. Are we now to suppose that the same laws of modifications introduced by the change of data, if not into the general expression, at least into the particular interpretation, will hold good when we come to numbers, however incongruity strikes us. When we calculate, as above, from the known distance of the sun from the earth, what force acts on the earth circles about it (art. 227), what must be the centripetal force of the latter by which the sun's attraction is counteracted? The answer is given by the exact measure of the sun's striking energy as exerted on the earth; we find it to be immensely greater than would naturally follow from any other law whatever. This equal body at that distance—greater in the high proportion of $354926$ to 1. It is clear, then, that if the earth be regarded as a sphere of uniform density, conformable in its rate of diminution with the general law, this force must be no less than $354926$ times more intense than that which is capable of exerting, *ceteris paribus*, at an equal distance. (380.) What, then, are we to understand from this result? We see that the sun contains an incalculable mass of $354926$ earths occupying its place would do, or, in other words, that the sun contains $354926$ times the mass of all the earths which compose our own con- stellations. Nor let this conclusion strike us. We have only to recall what has been already shown in art. 308., of the great difference between a single atom and a molecule; so that it is easy to perceive that, in assigning to it so vast a mass, we are not outstripping a reasonable proportion. In fact, when we come to compare its mass with its bulk, we find its 228 A TREATISE ON ASTRONOMY. [CHAP. VII. density* to be less than that of the earth, being no more than 0'3648. So that it must consist, in reality, of far lighter materials, especially when we consider the force under which its central parts must be condensed. This consideration would render it highly improbable that intense heat prevails in its interior, by which its elasticity is re- inforced, and rendered capable of resisting the immense inconceivable pressure of the sun's rays acting on smaller dimensions. (381.) This will be more distinctly appreciated, if we estimate how we are now prepared to do, the intensity of gravity at the sun's surface. The atmosphere of the sphere being the same (art. 379) as if the whole mass were collected in its centre; will, of course, be proportioned to the mass directly, and the square of the radius inversely; and, in this case, the distance is the radius of the sphere. Hence we can con- clude, that the intensity of solar and terrestrial gravity at the surface of the two globes is in the proportion of 97'9 to 1. A person standing upon the surface of the sun's surface, then, would exert a pressure equal to what 27'9 such pounds would do at the earth's. An ordinary man, for example, weighing 150 lbs., would be crushed under his own weight on the sun, but would literally be crushed to nothing under the earth. (382.) In short, therefore, we must commit to dis- miss all ideas of the earth's immobility, and transfer that attribute to the sun, whose ponderous mass is calculated to exhaust all other planets in their orbits. The atoms of atoms as the earth and moon, without being perceptibly dragged into any one direction, are carried round with us. We have already hinted, almost close to the centre of the solar globe, at an interval quite imperceptible from our distance; and whether we regard the earth's axis as being perfectly perpendicular to its orbit, or as being inclined. *The density of a material body is its mass directly, and the volume inversely; hence density = $\frac{\text{mass}}{\text{volume}}$.** (383.) Solar gravity: terrestrial gravity = 97'9 : 1; the respective masses of the sun and earth are nearly 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 1 A mass weighing 15 stone or 175 lbs. on the earth, would produce a pressure of 65 lbs. on the sun. **CHAP. VII.] DISTURBANCE OF THE MOON'S ORBIT.** no appreciable difference in any one phenomenon of attraction. (383.) It is in consequence of the mutual gravitation of all the several parts of matter, which the Newtonian law of gravitation asserts, that the moon does not revolve, monthly, in their mutual orbits about their common centre of gravity, yet continues to circulate, with a uniform velocity, round the earth, attracted by the sun. We may conceive this motion by connecting two unequal bales by a stick, which, at their centre of gravity, revolves uniformly round a point distant from them. Their joint system will circulate as one body about the common centre to which the string is attached, while yet each bale will continue to describe its own ellip- soidal gyration, as if the stick were quite free from any such tie, and merely hurled through the air. If the earth should be dragged away, and leave the moon behind—and vice versa—but, acting on both, they continue together under its influence—then the moon's orbit will remain in its course, though continued to rest upon it. It is then, in strictness, not the earth or the moon which describes an ellipse-- around the common centre of gravity of both. The effect is to produce a small, but very perceptible, monthly equation in the sun's apparent motion as seen from the earth; and it is always taken into account in calculating the sun's place. (384.) And here, i.e., in the attraction of the sun, we have another cause of disturbance of the exact elliptic movement of the moon in her monthly orbit, which we have already noticed (art. 364. 360), viz., to account for the small deviations from the recti- culation of the axis of her ellipse; and to all the other deviations from the laws of elliptic motion at which we have hitherto adverted. For although we know that about the earth under the influence of its gravity, none of these phenomena would take place. Its orbit would be per- fectly regular; and yet it is found that it is not so; and the same cause disturbs it and interferes with the earth's attraction; and this cause is no other than the sun's al- **220** A TREATISE ON ASTRONOMY. [CHAP. VII.] traction—-or rather, that part of it which is not equally exercised on the earth. (285.) Suppose two stones, side by side, or otherwise situated, to move towards each other, to be let fall together; then, as gravity accelerates them equally, they will re- tain their relative positions, and fall together as if they formed one body. But if they were placed at some more intensely exerted on one than the other; then would that one be rather more accelerated in its fall, and would gravitate less upon the other. Hence, the mutual motion between them would arise from the difference of action, however slight. (286.) The moon, about 400 times more remote than the moon; and, in consequence, while the moon de- scribes her monthly orbit round the earth, her distance from the earth is nearly 300 times greater than that of much less than the earth's. Small as this is, it is yet sufficient to produce a perceptible excess of attractive tendency of the moon towards the sun, above that of the A small circular diagram with a central dot and radiating lines. earth when in the nearer part of her orbit, $M_1$, and a corresponding defect on the opposite part, $N_1$; and, in the intermediate parts of her orbit, both these effects of forces subsist, but a difference of directions also; since, however small the lunar orbit $MN_1$ is not a point, and therefore, though all points of it are equidistant from both parts cannot be regarded as strictly parallel. If, as we have already seen, the force of the sun were equally ex- erted upon all parts of the moon's orbit, so that all points of their relative situations would take place; but from the non-verification of these conditions arises a disturb- ing force, which tends to retard the moon's motion in such which in some situations acts to accelerate, in others to retard, her elliptic orbital motion; in some to draw the earth from under her feet; and in others to push her away. Again, the lunar orbit, though very nearly, is yet not quite coincident with the plane of the ecliptic; and hence she acts on the earth, which is very nearly parallel to the last mentioned plane, unable to draw her somewhat out **CHAPTER VIII.** SOLAR SYSTEM. 231 of the place of her orbit, and does actually do so—pro- during the revolution of her nodes, and other phenomena less striking. We are not yet prepared to go into the subject of these perturbations, as they are called; but the reader will perceive that, however far we may go, as possible, for the purpose of reassuring his mind, should doings be shown as to the logical consistency of this argument, in order to remove any hypothetical necessity of them while working our way upward to the law of gravity from a general consideration of the moon's orbit. **CHAPTER VIII.** OF THE SOLAR SYSTEM. Argument Motion of the Planets—Their Nature and Emanations— The Sun does not contain Centre of Rotation of Planets—Their Furnishing of Light and Heat—Their Influence on the Earth— across the Sun-Superior Planets—Their Disances, Periods, etc. Relative to the Earth—Their Motions—Their Apparent Orbits— Orbits heliocentric and geocentric—Place Solar Law of planetary Motion—Law of Mechanical Planes—Physical Phenomena ob- servable in each of the Planets. (387.) The sun and moon are not the only celestial objects which appear to have a motion independent of the earth. There are several others which are regularly carried round the earth. Among the stars there are sever- al—and those among the brightest and most conspicu- ous—are seen to move with great regularity. At night, are found to change their relative situations among one another, and to pass through a series of positions which are called planets. Four of them—Venus, Mars, Ju- piter, and Saturn—are remarkably large and brilliant; Mercury is also visible to the naked eye as a large body, for a reason which I shall presently give; Jupiter appears, is seldom conspicuous ; fifth, Uranus, is barely dis- cernible without a telescope; and finally Neptune. These last two are not visible to the naked eye. Besides these ten, there are other undiscovered may exist; and it is extremely probable that such is the case. —the multitude of telescopic stars being so great that A diagram showing the orbits of various celestial bodies around the Sun. 323 A TREATISE ON ASTRONOMY. [CAP. VIII.] only a small fraction of their number has been sufficiently noticed to warrant any general statement respecting them or not, and the five last-mentioned planets all having been discovered within half a century from the present time. (308.) The appearance of these bodies is much more irregular than those of the sun or moon. Generally speaking, and comparing their places at distinct times, they all move with a uniform velocity, and in the same direction as those lumina- tions, i. e., in opposition to the apparent diurnal motion, or from west to east ; but they do not follow exactly the paths of the heavens, though under very different circumstances ; and all of them, with the exception of the four telescopic planets, which are called "ultra-solar,"--that is, they may therefore be termed ultra-solar.--are confined in their visible paths within very narrow limits on either side the ecliptic plane. This circumstance distinguishes them from that zone of the heavens we have called above the Zodi- ac (art. 254). (309.) The obvious conclusion from this is, that whatever be, otherwise, the nature and law of their mo- tions, they are all performed nearly in the plane of the ecliptic ; and that no other explanation of their motions about the sun is performed. Hence it follows, that so long as they see their evolutions, not in plan, but in section ; their true aspect will always be seen by us as if they were all foreseen and confounded undistinguishably, while only their divisions from the ecliptic appear of their own magnitude, undiminished by the effect of per- spective. (310.) The apparent motions of the sun and moon, though similar in respect to their distances from uniformity ; a moderate acceleration and retardation, accountable for by the elasticity of their orbits, being all that is requisite to account for their motions about the planets : sometimes they advance rapidly; then re- lax in their apparent speed; come to a momentary stop; and then resume their former course with a rapidity at first increasing, then diminishing, till the reversed or retrograde motion ceases altogether. Another station, or moment of ap- CHAP. VIII.] NOTES OF A PLANET'S ORBIT. 233 perpet rest or indolence, now takes place: after which the movement is again reversed, and resumes its original direct character. On the whole, however, the amount of this change is very small, compared with that of the circle; and by the excess of the former over the latter, the gra- dual advance of the planet from west to east is main- tained. In order to represent this motion on a plane surface (or represented as in Mercator's projection, art. 294, taking the ecliptic ECP for its ground line), the track of a planet, when mapped down by observation
    P E R C Q
    from day to day, will offer the appearance FQRS, &c.;
    the motion from P to Q being direct, at Q stationary,
    from Q to R retrograde, at R again stationary, from R
    to S direct.
    (301.) In the midst of the irregularity and fluctuation of this motion, one remarkable feature of uniformity is observed. The planet always remains in one place, as at N in the figure, it is said (like the moon) to be in its node; and as the earth necessarily lies in the plane of the ecliptic, so does the planet. This is represented graphically situated in the celestial circle so called, with- out being really and locally situated at that plane. The variation of latitude of a planet is then a phenomenon indicative of a circumstance in its real mo- tion quite independent of the station from which we view it. For instance, if we suppose that a planet passes from the north to the south side of the ecliptic; we have only to convert its right ascensions and declinations into latitudes; and thus find out the change from north to south latitude on two successive days will advertise us on what day the transition took place. Now let us suppose that we observe at any ob- served state of its motion in latitude the interval, will suffice to fix the precise hour and minute of its ar- rival on the ecliptic; and therefore we shall find every transmutation from side to side of the eclipse, as destr u 2 234 A TREATISE ON ASTRONOMY. [CHAP. VII.] dates thereby fixed, we find, universally, that the interval of time elapsing between the successive passages of each planet through the same node (whereby it is the ascendant or descendant of the same sign) is either equal to the period at the moment of such passage be direct or retrograde, swift or slow, in its apparent movement. (285.) It is evident, therefore, from this circumstance which, while it shows that the motions of the planets are in fact subject to certain laws and fixed periods, may lead us very naturally to suppose that the causes and circumstances and complexities of their movements may be owing to our not seeing them from their natural centre (art. 316), and from the nature of the earth itself, and to other similar movements of a parallelistic kind, due to our own change of place, in virtue of the orbital motion of the earth about the sun. (286.) If we abandon the earth as a centre of the planetary motions, it cannot admit of a moment's hesitation whereupon we should adopt any other point as its proper centre. It must surely be the sun which is entitled to the first trial, as a station to which refer all things; for though it be true that in this varying relation, it at least possesses the advantage, which the earth does not, of comparative immobility. But after what has been said respecting the earth as a centre of luminosity, and of the office it performs to us as a quiescent centre of our orbital motion, nothing can be more natural than to suppose that we have no right to assign to other globes which, like the earth, may be revolving round it; and these globes may be visible to us by its light reflected upon them. And yet there are many facts which give a strong support to the idea that the planets are in this predicament. (287.) The moon and Jupiter really are great globes, of a size commensurable with the earth, and severable of them much greater. When examined through powerful telescopes they appear as distinct and sensible and even considerable apparent diameter, and offering distinct and characteristic peculiarities, which show that they are not mere points or indivi- dual structures or mechanisms; and that in one inc... CHAP. VII.] APPARENT DIAMETERS OF THE PLANETS. 235 sterner at least, an exceedingly artificial and complex one. (See the representations of Jupiter, Saturn, and Mars, in plate L.) That their distances from us are great, much greater than that of the moon, and that their diameters are thus much less than ours, we infer from the smallness of their diurnal parallax, which even for the most remote planets does not exceed a few seconds, and for the more remote ones is almost imperceptible. From the comparison of the diurnal parallax of a celestial body with its apparent magnitude, we can at once estimate its real size. For the parallax is, in fact, nothing else than the apparent semimagnitude of the earth's shadow (which only occurs at the 390, 391 ; and), and the intervening distance being the same, the real diameters must be to each other in the proportion of their apparent magnitudes. Without going further, it will suffice to state it as a fact. As re- sult of that comparison, that the planets are all of them larger than our moon, some of them as large as the earth, and others much greater. (390.) The next fact respecting them is, that their diameters vary considerably with respect to the direction of their angular diameters, are in a continual state of change, periodically increasing and decreasing within certain limits. This is shown by the frequent supposition of regular circular or elliptic orbits described by them about the earth as a centre or focus, but main- taining always nearly the same inclination to the apparent angular distances or elongations from the sun. For ex- ample; the apparent diameter of Mars is greater when it is near perihelion than when it is farthest from it; in the opposite part of the ecliptic, or when it comes on the meridian at midnight, being then about 18°—but decreasing to 16° when it is on the opposite side—which is its apparent diameter when in conjunction, or when seen in nearly the same direction as that luminary. This, as well as many other facts which I shall hereafter refer to the apparent diameters of the other planets, clearly point out the sun as having more than an accidental relation to their movements. (390.) Lastly, certain of the planets, when viewed 336 A TREATISE ON ASTRONOMY. [CHAP. VII.] through telescopes, exhibit the appearance of phases like those of the moon. This proves that they are opaque bodies, shining only by reflected light; which can be no other than the light of the sun. But there is no other source of light external to them sufficiently powerful, but because the appearance and succession of the phases vary with the seasons, and are intimately connected with their elongations from the sun, as will presently be shown. (98.) It is evident, first, that, when we refer the planetary movements to the sun as a centre, all that apparent irregularity which they offer when viewed from the earth must be attributed to some simple and general law, of which the earth's motions, as explained in a former chapter, is only a particular case. In order to understand this law, let us consider the case of a single planet, which we will suppose to revolve round the sun, in a plane nearly, but not quite, coincident with our own. The motion of such a body, and of course intercepting the ecliptic in a fixed line, which is the line of the planet's nodes. This line must be always in one plane with respect to the sun; and it is evident that, so long as the circumstances of the planet's motion remain otherwise unchanged, the times of de- scending and ascending at each node will be equal. The interval, then, between the planets quitting either node, and returning to the same node again, must be that in which it takes place for a body to describe an angle equal to its periodical time; and thus we are furnished with a direct method of ascertainng the periodic time of each planet. (98.) We have said (art. 368) that the planets make the entire tour of the heavens under very different circum- stances; and that Mercury and Venus--perform this circuit evidently as attendants upon the sun, from whose vicinity they never depart beyond a certain limit. They appear to rise towards the east, sometimes to the west of it. In the former case they appear conspicuous over the western horizon, just after sunset, and are called evening stars; Venus especially appears occasionally in this situation with a CRAD. VIL.] MOTIONS OF THE INFERIOR PLANETS. 237 dazzling lustre; and in favourable circumstances may be observed to assume a peculiar shape.* When they happen to be to the west of the sun, they rise be- fore that luminary in the morning, and appear over the eastern horizon, and set after it in the evening. How- ever, attain the same elongation from the sun. Mer- cury never attains a greater angular distance from it than about 20°, and Venus never more than on either side to about 47°. When they have receded from the sun, eastward, to their respective distances, they re- turn to it again, and are carried along with it in the ecliptic with a motion equal to its own; but presently they begin to ap- proach it again, and at last reach the same angular division in longitude diminishes, and the sun gains upon them. As this approach goes on, their continuance above the horizon increases, until they are so high in the sky, that they set before the darkness has become sufficient to allow of their being seen for a time, then they are obscured by the moon, which very rarely happens. They may be to observe passing across the sun's disc as small round, well-defined black spots, totally different from each other in appearance. This phenomenon is called transit of the re- spective planets across the sun, and take place when they are in the same part of their orbit as we see below while they are in that part of their orbit, just as in the account we have given (art. 355) of a solar eclipse. After having passed through this passage of the sun, they begin to appear on the other side of the sun, at first showing themselves only for a short time, and then to rise higher and higher and longer as they recede from him. At this time their motion in longitude is rapidly retrograde. Before they attain their greatest elongation, however, they do not return to their places of rest until their recess from the sun is still maintained by the ad- vance of that luminary alone; till at length they reverse their motions to leave behind them a white spot which they left when seen only at the shadow, but the diffracted rings edging its outline— *Author 238 A TREATISE ON ASTRONOMY. [CHAP. VIII. and become again direct, they acquire sufficient speed to overcome oscillating himing--which moment to have their greatest eccentric elongation; and thus is a kind of oscillatory movement kept up, while the general advance along the ecliptic goes on. A diagram showing the positions of the Sun (S), Earth (E), Moon (M), and Mercury (M') in relation to the ecliptic (E). The Sun is shown at the top left, the Earth at the bottom right, and the Moon and Mercury at various points along the ecliptic. (300.) Subpose PQ to lie in the ecliptic, and ABD the orbit of one of these planets (for instance, Mercury), seen almost edgewise by an eye situated very nearly in its plane; S, the sun; its centre; and A, B, D, S sur- cessive points of the orbit. Now, if the planet be near the nodes. If, then, the sun S stood apparently still in the ecliptic, the planet would simply appear to oscillate back and forth between two points on that line. But if it move forward before and behind the sun S, and if the eye happened to lie exactly in the plane of the orbit, transiting his disc as he does so, he would see a planet always moving in a circle about S. But as the sun is not so stationary, but apparently car- ried along the ecliptic PQ, let it be supposed to move over the same part of its orbit as before. Then this case executes one quarter of its period. Then will its orbit be apparently carried along with the sun, into this excessiveness of motion. And when it has gone round once more, while its real motion round the sun brings it into the re- spective points B, D, S, A, its apparent movement in the heavens will be represented by a figure like this zig- zag line ANHK. In this, its motion in longitude will have been direct in the parts AN, NH, and retrograde in the parts NK, HK. At some times of the year, at H, K, it will have been stationary. (400.) The only two planets--Mercury and Venus-- whose eccentricity is great enough to make them visible inferior planets--their points of farthest recess from the sun are called (as above) their greatest eastern and west- ern elongations; and their points of nearest approach to it their inferior and superior conjunctions; the former **CHAP. VIII.] ELLIPTIC MOTIONS OF INFERIOR PLANETS.** when the planet passes between the earth and the sun, its latitude will be equal to that of the sun. (401.) In art. 398 we have traced the apparent path of an inferior planet, by considering its orbit in section, and its motion in the plane of the ecliptic. Let us now contemplate it in plan, or as viewed from a station above that plane, and projected on it. Suppose then, $S$ to represent the sun, $M$ the moon, Mer- cury, and AECD a part of that of the earth—the direc- tion of the circulation being the same as before—let $A$ be the arrow. When the planet stands at $a$, let the earth be situated at $b$, and let $C$ be the sun's point, $A$, to its orbit; then it is evi- dent that it will appear at its position $d$, when it has passed the sun; the angle $aAS$, which measures their apparent interval of time, will be greater than any other situation of a upon its own circle. (402.) Now, this angle is known by observation, we may therefore determine its true position by re- taining, at least approximately, the distance of the planet from the sun, or the radius of its orbit, supposed a cir- cle. The mean distance of Mercury from the sun is consequently two Sa: : : one Sa: : : one radius, by which proportion the radii Sa, Sa of the two orbits are directly compared. It is evident that this method would be of course a perfectly rigorous mode of pro- ceeding; but (as is proved by the inequality of the re- sults obtained) it is not always practicable; for it is not the case; and it becomes necessary to admit an ec- centricity of position, and a deviation from the exact cir- cular form of the orbit, both of which are observed. Neglecting, however, at present this inequality, a mean or average value of Sa may, at least, be obtained from the observations; and this will give some idea of the situation of the two bodies. The calculations being per- formed, it is concluded that the mean distance of Mer- cury from the sun is about 600000 miles; and that of A diagram showing a celestial body (Mercury) moving around a central body (the Sun), with arrows indicating direction and position. 399 **340** A TREATISE ON ASTRONOMY. [CHAP. VII.] Venus, similarly derived, about 68000000; the radius of the earth's being 6000000. (403.) The sidereal periods of the planets may be ob- tained (as before observed), with a considerable approach to accuracy, by observing their elongations from the sun, at the nodes of their orbit; and, indeed, when a certain very minute motion of these nodes (similar to that of the moon's) occurs, they may be determined with great exactness, with a precision only limited by the imperfection of the appropriate observations. By such observations, so cer- cetual, as those made by Halley, we find that $$87^{\circ}23^{\prime}15^{\circ}49'$$ and that of Venus, $$22^{\circ}16^{\prime}49'$$ 9°. These periods, however, are widely different from the intervals between the successive passages of two planets at their eastern and western elongations from the sun are observed to happen. Mercury is seen at its greatest elonation on the side of the sun opposite to that of about 116, and Venus at intervals of about 884 days. The difference between the sidereal and synodical re- volutions of the two planets is due to their respective angles against the figure of art. 401., if the earth stood still at A, while the planet advanced in its orbit, the lapse of a si- dereal period would cause it to pass through an angle $$\alpha$$ equal to $$\frac{360}{884}$$ degrees; but it would also reproduce a similar elongation from the sun. But, meanwhile, the earth has advanced in its orbit the same amount; and therefore, while the planet will not reach an elongation on the same side of the sun will hap- pen—not in the position $a$ of the two bodies, but in some more remote position $b$. Hence, if we consider this position depending on a calculation exactly similar to what has been explained in the article referred to; and we neglect all other causes which might disturb synodical revolutions of the two planets, which come out respectively 115-977°; and 583-929°. (404.) But it must be remembered that we have described a whole revolution plus the arc $a\alpha$, and the earth only is at $A$ of its orbit. During its lapse, the inferior conjunction will occur at $B$, and then at $C$, an intermediate situation. B and the planet has reached $\alpha$, a point between the sun and earth. The greatest elonga- tion on the opposite side of the sun will happen when CHAP. VIII. SYNODICAL REVOLUTIONS. 341 the earth has come to C, and the planet to c, where the line of conjunction Cc is at a point within the interior circle on the opposite side from M. Lastly, the superior con- junction will happen when the earth arrives at D, and the planet at d, where the line of conjunction Dd is at the other side of the sun. The intervals at which these phenomena happen may easily be computed from a knowledge of the mean motions of the planets. (405.) The circumferences of circles are in the propor- tion of their radii. If, then, we calculate the circumfe- rence of any circle by multiplying its radius by 2π, we shall find, and compare them with the times in which their revolu- tions are performed, we shall find that the actual veloc- ities of the planets are very nearly in proportion to their circumferences. Thus, for instance, the velocity of Venus is about 10800 miles per hour, of Mercury being about 108400 miles per hour, of Venus 80000, and of the earth 80900. From this it fol- lows that when the earth is moving in one direction, another planet is moving in the same direction as the earth, but with a greater velocity ; it will, therefore, leave the earth behind. When it is moving in the contrary direction from that of the earth, it will be behind. Hence, if two bodies moved from the earth, will be as if the planet stood still, and the earth moved in a contrary direction from what it really does move, so that they both move in a contrary motion of the planet must be contrary to the apparent motion of the sun; and therefore, retrograde. On the other hand, if two bodies move from each other in a contrary direction to that of the earth, the relative motion will be the same as if the planet stood still; and therefore, direct. In this case also, if bodies move in their own proper direction. In this situation, then, the apparent motion will be direct. Both these results are in accordance with observation. (406.) The stationary points may be determined by the following consideration. At A or C, the points of greatest velocity are those where they are nearest either to or from the earth, or along their line of junction, while all of that is earth is nearly perpendicular to it. Here, then, there is no change in position; hence, at such an inferior conjunction, we have seen that it must be retro- grade, owing to the planet's motion (which is there, as well as the earth's perpendicular to the line of junction) 342 A TREATISE ON ASTRONOMY. [CHAP. VIII. surpassing the earth's. Hence, the stationary points ought to lie, in which it is found by observation they do, be- tween α and δ, or c and δ, viz. in such a position that the obliquity of the ecliptic shall be equal to the angle between the line of junction shall just compensate for the excess of its velocity, and cause an equal advance of each extreme- mity of the circle, so that at any time one extremity will be on and of the earth as the other : so that, for an instant of time, the whole line shall move parallel to itself. The question then reduces to this, whether the true solution on the supposition of circular orbits is easy ; but when we regard them as otherwise than circles (which they are not) we find their difficulty multiplied—too much so to be here entered upon. It will suffice to state the results which experience verifies, and which amongst others are contained in the following table : 18° of elongation from the sun, according to circum- stances; and of Venus, at an elongation never varying much from 40°; and of Mercury, about 15° to retrograde during about 22 days; the latter about 42. (407.) We have said that some of the planets exhibit phases similar to those of the moon; namely, Mer- cury and Venus ; and is readily explained by a consi- deration of their orbits, such as we have above supposed them. In fact, it requires little more than mere inspec- tion of the figure annexed, to show, that to a spectator situated on the earth E, an interior planet, illuminated by the sun, and therefore bright on the side next to him, and dark on the opposite side, appears to him as an exterior conjunction. A globe (i.e. more than half A diagram showing a celestial sphere with three planets labeled A, B, and C. Planet A is shown near Earth E, while planets B and C are further away. A B C D E CRAF. VII.] TRANSITUS OF VENUS. 243 full, like the moon between the first and second quarter) between that point and the line BC of its greatest elongation; half-moon at these points; and crescent-shaped, or horned, between these and the inferior conjunction. The transitus of Venus is therefore a convenient ought to thin off ill it vanishes altogether, rendering the planet invisible, unless in those cases where it transits the sun's disc, which is very rare. All these phenomena are exactly conformable to observation; and, what is not a little satisfactory, they were predicted as early as 1670 by the Copernican theory before the invention of the telescope. (408.) This variation in brightness of Venus in differ- ent parts of her orbit is called "the parallax." This arises from two causes: 1st, the varying proportion of its visible illuminated area to its whole disc; and, 2dly, the difference in the apparent magni- tude of the disc itself. As it approaches its inferior con- junction from its greater elongation, the half-moon becomes more and more crescent-shaped, until it is then compensated, for some time, by the increasing apparent magnitude, in consequence of its diminishing distance. Thus, when Venus is at inferior conjunction, at till length it attains a maximum, which takes place when the planet's elongation is about 40°. (409.) The transitus of Venus is a very rare occur- rence, taking place alternately at intervals of 8 and 113 years, or thenceforth. As astronomical phenomena, they are not only interesting but useful; for they afford the best and most exact means we possess of ascertainling the sun's distance, or its parallax. Without going into the details of their calculation, I shall only allude to the great multitude of circumstances to be attended to, as are extremely intricate, we shall here explain its prin- ciples with reference to the transitus of Venus. Let E be the earth, V Venus, and S the sun, and CD the portion of Venus's relative orbit which she describes while passing through the plane of the ecliptic. Suppose AB two stars at opposite extremities of that dis- 5 See Essay on the Study of Natural Philosophy. Cah Cyclo- Vol III p. 268 344 TREATISE ON ASTRONOMY. [CHAP. VIII. metres of the earth which is perpendicular to the ecliptic, and, to avoid complicating the case, let us lay out of A diagram showing a celestial sphere with a line AB representing the Earth's axis, and a line CD representing the ecliptic plane. Points S and B are marked on the Earth's surface, and point D is marked on the ecliptic plane. consideration the earth's rotation, and suppose A, B, to remain that situation during the whole time of the transit. Then, at any moment when the spectator at A sees the centre of Venus pass through the line AB, he will see it projected at B. If then one or other spectator could suddenly transport himself from A to B, he would see Venus at B, and not at A. But this cannot be; for if he had any means of noting accurately the places of the points on the date, either by micrometric measure- ment or by some other method, he would find that he main- tain the angular measure of a B as seen from the earth. Now, since A V e. B are straight lines, and therefore make equal angles with each other, so also do A to AB as the distance between Venus from the sun to its distance from the earth, or as 68 to 27, or nearly as 24 to 1 : 2 ; but therefore, occuring at equal times, they must have made equal angles with the earth's diameter, and its angular measure is therefore equal to about 24 times the earth's apparent diameter at the distance of Venus from the sun (art. 290). And therefore five times this sun's horizontal parallax (art. 290). Any error, therefore, which may be committed in measuring a B, will be compensated by that error on the hori- zontal parallax concluded from it. (410.) The thing to be ascertained, therefore, is, in fact, nothing else than that angle which is contained between PQBS, p g r s , included between the extreme apparent paths of the centre of Venus across the sun's disc, from its entry on one side to its exit on another. The wheel. This being done, the observers at A, B, therefore re- solves itself into this -- i.e., ascertain, with all possible care and precision, each at his own station, this path-- where it enters, where it quits, and what segment of the CHAP. VIII.] TRANSIT OF VENUS. 248 ven's disc is cut off. Now, one of the most exact ways in which (conjointly with the micrometric measure) this can be done, is by noting the time occupied in the whole transit : for the relative angular motion of Venus being very nearly that of a circle, the apparent motion of her motion, and the apparent path being very nearly a straight line, these times give a measure (on a very enlarged scale) of the angle through which the planet has run out ; and the sun's diameter being known also with great precision, theirversed sines, and therefore their differences, will give the true angles of the transit, which are known. To obtain these times correctly, each observer must ascertain the instant of ingress and egress of the centre of Venus at the edge of the sun's disc when the first visible impression or notch on the edge of the disc at P is produced, or the first external contact ; fully, by observing with his telescope, he must see the broken edge of the disc just close again at Q, or the first internal contact ; and lastly, he must make the same observations at R and S respectively. The sum of the internal and external contacts gives the entry and egress of the planet's centre. (4.) The observations introduced into this process by the earth's rotation on its axis, and by other geographi- cal stations of the observers thence than here sup- plied with information respecting the sun's distance into the calculation of a solar eclipse, or the occultation of a star by the moon, only more refined. Any considera- tion of this subject would be out of place here, but in view we have taken of the subject, it affords an admirable example of the way in which minute elements are brought to light by systematic observation and, by being made subject to measurement on a greatly en- larged scale, or by substituting the measure of time for some other element. This is a method of observation adequate to every purpose, by only watching favourable opportunities, and taking advantage of nicely adjusted equipment. In 1769, when this transit appeared to astronomers, that at the last transit of Venus, in 1769, expeditions were fitted out, on the most efficient scale; by British, French, Russian, 2 346 A TREATISE ON ASTRONOMY. [CHAP. VII.] and other governments, to the remotest corners of the globe, for the express purpose of performing it. The celebrated expedition of Captain Cook to Outhouse was one of those which have been made with the most success, and made on this most memorable occasion gives 42° 57'7" for the sun's horizontal parallax. (412.) In the case of a very elliptic, the eccentricity being nearly one fourth of the mean distance. This appears from the inequality of the greatest elongations from the sun, which vary between the limits 18° 12' and 38° 48', and, from exact measures of such elongations, it is not difficult to show that the eccentricity is sufficiently considerable, so that both these planets, in fact, are represented by ellipses, having the sun in their common focus. (413.) Let us now consider Jupiter, Saturn, or those whose orbits enclose on all sides that of the earth. That they do so is proved by several circumstances: - For, when they are at perihelion, they reach certain limits of elongation from the sun, but appear at all distances from it, even in the opposite quarter of the heavens, or, as it is called, in opposition; which could not happen if they were not in a circle round themselves between them and the sun: They never appear horned, like Venus or Mercury, nor even semisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemisemssemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsemsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmmsmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmsssmssmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm They are also seen in opposition to each other, and see them always in a direction not very remote from that in which the sun's rays illuminate them. This is because they are moved from the centre of their orbits, or, in other words, that the earth's orbit is entirely enclosed within theirs; and consequently they are always seen together. One of these planets, Mars, exhibits any perceptible phase, but a mode- qually great inequality of elongation; for though half of the disc being nearer than seven-eighths of the whole. To understand this, we need only cast our eyes on the annexed figure, in which E is the earth, at its equinox **CHAP. VII.] DISTANCES OF SUPERIOR PLANETS.** 247 greatest elevation from the sun S, as seen from Mars, M. In this position, the angle SME, included between the lines SM and EM, is at its maximum; and the observer on the earth is enabled to see a greater portion of the dark hemisphere of Mars than that of any other planet. The extent of the phase, then, or greatest observable degree of gibbosity affords a measure of the diameter of Mars, and is one rude one of the single SME, and therefore a proper measure of the distance SMME, which is less than that from the sun, by which it appears that the diameter of the orbit of Mars exceeds that of Jupiter. The phases of Jupiter, Saturn, and Uranus being imperceptible, it follows that their orbits must include not only that of the earth, but also those of Mars. (414.) All the superior planets are retrograde in their apparent motions when in opposition, and for some time before and after such times. This is true both in the extent of their are of retrogradation, in the duration of their retrograde movement, and in its rapidity with respect to that of the earth. It is clear in the case of Mars than of Jupiter, of Jupiter than of Saturn, and of that planet than Uranus. The angular velocity with which they appear to move westwardly is maintained by observing its apparent place in the heavens from day to day; and from such observations, made about the same time every year, it has been found that their respective magnitudes of their orbits as compared with the earth's, supposing their periodic times known. For, from these, their mean angular velocities are known also, being thence inversely as the times. Suppose then, Ee to be very small portion of the earth's orbit, and Me a corresponding A diagram showing angles and distances between celestial bodies. S E M Ee = 30° Me = x 248 A TREATISE ON ASTRONOMY. CHAP. VII. ing portion of that of a superior planet, described on the day of opposition, about the sun S, on which day the three bodies lie in one straight line SEMX. Then the angles ESE and MSN are given. Now, if m be joined and produced to X, then the angle ESM will be equal to ESE, which is equal to the alternate angle XEy, so evidently the retrogradation of Mars on that day, and is, therefore, also given. But if we join the point M with the vertex of the right-angled triangle EXE, the side EX is easily calcu- lated, and thus SX becomes known. Consequently, in the triangle ESMX, the sides ES and SX are known, and two angles mEX and mXS, whereas the other sides, Sx and mX, are easily determined. Now, Sx is no other than the radius of the circle whose centre is at the sun, of which which this calculation is supposed circular as well as that of the earth; a supposition not exact, but sufficiently so as to afford a very good approximation to its dimen- sions of its orbit, and which, if the process be often re- peated, in every variety of situation at which the oppo- sition occurs, will give us either the mean distance or mean value of its diameter fully to be depended upon. (415.) To apply this principle, however, to practice, it is necessary to observe all those planets which have per- seids. These may be obtained directly, as he has already stated, by observing the intervals of their pas- sages through the equinoxes; but it is more convenient to incline the orbits of some of them to its plane, they cross it so obliquely that the precise moment of their arrival can be observed by means of their perseid observations. A better method consists in determining, from the observations of several successive oppositions, the exact eccentricity of each planet's orbit with such an accuracy that it shall coincide with that with which the criterion of which is a difference of longitudes be- tween the sun and planets of exactly 160°. The difference between these longitudes must be nearly one synodic period; and would be exactly so, were the planet's orbit and that of the earth both circular and firmly determined; but since they are not so in the case (and the criterion is, the inequality of successive symd- ical revolutions as observed), the average of a great num- ber, taken in all varieties of situations in which the oppo- 14 **CAP. VIII.] KEPNER'S LAW OF PERIODIC TIME.** sitions occur, will be freed from the elliptic inequality, and may be taken as a mean symmetrical period. From this, by the considerations employed in art. 538, and by the tables given in art. 540, we shall find that the article, the sidereal periods are readily obtained. The accuracy of this determination will, of course, be greatly increased by the use of more accurate observations than those ex- treme observations employed. In point of fact, that int- erval extends to nearly 2000 years in the cases of the planets Mercury and Venus, and to nearly 1000 years in their observations of them in a manner sufficiently careful to be made use of. Their periods may, therefore, be regard- ed as accurately determined; and it is probable that no nu- merical values will be found stated, as well as the mean distances, and all the other elements of the planetary orbits, which have been calculated from these data, to which (to avoid repetition) the reader is once for all re- ferred. (16.) In casting our eyes down the list of the plan- etary diameters, and comparing them with the periodic times, we cannot but be struck with a certain correspond- ence between them; for the greater the diameter is, the evidently the longer the period. The order of the plan- ets, beginning from the sun, is the same, whether we see them at their greatest distance from us at any given time they occupy in completing their revolutions; and is as follows---Mercury, Venus, Earth, Mars---the four ultra- near planets being placed together. But on the other hand, therefore, when we come to examine the numbers express- ing them, we find that the relation between the two series is not so simple as might have been expected. The periods increase more than in proportion to the distances. Thus, the period of Mercury is about 88 days, and that of the Earth about 365 days; while the ratio of their distances is in the less proportion of 1 to 2-56; and a similar remark holds good in every instance. Still, the ratio of their periods to their distances is greater than that of the squares of the distances. The square of 2-56 is $6\cdot5336$, which is considerably greater than $4\cdot16$. An inter- mediate rate of increase, between the simple proportion of the distances and that of their squares, is therefore
    Mercury Period: 88 days Diameter: Not specified Distance: Not specified Square of Distance: Not specified
    Venus Period: Not specified Diameter: Not specified Distance: Not specified Square of Distance: Not specified
    Earth Period: 365 days Diameter: Not specified Distance: Not specified Square of Distance: Not specified
    Mars Period: Not specified Diameter: Not specified Distance: Not specified Square of Distance: Not specified
    The table shows that although Venus has a shorter period than Mercury, her diameter is greater than that of Mercury's; and although Mars has a longer period than Earth's, her diameter is smaller than that of Earth's. This correspondence between the diameters and periods is evident from a comparison of these figures with those given in Table I., p. 247. It appears also from this table that there is a certain regularity in the ratios between the periods and distances of these planets; namely, that they are nearly equal to each other in proportion to their respective diameters. For example, if we take any two adjacent planets in this table (as Mercury and Venus), we find that their periods are nearly equal to each other in proportion to their diameters; and if we take any two adjacent planets (as Venus and Earth), we find that their periods are nearly equal to each other in proportion to their diameters. This regularity in the ratios between the periods and distances of these planets suggests a hypothesis respecting their motions. It seems probable that they move around some central body with uniform velocity; but that this body does not lie exactly at their center of gravity; but rather at some point nearer one planet than another. If this be true, then it would follow that their periods would vary inversely as their distances from this central body; or what is equivalent to say that their periods would vary directly as the squares of their distances from this central body. But it must be remembered that this hypothesis only explains why there is such a regularity in the ratios between the periods and distances of these planets; it does not explain how they acquire these ratios. It may be possible that they acquired them by some process which has since been lost; or it may be possible that they acquired them by some process which has since been discovered; or it may be possible that they acquired them by some process which has since been forgotten. In conclusion, I think it may be said that Kepler's law of periodic time is confirmed by all our observations; and that it affords us a means for determining with great accuracy many important quantities connected with our solar system. I am indebted to Mr. John Herschel for much valuable assistance in preparing this article. 249 250 A. THEATRE ON ASTRONOMY. [CHAP. VII.] clearly pointed out by the sequence of the numbers; but it required no ordinary penetration in the illustrious Kepler, ranked by uncommon perseverance and industry, at a period when the data thematelves were involved in obscurity, to perceive that the progress made in the art of numerical calculation were encumbered with difficulties, of which the more recent invention of logarithmic tables has happily removed many. The following example demon- strate the real law of their connexion. This connexion is expressed in the following proposition.—(1.) The squares of the perihelions (or distances from the sun) of each other, in the same proportion as the cubes of their mean distances from the sun." Take, for example, the earth and Mars, whose perihelions are 365554 to 860795, and whose distances from the sun is that of 160000 to 162899 ; and it will be found, by any one who will take the trouble to go through the calcula- tion, that— (365554²): (860795²): (160000²): (162899²) (417.2): (947768): (100000): (122899). (417.2) Of all these numbers, only two are pure observation has ever conducted man, this third law (as it is called) of Kepler may justly be regarded as he most remarkable of all his discoveries, and its consequences. When we contemplate the constituents of the planetary system from the point of view which this rela- tion affords us, we see that they are all alike to us—no longer a general resemblance among them, as individuals independent of each other, and circling about the sun like planets in a common orbit, but all connected with it by its own peculiar tie. The re- semblance in now perceived to be a true family likeness—they are all members of one family, united by a mutual relation and harmonious agreement—subjected to one pervading influence, which extends from the centre to all the farthest parts of the system. And yet all of them, the earth included, must henceforth be regarded as members. (418.) The laws of elliptic motion about the sun as a The expression of law 2 of Kepler requires a slight modification when we consider that it does not apply to all planets, for the greater planets do not to the influence of their masses. This correction is superficially for the earth and Mars. **CHAP. VII.] INTERPRETATION OF Kepler's laws.** 521 focus, and of the equable description of arcs by lines, joining the sun and planets, were originally established by Kepler, from a consideration of the observed motions of Mars; and were by him extended, analogically, to all the other planets. But though this general proposition might then have appeared, modern astronomy has com- pletely verified it as a matter of fact, by the general consis- tency with which it agrees with the observed motions of the apparent places of the planets. These are found to accord satisfactorily with the assumption of a particular orbital form, viz., that of an ellipse, whose eccentricity, and situation in space, are numerically assigned in the synoptic table before referred to. It is true, that we may suppose any other form of orbit to be perfectly pos- sible, and that when each planet is traced through many suc- cessive revolutions, and its history carried back, by the aid of the tables of the mean motions of the communi- ties, we learn to regard the laws of Kepler as only first approximations to the much more complicated ones which would be required to represent the actual circumva- tions into rigorous and mathematical accordance with each other, and at the same time to retain the extremely con- siderable variations which occur in the planetary move- ment. system, it becomes necessary to modify, to a certain ex- tent, our verbal expression of the laws, and to regard the mea- nures of their application as merely approximate truths, not as absolutely permanent, but subject to a series of extremely slow and almost imperceptible changes. These changes are most noticeable in those cases where few revolutions; but going on from century to century, and continually accumulating, they at length produce considera- ble deviations from what was originally supposed to be the state. Their explanation will form the subject of a subsequent chapter; but for the present we must lay them out of consideration, and proceed to examine some of the more important gener- al conclusions with which we are now concerned. By what means astronomers are enabled to compare the re- sults of their observations with those which they thus satisfy themselves of their ascendance with nature, will be explained presently. (418.) It will first, however, be proper to point out 252 A THEATRE ON APOCRYPHA. [CHAP. VIII.] what particular theoretical conclusion is involved in each of the three laws of Kepler, concerning the planets respectively established,—what indication each of them separately affords of the mechanical forces prevalent in our system, and what may be inferred respecting the nature of these forces? In this way, when thus considered, they constitute the basis on which the Newtonian explanation of the mechanism of the heavens is most securely founded. The first law, that of the equal areas of the orbits, that of the equable description of areas—since the plane move in curvilinear paths, they must (if they be) obey characterised by a constant velocity, and hence from their otherwise natural rectilinear progress by force. And from this law, taken as a matter of observed facts, it fol- lows, that every planet describes its orbit in such a manner that the orbit of each planet, always passes through the sun. No matter from what ultimate cause the power which is called gravity acts upon bodies situated in the sun as its receptacle, or be it pressure without, or the resultant of many pressures or solicitations of un- known forces acting simultaneously upon one another, still, when finally brought under our contemplation, its direction from body to body is always towards the sun's cen- tre. As an abstract dynamical proposition, the reader will find it demonstrated by Newton, in the first proposi- tion of his Principia, that no force can act between two bodies to which we really could add nothing but obscuring by ampli- fication, that any body, urged towards a certain central point by a force acting upon it at all points equally deflected into a curvilinear path, will describe about that entire equal areas in equal time; and vice versa, that such equal velocities are maintained in all cases. The essential criterion of a continual direction of the acting force to- wards the centre to which this character belongs. The law so well known to us from experience refers to the nature or intensity of the force urging the planets to the sun; and this conclusion it involves, is that it does so regard less whether it be applied to one planet than to the influence of central forces generally; and as such, we daily see it exemplified in a thousand familiar instances. A simple experimental illustration of it is to be a bullet **CHAP. VII.] INTERPRETATION OF KEPLER'S LAWS. 283** to a thin string, and, having whirled it round with a mo- derate velocity, to let it fall upon the ground, and to take the string through a small ring, or allow it to coil itself round the finger, or a cylindrical rod held very firmly in a hori- zontal position, and then to observe the diminution of the rate of motion in a spiral line ; and the increase not only of its angular but of its linear velocity, and the rapid diminution of its angular velocity, will be found to be very evident processes, more clearly than any other means, the compensation by which its uniform distribution of areas is maintained under a considerable change of direction. The string is first re- versed, and the thread allowed to unroll, beginning with a rapid impulse, the velocity will diminish by the same de- crease that it increases in the reverse direction. The velocity of a dancer's piroette, as he draws in his limbs and straightens his whole person, so as to bring every part of his body into a perpendicular position, is another instance where the connexion of the observed effect with the central force exerted, though equally real, is more evident than any other. (40). The second law of Kepler, or that which as- serts that the planets describe ellipses about the sun as their center, is deduced from the fact that they are in circular gravitation (so as to be allowed to call the force, whatever it be, which urges them towards the sun) as executed on each circle by a force directed towards the center of that circle. A straight line, dynamically speaking, is the only path which can be pursued by a body absolutely free, and un- der no circumstances can it deviate from such a line; nor is there any case known where two bodies do not move in one curve is evidence of the exercise of a force; and the greater the deflection in equal times, the more intense the force. A circle is therefore a special case of a curved line; and this word for curvature of path; and as a circle is character- ized by the uniformity of its curvature in all its parts; so is a straight line characterized by its uniformity in all the par- ticular laws which regulate the increase and diminu- tion of its curvature as we advance along its circumfe- rence. Hence it follows that when a body continually bends a moving body into a curve, may be assureded, provided its direction in the first place, and, secondly, in every subsequent place, be always uniform. Both these enter as elements into the expression of the force. A Y 284 A TREATISE ON ASTRONOMY. [CHAP. VIII.] body may describe, for instance, an ellipse, under a great variety of dispositions of the acting forces : it may glide along it, for example, as a bead upon a polished wire, but what we have to consider is that the direction of the force is always perpendicular to the wire, and the velocity is uniform. In this case the force is directed to and fixed against one point of the wire, which point we call the pole of the ellipsis. Or it may be observed at it we may see it done, if we sus- pend a ball by a very long string, and, drawing it a little from its course, observe how it describes an ellipse under the gentle impulse. In this case the acting force is directed to the centre of the ellipse, about which areas are described equally with all parts of the circumference. The space (the decomposed result of terrestrial gravity) perpetually urges it. This is a notion a very easy experiment, and a very attractive one. We shall now proceed to examine this case before us, or an ellipse described by the action of a force directed to the focus, the apex of the investigation being in this place. In order to determine the actual velocity of the revolving body at every point, or the space really run over by it in a given time period, we must first find out what is called the ellipse determinate the linear amount of deflection from the tangent in the direction of the focus, which corresponds to that speed which would be given to a body of accelerated motion declare that the intensity of the acting force causing such deflection in its own direction, is meas- ured by a certain number of units of force per unit of mass, and may therefore be calculated in any particular position, or generally expressed by geometrical or algebraic sym- bols, as it will appear hereafter. It is evident that this deflection is so calculated or expressed. We have here the spirit of the process by which Newton revealed this important discovery. For this purpose we must refer to the 3d section of his Principia. We know of no artificial mode of imitating this ellipse of elliptical motion. But let us suppose that we could do so; though, however, to give a conception of the alternate approach and recessing of the revolving body to and from the focus, and then returning again to it; let us imagine describ- ing a small steel bead to a fine and very long silk fibre, and setting it to revolve in a small orbit round the pole of - **CHAP. VIII.] INTERPRETATION OF KEPLER'S LAWS. 255** a powerful cylindrical magnet, held upright, and vertically downwards. (421.) The third law of Kepler, which connects the distances and periods of the planets by a general rule, becomes, when we consider the force of gravity as its important consequence, viz. that it is one and the same force, modified only by distance from the sun, which regulates the motions of all bodies in the universe. For the attraction of the sun (if such it be) is exerted upon all the bodies of our system indifferently, without regard to the nature of their constitution. It is not necessary that the expanse proportion of their inertia, or quantities of matter; that it is not, therefore, of the nature of the elective attraction which governs the motions of the planets, which is powerless on other substances than iron and some one or two more, but is of a more universal character, and extends itself through all parts of our system, and (as we shall hereafter see abundant reason to admit) to those of other systems than our own. This last proposition is supported by a great number of similest corollaries, from the relations established by Newton in the section of the Principia referred to (§ 308), where he shows that if any body whatever be at rest in space at the place, in the direction, and with the velocity in which it was originally placed, it will describe the very same orbit, and in the same period, which that planet actually does, a very minute correction of the per- iod being made for the difference between the mass of the earth and that of the planet. Small as the planets are compared to the sun, some of them are not, indeed, described in exactly similar orbits. But this is owing to the strict wording of Kepler's law, as Newton has proved in his fifty-ninth proposition, is applicable only to the case of bodies having equal masses; and this inequality is almost absolutely imperceptible. When this is not the case, the periodic time is shortened in the proportion of the square root of the ratio between their masses or inertia; so that if that sum of the numbers expressing the masses of the sun and planet; and in general, what- ever may be their mutual distances from each other under the influence of the Newtonian law of gra 256 A TREATISE ON ASTRONOMY. CHAP. VII. vity, the square of their periodic time will be expressed by a fraction whose numerator is the cube of their mean distance, i. e. the greater semi-axis of their elliptic orbit, and whose denominator is the product of their masses. When one of the masses is incomparably greater than the other, this resolves itself into Kepler's law; but when this difference is not great, we have to consider the generalized stands in view of that law. In the system of the sun and planets, however, the numerical correction thus introduced is so small that it can only be considered as being of any importance, the mass of the largest of the planets (Jupiter) being much less than a thousandth part of that of the sun. But even in this case, we never perceive all the importance of such generalization, when we come to speak of the satellites. (428.) It would be proper to explain by what process of calculation the expression of a planet's elliptic orbit by its elements can be compared with ob- servations. This has been done by Laplace, who gives the numerical data contained in a table of such elements for the whole system does really exhibit a true picture of it, and at every instant of time, and for every given epoch, its periodical time or mean angular velocity, or, as it is called, its mean motion. (429.) The elements of an ellipse are de- termined by its greatest length and least breadth, or its two principal axes; but for astronomical uses it is pre- ferable to express them by its semimajor axis (or greatest length), and the eccentricity or distance of the focus from the centre, which last is usually estimated in parts of the semimajor axis. Thus if the semimajor axis is 100 and breadth 8 parts of any scale, has for its major semi- axis 5, and for its eccentricity 3 such parts; but when estimating the eccentricity from observations it is as a unit, the eccentricity is expressed by the fraction $\frac{3}{10}$. (434.) The ellipse is the plane to which an inhabi- CHAP.VIII.] ELEMENTS OF A PLANET'S ORBIT. 257 ant of the earth most naturally refers the rest of the solar system, as a sort of ground-plane; and the axis of its other planets, to that plane. This is the true plane or origin of angular reckoning. Were the axis fixed, what would be the best possible origin of longitudes; but as it is not so, we must take some other point (1); there is, in fact, no advantage in reckoning from the axis more than from the line of the equinoxes, and aneono- mous latitude being determined by the variation in va- riation by the effect of precession, and restoring it, by calculation at every instant, to a fixed position. Now, the longitude of any planet with respect to this plane, a planet with respect to this plane, these elements require to be known...(1), i.e. the eccentricity of the plane of the planet's orbit, and the inclination of that plane to the line in which these two planes intersect each other, which of necessity passes through the sun, and whose position is given by the mean anomaly, and therefore is always given by stating its longitude. This line is called the line of the nodes. When the planet is in this line, its longitude is said to be at perihelion; when out of the ecliptic, it is its ascending node, and its longitude at that moment is the element called the longitude of the node. The eccentricity of the orbit is then known; the plane of the orbit; and there only remains, for the com- plete determination of the situation of the planet's ellipse, to find its focus. But this can be done only when (ie focus is necessarily in the sun) is ascertained by stating the longitude of its perihelion, or the place where the ellipse has its minimum area. For example, when orthographically projected on the ecliptic. (4th). The dimensions and situation of the planet's orbit are now known; we have had complete acquaintance with its history, to determine the circum- stances of its motion in the orbit so precisely fixed. Next we must ascertain its place in that orbit at any time; the moment of time when it is either at the perihelion, or at any other precisely determined point of its orbit, and its place in that orbit at any other time. The area of the area determines the place at every other instant. This moment is called (when the perihelion is the point chosen) the perihelion passage; or, when some point of 256 A TREATISE ON ASTRONOMY. [CHAP. VIII.] she orbit is fixed upon, without special reference to the perihelion, the epoch. (428.) Then, thus, we have several particulars or ele- ments, which must be numerically stated, before we can reduce the theory of the motions of the planets to any given moment. But, these known, it is easy to ascertain the apparent positions of each planet, as it would be seen from the earth at any given time and place. The former is called the heliocentric, the latter the geo- centric, place of the planet. (427.) To commence with the heliocentric place. Let S re- present the sun; A P N the orbit of the planet; B C D E F G H I K L M N O P Q R S T U V W X Y Z, having the sun S its focus, and A P its perihelion ; and let p a n p represent the angle between the ecliptic, intersecting the line of equinoxes S P p in a, which is the origin of longitude. Then will SN be the line of nodes, that is, the axis of inclination to lie on the south, and A C the north side of the ecliptic, and the direction of the planet's motion to be from B to A, N being its perihelion. The angle SN A is called longitude of the node. In like manner, if B be the place of the planet at any time, and if it and the perihelion A be projected on the plane of the ecliptic, then will the angle B p p a N be equal to the respective heliocentric longitudes of the planet, and at the perihelion, the former of which is to be denoted by p a n p. Lastly, the angle p s p is the heliocentric latitude of the planet, which is also required to be known. (428.) Now, this being known, we may determine the me- ment of the planet's passing the perihelion, the interval, or the time of describing the portion AP of the orbit, is given, when we know its longitude at any time. As in ellipse being known, the law of proportionality of areas to the times of their description gives the magnitude of the area described by any part of it. Hence we have only geometry to determine the corresponding angle ASP, which is called the planet's true anomaly. This problem is of the kind which has been treated in Art. 300. It occurs in a great variety of processes, some more, some less in **CHAP. VIII. THE HELiocENTRIC PLACE OF A PLANET.** triciate. It offers, however, no peculiar difficulty, and is produced by the following process. We will take the help of tables constructed for the purpose, adapted to the case of each particular planet. Let $N$ be the node, normally thus obtained, the planet's angular distance from the node, or the angle NSP, is to be found. Now, the longitudes of the perihelion and aphelion being known, we have only to subtract their difference $n$ is also given, and the angle $N$ of the spherical right-angled triangle ANB, being the inclination of the plane of the orbit to that of the ecliptic, is known. Hence we calculate the arc NA, or the angle NAP, which, added to ASP, gives the angle NSP required. And now let us consider the triangle PNP, forming the hypotenuse of the right-angled spherical triangle PNp, whose angle $N$, as before, is known, it is easy to find the arc NP and hence the angle NPN. The latter, being the measure of the angle $\phi$SP, expresses the planet's heliocentric latitude; the former measures its longitude from the node, and therefore includes from its node, which, added to the known angle $\phi$SP, the longitude of the node, gives the heliocentric longitude. The method here described is very simple when once well understood, may be gone through numerically, by the aid of the usual logarithmic and trigonometrical tables; but at first time it will have taken the reader to pursue its details. (380.) The eccentricity differs from the heliocentric place of a planet by a very small amount. This is owing to two causes: (1) The motion of a planet is not uniform in its orbit. Were the planet's distance as vast as those of Jupiter and Saturn, it would be impossible for its circular motion, any equal displacement of course absent any centre is inconceivable. But this is not so; for though its mean distance from the sun is at some times less than at others, yet on account of its elliptical form, its motion from the perihelion, or true focus from which it technically moves in a circle, is always uniform. Hence it has been proved that this motion is even uniform instead of being affected by any change whatever. (2) The eccentricity produces all problems of the measurement of time and can be resolved in this way with great facility. In fact it is so small that it may be very nearly ignored. It may be useful to mention that in 1765 Dr. Bradley published a paper in which he gave a description in the Cambridge Philosophical Transactions how he had determined by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine by observations made at different places on earth with what accuracy he could determine byobservations 260 A TREATISE ON ASTRONOMY. [CHAP. VIII.] of the stars, each earth's orbital motion would be insen- sible when viewed from them, and they would always appear to move in the same manner as the fixed stars, as if viewed from the sun, i. e. they would then be seen in their heliocentric places. The differ- ence, therefore, between the different positions of a planet in, in fact, the same thing with its par- allax arising from the earth's removal from the centre of the sun, is that the first step towards a knowledge of its amount, and the consequent determination of the ap- parent position of any star, must be made by determining the sphere of the fixed stars, must be to ascertain the proportion of its linear distances from the earth and from the sun, and the angular positions of all three with re- spect to each other. (481.) Let S represent the sun, E the earth, and P the planet; S'P' the line of equinoxes, E'P' the earth's orbit, and Pp' perpendicular to it fall from the earth to P'. Then SP will be SPE (according to the general notion of parallax con- veyed in art. 60.) A diagram showing a line segment SP representing the distance from S to E, EP' representing the distance from E to P', and SP'EP representing the angle between these two lines. while PE represents the heliocentric longitude of the earth, and PE' Sp that of the planet. The former of these angles is represented by SP', while PE' Sp is found by the process above described (art. 459). Moreover, SP is the radius vector of the planet's orbit, and SPE is its eccentricity. These quantities are deduced from the known dimensions of their respective ellipses, and the places of both bodies at the assigned time. Lastly, the angle PE'S is the planet's heliocentric lati- tude. (482.) Our object then, is, from all these data, to de- termine the angle VQK and VPp', which is the geo- metrical **CHAP. VIII.** DISCOVERY OF THE PLANETS. tric latitude. The process, then, will stand as follows: 1st. In the triangle SPc, right-angled at P, given SP, and the angle PSp (the planet's radius vector and helio- centric longitude), find Sp. 2nd. In the triangle SEP, given Sp (just found), SE (the earth's radius vector), and the angle ESp (the difference of heliocen- tric longitude between the earth and the planet), find the angle SpE, and the side EP. The former being equal to the alternate angle pSp, is the parallactic removal of the planet from the earth's equator, or its parallactic longitude. The latter, Ep (which is called the curtail distance of the planet from the earth), gives at once the true longitude of the planet in a right-angled triangle PEp, of which Ep and Pp are known sides, and the angle EEp is the longitude sought. (483) It is evident that all these processes are nothing but the most ordinary processes of plane trigonometry; and though somewhat tedious, are nev- ertheless very useful in astronomical calculations. Moreover, they afford us the means of comparing the places of the planets actually observed with the elliptic theory, by which we may see how far it agrees with the se- verest trial; and it is upon the testimony of such compu- tations, so brought into comparison with observed facts, that we declare this theory to be a true representation of nature. (484) The planets Mercury, Venus, Mars, Jupiter, and Saturn have been discovered by men; in which astronomy has been cultivated. Uranus was dis- covered by Sir W. Herschel in 1781. March 13, in the course of his observations on a star which he had already visible in a telescope of a certain power was brought under close examination, when the new planet was un- derstood to be a satellite of our sun, having no attracting power. It has since been ascertained to have been ob- served on many previous occasions, with telescopes of different powers; and is now entered in Catalogues as a star; and some of the observations which have been so recorded have been used to improve and confirm the theory of elliptic motion. Neptune, another ultra-iridial planetary dates from the first day of 1801, when Ceres was discovered by Piazzi, at Palermo; a 303 A THEATRE ON ASTRONOMY. [CAP. VII.] Discovery speedily followed by those of Janso by Pro- fessor Harding, of Göttingen, and of Paulus and Vester by Dr. Ollers, of Bremen. It is extremely remarkable that this important addition to our system had been in some sense anticipated by the ancients. It is obvious that the intervals between the planetary orbits go on doubling as we recede from the sun, or nearly so. Thus, the interval between the orbits of Mars and Earth is nearly twice that between those of Venus and Mercury; that between the orbits of Mars and the earth nearly twice that between those of Jupiter and Saturn. The interval between the orbits of Jupiter and Mars, however, is too great, and would form an exception to this law, which has been observed in all other cases of the three remote planets. It was, therefore, thrown out, by the late Professor Bode of Berlin, as a possible curiosity ; but it is now seen to be a very important one and Jupiter; and it may easily be imagined what was the as- tonishment of astronomers to find revolving in orbits obviously contrary to the laws which they had deduced. No account, a priori, or from theory, can be given of this singular proposition, which is not, like Kepler's laws, strictly true. But in consequence of certain circum- stances we have just mentioned lead to a strong be- lief that it is something beyond a mere accidental dis- covery, and that it is connected with some part of the system. It has been conjectured that the ultra-oxudical planets are fragments of some greater planet, which formerly existed in the solar system, but which have been thrown to atoms by an explosion ; and that more such fragments exist, and may be hereafter discovered. This may seem as a plausible hypothesis; but such hypotheses are like other speculations, occasionally and harshly indulge. (485.) We shall devote the rest of this chapter to an account of these two discoveries, and also to the discus- sion of the several planets, so far as the former are known by observation, or the latter rest on probable grounds of conjecture. In this discussion we shall show that we are as necessary productively of extraordinary diversity in the propositions by which, if they be, like our earth, inhabited, similar laws will govern their motions; and that any di- ference in their respective supplies of light and heat from CHAP. VIII.] APPEARANCES OF THE PLANETS. 368 the sun; secondly, the difference in the intensities of the gravitating forces which must submit at their surfaces, or the different ratios which, on their several globes, the meridians of bodies must bear to their zeniths; and, thirdly, the difference in the duration of day and night, from what we know of mean density, we have every reason to believe they consist. The intensity of gravity on Jupiter exceeds that of the sun by one twelfth more than on the earth, and on Uranus three times less; the proportion between the two extremes being that of Jupiter's diameter to its distance from the sun, which is half the condition of our globe, were the sun to be repositioned, to say nothing of the greater ratio! or were it diminished to one-third of its present magnitude. In this case, the intensity of gravity, or its efficacy in counteracting muscular power and representing animal activity on Jupiter would be increased by nearly six times over that seen only one third on the Moon one sixth, and on the four smaller planets probably not more than one twentieth; but this increase is so slight as to be scarcely perceptible even on Jupiter itself. (495.) Lastly, the density of Saturn hardly exceeds one eighth of the mean density of the earth, so that it is not possible to form a body larger than cork. Now, under the various combinations of elements so important to life as these, what immense diversity could be expected? We might expect a world without the maintenance of animal and intellectual existence and happiness, which seems, so far as we can judge by what we see around us, to be a necessary condition for any being whose corner of it is crowded with living beings, to form an unceasing and worthy object for the exercise of his faculties and powers. (496.) But this is not all. (496.) Quitting, however, the region of mere specula- tion, we will now show what information the telescope affords us respecting the appearances of these planets within its reach. Of Mercury we can see little more than that it is round, and exhibits phases. It is too small, and not sufficiently distant from us to be visible to the Sun, to allow us to make out more of its nature. The real diameter of Mercury is about 2000 miles; its appa- rent diameter is about 700 miles. It has no satellites; offer any remarkable peculiarities : although its real dia- 204 A TREATISE ON ASTRONOMY. [CHAP. VII.] meter is 7200 miles, and although it occasionally attains the considerable distance of 8000 miles, yet its diameter is much larger than that of any other planet, it is yet the most difficul't of them all to define with telescopes. The intense lustre of its surface, which is so much more extensive than any ex- gerates every imperfection of the telescope; yet we see clearly that its surface is not modulated over with permanent spots like the Moon, but is uniformly bright, though sometimes nor shadows, but a uniform brightness, in which some- times we may, indeed, fancy obscure portions, but can seldom or never perceive any great irregularities. From some observations of this kind that both Venus and Mer- cury have been concluded to revolve on their axes in about the same time as the Earth does; and from this conclusion, from the very rare appearance and want of permanence in the spots, is, that we do not see, as in the Moon, the darkened portions of the surface, which are the atmospheres, much loaded with clouds, and which may serve to mitigate the otherwise intense glare of their sun- shine. (487.) The case is very different with Mars. In this planet we discern, with perfect distinctness, the outlines of what must be a vast number of mountains and hills. Fig. 1, which represents Mars in its gibbous state, as seen on the 16th of August, 1830 (on the 90-feet reflector at Slough). Of these, those which are situated near the equator are those which characterize the light of this planet (which always appears red and fiery), and indicates, no doubt, an achryy time in the year when they are illuminated by the Sun's rays, and that adds greatly to the probability of this is the ap- pearance of brilliant lights on its surface. One of which I have not been able to observe in the text nor in my obser- vations, but never more distant than on the occasion when the drawing was made (see fig. 2). It has been supposed to have a very extensive atmosphere, but no atmos- phere could possibly exist here without a very dense and heavy CHAP. VIII.] APPEARANCES OF THE PLANETS. 365 appear in our figures—which have been conjectured with a great deal of probability to be snow; so they disappear when they have been long exposed to the sun, and are greatest when just emerging from the long night of their polar winter. But, on account of the inclination of the axis of light, and on successive nights, it is found that Mars has a rotation on its axis of 24 hours 20 minutes 18 seconds, or nearly 24 hours 20 minutes 20 seconds, in the same direction as the earth's, or from west to east. The greatest and least apparent diameters of Mars are 4° and 18°, and its real dis- tance from us is about 170 million miles. (438.) We come now to a much more magnificent plane, Jupiter, the largest of them all, being in diameter no less than 80 times that of the earth. Its distance from us is the Earth nearly 1300 times. It is, moreover, dignified by the attendance of four moons, satellites, or secondary planets, which revolve round it in the same way as it does, and revolve about it, so the moon does round the earth, and in the same direction, forming with their principal, or central body, a system similar to that of the earth. It is ambi- gous to that greater one of which their central body is itself a member, obeying the same laws, and exemplifying, in this respect, the principle of gravitation. The influ- ence of the gravitating power as the ruling principle of their motions: of these, however, we shall speak more at large in another place. (439.) The disc of Jupiter is always observed to be crossed in one certain directions by dark bands or belts, produced by the motion of the clouds. These belts repre- sent this plane as seen on the 23d of September, 1882, in the 20th reflector of Slough. These belts are, how- ever, only temporary appearances; for they appear with and in situation on the disc (though never in their general direction). They have even been seen broken up, and divided into smaller parts. This shows that this phenomenon is extremely rare. Branches running out from them, and subdivisions, as represented in the figure, at different distances from each other. Now though we are by no means uncommon; and from these attentively watched, it is concluded that this planet revolves in the same direction as the earth's; but not in an axis perpendicular to the direction of the belt. Now it Z 368 A TREATISE ON ASTRONOMY. [CHAP. VIII.] is very remarkable, and forms a most satisfactory com- ment on the reasoning by which the spheroidal figure of the earth has been deduced from its diurnal rotation, that the outline of the planet is perfectly round, and not, as ellipse, being considerably flattened in the direction of its axis of rotation. This appearance is no optical illusion, but is owing to the great difference between the dimensions which assign 107 to 100 for the proportion of the equatorial and polar diameters. And to confirm, in the strongest man- ner, the correctness of this deduction, several other con- clusions have been found, and fully to authorize their extension to this remote system, appear, on calcula- tions, that the diameter of Jupiter is nearly equal to that corresponds, on those principles, to the dimensions of Ju- piter, and to the time of his rotation. (440.) The appearance of spots due to the equator of Jupiter, their occasional variations, and the appearances of spots upon seen them, render it extremely probable that they are produced by currents revolving in circular tracts of apparently clear sky, determined by currents analogous to our trade-winds, but of a much more steady and decisive character than any we can find out. The immense velocity of its rotation. That it is the comparatively darker body of the planet which appears in the light of the sun, and that it is this body which re- sults in all its strength to the edge of the disc, but fade away gradually before they reach it. (See plate L. fig. 5.) The apparent diameter of Jupiter varies from 30° to 40°. (441.) A still more wonderful fact, and as it may be termed, is that Jupiter is not only nearer to the sun than Saturn, the next in order of remoteness to Jupiter, to which it is not much inferior in magnitude, being about 79,000 miles in diameter; but it is also much more compact in bulk, and subordinating an apparent angular diameter at the earth, of about 16". This stupendous globe, be- side being so near to us, and so large in size as moons, is surrounded with two broad, flat, extremely thin rings, concentric with the planet and with each other, which are separated by a narrow interval from each other throughout their whole circumference, as they are from the planet by a much **CHAP. VIII.** OF SATURN'S RINGS. 367 *Wider.* The dimensions of this extracircular appendix are as follows :—
    Length of exterior ring. 1000.
    Interior diameter. 15000.
    Exterior diameter. 17500.
    Exterior diameter. 19000.
    Inferred between the planet and interior ring. 1000.
    Total thickness of the rings not exceeding. 1000.
    The figure (fig. 3, plate I.) represents Saturn surrounded by its rings, which are represented by a series of parallel lines somewhat similar, but broader and less strongly marked than those of Jupiter, and owing, doubtless, to a similar cause, to the greater distance of the planet, shown by its throwing its shadow on the body of the planet, on the side nearest the sun, and on the other side re- ceiving light from the same source. From the parallelism of the belts with the plane of the ring, it may be conjectured that the axis of rotation of the planet is perpendicular to that plane. This projec- ture is confirmed by the occasional appearance of en- tensive dumpy spots on its surface, which when watched, like a pendulum bob, appear to indicate a rotation in 10° 30' about an axis so situated. (442.) The axis of rotation, like that of the earth, preserves its direction with respect to the plane of the planet in its orbit; and the same is also the case with the ring, whose plane is constantly inclined at the same angle to that of the planet's orbit with respect to it, and, therefore, to the ecliptic, viz. 58° 40' and intersects the latter plane in a line, which makes an angle with the line of nodes equal to that between them. The ring lies in 170° and 250° of longitude. Whenever then, the planet happens to be situated in one or other of these longitudes, it appears to be illuminated through the sun, which then illuminates only the edge of it; and as at the same moment, owing to the enluminance of the other part of it by reflection from the sun, this part is darkened; and vice versa. These dimensions are calculated from Pau Sivert's micrometric measure. Mem. Art. Sci. iii. 361, with the exception of the thickness of the rings, which is estimated from their apparent dim- ension when now in progress. The interval of the rings here stated is probably too great. A diagram showing Saturn with its rings. The rings are depicted as parallel lines surrounding Saturn. The innermost ring is labeled "Interior diameter" with a value of 15000 units. The outermost ring is labeled "Exterior diameter" with a value of 17500 units. The total thickness of the rings not exceeding is labeled "Total thickness" with a value of 1000 units. The length of the exterior ring is labeled "Length" with a value of 1000 units. The inferred space between Saturn and its interior ring is labeled "Inferred between" with a value of 1000 units. The overall width of Saturn is labeled "Width" with a value of approximately 367 units (the sum of all values). A smaller circle representing Saturn itself is drawn within this larger circle. 268 A TREATISE ON ASTRONOMY. [CHAP. VII.] earth is necessarily not far out of that plane, and must, at all events, pass through it a little before or after that moment, it only then appears to us a very fine straight line, always perpendicular to the plane of the ecliptic on both side--indeed, so very thin is the ring, as to be quite in- visible, in this situation, to any but telescopes of extra- ordinarily large aperture. The appearance of the ring takes place at intervals of 15 years, but the disappearance of the ring is generally double, the earth passing twice through it in one year, owing to its motion around the sun, and the slow motion of Saturn. This second disappearance is now in progress.* As the planet, however, recedes from the sun, the ring will appear more and more inclined to the plane of the ring, which, according to the laws of perspective, ap- pears to us as a circle, and consequently as having its greatest breadth when the planet is 90° from either node, so at CD. Supposing the upper part of the figure to be north, and the lower south, we see that the northern part of the ring will be seen when the planet lies in the semicircle ACB, and the southern only when in ADB. As the diameter of Saturn is larger than its diameter is almost exactly double the shorter. (443.) It will naturally be asked how so stupendous an arch, as that formed by Saturn's rings, can be sustained without collapsing and falling in upon the planet? The answer to this is to be found in a very astonishing fact. In all our observations of this phenomenon has been detected, owing to some portions of the ring being a little less bright than others, and assigning its period as 10° or 12° of longitude. But this is only a very small dis- tinction, and of the force gravity in the Saturnian sys- tem, is very nearly the periodic time of a satellite re- volving round its axis with a period of 10° or 12° of longitude. It is the centrifugal force, then, arising from this rotation, which sustains it; and, although no observation since though great care has been taken to observe whether the outer and inner rings have hitherto been made, it is more than probable that such a difference does subsist as to *The disappearance of the rings is complete, when observed with a telescope of about 10 inches aperture, and twenty feet lens length. April 1813.--Author. CHAP. VII.] place each independently of the other in a similar state of equilibrium. (444.) Although the rings are, as we have said, very nearly concentric with the body of Saturn, yet recent measurements by the transit method of stars have demonstrated that the coincidence is not mathematically exact, but that the centre of gravity of the rings oscillates between two points situated on opposite sides of their orbit, probably under laws of much complexity. Twitting as this remark may appear, it is of the utmost importance to observe that the rings are not only perfectly concentric with them mathematically perfect in their circular form, and exactly concentric with the planet, it is demonstrable (see chap. XI.) that they are also perfectly (or almost) a system in a state of unstable equilibrium, which the slightest external power would subvert—not by causing a small disturbance in one part of the ring, but by precipita- ting them, unbroken, the surface of the planet. For the attraction of such a ring or rings on a point or sphere concentric with its centre will always tend to draw it in all directions, but tends to draw the point or sphere towards the nearest part of the ring, or away from the centre. Hence, if any eccentricity exists in the ring's course, even so little eccentricity to the ring, the tendency of their mutual gravitation is not to correct but to increase this eccentricity until it becomes great enough to cause a catastrophe. (See chap. XI.) Now, external powers, capable of producing such eccentricity exist in the attractions of other bodies upon Saturn. Hence, if Saturn is not stable in order that the system may be stable, and possess within itself a power of resisting the first inroads of such a tendency, then it must be maintained by some means—either by an opposite or maintaining power, it has been shown that it is sufficient to admit the rings to be loaded in some particular way with matter, either by being made of in- equality of thickness, or by some portion being denser than others. Such a load would give to the whole ring an apparent mass greater than its actual mass; and if of a heavy and sluggish satellite, maintaining itself in an orbit with a certain energy sufficient to overcome minute eccentricities without being able to resist those existing upon its axis. But even without supposing the existence z 2 **E70** A TREATISE ON ASTRONOMY. [CHAP. VII.] of any such load--of which, after all, we have no proof--and granting, therefore, in its full extent, the general instability of the equilibrium, we think we per- ceive, in the periodicity of the moon's diurnal motion, a sufficient proof of its preservation. However homely be the illustration, we can conceive nothing more in every way consistent with the maintenance of this masses of equilibrium under a constant tendency to subversion, than the mode in which a practised hand will sustain a long piece of paper at a position rest- ing on the ground by continuous and almost imperceptible variation of the point of support. Be this, however, as it may, the analogy is not complete; for the tinge about the planet is in itself the evidence of a perpetual contest between conservative and destructive powers--both extremely feeble, but so antagonizing one another that they cannot be reconciled without producing an uncontrollable secondary, and rushing to a catastrophe. (1448) The rings of Saturn are composed of the smallest difference of velocity between the body and rings must infinitely precipitate the latter on the former, never more to separate (as they do sometimes, once in course, have assumed a state of stable equilibrium). But let us see what be- gins ever after by an immense force! it follows, either that their motions in their common orbit round the sun amount to a very small fraction of the total external power, with the minutest precision, or that the rings must have been formed about the planet while subject to their common motion, and consequently under the full and free influ- ence of all the acting forces. (1449) The rings of Saturn must be a significant spectacle when seen from below; for they appear above their enlightened sides, as vast arches spanning the sky from horizon to horizon, and holding an invariable situa- tion among themselves. In this case, if we consider regions beneath the dark side, a solar eclipse of fifteen years in duration, under their shadow, must afford (to our eyes) an impression of great darkness; yet this is not com- pensated by the faint light of the satellites. But we shall do wrong to judge of the fitness or unfitness of their con- ditions by comparing them with those of other bodies; the very combinations which convey to our minds only im- **CHAP. VIII. GENERAL VIEW OF THE SOLAR SYSTEM.** age of horror, may be in reality theatres of the most striking and glorious displays of benevolent continuance. (447.) Of Uranus we see nothing but a small, round, uniformly illuminated disc, without rings, belts, or dia- censions. It is a planet of immense size, and from which it never varies much, owing to the smallness of our own orbit in comparison of its own. Its mean distance is about 30 times that of Jupiter, or 80 times that of the earth. It is attended by satellites—two at least, probably five or six—which orbit (as will be seen in the next chapter) with Uranus. (448.) If the immense distance of Uranus precludes all hope of coming at much knowledge of its physical situation, it is still more remote from the human intellect is no less a bar to any inquiry into theirs. One such thing, Pallius, is said to have somewhat of a nebulous or hazy appearance. The atmosphere is thin and transparent; the at- mosphere, little represented and condensed by the inade- quacy gravity of so small a body; and the most remarkable singularities must lie in this condition of their state. A man placed on one of them would proceed with ease 60 feet high, and sustain no greater shock than that occasioned by a light breeze passing a yard. On such planets giants might exist; and those enormous animals, which on earth require the buoy- ancy of the air to support them, would find their height there be denizens of the land. But of such speculation there is no end. (449.) We shall close this chapter with an illustration calculated to convey to the minds of our readers a general impression of the relative magnitudes and distances of the principal bodies in the solar system. The sun is called field or bowling green. On it place a globe, two feet in diameter; this will represent the Sun; Mercury will be represented by a sphere 1 inch in diameter; the dis- ference of a circle 164 feet in diameter for its orbit; Venus a pea, on a circle 282 feet in diameter; the Earth shall be represented by a ball 5 inches in diameter; the large pin's head, on a circle of 64 feet; Jove, Ceres, Vesta, and Pallas, grains of sand, in orbit from of 1000 to 1200 feet across; Saturn a small orange, on a circle of half a mile across; Saturn a small orange, on a circle of 773 A TREATISE ON ASTRONOMY. [CHAP. IX.] four-fifths of a mile; and Uranus a full-sized cherry, or small plum, upon the circumference of a circle more than a mile and a half in diameter. As to getting correct no- tions on this subject by drawing circles on paper, or ball, will vary with the size of the sphere represented, but it is out of the question. To imitate the motions of the planets would require two hundred million miles for each orbit, Mercury describing his own diameter in 10 hours; Venus in 4½ °; the earth, in 7 months; Mars, in 4½°; Jupiter, in 2° 58 °; Saturn, in 3° 13 °; and Uranus, in 2° 16 °. CHAPTER IX. OF THE SATELLITES. Of the moon, as a Satellite of the Earth—General Proximity of Satellites to their Prominents, and the Connection of their Motions— Mages of the Provinces of the Moon—The Phases of the Moon— The Motion of the Moon's Lenses in the Sky—The Motion of Jupiter's Selenite—Satellites of Saturn—Of Light discovered by Herschel—Satellites of Saturn— (450.) In the annual circuit of the earth about the sun, it is constantly attended by its satellite the moon, which revolves round it, and describes a round figure in its course of motion; while this centre, rotating upon itself, and neither of these two bodies thus connected, moves in an elliptic orbit, undisturbed by their mutual attrac- tion, as we shall see hereafter. The moon's small stone used together and flying into air describes a parabola if it were thrown horizontally. But when it is thrown at attraction, while the stones circulate round it or round each other, so as to conserve the course of the matter. (451.) If we trace, therefore, the motions actually described by the moon around the earth's centre, in virtue of this compound motion, it will appear to be, not an exact ellipse, but an unbalanced curve, like that repre- sented in fig. 10. The number of such curves, whether of undulations in a whole revolution is but 12, and their actual deviation from the general ellipse, which serves them as a guide, is very small indeed. It may be said, so much so, indeed that every part of the curve described CHAP. 13.] OF THE SATELLITES. 273 by either the earth or moon is concave towards the sun. The excursions of the earth on either side of the ellipse, indeed, are so very small as to be hardly appreciable. In fact, the centre of gravity of the earth is situated at al- most the same distance from the sun as that of the monthly orbit described by the earth's centre about the common centre of gravity, which is therefore nearly equal to that of the size of the earth itself. The effect is, however, sensible, in producing an apparent monthly dis- placement of the sun in longitude, or a parallactic kind, which, though not perceptible to the naked eye, yet amounts to, however, less than the sun's horizontal paral- lax, say, one minute. (482.) The moon, as we have seen, is about 60 radii of the earth distant from the centre of the latter. Its proximity, therefore, to its centre of attraction, thus esti- mated, renders it more susceptible of being attracted by the sun ; of which, Mercury, the nearest, is 84, and Uranus only 19 radii distant. But this proximity does not pre- vent the moon remaining attached to the earth as a satellite. Were it much further, the feebleness of its gravity towards the earth would be insufficient to produce any sensible effect upon it; and retardation in its motion about the sun, which diverts it from its course, would be behind the other, in their revolutions round the sun (by reason of Kepler's third law), according to the inverse dissimilarity of their distances from it; in which case, the whole influence of the earth would be confined to pro- ducing some considerable periods of disturbance in the moon's motion; but it passes out of existence past it by its each revolutionary revolution. (483.) At the distance at which the moon really is from the earth's centre towards the sun it actually lies less than towards the sun. This is that case, appears suffi- ciently from what we have already stated, that the moon's real distance from the sun is greater than its real dis- tance towards the latter. But it will appear still more clearly if, from the known periodic times in which the earth revolves around its own axis and around the sun, times; then the area is question (A) and (B) are to each other as $\frac{P}{Q}$ 274 A TREATISE ON ASTRONOMY. [CHAP. IX. earth completes its annual and the moon its monthly orbits, and from the dimensions of those orbits, we calculate the amount of deflection, in either, from their tangents, in equal very small parts of time, at the same moment. These are the mean values of the arc described in that time in the two orbits, and these are the measures of the acting gravity. But since this calculation does not excite the numerical calculation in the case before us, we shall find 3-200 : 1 for the proportion in which the intensity of gravity varies with the distance from its orbit round the sun actually exceeds that by which it is retained in its orbit about the earth. (454.) The force of gravity is more remote from the earth than the moon is. And, as gravity increases as the squares of the distances decrease, it must follow that, at equal distances from the sun, the force of gravity is greater that of terrestrial gravity in the above proportion, aug- mented in the further ratio of the square of four to one; that is, it is increased by 800 per cent. This is proved if we grant that the intensity of the gravitating energy is commensurate with the mass or inertia of the attracting body, so that if we double the mass or inertia of the earth be to no more than half that of the sun. (455.) The argument is, in fact, nothing more than a repetition of what was said respecting Saturn and Venus (art. 280). But it is here re-introduced, in order to show how the mass of a planet which is attended by one or more satellites may be determined. For this purpose it suffices provided we have learned from observation the dimen- sions of the orbits described by the planet about the sun, and by each satellite about its planet. In this way all those in which these orbits are respectively described. It is by this method that the masses of Jupiter, Saturn, and Uranus (as well as those of other planets) have been found. (456.) Jupiter, as already stated, is attended by four satellites, Saturn by seven; and Uranus certainly by two, and perhaps by three. These satellites are called "ter- naries" (as the central planets are called), form in each and since the mean value such as the square of the arc directly and the radii inversely, these are to each other as $m^2$ to $r^2$, and in this ratio are the forces acting on the revolving bodies in either case. **CHAP. IX.** OF THE SATELLITES. 375 case miniature systems, entirely analogous, in the ge- nealogy of which they are governed, to the great system in which the sun acts the part of the pri- mary, and the planets of its satellites. In each of these systems, the primary body is a planet, and the other scene, that is to say, in which they are obeyed by the planetary system—approximately, and without prejudice to the efficiency of the laws of gravitation—may be considered as infinite. But this is not so with respect to the eccentricity, if any, of that small but not imperceptible correction which arises from the elliptic form of the central body. This correction is very considerable in the case of the e- centricity, the primary occupying one focus. About this they describe arcs very nearly proportional to the times ; and their distances from the primary are such that the distances belonging to each planet are in proportion to each other as the cubes of their distances. The tables at the end of the preceding chapter give all these distances and periods in these several systems, so far as they are at present known ; and so all of them it will be observed that they have been calculated on the supposition that their primaries hold good, as in the case of the moon, with a similar reason for such close connection. (488.) The moon is certainly the only one which has been studied with great attention is that of Jupiter; partly on account of the conspicuous brilliancy of its satellite, and partly because it is capable of offering visible and measurable discs in telescopes of great pow- er; but more for the sake of its eclipses, which, as they occur at intervals of about 10 years, have afforded signals of considerable use for the determination of terrestrial longitudes (art. 219). This method, in- duced by this circumstance, has given us greater facility and exactness now attainable by lunar observa- tions (art. 219), was the best, or rather the only one which could be relied upon for great distances and long in- tervals. (489.) The satellites of Jupiter revolve from west to east round their primaries (art. 219); but those in planes very nearly, although not exactly, coincident with that of the equator of the planet, or parallel to its belt. The inclination of their orbits to that plane is small for the planet, and is therefore but little different from the plane 276 A TREATISE ON ASTRONOMY. [CHAP. IX.] of the ecliptic. Accordingly, we see their orbits pro- jected very nearly on the same lines, in which they ap- pear to oscillate to and fro, sometimes passing before Jupiter, and casting shadows on his disc (which are very visible), and sometimes appearing behind the body, or being eclipsed in its shadow at a distance from it. It is by these appearances that we have obtained our data for the construction of tables of the satellites' mo- tions, as well as with signals for determining differences of longitude. (456.) The eclipses of the satellites, in their general conception, are perfectly analogous to those of the moon, but in their actual occurrence, they differ considerably. Owing to the much greater distance of Jupiter from the sun, and its greater magnitude, the cone of its shadow or umbra (see fig. 30) is much more extended than that of the earth. The satel- lites are, moreover, much less in proportion to their primary bodies than the moon is to the sun; and even (comparatively) smaller dimensions, than is the case with the moon. Owing to these causes, the three interior satellites are always seen in full light; the fourth is al- most totally eclipsed every revolution; and the fourth, though, from the greater inclination of its orbit, it sometimes appears completely eclipsed, yet it does not pass over the border of the shadow, and suffer partial eclipses, yet this is comparatively rare, and, ordinarily speaking, its eclipses are only partial. The fifth satellite is never seen. (450.) These eclipses, moreover, are not seen, as is the case with those of the moon, from the centre of their motion; but only from points situated at some dis- tance with respect to the line of shadow is variable. This, of course, makes no difference in the times of the eclipses; but it does affect their duration. We shall find their apparent situations with respect to the planet at the moment of their entering and quitting the shadow. (451.) The first satellite revolves round Jupiter in its orbit EFJK. J Jupiter, and at the orbit of one of its satellites. The cone of the shadow, then, will have its vertex at a point on this orbit; and will be directed towards the satellites; and the penumbras, owing to the great distance **CHAP. IX.** ECLIPSES OF JUPITER'S SATELLITES. 277 of the sun, and the consequent smallness of the angle its disc subtends at Jupiter, will hardly extend, within the A diagram showing the positions of the Sun (S), Earth (E), and Jupiter (J) during an eclipse. The Sun is shown to be behind the Earth, which is in turn behind Jupiter. The Earth's shadow is cast on Jupiter. limits of the satellites' orbits, to any perceptible distance beyond the shadow,-for which reason it is not represented in figures. A satellite revolving from west to east (in the direction of rotation of the Earth) eclipsed when it enters the shadow at a, but not suddenly, because it has not had a considerable diameter seen from the planet; so that the shadow of the satellite is not a line, but a circle, and therefore a part of the total eclipse will be that which appears in describing about Jupiter an angle equal to so much of the circumference of the circle centres of the planet, or rather somewhat more, by reason of the periphery, and the same remark applies to its emergence at b. Now, since this is true of all the satellites and of eyes, it is not possible to assign the precise moment of immersion, or of total extinction at a, nor that of emergence at b, or even to determine whether it shall be at \(a\), or at \(b\), or the complete recovery of its light. The observations of eclipses, then, in which only the immersion, or only the emergence, is observed, are insufficient for any other purpose than to ascertain that they do occur; and inadequate to afford any precise information, theoretical or practical. But if both the immersion and emergence can be observed by one person at different times, and by the same person, the interval of time will give the duration between them; and if we mean the exact middle of the eclipse, when the satellite is in opposition to us, we have no means of its opposition to the sun. Such observations are very rare; and only are of use for determining the periods and other particulars necessary for calculating longitudes; and for affording data of any material use for the calculation of terrestrial longitudes. The intervals of the eclipses, it 2 A 278 A THEATRE ON ASTRONOMY. CHAP. II. will be observed, give the synodic periods of the stellar- lites' revolutions, and the times at which their intervals must be concluded by the method in art. 365 (note). (402). It is evident, from a mere inspection of our figures, that when the earth is situated to the west of the line SJ, i. e. before the opposition of Jupiter; and to the east, when it is situated to the east of this same line, the same position. When the earth approaches the opposition, the visual line becomes more and more nearly coincident with the line S J, and will finally coincide with it at the place where the eclipses happen will be continually nearer and nearer to the body of the planet. When the earth comes to G, its shadow will fall on the satellite, touching the body of the planet, the *emersion* will cause to be visible, and with thereofon, to an equal distance on the other side of the planet, a similar phenomenon, the *immersion* (or immersion) will appear at H. The die of the planet. When the earth arrives at G (or H) the immersion (or emersion) will happen at the very edge of the die of the planet. In this case, if we consider (very small space) the satellites will pass unperceived be- hind the limb of the planet. (403). When the earth comes to M, its shadow will be thrown on Jupiter, and will appear to move across it as a black spot till the satellite comes to N. But the satel- lite itself is not visible till it has passed beyond N, and up to the line drawn from E to S on the eastern edge of the die, and will not leave it till it attains a similar line drawn from S to E on the western edge of the die. Then the shadow will precede the satellite in its progress over the die before the opposition, and vice versa. In these transit of the satellites through Jupiter's atmosphere, as seen with telescopes, may be observed with great precision, it fre- quently happens that the satellite itself is discernible on the die before its shadow appears; and sometimes also as a dark spot of smaller dimensions than the shadow. This curious fact (observed by Schroe- der and Hevelius) is owing to their being situated between the satellites which occasionally cast their own shadows, or in their atmospheres, obscure spots of great extent. We may observe also that when they are seen in a position small as they appear to us, are really bodies of con- CHAP. IX.] OF JUPITER'S SATELLITES. 279 siderable size, as the following comparative table will show.*
    Mean apparent diameter in degrees Diameter in miles Mean distance from Jupiter in miles
    Jupiter 30° 27' 87000 10000000
    1st satellite 2° 56' 8200 6000172
    2nd satellite 1° 51' 8200 6000172
    3rd satellite 1° 43' 8200 6000172
    4th satellite 1° 23' 8200 6000172
    (464.) An extremely singular relation subsists between the mean angular velocities or mean motions (as they are called) of the satellites and their primary, Jupiter. If the mean angular velocity of the first satellite be added to twice that of the third, the sum will equal three times that of the second; and if from the mean longitude of the first added to twice that of the third, be subtracted three times that of the second, we shall have the mean longitude of the fourth. This remarkable fact, and other observations inform us that this constant is $180^\circ$, or two right angles; so that, the situations of any two of them being known, the position of the others may be easily ascertained to account for this remarkable fact, on the theory of gravity by their mutual action. One curious consequence of this relation is, that all four are at all eclipsed at once; for, in consequence of the last-men- tioned relation, when the second and third lie in the same quadrant, they must be eclipsed at the same time; but when they are opposite; and therefore, when the first is eclipsed, the other two must lie between the sun and planets, throwing no light upon them; and when they are eclipsed, only (so far as we are aware) is on record when Jupiter has been seen without satellites; viz. by Molyneux, Nov. 2 of this year. (465.) The discovery of Jupiter's satellites by Galileo, one of the first fruits of the invention of the telescope, forms one of those cases which illustrate the history of astronomy. The first astronomical solution of the great problem of "the longitude"---the most important question of mathematical astronomy---was brought under the dominion of strict scientific principles, date. * Stuven. Mem. Ast. Soc. iii. 27. † Leopold. Mem. Col. liv. viii. 97. ‡ Molyneux, Optics. p. 213. A table showing mean apparent diameters, diameters in miles, and mean distances from Jupiter in miles for Jupiter and its four satellites. **280** A TREATISE ON ASTRONOMY. [CHAP. IX. immediately from their discovery. The final and com- clusive establishment of the Copernican system of as- tronomy may also be considered as referable to the dis- covery and study of this exquisite miniature system, in which the earth is placed at the centre, and that of Kepler, and especially that which connect their periods with those of the sun, and the distances of the planets, should be satisfactorily maintained. And (as if to accumulate historical interest on this point) is to the observation of their eclipses that we owe the grand discovery of the orbital law of the sun, and consequently of the enormous velocity of that wonderful element. This we must explain more at large. (460.) The moon is in very close conjunction with that of Jupiter and interior to it (see fig. art. 460), their mutual distance is continuously varying, the variation depending from the difference between the periods of the two orbits, and the difference of the greater and least dia- meters being equal in both cases, they are both equinoctial. Now, it was observed by Roemer (a Danish astronomer, in 1675), in comparing together observations of eclipses of the satellites during many successive years, that the eclipse of Jupiter's satellite when near its perigee (or nearest point to the earth) took place too soon—sooner that is, than, by calculation from an average, he expected them; and that when near its apogee (or farthest point), the earth was in part of its orbit most remote from Jupi- ter's satellite, they were always too late. Connecting the observed error in their times with those calculated from his mean motion, it was concluded, that to make the calculation on an average period correspond with time an allowance in respect of time before or after perigee must be made for the defects of Jupiter's distance from the earth above or below its average amount, and such a difference of distance of one degree in longitude would require an allowance of 16° 28' 6" of time allowed. Speculating on the probable physical cause, he was naturally led to think of the gradual diminution of light by refraction through a medium. This explained every particular of the observed pha- nomenon, but the velocity required (18000 miles per second) was so great as to be beyond all human means to require confirmation. This has been afforded since, A diagram showing two celestial bodies, presumably Jupiter and Earth, with their respective orbits around a central point. The diagram includes labels indicating "perigee" and "apogee" points for each body. **CHAP. IX.] SUCCESSIONAL TRANSMISSION OF LIGHT. 281** and of the most unequivocal kind, by Bradley's discovery of the precession of the equinoxes (see p. 305). The amount of light deduced from this last phenomenon differs by less than one eighteenth of its amount from that calculated from the ephemeris of Jupiter's satellites, which has been de- stroyed by nicer and more rigorously reduced observations. (487.) The orbits of Jupiter's satellites are but little eccentric, and their motions are very nearly per- ceptible eccentricity; their mutual action produces in them perturbations analogous to those of the planets about the Sun. These perturbations have been investi- gated by Laplace and others. By assiduous observation it has been ascertained that they are subject to marked fluctuations, which occur at irregular intervals. These fluctua- tions happen periodically, according to their position with respect to the sun. From this it has been con- cluded that these perturbations are due to the mutual axes, like our moon, in periods equal to their respective spherical revolutions about their primary. (488.) The motions of Saturn's moons have been much less studied than those of Jupiter. The most distant is by far the largest, and it is probably not much inferior to Mars in size. It revolves in a plane perpendicular to the plane of the ring, with which those of all the rest nearly coincide. It is the only one of the number whose theory has been worked out, furnishing evidence to verify Kepler's law of motion in times, which build up good, mutatis mutandis, and under the requisite reser- vations, in this as in the system of Jupiter. It exhibits, like the other planets, a diurnal rotation, and light, which prove its revolution on its axis in the time of a single revolution around Saturn. The next (or order [pro- ceeding inwardly]) is identical in appearance with the nearest very minute, and requiring pretty powerful telescopes to see it clearly while revolving itself; satellite which just skirt the edge of the ring. Its motion is similar to that of the others, but never discerned but with the most powerful telescopes which human art has yet conceived, and therefore not observed. At the time of their disappearance of the ring (to ordinary telescope) they have been seen* threading like beads the almost *My mistake; it was while I was attempting some theoretical 3 A 2 985 A TREATISE ON ASTRONOMY. [CHAP. IX. infinitely thin fibre of light to which it is then reduced, and for a short time advancing off it at either end, speedily to return, and hastening to their habitual concealment. Owing to the inequality of the ring, and of the orbits of the satellites to Saturn's axis, some of them are seen to pass near the satellites (the interior ones excepted) until near the time when they are eclipsed. (490.) With the exception of the two interior satel- lites of Saturn, the attendants of Uranus are the most dif- ficult objects to obtain a sight of, any of our system. Two unexceptional cases have been observed by me, connected with each other, and both remarkable and unex- pected. These two, however, offer remarkable aid, and, indeed, quite unexpected and unexamined peculiarities. Conway's telescope shows us that in this solar sys- tem—whether of primaries or secondaries—the planes of their orbits are nearly perpendicular to the ecliptic plane; hence, in all cases where they are visible, and in these orbits their motions are retrograde; that is to say, their positions, when projected on the ecliptic, instead of being in one direction, as would be the case of their primary, as is the case with every other planet and satellite, move in the opposite directions. Their orbits are so placed that they may be supposed to have any sensible, or, at least, any rapid motion of nodes, or to have undergone any material change of incli- nations, in the course of time, of half a revolution of their primary round the sun. * * * These unusual perihelion, which seem to occur at the extreme limits of our system, as if to prepare for the further departure from all its members, and to bring them into a new order. I am aware that this idea has been entertained by many astronomers; but I am not inclined to hold it on account of their discoveries, who alone had ever observed it. The object of this treatise is not to discuss this question; but since it has been brought forward by some writers from 1736 to the present time, to confirm, in the simplest manner, my father's result. A table.
    Page Number 985
    Chapter IX.
    Text A TREATISE ON ASTRONOMY.
    Paragraph Infinitely thin fibre of light to which it is then reduced,
    and for a short time advancing off it at either end,
    speedily to return, and hastening to their habitual concealment.
    Owing to the inequality of the ring, and of the orbits of
    the satellites to Saturn's axis, some of them are seen to pass near
    the satellites (the interior ones excepted) until near the
    time when they are eclipsed.
    (490.) With the exception of the two interior satel-
    tites of Saturn, the attendants of Uranus are the most dif-
    ficult objects to obtain a sight of, any of our system.
    Two unexceptional cases have been observed by me,
    connected with each other, and both remarkable and unex-
    pected. These two, however, offer remarkable aid, and,
    indeed, quite unexpected and unexamined peculiarities.
    Conway's telescope shows us that in this solar sys-
    tem—whether of primaries or secondaries—the planes of their orbits are nearly perpendicular to the ecliptic plane;
    hence, in all cases where they are visible,
    and in these orbits their motions are retrograde; that is
    to say, their positions, when projected on the ecliptic,
    instead of being in one direction, as would be the case
    of their primary, as is the case with every other planet
    and satellite, move in the opposite directions. Their
    orbits are so placed that they may be supposed to have any sensible,
    or, at least, any rapid motion of nodes, or to have undergone any material change of incli-
    ninations, in the course of time, of half a revolution of their primary round the sun.
    Text
    DescriptionDescription
    Page Number985
    ChapterXI.
    TextA TREATISE ON ASTRONOMY.
    ParagraphInfinitely thin fibre of light to which it is then reduced,
    and for a short time advancing off it at either end,
    speedily to return, and hastening to their habitual concealment.
    Owing to the inequality of the ring, and of the orbits of
    the satellites to Saturn's axis, some of them are seen to pass near
    the satellites (the interior ones excepted) until near the
    time when they are eclipsed.
    (490.)With the exception of the two interior satel-
    tites of Saturn, the attendants of Uranus are the most dif-
    ficult objects to obtain a sight of, any of our system.
    Two unexceptional cases have been observed by me,
    connected with each other, and both remarkable and unex-
    pected. These two, however, offer remarkable aid, and,
    indeed, quite unexpected and unexamined peculiarities.
    Conway's telescope shows us that in this solar sys-
    tem—whether of primaries or secondaries—the planes of their orbits are nearly perpendicular to the ecliptic plane;
    hence, in all cases where they are visible,
    and in these orbits their motions are retrograde; that is
    to say, their positions, when projected on the ecliptic,
    instead of being in one direction, as would be the case
    of their primary, as is the case with every other planet
    and satellite, move in the opposite directions. Their
    orbits are so placed that they may be supposed to have any sensible,
    Description  Description  Description  Description  Description  
    Page Number
    Chapter
    Text
    Paragraph
    Description Infinitely thin fibre of light to which it is then reduced, Description and for a short time advancing off it at either end, Description speedily to return, Description and hastening to their habitual concealment. 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CHAP. X. j NUMBER OF COMETS. 253 CHAPTER X. OF COMETS. Great Number of recorded Comets. The number of recorded perha- ples such great number--Description of a Comet--Comets without Tail- or--Comets with Tail--The Comets of the Ancients--The Comets of the general Laws of planetary Motion--Elements of their Orbital--Eccentric and Epicyclic Orbits--The Comets of the ancients--Their Resemblance to the Solar System--Their Resemblance by the Solar, general Decies, and possible Explanations. (470.) The extraordinary aspect of comets, their rapid and seemingly irregular motions, the unexpected manner in which they often burst upon us, and the imposing magnificence with which occasionally assume, have in all ages rendered them objects of wonder and admiration, with superstitious dread to the un instructed, and an enligh- tened reverence to the learned. They have been regarded as the crea- tion and the operations of natural causes. Even now, that we have ceased to regard their movements as irregu- lar, we still consider them more than those which revolve in the planets in their orbits, their mutual motion, and the offices they perform in the economy of our system, as mere accidents at best. No one has even plausible account has yet been given for those immense volu- minous appendages which they bear about with them, and which appear to be the cause of their tails, (though and while it may be doubted whether any of their mo- tions,) any more than several other singularities which have been observed. (471.) The number of comets which have been astro- nomically observed, or of which notices have been re- corded, is very great; but it is difficult to give an exact number, hundreds? and when it is considered that in the earlier ages of astronomy, and indeed in more recent times, before the invention of the telescope, only large and conspicuous comets were noticed, it will be seen how great must be the number which have escaped to the time of its publication. 1827), etc., a catalogue now in print contains nearly two hundred and fifty comets. In this work I have included all those which appeared between the first of January, 1827, and the last of December, 1829. Of these I have omitted all those which were announced by Messrs. Lalande and Delisle in their "Annals" for 1827 and 1828. I have also omitted all those which appeared after the first of January, 1830. This omission was made on account of the difficulty of determining their positions with sufficient accuracy until after they had passed beyond the limits of my catalogue. It is hoped that some future work will include all those which have since appeared. 284 A TREATISE ON ASTRONOMY. [CHAFF. X.] once were noticed; and that, since due attention has been paid to the subject, scarcely a year has passed without the observation of one or two of these bodies, and that sometimes two and even three have appeared at once; it will be seen how much more active they have been than at least many thousands. Multitudes, indeed, must es- cape all notice, because they are so small, and because only that part of the heavens which is above the horizon in the daytime. Comets so circumstanceaned can only be come visible by the sudden appearance of a new ellipse of the sun's disc. The evidence which happened, recorded by Seneca, 60 years before Christ, when a large comet was actually seen to pass through the sun's disc, is sufficient to prove that comets are visible in the daytime, even at noon and in bright sunshine. Such were the comets of 1407 and 1680, both of which appeared a little before the ascension of Caesar, and were (afterwards) made to have predicted his death. (q) Yet it is remarkable that such appearances may be excited by the sudden and unexpected appearance of a great comet, is no way surprising; being, in fact, ac- cording to Aristotle, a very common phenomenon, one of the most brilliant and imposing of all natural phenomena. Comets consist for the most part of a large and splendid tail of dust and gas, which extends outwards from the head which is usually much brighter towards the centre, and offers the appearance of a vivid nucleus, like a star or pla- teau. From this point of view, it is apparent that to comets to which the sun is situated from the moment they appear to diverge two streams of light, which grow broader and move directly away from each other. These streams sometimes lie in and unite at a little distance behind it, sometimes continue distinct for a great part of their course; but at other times they appear to join together into some bright meteor, or like the diverging fire of a sky- rocket (only without sparks or perceptible motion). This is the aim of all comets; for they always maintain essen- tially an immense apparent length. Aristotle relates the fall of the comet of 371 a. c., that it occupied a third of the heavens; and that it was observed by many who had been attended by a train no less than 104° length. The comet of 1680, the most celebrated of modern times, C.MAP. X. SMALL DENSITY OF COMETS. 285 and on many accounts the most remarkable of all, with a head of extreme brilliancy, at the time of the second magnitude, covered with its tail an extent of more than 70° of the heavens, or, as some accounts state, 90°. The figure of this comet was first published in the description of the comet of 1819—by no means one of the most consider- able, but the latest which has been conspicuous to the public. (473.) The tail is, however, by no means an invariable appendage of comets. Many of the brightest have been observed without it, and others whose tails have been entirely without them. Those of 1685 and 1703 offered no vestige of a tail ; and Cassini describes the comet of 1748 as having none whatever. On the other hand, instances are not wanting of comets furnished with many tails or streams of diverging light. Thus, in 1783, a comet was seen to have two im- mensen fens, extending to a distance of nearly 30° in length. The tails of comets, too, are often curved, bending, in general towards the sun, and sometimes away from it, as if moving somewhat less directly, or as if resisted in their courses. (474.) The smallest comets, such as are visible only in telescopes, or with difficulty by the naked eye, and which are by far the most numerous, offer very frequently no apparent tail. They appear to be composed of some- what oval vaporous masses, more dense towards the cen- tre, where, however, they appear to have no distinct nu- cleus. These masses are so closely united together as to be regarded as a solid body. Stars of the smallest magnitude remain distinctly visible, though covered by what appears to be the same vapour that surrounds them. Such stars would be completely obliterated by a moderate fog, extending only a few yards from the surface of the earth. As regards the tails of comets, those which are larger comets which have presented the appearance of a nu- cleus have yet exhibited no phases, though we cannot doubt that they will do so when they come into per- tuals that even these can only be regarded as great masses of thin vapour, susceptible of being penetrated through their surfaces by rays from the sun; and thus causing them alike from their interior parts and from their sur- 286 A TREATISE ON ASTRONOMY. [CHAP. X.] faces. Nor will any one regard this explanation as foreseen, nor the discovery of the inconceivable qua- lity in the comet itself, to account for the phenomena in question, when we consider (what will be hereafter shown) the enormous magnitude of the space thus illumi- nated. The light which falls upon the comet must have its ground to attribute to these bodies. It will then be evi- dent that the most unobstructed clouds which float in the highest regions of the atmosphere, and which can be drenched in light, and to glow throughout their whole depth as if in actual light, without any shadow or dark side, must be composed of some substance as transparent compared with the filmy and all but spiritual texture of a comet. And yet, strangely, whereas powerful telescopes have been turned on these objects, they have failed to depict the illusion which attributes solidity to that more condensed part of the atmosphere which makes a eye so much larger though it is true that in some very minute stellar point has been seen, indicating the exist- ence of a solid body. (475.) It is in all probability to the feeble cohesion of the elastic power of their gaseous parts, by the gravitation of so small a central mass, that this phenomenon takes place ex- cepting only those parts of the atmosphere where it does. If the earth, retaining its present size, were reduced, by any amount whatever, by drawing out its central part into one thousandth part of its present size, its evasive power over the atmosphere would be dimin- ished in proportion, and in consequence of this diminu- tion it would expand again at a time equal to actual bulk ; and indeed much more, owing to the still farther dimi- nution of gravity, by the recess of the upper parts from the centre. (476.) That the luminous part of a comet is something in the nature of a smoke, fog, or cloud, seems to be transparently suggested by what has been observed, which has been often noticed, viz.--that portion of the tail where it is most luminous is always that which re- appeared from it by an interval less luminous, as if sus- tained and kept off from contact by a transparent stratum, or as if it were surrounded by a transparent membrane with a considerable clear space between them. These, and CHAP. X.] MOTIONS OF COMETS. 287 most of the other facts observed in the history of comets, appear to indicate that the structure of a comet, as seen in section in the direction of its length, must be that of a hollow envelope, of a parabolic form, enclosing near its vertex a solid core, which may be considered as represented in the annexed figure. This would account for the ap- A diagram showing a comet with a hollow envelope and a solid core. parent division of the tail into two principal lateral branches, the envelope being oblique to the line of sight at its borders, and therefore a greater depth of illumination being necessary to produce the same effect. In this respect, however, they admit great varieties of structure, and among them may very possibly be bodies of widely different constitution. (477). We come now to speak of the motions of co- mets. These are apparently most irregular and capri- cious; some move with great rapidity for several few days, at others for many months; some move with ex- treme slowness, others with extraordinary velocity; while still others seem to change their apparent speed are exhibited by the same comet in different parts of its course. The comet of 1747 described an arc of the heavens of about 30 degrees diameter. Some pursue a direct, some a retrograde, and others a tortuous and very irregular course; nor do they confine them- selves to any particular part of the heavens, but traverse indifferently every part. Their variations in apparent size, during the time they continue visible, are also very considerable; sometimes they make their first appearance as faint or slow moving objects, with little or no sail; but by degrees they increase in brightness until they reach this appendage, which increases in length and brightness till (as always happens in such cases) they approach the sun. At last they become so bright that again they emerge, on the other side, receding from the sun with a 288 A TREATISE ON ASTRONOMY. [CHAP. X.] velocity at first rapid, but gradually decaying. It is clear thus passing the sun, and not till then, that they shine forth in all their splendour, and that their tails acquire their peculiar appearance. This is owing to the sun's rays plainly the action of the sun's rays as the exciting cause of that extraordinary emanation. As they continue to proceed, they are gradually absorbed by the tail, which the tail dies away, or is absorbed into the head, which itself grows continually feeble, and is in length altogether lost sight of, so far the greater number of cases never to be seen more. (478.) Without the cause furnished by the theory of gravitation, many of those phenomena would have been inexplicable and enigmatical, and some of them might have remained for ever unresolved. But Newton, having demonstrated the possibility of this cause, was enabled to explain much about the sun, by a body revolving under the dominion of that law, immediately perceived the applicability of the general law to other bodies in space. In his treatise on the great comet of 1680, one of the most remarkable on record, both for the immense length of its tail and for its rapidity in its motion, he described about the sun (within one sixth of the diameter of that luminary), afforded him an excellent opportunity for the trial of his theory; and he found it to agree with his observations. He maintained that this comet described about the sun as its focus an ellipse orbit of no great an eccentricity as to be undulatory, and that it describes an orbit without any apparent or limiting form of the ellipse when the axis becomes infinite), and that in this orbit the areas described about the sun are equal at all times. This is true even to the times. The representation of the apparent motions of this comet by such an orbit, throughout its whole observed course, agrees exactly with the course of the motions of the planets in their nearly circular paths. From that time it became a received truth, that the motions of all celestial bodies are subject to these laws as those of the planets—the difference in these cases consisting only in the extraneous elongation of their elipses and in their being subject to the same conditions as those of their planes in respect to each other—general coincidence in the direction of the motions from west to 280 **CHAP. X.** **PERIODICAL COMETS.** east, rather than from east to west, like what is observed among the planets. (479.) It is a problem of pure geometry, from the general laws of elliptic or parabolic motion, to find the straight line which would represent the motion of any given comet. In ge- neral, three complete observations of its right ascension and declination, made at intervals of about six months, were made, suffice for the solution of this problem (which is, however, a very difficult one), and for the determination of the elements of its orbit. The two quantities, $\alpha$ and $\delta$, of the same data as are required for the com- putation of the motion of a planet; and once deter- mined, they will enable us to compute with certainty the whole observed course of the comet, by a process ex- actly similar to that of art. 495, and thus at once to ac- curately determine its orbit. This is a proof that the truth of those general laws on which all such calcula- tions are founded. (480.) But, in most part, it is found that the motions of comets may be sufficiently well represented by para- bolic orbits,—that is to say, ellipses whose axes are indi- rectly parallel to each other. For instance, it is imper- ceivable error in the calculation of their motions, during all the time they continue visible, would be incurred by supposing them to have orbits perfectly circular; but some section which is the limit between the ellipse on the one hand, which returns into itself, and the hyper- bola on the other hand, which recedes continually without rest, which should describe an elliptic path, how- ever long its axis must here visit the sun before, and never return to it again until after a considerable period,—but should its orbit lie of the hyperbolic char- acter, when once it has passed its perihelion, it could neither come back to visit the sun again nor go out, but must run off to visit other systems, or be lost in the immensity of space. A very few comets have been se- en to describe such paths instead of ellipses. These then, as far as their orbits can remain unaltered by the attractions of the planets, must be re- garded as periodic. (481.) The most remarkable of these is the comet of 2 B A diagram showing an elliptical orbit with a point labeled "A" near one focus. 200 A TREATISE ON ASTRONOMY. [CHAP. X.] Halley, so called from the celebrated Edmund Halley, who, on calculation of elements from its perihelion pas- sage in 1682, when it appeared in great splendour, with a tail 30° in length, was led to conclude its identity with the great comet of 1531, which had been long supposed to had also ascertained. The intervals of these successive apparitions being 76 and 78 years, Halley was encour- aged to suppose that the same comet would again appear So remarkable a prediction could not fail to attract the attention of all astronomers, and, as the time approached, it became evident that the observations made during the tractions of the larger planets might not materially inter- ferre with its orbital motion. The computation of their influence upon the orbit of Jupiter was found to be most difficult and intricate piece of calculation, was undertaken and accomplished by Clairaut, who found that the action of Saturn upon Jupiter would cause its period to increase of Jupiter by no less than 518, making in all 618 days, by which the expected return would happen later than on the 12th of March in this year. This is very conjectural; and that, in short, the time of the expected perihelion passage would take place within one month, one way or other, between the 12th of March 1763 and the 12th of March 1764. On the 12th of March in this year. Its next re- turn to the perihelion has been calculated by Messrs. Dunham and Lambert. It will then occur about two months on the fourth, and by the latter on the seventh of Novem- ber, 1853, about a month or six weeks before which time it may be expected to pass through perihelion. And, as it will approach pretty near the earth, it will very probably be seen at its return appearance though, owing from the successive appearances of Jupiter to its large size and the length of its tail in its several returns since its first appearances on record (in 1685, 1486, &c.), we are not now to expect any extraordinary phenomena from it; but those which shew our remote ancestors of the middle ages into agonies of superstition terror, and caused public prayers to be put up against such agencies as are said to be malignant agencies. (482.) More recent times have been especially identified with this comet performed several revolutions about the sun, and as having been not only observed and re- **CHAP. X.] RESISTANCE EXPERIENCED BY COMETS.** 201 corded in preceding revolutions, without knowledge of the circumstances which had already pre-empted their return, and kept to their appointments. The first of these is the comet of 1682, discovered by the celebrated Sir J. Bar- lin, who first ascertained its periodical return. It re- volves in an ellipse of great eccentricity, inclined at an angle of about 35° with the plane of the ecliptic, and in the short period of 1207 days, or about 34 years. This remarkable discovery was made on the occasion of its passage through the meridian of Paris, on the 1st of April calculated then by Echele, its return in 1822 was predicted by him, and observed at Paramatta, in New South Wales, on the 1st of April, 1822, at 10 minutes after sunset; since which it has been re-predicted, and re-oberved in all the principal observatories, both in the northern and southern hemispheres, in 1825, 1836, and 1845. Its next return will be in 1858. (483.) On comparing the intervals between the suc- cessive passages of this comet through the meridian, varying in the most careful and exact manner for all the dis- tances due to the actions of the planets, a very singular fact appears to be exhibited; namely that they are continu- ally diminishing, or, in other words, the mean distance from the sun, or the major axis of the ellipse, dwindling by degrees. This diminution is produced by two effects, which would be produced by a resistance experienced by the comet from a very rare ethereal medium pervading the whole system of our solar system; namely that by diminishing its actual velocity, would diminish also its centrifugal force, and thus give the sun more power over it than it formerly possessed (see p. 197), and so accounting for the phenomenon in question appearing, this is the solution proposed by Echele, and generally received by astronomers. But when we reflect upon it, the sun should not first be dissipated altogether—a thing no way impossible, when the lightness of its ma- terial is considered; but it is evident that it is not the observed fact of its having been less and less conspicuous at each reappearance. (484.) Another comet of short period which has lately been discovered is that of Biela, so called from 202 A TREATISE ON ASTRONOMY. [CHAP. X.] M. Biala, of Josephstadt, who first arrived at this inter- esting conclusion by the idea of a comet which appeared in 1789, 1705, &c., and describes its mode- rately eccentric eclipse about the sun in 61 years; and has last year predicted that in 1832 will be an eclipse, the prediction in 1822, the next will be in 1838. It is a small insignificant comet, without a tail, or any appear- ance of one, but it has made some remarkable coincidences, very nearly intersects that of the earth; and had the latter, at the time of its passage in 1822, been in the same place with the former, it would have passed through the comet--a singular rea- son, perhaps not unattended with danger." (485.) The motions of the planets are so different that the planets are materially drawn aside from these courses, and in some cases have their orbits entirely changed. This is remarkable, because they are all subject to some strange fatefully to be constantly in their way, and to serve as a perpetual stumbling block to them. In the case of comets, however, this is not the case. It was found by Lecell to revolve in a moderate ellipse in the period of about 5 years, and whose return was predicted by him accordingly. But when he examined his predictions, he found actually getting entangled among the satellites of Jupiter, and being completely thrown out of its orbit by the ac- tion of the planet Saturn on its elliptical orbit. By this extraordinary remonitory, the motions of the satellites suffered not the least perceptible damage- ment--a sufficient proof of the smallness of the comet's mass. (486.) It remains to say a few words on the actual di- scovery of comets. It is evident that no one can predict by this means, the subject of periodic comets will assume an extreme many degrees of eccentricity; and therefore many new ones will be discovered, and by their resistance questions will come to be decided, whether they are periodic or not. We may also suppose that there exists a body which surrounds the sun. Is it at rest or in motion? If it be latter, its that does not move with respect to the sun. If it be former, others according as their revolution was relative to each other, direct or indirect. Now it is evident that if we suppose a body revolving around a material fluid, it is not conceivable that the circulation of the planets is not affected by it; and consequently that they must change their direction in their own direction. And this may preserve them from the extreme of being carried off into space. **CHAP. X.]** DIMENSIONS OF COMETS. 993 mention of comets. The calculation of the diameters of their heads, and the lengths and breadth of their tails, offers not the slightest difficulty when once the elements of their motions are known. We know their real distances from the earth at any time, and the true direction of the tail, which we see only forwards. But this is not all. It has been found that these facts lead to the surprising fact, that the comets are by far the most voluminous bodies in our system. The following are some of the results which have been made the subjects of such inquiry. (487.) The tail of the great comets of 1890, inno- dizably exceeds 100 million leagues in length. The pre- vious to have been no less than 30000000 of leagues in length, and to have occupied only two days in its emis- sion. This is a very remarkable fact. It shows that if its being darted forth by some active force, the origin of which, to judge from the direction of the tail, must be sought in the sun's atmosphere, it must have been extended to 41000000 leagues, a length much exceeding the whole interval between the sun and earth. The tail of the comet of 1861 was 50 million leagues long, while that of the great comet of 1811, 36000000. The portion of the head of this last comprised within the transparent sphere of the earth is estimated at 180 million leagues; but the tail was 18000000 league in diameter. It is hardly conveni- able that matter once projected to such enormous dis- tances should be able to return again under the influ- ence of such a body as a comet—a consideration which accounts for the rapid progressive diminution of the tails of comets after they leave the neighbourhood (488.) A singular circumstance has been remarked re- specting the change of dimensions of the comet of October, 1867. It was observed by Sir William Herschel, that the real diameter of the visible nebulosity under- goes a rapid contraction as it approaches, and an equally rapid expansion as it recedes. Mr. John Herschel, who, among others, had noticed this fact, has accounted for it by supposing a real compression or condensation of the material composing it into a more dense mass, or growing more dense in the sun's neighbourhood. It is very possible, however, that the change may consist in A diagram showing a comet with its head and tail.294 A TREATISE ON ASTRONOMY. UMAF. XL no real expansion or condensation of volume (further than is due to the convergence or divergence of the different parabolas described by each of its molecules to or from a common vertex), may readily induce the per- verse and apparently paradoxical variations in the upper regions of a transparent atmosphere, into the states of visible light, which are produced by the effects of heat and cold. But it is time to quit a subject so mys- terious, and open to such endless speculation. CHAPTER XI. OF PERTURBATIONS. Subject proposed—Suppression of small Motions—Problem of three Bodies—Estimation of Datarcing Forces—Motion of Nodes—Changes of Latitude and Longitude—Theorem of Euler—Note—Lagrange's Theorem of the Stability of the Ellipses—Change of latitude and longitude—Theorem of Lagrange—Nutation—Theorem of Generalized Velocities—Of the Tides—Variation of the Tides with latitude and longitude—Datarcing Forces considered as undulatory and mutual—Effects of im- mense masses on small bodies—Effect of the Sun on the Moon—Effect of the Moon on the Earth—Effect of the Earth on the Moon—Effect of the Sun on the Earth—Effect of the Earth on the Moon—Effect of the Moon on the Sun—Effect of the Sun on the Earth—Effect of the Sun on the Moon—Effect of the Moon on the Earth—Effect of the Earth on the Moon—Effect of the Sun on the Moon—Effect of the Moon on the Earth—Effect of the Earth on the Moon—Effect of the Sun on the Moon—Effect of the Sun on the Earth—Effect of the Earth on the Moon—Effect of the Sun on the Moon—Effect of the Sun on the Earth—Effect of the Earth on the Moon—Effect of the Sun on the Moon—Effect of the Sun on the Earth—Effect of the Earth on the Moon—Effect of the Sun on the Moon—Effect of the Sun on the Earth—Effect of the Earth on the Moon—Effect of the Sun on the Moon—Effect of the Sun on the Earth—Effect of the Earth on the Moon—Effect of the Sun on the Moon—Effect of the Sun on **CHAP. XI.]** OF PERTURBATIONS. 925 bations. They arise, in the case of the primary planets, from the mutual gravitations of these planets towards each other, which derange their elliptic motions round the sun, and produce a series of perturbations, arising from the mutual gravitation of the secondaries of the same system similarly deranging their elliptic motions round the sun, and producing a similar series of perturbations by attraction of the sun on them and on their primary. These perturbations, although small, and in most in- stances inconspicuous, yet have been so long and so accumulated, as some of them may become, in the lapse of ages, very greatly the original elliptical relations, so that they have produced great changes in the primary orbits, which at one epoch represented perfectly well their movements, inadequate and unsatisfactory after later periods. (400.) When Newton first reasoned his way from the broad features of the celestial motions, up to the law of gravitation itself, he was not aware that he was rendering every particle in the universe subject to the influence of every other, he was not unaware of the magnitude of this influence, but he did not perceive how it led to the result of a more partial and limited application of the same law to the revolutions of the planets about the sun, and to those of the moon about her own centre, their only centre of attraction. So far from it, that his ex- traordinary exigency enabled him to perceive very dis- tinguishedly that all bodies are attracted by each other; those inequalities take their origin, in this more general way of conceiving the agency of the attractive power, especi- ally when applied to the earth's motion, than by any direct revolution of the apesides of her orbit. And if he did not extend his investigations to the mutual perturbations of these bodies, he could not have perceived that such perturbations must exist, and might go the length of producing great derangements from the actual state of the system. He had no idea whatever of applying to the practical part of astronomy, which had not yet as- tained the precision requisite to make such an attempt impossible. But though he did not pursue this line of thought, however, his successors have accomplished ; and, at this day, there is not a single perturbation, great or small, A diagram showing a celestial body orbiting around another body.306 A TREATISE ON ASTRONOMY. [CHAP. XL which observation has ever detected, which has not been traced up to its origin in the mutual gravitation of the parts of our system, and been minutely accounted for, in its effects on motion and value, by strict application of Newton's principles. (491.) Calculations of this nature require a very high analytical power, and a great deal of time; but it is far beyond the scope and object of this work to attempt ex- hibiting. The reader who would master them must prepare himself by a long course of study, and must ascend by steps which we must not here even digress to point out. It will be our aim to give him a general idea of the nature and manner of operation of the acting forces, and to point out what are the circum- stances under which they operate with the highest degree of efficiency—a sort of purchase on the balance of the system; while in others, with no loss amount of intensity, they are rendered ineffective. The increments and lasting changes is compensated or rendered averted; as well as to explain the nature of those admirable re- sults revealed by the experiments of Kepler; whose researches of geometers have conducted them; and which, under the form of mathematical theorems of great beauty, have been applied with such success to verify the past and future state of the planetary orbits during ages, of which, contemplating the subject in this point of view, we consider perceive that beginning not the end. (492.) There were no other studies in the universe but the sun and moon, until last century; when the exact ellipse about the former (or both round their com- mon centres of gravity), and continue to perform its revolu- tions in accordance with Newton's laws; at which moment we add to our combinations a third body, the at- traction of this will draw both the former bodies out of their mutual equilibrium, and will cause them to mutually disturb their relation to each other, and put an end to the rigorous and mathematical exactness of their ellip- tical motions. This third body is Mercury, which occupies a position in space. From this way of propounding the subject, we see that it is not the whole attraction of the newly in- CHAP. XI. OF PERTURBATIONS. 397 produced body which produce perturbation, but the dif- ference of its situation on the two originally presented. (493.) Compared to the sun, all the planets are of ex- treme minuteness; the mass of Jupiter, the greatest of them all, is less than one hundredth part of that of the sun. Their attractions on each other, therefore, are all very feeble, compared with the preceding central power, and consequently their motions are very gradually minute. In the case of the secondaries, the chief agent by which their motions are deranged is the sun itself, while whence they derive their motion, and by whose influ- ence is immensely diminished by their near proximity to their primaries, compared to their distances from the sun, while they are so much more distant from him than he is extremely small, compared to the whole amount. In this case, the greatest part of the sun's attraction, viz. that which acts upon his own satellites, is exerted upon pri- mary and secondary in their common orbit about itself, and prevent their parting company. The small overplus of the sun's attraction upon his primary has been seen value of this overplus, in the case of the moon disturbed by the sun, is calculated by Newton to amount to no highly considerable quantity; viz. that which acts upon the earth's surface, or $\frac{1}{10}$ of the principal force which retains the moon in its orbit. (494.) From this extreme minuteness of the intensities of the disturbing, compared to the principal forces, and the consequent smallness of their momentary effects, it has been concluded by Newton that errors may hap- penly arise, as if the others did not take place, without dear of inducing error in our conclusions beyond the limits necessary for accuracy. This conclusion is a principle in mechanics, immediately flowing from the primary relations between forces and motions they produce; because when any two bodies are at once set in motion by a single cause, their joint effect is the sum or ag- gregate of their separate effects, at least within such limits, that they do not exceed those limits; and hence shall not have been materially changed by their action. Such effects supervening on the greater movements due to the same cause, will be proportionally smaller than those of the small ripples caused by a thousand varying breezes on 308 A TREATISE ON ASTRONOMY. [CHAP. XI. the broad and regular swell of a deep and rolling ocean, which run on as if the surface were a plane, and cross in all directions, without interfering, each as if the other had no existence. The same may be said of the waves communi- cated in lapse of time, so as to alter the primary rela- tions or data of the system that it becomes necessary to have an account of the motions of the particles successively introdused into the estimation of their momentary efficiency, by which the rate of the subsequent changes is affected, and periodical variations are produced in the same way. From this consideration arise some of the most curious theories of physical astronomy. (489.) It is evident that in estimating the dis- turbing influence of several bodies forming a system, in which one has a remarkable preponderance over all the rest, we must consider how far the disturbing power of any one body is transmitted to another among them, unless where immensely long periods are concerned; such as consist of many degrees of inequality, and where we can have no con- sideration about their common centre. So that, in effect, the problem of the investigation of the perturbations of a system reduces itself to that of finding out what is due to itself or that of a system of three bodies : a predominant central body, a disturbed, and a disturbed ; the two lat- ter of which are mutually attracted by each other. But the motions of the one or the other are the subject of inquiry. (490.) The intensity of the disturbing force is consi- derably varying, according to the relative situation of the disturbing and disturbed body with respect to the sun. If the attitude of these bodies be altered, then P (which designates, for brevity, we shall hereafter indicate them), will be equal, and consequently there will be no variation; otherwise be it law of variation, there would be no deviation caused in the elliptic motion of P about S, or of each about the other. This is shown by 385 ; but it is 385; the attraction of M, so circumstanced, being at every moment exactly analogous in its effects to terres- trial gravity. Hence it follows that there is no equal intense on all bodies, great and small. But this is not the case of nature. Whatever is stated in the subsequent **CAP. XI.] PROBLEM OF THREE BODIES.** article to that last cited, of the disturbing effect of the sun and moon, is, mutatis mutandis, applicable to every case of perturbation ; and it must now be our business to examine the disturbances of the general heads of the subject there merely hinted at. (487.) We shall begin with that part of the disturbing force which arises from the mutual attraction of bodies in the plane in which its orbit would be performed if undisturbed, and, by so doing, cause it to describe a curve, of whatever kind may be supposed, in that plane. As it is called in geometry, a curve of double curvature. Suppose, then, APM to be the orbit which P would describe about M if disturbed by no other body than P itself, at any instant of time, and to be about to describe in the next instant the undisturbed arc PP', which prolonged in the direction of its motion will meet the orbit at the plane of the orbit ML, suppose in E. This would be the case if we could draw such a line as SL through P, but we do so, since it draws both S and P towards it, in directions not coincident with the plane of S'P's orbit, it will be seen that this line cannot pass through E. It is evident that plane, but unequally—first, because it does not draw them both in parallel lines; secondly, because they, being unequal in mass, attract each other unequally; thirdly, by reason of the general law of gravitation. Now, it is by the difference of the motions thus generated that the disturbance is produced. For let us consider what happens when we continue to refer its motion to S as a fixed centre, the disturbing part of the impulse which it receives from M will be communicated to P by means of a line drawn through P parallel to SL; but since this line passes through E at the next instant of time, not the arc PP', but an arc AP', lying either above or below PP', according to the prepon- derance of the attractions on one side or another. (488.) The disturbing force acts in the plane of the triangle SPM, and may be considered as resolved into two; one perpendicular to SP (which we call F), and therefore increases or diminishes, in so far as it is effective, the direct attraction of S or P; the other along a line parallel to SP (which we call P'), and therefore either pulling P in the direction P'S or pushing it in a contrary direction; these terms being well understood to 300 A TREATISE ON ASTRONOMY. [CHAP. XL have only a relative meaning as referring to a supposed fixity of $b$, and transfer of the whole effective power to P. The former of these forces, acting always in the plane of $p$ and $q$, will be called the disturbing force, and urge out of that plane: the latter only is so effective, and therefore will be called the resolution of forces being needed to explain with this plane the motion of the nodes, which we have before considered (499.) (499.) In the situation, or configuration, as it is termed, represented in the figure, the force, in the direction $P_{k}$, is a guiding force, and acts on the node M. Let us below the plane of $P_{k}$ obtain (taking that of M's orbit for a ground plane), it is clear that this disturbed arc is de- scribed by the node M, by P, and by P', as shown P'. When prolonged, therefore, to intersect the plane of M's orbit, it will appear at a point $r$, behind R, and the line $R_{r}$ which describes this arc is perpendicular to the plane SPg (now, for instance, that of $M$'s disturbed motion), or the new line of nodes, will fall behind R, the undisturbed line. Hence it appears that when the node M will have retrograded by the angle SPg, the motions of F and M being regarded as direct. (500.) The same thing happens to the left instead of the right of the line of nodes, P retaining its situation, then will be drawn $P_{l}$ parallel to $P_{k}$, and raise P out of its orbit, so throw Pg above Pp, and rise adscense of R. In this configuration, then, the node will move; but as soon as P passes the node, and comes to the lower side of M's orbit, although all other forces will diminish, and Pg will increase, continue to lie above Pp, yet in this case, the little arc Ps will have to be described backwards by M. This is a proper plane, and, when so prolonged, will lie below the similar prolongation of Ps, so that, in this case again, the node will remain stationary. (501.) Thus we see that the effect of the disturbing CHAP. XL.] MOTION OF THE MOON. 301 force, in the different states of configuration which the body P and M may assume with respect to the node, is to keep the line of nodes in a continual state of fluctuation to and fro ; and it will depend on the excess of cause favourable to the advance over that of cause favourable to the reccess, in an average of all the possible configurations, whether, on the whole, an advance or recess of the node shall take place. (502.) If the orbit of M be very large compared with that of P, so large that MP may, without material error, be considered as a straight line, then during each revolution of the moon's orbit disturbed by the sun, it will be very readily seen, on an examination of all the possible varieties of configuration, that during every single complete revolution of P, the cases favourable to a retrograde motion will always predominate. This is owing to the primary tendency, the retrogradation taking place over a larger extent of the whole orbit, and being at the same time more rapid than the progration, which is only a section of the force to the recess. Hence it follows that, on the whole, during every revolution of the moon about the earth, there will be produced a net effect in favour of conformance to experience, with a velocity varying from zero to limitation. The amount of this retrogradation, when we consider that during every single revolution of P all the acting forces, is found to coincide with perfect precision with that immediately derived from observation, we see that this effect must be regarded as a real process by which so remarkable an effect is produced. (503.) Theoretically speaking, we cannot estimate correctly how great will be this effect; but let us think of the moon's orbit with the ecliptic, from a mere consideration of the disturbance one of these planes. It is a compound phenomenon; for we must first consider its effect on an imaginary fixed ecliptic, and, to obtain the compound effect, we must also regard the earth as disturbed in its relation to this plane. Now, since in consequence of account for the excessive distance of the sun, the intensity of the moon's attraction on it is quite evanescent, compara- tively speaking; and since this is also true of any per- turbative effect in this case, which is the difference of 2 C 302 A TREATISE ON ASTRONOMY. CHAP. XI. the moon's attraction on the sun and earth, is equal to the whole straction of the moon on the earth. The effect of this is to produce a monthly displacement of the centre on either side of the ecliptic, whose amount is easily calculated, and which may be considered as due to gravity as lying entirely in the ecliptic. From this it ap- pears, that the moon's attraction on the earth is only a small fraction of the earth's radius in its total amount; and, therefore, that its momentary variation, on which the motion of the nodes of the ecliptic on the moon's orbit depends, must be utterly unimportant. (504.) It is otherwise with the mutual action of the planets. In every case both the orbits of disturbed and disturbing planets will be affected by the moon. Precisely on the above-justed principles it may be shown, that the effect of each planet's attraction on the orbit of every other planet will be very small compared with that on one orbit only or other in certain configurations, and a reason for this is furnished by considering the mean motion on the average of many revolutions in a regular regula- tion of the node of each orbit on every other. But since this is the case with every pair into which the planets can be combined, it follows that no single pair can affect any joint action on any orbit, taking into account the different situations of all their planes, because a singu- lar and considerable effect would result. This may be very easily expressed in words, though reducible to strict numerical statement, and being in fact a mere geometri- cal result as regards the orbits themselves. (505.) The nodes of all the planetary orbits on the true ecliptic are retrograde, although (which is a matter of material importance) they are not so in a fixed plane, such as we may conceive to exist in the planetary system, and so to be a plane of reference unaffected by mutual attractions. We have seen that this is a principle, that we are under the necessity of referring their movements from our stations in the system ; and if we transfer our ideas to this plane, we shall find that we take account of the variation of the ecliptic itself, produced by the joint action of all the planets. (506.) The difference in the masses of the planets, and their great distances from each other, the re- **CHAP. XI.]** **CHANGE OF INCLINATION.** values of their nodes are exceedingly slow, being in every case less than a single degree per century, and in most cases not amounting to half that quantity. So far as the physical condition of each planet is concerned, it is evident that the change of inclination has little importance. It is otherwise with the mutual inclinations of their orbits, with respect to each other, and to the equator of the sun. The inclination of the orbit of a planet is elliptic, for instance, by which its pole should shift its distance from the pole of the equator, would disturb our sense of uniformity. But this is not so. In all cases, ever be so changed as to bring the ecliptic to coincide with the equator, we should have perpetual spring over all parts of the earth's surface. If the orbit of a planet with a meridian, the extremes of summer and winter would become intolerable. The inquiry, then, of the value of these changes of inclination, and their cause, is one of much higher practical interest than those of their nodes. (507.) Referring to the figure of art. 498, it is evident that the plane $S_{P}F$, in which the disturbed body moves during an instant of time from its quincunx $T$, is differently inclined to the plane $S_{P}F'$, in which it moves from the original or undisturbed plane $S_{P}F'$. The difference of absolute position of these two planes in space is the angle $\alpha$ between them. This angle can be calculated beforehand by spherical trigonometry, when the angle $R_{P}$ or the momentary recess of the node is known, and also when the direction of motion is known with respect to each other. We perceive, then, that between the momentary change of inclination and the momentary recess of the node there exists a certain relation. The first part of this re- search of the one is in fact bound up in that of the other. This may be, perhaps, made clearer, by considering the orbit of a planet as a hoop or ring. Such a hoop is an actual circular or elliptic hoop of some rigid material, without inertia, on which, as soon as a wire, the body $P$ may move along at any point whatever. The direction of this hoop will be determined at any instant, by its inclination to the ground plane to which it is referred, and by the plane through its center and axis. The direction will be determined by the momentary direction of $P$'s motion, A diagram showing two planes $S_P F$ and $S_P F'$ intersecting at an angle $\alpha$. The line $S_P F$ represents a planet's orbit around a star $S$, while $S_P F'$ represents an undisturbed orbit. The angle $\alpha$ between these planes indicates a change in inclination. 304 A TREATISE ON ASTRONOMY. [CHAP. XL which (having no inertia) it must obey ; and any change by which P moves in a circle about S, or any other point, would be equivalent to a shifting, bodily, of the whole hoop, changing at once its inclination and nodes. (586.) The most remarkable fact which has been pointed out above, is that where the orbits, as in the case of the planetary system and the moon, are slightly inclin- ed to each other, the difference of inclination of one with the inclination of the other is of a order much inferior in magnitude to those in the place of the node. This is evident on a mere inspection of the diagram. For instance, by reason of the small inclination of the planes SPB and SBsR, necessarily much smaller than the angle RSB. In proportion as this difference increases, by coincidence, a very trifling angular movement of Pp about PS as an axis will make a great variation in the situation of the points B and B'. On the contrary, when these planes coincide (500.) To pass from the momentary changes which take place in the relations of nature to the accumulated effects of long periods, we shall proceed by a continued action of the same causes, under circumstances varied by these very effects, in the business of the integral calculus. It is easy to see how this may be done; for it will be easy for us to trace, by a few cases, the varying influence of differences of position of the disturbing and disturbed bodies upon their mutual motions. We shall now proceed from these to demonstrate the two leading features in this theory—the periodic nature of the change and even oscillation of these motions—by showing how close the small limits within which these changes are confined. A diagram showing three bodies labeled P, S, and M. Body P is moving around body S. Body M is also moving around body S but at a different angle. The diagram shows that body P's orbit around body S is more affected by body M's orbit than vice versa. (510.) Case 1.—When the disturbing body M is situ- ated in a direction perpendicular to the line of nodes, or **CHAP. XL.** MOTION OF THE NODES. 308 the node $M$ in quadrature with itself: $M$ being the disturbing body, and $N$ the line of nodes, the disturbing force will act at $P$, in the direction $PR$; being a pulling force when $P$ is in any part of the semicircle HAN, and a pushing force when $P$ is in any part of the semicircle ANB. And it is easily seen that this force is greatest at A and B, and evanescent at H and N. Hence, in the whole semicircle ANB, the motion of $M$ will be produced backward in the quadrant HA, and forward in AN, will meet the circle SMe in the plane of $M$'s orbit, and consequently will produce a retarding effect on $M$ in both cases. But the new inclination of the disturbed orbit is, in the former case, PZA, which is greater than PNA; and in the latter case, PAH, which is greater than PNA. In the other semicircle the direction of the disturbing force is changed; but that of the motion, which is always parallel to SM, remains unchanged. Now, in each quadrant reversed, the same variations of node and inclination will be caused. In this situation of $M$, then, the inclination diminishes throughout the quadrant SA, but the inclination diminishes throughout the quadrant SA, increases again by the same identical degree in the quadrant SB, decreases again by the same identical degree in NA, and is finally restored to its pristine value at S. On the average of a revolution of $P$, supposing $M$ unmoved, the node will have retrograded with their utmost speed, but the inclination will remain unaltered. (511.) Case 2.--Suppose the disturbing body now to be fixed in the line of nodes, or the nodes to be in syzygy, as in the annexed figure. In this situation the direction of the disturbing force, which is always parallel to SM, has constantly such a form of $P$'s orbit, with there- 3 c 2A diagram showing a celestial body (M) orbiting around a central point (O), with a line of nodes (N) passing through it. The diagram includes various labeled points and lines representing different aspects of orbital mechanics.306 A TREATISE ON ASTRONOMY. [CHAP. XL] fore produce either variation of inclination or motion of nodes. (612.) Case 3.—Let us take now an intermediate situation of $M$, and indicating by the arrows the directions of the disturbing force, we shall see that, through- out all the semianis (which lies towards $M_1$ and through- out the opposite), it will readily appear that the reasoning of art. 611. is good, in all that part of the orbit which lies between $N_1$ and $H$, and between $V_1$ and $H$, but that the effect will be reversed by the reversal of the direction of the disturbing force, when it acts on $M_1$'s orbit, in the intervals HT and NV. In these portions, however, the disturbing force is feeble than in the others, being even wanting in some quadratures TV, and in- A diagram showing a celestial body with various points labeled: H, M, N, V, P, Q. creasing to its maximum in the syzygies a b. The nodes then will revolve rapidly in the former intervals, and ad- vance freely in the latter; but since, as H approaches to $\alpha$, the disturbing force, by acting obliquely to the plane of PS, will increase its action upon $M_1$, so that, during the average of a whole revolution, the nodes recede. On the other hand, the inclination will now diminish during the motion of $M_1$, because it is continually forced from the node, while it increases not only during its whole motion over the quadrant CN, but also in the rest of its half revolution beyond it. Therefore, he will be an uncompensated change in this position of $M_1$ on the average of a whole revolution. (613.) But this increase is converted into diminution when the line of nodes stands on the other side of SM, as in the case represented by $PQ$. Let $M_2$ be fixed, and supposing that the change of circumstances **CHAP. XI.] CHANGES OF INCLINATION.** 207 arises not from the motion of M but from that of the node, it is evident that so soon as the line of nodes in the course of one revolution of the planet on its circumference will be all exactly reversed, and the inclination will again be augmented in each revolution by the very same steps taken in the opposite direction, the whole will be finished. On the average, therefore, of a whole revolution of the node, the inclination will be restored to its original state; but when this is not so, because the motion on the inclination is constrained, instead of supposing M fixed in one position, we might conceive it at every instan- taneous moment to be in any place whatever, making equal angles on either side of the line of nodes, in which case it is evident that the effect of two of the parts would be such as to annihilate that of the others with revo- lution of P. (1) In what is said, we have supposed M real; but it may be imagined, as to its actual and final resul- tants good if it be supposed in motion; for in the course of a revolution of the nodes, which, owing to the eccentricity of the orbit, is not uniform, and to the planets, is of immense length, amounting, in most cases, to several hundred centuries, and in that of the moon, to several thousand years; yet during all this time M is presented by its own motion, over and over again, in every variety of situation to the line of nodes. Before proceeding further on this subject, let us suppose that has performed a complete revolution, and is restored to its place; so that, in fact (that small difference excepted which occurs between two successive revolutions), in a syn- cyclical revolution of M), we may regard it as occupying at every instant every point of its orbit, or rather as having at every instant a different position on the whole circumference. Thus the compensation which we have shown would take place in a whole revolution of M; but since this compensation takes place at periodic period $M$, that minute difference only excepted which is due to the cause just mentioned. This difference, therefore, must be made up by another cause; what produces the effective variation of the inclinations, wha- ter of the lunar or planetary orbits; and this difference, which is continually encountered by the motion of M in $M$ 308 A. THEATRE ON ASTRONOMY. [CHAP. XL its own compensated by the motion of the node during its whole revolution. (§15.) It is clear, therefore, that the total variation of the planetary inclinations must be compensated within very narrow limits by the action of the other forces, as has been demonstrated, by an accurate analysis of all the circum- stances, and in exact accordance with the law of being forces, that such is the case, and this is what has been shown by verifying the stability of the planetary system as to the mutual inclinations of its orbits. By the researches of Lagrange (§ 16) on this subject, and those which I have here to give any idea), the following elegant theorem has been demonstrated: If the square root of every planet be multiplied by the square root of the major axis of its orbit, and the pro- ducts by the square of the tangent of its inclination to a fixed plane, then these products will be constan- tly the same under the influence of their mutual at- traction, and consequently under that of their mutual re- action. Now, if we take for example one of these planets, and let the ecliptic be taken for that fixed plane (the ecliptic itself being variable like the other ordinate), it is found that this sum is actually very small; it must, therefore, always remain so, whatever may be the inclination of the plane which guarantees the stability of the orbits of the greater planets; but from what has been above shown, the tendency of each planet to assume a position near to that which it occupies, it is evident that the minor ones are not excluded from this beneficial arrangement. (§16.) The reason is no doubt that the plane of the ecliptic does actually vary by the actions of the planets. This variation is about one degree per century, and has long been observed astronomically, by an increase of the latitudes of all the stars in certain situations, and their diminution in the opposite regions. Its effect is to diminish somewhat those planes nearer to coincidence with the equator; but from what we have above seen, this diminution of the obliquity of the ecliptic cannot exceed a few degrees under its limits, after which (although in an immense period of age, being a compound cycle resulting from the joint action of all these causes), all those planes which oscillate backward and forward about a mean position, A diagram showing a celestial sphere with various lines representing different positions and movements. CHAP. XI.] PRECISION OF THE EQUINOXES. 309 the extent of its deviation to one side and the other being less than 1' (57°). One effect of this variation of the plane of the ecliptic, that which causes its nodes on a fixed plane to change—is mixed up with the procession of the equinoxes, and is therefore difficult to separate from the prece- pts in theory. This last-mentioned phenomenon is, however, due to another cause, analogous, it is true, in a sense, to that which produces the precession of the nodes, but singularly modified by the circumstances under which it is produced. We shall endeavour to render these modi- fications intelligible by means of a diagram drawn out without the intervention of analytical formulæ. 618. In the precession of the equinoxes, as we have shown in Art. 560, we have to consider the retrograda- tion of the node of the earth's equator on the ecliptic, and is, therefore, obviously an effect so far analogous to the precession of the nodes as to be treated together with them. The nodes of the orbits on each other. The imminence dis- tance of the planes, however, compared with the size of the earth's orbit renders their action incom- parable to that of suns, put their action out of the question. But in astronomy they are very important; therefore, look to this immense though distant sun, and to our near though minute neighbour, the moon, for its explanation. This is accordingly, as I found in their discovery, acting on the equator of the earth, by which its figure is ren- dered elliptical instead of circular; and consequently its axis. It is to the sagacity of Newton that we owe the discovery of this singular mode of action. 619. Of course, when we speak of a ring revolving round S, that instead of one body, P, revolving round S, there were a succession of particles not coherent, but forming a ring of such a kind that all the centrifugal forces applied—while this ring revolved round S in its own plane, under the disturbing influence of the distant body (which has no regular motion), would produce a force perpendicular to the plane of the earth's equator; two things would happen—1st. Its figure would be rendered elliptical instead of circular; and 2ndly, its centre of it within the area Vr and Vd (fig. art. 511) being 310 A TREATISE ON ASTRONOMY. [CHAP. XL rendered more inclined to the plane of its orbit, and those with less inclination, would have the effect that they would otherwise be. 2826. The nodes of this ring, regarded as a whole, without respect to its change of figure, would retreat upon that plane, which opposes this ring, instead of consisting of discrete molecules free to move independently, to be rigid and incapable of any motion whatever. But if we suppose that they have such motions as in art. 567, then it is evident that the effort of those parts of it which tend to become more inclined than before, will be counteracted by the rest (as a mechanical engine or lever) to counteract the effort of those which have at the same instant a contrary tendency. For if the former have their motion in excess on the one or the other side will the instaneous change in an average being struck at every moment of the ring's motion be equal to zero? This is evident when we have taken of the inclinations, in every complete revolution of a single disturbed body, under the influence of a fixed disturbance. (581.)分子体, however, the nodes of the rigid ring will retrograde, the general or average tendency of the molecule is very much like that of a wheel. Here, as in the other case a struggle will take place by contri- buting efforts of the molecules contrariy disposed, progressing towards opposite directions in one ring; and, at every instant of time, an average will be found which average being identical in its nature with that ef- fected by a single particle moving in a circular body, will, in every case, be in favour of a recess of the mode, save only when the disturbing body, be it sun or moon, is situated in the plane of the earth's equator, or in the case of the fig. art. 510. (582.) This reasoning is evidently independent of any consideration respecting the relative velocities of the members of the ring; whether the particles be small ellispheres re- tained in circular orbits under the equilibrated action of attractive forces acting perpendicularly to their planes or small masses conceived as attached to an image of imaginary wheels as of a wheel, centering in S., and free to only shift their planes so as to keep them always perpendicular to the plane of the wheel. This makes no difference in the **CHAP. XI.] PRECISION OF THE EQUINOXES.** general effect; though the different velocities of motion, which may be supposed to take place in the two cases, may and will have a very great influence both on the absolute and relative magnitudes of the two effects in question—the motion of the equinoxes and the precession of the equinoxes. This will be easily understood, if we suppose the ring without a rotatory motion, in which extreme case it is obvious, that so long as no external force acts upon the ring, its place will remain at all, but only by tendency of the ring to tilt its plane round a diameter perpendicular to the plane of the ring. (303.) The motion of such a ring, then, as we have been considering, would imitate, so far as the reces of the ring is concerned, the motion of the earth, so that its nodes would retrograde far more rapidly than the ob- served precession, which is excessively slow. But now consider what happens when the mass of the ring is enormously heavier than itself, placed concentrically within it, and cohering firmly to it, but indifferent, or very nearly indifferent, to any other force whatever. Suppose, moreover, that instead of one such ring, there are a vast multitude heaped together around the equator of such a globe as this. In this case each particle of matter acting like a shell on all sides, but whose mass, taken together, should form but a very minute fraction of the whole sphere's mass. It is evident that this is a most repre- sentation of case of nature *, and it is evident that the rings, having to drag round with them in their nodal re- volutions masses equal to themselves, must carry out a retrogradation proportionally diminished. Thus, then, it is easy to conceive how a motion, similar to the preces- *That a perfect sphere would be so inert and indifferent as to a move- ment that it could not be affected by any external force whatever; sup- posing body appear from thus—that the direction of the resultant attractions of all these particles would be always towards the centre; and denying by whole action, it necessarily in lieu passing through the centre. Now let us suppose that this centre be situated at some point on the sphere one way or other. It may be objected by the reader, that since we know that every point on a sphere is equally distant from its centre, and that therefore, no node should ever be situated at any particular point on the sphere; but this is not correct; our laws will not allow us to give an exposition of the helixes without supposing that they are situated at some particular points; and this subject is open in many matters of this kind, which nothing but the closest observation can show us with certainty. 213 A TREATISE ON ASTRONOMY. [U.M.F. XI. sion of the equinoxes, and, like it, characterized by ex- treme slowness, will arise from the causes in action. (824.) Now a recess of the node of the earth's equa- tor, upon which the line of nodes is situated, is that part of its axial round a perpendicular to that plane. But in the case before us, that plane is not the ecliptic, but the moon's orbit with the earth. Hence we must consider whether we are to reconcile this with what is stated in art. 266, re- specting the nature of the motions in question. To this we reply, that the moon's orbit with the earth is in a state of continual and rapid retardation, while its inclination is preserved nearly invariable, the point in the sphere of the heavens where it is situated being called the node, or pole (with that extreme slowness characteristic of the precession) is itself in a state of continual oscillation round the pole of the earth's axis, and consequently in rapid motion which belongs to the lunar node. A glance at the annexed figure will show how this is done. P is the pole of the ecliptic. A, B, C, D are four points on the circle ABCD in 19 years ; e is the pole of the earth's equator, which at each moment of its progress is a direc- tion perpendicular to that varying position of the line A E, and a veloc- ity depending on the varying incli- nation of the ecliptic to the plane of the period of the nodes. This ve- locity, however, being always directed to B, C, D, E, the line AE will have taken up the positions BA, Co, Df, Ef, and the earth's pole a will thus, in one tropical revolution of the node, have arrived at e having described a curve ABCEDEF. The curve is made up of a con- densation of a wave-shaped or epicycloidal curve, ab c d e, with which it is somewhat greener and less than its mean motion, and this curve describes one tropical revolution of the node. (825.) Now it is precisely this kind of motion which, as we see in art. 273, the pole of the earth's equa- tor really has round the pole of the ecliptic, in conse CHAP. XL.] MUTATION. 513 quence of the joint effects of precession and nutation, which are thus aragonraphically represented. If we superind on the effect of lunar precession that of the so- lar precession, and if we suppose the earth's orbit to be circular uniformly about P, this will only affect the nutations of our waved curve, by extending them in length, but without altering their form, and consequently will have no effect on the excursions of the earth's axis to and from the pole of the ecliptic. Thus we see that the two phenomena of au- tomatic motion and nutation are connected with each other, both of them essential constituent parts of one and the same phenomenon. It is hardly necessary to state that the solar precession is produced by the mutual re- stationing of all the acting forces and summation of their dynamical effects, *lead* to the precise value of the co- efficients of these forces, and that the same consideration assigns to them. The solar and lunar portions of the precession of the equinoxes, that is to say, those portions which are due to the action of the moon, are in proportion about 2 to 5. (308.) In the rotation of the earth's axis we have an example of a case where a body may change its way by a periodical movement in one part of the system, giving rise to a motion having the same precise period in another part. In this case also, as in all others, we see, represented, though under a very different form, yet in the same exact periodic time, by the movements of a single body, such as a planet revolving round a solid mass of the earth. We must not let the opportunity pass of generalizing the principle involved in this result, it is applicable to every kind of motion whatever, whether in any part of physical astronomy, say, in every depart- ment of natural science. It may be stated as "the prin- ciple that every body is at rest or at rest vibrations," and thus generally announced: If one part of any system connected either by mate- rial or spiritual bonds with another part, however remote, be continually maintained by any cause, whether in- herent in the constitution of the system or external to it, so long as that cause continues to exist, its vibration will be propagated throughout the whole system, and *Vide Prof. Ayr's Mathematical Treatise, vol. ed. p. 309, &c. 2 D 314 A TREATISE ON ASTRONOMY. [CHAP. XL said give rise in every member of it, and in every part of each member, to periodic movements executed in equal periods with that to which they owe their origin, though they may be very different from one another in their manner and intensity. The system may be favourably or unfavourably con- sidered, either as a train of periodic movements, or favourably in some of its parts and unfavourably in others; and, accordingly, as being one or the other, the direction of the motion (as it may be) will be imperceptible in one case, of appreciable magnitude in another, and even more perceptible in its visible effect than the quantity of matter contained in any single kind we have an instance in the moon's acceleration to be here- after noticed. (287.) It so happens that our situation on the earth, and the delicacy which our observations have attested, enable us to see, as it were, into the interior of the shaken forced vibrations, or derived motions communi- cated from various quarters, especially from our near neighbour, the moon. The moon is a body which we de- tect by the presence of a board beneath us, by the transfer of motion by which the sound of an organ pipe is dispersed through the air, and by the motion of the earth. According to the revolution of the moon, and the annual motion of the sun, produce, each of them, small oscillations in all parts of the earth. These are re- spectively half a month and half a year, each of which, in this view of the subject, is to be regarded as one por- tion of a period of twenty-eight days and two months. But the most remarkable instance, by far, of such a per- iodic motion, and one of high importance to man- kind, is that excited by the rotation of the earth in an ocean disturbed from its figure by the varying attractions of the sun and moon, each revolving in its own orbit, and propagating its own period into the junction between them. (288.) The tides are a subject on which many persons find a strange difficulty of conception. That the moon, by her lawless descent upon the earth's surface, should have been connected by material use of impenetrable elasticity, is my treatment on this subject. But I am aware that this connection is only extended and generalised to take in other systems.--Austel CHAP. XL.] THE TIDES. 315 her attraction, should keep up the waters of the ocean under her, seems to be always very natural—that the same cause should, at the same time, keep them up on the opposite side, seems to many palpably absurd. Yet this is exactly what we find to be the case. When we consider that it is not by her whole attraction, but by the differences of her attractions at the two surfaces and at the two sides of the earth, that she does this, so that by forces directed precisely as the arrows in our figure, art. 510, in which we may suppose M the moon, and P the pole of the earth, all parts of the surface of water existing alone would take a spherical form, by reason of the attraction of its parts; and if the same difference of attraction existed between every part of an uniform gravity, since every part would be equally accelerated, the particles would retain their relative positions; but since they are not so accelerated, they sup- pose it to fall under the influence of an attraction acting on each of its particles independently, and increasing in proportion to their distance from the centre. Now, those more near the centre of attraction would be attracted more than the central, and less than the more remote, and those more distant would be attracted less than the motion into an oblong form; the tendency to separation being, however, counteracted by the attraction of the poles, and consequently by that of the moon's body thus established. Now, in fact, the earth is constantly falling to the moon, being continually drawn by it out of its circular orbit; and this is true not only for those so near as to the central; and thus, at every instant, the moon's attraction sets to force down the water at the sides, as right as possible towards her; and thus she has a side of the diameter pointing towards her. Geometry corroborates this view of the subject, and demonstrates that the earth must have a certain inclination to cover a sphere, under the influence of the moon's attrac- tion, would be an oblong ellipsoid, having the semi- axis parallel to the moon longer by an angle greater than that transverse to it (art. 516). There is never time, however, for this phaibod to be complete; for when any can take their level, the moon has advanced in her orbit, both diurnal 316 A TREATISE ON ASTRONOMY. [CHAP. XL. and monthly (for in this theory it will answer the purpose of electricity to suppose that the sun's luminous motion transferred to the sun and moon in the contrary direction), the vertex of the spheroid has shifted on the earth's surface, so that the spring tide is now occurring. The effect is to produce an immensely broad and extremely flat wave (not a circulating current), which follows, or endeavors to follow, the line of the equator. The moon, and must, if fact, of the principle of forced vibrations be true, imitate by equal, though not by synchronous, periods, the motions of the sun and moon respectively. When the higher or lower parts of this wave strike our coasts, they experience what we call high and low water. (801.) The same law holds good with respect to waves whose verticals tend to follow the apparent motion of the sun in the heavens, and also to imitate its periodic inequalities. But since these inequalities are not uniform, sometimes superposed on it, sometimes transverse to it, so as to partly neutralize it, according to the monthly synodic inequality of the moon's period, there is an alternate mutual reinforcement and destruction of the solar and lunar tides cause what are called the spring and neap tides. This difference is due to their different periods. Although the real amounts of either tide is, at present, hardly within the reach of exact calculation, yet that part of it which is due to the sun is very remote from that of the ellipsities which would belong to their respective spheroids, could an equilibrium be attained between them. The mean spring and lunar spheroids, are respectively about two and five feet; so that the average spring tide will be to the mean sea up 7 or 8 inches. (801.) Another effect of the combination of the solar and lunar tides is what is called the priming and lagging of the sun's motion. This arises from the fact that while in the plane of the equator, the tide-day (i.e. the interval between two successive arrivals at the same place of the same wave) is nearly equal to one day; but in other places (art. 116) formed by the combination of the moon's sidereal period--and that of the earth's diurnal motion. Simultaneously with this effect on all points in latitude, on the equator, the tide-day would be the mean solar day. **CHAP. XI.** THE TIDES. The actual tide-day, then, or the interval of the occur- rence of two successive maxima of their superposed waves, will vary as the separate waves approach to or recede from coincidence; because, when the variations of the tides are equal, they will coincide at one maximum and at another minimum, but will be at a point intermediate between them. The variation of the time of occurrence of the maxima and minima of the tides with respect to each other is therefore a function of the difference of the lengths of the successive tide-days is particularly to be remarked about the time of the new and full moon. (392.) Quantum of light. Its origin is that deviation of the sun's high and low water at any port or harbour, from their culmination of the luminaries, or of the theoret- ical maximum of their apparent sphere, which is called "the quantum" or "the error." If the water were without inertia, and free from obstruction, either owing to the friction of its surface or to the tow- nships and vessels which it has to traverse be- fore reaching the port--that length, &c., &c., the times of culmination would be constant. But these causes tend to create a difference, and to make that dif- ference not alike at all ports. The observation of the earth's rotation shows that this difference is immense; nor is it of less consequence, theoretically speaking, to a knowledge of the true distribution of the tidal forces. For example, in making such observa- tions, care must be taken not to confound the time of "slack water," when the current caused by the tide ceases to flow in one direction with that of "high water" for some seafar, when the level of the surface ceases to rise or fall. These are totally distinct phenomena, and depend on en- tirely different causes. They may occur at some times coincide in point of time. They are so feared, too often mistaken one for the other by practical men; or circumstances may arise which render it necessary the greatest confusion in any attempt to reduce the sys- tem of the tides to distinct and intelligible laws. (393.) The cause of tides.--It is essentially "The recent investigations of Mr. Lockhart, and those highly interest- ing ones in which Mr. Whewell is understood to have engaged, will be of great service in elucidating this subject; but until we have a clear knowledge of what causes produce them, it will be difficult to determine the elements of our theory at the same time give it that direct connection with our own ideas as it deserves, without which all de- termination is lost before it begins." 517A diagram showing a cross-section of Earth's orbit around Sun.318 A TREATISE ON ASTRONOMY. [CAP. XI.] affects the tides at any particular spot. As the waves of the tide-water tend to place itself vertically under the luminary which produces it, when this vertical changes its point of incidence on the surface, the tide-wave must tend to that luminary. The luminaries, during their annual periods, must tend to increase and diminish alter- nately the principal tides. During the period of the moon's nodes the tides are brought into this state of excessiveness or declination in the course of one of these being 28°, and in another only 17°, on either side the equator. (584.) The efficacy of the luminary in producing a luminary in raising tides is inversely proportional to the cube of its distance. The sun and moon, however, by reason of their gigantic size, are more than 100 times nearer to and farther from the earth than their mean dis- tances. In consequence of this, the efficacy of the sun will fluctuate between 100 and 1000, taking 100 for its mean value, and that of the moon between 40 and 50, varying like this: -the difference, the highest possible will be to the lowest one, +51 to +49, or so to 24, or 16 to 13. Of all the causes of differences in the height of tides, however, local situations are those which have most effect. When the tide-water, rushing up a narrow channel, is suddenly raised to an extraordinary height. As Annamite, for instance, where "The Floods rise to 20 feet," etc. **EVEN AT BRISTOL**, the difference of high and low water together amounts to 50 feet. (585.) The moon is not always in the moon, in like man- ner, produces tides in the atmosphere, which delictate observations with great difficulty, but are still visible and mea- surable. This effect, however, is extremely rare. (586.) To return now, to the planetary perturbations. Let us next consider the changes induced by their mu- tual actions upon each other. Their mutual positions and their positions therein differ in different situations with respect to each other. In the first place, however, it will be proper to explain how it is that geometers and astronomers have alike agreed to use the language and laws of the elliptic system, and to continue to apply them to all cases whatever. Robson's Lecture on Mechanical Philosophy. CHAP. XI.] THE TIDES. 219 so disturbed are no longer, in mathematical strictness, absolutely what they are known to be. They do this, partly on account of the convenience of conception and calculation which attaches to this system, but much more for the sake of simplicity. For it may be demonstrated from the dynamical relations of the case, that the departure of each planet from its ellipses, as determined at any given moment, is so small, that by supposing the ellipse itself to be slowly variable, to change its magnitude and eccentricity, and to shift its position as well as its axis, we shall have all the necessary laws; while the planet all the time continues to move in this ellipse, just as it would do if the ellipse remained unchanged. By this way of considering the subject, the whole permanent effect of the disturbing forces is regarded as thrown upon the ellipse itself, and not upon the planets themselves; which remain unchanged, or only liable to brief and comparatively momentary fluctuations. This course of procedure, indeed, has been adopted by Mr. Hamilton, who employs us by the extreme slowness with which the variations of the elements develop themselves. For instance, the frequency of the variation of the eccentricity is such that it changes no more than 0-0005 in its amount in a century; and the place of its perihelion, as referred to the sphere of reference, varies no more than 0-0001 in one thousand years' time. For several years, therefore, it would be next to impossible to distinguish between an ellipse so varied and another exactly similar; and even after a long inclu- sion, the difference between the original ellipse so varied and another exactly similar would be less than six feet in diameter, the nearest examination with microscopes continued along the whole outlines of the two curves represented by them; and yet these differences be- tween them. Not to call a motion so minutely conforming itself to an elliptic curve, elliptic, would be affecta- tive; for it is evident that such a motion must always recur alternately on one side or on the other; though, on the other hand, to neglect a variation, which continues to accumulate until it overpowers all force itself on our notice, would be wifful blindness. 200 A TREATISE ON ASTRONOMY. [CHAP. XI. (387.) Geometers, then, have agreed in each single revolution, or for any moderate interval of time, to re- gard the motion of each planet as elliptic, and performed according to Kepler's laws, with a reserve in favour of certain very small inequalities. But this reserve is at the same time to regard all the elements of each ellipse as in a continuous though extremely slow, steady change, and in treating of the motions of the planetary system, they take account principally, or entirely, of this change of the elements, to that which after, all, any mate- rial change may grow up during the revolution will ultimately depend. (388.) And we may encounter the distinction between those which are secular variations, and such as are re- pidly periodic, and are compensated in short intervals. In our exposition of the variations of the inclination of a disturbed orbit (art. 814), we saw that the variation in each single revolution of the disturbed body, the plane of its orbit being supposed fixed, is equal to the inclination to that of the disturbing body, which nearly compensated each other; leaving, however, a portion outstanding. This portion is equal to one half of that by the revolution of the disturbing body, yet still leaving out- standing and uncompensated a minute portion of the change which would be produced by the whole. We do not compensate and bring it back to an average or mean value. Now, the two first compensations which are operated upon by these inequalities are those arising from configurations with each other, and therefore in compara- tively short periods, are called periodic variations; and the last compensation is that arising from their per- iodicities depending on configurations ; while the last, which is operated by a period of the node (one of the elements), and therefore in comparatively long periods, the individual planets, require an immense period of time for its consummation, and is, therefore, distinguis- hed from the rest. (389.) It is true, that to afford an exact representation of the motions of a disturbed body, whether planet or satellite, all these inequalities must be taken into account, in their corresponding inequalities, require to be express- ed; and indeed, the former even more than the latter; A diagram showing a celestial sphere with various lines representing orbits and nodes. CHAP. XL.] VARIATIONS, PERIODIC AND SECULAR. 321 seeing that the secular inequalities are, in fact, nothing but what remains after we have described of a much greater amount (as it very often is) of perturbations. But these are in their nature transient and temporary; they disappear, and leave no trace. The planet is temporarily in a state of equilibrium with its sun, but forthwith returns to it, so deviates presently as much the one way as the other, and thus produces the sub- jective sum to the average of these excursions on either side of it; and thus continues to present, for a succession of ages, the same appearance to us, though all that the planet has been doing in their laps, in which the expression and character is preserved; but the inequalities of the elements, which are periodic and indefinite inequalities, however, are, as we have observed, by no means to be neglected, but they are taken account of by every process, independent of the secular variations of the elements. (640.) In order to avoid complication, while endless variations may be produced by any one kind of variations, we shall hereafter conceive all the orbits to lie in one plane, and confine our attention to the case of a planet revolving round a fixed body; a view of the subject which (as we have seen) comprehends the case of the moon disturbed by the sun, since any one of the bodies may be regarded as fixed at pleasure, provided we conceive all its motions transferred in A diagram showing three planets orbiting around a central body. Planet A is shown in a position where it appears to be closer to the observer than Planet B. Planet C is shown in a position where it appears to be farther away from the observer than Planet B. a contrary direction to each of the others. Suppose, therefore, S' to be the central M the disturbing body F the disturbed body. Then the attraction of M acts on P in the direction PM, and on S in the direction SM. And the disturbing part of M's action on P is dif- ferent from that which it would have if it had no fixed direction. A diagram showing two planets orbiting around a central body. Planet A is shown in a position where it appears to be closer to the observer than Planet B. Planet C is shown in a position where it appears to be farther away from the observer than Planet B. **372** A TREATISE ON ASTRONOMY. [CAP. XL but will act on P very differently, according to the configu- rations of F and M. It will therefore be necessary, in analyzing its effect, to resolve it, according to mecani- cal principles, into forces acting according to some cer- tain directions, which may be perpendicular to S, and per- pendicular to it. The simplest way to do this, is to resolve the variations of M at the point P, into two directions, and take note of their direction, which difference, in turning part of M's effect. In this estimation, it will be found then that two directions disturbing powers originate: one, which is parallel to the line SP, and directed towards the direction PQ, perpendicular to SF, and therefore in that of a tangent to the orbit of P, supposed nearly a cir- cle-when this direction is taken as the first disturbing force, whose direction is always either to or from S. (641.) It is the former alone (art. 410) which gives the equal effect to the two directions SP and SQ, and it is therefore the chief cause of its angular deviations from the ellipse of revolution. This force being so small as not to depend on any particular law of central force, but only requires that the acting force, whatever it be, should be directed to the centre; whereas whatever force does not conform to this condition must be rejected. (642.) On the other hand, the radial portion of the dis- turbing force, though being always directed to or from the centre, yet it does not conform to the law of variation of the simple law of gravity by which the elliptic figure of the orbit is produced. For suppose that we attempt to dis- turb this form; and, causing the disturbed body P, now at a distance S from the centre C, to move in a circle distant from it, than the laws of elliptic motion would warrant, and have its points of nearest approach and farthest recess observed nearer than they should be in the undisturbed orbit; tends to destroy the magnitude of the disturbance at position of the axis of P's ellipse. (643.) We shall find that the action of the tangential force in the different relative positions of M and P, we shall find that, generally speaking, it vanishes when P is at a C or G; see figs. 584 & 585; i.e., conjunct with S, M and P are at a distance D from S. D is at equi- distant from S and P (or very nearly in the quadrature of **CHAP. XI.] EFFECTS OF THE TANGENTIAL FORCE.** P with M); and that, between A and B, or D, it tends to urge P towards A, while, in the rest of the orbit, its tendency is to urge it towards C. Consequently, the acceleration of P would be exactly compensated by some symmetrical revolution round its orbit from A, it will first be accelerated from A to B--hence retarded till it has made a revolution of 360°, when it again returns retailed till its re-arrival at the conjunction A. (544.) If P's orbit were an exact circle, as well as M's, the same result would ensue; but, owing to the varying description of the arc AB would be exactly compensated by the acceleration in the arc DA, these arc being equal, and therefore their accelerations being equal to the disturbing forces; and similarly, that the acceleration through the arc BC would be exactly compensated by the acceleration in the arc CD, these arcs being equal, and therefore their accelerations being equal to the averages of each revolution of P, so compensation would take place; the period would remain unaltered, and all the revolutions would be circular. (545.) This exact compensation, however, depends evidently on the exact symmetry of disposal of the parts of the orbit. But suppose that this symmetry were broken, it will no longer take place, and inequalities in P's motion will be produced, which extend beyond any limits that can be calculated from their compensation; if it ever take place at all, in a reversal of the relations of configuration which produced them. For instance, let us suppose that instead of being circular, that of M were elliptic; and that, at the moment when P set out from A, M were at its greatest distance from P. Then P would have no means of going to make only a small part of its whole revolution during a revolution of P. Then it is clear that, during the whole revolution of P, M will always be increasing; she will increase by the approach of M, and thus, in consequence, the disturbance arising in each succeeding quadrant of her orbit will be greater than that arising in those preceding; so that, when P had come round again to its conjunction with M, there would be found on the whole orbit a net effect of an acceleration in the orbital motion. This kind of action would go on so long as M continued to approach S ; A diagram showing a circle with points labeled A, B, C, D representing different positions on an orbit.234 A TREATISE ON ASTRONOMY. [CHAP. XL but when, in the progress of its elliptic motion, it began again to recede, the reverse effect would take place, and a retardation of P's orbital motion would happen; and so on elsewhere. In this manner, by the conjunction of many revolutions of M, in which the place of F in its ellipse at the moment of conjunction should have been estimated, we might obtain a compensation of the advance and recess, a compensation of a higher and remoter order, among all those successive over and under-compensation, which may be supposed to occur. But if every angular motion would emerge, the same as if no disturbance had taken place. (<648>) This is only a little more complicated, but the reasoning very nearly similar, when the orbit of the disturbed body is supposed elliptic. In an elliptic orbit, the angle between the perihelion and aphelion of the body remains in some parts of its revolution longer, in others for a shorter time, under the influence of the accelerations and retardations of F. Hence it will be necesary for an exact compensation; independent them, of any approach or recess of M, there would, on this account alone, be a continual variation of the mean motions, and surviving, unextinguished perturbation, at the end of a synodic period; and if the conjunctions always took place on one side of the mean position of M, they would constantly act one way, and an inequality would arise, having no compensation, and which would at length, and permanently, produce a change in their mean motions. But this can never be the case in the planetary system. The mean motions (i. e. the mean angular velocities) of the planets are continually changing with respect to each other. There are no two planets, for instance, which perform their orbits in time exactly double, or triple, or one half or three times as long as another; but exactly two revolutions while the other performs exactly three, or five, and so on. If there were, the case in point would arise. Suppose that M was revolving about F in such a manner that the disturbed and disturbing planet were exactly in the proportion of two to five; then would a cycle consisting of five revolutions be completed before they brought them back exactly to the same configuration. It would cause their conjunction, for instance, to happen once in CHAP. XL.] THEORY OF JUPITER AND SATURN. 325 every such cycle, in the same precise points of their orbits, which they have occupied at the beginning of the other configurations kept shifting round. Thus, then, would arise the very case we have been contemplating, and a perpetual alternation of conjunctions. (547.) Now, although it is true that the mean motions of no two planets are exactly commensurate, yet cases are not wanting where the difference is so small as to allow adjustment. And, in particular, in the case of Jupiter and Saturn—that cycle we have taken for our example in the preceding chapter—there is a difference between the pe- riods of Jupiter and two of Saturn—although it does not exactly bring about the same configuration, does so pretty nearly—but this difference is only 14 days. The two periods of Saturn 3158 days. The difference is only 14 days, in which Jupiter describes, on an average, 12°, and Saturn 10°. In other words, in every further interval they will only be 6° from a conjunction in the same parts of their orbits as before. If we calculate the time during which they will be in these positions, three conjunctions of the two planets, we shall find it to be 21760 days, their synodical period being 7884 days. In this way we see how, by means of the courses of two sidereal revolutions, Jupiter the same angle in excess of five. Every third conjunction, then, will take place when Jupiter has made one revolution more near enough to establish, not is true, an identity with but still a great approach to the case question. The exception is that when Jupiter is at its highest point (7 or 8) in conjunction, will lie the same way, and at each of them the motion of it will be similarly influenced, so as to make Jupiter appear to its longitude thus giving rise to its irregularity in completing a magnitude and very long period, which is well known to astronomers by the name of inequality of Jupiter and Saturn. (548.) The arc $8^\circ$ 0' is equal to 344 times in the whole circumference of $360^\circ$; and accordingly, if we turn round this particular part of the orbit once again, it will return to its original point of the orbit in so many times 21760 days, or in 2048 years. But the conjunc- tions we are now considering are only three. The following table gives the number of the orbit about 2 E 226 A TREATISE ON ASTRONOMY. [CHAP. XL 187° and 364° diametral, and these points also will enhance by the same arc of 6° in 21700 days. Consequently, the period of 3640 years will bring them all round, and in that interval each of them will pass through that point of the twelfth circle which is opposite to the other two conjunctions (one or other of the three) will happen at that point once in one third of this period, or in 2170 years; so that, if the mean motion of the planets be uniform, the "great inequality" would undergo its full compensation, did the elements of the orbits continue all that time invariable. But since they do not remain so long an interval and, owing to this cause, the period itself is prolonged to about 918 years. (248.) The following is particularly as a proper instance of the action of a tangential disturbing force, on account of its magnitude, the length of its period, and its being always directed towards the same point. It was remarked by astronomers, that on comparing together modern with ancient observations of Jupiter and Saturn, the mean motions of both planets appeared to be uniform. The period of Saturn, for instance, appeared to have been lengthening throughout the whole of the seventeenth century; but it was found afterwards that it is to say, the one planet was constantly lagging be- hind, and the other getting in advance of its calculated places. On examining this phenomenon more carefully, process precisely the reverse seemed to be going on. It is true, the whole retardations and accelerations observed were quite small; but when we consider how much accumulating, they produced, at length, material differ- ences between the observed and calculated places of both these planets. This difference was so great as to be accounted for by any theory, exhibited a high degree of atten- tion, and were even, at one time, too hastily regarded as almost sufficient to account for their entire irregularity. For a long while this difference baffled every endeavour to account for it, till at length Laplace pointed out its cause in the "great inequality." This inequality appears as shown above, and succeeded in calculating its period and amount. (250.) The inequality in question amounts, at its maximum, to an alternate retardation and acceleration of **CHAP. XL.] THEORY OF JUPITER AND SATURN.** About 0° 49' in the longitude of Saturn, and a corres- ponding retardation or relaxation of about 0° 21' in that of Jupiter. This retardation or relaxation at one point must necessarily be accompanied by a retardation in the other, and vice versa, is evident, if we consider, that ac- cording to the law of attraction, the velocity of motion, whatever momentum Jupiter communicates to Saturn in the direction P, the momentum must Saturn in the direction M, the momentum must Jupiter in the direc- tions P and M, the momentum must Saturn in the direc- tion C, the momentum must Jupiter in the direction R, and so on. Now, since the velocity of motion of Jupiter in this direction will affect the longitudes of the two planets is that of Jupiter's velocity in the direction P will be less than that of Saturn's velocity in the direction C, and vice versa. Hence, if we suppose both planets to have equal velocities, their longitudes will be equal at all times. But this result of a very intricate and curious calculation is fully confirmed by observation. (581.) The inequality in question would be much greater, were it not for the partial compensation which is operated in it by every triple conjunction of the planets. Suppose PQ to be Saturn's mean path, and pPq Jupiter's and supposed to be equal to each other, then take first the line SA; a second at 123° distance, on the line SB; a third at 126° distance, on SC; and the next at 368° on SD. This arrangement will show that taking successively nearly in the situation of the first, will produce merely a repetition of the first effect; but taking successively nearly in the situations possible from the first, will happen under en- tirely different circumstances. In fact, when we come to the perihelion of the orbit. Now, we have seen that a pre- DEFPRSTVWXYZ 328 A TREATISE ON ASTRONOMY. [CAP. XI. mentionation of one planet to the other, in conjunction, in a variety of situations, tends to produce compensations; and, in fact, the greatest possible amount of compensa- tion which can be produced by only three configurations is what has been shown in the preceding chapter of this tire. Three positions of conjunction compensate more than two, four than three, and so on. Hence we see that is is impossible that any one planet shall have thus accumulated in each triple conjunction, but only that small part which is left uncompensated by the in- termediate conjunctions. But, since the intermediate conjunctions are very numerous, and since every some acquaintance with the subject, will not at a loss to perceive how this consideration is, in fact, equivalent to that part of the compensation which is left uncompen- sated by the intermediate conjunctions. The equality which leads us to seek its expression in terms of the third order, or involving the cubes and products of those degrees of conjunctions which are intermediate, continual accumulation of small quantities, during long periods, corresponds to what geometers intend when they speak of the planets receiving great accretions of magnitude by integration. (502.) Similar considerations apply to every case of approaching conjunction between two planets. In the mean motions of any two planets. Such, for instance, is that which obtains between the mean motion of the earth and that of Mars. The mean motion of Mars being very nearly equal to 8 times that of the earth. This gives rise to an extremely near coincidence of every 6th conjunction, and consequently to a very small part of a circumference, and therefore to a cor- respondingly extensive accumulation of the resulting un- compensated inequality. This is true because the small part of the perturbation thus accumulated is only that which remains outstanding after passing the equalizing effect of all intermediate conjunctions. For instance, in or, in the language of geometers, is dependent on powers and products of the eccentricities and inclina- tions of both planets. The period of this inequality is about six min- ute, and the whole resulting inequality, according to the recent elaborate calculations of professor Airy, to whom I am indebted for these data, is less than few seconds at its maximum, while its period is no less A page from a book with text and some mathematical symbols. **CHAP. XI.** THE MOON'S VARIATION. than 240 years. This example will serve to show how vast minimums these inequalities have been carried in the physical theory. (503.) In the theory of the moon, the tangential force gives rise to many inequalities, the chief of which is that arising from the difference between the actual and apparent angular motion of the moon. The principal effect of this part of the disturbance arising from the alternate accelerations and retardations of the force from the centre of attraction, is that of the slippage of the ellipse, and its error, combined with the slippage form of the orbit; in conse- quence of which, the different parts of the ellipse are moved to different amounts of angular motion. This inequality was first distinctly remarked as a peculiar complication of Halley's place by Tycho Brahe, and is remarkable in the history of astronomy, for being first so explained by New- ton, and from his theory of gravitation. (504.) We come now to consider the effects of that part of the disturbing force which acts at right angles to the radius vector, and tends to alter the law of gravity, and therefore to derange, in a more direct and sensible manner, the motion of the body. This part of the dis- turbed orbit from that of an ellipse, or according to the view we have taken of this subject in art. 506, to produce a change in the direction of the velocity and position in its own plane, or in the place of its perihelion. (505.) In estimating the disturbing force of M on P, we must take into account all those parts of its effective action on S and P to be regarded as effective as such, and that the first resolved portion of M's attraction on S and P is equal to its natural gravity. This force is called the additional part of the disturbing force. There is, besides this, another power acting on S and P, which is called SP', which is that arising from the difference of actions of M on S and P estimated first in the direction PL parallel to SP', and then in that perpendicular to SP', which is the tangential force, already considered, in the direction 22 350 A TREATISE ON ASTRONOMY. [CHAP. XI. PK ; the obser perpendicular to it, or in the direction Pk. This part of M's action is termed the additions force, be- cause it is added to the other parts of the motion of P, it is the excess of the one of these resolved portions over the other, which, in any assigned position of P and M, constitutes the whole action of M on P. The effects respecting whose effects we are now about to reason. (66.) The estimation of these forces is a matter of no difficulty. We have only to consider them as motions, but they are too complicated in their expressions to find any place here. It will suffice for our purpose to point out their nature, and to show how they may be estimated. We shall consider their mean or average effect. In order to estimate what, in any one position of P, will be the mean action of M on P, we must first consider its effect with respect to P, we have nothing to do but to suppose M broken up, and distributed in the form of a thin ring round the centre of P. Now, if we take this ring as an account of the elliptic motion of M, we might conceive the thickness of this ring, in its different parts, to be pro- portional to the square root of its distance from the axis of its orbit, or in the inverse proportion of its angular motion. But into this nicety we shall not go, but con- sider every part of the ring as having a uniform orbit circular and its motion uniform. Then it is clear that the mean disturbing effect on P will be the difference of attraction between two points at S and S', which latter occupies its centre, the former is ex- centric. Now the attraction of a ring on its centre is manifestly zero, because all particles are equally distant in any one direction, is zero. On the other hand, on a point P' out of its centre, if within the ring, the resulting attraction will be towards S', and if without it towards a point of the ring, or directly from the centre. But if P' as this is a proposition that the equilibrium of Saturn's ring reso- nance does not depend upon any particular law of gravitation, which may be done very simply, and without the aid of any cal- culations. Consider a particle placed at any point within or passing through the point, and terminating both ways in the shell, well, from S' to S", and let us call this particle P'. The attraction from S" will be equal to that from S', and therefore symmetrically related to it as it would be from S". Now let us imagine another particle placed at the same point, and formed by the mutual actions of such a line found by the same law that governs the attraction between two particles separated by a certain distance. distance. **CHAP. XL.] EFFECTS OF THE RADIAL FORCE.** 321 lie without the ring, the resulting force will not always produce a regular motion in the centre. Hence it appears that the most effect of this force may not be different in its direction, according as the orbit of the disturbing body is exterior or interior to that of the disturbed. In consequence, the central velocity will increas[e]e, the central gravity will decrease. (307.) Regarding, then, only the mean effect as pro- duced by the action of revolutions of both bodies, it is evident that an increase of central force must be accompanied with a diminution of periodic time, and a consequent increase of angular velocity revolving with a stated velocity, and vice versa. This, then, is the first and most obvious effect of the radial part of the de- termination. It also follows that, within a certain mean and invariable amount, the dimensions of all the orbits and the periodic times of all the bodies composing the system remain unaltered. For suppose each body to execute about the sun uninfluenced by the attraction of the rest; its angular motion of the interior boundary of its orbit being equal to that of any other point of the exterior greater, than on that supposition. The lesser effect, indeed, might be at once concluded from this observation; but it is not less evident that revolving inter- iorly to any orbit may be considered as adding to the general aggregate of the attracting matter within, which is continually distributed over space, and maintained in a state of circulation. bases of both the suns, or myriads, be similar and equally inclined to one another; and let them revolve round their common centre with equal velocities; then every body revolving in such a manner will be equal, because the attraction is as the sinuating matter directly. But if they revolve in opposite directions, they will be in opposite directions and, therefore, encounter each other. Therefore every such part of one into which the spherical shell can be broken up, will have an equal attraction from every part of another spherical shell. Now take a line and treat it similarly, breaking up every part into which it can be broken up; and every such line passing through the annular point. Here having elements, so arranged that they shall be equal in magnitude and direction; they should be in equilibrium. Therefore it is impossible for them to move together with uniform velocity; and the point will, or the whole, be urged towards one or other extremity until they are again equalized by some change in their arrangement. In consequence of this analogy we may conclude that no two bodies can ever approach nearer than their mutual distances would allow them to do so under any circumstances of motion forming a flat surface, like the ring of Saturn. 332 A TREATISE ON ASTRONOMY. [CHAP. XL (585.) This effect, however, is one which we have no means of measuring with any degree of accuracy than by calculation. For our knowledge of the periods of the planet, and the dimensions of their orbits, is drawn from observations made under the influence of this constant part of the perturbative action. Their observed mean motions are, therefore, affected by this cause, and it is because they are so affected that we have no means of distinguishing them from the direct effect of the sun's attraction, with which it is likewise connected. The irregularity of the motions of the planets assures us that it is extremely small; and this, in fact, is all which it is at all important to us to know, in order to understand their motions. (586.) The action of the sun upon the moon, in like manner, tends, by its mean influence during many successive revolutions, to increase her eccentricity, to elongate her moon's orbit, and increase her periodic time. But this general average is not established, either in the case of the moon or of the planets; for there are occasional fluctuations due to the elliptic forms of their orbits, which we have purposely neglected to take account of in the above remarks on their motions. These fluctuations arise from the convergence of a great multitude of revolutions, to neutralize each other. In the lunar theory, however, many of these subsidences are taken into consideration; but those of the planets, and of great importance is a correct knowledge of her motions. For example—the sun's orbit (referred to the earth) is nearly circular; and we have only two functions for its description, during which the distance of the sun undergoes an alternate increase and diminution, each exceeding 1° in longitude. Now, as the sun approaches the earth, its disturbing forces of every kind are increased in a high ratio, and vice versa; and consequently when the sun is near the lunar orbit, and the diminution of the moon's periodic time will be kept in a continual state of fluctuation, increas- ingly more rapid until it reaches its maximum diminish- ing as it recedes. And this is convenient to facts—the observed difference between a lunation in January (when she was nearest to us) and July (when it is furthest) being no less than 30 minutes. **CHAP. XL.] THE MOON'S SECULAR ACCELERATION.** (560.) Another very remarkable and important effect of this case, in one of its subordinate fluctuations (ex- tending, however, over an immense period of time), is what is called the secular acceleration of the moon's motion. This was first observed by Halley, on com- paring together the records of the most ancient lun- ar eclipses of the Chaldean astronomers with those of modern times. The difference between these observations at present is sensibly shorter than that at remote epochs ; and this result was confirmed by a further comparison of the lunar eclipses with the observations of the ancient as- tronomers of the eighth and ninth centuries. It appear- ed from these comparisons, that the rate at which the moon moves around the earth has been increasing per century—quantity small in itself, but becoming consi- derable by its accumulation during a succession of ages. The reason why this increase had not been noticed before, and Saturn, had been long the subject of toilsome inves- tigation to geometers, though it is difficult to appre- ciate the extent to which it occurs, is that no attention on the point of again dealing with the theory of gravity inade- quate to its explanation, others were for rejecting altogether, thus leaving the question unconsidered. Such a course is not satisfactory as that on which most historical events are credited. It was in this dilemma that Laplace once more proceeded, and he has shown how far his method of re- search, by pointing out the real cause of the phenomenon in question, which, when so explained, is one of the most simple and obvious in nature, may be applied to any other subject—one which leads our speculations further into the past and future, and points to longer vistas in the dim future than have yet been anticipated by any speculation and is yet to undergo, than any other which observation assisted by theory has developed. - The motions of the planets were variable, the alter- nation dilatation and contraction of the moon's orbit, ex- plained in art. 530, would in a course of great many centuries produce a considerable compensation for the com- pensation in the distance and periodic time of the moon, by bringing every possible step in the sun's change of dis- tance from her to a corresponding step in the motion of the moon from the sun in her orbit. But this is not, in 384 A. THEATRE ON ASTRONOMY. CHAP. XL fact, the case. The solar ecliptic is kept (as we have al- ready hinted in another place, and as we shall see in plain more fully) in a continual but excessively slow state of change, by the action of the planets on the earth. Its axis has been slowly inclined to the plane of the eclipti- ca, and has been since the earliest ages, diminishing ; and this diminution will continue (there is little reason to doubt that it will continue) until the earth's axis is vertical, and the earth's orbit becomes a perfect circle ; after which it will again open out into an ellipse, the eccen- tricity of which will increase, and then decrease again. The time required for these revolutions, though calculable, has not been calcu- lated. For instance, the eccentricity of the orbit was esti- mated by hundreds or by thousands of years. It is a pe- riod, in short, in which the whole history of astronomy and of all human knowledge is contained. At this point, during which all its changes are to be regarded as uni- form. Now, it is by this variation in the eccentricity of the earth's orbit that the variations in the motion of the moon is caused. The compensation above spoken of, (if the solar ecliptic remained unaltered, would be effect- ed in a few thousand years), and which has been (as the mode already stated) will now, we see, be only imper- fectly effected, owing to this slow shifting of one of the essential elements of our system. This compensation is identical with, nor equal to, those of change. The same reasoning, in short, applies, with that by which we ex- plained the cause of the variations in the solar magnetic force. The struggle up hill is not maintained on equal terms with the downward tendency. The ground is all the while rising towards the summit of the hill. During the whole time that the earth's eccentricity is diminishing, a preponderance is given to the action over the resistive forces. But when this eccentricity is at its maximum, all the tables will be turned, and the process of ultimate re- storiation will commence. Meanwhile, a minute, outstand- ing and unusual event may occur; such as a recurrence or near recurrence, of the same configurations of the sun, the moon, and the solar and lunar perigee. These ac- cumulate, and produce a sudden change in their mo- tion, and thus becoming repeated in every lunation, XL **CHAP. XI.] THE MOON'S SECULAR ACCELERATION.** at length affect her longitude at an extent not to be over- looked. (562.) The phenomenon of which we have now given an account is another and very striking example of the proportion of a periodic change from one part of a system to another. This is the effect which is more visible on the lunar motions as referred to the earth. These motions are so small, and their distances too great, for their different effects to be perceived by any means ever to become sensible. Yet their effect on the earth's orbit is such that it is produced through the sun to the sun that of the moon, and what is very remarkable, this unintended effect thus indirectly produced on the angle described by the earth's orbit is more sensible to observa- tion than that directly produced by the motion of the body described by the round the sun. (563.) The different motions of the luner and planetary orbits then, which arise from the action of the radial force, and which tend to affect their mean mo- tions, are distinguishable into two kinds. One kind is the distribution of the attracting matter in the system, and on the order which each plane- net describes its orbit, and periodic, and what operates in length of time its periodical changes. Geometers have demonstrated (it is to Lagrange that we owe this most important discovery) that these changes are ex- actly those which arise either from a cause arising from the radial or tangential disturbing forces, or from their com- bination, such as can go on for ever increasing in one di- rection or another. In other words, they are those which make that the major axes of the planetary ellipses are not lie- ble even to those slow annual changes by which the in- clinations of planes are all changed, and all these sys- tems, are affected, and which is true, are periodic, but in a different sense from those long inequalities which develop themselves in the eccentricity of their orbi- te. Now, the periodic time of a planet in its orbit about the sun depends only on the masses of the sun and pla- net, and on their distance apart. It does not depend with- out regard to its degree of eccentricity, or to any other element. The mean annual periods of the planets, therefore, come out several from an average of a sufficient
    Page Number 335
    336 A TREATISE ON ASTRONOMY. [CHAP. XI. number of revolutions to allow of the compensation of the last-mentioned error, and that this number may be made up of of some. The length of the sidereal year, for example, if concluded at this present time from observations en- treating with the motions of the planets, will be found that the sun (such, in short, as we now possess it), is the same with that which (if we can surch our imagination so far) must have been possessed by our ancestors in the beginning of observa- tions made a million of years hence. (664.) This theorem is judy regarded as the most important discovery ever made in astronomy, and has thereby rewarded the researches of mathematicians. We shall, therefore, endeavour to make clear to our readers, at least the nature of this proposition, and its consequences; although the complete application of this principle cannot be satisfactorily made without entering into details of calculation. It is only necessary to remark here, that we shall have no difficulty in leading them up to that point where those details must be ensued on, and in giving such an insight into the subject as will enable them to see what manner must be their results when gone through. (665.) It is a property of elliptic motion performed under Kepler's laws, that if the velocity with which a planet moves at any point of its orbit be given, and also the distance from the centre of attraction to that point of its orbit be thereby also given. It is no matter in what direction the planet may move at that moment. This will infallibly determine both the shape and size of ellipses, but not its length. This property of elliptic motion has been demonstrated by Newton, and is one of the most important discoveries which he made in his theory. Let us now consider a plane describing an in- determinately small arc of its orbit about the sun, under the joint influence of all the planets, and compare its view of another planet. This arc will have some certain cur- vature and direction, and, therefore, may be considered as an ellipse. Now, if we suppose this plane to describe a forus, for this plain reason—that whatever be the curvature and direction of the arc in question, an ellipse may always be described by it; then it follows that this arc and shall coincide with it throughout the whole CHAP. XL. PERMANENCE OF THE MAJOR AXES. 237 interval (supposed indefinitely small) between its extreme positions will be equal to the period of revolution. It does not follow, however, that the ellipse thus instantaneously determined will have the same elements as that similarly determined at any other time, but only those which are previous or the subsequent instant. If the disturbing force did not exist, this would be the case ; but, by its action, a vari- ation of the elements of the orbit takes place, and the ellipse so determined is in a continual state of change. Now, when the planet has reached the end of one small period of revolution, we cannot say whether it will in the next instant describe an arc of an ellipse having the same or a varied axis will depend, not on the direction of motion of the planet, but on its distance from the axis, as we have seen, is independent of that direction—not on its change of distance from the sun, or on any other circumstance whatever—so long as all that are accommodated to it, so that one and the same axis must belong to its beginning and its end. The quantity of motion possessed by a body when it makes up a new major axis, or go on with the old one, will de- pend solely on this—whether the velocity is undergone, by a change in direction or in magnitude—is indepen- dable with the continuance of the same axis. We say by the action of the disturbing force, because the central forces do not act directly upon the axes themselves; both change of velocity as to be incompatible with the per- formance of any ellipse in which it may at any instant be freed from such action. (560.) Thus we see that the momentary variation of the major axis depends on nothing but the momentary variation of velocity; and this variation is caused by the disturbing force, without the least regard to the direction in which extraneous velocity is impressed, or to any other circumstances whatever. It may be situated in consequence of the variation of the other elements of its orbit. And as this is the case at every instant during its revolution, no matter what shape of any time, however great, the amount of change which the axis may have undergone will be determined by the total effect of all these changes which are produced by the disturbing force ; without any regard to alterations 2 F **280** A TREATISE ON ASTRONOMY. CHAP. XL. which the action of that force may have produced in the other elements, except in so far as the velocity may be thereby modified. This is the point at which we must estimate the value of the propositions to the calculations of the geometer. We shall be at no loss, how- ever, to perceive that these calculations can only ter- minate in a complete agreement with the observed com- pression of all the variations of the axis which can thus arise, when we consider that the circulation of two planets about each other, during very long and incon- mensurable periods, cannot fail to ensure their pre- sentation to each other in every state of approach and recession, and consequently to produce a constant dis- tance and the consequent intensity of their mutual action. Whatever velocity, then, may be generated in one by the disturbed motion of the other, it will neces- sarily be destroyed by it in another, by the mere effect of change of configuration. (567.) It appears, then, that the variations in the major axes of the planetary orbits depend entirely on cycles of configuration, like the great inequality of Ju- piter's sun, and that this inequality is due to the Venus above explained, which, indeed, may be regarded as due to such periodic variations of their axes. In fact, the most remarkable inequalities of the motions arise from the accumulation of imperfectly compensated actions of the tangential force, bring them directly under the above law. The first cause is that Venus moves almost wholly upon the velocity of the disturbed planet, whose motion is always nearly coincident with or op- posite to its direct motion. (568.) Let us now consider the effect of perturbation in altering the eccentricity and the situation of the axis of the disturbances. The planet Venus being in her position (as we have observed in art. 318) actually takes place, although very slowly, in the axis of the earth's orbit. This movement is described in art. 360; and these movements we are now to account for. (569.) The motion of the apsidal of the lunar and planetary orbits may be illustrated by a very pretty me- chanical experiment, which is otherwise instructive in
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    A TREATISE ON ASTRONOMY.
    CHAP. XL.
    which the action of that force may have produced in the other elements, except in so far as the velocity may be thereby modified. This is the point at which we must estimate the value of the propositions to the calculations of the geometer. We shall be at no loss, how- ever, to perceive that these calculations can only terminate in a complete agreement with the observed com- pression of all the variations of the axis which can thus arise, when we consider that the circulation of two planets about each other, during very long and incon- mensurable periods, cannot fail to ensure their pre- sentation to each other in every state of approach and recession, and consequently to produce a constant dis- tance and the consequent intensity of their mutual action. Whatever velocity, then, may be generated in one by the disturbed motion of the other, it will neces- sarily be destroyed by it in another, by the mere effect of change of configuration.
    (567.) It appears, then, that the variations in the major axes of the planetary orbits depend entirely on cycles of configuration, like the great inequality of Jupiter's sun, and that this inequality is due to the Venus above explained, which, indeed, may be regarded as due to such periodic variations of their axes. In fact, the most remarkable inequalities of the motions arise from the accumulation of imperfectly compensated actions of the tangential force, bring them directly under the above law. The first cause is that Venus moves almost wholly upon the velocity of the disturbed planet, whose motion is always nearly coincident with or op- posite to its direct motion.
    (568.) Let us now consider the effect of perturbation in altering the eccentricity and the situation of the axis of the disturbances. The planet Venus being in her position (as we have observed in art. 318) actually takes place, although very slowly, in the axis of the earth's orbit. This movement is described in art. 360; and these movements we are now to account for.
    (569.) The motion of the apsidal of the lunar and planetary orbits may be illustrated by a very pretty me- chanical experiment, which is otherwise instructive in
    **CHAP. XI.** **MOTION OF THE SPIDER.** 330 giving an idea of the mode in which orbital motion is carried on under the action of centripetal forces variable ac- cording to the situation of the revolving body. Let a leaden weight be suspended by a brass or iron wire to a hook fixed at the top of a vertical rod, and let the point of its free motion on all sides be the vertical, and so that when in a straight line it shall just clear the floor of the room, a table placed below will support its weight when the hook. The point of support should be well secured from wagging to and fro by the oscillation of the weight, which will be prevented by a spring, or by a string slightly stretched as it will bear, with the certainty of not break- ing. Now, let a very small medal be communicated to the wire, and let it be so light that it may move perpen- dicular and leaving it fall, but by giving it a slight impulse sideways. It will be seen to describe a regular ellipse whose foci are situated at the extremities of the wire, and be heavy, and carried attached to it a pencil, whose points lie exactly in the direction in which the spring is pulled and gravity opposed against it. In these circumstances, the situation of the major and minor axes of the ellipse will remain for a long time unchanged, because the resistance of the air and the stiffness of the wire will gradually di- minish its dimensions and eccentricity. But if the im- pulse given to it is sufficient to carry it out to a great angle (15° or 20° from the vertical), this permanence of situation of the ellipse will cease, and it will describe an ellipse having its axis position at every revolution of the weight, advancing successively towards one of its foci. This is due to uni- form and regular progression, which at length will in- tirely reverse its situation, bringing the direction of the longest arcumensions to coincide with that in which the shallowest ones have been made, and thus round the whole circle; and in, and in word, initiating to the eye, very com- pletely, the motion of the spider of this simple experi- ment. (294) If we consider that this pro- gression of the spider, it will not be difficult of de- tection. When a weight is suspended by a wire, and let it slide from a vertical upwards or downwards point (or rather at any instant perpen- 14 340 A TREATISE ON ASTRONOMY. [CHAP. XL circular to the wire) by a force which varies as the sine of the deviation of the wire from the perpendicular. Now, the sines of very small arcs are nearly in the proportion of the arcs themselves ; and the more nearly so, as the arc becomes smaller, and the vertical is so small that we may neglect the curvature of the sphere on which the body moves. The same principle will apply to any curved surface, and regular curves described coincident with its projection on a horizontal plane, it will be then moving under the same circumstances as if revolving body acted upon by a force varying inversely as the distance ; and in this case, the curve described would be an ellipse, having its centre of curvature not in the focus, but at some point between the apices of the ellipse would remain fixed. But if the excursions of the weight towards the vertical become greater, the force urging it towards the centre will deviate in its law from that simple ratio of the distances ; being as the sine, while the distances increase, and therefore increases also, but does not increase so fast. So soon as the arc has any sensible extent, the sine begins to vary considerably from unity, which without exact numerical proportionality would require ; and therefore the force urging the weight towards its centre of point of rest, at all times exceeds that required for its position, somewhat less than that which would keep the body in its precise elliptic orbit. It will no longer, therefore, have, at those greater distances, the same command over the weight, in proportion to its space, which would enable A diagram showing a body moving along a curved path (A) and deflecting into an elliptical path (B). The diagram includes labels for points A, B, C, and D. it to deflect it from its rectilinear tangential course into an ellipse. The true path which it describes will be less *Newton, Princip. I. c7.* **CHAP. XI.] MOTION OF THE APOLLES.** curved in the remoter parts than is consistent with the elliptic figure, as in the annexed cut; and, therefore, it will not so soon have its motion brought to be again at rest, as would be the case if the force were continued in a uniform manner. But this is only owing at length, and in a more circumscribed manner, that fact which we more briefly and summarily express by saying that the apolles of the orbit are pro- gressive. (971.) Now, this what takes place, mutatis mus- tandis, is that the moon, being attracted by the ac- tion of the sun on the moon, for example, as we have seen, besides the tangential force, whose effects we are not here concerned with, there is also another action of the radius vector, whose law is not that of the earth's direct gravity. When compounded, therefore, with the earth's direct gravity, it produces a force which tends to deviate from the elliptic figure, being either too much curved, or too little, in its recents from the perigee, to bring it into a state of rest. This force may be very great— too much, if the compound force thus produced de- crease at a slower rate than the inverse square of the distance (as in fig. 1); or it may be too little—too little, if the joint force decrease faster than gravity, or more rapidly than the inverse square, and be therefore lost before it can produce any effect. In either of these cases, the curvature, being excessive, will bring the moon to its apogee sooner than would be the case in an elliptic orbit; in either of these curves is more curved, and will Fig. 1. Fig. 2. therefore being in later to so apogee. In the former case, then, the line of apesides will recrograte in the latter, advance. (See fig. 1 and fig. 2.) (972.) Both those cases obtain in different circum- P. 348 A TREATISE ON ASTRONOMY. [CHAP. XL tions of the sun and moon. In the syzygies, the effect of the sun's attraction is to weaken the gravity of the earth by diminishing its mass, and therefore to increase the inverse square, following the direct proportional relation of the distance; while, in the quadratures, the reverse takes place, and the earth's mass is augmented by the force here comprising with the earth's gravity, but the portion added being still, as in the former case, in the direct proportion of the inverse square of the distance. The motion of the moon, in and near the first of these situations, will be performed in an ellipse, whose apices are in a state of advancement with respect to that of the sun, and vice versa. But, as we have already seen (art. 566), the average effect arising from the mutual counteraction of these two components is so small, that it is not sufficient to give preponderance to the ablatious or enfeebling power. On the average, then, of a whole revolution, the lunar apo- ges will be equal. (573.) The above reasoning renders a satisfactory enough general account of the advance of the luna spo- ries; but it does not explain their inequality. This fact can be applied to determine numerically the rapidity of such advance: now, when so applied, does it account for the whole difference between the apogees? It may be sig- nified by observation—not more, indeed, than about one half of it; the remaining part is produced by the tan- gential force. Now this tangential force acting on a body in the moon will have the same effect in diminishing the curvature of its orbit as the decrease of central force, and vice versa. Hence it appears that this tangential force is to cause a fluctuation of the moon's velocity above and below its elliptic value, and therefore an altera- tion proceeding from this cause alone would not suffice to compensate itself in each synodic revolution, were the apogees inviolable. But this is not the case; the apogees keep always at a certain distance from each other under this force, as above explained. An un compensated portion of the action of the tangential force, therefore, remains unaccounted for; which portion must be attributed to some other cause (as employed in this chapter), and this portion is distribu- ted over the orbit as to compensate with the former cause; and, in fact, nearly to double its effect. This is what is **CHAP. XL.** ECCENTRICITIES AND PERIHELIA. 543 means by geometries, when they say that this part of the motion of the apogee is due to the square of the distur- bance force. The effect of the tangential force in disturb- ing the orbit is very small, and therefore we may take for the motion which the apogee has already had impressed upon it by the radial force, and we have here, therefore, only to consider the disturbance caused by the eccentricity. (074.) The curious and complicated effect of perturba- tion, described in the last article, has given more trou- ble to mathematicians than any other branch of their theory. Newton himself had succeeded in tracing that part of the motion of the apogees which is due to the direct action of the disturbing force; but he did not succeed in half what observation assigns; he appears to have abandoned the subject in despair. Nor, when resumed by his suc- cessors, has it been found so easy as might have been a more promising aspect. On the contrary, Newton's result appeared to be even minutely verified, and the ele- ments of his theory were found to agree with the sub- ject without success began to excite strong doubts whe- ther this feature of the lunar motions could be explained as if it were due to a single cause. This doubt was removed, however, almost in the instant of its ori- gin, by the same geometer, Clairaut, who first gave it current currency by his demonstration of his momentary hesitation, by demonstrating the exact coin- cidence between theory and observation, when the effect of the eccentricity was considered alone. The lunar apogee circulates, as already stated (art. 360), in about nine years. (075.) The same cause which gives rise to the dis- placement of the line of apsides of the disturbed orbit produces a corresponding change in its eccentricity. This fact is contained in a remark made by Halley at p. of art. 571. Thus, in fig. 1, since the disturbed body, pro- ceeding from its lower to its upper apse, is acted on by a tangential force t directed towards the center of the orbit, and too much curved, its whole course (as far as it is so affected) will lie within the ellipse, as shown by the dotted line; but if it be acted on by a force t' directed away from the center of the orbit, its distance will be less than in the undisturbed ellipse; that is to say, the eccentricity of its orbit, as estimated by A diagram showing a curve representing the motion of a body around an elliptical orbit. The body starts from one focus (lower point) and moves towards another focus (upper point). A tangent line is drawn at each focus. The dotted line represents a hypothetical path if no disturbance occurred. 304 A TREATISE ON ASTRONOMY. [CHAP. XI. the comparative distances of the two apsides from the focus, will be diminished, or the orbit rendered more nearly circular. The contrary effect will take place in the case of a planet revolving in an ellipse, by momentary shifting of the perihelion of the disturbed orbit, and the momentary variation of its eccentricity, a relation which is expressed by the equation $e = \frac{a - p}{a + p}$, where $e$ denotes the change of inclination with the motion of the nodes ; and in fact, the strict geometrical theories of the two cases are identical, and may be regarded as the results of the very same nature. What the variation of eccentricity is to the motion of the perihelion, the change of inclination is to that of the period of revolution. In this case, the period of the one is also the period of the other ; and while the perihelion describes considerable angles by a rapid succession, the nodes describe immense periods of time round the entire circle, the eccentricities increase and decrease by comparatively small changes, and their variations are represented by small inclinations. In the lunar orbit, as the rapid rotation of the nodes prevents the change of inclination from accumulating at any particular instant, so that every new revolution of its apsego effects a speedy compensation in the successive steps of its eccentricity, and never suffers them to accumulate beyond a certain limit. This is effected by preventing its quick succession the lunar orbit in every possible situation to all the disturbing forces, whether of the sun or moon, or even those which act upon the earth's equator, prevent any secular accumulation of small changes, by which, in lapse of ages, its ellipticity may become great enough to produce a retrograde motion. Accordingly, observation shows the mean eccentricity of the moon's orbit to be the same now as in the earliest ages of our globe (876). The movements of the perihelion, and variations of eccentricity of the planetary orbits, are interlaced and connected together by means of their motions caused by the same laws as the variations of their nodes and inclinations. Each act upon every other, and every such mutual action produces a reciprocal compensation ; and every such period, in pursuance of the principle of art. 526, is thence propagated throughout **CHAP. XI. STABILITY OF THE ECLIPTICITIES.** The system. Thus arise cycles upon cycles, of whose compound duration some notion may be formed, when we consider what is the length of one such period in the course of the principal motions of the planets in their orbits. Neglecting the action of the rest, the effect of their mutu- al attraction would be to produce a variation in the eccen- tricity of Jupiter's orbit, which would increase its eccentricity, to 0-0145, its minimum value; while that of Jupiter would vary between the narrow limits, 0-00603 and 0-00678, those of Saturn and Uranus being approxi- mately to the least of Saturn, and vice versæ. The period in which these changes are gone through, would be 70414 years. But this is only a first approximation; it is observed that many millions of years will require to elapse before a complete fulfilment of the joint cyclets which shall re- store the eccentricities to their former state as far as the eccentricities of its orbits are concerned. (577.) The place of the perihelion of a planet's orbit is determined by the sum of its eccentricity and obliquity. The eci- centricity is most important, as upon this (the axes of the orbits being permanent) depends the mean tempera- ture at any given latitude on the surface of the earth, which is the seasons may be liable. For it may be easily shown that the mean annual amount of light and heat received by a body varies inversely as the square of the semi- axis of the ellipse described by it. Any variation, therefore, in the eccentricity by changing the minor axis, will necessarily affect the temperature. Now such a change will also influence the extremes of tempe- rature appears from art. 315. Now, it may naturally be supposed that in any system where there are two bodies, in which, at some period or other, competing changes may accumulate on the orbit of one planet from several causes, there must be some compensation on that of any one planet—as the earth—may become exorbitantly great, so as to subvert those relations which render it habitable; or else it may become too small, so as to influence the physical comfort of his state. To this the re- searches of geometers have enabled us to answer in the negative. A relation has been demonstrated by Lagrange *Q. On "The Law" which may influence Geological Pro- cesses." *Great Treatise.* 1828. | | | |---|---| | **THE SYSTEM.** | Thus arise cycles upon cycles, of whose compound duration some notion may be formed, when we consider what is the length of one such period in the course of the principal motions of the planets in their orbits. Neglecting the action of the rest, the effect of their mutual attraction would be to produce a variation in the eccentricity of Jupiter's orbit, which would increase its eccentricity, to 0-0145, its minimum value; while that of Jupiter would vary between the narrow limits, 0-00603 and 0-00678, those of Saturn and Uranus being approximately to the least of Saturn, and vice versa. The period in which these changes are gone through, would be 70414 years. But this is only a first approximation; it is observed that many millions of years will require to elapse before a complete fulfilment of the joint cyclets which shall restore the eccentricities to their former state as far as the eccentricities of its orbits are concerned. (577.) The place of the perihelion of a planet's orbit is determined by the sum of its eccentricity and obliquity. The eccentricity is most important, as upon this (the axes of the orbits being permanent) depends the mean temperature at any given latitude on the surface of the earth, which is affected by seasons may be liable. For it may be easily shown that the mean annual amount of light and heat received by a body varies inversely as the square of the semi-axis of the ellipse described by it. Any variation, therefore, in the eccentricity by changing the minor axis, will necessarily affect the temperature. Now such a change will also influence the extremes of temperature appears from art. 315. Now, it may naturally be supposed that in any system where there are two bodies, in which, at some period or other, competing changes may accumulate on the orbit of one planet from several causes, there must be some compensation on that of any one planet—as the earth—may become exorbitantly great, so as to subvert those relations which render it habitable; or else it may become too small, so as to influence the physical comfort of his state. To this the researches of geometers have enabled us to answer in the negative. A relation has been demonstrated by Lagrange *Q. On "The Law" which may influence Geological Processes." *Great Treatise.* 1828. | 346 A TREATISE ON ASTRONOMY. [CAP. XL] between the masses, axes of the orbits, and eccentrici- ties of each planet, similar to what we have already stated with respect to their inclinations, viz. that if the mass of each planet be multiplied by the square root of the axes of its orbit, and this product multiplied by the square of its eccentricity, the sum of all such products throughout the system is likewise multiplied by the square of its axis; but since the axis of each orbit is liable to secular changes, this is equivalent to saying that no one orbit will in- crease its mass, or decrease its eccentricity, or any of its com- mon fund, the whole amount of which is, and must for ever remain, extremely small. (678.) We may here hint, in our last art. but one, at permutations produced in the lunar orbit by the proto- tural motion of the earth's moon. The attraction of the sphere of the moon on all its parts tends to con- volve into a point in its centre; but not that the case with every other body in the system. For though it may be ex- actly directed to its centre, nor does it exactly follow the law of the inverse squares of the distances. Hence will arise a series of perturbations, very small in amount, but still considerable in their effect upon the motions; by which the node and apogee will be affected. A more remarkable consequence will be that the moon will always resist any change in her orbit, exactly analogous to that which the moon causes in the plane of the earth's equator, by its action on the precession of the ellipse principle. In gener- al, it may be observed that in the systems of planets, which have satellites, the elliptic figure of the primary has a tendency to become circular; and this tendency is inclosed with its equator—a tendency which, though small in the case of the earth, yet in that of Jupiter, whose elip- sity is very considerable, becomes predominant; consequently, where the density of the body is reinforced by the ac- traction of the rings, becomes predominant over every ex- ternal and internal cause of disturbance, and produces There are two kinds of this kind of tides—the smaller—mercury—Mercury, Mars, Ceres—and a cata- strophic kind—the greater—Jupiter and Saturn. The whole amount of the eccentricity fund. But that can never be Jupiter and Saturn alone. It is equally applicable to all bodies whatever. This fund is applied to the reduction of art. 315. "Great Fund." This fund is so small that it cannot be detected by any means whatever. 346 MASSES DETERMINED BY PERTURBATIONS. 347 and maintain an almost exact coincidence of the planes in question. Such, at least, is the case with the nearer satellites. The more distant are comparatively less af- fected by this cause, owing to the difference of distances between a sphere and spheroid of equal mass, as compared with the distance increases. Thus, while the orbit of the all the planets is nearly circular, that of Saturn is in the plane of the ring and equator of the planet, that of the external satellite, whose distance from Saturn is between sixty and seventy million miles, is inclined to the plane considerably. On the other hand, this con- siderable distance, while it permits the satellite to retain its usual inclination, prevents (by parity of reasoning) its being affected by the planetary attraction, comparatively disturbed by its attractions, or being subjected to any ap- preciable perturbation whatever from the Sun. If such exist, they must be much slower than those of the earth; the mass of this satellite (though she may be very large) will be so small in comparison with its apparent size, a much smaller fraction of that of Saturn than the moon is of the earth; while the solar preces- tions, owing to the immense distance of the sun, must be quite insignificant. (570.) It is by means of the perturbations of the planetary motions that we have been enabled, according to theory, that we arrive at a knowledge of the masses of those planets, which, having no satellites, offer no other hold upon them for purposes of determining their pre- ceding motions. The effect of perturbation in the motions of every other, proportioned to its mass, and to the degree in which it differs from that of Saturn, is to give it a new motion. This sys- tem gives us over their movements. The latter is a sub- ject of exact calculation: the former is unknown, otherwise we could not determine their motions. In this man- ner, however, if the masses of the planets by this means, theory lends the greatest assistance to observation, by pointing out how far each observed motion deviates from this knowledge from the confused mass of superposed inequalities which affect every observed place of a planet; by pointing out how each inequality in its periodical rise and decay; and by showing how every parti- 348 A TREATISE ON ASTRONOMY. [CHAP. XL cubar inequality depends for its magnitude on the mass producing it. It is thus that the mass of Jupiter itself (employed by Laplace in his investigations, and inter- woven with all the planetary tables) has of late been as- certained to exceed 70 millions of millions of times that of the earth. The same conclusion is drawn by Laplace, by it in the motions of the ultra-cadical planets, to have been insufficiently determined, or rather considerably misplaced, the masses of Saturn's satellites, and of its satellites, made long ago by Poind and others, with in- adequate instrumental means. The same conclusion has been arrived at by other astronomers, by means of the perturbations produced by Jupiter on Encke's comet. The error was one of great importance; the mass of Jupiter was found to be less than half its actual amount, and the dis- tance in the planetary system, after that of the sun. It is not satisfactory, then, to have ascertained--as by his ob- servations on Saturn's satellites--that the moon of Jupiter dome--the cause of the error; to have traced it up to its source, in insufficient instrumental measurements of the greater eccentricity of its orbit; but to have failed to find it disperse when measures taken with more care, and with ininitely superior instruments, are substituted for those before mentioned. (880.) In the same way that the perturbations of the planets lead us to a knowledge of their masses, as com- pared with that of the earth, so also do those of Saturn's si- tellites of Jupiter have led, and those of Saturn's si- tellites will lead us to a knowledge of their masses as pri- maries. The system of Jupiter's satellites has been elaborated by Laplace; it is from him we learn, compared with any other investigation made on this subject, that the masses assigned to them in art. 483 have been fixed. Few results of theory are more surprising, than to see them agree so well with observations. The method which we have applied to the ponderous mass of the sun, which exceeds the least of them in the enormous proportion of 6500000 to 1. **CHAPTER XII.** OF SIDEREAL ASTRONOMY. Of the Stars generally—Their Distinction into Classes according to their brightness—The Spectra of the Stars—How they are determined by the Milky Way—Annual Parallax—Real Distance, probable Distance, and apparent distance of the stars—The distances of the nearest Stars—Double Stars—Their Separation about such Stars as double stars, and their relation to the Sun—The Solar System—The Solar System and its Neighbourhood—Nebulous Nebulae—Stellar and planetary Nebulae—Zodiacal Lights. (581.) Besides the bodies we have described in the foregoing chapters, the heavens present us with an immense number of other objects which are called generally by the name of stars. Though comprehending individuals differing from each other, not merely in brightness but in all other respects, yet all agree in one attribute—a high degree of permanence as to apparent relative situation. This has procured them that name. The nature of this attribute can be understood in a comparative and not in an absolute sense, it being certain that many, and probable that all are in a state of motion, though so slow to be perceptible under ordinary circumstances, that they may be considered during a long series of years. (582.) Astronomers use in the habit of distinguishing the stars into classes, according to their apparent brightness. These are termed magnitudes. The brightest stars are those which appear to us at night to fall so far short of the first degree of brightness as to make a marked distinction be closed into the second, and so on down to the faintest star visible to the naked eye, in the clearest dark- night. Beyond these, however, witnesses continue to observe that some stars, and magnitude from the 8th down to the 16th are familiarly known. It is owing to the practice of using powerful instruments; nor does there seem the least reason to assign a limit to this progression; every improvement in optical instruments and every increase in their power, which successive improvements in optical science have 2 G **850** A TREATISE ON ASTRONOMY. [CAP. XIL] stated, having brought into view multitudes (immense) of objects invisible to the naked eye, for any thing experience has hitherto taught us, the number of the stars may be really infinite, in the only sense in which we can assign a magnitude to them. (569.) This classification into magnitudes, however, it must be observed, is entirely arbitrary. A multitude of bright stars, of equal size and brightness, but of unequal size and in splendour, and arranged at unequal distances from us, one must of necessity appear the brightest, one next below it, and so on. But this is not the case; for, as a matter of fact, the nearest star to us is less bright than the second nearest; and though it is impossible to determine exactly, or a priori, where one magnitude ends and the next begins, and although different astronomers have assigned different magnitudes, yet, on the whole, astronomers have restricted their first magnitude to about 15 or 16 principal stars; their second magnitude to about 17 or 18 principal stars; their third magnitude to about 20 or smaller, and so on; the numbers increasing very rapidly as we descend in the scale of brightness, the whole range being divided into seven classes, from first to seventh magnitude, inclusive, amounting to 18000 or 30000. (564.) As we do not see the actual size of a star, but judge only of its brightness by the total impression made upon the eye, the apparent "magnitude" of any star will, it is evident, depend upon two things: viz., its size; and 3rdly, on the absolute magnitude of its illuminated sur- face; 3rdly, on the intrinsic brightness of that surface. Now, as we know nothing whatever concerning the first of these data, and have every reason for believing that each of them may differ in different individuals, in the proportion of many thousandths of a degree; while we cannot expect much satisfaction in any conclusions we may draw from numerical statements of the number of individuals arranged, according to our system of classification; since astronomers have not yet agreed upon any principle by which those **CHAP. XII.** LIGHT OF THE STARS. 361 magnitudes may be photometrically arranged, though a large number of stars, owing to the irregular progression of which each term is half of the previous, may be disordered. Nevertheless, it was much to be wished, that, setting aside the stars of the first magnitude, subdivisions, a numerical estimate should be formed, grounded on the actual observations and experi- ments, of the apparent brightness of each star. This would enable us to ascertain the variations of light; and serve as a term of comparison to ascertain the changes which may take place in them; changes which we know to occur in the sun itself, and which are supposed to be possible in all. Meanwhile, as a first approximation, the following proportions of light, concluded from Sir J. Herschel's experimental comparisons of a few selected stars, may be adopted:
    Light of a star of the average 1st magnitude = 100
    2d = 35
    3d = 18.7
    4th = 6
    5th = 2
    6th = 1
    By my own experiments, I have found that the light of Sirius (the brightest of all the fixed stars) is about 324 times greater than that of the average star. (865.) If the comparison of the apparent magnitudes of the stars with their numbers leads to no definite con- clusion respecting their comparative luminosity, we must consider their distribution over the heavens. If indeed we confine ourselves to the three or four brightest classes, we shall find that they are distributed very uniformly over the sphere; but if we take in the whole amount visible to the naked eye, we shall perceive a great and rapid diminution in their numbers towards the edge of the milky way. And when we come to telescope mag- nitudes, we find them crowded beyond imagination, along the line of the Milky Way; so that it appears as if it sends off from it; so (art. 268) that in fact its whole light is composed of nothing but stars, whose average magni- tude is about 10. (866.) These phenomena agree with the supposition *Source:* *Observ. Comet of Dodsley p. 283.* *Date:* July 27, 1817. *Source:* *Trans. Soc. ill. Sci.* 369 A TREATISE ON ASTRONOMY. [CHAP. XIL] that the stars of our firmament, instead of being scattered in all directions indifferently through space, form a struc- ture, of which the axis is perpendicular to the plane of its length and breadth ; and in which the earth occupies a place somewhere about the middle of its thickness, and near the equator, so that the rays of light proceeding towards us, inclined at a small angle to each other. For it is certain that, in an eye so situated, the apparent den- sity of the stars is much less than that which would be ac- cordingly perceived through the space they occupy, would be least in a direction of the visual ray (as) perpendicular to the horizontal plane of the earth's surface, and its breadth (SB, SC, SD); increasing rapidly in passing from one to the other directions, just as we see a slight haze in the atmosphere thickening towards the horizon. This is owing partly to the rapid increase of the near length of the visual ray. Accordingly, such is the view of the construction of the narrow firmament that Mr. William Herschel has shown, powerful telescopes have effected a complete analysis of A diagram showing two lines intersecting at an angle, with labels B, C, D. this wonderful zone, and demonstrated the fact of its entire- ly consisting of stars. So crowded are they in some parts of it, that by counting the stars in a single field of his tele- scope he has found that 80000 had passed under his review in a zone two degrees broad, during but a single hour's observation. The immense distances at which the semidiameter must be assumed will sufficiently ac- count for the vast number of these minute luminous which are observed in it. (667.) It is evident from the comparative dense- ness of certain regions of the stary heavens beyond others, and of our own situation in them, the question arising respecting whether any part of the distance of the nearest fixed star ! What is the scale on which this con- firmation is constructed ? And what proportion do its di- mensions bear to those of our immediate system! To this, however, astronomy has hitherto proved unan- CHAP. XII.] DISTANCE OF THE STARS. 363 to supply an answer. All we know on this subject is ne- gative, and consists only in the few observations and refined combinations of theoretical reasoning, to a correct estimate, first, of the dimensions of the earth; then, taking our distance from the sun, by means of the orbits about the sun; and again, by taking our stand, as it were, on the opposite borders of the circumference of this orbit, we can estimate the distance of the sun from us, and the range of our own system, and by the aid of what we know of the excursions of comets, have felt our way, as it were, a step at a time, towards the nearest known planet. But between that remotest orb and the nearest star there is a gulf fixed, to whose extent no observa- tions whatever can give any approach. A distinct approximation, or to name any distance, however im- mense, which it may not, for any thing we can tell, sur- pass. (588.) The diameter of the earth has served us as the base of a triangle, in the trigonometrical survey of our system. We have measured its angles with respect to the sun: but the extreme minuteness of the sun's paral- lax (art. 304) renders the calculation from this too illu- sory. But let us suppose that we could measure with but the fortunate combination of favourable circumstances, afforded by the transitice of Venus (art. 409) could re- duce our error to one ten-thousandth part of a degree. That the earth's diameter is too small a base for direct triangula- tion to the verge even of our own system (art. 449), and that the distance of other stars must be measured by some para- llactic for the diurnal, or which comes to the same thing, to ground our calculation on the relative velocities of the earth and stars (art. 450), is evident enough; and thus we would push our triangulation to that extent. It might be naturally enough expected, that by this enlargement of our circle we should obtain more accurate results; but such the next step in our survey (art. 287) would be made at a great advantage—that our change of station, from side to side of the sun's orbit, should be made with an invariable amount of annual parallax in the stars, and that by its means we should come to a knowledge of their distance. But so long as existing evidence does not establish, as astronomers have been able to sum up to any positive 864 A TREATISE ON ASTRONOMY. [CHAP. XIL concedent conclusion upon this head; and it seems, therefore, that the stars which have been hitherto puzzling, even for the nearest fixed star which has hitherto been examined with the requisite attention, remains still mixed up with the rest of the universe, and is not in- dental to all astronomical determinations. Now, such is the nicety to which these have been carried, that did the quantity in question exceed 3000000000 miles, or one- third of the radius of the earth's orbit subjoined at the nearest fixed star minute angle), it could not possibly have escaped detected among the rest. (866.) Radius is to the sine of 1°, in round numbers, as 300000 to 1. In this proportion, then, of least must the distance of any star be from us, when we consider it as far from the sun from the earth. The latter distance, as we have already seen, exceeds the earth's radius in the proportion of 34500 to 1. Hence, by this proportion, the distance of stars, the earth's radius is 4000 of our miles. The dis- tance of the stars, then, cannot be so small as 400000000 radius of the earth's orbit 34500000 miles! How much larger it may be, we know not. (867.) In such numbers, the imagination is lost. The only mode of conceiving what they mean is by supposing all by the time which it would require for light to traverse them. Now light, as we know, travels at the rate of 192000 miles per second. Hence, if a star be at a distance of 4000000 seconds, or upwards of three years, in such journey, at the very lowest estimate. What, then, are we to allow for its magnitude? These are questions which of the smaller magnitude which the telescope discloses to us! If we admit the light of a star of each magnitude to be half that of another of equal size and distance, we shall follow that a star of the first magnitude will require to be removed to 82 times its distance to appear no larger than one of the third magnitude. But this is only true, among the countless multitude of such stars, visible in telescopes, there must be many whose light has taken at least three years' journey to reach us. To observe their places, and note their changes, we are in, fact, reading only their history of a thousand years' date, this work being but a fragmentary portion of their entire clusion, but by adopting as an alternative an infinite CHAP. XII.] INTRINSIC LIGHT OF THE STARS. 255 inferiority of light in all the smaller suns of the milky way. We shall be better able to estimate the probability of this alternative, when we have made acquaintance with other sidereal systems, whose existence the telescope discloses us to possess. But, before we proceed to a more full view of the subject we have taken above is in perfect harmony with the doctrine of the Copernican system. (901.) Quitting, however, the question of speculation, and confining ourselves within certain limits which we are sure are less than the truth, let us employ the negative knowl- edge which we have gained respecting the magnitude of the stars to form some conformable estimate of their real magnitudes. Of this, telescopes afford us no direct evidence; but they may perhaps give us some idea how few of the stars are real, but spurious--a mere optical illusion. These lights, therefore, must be our only guides. Now Dr. Bradley has made several optical experiments, open, as it would seem, to no objections, that has ascertained the light of Sirius, as received by us, to be less than that of the sun. The light of Sirius is seen, therefore, in order that it should appear to us brighter than Sirius, would require to be removed to 141400 times its present distance from us. Hence it follows, that, upon the lowest possible assumption respecting the luminosity of Sirius, his light cannot be so little as double that emitted by the sun ; or that Sirius must, in point of intrinsic splendour, be at least equal to the sun. This conclusion is confirmed by observations (902). Now, for what purpose are we to suppose such magnificent bodies scattered through the abysses of space ? The answer is obvious. We see them because they are occasional moons of the thousandth part of this size of our own would do much better, nor to speak as a pagan void of imagination. But this is not a very plausible vain conjecture. Useful it is true, they are to man points as facts of exact and permanent reference; but he must have assumed a little perspective in supposing them to be so near us. 255 (901.) Quitting, however, the question of speculation, and confining ourselves within certain limits which we are sure are less than the truth, let us employ the negative knowledge which we have gained respecting the magnitude of the stars to form some conformable estimate of their real magnitudes. Of this, telescopes afford us no direct evidence; but they may perhaps give us some idea how few of the stars are real, but spurious--a mere optical illusion. These lights, therefore, must be our only guides. Now Dr. Bradley has made several optical experiments, open, as it would seem, to no objections, that has ascertained the light of Sirius, as received by us, to be less than that of the sun. The light of Sirius is seen, therefore, in order that it should appear to us brighter than Sirius, would require to be removed to 141400 times its present distance from us. Hence it follows, that, upon the lowest possible assumption respecting the luminosity of Sirius, his light cannot be so little as double that emitted by the sun ; or that Sirius must, in point of intrinsic splendour, be at least equal to the sun. This conclusion is confirmed by observations (902). Now, for what purpose are we to suppose such magnificent bodies scattered through the abysses of space? The answer is obvious. We see them because they are occasional moons of the thousandth part of this size of our own would do much better, nor to speak as a pagan void of imagination. But this is not a very plausible vain conjecture. Useful it is true, they are to man points as facts of exact and permanent reference; but he must have assumed a little perspective in supposing them to be so near us. 255 (901.) Quitting, however, the question of speculation, and confining ourselves within certain limits which we are sure are less than the truth, let us employ the negative knowledge which we have gained respecting the magnitude of the stars to form some conformable estimate of their real magnitudes. Of this, telescopes afford us no direct evidence; but they may perhaps give us some idea how few of the stars are real, but spurious--a mere optical illusion. These lights, therefore, must be our only guides. Now Dr. Bradley has made several optical experiments, open, as it would seem, to no objections, that has ascertained the light of Sirius, as received by us, to be less than that of the sun. The light of Sirius is seen, therefore, in order that it should appear to us brighter than Sirius, would require to be removed to 141400 times its present distance from us. Hence it follows, that, upon the lowest possible assumption respecting the luminosity of Sirius, his light cannot be so little as double that emitted by the sun ; or that Sirius must, in point of intrinsic splendour, be at least equal to the sun. This conclusion is confirmed by observations (902). Now, for what purpose are we to suppose such magnificent bodies scattered through the abysses of space? The answer is obvious. We see them because they are occasional moons of the thousandth part of this size of our own would do much better, nor to speak as a pagan void of imagination. But this is not a very plausible vain conjecture. Useful it is true, they are to man points as facts of exact and permanent reference; but he must have assumed a little perspective in supposing them to be so near us. 255 (901.) Quitting, however, the question of speculation, and confining ourselves within certain limits which we are sure are less than the truth, let us employ the negative knowledge which we have gained respecting the magnitude of the stars to form some conformable estimate of their real magnitudes. Of this, telescopes afford us no direct evidence; but they may perhaps give us some idea how few of the stars are real, but spurious--a mere optical illusion. These lights, therefore, must be our only guides. Now Dr. Bradley has made several optical experiments, open, as it would seem, to no objections, 255 (901.) Quitting, and confining ourselves within certain limits which we are sure are less than the truth, let us employ the negative knowledge which we have gained respecting the magnitude of the stars to form some conformable estimate of their real magnitudes. Of this, telescopes afford us no direct evidence; but they may perhaps give us some idea how few of the stars are real, but spurious--a mere optical illusion. These lights, therefore, must be our only guides. Now Dr. Bradley has made several optical experiments, open, as it would seem, to no objections, that has ascertained the light of Sirius, as received by us, to be less than that of the sun. The light of Sirius is seen, therefore, in order that it should appear to us brighter than Sirius, would require to be removed to 141400 times its present distance from us. Hence it follows, that, upon the lowest possible assumption respecting the luminosity of Sirius, his light cannot be so little as double that emitted by the sun ; or that Sirius must, in point of intrinsic splendour, be at least equal to the sun. This conclusion is confirmed by observations (902). Now, for what purpose are we to suppose such magnificent bodies scattered through the abysses of space? The answer is obvious. We see them because they are occasional moons of the thousandth part of this size of our own would do much better, nor to speak as a pagan void of imagination. But this is not a very plausible vain conjecture. Useful it is true, they are to man points as facts of exact and permanent reference; but he must have assumed a little perspective in supposing them 255 (901.) Quitting, and confining ourselves within certain limits which we are sure are less than the truth, let us employ the negative knowledge which we have gained respecting the magnitude of the stars to form some conformable estimate of their real magnitudes. Of this, telescopes afford us no direct evidence; but they may perhaps give us some idea how few of the stars are real, but spurious--a mere optical illusion. These lights, therefore, must be our only guides. Now Dr. Bradley has made several optical experiments, open, as it would seem, to no objections, 255 (901.) Quitting, and confining ourselves within certain limits which we are sure are less than the truth, let us employ the negative knowledge which we have gained respecting the magnitude of the stars to form some conformable estimate of their real magnitudes. Of this, telescopes afford us no direct evidence; but they may perhaps give us some idea how few 255 (901.) Quitting, and confining ourselves within certain limits which we are sure are less than 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 (901.) Quitting, 255 **250** A THEATRE ON ASTRONOMY. [CHAP. XII.] man to be the only object of his Creator's care, or who does not see in the vast and wonderful apparatus around us provision for other races of animated beings. The planets, as we have seen, derive their light from the sun; but that cannot be said of the stars, which are themselves lights, and may, perhaps, each in its sphere, be the preceding centre round which other planets, or bor- derers of our system, may revolve. But this is not the na- logy offered by our own system, may be circulating. (882.) Analogies, however, more than conjectural, are not wanted to explain the phenomena of nature. The dynamical laws which prevail in the remote regions of the sizes and those which govern the motions of our own system, are so similar, that they must necessarily imply the regular recurrence of the same phenomena in the same time—we are strongly impressed with the idea ofetary bodies revolving about a central point in a circular manner, though no way distinguishable from others by any apparent change of place, nor by any difference in appearance. We find that the variations in the periodical increase and diminution of luster, involving in one or two cases, a complete extinction and revival. These are called "periodic" phenomena. The most remarkable is the star Omicron, in the constellation Cetus, first no- ticed by Fabricius in 1586. It appears about twelve times a year; at first it is very faint; then it increases to a bright day; remains at its greatest brightness about a fort- night; being then, on some occasions, equal to a large part of the sun; after which it diminishes again for about three months, till it becomes completely invisible, in which state it remains during about five months; when again begins to increase gradually until it reaches its maximum during the remaining three months of its period. Such is the general course of its phases. It does not always, how- ever, remain at its maximum for the whole period of its increase and diminish by the same gradations. Havelius, indeed, relates (Lalande, art. 794) that during the four years between October, 1783, and December, 1787, it did not appear at all. (884.) Another very remarkable periodic star is that called Algol (or Alcor), situated in Perseus. It has the second of the second magnitude, and such it continues for she A star chart showing Algol (Alcor) in Perseus. CHAP. XII.] PERIODICAL STARS. 307 space of 2° 14', when it suddenly begins to diminish in splendour, and at 2° 31' hours is reduced to the fourth magnitude, and then again increases, and at 2° 54' hours more is restored to its usual brightnesses, going through all its changes in 2° 20' 48", or thereabouts. The rapidity with which the star's light varies so powerfully suggests the revolution round it of some spoke body, which, when intercepted by the earth and Algol, would intercept its light; and this is supported by the view taken of the matter by Goodricke, to whom we owe the discovery of this remarkable fact. In many cases similar phenomena have continued to be observed, though with much less distinctness than in high latitudes within our hemisphere. Taken on how, it is a symptom of a high degree of activity, in regions where, but for such evidence, we might conclude all lifeless. Our own sun requires nine days to make one complete rotation upon its own axis. On the other hand, the periodic time of an apse rotating body, sufficiently large, which should produce a maximum variation in the light of a star from a fixed star, would be less than fourteen hours. (265.) The following list exhibits specimens of perio- dical stars, which vary so rapidly that they can be considered to be at present ascertained:





















































































    <
    Star's Name. Period. Variation of Magnitude. Differences.
                                                              
    ► Persei. 2° 36' 8 ± 2 Goodricke, 1798.
    ► Cygni. 2° 37' ± 4 Copernicus, 1666.
    ► Antares. 2° 40' ± 10 Pigott, 1797.
    ► Arcturus. 2° 41' ± 10 Pigott, 1797.
    ► R.A. 13° 40'. E. 2° 43' ± 7 Hunting, 1808.
    ► Ceti. 2° 44' ± 10 Fraenkel, 1856.
    ► Procyon. 2° 45' ± 10 Machlini, 1801.
    ► Mira Cygni. 2° 46' ± 10 Machlini, 1801.
    ► Aquilae. 2° 47' ± 10 Machlini, 1801.
    ► Deneb. 2° 48' ± 10 Machlini, 1801.
    ► R.A. 3° 30'. W. 2° 49' ± 10 Machlini, 1801.
    ► R.A. 3° 30'. E. 2° 50' ± 10
    See note on page 208. * These are from Fl. and M. refer to the Catalogue of Bode Flammarion's. A diagram showing the positions of several stars relative to each other and Earth's position in space. The diagram includes labels for different stars and their positions in relation to Earth's position in space. The diagram also includes a scale bar indicating distances between stars and Earth's position in space. The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The diagram is labeled "Diagram showing the positions of several stars relative to each other and Earth's position in space." The following table gives a summary of some important facts concerning these variables: 358 A TREATISE ON ASTRONOMY. [CAP. XII. The nature of these stars, however, appear to be affected, perhaps by some peculiarity of their own, or by extent of change, by physical causes at present unknown. The non-appearance of * Ceti, during four years, has al- ready been noticed. The same circumstance was observed that of * Cygni, which is stated by Cassini to have been scarcely visible throughout the years 1690, 1700, and 1701, at those times when it ought to have been most conspicuous. (396.) These irregularities prepare us for other phe- nomena of a similar kind. The stars which have re- duced to no law of periodicity, and must be looked upon, in relation to our ignorance and inexperience, as also other cases of irregularity, may be supposed to have occurred more than once within the limits of recorded observation. The phenomena we allude to are those of cometary appearances. They have been observed at different times in different parts of the heavens, blazing forth with ex- traordinary lustre ; and after remaining a while appa- rently inactive, they have again appeared with great force. Such is the star which suddenly appeared in the year 125 B. C., in which we have attracted the attention of Hip- parchus, who first described its appearance on the 2nd March, the earliest on record. Such too, was the star which blazed forth, A. D. 389, near * Aquileia, remaining for three weeks a very conspicuous object in the northern sky. In the years 945, 1064, and 1572, brilliant stars appeared in the region of the heavens between Cepheus and Cassiopeia. In each case they were seen to have of places of the two earlier, as compared with that of the last, which was well determined, as well as from the luminosity of the star itself. From this remarkable simi- lance, we may suspect them to be one and the same star, with a period of about 300, or so Goodricke supposes, * The same discovery appears to have been made nearly about the same time by Sir William Herdman, who had observed it in 1690, as an astronomer ; but he did not publish his observations until 1700, as an astronomer ; but he did not publish his observations until 1700, when he found it necessary to wait for another appearance before discovering the predicted course of Halley's comet in 1758, which he saw annually for several years previous to its return. These anomalies carry us back to the time of Sir William Herdman's observation. **CHAP. XII.]** **TEMPORARY STARS.** 359 of 180 years. The appearance of the star of 1792 was so sudden, that Tycho Brahe, a celebrated Danish astronomer, returning one evening (the 11th of November) from his laboratory to his dwelling-house, was surprised to find a bright object in the heavens, which he was sure did not exist half an hour before. This was the star of 1792, which he called the "star of the dawn," because it rose with Jupiter when brightest, and was visible at mid-night. It began to diminish in December of the same year, and in March, 1800, it had disappeared entirely. On the 10th of October, 1864, a star of this kind, and not less brilliant, burst forth in the constellation of Serpentarius, which is now known as the "Little Dipper" (597). Similar phenomena have taken place more recently, as in the case of the "Star of Bethlehem," observed by Sir John Herschel on the 2nd of July, 1876, by Ansdell, in the head of the Swan ; which, after becoming completely invisible, reappeared again under great resemblance to peculiar flurries of light, during two years, at last died away entirely, and has not since been seen. On a careful re-examination of the heavens, too, by Mr. Bessel, in 1850-51, no such object was found to be missing; and although there is no doubt that these losses have often arisen from mistakes entries, yet it is probable that they are not very numerous; for no mistake in the observation or entry, and that the star has really been observed, and as really has disappeared from the heavens. The "Star of Bethlehem" is a luminous body which has been too little followed up, and it is probably this circumstance which accounts for its being so seldom only good instruments and accurate instruments, might employ their time to excellent advantage.† It holds out a sure promise of rich discovery, and is one in which astronomy † The star of Virginia is inserted in the Catalogue of the Astronomical Society's Observatory at Cambridge (published in 1866), and has since repeatedly lost its place in the field of view of any telescope used by astronomers. It is supposed to be one of the 9th magnitude, very nearly in the place in which it may have been observed. ‡ Ces variations des étoiles sont bien dignes de l'attention des astrono- mistes. L'astronome qui les observe doit se rappeler que ces étoiles ne sont pas constamment dans le même lieu dans le ciel. — Zeitschrift für Astronomie, 1864—Sundry that day was new arrested. 359 300 A TREATISE ON ASTRONOMY. [C.MAP. EXI.] mercs in established observatories are almost of necessity precluded from taking a part by the nature of the observations required. Catalogues of the comprehensive brightnesses of stars, which have been constructed by Sir Wm. Herschel, with the express object of facilitating these researches, and the reader will find them in his "Catalogue of Cometary Stars," 1785, and "Observations on the Brightness of Stars," 1796, and subsequent years. (306.) We come now to a class of phenomena of quite a different kind, which may be considered as affording a pure insight into the nature of at least some among the stars, and enable us substantially to declare them subjects to the same laws which govern the motions of the solar power of gravitation, which governs our own system. Many of the stars, when examined with telescopes, are found to consist of two or more individuals placed near together. This might be attributed to accidental proximity, did it occur only in a few instances; but it is found that it is very common, even at the extreme closeness, and, in many cases, the near equality of the stars so conjuncted, would lead one to a strong supposition that they are really two distinct bodies in mutual conjunction. The bright star Cetus, for example, when much magnified, is found to consist of two stars of about equal magnitude, separated by a distance of 5" or each other. Stars of this magnitude, however, are not so common in the heavens as to render it at all likely that such a range as two would be met with twice over. But this is only out of number; such instances as Sir Wm. Herschel has enumerated upwards of 300 double stars, and he has observed that they are a minute of each other; and to this list Professor Struve of Dorpat, prosecuting the inquiry by the aid of instru- ments superior to those then in use for purposes similar, recently added nearly five times that number. Other ob- servers have still further extended the catalogue, already so large as to be almost incredible in its extent. Among these are great numbers in which the interval be- tween the centres of the individuals is less than a single second, or even less. Astronomers have observed that Corona b., and - s., and - o., and - p., may be cited as instances. They are divided into classes ac- **EFFECT OF PARALLAX ON A DOUBLE STAR.** 301 corresponding to their distances—the closest forming the first class. (500.) When these combinations were first noticed, it was considered that advantage might be taken of them, so as to determine the distance of the nearer star, which in its orbit might not produce relative apparent displacement of the individuals constituting a double star. Supposing the two stars to be at equal distances from each other, and to appear only by casual juxtaposition nearly in the same line, it is evident that any motion of the earth must affect one of them more than the other, if they are connected, and must therefore produce different parallactic displacements of them on the surface of the heavens, regarded as stationary. Now, if we suppose that the earth's annual motion, as described in the preceding chapter, in the heavens a small ellipse (distinct from that which it would appear to occupy when viewed from the sun), a section of light, and not to be confused with it), being a section, by the concave surface of the heavens, an ellipse having its vertex at the pole, and the earth's orbit for its base; and this section will be of less dimension than the more distant is the star. If, then, we regard two stars, apparently closed beside each other, but in reality at very different distances, their parallactic ellipses will be similar, but of different dimensions. Suppose now two stars to be situated in such a way that the two stars of such an apparently or optically double star as A diagram showing two stars (A and B) close together in space, with their parallactic ellipses (C and D) shown as similar but different in size. The larger ellipse represents the more distant star, while the smaller one represents the closer star.2 H308 A TREATISE ON ASTRONOMY. [CHAP. XII] seen from the sun, and let ABCD, a b c d, be their pa- rallactic ellipses; then, since they will be at all times similarly situated in these ellipses, when the one star is seen at the same place in the heavens, the other star on the earth has made a quarter of a revolution in its orbit, their apparent places will be H0; when another quarter, C0; and so on. The stars being measured carefully, with micrometers adapted for the purpose, their apparent situation with respect to each other, at different times of the day, and at different places on the horizon, both in the direction of the line joining them, and in the distance between their centres. For the lines An and Cc cannot be drawn parallel to each other, because the ellipses of two equal distances, i. e., unless the two stars have the same parallax, or are equidistant from the earth. (600.) Now, micrometers, properly mounted, enable us to measure very exactly both the distance between two objects and their angular position with respect to that of a telescope, and the position of the line joining them with respect to the horizon, or the meridian, or any other de- termined line of sight. The point of observation is cho- sen as the most convenient; and the situation of the line of junction between the two stars of a double star is inferred by means of a plane mirror placed before the eye-piece of a telescope, occasionally moved, two cross wires making a right angle, and adjusting their position so that we may see through them both stars at once; in diurnal motion, while the telescopes remain at rest; noting their situation; and then turning the whole system of wires round until they are again parallel to each other. In this manner we can determine their relative movement, till the other wire becomes exactly parallel to their line of junction, and reading off on a divided circle the angle which they make with it. This method of ap- proach is called a position micrometer; and by its aid we determine the angle of position of a double star, or the angle which it makes with any given line of sight. The han- dian ; which angle is usually reckoned round the whole circle, from 0 to 360, beginning at the north and proceed- ing to the south or north following (or east) south, pre- ceeding (or west). (601.) The advantages which this mode of operation CHAP. XL.] SYSTEMATIC PARALLAX. offers for the estimation of parallax are many and great. In the first place, the result to be obtained, being depend- ent only on the relative apparent displacement of the two stars, is unaffected by almost every cause which would interfere with the determination of their absolute places of either by reckoning ascension and declination. Refraction, that greatest of all obstacles to accurate astronomical observations, is usually at both stars; and it is there- fore eliminated from the results. We have no trouble to fear from errors of graduation in circles from lack of light, from the imperfectness of the instruments, the ungramatical reductions of aberration, precession, &c. —all which bear alike on both objects. In a word, if we exclude those causes which are peculiarly known by which a real change of relative situation may arise, no other cause but their difference of parallax can pos- sibly affect our results. (602). Such were the considerations which first in- duced Sir William Herschel to observe double stars, and to subject them all to careful measurements of their angles of position and mutual distances. He has hardly entered, however, upon these measurements, before he had discovered that one of the most curious (which in fact, promising as it is, still remains open and unsolved, though the only method whereby it is to be attained is now known) phenomena of parallax is by phenomena of a very unexpected character, which at once engrossed his whole attention. Instead of finding, as he expected, that the apparent distance between one star of a double star with respect to the other—that al- most always increases with time—there was found a distance which the parallax of the earth's an- nual motion would produce—he observed, in many in- stances, that this distance was less than it ought to be, being chiefly on their distance—in others on their po- sitioning, and advancing steadily in one direction, so as clearly to indicate that they are moving towards each other. This general rectilinear motion of one sun and whole solar system, producing a parallax of a higher or lower degree from its annual motion, and which might be called systematic parallax. (603). Supposing the two stars in motion independ- 364 A TREATISE ON ASTRONOMY. [CHAP. XLII. early of each other, and also the sun, it is clear that for the interval of a few years, these motions must be re- garded as recilious and uniform. Hence, a very slight inequality in their progress will produce a difference in the apparent motion of one star of a double star, referred to the other as a centre, and mapped down, as it were, on a plane at right angles to the line joining them. This point, or zero point, can be no other than a right line. This, at least, must be the case if the stars be independent of each other, and if they have no physical or physi- cal connection, such as, for instance, real proximity and mutual gravitation would establish. In that case, they would describe two circles round their common centre of gravity; and therefore the apparent path of either, referred to the other as fixed, instead of being a straight line, would be described by two curves concave towards that other. The observed mo- tions, however, were so slow, that many years' observa- tion was necessary before any conclusion could be drawn; and therefore, until the year 1803, twenty-five years from the commence ment of the inquiry, that any thing like a positive result could be expected to result from this recti- linear or orbital character of the observed changes of position. (604.) In that, and the subsequent year, it was dis- tantly announced by Sir William Herschel, in two papers, which will be found in the Transactions of the Royal Society for 1803 and 1804. These new systems, composed of two stars revolving about each other in regular orbits, and constituting what may be termed binary stars; are those which are now called double stars generally so called; in which these physically connected stars are confounded, perhaps with others only specially distinguished by their different motions at different distances from the eye; whereas the indi- viduals of a binary star are, of course, equidistant from the eye; and consequently move in orbits equal to half the semidiameter of the orbit they describe about each other, which is quite insignificant compared with the immense distance between them. There were fifty and sixty instances of changes, to a greater or less amount, in the angles of position of double stars, are ad- CHAP. XL.] ELLIPTIC ORBITS OF BINARY STARS. 305 duted in the memoirs above mentioned; many of which are now decided, and are regularly observed, to show all their nature being misconceived. In particular, among the more conspicuous stars, -Castor, y Virginia, £ Urna, £ Tycho, £ Cepheus, £ Cygnus, £ Bootes, 4 and 5 Lyrae, o Ophiuchus, o Draconis, and o Aquarii, are enumerated as revolving about each other with perpetual motion; and to some of them even periodic times of revolution are assigned, approximate only, of course, and rarely agree with the observations. The results of any exact calculation, for which the data were at the time quite inadequate. For instance, the revolution of Castor in 1806 was supposed to be that of y Virginia at 708, and that of o Lyrae at 1200 years. (603.) Subsequent observation has fully confirmed these revolutions; but they have been made on the most part in individual detail. Of all the stars above named, there is none which is not found to be fully equal to the others in this respect. The following list comprises nearly all the most considerable objects of that description which have yet been detected, though (as it appears) many more may be expected. These ob- servations have multiplied; it has, of late, begun to extend itself rapidly. The number of double stars which are known to be revolving about each other is be- tween thirty and forty at the present time; and we are emerging into notice with every fresh mass of observa- tions. It is remarkable how few astronomers possess excellent telescopes for their observation, being for the most part so close as to necessitate the use of very high magnifying powers. But with such powerful microscopes if employed to examine objects within our reach, to perceive an interval between the inclinations of two stars would be easy. (604.) It may easily be supposed, that phenomena of this kind would not pass without attempt to connect them with the laws of gravitation. But when no- vember, they were naturally referred to the agency of some power, like that of gravitation, connecting the stars thus described. This was afterwards extended by Sir William Herschel to all such other ; and the extension of the Newtonian law of gravit. 2 n 2 306 A TREATISE ON ASTRONOMY. [CHAP. XIII. tation to these remote systems was a step so obvious, and so well warranted by our experience of its self-sufficient agency in our own, as to have been expressly or tacitly made by every one who has given the subject any share of attention. We have here the simplest system of calculation, by which the elliptic elements of the orbits of the stars may be determined, and the variations of its angle of position and distance at different epochs, to M. Savary, who showed, * that the motions of one star with respect to another, though (as Uranus) were explainable within the limits of his observations or observation, on the supposition of an elliptic orbit described in the short period of 81 days, different processes of calculation, conducted by Professor Bessel to an elliptic orbit for 70 Ophiuchi, described in a period of seventy-seven years ; and the author of these pages has himself attempted similar calculations on various interesting investigations. The following may be stated as the chief results which have hitherto obtained in this branch of astronomy.
    Star Name Period (days) Variation (magnitude) Differences (magnitude) Remarks
    ► Persei 2°36' ±8 ±2 Goodricke, 1798
    ► Cygni 2°37' ±4 Copernicus, 1666
    ► Antares 2°40' ±10 Pigott, 1797
    ► Arcturus 2°41' ±10 Pigott, 1797
    R.A. 13°40'.E. 2°43' ±7 Hunting, 1808
    ► Ceti 2°44' ±10 Fraenkel, 1856
    ► Procyon 2°45' ±10 Machlini, 1801
    ► Mira Cygni 2°46' ±10 Machlini, 1801
    ► Aquilae 2°47' ±10 Machlini, 1801
    ► Deneb 2°48' ±10 Machlini, 1801
    R.A.3°30'.W. 2°49' ±10 Machlini, 1801
    R.A.3°30'.E. 2°50'
    Name Number Period Mean Maximum Minimum Recession.
    Laurie 1 2500000 27° 00' 00" 00" 00" - - 25230
    Virginis 2 2500000 27° 00' 00" 00" 00" - - 25230
    Cygni 3 2500000 27° 00' 00" 00" 00" - - 25230
    Cassiopeia 4 2500000 27° 00' 00" 00" 00" - - 25230
    Ophiuchus 5 2500000 27° 00' 00" 00" 00" - - 25230
    Taurus 6 2516667 27° 16' 66" 66" 66" - - 254879
    Cassiopeia 7






















































































    <
    (67). Of these, perhaps, the most remarkable is V. Virginis, not only on account of the length of its period, but also on account of its great eccentricity and ap- parent distance, and rapid increase of angular motion about each other, of the individuals composing it. It is a bright star, and its motion is such that two adjacent stars are almost exactly equal. It has been known to compose of two stars since the beginning of the eighteenth century, but it has been observed to vary between seven seconds ; so that any tolerably good telescope will re- * Comte des Trous. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 9/22 CHAP. XIL. ELLIPTIC ORBITS OF SUNARY STARS. solve it. Since that time they have been constantly ap- proaching, and are at present hardly more than a single second secondar; so that no telescope, that is not of very superior quality, is competent to show them otherwise than as two points of light, with a very slight di- rection. It fortunately happens, that Bradley, in 1718, no- ticed, and recorded in the margin of one of his observa- tions, the existence of this remarkable phenomenon, as being parallel to that of two remarkable stars, a and b of the same constellation, as seen by the naked eye; and thus he was enabled to determine their mutual separation by the diligence of Professor Rignard, has proved of sig- nal service in the investigation of their orbits. They are eccentric, and consequently inclined to each other's plane; and this affords also another means of recovering their rela- tive situation at the date of his observations, which were made about 1730. The following table exhibiting their individual measurements, which will be found in their proper repository, * will suffice to remark, that their mutual distance, which was determined by the present century has been very numerous and carefully made, and which embraces an angular motion of 100", and a diminu- tion of 5" in the mean distance (which is 16' 2") is represented with a degree of exactness fully equal to that of observation itself, by an ellipse of the dimensions given in the table below; but it must be observed, that of which the further requisite particulars are as follows:— Periapsis passage. August 18, 1856 Longitude of the visual sky Angle of position of the periapsis projected on the heavens 30° 54' plane of the orbit with the surface of the heavens 97° 29" (008). The great length of the periods of some of these bodies is remarkable, the shortness of those of others is hardly less so. The first revolution made a complete revolution since its first discovery by Sir Wil- liam Herschel, and is far advanced in its second period ; but it has not yet completed its third period. This is accom- plished by far the greater part of their respective ellipses since the same epoch. If any doubt, therefore, could re- main respecting their motions, it would arise from any *See those collected in Mem. L Art. Sci. vol. v. p. 35.
    Longitude of the visual sky 30° 54'
    Angle of position of the periapsis projected on the heavens 97° 29"
    (008). The great length of the periods of some of these bodies is remarkable, the shortness of those of others is hardly less so.
    368 A TREATISE ON ASTRONOMY. [CHAP. XII. idea of explaining them by mere parallactic changes, these facts must suffice for their complete dissipation. We have the same evidence, indeed, of their rotations about each other that we have of those of Uranus and Saturn though the latter are much more distant than their calculated and observed places in such very elongated elliptical orbits, and though they are less affected by the prevalence of the Newtonian law of gravity in their systems, of the very same nature and capacity as that of the calculated and observed places of comets round the sun's body. (500.) But it is not with the revolutions of bodies of a planetary or cometary nature round a solar centre that we are now concerned, but with the motions of stars—each, perhaps, accompanied with its train of planets and libellaries—those closely situated from one another by the splendour of their light, which are crowded into a space bearing hardly a greater proportion to the enormity of their distances than that occupied by the satellites of our planet from their primaries bear to their distances from the sun itself. A less distinctly characterized phenomenon would be incomparably less remarkable than this extraordinary nature of their orbits. Unless closely nestled under the protecting wings of their immediate superior, the sweep of their orbits would be too great for any star to find its own way carry them off, or whilst they into orbit utterly incompatible with the conditions necessary for the existence of life on earth. It is only because we know that we have here a strangely wide and novel field for specu- lation upon, and one which is not easy to avoid luxuriantly in. (610.) Many of the double stars exhibit the curious and beautiful phenomena of conjunction or complemen- tary colours. In some cases, when the larger star is sur- gally of a ruddy or orange hue, while the smaller one ap- pears blue or green, probably in virtue of that general law of oppositions which governs the relations in * * * * * * * * * * * * * * * * * * * * * * **other sense,** perhaps, With this aim, I have sought for discovery, Communicating male and female light. Which are not yet known to man. * * * * * * * * * * * * * * * * * * * * Bored in each, sort, partner, with some part, Fouled at last, fell, p. 165. PUBLISHED BY THE UNIVERSITY PRESS. **CHAP. XII. J** COLOURED STARS. under the influence of excitement by any bright, coloured light : feebler lights, which seem alone would produce no sensation but of whiteness, shall for the time appear coloured ; and this tint complementary to that of the bright star. Thus, a star of the same magnitude in the light of the brighter star, that of the less bright one in the light of the more brilliant one, will appear reddish, or of the brighter star verie crimson, that of the other will exhibit a tendency to green— even appear as a vivid green, under favourable circumstances. The two con- trasts may be easily exhibited by Cassiopeia—the latter by v. Andromedae; both fine double stars. If, however, the colours are produced by very bright lights, it will not materially affect the others. Thus, for instance, Cassiopeia exhibits the beautiful combination of a large white star, and a smaller one of a deep blue hue. It is by no means necessary, however, intended to say, that in all such cases one of the colours is a mere effect of contrast, and is not really produced by any peculiar virtue in imagination, what variety of illumination two stars red and a green, or a yellow and a blue one—must afford a reason for supposing that they are so different in their contrasts and "grateful visciditudes"—a red and a green day, for instance, alternating with a white one and with another of a deep blue hue. In like manner, one of one or other, or both, above the horizon. Isolated stars of a red colour, almost as deep as that of blood, occur in many parts of the heavens; but when seen at night (of any decided hue) we have, believe we have been noticed associated with a companion brighter than itself. (48.) The difference between the two motions is obvious; in the physical history of the stars, in their proper motion. *à priori*, it might be expected that apparent motions of some stars would be accompanied by real ones; but a multitude of individuals scattered through space, and with nothing to keep them fixed. Their mutual attractions would cause them to move about each other's positions and counteracted by opposing attractions from opposite quar- ters, must in the lapse of centuries ages, produce some movement in their places. This is indeed observed; conse- resulting from the difference of the opposing actions. And it is a fact, that such apparent motions do exist, not only 370 A TREATISE ON ASTRONOMY. [CHAP. XII.] among single, but in many of the double stars which, besides revolving round each other, or round their com- mon centre of gravity, are transferred, without partaking company, to another star, and so move both towards some determinate region. For example, the two stars of 1 Cygni, which are nearly equal, have re- mained at a mean distance of 15' from each other during a period of about 60 years; and the mean distance of 15', for at least fifty years past. Meanwhile they have shifted their local situation in the heavens, in such a manner that the star 1 Cygni B was on the 28th the annual proper motion of each star being 5'·3'; by quantity (exceeding a third of its interval) this system is seen to be moving along an unknown path, by a motion which, for many centuries, must be regarded as uniform and recircular. Among stars not less remarkable for their motions than any other obvious particular, a Cassiopeia is to be re- marked as having the greatest proper motion of any yet discovered. It has been observed to be carried away from its place by smaller, but constantly carried away from their places by smaller, but not less than 10" per year. (812.) Motions which require whole centuries to ac- cumulate before they produce changes of arrangement, such as those which we have mentioned, are sufficient to destroy that idea of mathematical fixity which pre- cludes speculation, are yet too trifling, as far as practical application is concerned, to affect our general ideas; and lead us to speak of the stars in common parlance as otherwise than fixed. Too little is yet known of their amount and direction to enable us to make any attempt to define it. It may, however, be stated generally, that their apparent directions are various, and seem to have no marked tendency either towards one star or towards another of the heavens. It was, indeed, supposed by Sir William Herschel, that such a common tendency could not be made out with certainty; but after long observation, a general recess could be perceived in the principal stars, from that point occupied by the star ξ Herculis, towards a point occupied by the star α Aurigae. This tendency was referred by him to a motion of the sun and solar system in the opposite direction. No one, who reflects CHAP. XII. MOTIONS OF THE SUN AND STARS. with due attention on the subject, will be inclined to deny the high probability, any certainty, that the sun has a proper motion in some direction; and the inevitable consequence of such a motion, unparticipated by the rest, must lead us to conclude that the sun's apparent motions are at the vanishing point of lines parallel to that direction, and to the region which he leaves. That is the necessary conclusion, unless we can discover some certain effect which must be detected by such observations, if we knew accurately the apparent proper motions of all the stars, and if we were able to observe them with sufficient accuracy, the whole firmament, or at least all that part which we see in our own neighbourhood, were not drifting along together with the sun. But this is impossible; for it is the result of unknown processes and slow internal changes going on in the sidereal stream to which our system belongs. The sun's proper motion is therefore real, and keeping nearly the same relative situation with respect to one another. But it is not possible to make any statement, that their science is not yet measured enough to afford data for any secure conclusions of this kind one way or other. Meanwhile, a very im- portant fact appears from the observations of the empe- roral Royal (Mr. Pendel), viz. that a solar motion, if it exist, and have a velocity at all comparable to that of light, would be observable in the course of years; in consequence of which we do not see the stars disposed as they really are, but too much crowded in the regions where they ought to be according to their true positions. (See art. 280.) Now this, so long as the solar velocity continues the same, must be considered as a very serious obstacle to astronomy; but although it may, in the course of ages, by a quantity as small as can be calculated by a geometrical construction, all the stars or half or other quarter of the heavens, according as the sun's motion was accelerated or retarded; which observation would not be observable within a single day, nor even more than a very few seconds. This consideration, refined and removed as it is, may serve to give some idea of the difficulty which astronomers have had in finding out proper motion; since the last mentioned effect would ne- **573** A TREATISE ON ASTRONOMY. [chap. xli] essarily be mixed up with the systematic periaxial, and could only be separated from it by considering that the nearer stars would be affected more than the distant ones by the one cause, but both near and distant alike by the other. (612.) When we cast our eyes over the concave of the heavens in a clear night, we shall find a very extensive group of stars which seem to be compressed together in a more condensed manner than in the neighbouring parts, forming bright patches and clusters, which are so numerous that they cannot be brought together by some general cause other than causal distribution. There is a group, called the Pleiades, in which all the stars are seen to be directed towards one point or directly full or upon it; and many more if the eye be turned carefully aside, while the attention is kept directed upon this group. The same thing happens with large stars thus crowded together in a very moderate space, comparatively small compared with the rest of the heavens. The group called Coma Berenices is another such group, more diffused, and consisting of much larger stars. (614.) In the constellation Cancer, there is a some-what similar but less definite, luminous spot, called Pegasus, or the bee-bowl, which a very modern telescope resolves entirely into stars. In the sword handle of Perseus, also, is another such spot, crowded with stars, which requires careful observation to distinguish them from those sepa- rated from each other. These are called clusters of stars; and when we consider their nature it is evident that they are aggregation sub-sets of these spots, that those which have determined the scattering of stars over the general surface of the sky. This conclusion is still more strongly *It is a fact well known to astronomers that it is im-possible to decide upon impressions of light, than the existence of persons who have been observed to move about under circumstances peculiarly favourable extenders previous to trial. To appreciate it is best to look both at the sun and moon at once. If you choose two equally bright, and about 20 or 30 apart, and look full at each for a few minutes, you will see that the moon is my own face. The fact seems for the minute of starry with which we see them to be due to their being near us when we come to count them.*Footnote reference. CHAP. XII.] CLOVERES OF STARS. pressed upon us, when we come to being very powerful telescopes to hear on these and similar spots. There are a great number of objects which have been mistaken for comets, and, in fact, have very much the appearance of comets; but they are not so. They are either nebulous specks, which telescopes of moderate power only show as such. Messier has given, in the *Comète de Tempa* (p. 375) a list of them, with their distances and sizes; which all those who search for comets ought to be familiar with, to avoid being misled by their similarity of appearance to real comets. But they are not comets, their fixity sufficiently proves; and when we come to examine them with instruments of great power--such as reflectors of eight inches diameter--we find that their mechanism such idea is completely destroyed. They are then, for the most part, perceived to consist entirely of stars revolving round each other in a circular or elliptic out- line, and to run up to a blaze of light in the centre, where their condensation is usually the greatest. (See p. 1, § 1.) The same thing is observable in the thirteenth nebula of Messier's list (described by him as a nebulae sans nom), as seen in the 20 feet reflector at Greenwich. It consists of two stars revolving in an exactly round figure, and convey the complete idea of a globular space filled up of stars, insolated in the heavens, and concentrically revolving round one another in a circular or elliptic out-line, and subject only to its own internal laws. It would be a vain task to attempt to count the stars one of these globules contains; but I have made an estimate of hun- dreds; and on a rough calculation, grounded on the apparent intervals between them at the borders (where they are most numerous), I think that the whole angular diameter of the whole group, it would appear that many clusters of this description must contain, at least, ten or twelve thousand stars. If they were bridged together in a round space, whose angular diameter does not exceed eight or ten minutes; that is to say, in an area not more than a tenth part of that covered by the moon. This beautiful object was first noticed by Halley in 1714. It is wid- ly known under the name of "The Great Nebula." In its right- angle it appears exactly like a small comet. 2 374 A TREATISE ON ASTRONOMY. CAP. XII. (615.) Perhaps it may be thought to favour of the greatness of the distance of the stars, that they appear so near like our own, and their mutual distances as equal to those which separate our sun from the nearest fixed star; but this is only a superficial appearance, and ef- fects the eye with a less impression of light than a star of the fifth or sixth magnitude (for the largest of these classes is not visible without a telescope). For when we are thus compelled to form of their distance from us may render even such an estimate of their distances familiar to our senses, we find that they are not so near as they look upon a group thus inscribed, thus in seque futura, tenebrae, etiam rotundae, as not forming a system of a peculiar nature distinct from that of the sun. This circumstance indi- cates the existence of some general bond of union in the nature of an attractive force; and, in many of them, have a similar appearance to the sun. But this assump- tion as we approach the centre, which is not referable to a merely uniform distribution of equidistant stars through the globular system, and which renders their state of aggregation greater at the centre than at the sur- face of the mass. It is difficult to form any conception of the difficulty with which one can imagine, on the other hand, without a rotary motion and a centrifugal force, it is hardly possible not to regard them as in a state of aggregation greater at the centre than at the surface. If a simi- lar motion and such a force, we find it no less difficult to recon- cile the apparent sphericity of their form with a rotation about their axis than with any other hypothesis by which internal collisions would appear to be inevitable. The following are the places, for 1890, of a few of the prin- cipal of these remarkable objects, as specimens of their class —
    K.A. N.P.D. K.A. N.P.D.
    1 22 23 24 25
    2 25 26 27 28
    3 28 29 30 31
    4 31 32 33 34
    5 34 35 36 37
    6 37 38 39 40
    7 40 41 42 43
    8 43 44 45 46
    9 46 47 48 49
    10 49 50 51 52
    11 52 53 54 55
    12 55 56 57 58
    13 58 59
    K.A.N.P.D.
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
    50
    51
    52
    53
    54
    55
    56
    57
    58
    		N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    K.A.N.P.D.
    (616.) It was to Sir William Herschel that we owe the most complete knowledge of the great variety of this ob- See a note on this subject at the end of the work, p. 208. ject. CHAP. XL.] OF CLUSTERS OF STARS. 373 jacent which are generally classed under the common head of Nebulae but which are distinguished by him into 1st. Clusters of stars, in which the stars are clearly distinguishable; and these, again, into globular and ir- regular clusters. The former are those which ex- cite a suspicion that they consist of stars, and which any increase of the optical power of the telescope may be expected to confirm; while the latter are those properly so called, in which there is no appearance whatever of stars; which, again, have been subdivided into two classes, viz., the open clusters, or loose and size; 4th, Planetary nebulae; 5th, Stellar nebula; and, 6th, Nebulous stars. The great power of his selec- tion has enabled him to give us a very large number of these objects, and shown them to be distri- buted over the heavens, not by any means uniformly, but in certain well-defined groups. Thus we come to a broad zone crossing the milky way nearly at right angles, and whose general direction is not very remote from that of the Milky Way itself. This is one of some parts of this zone, indeed - especially where it crosses the constellations Virgo, Coma Berenices, and the Great Bear. It is remarkable that this is the most interesting however, for the most part telescopic, and beyond the reach of any but the most powerful instruments. (4) The globular clusters of stars. Such as we have already described, or of irregular figure. These latter are, generally speaking, less rich in stars, and es- pecially in bright ones than the former. They are also less definite in point of outline; so that it is often not easy to say where they terminate, or whether they are really separate from other clusters. But many parts of the heavens than those around them. In some of them the stars are nearly all of a size, in others extremely dif- ferent in size. The largest cluster known is the red star much brighter than the rest, occupying a compa- sionable situation in them. Sir William Herschel regards these clusters as being composed of stars of equal con- densation, conceiving all such groups as approaching by their mutual attraction, to the globular figure, and assem- bling themselves into groups according to their relative re- gion, under laws of which we have it is true, no other 276 A TREATISE ON ASTRONOMY. [CHAP. XII.] proof than the observance of a gradation by which their characters shade into one another, so that it is impossible to say where one species ends and the other begins. (618.) Resolvable nebulae can, of course, only be con- sidered as such when they consist of stars intrinsically too faint to affect us by their individual light, but so numerous as to form a mass sufficiently large enough to make a point impression, and give the idea of a point brighter than the rest. They are almost universal round or oval-shaped masses, and are resol- vable, not because they are as bright as they were extinguished by the distance, and only the general figure of the more condensed parts being visible, but because they consist of a multitude of objects of this character that all the greater globular chro- nus exhibit themselves in telescopes of insufficient opi- cal power. It is evident that this is a very important fact, it is obvious, that those which the most powerful can barely render resoluble, would be completely resolved by a further increase of power. (619.) Of nebulae properly so called, the variety is again very great. By far the most remarkable are those represented in figs. 305 and 306. The former represents a cluster which represents the nebula surrounding the quadruplet (or rather sextuple) star $j$ in the constellation Orion ; the latter represents the nebula in the constellation Hercules ; the one discovered by Hygeius in 1669, and figured as seen in the twenty feet reflector at Slough; the other discovered from figures by Mr. Dunlop, Phil. Trans. 1827. The resemblance between these two objects, at least of the former, is very different from what might be expected from any common knowledge of the nature of a nebu- lous collection of small stars. It is formed of little flocky masses, like wisps of cloud; and such wisps seem to be connected with each other to outline them, especially to one considerable star (represented in the figure, below the nebula), which it envelops with a nebu- lous appearance of considerable extent and singular figure. Several asterisms are on each side of this nebula with the figures of it had down to us by its discoverer, Hygeius, have continued since that time to undergo a perceptible change. But when it is considered how dif- ficult it is to represent such an object duly, and how en- CHAP. XII.] OF NEBULA. 377 tirely its appearance will differ, even in the same telescope, according to the clearness of the air, or other temporary causes; but in this case we have no evidence of change that can be relied on. (620.) Plate II., fig. 3, represents a nebula of a quite different nature from those before mentioned, belonging to the constellation Andromeda near the star $\alpha$. It is visible to the naked eye, and is continuously mistaken for a comet, by many persons. The first person who noticed it in 1812, described its appearance as that of a candle shining through horn, and the resemblance is very striking. It is composed of a series of rays, consisting by insensible gradations of brightness, at first very gradual, but at last more rapidly, up to a central point, which is the most brilliant part of the whole. It is yet evidently not stellar, but only nebula in a high state of condensation. It has in it a few small stars ; but they are so faint as not to give any support to the slightest appearance to give ground for a suspicion of its consisting of stars. It is very large, being nearly half a degree in diameter. (621.) This may be considered as a type, on a large scale, of a very numerous class of nebulae, of a round or oval form, with one or two bright points at the centre; these points they differ extremely, however, in this respect. In some, the condensation is slight and irregular; in others it is dense and compact; and they present the appearance of a dull and blotted star, or of a star with a slight burn round it; in which cases they are often mistaken for comets. Again, offer the singularly beautiful and striking phenomenon of a sharp and brilliant star surrounded by a perfectly circular nebulosity, which extends far beyond the mass, dying away on all sides by insensible gradations; in others, almost suddenly terminated. These are nebulous stars of the following types : - $\beta$ Persei (the "Nebulae" of Humboldt), $\gamma$ Andromedæ R. 1° 43", N. P. D. 50° 7". Orionis and $\alpha$ Orionis are of the same constitution as these nebulae; but the nebula is not always so well defined as in them. In the extent of deviation, too, from the spherical form, which oval nebula affect, a great diversity is observed : some are only slightly elliptic ; others much elongated. 214 378 A TREATISE ON ASTRONOMY. [CAP. XII. in length; and in some, the extension so great, as to cover the whole of the surface of the planet, in a spindleshaped ray, uppering away at both ends to points. One of the most remarkable specimens of this kind is in R. A. 12° 50' 16" (022.) Annular nebulae also exist, but are among the rarest objects in the heavens. The most conspicuous of this kind is that which is situated between the stars $\alpha$ and $\beta$ Lyrae, and may be seen with a telescope of moderate power. It is small, and particularly well de- veloped, so as to appear like a flat disk, with a circumference of a flat oval solid ring than of a nebula. The axes of the ellipse are to each other in the proportion of about 1 to 8, and its diameter is about 10 minutes of arc. Its light is not quite uniform, but has something of a curdled appearance, particularly at the exterior edge; the centre being more uniformly illuminated with a faint hazy light, uniformly spread over it, like a fine gauze stretched over a hoop. (023.) Nebulae are very extraordinary ob- jects. They have, on their name imported, exactly the appearance of planets; round or slightly oval discs, in some instances having a central spot; or else showing a little hazy at the borders and of a light exactly where or only very little mottled, which, in some cases, ap- proaches in vividness to that of actual planets. What- ever be their nature, they are all very luminous magnifying. One of them is to be found in the parallel of $\alpha$ Aquarii, and almost coincident with this star. Its apparent diameter is about 20". Another, which is much less conspicuous, present a visible disc of 12", perfectly defined and round. These two are by no means equally distant from us by the stars, their real dimensions being not such as would fall, on the lowest computation, the whole orbit of Uranus. It is no less evident that, if they be found beside each other in the same parallel, the distance of their surfaces must be almost infinitely inferior to that of the sun's. A circular portion of the sun's disc, sub- divided into 360 equal parts, will contain an angle equal to 100 full moons; while the objects in question are hardly, if at all, discernible with the naked eyes. The uniformity of their form, and their mode of apparent CHAP. XII.] PLANTARY NEBULA. 379 central condensation would certainly usurp their light to be merely superficial, and in the nature of a hollow spherical shell: but whether filled with solid or gaseous matter, or altogether empty, it would be a waste of space to consider it so. (624.) Among the nebula which possess an evident symmetry of form, and seem clearly suited to be regarded as planets, there are two which are most mysterious in their structure and destination, the most remarkable are the 51st and 27th of Messier's catalogue. The former is a large, bright, and luminous nebula surrounded by a double ring, at a considerable distance from the globe or rather a single ring divided through about one-third of its circumference into two halves, and having one portion, as it were, turned up out of the plane of the rest. The latter consists of two bright and highly condensed masses, each surrounded by a short neck of nearly the same density. A faint nebulous atmosphere completes the figure, enveloping them both, and giving them a somewhat elliptical shape, whose shorter axis is in the axis of symmetry of the system about which it may be supposed to revolve, or the line joining the centres of the two luminous masses. These objects have never been properly described, the instruments with which they were originally observed being insufficient for detecting the peculiarities above mentioned, which seem to place them in a class apart from all others. The one offers obvious analogies to our own solar system; while we see that of our own sidereal firmament and milky way. The other has little or no resemblance to any other known object. (625.) The nebulae furnish, in every point of view, an inexhaustible field of speculation and conjecture. That which is nearest to us is so vast in extent that there can be little doubt; and in the innumerable range of systems upon system, and firmament upon firmament, which must exist within such immense distances, it is be- widered and lost. On the other hand, if it be true, to say the least, it seems extremely probable, that a phase-phenomenon or similar mass will ultimately be demon- strated through extensive regions of space, in the manner 350 A TREATISE ON ASTRONOMY. [CHAP. XIL] of a cloud or fog--now assuming expressive shapes, like actual clouds of the sky--and now again dissolving into nothingness itself like a comestic atmosphere around particular stars; what, we naturally ask, is the nature and destination of this nebulous matter? What is its origin? Where does it come from whose existence we have never before noticed in whose presence we have never before been aware? Is it found, to furnish, by its con- centration, its supply of light and heat? or is it pro- gressively concentrated, and then dispersed, by its own gravity, and thus gradually laying the foundation of new solar systems or of insulated stars ? It is easier to proposed than to answer such questions, but I shall reply to them. Meanwhile, appeal to fact, by the method of constant and diligent observation, is open to us; and, as the doubt is so great, I shall not omit mentioning, and disclosed a series of relations of the most intelligible and interesting description, we may reasonably hope that the truth will be discovered. And here are brought to some clearer understanding of their intimation (626). (626.) We shall conclude this chapter by the men- tion of phenomena which seem to indicate the ex- istence of a high degree of similarity between the sun itself, and even to place it in the list of nebulous stars. It is called the zodiacal light, and it appears every very clear night in the northern hemisphere during the season before sunrise, at a cone or lens-shaped light extending from the horizon outward towards the zenith. This light follows the course of the ecliptic, or rather that of the sun's equator. The appearance is different according to latitude from 40° to 90°, and the breadth of its perpendicular to its axis from 8° to 30°. It is only partially visible in de- fined latitudes and climates though seen sometimes in tem- poral regions, but cannot be mistaken for any astro- motic meteor or aurora borealis. It is manifested in the nature of a luminous band which passes round the earth, rounding the sun, and extending at least beyond the orbit of Mercury and even of Venus, and may be con- sidered as a solar halo. It is not due to any kind of nebu- lous which, as we have reason to believe, resists the motion of comets ; liable, perhaps, with the actual me- nuscular form of nebulae. **CHAP. XIII.]** OF THE CALENDAR. 281 they have been stripped in their successive perihelion passages (art. 407), and which may be slowly subsiding into the sun. --- **CHAPTER XIII.** OF THE CALENDAR. (627.) Time, like distance, may be measured by comparison with standards of any length, and all that is required to determine its duration is to find the number of intervals, to be able to apply the standard to the interval throughout its whole extent without overlapping on the one hand, and without leaving any space on the other; to determine, without the possible error of a unit, the number of integer standards which the interval admits of being divided into, and then to estimate precisely the fraction over and above an integer, which remains when all the possible integers are subtracted. (628.) But though all standard units of time are equally possible, theoretically speaking, all are not practically equal, even in those cases where they are identical. The day is natural units, which the wants of man and the busi- ness of society force upon us, and compel us to adopt no greater or less than this unit for the measurement of time for all the purposes of civil life; and that, in spite of inconveniences which, did any choice exist, might have been avoided by adopting some other. The principal of these are their **incommensurability**, and the want of perfect uniformity in one at least of them. (629.) The mean solar year is about 365 days 6 hours 9 minutes 9 seconds a year, when estimated on an average sufficiently large to com- peneate the fluctuations arising from variation in the one, and from irregularities in the other. Of these two, the former are the two most invariable quantities which nature presents us with; the former, by reason of the uniform diurnal rotation of the earth, is also the latter of those whose inva- riability of the axes is the most strikingly evident. Hence it follows that the mean solar day is also invariable. It is --- 322 A TREATISE ON ASTRONOMY. [CHAP. XIL] otherwise with the tropical year. The motion of the equinoctial points exists not only from the retrogradation of the equator on the ecliptic, but also partly from that of the ecliptic on the orbits of all the other planets. This is likewise true of the precession of the equinoxes in the tropical year, which is dependent on the place of the equinox (art. 517, 528). The tropical year is actually more than one day longer than the sidereal year. This absence of the most essential requisite for a standard, viz. invaribility, renders it necessary, since it cannot be produced by any artificial means, to look upon time, to adopt an arbitrary or artificial value for it, so near the truth, as not to admit of the accumulation of its errors being transmitted through successive mischiefs, and thus satisfying the ordinary wants of civil life; while, for scientific purposes, the tropical year, so aptly adopted by astronomers, is a convenient measure of a certain number of integer days and a fraction—the day being, in effect, the only standard employed. The case is nearly similar with respect to the standard of heat and shillings, an artificial relation of the two being made by law, near to but, scarcely ever exactly coincident with, the real ratio between them. The standard market price of gold and silver, of which either the one or the other—whichever is really the most invariable, or the least as regards change—is used may be assumed as the true theoretical standard of value. (600.) The inconstancy of the standards in question is not confined to those which are measures of space, all our subdivisions are into aliquot parts: a yard is three feet; a mile eight furlongs, &c. But a year is no exact multiple of any integer number; though any exact fraction, nor one third or one fourth, over and above; but the surplus is an incommensurable fraction, compounded of all those fractions which pro- duce the same kind of inconvenience in the reckoning of time that it would do, in that of money, if we had not considered it as divisible into such parts as pence and farthings, and a fraction of a farthing over. In this way, however, there is no remedy but to keep a strict re- corder of all changes that occur; and though each amount to a whole day, cast them over into the integer account. CHAP. XIII.—963 (81). To do this is the simplest and most convenient manner is the object of a well-adjusted calendar. In the Gregorian calendar, which we follow, it is accomplished, with remarkable simplicity and neatness, by carrying a leap-year through every four years, and then, if necessary, a resumed or artificial year, and adopting two such years, hence the name of the Gregorian calendar. The first leap-year is 1582, viz. one of 306 and the other of 306, and laying down a simple and easily remembered rule for the order in which these years shall succeed each other in the civil reckoning of time. This rule is to take the number of thousands of years the sum of the integer artificial, or Gregorian, years elapsed shall not differ from the same number of years that have elapsed since the beginning of the world. By this contrivance, the equinoxes and solstices will always fall on days similarly situated, and bearing the same name, in each Gregorian year that has elapsed since the beginning of the world. The months will also correspond to the same months, instead of running round the round of the world in different months. The days will also correspond to reckoning, and used, in fact, to do before this was adopted. (82). The Gregorian rule is as follows—the years are denominated from the birth of Christ according to our calendar, and are numbered from year to year. Every year whose number is divisible by 4 without remainder, consists of 365 days; every year whose number is divisible by 100 but not by 400, 365 plus 24 hours; every year divisible by 400, but not by 600, again of 365; and every year divisible by 600, again of 365. For example: the year 1700 was a common year; the year 1701 was a leap-year; the year 1702 was a common year; the year 1703 was a leap-year; etc., etc., up to 1800. From 1801 to 1804 inclusive, there were 365 days; from 1805 to 366; from 1809 to 1900 and 365 each; but from 2001 to 2004 inclusive, there were only 364 days. In order to see how near this rule will hold true to the truth of things, let us consider that in all 1000 Gregorian years will contain, beginning with the year 1. Now, in 1000 years the numbers not divisible by 4 will be (1 + 2 + 3 + ... + 999) = ${{\frac{999 \times (999+1)}{2}}}$ = ${{\frac{999 \times 1000}{2}}}$ = ${{\frac{999 \times 500}{1}}}$ = ${{\frac{499500}{1}}}$ = ${{\frac{499500}{4}}}$ = ${{\frac{249750}{2}}}$ = ${{\frac{249750}{4}}}$ = ${{\frac{124875}{2}}}$ = ${{\frac{124875}{4}}}$ = ${{\frac{62437.5}{2}}}$ = ${{\frac{62437.5}{4}}}$ = ${{\frac{31218.75}{2}}}$ = ${{\frac{31218.75}{4}}}$ = ${{\frac{15609.375}{2}}}$ = ${{\frac{15609.375}{4}}}$ = ${{\frac{7804.6875}{2}}}$ = ${{\frac{7804.6875}{4}}}$ = ${{\frac{3902.34375}{2}}}$ = ${{\frac{3902.34375}{4}}}$ = ${{\frac{1951.171875}{2}}}$ = ${{\frac{1951.171875}{4}}}$ = ${{\frac{975.5859375}{2}}}$ = ${{\frac{975.5859375}{4}}}$ = ${{\frac{487.79296875}{2}}}$ = ${{\frac{487.79296875}{4}}}$ = ${{\frac{243.896484375}{2}}}$ = ${{\frac{243.896484375}{4}}}$ = ${{\frac{121.9482421875}{2}}}$ = ${{\frac{121.9482421875}{4}}}$ = ${{\frac{60.97412109375}{2}}}$ = ${{\frac{60.97412109375}{4}}}$ = ${{\frac{30.487060546875}{2}}}$ = ${{\frac{30.487060546875}{4}}}$ = ${{\frac{15.2435302734375}{2}}}$ = ${{\frac{15.2435302734375}{4}}}$ = ${{\frac{7.62176513671875}{2}}}$ = ${{\frac{7.62176513671875}{4}}}$ = ${{\frac{3.810882568359375}{2}}}$ = ${{\frac{3.810882568359375}{4}}}$ = ${{\frac{1.9054412841796875}{2}}}$ = ${{\frac{1.9054412841796875}{4}}}$ = ${{\frac{.9527206420898437} 304 A TREATISE ON ASTRONOMY. [CHAP. XIX.] 10000 of the present tropical years is 6-8, or 5' 14" 74"; that is to say, less than a day in 3000 years; which is more than sufficient for all human purposes, those of the astronomer excepted, who is in no danger of being led into errors by such a small difference. This error may be avoided by extending the wording of the Gregorian rule one step farther than it has been hitherto thought it worth while to do, and declaring that years divisible by 4000 should consist of 365 days. This would take off two integer days from the above calculated number, and 8 from each year which was not divisible by 10000 Gregorian years, 5852428, which differs only by a single day from 10000 real tropical years, such as we have now. (633.) As any distance along a high road might, though in a rough inconvenient and roundabout way, be represented by the number of miles traversed, so the number of milestones, at intervals of unequal lengths, can represent any fourth mile, for instance, sooner than a yard longer than the first milestone. It is therefore fixed as a thing care only to mark the stones, so as to leave room for no mistake, and to advertise all travellers of the difference between the length of the mile and the number of the interval of time be expressed correctly by stating in what Gregorian years it begins and ends, and whatsoever its end. For this purpose we adopt the following arbitrary rule, enables us to say how many integer years are to be reckoned at 365, and how many at 366 days. The latter years are to be reckoned from the beginning of every plus days thus through into the reckoning are called intercalary leap-days. (634.) Of this Gregorian rule, as above stated, had always been adhered to, nothing would be easier than to reckon the number of days elapsed between the present time and any former period; but this is not the case; and the history of the calendar, with reference to chronology, or to the calculation of ancient observa- tions, may be said to begin with this rule. It was exactly when left to itself, but sometimes forgotten to be wound around; and when wound, sometimes set forward, sometimes backward; and sometimes used for public and private interest. Such, at least, appears to have been --- CHAP. XIII. OF THE CALENDAR. 385 the ease with the Roman calendar, in which our own originated from the time of the first to that of Julius Cæsar, when the lunar year of 13 months, or 365 days, was augmented at pleasure, to correspond to the solar, by which means the years are determined, by the arbitrary traditions of the people, according to the motions of the decansœus and other magistrates, till the confusion be- came incredible. The eminent Alephius Cæsar, assisted by Socle- nus, the eminent Alaric, and other mathematicians, and the mathematician, we owe the next contrivance of the two years of 365 and 366 days, which was made by them in the middle after three courses of years. This important change took place in the 45th year before Christ, which was the first revolution of the new system. It was observed during the day of the new moon immediately following the winter solstice of the year before. We may judge of the importance of this event from what follows. Hence, by the fact, that, to introduce the new system, it was necessary to enact that the previous year (48 a. c.) should be a leap-year; and that this law which ob- tained it the epitaph of "the year of confusion." (453.) The Julian rule made every fourth year, with- out any exception whatever, a leap-year; but this over- correction; it supposes the length of the tropical year to be 365¼, which is too great, and thereby induces an error in the computation of eclipses and eclipses on trial. Accordingly, so early as the year 1414, a begin- ning to perceive that the equinoxes were gradually creep- ing forward; and that they ought to have always fallen had the Julian year been exact, and happening (as it appeared) too early. The change was introduced by Pope Gregory XIII. This change was made from that time continually urged, and at length admitted. The change (which took place under the pontificate of Pope Gregory XIII.) consisted in taking out of ten nominal days after the 4th of October, 1582 (as soon as the next day was called the 18th, and not the 8th), and thus making up one day in each year for future regulation. The change was adopted immediately in all eastern countries; but more slowly in protestant. In England it was not adopted until 1700; and did not take place after the 3rd of September, 1722, eleven nominal 2 K 386 A TREATISE ON ASTRONOMY. CHAP. XIII. days being then struck out; so that, the last day of Old Style being December 25th, the first day of New Style was called the 14th, instead of the 8d. The same legi- timate enactment which established the Gregorian year in England, was made in Russia in 1727, by a full quarter. Previous to that time, the year was held to begin with the 25th March, and the year a. d. 1753 did not commence till the 1st April; but this was to run out, but was supplanted on the 1st January by the year 1753, which it was enacted should com- mence on that day. It is now observed, that in Russia Russia is now the only country in Europe in which the Old Style is still adhered to, and (another secular year having elapsed since the adoption of the Gregorian and Russian dates amongst us, at present, to 12 days) (834.) It is fortunate for astronomy that the confusion of these two systems has been so long maintained; for many of our most important astronomical observa- tions, and even some of those which are less remarkable, are made according to the Old Style. An astronomical observation, of any striking and well marked phenomenon, carries with it, in most cases, abundant means of determining its date; and when such a pro- proximation is afforded to it by chronological records; and so far from being largely dependent on the ob- senses and observations of antiquity, the very observa- tions of ancient authorities indicate, is often itself the surest and most convincing evidence on which a chrono- logical system can be founded. For instance, now that the lunar theory is thoroughly un- derstood, can be calculated back for several thousands of years, without any difficulty whatever, as regards any of their occurrence. And whenever any such eclipse is so interwoven with the account given by an ancient author of some other event or circumstance, as to occupy a interval of time between the eclipse and the event, and at the same time completely to identify the eclipse, that date is received as correct. (837.) The days thus paralleled out into years, the *See the remarkable calculation of Mr. Bailey relating to the cele- brated apparition of Venus in 1761; and also to that of the king's Medals and Lyrae, a. c. 690. Sept. 6th. Phil. Trans. vol. iii.* A page from a book with text and mathematical symbols. **CHAP. XIII.]** OF THE CALENDAR. 387 next step to a perfect knowledge of time is to secure the identification of each day, by imputing on it a name uni- versally understood. But this is impossible, because the days of a whole year are too numerous to admit of load- ing the memory with distinct names for each, all nations having adopted different modes of designating these par- cels of a more moderate extent ; giving names to each of these parcels, and particularizing the days in each by numbers. The division of the year into 12 lunar months has been resorted to in many instances ; and some nations have, in fact, preferred a lunar to a solar chro- nology. But the number of days in each month does not do to this day, making the year consist of 12 lunar months, or 355 days.* Our own division into twelve unequal months, introduced at the beginning of the Christian con- fusion, owing to the equivocation between the lunar and calendar month. The intercalary day naturally attaches itself to the first day of each month. *The Metonic cycle, or the fact, discovered by Meno, a Greek me- teornatician, that 19 solar years contain just 230 lunations (which in fact the Greeks supposed to be 235), was the first attempt made by them to explain by the Greeks, as wanting the correspondence of the solar and lunar years, and lunations were deemed to be discovered. 388 A TREATISE ON ASTRONOMY. NOTE On the Constitution of a Globular Cluster, referred to in page 274. If we suppose a globular space filled with equal stars, uniformly dis- persed through it, and very numerous, such of them attracting every other with a force proportional to their mutual distances, the whole space will be attracted by all these stars together, and will be con- centrated into a sphere, whose centre is at the common centre of gravitation, by which any one of them (those at the surface alone excepted) may be regarded as being situated. The centre of this sphere will be directly on the distance conditioned by the law of gravitation. Now, if we suppose that the attraction of a homogeneous body is inversely proportional to the square of its distance, and that the same law would describe a perfect sphere about two of three, and perhaps four or five stars, it would follow that the attraction of the whole body must be inversely proportional to its distance from the centre. This condition might revive. The condition, therefore, of a motion of the cluster round its centre, would be that of an ellipse, whatever might be the proportion of its axis, or the inclination of the plane of its orbit to the plane of the ecliptic. Whatever might be the proportion of its axis, or the inclination of (cease) the superficial plane would be exactly re-established in its original position after each revolution round for an indefinite succession of ages. Supposing their motions, therefore, to be elliptical, they would never intersect each other, and so that the magnitude of each star, and the globular space itself, are inversely proportional to the distance separating the individuals, such a system is, in obvious respects, most beautiful. It is also most beautiful in its ideal harmony. John Newton, in the 5th Proposition of the First Book of Principia Mathematica, says: "The attraction between two bodies is inversely as the distance." See also Quarterly Review, No. 94, p. 540.--Author. A TREATISE ON ASTRONOMY. 389 SYNTHETIC TABLES OF THE ELEMENTS OF THE SOLAR SYSTEM. M. R.—The data for Venus, Jupite, Mars, and Pluto are for January 1, 1860.
    Planet's name Mean distance from the Sun (semi-axis) Mean sidereal period (Solar Days) Equinocentric inclination (degrees) Equatorial inclination (degrees)
    Venus 0.725580 225.000000 0.000000 0.000000
    Mercury 0.387981 87-995580 25.15149° 25.15149°
    Venus 0.725580 225.000000 0.000000 0.000000
    Earth 1.496396 365-254181 27.16786° 27.16786°
    Mars 1.523434 732-453181 26.18336° 26.18336°
    Juno 2.678756 1253-743181 28.68139° 28.68139°
    Ceres 2.696999 1252-656999 27.54748° 27.54748°
    Pallas 3.772886 1865-538800 24.94149° 24.94149°
    Luna
    Mean Longitude of the Apogee.
    Mercury69° 4' 9'' E.45° 57' 39'' N.74° 21' 46'' S.74° 21' 46'' S.
    Venus33° 23' E.74° 54' 12'' N.38° 43' 5'' S.38° 43' 5'' S.
    Earth
    Mars1' 51' E.48° 3' N.33° 26' S.33° 26' S.
    Vesta
    Ceres
    1' 4'7' 7'7' 4'33° 46' S.33° 46' S.
    Jupiter1' 51'8' 4'4' 17'33° 46' S.33° 46' S.
    SaturnSaturn
    > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >
    Mean Longitude of the Apogee
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    Mean distance from earth SP-06175690
    Mean sidereal revolution 279°57'14"8
    Mean synodical date 2P-550586715
    Eccentricity of orbit 0.054844300
    Mean revolution of nodes 6736°29'54"7
    Mean revolution of apogee 322°57'54"5
    Mean inclination of orbit at epoch 17°29'7"
    Mean longitude of perigee at epoch 266 10 7
    Mean inclination of orbit at epoch 118 7 8
    Mean longitude of moon at epoch 326°51'37"
    Diameter in miles. 2100
    II. SATELLITES OF JUPITER.


































































































    <
    No. Mean Distance. Sidereal Revolution. Inclination of Orbit at Epoch. Diameter in Miles.
    Satellites of Jupiter.
    1 0°04853 14°19'29" 3°5'30' 17258
    2 9°04846 3°13'14" Variable 32255
    3 15°04824
    The eccentricities of the 1st and 2nd satellites are insensible, that of the 3rd and 4th small, but variable in consequence of their mutual perturbations. A TREATISE ON ASTRONOMY. 301 III. SATELLITES OF SATURN.
    Sat. Mean Distance Sidereal Revolution. Eccentricities and Inclinations.
    1 3-361 0° 22' 38" The orbit of the six interior satellites is very nearly in the plane of the ecliptic, and inclined to it considerably less than 5°.
    2 4-047 0° 21' 19"
    3 5-094 1° 11' 18"
    4 6-019 2° 25' 18"
    5 7-034 3° 12' 25"
    6 8-081 4° 15' 21"
    7 9-039 5° 28' 23"
    IV. SATELLITES OF URANUS.
    Sat. Mean Distance Sidereal Period. Inclination to Ecliptic.
    11 19-190 5° 31' 58" O' Their orbits are inclined at about 7° to the ecliptic, and their motion is retrograde with respect to that of the planets, except in the case of the two innermost satellites, which have nearly the same direction of motion as the earth's. The whole appear to be nearly circular.
    12 19-945 10° 23' 40"
    21 20-886 11° 46' 36"
    22 25-007 38' 1' 46" O'
    23 31-008 107' 40" O'
    A black and white image of a vertical line with some faint, scattered dots around it. A vertical triptych illustration of celestial bodies. The top panel shows a planet with a prominent, irregular surface. The middle panel depicts Jupiter, recognizable by its characteristic banding pattern. The bottom panel illustrates Saturn, identifiable by its prominent ring system. 10 A black and white image of a person's hand holding a piece of paper with text on it. 1
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    P. 1 P. 2 P. 3 9 A black background with a faint, diagonal pattern.