Transcriber’s Notes

Obvious typographical errors have been silently corrected. Variations in hyphenation has been standardised but all other spelling and punctuation remains unchanged.

Ditto marks, in the table of contents and illustrations, have been replaced by the text. Blank pages and their page numbers have been omitted.

The cover was created by the transcriber and is placed in the public domain.

The moon. From a photograph taken at the Lick Observatory.

THE UNIVERSITY SERIES

A Short History
of
Astronomy

By ARTHUR BERRY, M.A.
FELLOW AND ASSISTANT TUTOR OF KING’S COLLEGE, CAMBRIDGE;
FELLOW OF UNIVERSITY COLLEGE, LONDON

Wagner. Verzeiht! es ist ein gross Ergetzen

Sich in den Geist der Zeiten zu versetzen.

Zu schauen wie vor uns ein weiser Mann gebracht,

Und wie wir’s dann zuletzt so herrlich weit gebracht.

Faust. O ja, bis an die Sterne weit!

Goethe’s Faust.

NEW YORK
CHARLES SCRIBNER’S SONS
1899

[PREFACE.]

I have tried to give in this book an outline of the history of astronomy from the earliest historical times to the present day, and to present it in a form which shall be intelligible to a reader who has no special knowledge of either astronomy or mathematics, and has only an ordinary educated person’s power of following scientific reasoning.

In order to accomplish my object within the limits of one small volume it has been necessary to pay the strictest attention to compression; this has been effected to some extent by the omission of all but the scantiest treatment of several branches of the subject which would figure prominently in a book written on a different plan or on a different scale. I have deliberately abstained from giving any connected account of the astronomy of the Egyptians, Chaldaeans, Chinese, and others to whom the early development of astronomy is usually attributed. On the one hand, it does not appear to me possible to form an independent opinion on the subject without a first-hand knowledge of the documents and inscriptions from which our information is derived; and on the other, the various Oriental scholars who have this knowledge still differ so widely from one another in the interpretations that they give that it appears premature to embody their results in the dogmatic form of a textbook. It has also seemed advisable to lighten the book by omitting—except in a very few simple and important cases—all accounts of astronomical instruments; I do not remember ever to have derived any pleasure or profit from a written description of a scientific instrument before seeing the instrument itself, or one very similar to it, and I have abstained from attempting to give to my readers what I have never succeeded in obtaining myself. The aim of the book has also necessitated the omission of a number of important astronomical discoveries, which find their natural expression in the technical language of mathematics. I have on this account only been able to describe in the briefest and most general way the wonderful and beautiful superstructure which several generations of mathematicians have erected on the foundations laid by Newton. For the same reason I have been compelled occasionally to occupy a good deal of space in stating in ordinary English what might have been expressed much more briefly, as well as more clearly, by an algebraical formula: for the benefit of such mathematicians as may happen to read the book I have added a few mathematical footnotes; otherwise I have tried to abstain scrupulously from the use of any mathematics beyond simple arithmetic and a few technical terms which are explained in the text. A good deal of space has also been saved by the total omission of, or the briefest possible reference to, a very large number of astronomical facts which do not bear on any well-established general theory; and for similar reasons I have generally abstained from noticing speculative theories which have not yet been established or refuted. In particular, for these and for other reasons (stated more fully at the beginning of chapter XIII.), I have dealt in the briefest possible way with the immense mass of observations which modern astronomy has accumulated; it would, for example, have been easy to have filled one or more volumes with an account of observations of sun-spots made during the last half-century, and of theories based on them, but I have in fact only given a page or two to the subject.

I have given short biographical sketches of leading astronomers (other than living ones), whenever the material existed, and have attempted in this way to make their personalities and surroundings tolerably vivid; but I have tried to resist the temptation of filling up space with merely picturesque details having no real bearing on scientific progress. The trial of Kepler’s mother for witchcraft is probably quite as interesting as that of Galilei before the Inquisition, but I have entirely omitted the first and given a good deal of space to the second, because, while the former appeared to be chiefly of curious interest, the latter appeared to me to be not merely a striking incident in the life of a great astronomer, but a part of the history of astronomical thought. I have also inserted a large number of dates, as they occupy very little space, and may be found useful by some readers, while they can be ignored with great ease by others; to facilitate reference the dates of birth and death (when known) of every astronomer of note mentioned in the book (other than living ones) have been put into the Index of Names.

I have not scrupled to give a good deal of space to descriptions of such obsolete theories as appeared to me to form an integral part of astronomical progress. One of the reasons why the history of a science is worth studying is that it sheds light on the processes whereby a scientific theory is formed in order to account for certain facts, and then undergoes successive modifications as new facts are gradually brought to bear on it, and is perhaps finally abandoned when its discrepancies with facts can no longer be explained or concealed. For example, no modern astronomer as such need be concerned with the Greek scheme of epicycles, but the history of its invention, of its gradual perfection as fresh observations were obtained, of its subsequent failure to stand more stringent tests, and of its final abandonment in favour of a more satisfactory theory, is, I think, a valuable and interesting object-lesson in scientific method. I have at any rate written this book with that conviction, and have decided very largely from that point of view what to omit and what to include.

The book makes no claim to be an original contribution to the subject; it is written largely from second-hand sources, of which, however, many are not very accessible to the general reader. Particulars of the authorities which have been used are given in an appendix.

It remains gratefully to acknowledge the help that I have received in my work. Mr. W. W. Rouse Ball, Tutor of Trinity College, whose great knowledge of the history of mathematics—a subject very closely connected with astronomy—has made his criticisms of special value, has been kind enough to read the proofs, and has thereby saved me from several errors; he has also given me valuable information with regard to portraits of astronomers. Miss H. M. Johnson has undertaken the laborious and tedious task of reading the whole book in manuscript as well as in proof, and of verifying the cross-references. Miss F. Hardcastle, of Girton College, has also read the proofs, and verified most of the numerical calculations, as well as the cross-references. To both I am indebted for the detection of a large number of obscurities in expression, as well as of clerical and other errors and of misprints. Miss Johnson has also saved me much time by making the Index of Names, and Miss Hardcastle has rendered me a further service of great value by drawing a considerable number of the diagrams. I am also indebted to Mr. C. E. Inglis, of this College, for fig. 81; and I have to thank Mr. W. H. Wesley, of the Royal Astronomical Society, for various references to the literature of the subject, and in particular for help in obtaining access to various illustrations.

I am further indebted to the following bodies and individual astronomers for permission to reproduce photographs and drawings, and in some cases also for the gift of copies of the originals: the Council of the Royal Society, the Council of the Royal Astronomical Society, the Director of the Lick Observatory, the Director of the Instituto Geographico-Militare of Florence, Professor Barnard, Major Darwin, Dr. Gill, M. Janssen, M. Loewy, Mr. E. W. Maunder, Mr. H. Pain, Professor E. C. Pickering, Dr. Schuster, Dr. Max Wolf.

ARTHUR BERRY.

King’s College, Cambridge

CONTENTS.

LIST OF ILLUSTRATIONS.

A SHORT HISTORY OF ASTRONOMY.

[CHAPTER I.]
PRIMITIVE ASTRONOMY.

“The never-wearied Sun, the Moon exactly round,

And all those Stars with which the brows of ample heaven are crowned,

Orion, all the Pleiades, and those seven Atlas got,

The close beamed Hyades, the Bear, surnam’d the Chariot,

That turns about heaven’s axle tree, holds ope a constant eye

Upon Orion, and of all the cressets in the sky

His golden forehead never bows to th’ Ocean empery.”

The Iliad (Chapman’s translation).

1. Astronomy is the science which treats of the sun, the moon, the stars, and other objects such as comets which are seen in the sky. It deals to some extent also with the earth, but only in so far as it has properties in common with the heavenly bodies. In early times astronomy was concerned almost entirely with the observed motions of the heavenly bodies. At a later stage astronomers were able to discover the distances and sizes of many of the heavenly bodies, and to weigh some of them; and more recently they have acquired a considerable amount of knowledge as to their nature and the material of which they are made.

2. We know nothing of the beginnings of astronomy, and can only conjecture how certain of the simpler facts of the science—particularly those with a direct influence on human life and comfort—gradually became familiar to early mankind, very much as they are familiar to modern savages.

With these facts it is convenient to begin, taking them in the order in which they most readily present themselves to any ordinary observer.

3. The sun is daily seen to rise in the eastern part of the sky, to travel across the sky, to reach its highest position in the south in the middle of the day, then to sink, and finally to set in the western part of the sky. But its daily path across the sky is not always the same: the points of the horizon at which it rises and sets, its height in the sky at midday, and the time from sunrise to sunset, all go through a series of changes, which are accompanied by changes in the weather, in vegetation, etc.; and we are thus able to recognise the existence of the seasons, and their recurrence after a certain interval of time which is known as a year.

4. But while the sun always appears as a bright circular disc, the next most conspicuous of the heavenly bodies, the moon, undergoes changes of form which readily strike the observer, and are at once seen to take place in a regular order and at about the same intervals of time. A little more care, however, is necessary in order to observe the connection between the form of the moon and her position in the sky with respect to the sun. Thus when the moon is first visible soon after sunset near the place where the sun has set, her form is a thin crescent (cf. fig. 11 on p. 31), the hollow side being turned away from the sun, and she sets soon after the sun. Next night the moon is farther from the sun, the crescent is thicker, and she sets later; and so on, until after rather less than a week from the first appearance of the crescent, she appears as a semicircular disc, with the flat side turned away from the sun. The semicircle enlarges, and after another week has grown into a complete disc; the moon is now nearly in the opposite direction to the sun, and therefore rises about at sunset and sets about at sunrise. She then begins to approach the sun on the other side, rising before it and setting in the daytime; her size again diminishes, until after another week she is again semicircular, the flat side being still turned away from the sun, but being now turned towards the west instead of towards the east. The semicircle then becomes a gradually diminishing crescent, and the time of rising approaches the time of sunrise, until the moon becomes altogether invisible. After two or three nights the new moon reappears, and the whole series of changes is repeated. The different forms thus assumed by the moon are now known as her phases; the time occupied by this series of changes, the month, would naturally suggest itself as a convenient measure of time; and the day, month, and year would thus form the basis of a rough system of time-measurement.

5. From a few observations of the stars it could also clearly be seen that they too, like the sun and moon, changed their positions in the sky, those towards the east being seen to rise, and those towards the west to sink and finally set, while others moved across the sky from east to west, and those in a certain northern part of the sky, though also in motion, were never seen either to rise or set. Although anything like a complete classification of the stars belongs to a more advanced stage of the subject, a few star groups could easily be recognised, and their position in the sky could be used as a rough means of measuring time at night, just as the position of the sun to indicate the time of day.

6. To these rudimentary notions important additions were made when rather more careful and prolonged observations became possible, and some little thought was devoted to their interpretation.

Several peoples who reached a high stage of civilisation at an early period claim to have made important progress in astronomy. Greek traditions assign considerable astronomical knowledge to Egyptian priests who lived some thousands of years B.C., and some of the peculiarities of the pyramids which were built at some such period are at any rate plausibly interpreted as evidence of pretty accurate astronomical observations; Chinese records describe observations supposed to have been made in the 25th century B.C.; some of the Indian sacred books refer to astronomical knowledge acquired several centuries before this time; and the first observations of the Chaldaean priests of Babylon have been attributed to times not much later.

On the other hand, the earliest recorded astronomical observation the authenticity of which may be accepted without scruple belongs only to the 8th century B.C.

For the purposes of this book it is not worth while to make any attempt to disentangle from the mass of doubtful tradition and conjectural interpretation of inscriptions, bearing on this early astronomy, the few facts which lie embedded therein; and we may proceed at once to give some account of the astronomical knowledge, other than that already dealt with, which is discovered in the possession of the earliest really historical astronomers—the Greeks—at the beginning of their scientific history, leaving it an open question what portions of it were derived from Egyptians, Chaldaeans, their own ancestors, or other sources.

7. If an observer looks at the stars on any clear night he sees an apparently innumerable[1] host of them, which seem to lie on a portion of a spherical surface, of which he is the centre. This spherical surface is commonly spoken of as the sky, and is known to astronomy as the celestial sphere. The visible part of this sphere is bounded by the earth, so that only half can be seen at once; but only the slightest effort of the imagination is required to think of the other half as lying below the earth, and containing other stars, as well as the sun. This sphere appears to the observer to be very large, though he is incapable of forming any precise estimate of its size.[2]

Most of us at the present day have been taught in childhood that the stars are at different distances, and that this sphere has in consequence no real existence. The early peoples had no knowledge of this, and for them the celestial sphere really existed, and was often thought to be a solid sphere of crystal.

Fig. 1.—The celestial sphere.

Moreover modern astronomers, as well as ancient, find it convenient for very many purposes to make use of this sphere, though it has no material existence, as a means of representing the directions in which the heavenly bodies are seen and their motions. For all that direct observation can tell us about the position of such an object as a star is its direction; its distance can only be ascertained by indirect methods, if at all. If we draw a sphere, and suppose the observer’s eye placed at its centre O (fig. 1), and then draw a straight line from O to a star S, meeting the surface of the sphere in the point s; then the star appears exactly in the same position as if it were at s, nor would its apparent position be changed if it were placed at any other point, such as S′ or S″, on this same line. When we speak, therefore, of a star as being at a point s on the celestial sphere, all that we mean is that it is in the same direction as the point s, or, in other words, that it is situated somewhere on the straight line through O and S. The advantages of this method of representing the position of a star become evident when we wish to compare the positions of several stars. The difference of direction of two stars is the angle between the lines drawn from the eye to the stars; e.g., if the stars are R, S, it is the angle R O S. Similarly the difference of direction of another pair of stars, P, Q, is the angle P O Q. The two stars P and Q appear nearer together than do R and S, or farther apart, according as the angle P O Q is less or greater than the angle R O S. But if we represent the stars by the corresponding points p, q, r, s on the celestial sphere, then (by an obvious property of the sphere) the angle P O Q (which is the same as p O q) is less or greater than the angle R O S (or r O s) according as the arc joining p q on the sphere is less or greater than the arc joining r s, and in the same proportion; if, for example, the angle R O S is twice as great as the angle P O Q, so also is the arc p q twice as great as the arc r s. We may therefore, in all questions relating only to the directions of the stars, replace the angle between the directions of two stars by the arc joining the corresponding points on the celestial sphere, or, in other words, by the distance between these points on the celestial sphere. But such arcs on a sphere are easier both to estimate by eye and to treat geometrically than angles, and the use of the celestial sphere is therefore of great value, apart from its historical origin. It is important to note that this apparent distance of two stars, i.e. their distance from one another on the celestial sphere, is an entirely different thing from their actual distance from one another in space. In the figure, for example, Q is actually much nearer to S than it is to P, but the apparent distance measured by the arc q s is several times greater than q p. The apparent distance of two points on the celestial sphere is measured numerically by the angle between the lines joining the eye to the two points, expressed in degrees, minutes, and seconds.[3]

We might of course agree to regard the celestial sphere as of a particular size, and then express the distance between two points on it in miles, feet, or inches; but it is practically very inconvenient to do so. To say, as some people occasionally do, that the distance between two stars is so many feet is meaningless, unless the supposed size of the celestial sphere is given at the same time.

It has already been pointed out that the observer is always at the centre of the celestial sphere; this remains true even if he moves to another place. A sphere has, however, only one centre, and therefore if the sphere remains fixed the observer cannot move about and yet always remain at the centre. The old astronomers met this difficulty by supposing that the celestial sphere was so large that any possible motion of the observer would be insignificant in comparison with the radius of the sphere and could be neglected. It is often more convenient—when we are using the sphere as a mere geometrical device for representing the position of the stars—to regard the sphere as moving with the observer, so that he always remains at the centre.

8. Although the stars all appear to move across the sky ([§ 5]), and their rates of motion differ, yet the distance between any two stars remains unchanged, and they were consequently regarded as being attached to the celestial sphere. Moreover a little careful observation would have shown that the motions of the stars in different parts of the sky, though at first sight very different, were just such as would have been produced by the celestial sphere—with the stars attached to it—turning about an axis passing through the centre and through a point in the northern sky close to the familiar pole-star. This point is called the pole. As, however, a straight line drawn through the centre of a sphere meets it in two points, the axis of the celestial sphere meets it again in a second point, opposite the first, lying in a part of the celestial sphere which is permanently below the horizon. This second point is also called a pole; and if the two poles have to be distinguished, the one mentioned first is called the north pole, and the other the south pole. The direction of the rotation of the celestial sphere about its axis is such that stars near the north pole are seen to move round it in circles in the direction opposite to that in which the hands of a clock move; the motion is uniform, and a complete revolution is performed in four minutes less than twenty-four hours; so that the position of any star in the sky at twelve o’clock to-night is the same as its position at four minutes to twelve to-morrow night.

The moon, like the stars, shares this motion of the celestial sphere and so also does the sun, though this is more difficult to recognise owing to the fact that the sun and stars are not seen together.

As other motions of the celestial bodies have to be dealt with, the general motion just described may be conveniently referred to as the daily motion or daily rotation of the celestial sphere.

9. A further study of the daily motion would lead to the recognition of certain important circles of the celestial sphere.

Each star describes in its daily motion a circle, the size of which depends on its distance from the poles. Fig. 2 shews the paths described by a number of stars near the pole, recorded photographically, during part of a night. The pole-star describes so small a circle that its motion can only with difficulty be detected with the naked eye, stars a little farther off the pole describe larger circles, and so on, until we come to stars half-way between the two poles, which describe the largest circle which can be drawn on the celestial sphere. The circle on which these stars lie and which is described by any one of them daily is called the equator. By looking at a diagram such as fig. 3, or, better still, by looking at an actual globe, it can easily be seen that half the equator (E Q W) lies above and half (the dotted part, W R E) below the horizon, and that in consequence a star, such as s, lying on the equator, is in its daily motion as long a time above the horizon as below. If a star, such as S, lies on the north side of the equator, i.e. on the side on which the north pole P lies, more than half of its daily path lies above the horizon and less than half (as shewn by the dotted line) lies below; and if a star is near enough to the north pole (more precisely, if it is nearer to the north pole than the nearest point, K, of the horizon), as σ, it never sets, but remains continually above the horizon. Such a star is called a (northern) circumpolar star. On the other hand, less than half of the daily path of a star on the south side of the equator, as S′, is above the horizon, and a star, such as σ′, the distance of which from the north pole is greater than the distance of the farthest point, H, of the horizon, or which is nearer than H to the south pole, remains continually below the horizon.

Fig. 2.—The paths of circumpolar stars, shewing their movement during seven hours. From a photograph by Mr. H. Pain. The thickest line is the path of the pole star.

To face p. 8.

10. A slight familiarity with the stars is enough to shew any one that the same stars are not always visible at the same time of night. Rather more careful observation, carried out for a considerable time, is necessary in order to see that the aspect of the sky changes in a regular way from night to night, and that after the lapse of a year the same stars become again visible at the same time. The explanation of these changes as due to the motion of the sun on the celestial sphere is more difficult, and the unknown discoverer of this fact certainly made one of the most important steps in early astronomy.

Fig. 3.—The circles of the celestial sphere.

If an observer notices soon after sunset a star somewhere in the west, and looks for it again a few evenings later at about the same time, he finds it lower down and nearer to the sun; a few evenings later still it is invisible, while its place has now been taken by some other star which was at first farther east in the sky. This star can in turn be observed to approach the sun evening by evening. Or if the stars visible after sunset low down in the east are noticed a few days later, they are found to be higher up in the sky, and their place is taken by other stars at first too low down to be seen. Such observations of stars rising or setting about sunrise or sunset shewed to early observers that the stars were gradually changing their position with respect to the sun, or that the sun was changing its position with respect to the stars.

The changes just described, coupled with the fact that the stars do not change their positions with respect to one another, shew that the stars as a whole perform their daily revolution rather more rapidly than the sun, and at such a rate that they gain on it one complete revolution in the course of the year. This can be expressed otherwise in the form that the stars are all moving westward on the celestial sphere, relatively to the sun, so that stars on the east are continually approaching and those on the west continually receding from the sun. But, again, the same facts can be expressed with equal accuracy and greater simplicity if we regard the stars as fixed on the celestial sphere, and the sun as moving on it from west to east among them (that is, in the direction opposite to that of the daily motion), and at such a rate as to complete a circuit of the celestial sphere and to return to the same position after a year.

This annual motion of the sun is, however, readily seen not to be merely a motion from west to east, for if so the sun would always rise and set at the same points of the horizon, as a star does, and its midday height in the sky and the time from sunrise to sunset would always be the same. We have already seen that if a star lies on the equator half of its daily path is above the horizon, if the star is north of the equator more than half, and if south of the equator less than half; and what is true of a star is true for the same reason of any body sharing the daily motion of the celestial sphere. During the summer months therefore (March to September), when the day is longer than the night, and more than half of the sun’s daily path is above the horizon, the sun must be north of the equator, and during the winter months (September to March) the sun must be south of the equator. The change in the sun’s distance from the pole is also evident from the fact that in the winter months the sun is on the whole lower down in the sky than in summer, and that in particular its midday height is less.

11. The sun’s path on the celestial sphere is therefore oblique to the equator, lying partly on one side of it and partly on the other. A good deal of careful observation of the kind we have been describing must, however, have been necessary before it was ascertained that the sun’s annual path on the celestial sphere (see fig. 4) is a great circle (that is, a circle having its centre at the centre of the sphere). This great circle is now called the ecliptic (because eclipses take place only when the moon is in or near it), and the angle at which it cuts the equator is called the obliquity of the ecliptic. The Chinese claim to have measured the obliquity in 1100 B.C., and to have found the remarkably accurate value 23° 52′ (cf. chapter II., [§ 35]). The truth of this statement may reasonably be doubted, but on the other hand the statement of some late Greek writers that either Pythagoras or Anaximander (6th century B.C.) was the first to discover the obliquity of the ecliptic is almost certainly wrong. It must have been known with reasonable accuracy to both Chaldaeans and Egyptians long before.

Fig. 4.—The equator and the ecliptic.

When the sun crosses the equator the day is equal to the night, and the times when this occurs are consequently known as the equinoxes, the vernal equinox occurring when the sun crosses the equator from south to north (about March 21st), and the autumnal equinox when it crosses back (about September 23rd). The points on the celestial sphere where the sun crosses the equator (A, C in fig. 4), i.e. where ecliptic and equator cross one another, are called the equinoctial points, occasionally also the equinoxes.

After the vernal equinox the sun in its path along the ecliptic recedes from the equator towards the north, until it reaches, about three months afterwards, its greatest distance from the equator, and then approaches the equator again. The time when the sun is at its greatest distance from the equator on the north side is called the summer solstice, because then the northward motion of the sun is arrested and it temporarily appears to stand still. Similarly the sun is at its greatest distance from the equator towards the south at the winter solstice. The points on the ecliptic (B, D in fig. 4) where the sun is at the solstices are called the solstitial points, and are half-way between the equinoctial points.

12. The earliest observers probably noticed particular groups of stars remarkable for their form or for the presence of bright stars among them, and occupied their fancy by tracing resemblances between them and familiar objects, etc. We have thus at a very early period a rough attempt at dividing the stars into groups called constellations and at naming the latter.

In some cases the stars regarded as belonging to a constellation form a well-marked group on the sky, sufficiently separated from other stars to be conveniently classed together, although the resemblance which the group bears to the object after which it is named is often very slight. The seven bright stars of the Great Bear, for example, form a group which any observer would very soon notice and naturally make into a constellation, but the resemblance to a bear of these and the fainter stars of the constellation is sufficiently remote (see fig. 5), and as a matter of fact this part of the Bear has also been called a Waggon and is in America familiarly known as the Dipper; another constellation has sometimes been called the Lyre and sometimes also the Vulture. In very many cases the choice of stars seems to have been made in such an arbitrary manner, as to suggest that some fanciful figure was first imagined and that stars were then selected so as to represent it in some rough sort of way. In fact, as Sir John Herschel remarks, “The constellations seem to have been purposely named and delineated to cause as much confusion and inconvenience as possible. Innumerable snakes twine through long and contorted areas of the heavens where no memory can follow them; bears, lions, and fishes, large and small, confuse all nomenclature.” (Outlines of Astronomy, § 301.)

Fig. 5.—The Great Bear. From Bayer’s Uranometria (1603). [

To face p. 12.

The constellations as we now have them are, with the exception of a certain number (chiefly in the southern skies) which have been added in modern times, substantially those which existed in early Greek astronomy; and such information as we possess of the Chaldaean and Egyptian constellations shews resemblances indicating that the Greeks borrowed some of them. The names, as far as they are not those of animals or common objects (Bear, Serpent, Lyre, etc.), are largely taken from characters in the Greek mythology (Hercules, Perseus, Orion, etc.). The constellation Berenice’s Hair, named after an Egyptian queen of the 3rd century B.C., is one of the few which commemorate a historical personage.[4]

13. Among the constellations which first received names were those through which the sun passes in its annual circuit of the celestial sphere, that is those through which the ecliptic passes. The moon’s monthly path is also a great circle, never differing very much from the ecliptic, and the paths of the planets ([§ 14]) are such that they also are never far from the ecliptic. Consequently the sun, the moon, and the five planets were always to be found within a region of the sky extending about 8° on each side of the ecliptic. This strip of the celestial sphere was called the zodiac, because the constellations in it were (with one exception) named after living things (Greek ζῷον, an animal); it was divided into twelve equal parts, the signs of the zodiac, through one of which the sun passed every month, so that the position of the sun at any time could be roughly described by stating in what “sign” it was. The stars in each “sign” were formed into a constellation, the “sign” and the constellation each receiving the same name. Thus arose twelve zodiacal constellations, the names of which have come down to us with unimportant changes from early Greek times.[5] Owing, however, to an alteration of the position of the equator, and consequently of the equinoctial points, the sign Aries, which was defined by Hipparchus in the second century B.C. (see chapter II., [§ 42]) as beginning at the vernal equinoctial point, no longer contains the constellation Aries, but the preceding one, Pisces: and there is a corresponding change throughout the zodiac. The more precise numerical methods of modern astronomy have, however, rendered the signs of the zodiac almost obsolete: but the first point of Aries (♈), and the first point of Libra (♎), are still the recognised names for the equinoctial points.

In some cases individual stars also received special names, or were called after the part of the constellation in which they were situated, e.g. Sirius, the Eye of the Bull, the Heart of the Lion, etc.; but the majority of the present names of single stars are of Arabic origin (chapter III., [§ 64]).

14. We have seen that the stars, as a whole, retain invariable positions on the celestial sphere,[6] whereas the sun and moon change their positions. It was, however, discovered in prehistoric times that five bodies, at first sight barely distinguishable from the other stars, also changed their places. These five—Mercury, Venus, Mars, Jupiter, and Saturn—with the sun and moon, were called planets,[7] or wanderers, as distinguished from the fixed stars. Mercury is never seen except occasionally near the horizon just after sunset or before sunrise, and in a climate like ours requires a good deal of looking for; and it is rather remarkable that no record of its discovery should exist. Venus is conspicuous as the Evening Star or as the Morning Star. The discovery of the identity of the Evening and Morning Stars is attributed to Pythagoras (6th century B.C.), but must almost certainly have been made earlier, though the Homeric poems contain references to both, without any indication of their identity. Jupiter is at times as conspicuous as Venus at her brightest, while Mars and Saturn, when well situated, rank with the brightest of the fixed stars.

The paths of the planets on the celestial sphere are, as we have seen ([§ 13]), never very far from the ecliptic; but whereas the sun and moon move continuously along their paths from west to east, the motion of a planet is sometimes from west to east, or direct, and sometimes from east to west, or retrograde. If we begin to watch a planet when it is moving eastwards among the stars, we find that after a time the motion becomes slower and slower, until the planet hardly seems to move at all, and then begins to move with gradually increasing speed in the opposite direction; after a time this westward motion becomes slower and then ceases, and the planet then begins to move eastwards again, at first slowly and then faster, until it returns to its original condition, and the changes are repeated. When the planet is just reversing its motion it is said to be stationary, and its position then is called a stationary point. The time during which a planet’s motion is retrograde is, however, always considerably less than that during which it is direct; Jupiter’s motion, for example, is direct for about 39 weeks and retrograde for 17, while Mercury’s direct motion lasts 13 or 14 weeks and the retrograde motion only about 3 weeks (see figs. 6, 7). On the whole the planets advance from west to east and describe circuits round the celestial sphere in periods which are different for each planet. The explanation of these irregularities in the planetary motions was long one of the great difficulties of astronomy.

Fig. 6.—The apparent path of Jupiter from Oct. 28, 1897, to Sept. 3, 1898. The dates printed in the diagram shew the positions of Jupiter.

15. The idea that some of the heavenly bodies are nearer to the earth than others must have been suggested by eclipses ([§ 17]) and occultations, i.e. passages of the moon over a planet or fixed star. In this way the moon would be recognised as nearer than any of the other celestial bodies. No direct means being available for determining the distances, rapidity of motion was employed as a test of probable nearness. Now Saturn returns to the same place among the stars in about 29-1∕2 years, Jupiter in 12 years, Mars in 2 years, the sun in one year, Venus in 225 days, Mercury in 88 days, and the moon in 27 days; and this order was usually taken to be the order of distance, Saturn being the most distant, the moon the nearest. The stars being seen above us it was natural to think of the most distant celestial bodies as being the highest, and accordingly Saturn, Jupiter, and Mars being beyond the sun were called superior planets, as distinguished from the two inferior planets Venus and Mercury. This division corresponds also to a difference in the observed motions, as Venus and Mercury seem to accompany the sun in its annual journey, being never more than about 47 and 29° respectively distant from it, on either side; while the other planets are not thus restricted in their motions.

Fig. 7.—The apparent path of Mercury from Aug. 1 to Oct. 3, 1898. The dates printed in capital letters shew the positions of the sun; the other dates shew those of Mercury.

16. One of the purposes to which applications of astronomical knowledge was first applied was to the measurement of time. As the alternate appearance and disappearance of the sun, bringing with it light and heat, is the most obvious of astronomical facts, so the day is the simplest unit of time.[8] Some of the early civilised nations divided the time from sunrise to sunset and also the night each into 12 equal hours. According to this arrangement a day-hour was in summer longer than a night-hour and in winter shorter, and the length of an hour varied during the year. At Babylon, for example, where this arrangement existed, the length of a day-hour was at midsummer about half as long again as in midwinter, and in London it would be about twice as long. It was therefore a great improvement when the Greeks, in comparatively late times, divided the whole day into 24 equal hours. Other early nations divided the same period into 12 double hours, and others again into 60 hours.

The next most obvious unit of time is the lunar month, or period during which the moon goes through her phases. A third independent unit is the year. Although the year is for ordinary life much more important than the month, yet as it is much longer and any one time of year is harder to recognise than a particular phase of the moon, the length of the year is more difficult to determine, and the earliest known systems of time-measurement were accordingly based on the month, not on the year. The month was found to be nearly equal to 29-1∕2 days, and as a period consisting of an exact number of days was obviously convenient for most ordinary purposes, months of 29 or 30 days were used, and subsequently the calendar was brought into closer accord with the moon by the use of months containing alternately 29 and 30 days (cf. chapter II., [§ 19]).

Both Chaldaeans and Egyptians appear to have known that the year consisted of about 365-1∕4 days; and the latter, for whom the importance of the year was emphasised by the rising and falling of the Nile, were probably the first nation to use the year in preference to the month as a measure of time. They chose a year of 365 days.

The origin of the week is quite different from that of the month or year, and rests on certain astrological ideas about the planets. To each hour of the day one of the seven planets (sun and moon included) was assigned as a “ruler,” and each day named after the planet which ruled its first hour. The planets being taken in the order already given ([§ 15]), Saturn ruled the first hour of the first day, and therefore also the 8th, 15th, and 22nd hours of the first day, the 5th, 12th, and 19th of the second day, and so on; Jupiter ruled the 2nd, 9th, 16th, and 23rd hours of the first day, and subsequently the 1st hour of the 6th day. In this way the first hours of successive days fell respectively to Saturn, the Sun, the Moon, Mars, Mercury, Jupiter, and Venus. The first three are easily recognised in our Saturday, Sunday, and Monday; in the other days the names of the Roman gods have been replaced by their supposed Teutonic equivalents—Mercury by Wodan, Mars by Thues, Jupiter by Thor, Venus by Freia.[9]

17. Eclipses of the sun and moon must from very early times have excited great interest, mingled with superstitious terror, and the hope of acquiring some knowledge of them was probably an important stimulus to early astronomical work. That eclipses of the sun only take place at new moon, and those of the moon only at full moon, must have been noticed after very little observation; that eclipses of the sun are caused by the passage of the moon in front of it must have been only a little less obvious; but the discovery that eclipses of the moon are caused by the earth’s shadow was probably made much later. In fact even in the time of Anaxagoras (5th century B.C.) the idea was so unfamiliar to the Athenian public as to be regarded as blasphemous.

One of the most remarkable of the Chaldaean contributions to astronomy was the discovery (made at any rate several centuries B.C.) of the recurrence of eclipses after a period, known as the saros, consisting of 6,585 days (or eighteen of our years and ten or eleven days, according as five or four leap-years are included). It is probable that the discovery was made, not by calculations based on knowledge of the motions of the sun and moon, but by mere study of the dates on which eclipses were recorded to have taken place. As, however, an eclipse of the sun (unlike an eclipse of the moon) is only visible over a small part of the surface of the earth, and eclipses of the sun occurring at intervals of eighteen years are not generally visible at the same place, it is not at all easy to see how the Chaldaeans could have established their cycle for this case, nor is it in fact clear that the saros was supposed to apply to solar as well as to lunar eclipses. The saros may be illustrated in modern times by the eclipses of the sun which took place on July 18th, 1860, on July 29th, 1878, and on August 9th, 1896; but the first was visible in Southern Europe, the second in North America, and the third in Northern Europe and Asia.

18. To the Chaldaeans may be assigned also the doubtful honour of having been among the first to develop astrology, the false science which has professed to ascertain the influence of the stars on human affairs, to predict by celestial observations wars, famines, and pestilences, and to discover the fate of individuals from the positions of the stars at their birth. A belief in some form of astrology has always prevailed in oriental countries; it flourished at times among the Greeks and the Romans; it formed an important part of the thought of the Middle Ages, and is not even quite extinct among ourselves at the present day.[10] It should, however, be remembered that if the history of astrology is a painful one, owing to the numerous illustrations which it affords of human credulity and knavery, the belief in it has undoubtedly been a powerful stimulus to genuine astronomical study (cf. chapter III., [§ 56], and chapter V., §§ 99, 100)

[CHAPTER II.]
GREEK ASTRONOMY.

“The astronomer discovers that geometry, a pure abstraction of the human mind, is the measure of planetary motion.”

Emerson.

19. In the earlier period of Greek history one of the chief functions expected of astronomers was the proper regulation of the calendar. The Greeks, like earlier nations, began with a calendar based on the moon. In the time of Hesiod a year consisting of 12 months of 30 days was in common use; at a later date a year made up of 6 full months of 30 days and 6 empty months of 29 days was introduced. To Solon is attributed the merit of having introduced at Athens, about 594 B.C., the practice of adding to every alternate year a “full” month. Thus a period of two years would contain 13 months of 30 days and 12 of 29 days, or 738 days in all, distributed among 25 months, giving, for the average length of the year and month, 369 days and about 29-1∕2 days respectively. This arrangement was further improved by the introduction, probably during the 5th century B.C., of the octaeteris, or eight-year cycle, in three of the years of which an additional “full” month was introduced, while the remaining years consisted as before of 6 “full” and 6 “empty” months. By this arrangement the average length of the year was reduced to 365-1∕4 days, that of the month remaining nearly unchanged. As, however, the Greeks laid some stress on beginning the month when the new moon was first visible, it was necessary to make from time to time arbitrary alterations in the calendar, and considerable confusion resulted, of which Aristophanes makes the Moon complain in his play The Clouds, acted in 423 B.C.:

“Yet you will not mark your days

As she bids you, but confuse them, jumbling them all sorts of ways.

And, she says, the Gods in chorus shower reproaches on her head,

When, in bitter disappointment, they go supperless to bed.

Not obtaining festal banquets, duly on the festal day.”

20. A little later, the astronomer Meton (born about 460 B.C.) made the discovery that the length of 19 years is very nearly equal to that of 235 lunar months (the difference being in fact less than a day), and he devised accordingly an arrangement of 12 years of 12 months and 7 of 13 months, 125 of the months in the whole cycle being “full” and the others “empty.” Nearly a century later Callippus made a slight improvement, by substituting in every fourth period of 19 years a “full” month for one of the “empty” ones. Whether Meton’s cycle, as it is called, was introduced for the civil calendar or not is uncertain, but if not it was used as a standard by reference to which the actual calendar was from time to time adjusted. The use of this cycle seems to have soon spread to other parts of Greece, and it is the basis of the present ecclesiastical rule for fixing Easter. The difficulty of ensuring satisfactory correspondence between the civil calendar and the actual motions of the sun and moon led to the practice of publishing from time to time tables (παραπήγματα) not unlike our modern almanacks, giving for a series of years the dates of the phases of the moon, and the rising and setting of some of the fixed stars, together with predictions of the weather. Owing to the same cause the early writers on agriculture (e.g. Hesiod) fixed the dates for agricultural operations, not by the calendar, but by the times of the rising and setting of constellations, i.e. the times when they first became visible before sunrise or were last visible immediately after sunset—a practice which was continued long after the establishment of a fairly satisfactory calendar, and was apparently by no means extinct in the time of Galen (2nd century A.D.).

21. The Roman calendar was in early times even more confused than the Greek. There appears to have been at one time a year of either 304 or 354 days; tradition assigned to Numa the introduction of a cycle of four years, which brought the calendar into fair agreement with the sun, but made the average length of the month considerably too short. Instead, however, of introducing further refinements the Romans cut the knot by entrusting to the ecclesiastical authorities the adjustment of the calendar from time to time, so as to make it agree with the sun and moon. According to one account, the first day of each month was proclaimed by a crier. Owing either to ignorance, or, as was alleged, to political and commercial favouritism, the priests allowed the calendar to fall into a state of great confusion, so that, as Voltaire remarked, “les généraux romains triomphaient toujours, mais ils ne savaient pas quel jour ils triomphaient.”

A satisfactory reform of the calendar was finally effected by Julius Caesar during the short period of his supremacy at Rome, under the advice of an Alexandrine astronomer Sosigenes. The error in the calendar had mounted up to such an extent, that it was found necessary, in order to correct it, to interpolate three additional months in a single year (46 B.C.), bringing the total number of days in that year up to 445. For the future the year was to be independent of the moon; the ordinary year was to consist of 365 days, an extra day being added to February every fourth year (our leap-year), so that the average length of the year would be 365-1∕4 days.

The new system began with the year 45 B.C., and soon spread, under the name of the Julian Calendar, over the civilised world.

22. To avoid returning to the subject, it may be convenient to deal here with the only later reform of any importance.

The difference between the average length of the year as fixed by Julius Caesar and the true year is so small as only to amount to about one day in 128 years. By the latter half of the 16th century the date of the vernal equinox was therefore about ten days earlier than it was at the time of the Council of Nice (A.D. 325), at which rules for the observance of Easter had been fixed. Pope Gregory XIII. introduced therefore, in 1582, a slight change;, ten days were omitted from that year, and it was arranged to omit for the future three leap-years in four centuries (viz. in 1700, 1800, 1900, 2100, etc., the years 1600, 2000, 2400, etc., remaining leap-years). The Gregorian Calendar, or New Style, as it was commonly called, was not adopted in England till 1752, when 11 days had to be omitted; and has not yet been adopted in Russia and Greece, the dates there being now 12 days behind those of Western Europe.

23. While their oriental predecessors had confined themselves chiefly to astronomical observations, the earlier Greek philosophers appear to have made next to no observations of importance, and to have been far more interested in inquiring into causes of phenomena. Thales, the founder of the Ionian school, was credited by later writers with the introduction of Egyptian astronomy into Greece, at about the end of the 7th century B.C.; but both Thales and the majority of his immediate successors appear to have added little or nothing to astronomy, except some rather vague speculations as to the form of the earth and its relation to the rest of the world. On the other hand, some real progress seems to have been made by Pythagoras[11] and his followers. Pythagoras taught that the earth, in common with the heavenly bodies, is a sphere, and that it rests without requiring support in the middle of the universe. Whether he had any real evidence in support of these views is doubtful, but it is at any rate a reasonable conjecture that he knew the moon to be bright because the sun shines on it, and the phases to be caused by the greater or less amount of the illuminated half turned towards us; and the curved form of the boundary between the bright and dark portions of the moon was correctly interpreted by him as evidence that the moon was spherical, and not a flat disc, as it appears at first sight. Analogy would then probably suggest that the earth also was spherical. However this may be, the belief in the spherical form of the earth never disappeared from Greek thought, and was in later times an established part of Greek systems, whence it has been handed down, almost unchanged, to modern times. This belief is thus 2,000 years older than the belief in the rotation of the earth and its revolution round the sun (chapter IV.), doctrines which we are sometimes inclined to couple with it as the foundations of modern astronomy.

In Pythagoras occurs also, perhaps for the first time, an idea which had an extremely important influence on ancient and mediaeval astronomy. Not only were the stars supposed to be attached to a crystal sphere, which revolved daily on an axis through the earth, but each of the seven planets (the sun and moon being included) moved on a sphere of its own. The distances of these spheres from the earth were fixed in accordance with certain speculative notions of Pythagoras as to numbers and music; hence the spheres as they revolved produced harmonious sounds which specially gifted persons might at times hear: this is the origin of the idea of the music of the spheres which recurs continually in mediaeval speculation and is found occasionally in modern literature. At a later stage these spheres of Pythagoras were developed into a scientific representation of the motions of the celestial bodies, which remained the basis of astronomy till the time of Kepler (chapter VII.).

24. The Pythagorean Philolaus, who lived about a century later than his master, introduced for the first time the idea of the motion of the earth: he appears to have regarded the earth, as well as the sun, moon, and five planets, as revolving round some central fire, the earth rotating on its own axis as it revolved, apparently in order to ensure that the central fire should always remain invisible to the inhabitants of the known parts of the earth. That the scheme was a purely fanciful one, and entirely different from the modern doctrine of the motion of the earth, with which later writers confused it, is sufficiently shewn by the invention as part of the scheme of a purely imaginary body, the counter-earth ([Greek: ἀντιχθών]), which brought the number of moving bodies up to ten, a sacred Pythagorean number. The suggestion of such an important idea as that of the motion of the earth, an idea so repugnant to uninstructed common sense, although presented in such a crude form, without any of the evidence required to win general assent, was, however, undoubtedly a valuable contribution to astronomical thought. It is well worth notice that Coppernicus in the great book which is the foundation of modern astronomy (chapter IV., [§ 75]) especially quotes Philolaus and other Pythagoreans as authorities for his doctrine of the motion of the earth.

Three other Pythagoreans, belonging to the end of the 6th century and to the 5th century B.C., Hicetas of Syracuse, Heraclitus, and Ecphantus, are explicitly mentioned by later writers as having believed in the rotation of the earth.

An obscure passage in one of Plato’s dialogues (the Timaeus) has been interpreted by many ancient and modern commentators as implying a belief in the rotation of the earth, and Plutarch also tells us, partly on the authority of Theophrastus, that Plato in old age adopted the belief that the centre of the universe was not occupied by the earth but by some better body.[12]

Almost the only scientific Greek astronomer who believed in the motion of the earth was Aristarchus of Samos, who lived in the first half of the 3rd century B.C., and is best known by his measurements of the distances of the sun and moon ([§ 32]). He held that the sun and fixed stars were motionless, the sun being in the centre of the sphere on which the latter lay, and that the earth not only rotated on its axis, but also described an orbit round the sun. Seleucus of Seleucia, who belonged to the middle of the 2nd century B.C., also held a similar opinion. Unfortunately we know nothing of the grounds of this belief in either case, and their views appear to have found little favour among their contemporaries or successors.

It may also be mentioned in this connection that Aristotle ([§ 27]) clearly realised that the apparent daily motion of the stars could be explained by a motion either of the stars or of the earth, but that he rejected the latter explanation.

25. Plato (about 428-347 B.C.) devoted no dialogue especially to astronomy, but made a good many references to the subject in various places. He condemned any careful study of the actual celestial motions as degrading rather than elevating, and apparently regarded the subject as worthy of attention chiefly on account of its connection with geometry, and because the actual celestial motions suggested ideal motions of greater beauty and interest. This view of astronomy he contrasts with the popular conception, according to which the subject was useful chiefly for giving to the agriculturist, the navigator, and others a knowledge of times and seasons.[13] At the end of the same dialogue he gives a short account of the celestial bodies, according to which the sun, moon, planets, and fixed stars revolve on eight concentric and closely fitting wheels or circles round an axis passing through the earth. Beginning with the body nearest to the earth, the order is Moon, Sun, Mercury, Venus, Mars, Jupiter, Saturn, stars. The Sun, Mercury, and Venus are said to perform their revolutions in the same time, while the other planets move more slowly, statements which shew that Plato was at any rate aware that the motions of Venus and Mercury are different from those of the other planets. He also states that the moon shines by reflected light received from the sun.

Plato is said to have suggested to his pupils as a worthy problem the explanation of the celestial motions by means of a combination of uniform circular or spherical motions. Anything like an accurate theory of the celestial motions, agreeing with actual observation, such as Hipparchus and Ptolemy afterwards constructed with fair success, would hardly seem to be in accordance with Plato’s ideas of the true astronomy, but he may well have wished to see established some simple and harmonious geometrical scheme which would not be altogether at variance with known facts.

26. Acting to some extent on this idea of Plato’s, Eudoxus of Cnidus (about 409-356 B.C.) attempted to explain the most obvious peculiarities of the celestial motions by means of a combination of uniform circular motions. He may be regarded as representative of the transition from speculative to scientific Greek astronomy. As in the schemes of several of his predecessors, the fixed stars lie on a sphere which revolves daily about an axis through the earth; the motion of each of the other bodies is produced by a combination of other spheres, the centre of each sphere lying on the surface of the preceding one. For the sun and moon three spheres were in each case necessary: one to produce the daily motion, shared by all the celestial bodies; one to produce the annual or monthly motion in the opposite direction along the ecliptic; and a third, with its axis inclined to the axis of the preceding, to produce the smaller motion to and from the ecliptic. Eudoxus evidently was well aware that the moon’s path is not coincident with the ecliptic, and even that its path is not always the same, but changes continuously, so that the third sphere was in this case necessary; on the other hand, he could not possibly have been acquainted with the minute deviations of the sun from the ecliptic with which modern astronomy deals. Either therefore he used erroneous observations, or, as is more probable, the sun’s third sphere was introduced to explain a purely imaginary motion conjectured to exist by “analogy” with the known motion of the moon. For each of the five planets four spheres were necessary, the additional one serving to produce the variations in the speed of the motion and the reversal of the direction of motion along the ecliptic (chapter I., [§ 14], and below, [§ 51]). Thus the celestial motions were to some extent explained by means of a system of 27 spheres, 1 for the stars, 6 for the sun and moon, 20 for the planets. There is no clear evidence that Eudoxus made any serious attempt to arrange either the size or the time of revolution of the spheres so as to produce any precise agreement with the observed motions of the celestial bodies, though he knew with considerable accuracy the time required by each planet to return to the same position with respect to the sun; in other words, his scheme represented the celestial motions qualitatively but not quantitatively. On the other hand, there is no reason to suppose that Eudoxus regarded his spheres (with the possible exception of the sphere of the fixed stars) as material; his known devotion to mathematics renders it probable that in his eyes (as in those of most of the scientific Greek astronomers who succeeded him) the spheres were mere geometrical figures, useful as a means of resolving highly complicated motions into simpler elements. Eudoxus was also the first Greek recorded to have had an observatory, which was at Cnidus, but we have few details as to the instruments used or as to the observations made. We owe, however, to him the first systematic description of the constellations (see below, [§ 42]), though it was probably based, to a large extent, on rough observations borrowed from his Greek predecessors or from the Egyptians. He was also an accomplished mathematician, and skilled in various other branches of learning.

Shortly afterwards Callippus ([§ 20]) further developed Eudoxus’s scheme of revolving spheres by adding, for reasons not known to us, two spheres each for the sun and moon and one each for Venus, Mercury, and Mars, thus bringing the total number up to 34.

27. We have a tolerably full account of the astronomical views of Aristotle (384-322 B.C.), both by means of incidental references, and by two treatises—the Meteorologica and the De Coelo—though another book of his, dealing specially with the subject, has unfortunately been lost. He adopted the planetary scheme of Eudoxus and Callippus, but imagined on “metaphysical grounds” that the spheres would have certain disturbing effects on one another, and to counteract these found it necessary to add 22 fresh spheres, making 56 in all. At the same time he treated the spheres as material bodies, thus converting an ingenious and beautiful geometrical scheme into a confused mechanism.[14] Aristotle’s spheres were, however, not adopted by the leading Greek astronomers who succeeded him, the systems of Hipparchus and Ptolemy being geometrical schemes based on ideas more like those of Eudoxus.

Fig. 8.—The phases of the moon.

Fig. 9.—The phases of the moon.

28. Aristotle, in common with other philosophers of his time, believed the heavens and the heavenly bodies to be spherical. In the case of the moon he supports this belief by the argument attributed to Pythagoras ([§ 23]), namely that the observed appearances of the moon in its several phases are those which would be assumed by a spherical body of which one half only is illuminated by the sun. Thus the visible portion of the moon is bounded by two planes passing nearly through its centre, perpendicular respectively to the lines joining the centre of the moon to those of the sun and earth. In the accompanying diagram, which represents a section through the centres of the sun (S), earth (E), and moon (M), A B C D representing on a much enlarged scale a section of the moon itself, the portion D A B which is turned away from the sun is dark, while the portion A D C, being turned away from the observer on the earth, is in any case invisible to him. The part of the moon which appears bright is therefore that of which B C is a section, or the portion represented by F B G C in fig. 9 (which represents the complete moon), which consequently appears to the eye as bounded by a semicircle F C G, and a portion F B G of an oval curve (actually an ellipse). The breadth of this bright surface clearly varies with the relative positions of sun, moon, and earth; so that in the course of a month, during which the moon assumes successively the positions relative to sun and earth represented by 1, 2, 3, 4, 5, 6, 7, 8 in fig. 10, its appearances are those represented by the corresponding numbers in fig. 11, the moon thus passing through the familiar phases of crescent, half full, gibbous, full moon, and gibbous, half full, crescent again.[15]

Fig. 10.—The phases of the moon.

Aristotle then argues that as one heavenly body is spherical, the others must be so also, and supports this conclusion by another argument, equally inconclusive to us, that a spherical form is appropriate to bodies moving as the heavenly bodies appear to do.

Fig. 11.—The phases of the moon.

29. His proofs that the earth is spherical are more interesting. After discussing and rejecting various other suggested forms, he points out that an eclipse of the moon is caused by the shadow of the earth cast by the sun, and argues from the circular form of the boundary of the shadow as seen on the face of the moon during the progress of the eclipse, or in a partial eclipse, that the earth must be spherical; for otherwise it would cast a shadow of a different shape. A second reason for the spherical form of the earth is that when we move north and south the stars change their positions with respect to the horizon, while some even disappear and fresh ones take their place. This shows that the direction of the stars has changed as compared with the observer’s horizon; hence, the actual direction of the stars being imperceptibly affected by any motion of the observer on the earth, the horizons at two places, north and south of one another, are in different directions, and the earth is therefore curved. For example, if a star is visible to an observer at A (fig. 12), while to an observer at B it is at the same time invisible, i.e. hidden by the earth, the surface of the earth at A must be in a different direction from that at B. Aristotle quotes further, in confirmation of the roundness of the earth, that travellers from the far East and the far West (practically India and Morocco) alike reported the presence of elephants, whence it may be inferred that the two regions in question are not very far apart. He also makes use of some rather obscure arguments of an a priori character.

Fig. 12.—The curvature of the earth.

There can be but little doubt that the readiness with which Aristotle, as well as other Greeks, admitted the spherical form of the earth and of the heavenly bodies, was due to the affection which the Greeks always seem to have had for the circle and sphere as being “perfect,” i.e. perfectly symmetrical figures.

30. Aristotle argues against the possibility of the revolution of the earth round the sun, on the ground that this motion, if it existed, ought to produce a corresponding apparent motion of the stars. We have here the first appearance of one of the most serious of the many objections ever brought against the belief in the motion of the earth, an objection really only finally disposed of during the present century by the discovery that such a motion of the stars can be seen in a few cases, though owing to the almost inconceivably great distance of the stars the motion is imperceptible except by extremely refined methods of observation (cf. chapter XIII., [§§ 278], 279). The question of the distances of the several celestial bodies is also discussed, and Aristotle arrives at the conclusion that the planets are farther off than the sun and moon, supporting his view by his observation of an occultation of Mars by the moon (i.e. a passage of the moon in front of Mars), and by the fact that similar observations had been made in the case of other planets by Egyptians and Babylonians. It is, however, difficult to see why he placed the planets beyond the sun, as he must have known that the intense brilliancy of the sun renders planets invisible in its neighbourhood, and that no occultations of planets by the sun could really have been seen even if they had been reported to have taken place. He quotes also, as an opinion of “the mathematicians,” that the stars must be at least nine times as far off as the sun.

There are also in Aristotle’s writings a number of astronomical speculations, founded on no solid evidence and of little value; thus among other questions he discusses the nature of comets, of the Milky Way, and of the stars, why the stars twinkle, and the causes which produce the various celestial motions.

In astronomy, as in other subjects, Aristotle appears to have collected and systematised the best knowledge of the time; but his original contributions are not only not comparable with his contributions to the mental and moral sciences, but are inferior in value to his work in other natural sciences, e.g. Natural History. Unfortunately the Greek astronomy of his time, still in an undeveloped state, was as it were crystallised in his writings, and his great authority was invoked, centuries afterwards, by comparatively unintelligent or ignorant disciples in support of doctrines which were plausible enough in his time, but which subsequent research was shewing to be untenable. The advice which he gives to his readers at the beginning of his exposition of the planetary motions, to compare his views with those which they arrived at themselves or met with elsewhere, might with advantage have been noted and followed by many of the so-called Aristotelians of the Middle Ages and of the Renaissance.[16]

31. After the time of Aristotle the centre of Greek scientific thought moved to Alexandria. Founded by Alexander the Great (who was for a time a pupil of Aristotle) in 332 B.C., Alexandria was the capital of Egypt during the reigns of the successive Ptolemies. These kings, especially the second of them, surnamed Philadelphos, were patrons of learning; they founded the famous Museum, which contained a magnificent library as well as an observatory, and Alexandria soon became the home of a distinguished body of mathematicians and astronomers. During the next five centuries the only astronomers of importance, with the great exception of Hipparchus ([§ 37]), were Alexandrines.

Fig. 13.—The method of Aristarchus for comparing the distances of the sun and moon.

32. Among the earlier members of the Alexandrine school were Aristarchus of Samos, Aristyllus, and Timocharis, three nearly contemporary astronomers belonging to the first half of the 3rd century B.C. The views of Aristarchus on the motion of the earth have already been mentioned ([§ 24]). A treatise of his On the Magnitudes and Distances of the Sun and Moon is still extant: he there gives an extremely ingenious method for ascertaining the comparative distances of the sun and moon. If, in the figure, E, S, and M denote respectively the centres of the earth, sun, and moon, the moon evidently appears to an observer at E half full when the angle E M S is a right angle. If when this is the case the angular distance between the centres of the sun and moon, i.e. the angle M E S, is measured, two angles of the triangle M E S are known; its shape is therefore completely determined, and the ratio of its sides E M, E S can be calculated without much difficulty. In fact, it being known (by a well-known result in elementary geometry) that the angles at E and S are together equal to a right angle, the angle at S is obtained by subtracting the angle S E M from a right angle. Aristarchus made the angle at S about 3°, and hence calculated that the distance of the sun was from 18 to 20 times that of the moon, whereas, in fact, the sun is about 400 times as distant as the moon. The enormous error is due to the difficulty of determining with sufficient accuracy the moment when the moon is half full: the boundary separating the bright and dark parts of the moon’s face is in reality (owing to the irregularities on the surface of the moon) an ill-defined and broken line (cf. fig. 53 and the frontispiece), so that the observation on which Aristarchus based his work could not have been made with any accuracy even with our modern instruments, much less with those available in his time. Aristarchus further estimated the apparent sizes of the sun and moon to be about equal (as is shewn, for example, at an eclipse of the sun, when the moon sometimes rather more than hides the surface of the sun and sometimes does not quite cover it), and inferred correctly that the real diameters of the sun and moon were in proportion to their distances. By a method based on eclipse observations which was afterwards developed by Hipparchus ([§ 41]), 1∕3 that of the earth, a result very near to the truth; and the same method supplied data from which the distance of the moon could at once have been expressed in terms of the radius of the earth, but his work was spoilt at this point by a grossly inaccurate estimate of the apparent size of the moon (2° instead of 1∕2°), and his conclusions seem to contradict one another. He appears also to have believed the distance of the fixed stars to be immeasurably great as compared with that of the sun. Both his speculative opinions and his actual results mark therefore a decided advance in astronomy.

Timocharis and Aristyllus were the first to ascertain and to record the positions of the chief stars, by means of numerical measurements of their distances from fixed positions on the sky; they may thus be regarded as the authors of the first real star catalogue, earlier astronomers having only attempted to fix the position of the stars by more or less vague verbal descriptions. They also made a number of valuable observations of the planets, the sun, etc., of which succeeding astronomers, notably Hipparchus and Ptolemy, were able to make good use.

Fig. 14.—The equator and the ecliptic.

33. Among the important contributions of the Greeks to astronomy must be placed the development, chiefly from the mathematical point of view, of the consequences of the rotation of the celestial sphere and of some of the simpler motions of the celestial bodies, a development the individual steps of which it is difficult to trace. We have, however, a series of minor treatises or textbooks, written for the most part during the Alexandrine period, dealing with this branch of the subject (known generally as Spherics, or the Doctrine of the Sphere), of which the Phenomena of the famous geometer Euclid (about 300 B.C.) is a good example. In addition to the points and circles of the sphere already mentioned (chapter I., [§§ 8-11]), we now find explicitly recognised the horizon, or the great circle in which a horizontal plane through the observer meets the celestial sphere, and its pole,[17] the zenith,[18] or point on the celestial sphere vertically above the observer; the verticals, or great circles through the zenith, meeting the horizon at right angles; and the declination circles, which pass through the north and south poles and cut the equator at right angles. Another important great circle was the meridian, passing through the zenith and the poles. The well-known Milky Way had been noticed, and was regarded as forming another great circle. There are also traces of the two chief methods in common use at the present day of indicating the position of a star on the celestial sphere, namely, by reference either to the equator or to the ecliptic. If through a star S we draw on the sphere a portion of a great circle S N, cutting the ecliptic ♈ N at right angles in N, and another great circle (a declination circle) cutting the equator at M, and if ♈ be the first point of Aries ([§ 13]), where the ecliptic crosses the equator, then the position of the star is completely defined either by the lengths of the arcs ♈ N, N S, which are called the celestial longitude and latitude respectively, or by the arcs ♈ M, M S, called respectively the right ascension and declination.[19] For some purposes it is more convenient to find the position of the star by the first method, i.e. by reference to the ecliptic; for other purposes in the second way, by making use of the equator.

34. One of the applications of Spherics was to the construction of sun-dials, which were supposed to have been originally introduced into Greece from Babylon, but which were much improved by the Greeks, and extensively used both in Greek and in mediaeval times. The proper graduation of sun-dials placed in various positions, horizontal, vertical, and oblique, required considerable mathematical skill. Much attention was also given to the time of the rising and setting of the various constellations, and to similar questions.

35. The discovery of the spherical form of the earth led to a scientific treatment of the differences between the seasons in different parts of the earth, and to a corresponding division of the earth into zones. We have already seen that the height of the pole above the horizon varies in different places, and that it was recognised that, if a traveller were to go far enough north, he would find the pole to coincide with the zenith, whereas by going south he would reach a region (not very far beyond the limits of actual Greek travel) where the pole would be on the horizon and the equator consequently pass through the zenith; in regions still farther south the north pole would be permanently invisible, and the south pole would appear above the horizon.

Fig. 15.—The equator, the horizon, and the meridian.

Further, if in the figure H E K W represents the horizon, meeting the equator Q E R W in the east and west points E W, and the meridian H Q Z P K in the south and north points H and K, Z being the zenith and P the pole, then it is easily seen that Q Z is equal to P K, the height of the pole above the horizon. Any celestial body, therefore, the distance of which from the equator towards the north (declination) is less than P K, will cross the meridian to the south of the zenith, whereas if its declination be greater than P K, it will cross to the north of the zenith. Now the greatest distance of the sun from the equator is equal to the angle between the ecliptic and the equator, or about 23-1∕2°, Consequently at places at which the height of the pole is less than 23-1∕2° the sun will, during part of the year, cast shadows at midday towards the south. This was known actually to be the case not very far south of Alexandria. It was similarly recognised that on the other side of the equator there must be a region in which the sun ordinarily cast shadows towards the south, but occasionally towards the north. These two regions are the torrid zones of modern geographers.

Again, if the distance of the sun from the equator is 23-1∕2°, its distance from the pole is 66-1∕2°; therefore in regions so far north that the height P K of the north pole is more than 66-1∕2°, the sun passes in summer into the region of the circumpolar stars which never set (chapter I., [§ 9]), and therefore during a portion of the summer the sun remains continuously above the horizon. Similarly in the same regions the sun is in winter so near the south pole that for a time it remains continuously below the horizon. Regions in which this occurs (our Arctic regions) were unknown to Greek travellers, but their existence was clearly indicated by the astronomers.

Fig. 16.—The measurement of the earth.

36. To Eratosthenes (276 B.C. to 195 or 196 B.C.), another member of the Alexandrine school, we owe one of the first scientific estimates of the size of the earth. He found that at the summer solstice the angular distance of the sun from the zenith at Alexandria was at midday 1∕50th of a complete circumference, or about 7°, whereas at Syene in Upper Egypt the sun was known to be vertical at the same time. From this he inferred, assuming Syene to be due south of Alexandria, that the distance from Syene to Alexandria was also 1∕50th of the circumference of the earth. Thus if in the figure S denotes the sun, A and B Alexandria and Syene respectively, C the centre of the earth, and A Z the direction of the zenith at Alexandria, Eratosthenes estimated the angle S A Z, which, owing to the great distance of S, is sensibly equal to the angle S C A, to be 7°, and hence inferred that the arc A B was to the circumference of the earth in the proportion of 7° to 360° or 1 to 50. The distance between Alexandria and Syene being known to be 5,000 stadia, Eratosthenes thus arrived at 250,000 stadia as an estimate of the circumference of the earth, a number altered into 252,000 in order to give an exact number of stadia (700) for each degree on the earth. It is evident that the data employed were rough, though the principle of the method is perfectly sound; it is, however, difficult to estimate the correctness of the result on account of the uncertainty as to the value of the stadium used. If, as seems probable, it was the common Olympic stadium, the result is about 20 per cent. too great, but according to another interpretation[20] the result is less than 1 per cent. in error (cf. chapter X., [§ 221]).

Another measurement due to Eratosthenes was that of the obliquity of the ecliptic, which he estimated at 22∕83 of a right angle, or 23° 51′, the error in which is only about 7′.

37. An immense advance in astronomy was made by Hipparchus, whom all competent critics have agreed to rank far above any other astronomer of the ancient world, and who must stand side by side with the greatest astronomers of all time. Unfortunately only one unimportant book of his has been preserved, and our knowledge of his work is derived almost entirely from the writings of his great admirer and disciple Ptolemy, who lived nearly three centuries later (§§ 46 seqq.). We have also scarcely any information about his life. He was born either at Nicaea in Bithynia or in Rhodes, in which island he erected an observatory and did most of his work. There is no evidence that he belonged to the Alexandrine school, though he probably visited Alexandria and may have made some observations there. Ptolemy mentions observations made by him in 146 B.C., 126 B.C., and at many intermediate dates, as well as a rather doubtful one of 161 B.C. The period of his greatest activity must therefore have been about the middle of the 2nd century B.C.

Apart from individual astronomical discoveries, his chief services to astronomy may be put under four heads. He invented or greatly developed a special branch of mathematics,[21] which enabled processes of numerical calculation to be applied to geometrical figures, whether in a plane or on a sphere. He made an extensive series of observations, taken with all the accuracy that his instruments would permit. He systematically and critically made use of old observations for comparison with later ones so as to discover astronomical changes too slow to be detected within a single lifetime. Finally, he systematically employed a particular geometrical scheme (that of eccentrics, and to a less extent that of epicycles) for the representation of the motions of the sun and moon.

38. The merit of suggesting that the motions of the heavenly bodies could be represented more simply by combinations of uniform circular motions than by the revolving spheres of Eudoxus and his school ([§ 26]) is generally attributed to the great Alexandrine mathematician Apollonius of Perga, who lived in the latter half of the 3rd century B.C., but there is no clear evidence that he worked out a system in any detail.

On account of the important part that this idea played in astronomy for nearly 2,000 years, it may be worth while to examine in some detail Hipparchus’s theory of the sun, the simplest and most successful application of the idea.

We have already seen (chapter I., [§ 10]) that, in addition to the daily motion (from east to west) which it shares with the rest of the celestial bodies, and of which we need here take no further account, the sun has also an annual motion on the celestial sphere in the reverse direction (from west to east) in a path oblique to the equator, which was early recognised as a great circle, called the ecliptic. It must be remembered further that the celestial sphere, on which the sun appears to lie, is a mere geometrical fiction introduced for convenience; all that direct observation gives is the change in the sun’s direction, and therefore the sun may consistently be supposed to move in such a way as to vary its distance from the earth in any arbitrary manner, provided only that the alterations in the apparent size of the sun, caused by the variations in its distance, agree with those observed, or that at any rate the differences are not great enough to be perceptible. It was, moreover, known (probably long before the time of Hipparchus) that the sun’s apparent motion in the ecliptic is not quite uniform, the motion at some times of the year being slightly more rapid than at others.

Supposing that we had such a complete set of observations of the motion of the sun, that we knew its position from day to day, how should we set to work to record and describe its motion? For practical purposes nothing could be more satisfactory than the method adopted in our almanacks, of giving from day to day the position of the sun; after observations extending over a few years it would not be difficult to verify that the motion of the sun is (after allowing for the irregularities of our calendar) from year to year the same, and to predict in this way the place of the sun from day to day in future years.

But it is clear that such a description would not only be long, but would be felt as unsatisfactory by any one who approached the question from the point of view of intellectual curiosity or scientific interest. Such a person would feel that these detailed facts ought to be capable of being exhibited as consequences of some simpler general statement.

A modern astronomer would effect this by expressing the motion of the sun by means of an algebraical formula, i.e. he would represent the velocity of the sun or its distance from some fixed point in its path by some symbolic expression representing a quantity undergoing changes with the time in a certain definite way, and enabling an expert to compute with ease the required position of the sun at any assigned instant.[22]

The Greeks, however, had not the requisite algebraical knowledge for such a method of representation, and Hipparchus, like his predecessors, made use of a geometrical representation of the required variations in the sun’s motion in the ecliptic, a method of representation which is in some respects more intelligible and vivid than the use of algebra, but which becomes unmanageable in complicated cases. It runs moreover the risk of being taken for a mechanism. The circle, being the simplest curve known, would naturally be thought of, and as any motion other than a uniform motion would itself require a special representation, the idea of Apollonius, adopted by Hipparchus, was to devise a proper combination of uniform circular motions.

39. The simplest device that was found to be satisfactory in the case of the sun was the use of the eccentric, i.e. a circle the centre of which (C) does not coincide with the position of the observer on the earth (E). If in fig. 17 a point, S, describes the eccentric circle A F G B uniformly, so that it always passes over equal arcs of the circle in equal times and the angle A C S increases uniformly, then it is evident that the angle A E S, or the apparent distance of S from A, does not increase uniformly. When S is near the point A, which is farthest from the earth and hence called the apogee, it appears on account of its greater distance from the observer to move more slowly than when near F or G; and it appears to move fastest when near B, the point nearest to E, hence called the perigee. Thus the motion of S varies in the same sort of way as the motion of the sun as actually observed. Before, however, the eccentric could be considered as satisfactory, it was necessary to show that it was possible to choose the direction of the line B E C A (the line of apses) which determines the positions of the sun when moving fastest and when moving most slowly, and the magnitude of the ratio of E C to the radius C A of the circle (the eccentricity), so as to make the calculated positions of the sun in various parts of its path differ from the observed positions at the corresponding times of year by quantities so small that they might fairly be attributed to errors of observation.

Fig. 17.—The eccentric.

This problem was much more difficult than might at first sight appear, on account of the great difficulty experienced in Greek times and long afterwards in getting satisfactory observations of the sun. As the sun and stars are not visible at the same time, it is not possible to measure directly the distance of the sun from neighbouring stars and so to fix its place on the celestial sphere. But it is possible, by measuring the length of the shadow cast by a rod at midday, to ascertain with fair accuracy the height of the sun above the horizon, and hence to deduce its distance from the equator, or the declination (figs. 3, 14). This one quantity does not suffice to fix the sun’s position, but if also the sun’s right ascension ([§ 33]), or its distance east and west from the stars, can be accurately ascertained, its place on the celestial sphere is completely determined. The methods available for determining this second quantity were, however, very imperfect. One method was to note the time between the passage of the sun across some fixed position in the sky (e.g. the meridian), and the passage of a star across the same place, and thus to ascertain the angular distance between them (the celestial sphere being known to turn through 15° in an hour), a method which with modern clocks is extremely accurate, but with the rough water-clocks or sand-glasses of former times was very uncertain. In another method the moon was used as a connecting link between sun and stars, her position relative to the latter being observed by night, and with respect to the former by day; but owing to the rapid motion of the moon in the interval between the two observations, this method also was not susceptible of much accuracy.

Fig. 18.—The position of the sun’s apogee.

In the case of the particular problem of the determination of the line of apses, Hipparchus made use of another method, and his skill is shewn in a striking manner by his recognition that both the eccentricity and position of the apse line could be determined from a knowledge of the lengths of two of the seasons of the year, i.e. of the intervals into which the year is divided by the solstices and the equinoxes ([§ 11]). By means of his own observations, and of others made by his predecessors, he ascertained the length of the spring (from the vernal equinox to the summer solstice) to be 94 days, and that of the summer (summer solstice to autumnal equinox) to be 92-1∕2 days, the length of the year being 365-1∕4 days. As the sun moves in each season through the same angular distance, a right angle, and as the spring and summer make together more than half the year, and the spring is longer than the summer, it follows that the sun must, on the whole, be moving more slowly during the spring than in any other season, and that it must therefore pass through the apogee in the spring. If, therefore, in fig. 18, we draw two perpendicular lines Q E S, P E R to represent the directions of the sun at the solstices and equinoxes, P corresponding to the vernal equinox and R to the autumnal equinox, the apogee must lie at some point A between P and Q. So much can be seen without any mathematics: the actual calculation of the position of A and of the eccentricity is a matter of some complexity. The angle P E A was found to be about 65°, so that the sun would pass through its apogee about the beginning of June; and the eccentricity was estimated at 1∕24.

The motion being thus represented geometrically, it became merely a matter of not very difficult calculation to construct a table from which the position of the sun for any day in the year could be easily deduced. This was done by computing the so-called equation of the centre, the angle C S E of fig. 17, which is the excess of the actual longitude of the sun over the longitude which it would have had if moving uniformly.

Owing to the imperfection of the observations used (Hipparchus estimated that the times of the equinoxes and solstices could only be relied upon to within about half a day), the actual results obtained were not, according to modern ideas, very accurate, but the theory represented the sun’s motion with an accuracy about as great as that of the observations. It is worth noticing that with the same theory, but with an improved value of the eccentricity, the motion of the sun can be represented so accurately that the error never exceeds about 1′, a quantity insensible to the naked eye.

The theory of Hipparchus represents the variations in the distance of the sun with much less accuracy, and whereas in fact the angular diameter of the sun varies by about 1∕30th part of itself, or by about 1′ in the course of the year, this variation according to Hipparchus should be about twice as great. But this error would also have been quite imperceptible with his instruments.

Fig. 19.—The epicycle and the deferent.

Hipparchus saw that the motion of the sun could equally well be represented by the other device suggested by Apollonius, the epicycle. The body the motion of which is to be represented is supposed to move uniformly round the circumference of one circle, called the epicycle, the centre of which in turn moves on another circle called the deferent. It is in fact evident that if a circle equal to the eccentric, but with its centre at E (fig. 19), be taken as the deferent, and if S′ be taken on this so that E S′ is parallel to C S, then S′ S is parallel and equal to E C; and that therefore the sun S, moving uniformly on the eccentric, may equally well be regarded as lying on a circle of radius S′ S, the centre S′ of which moves on the deferent. The two constructions lead in fact in this particular problem to exactly the same result, and Hipparchus chose the eccentric as being the simpler.

40. The motion of the moon being much more complicated than that of the sun has always presented difficulties to astronomers,[23] and Hipparchus required for it a more elaborate construction. Some further description of the moon’s motion is, however, necessary before discussing his theory.

We have already spoken (chapter I., [§ 16]) of the lunar month as the period during which the moon returns to the same position with respect to the sun; more precisely this period (about 29-1∕2 days) is spoken of as a lunation or synodic month: as, however, the sun moves eastward on the celestial sphere like the moon but more slowly, the moon returns to the same position with respect to the stars in a somewhat shorter time; this period (about 27 days 8 hours) is known as the sidereal month. Again, the moon’s path on the celestial sphere is slightly inclined to the ecliptic, and may be regarded approximately as a great circle cutting the ecliptic in two nodes, at an angle which Hipparchus was probably the first to fix definitely at about 5°. Moreover, the moon’s path is always changing in such a way that, the inclination to the ecliptic remaining nearly constant (but cf. chapter V., [§ 111]), the nodes move slowly backwards (from east to west) along the ecliptic, performing a complete revolution in about 19 years. It is therefore convenient to give a special name, the draconitic month,[24] to the period (about 27 days 5 hours) during which the moon returns to the same position with respect to the nodes.

Again, the motion of the moon, like that of the sun, is not uniform, the variations being greater than in the case of the sun. Hipparchus appears to have been the first to discover that the part of the moon’s path in which the motion is most rapid is not always in the same position on the celestial sphere, but moves continuously; or, in other words, that the line of apses ([§ 39]) of the moon’s path moves. The motion is an advance, and a complete circuit is described in about nine years. Hence arises a fourth kind of month, the anomalistic month, which is the period in which the moon returns to apogee or perigee.

To Hipparchus is due the credit of fixing with greater exactitude than before the lengths of each of these months. In order to determine them with accuracy he recognised the importance of comparing observations of the moon taken at as great a distance of time as possible, and saw that the most satisfactory results could be obtained by using Chaldaean and other eclipse observations, which, as eclipses only take place near the moon’s nodes, were simultaneous records of the position of the moon, the nodes, and the sun.

To represent this complicated set of motions, Hipparchus used, as in the case of the sun, an eccentric, the centre of which described a circle round the earth in about nine years (corresponding to the motion of the apses), the plane of the eccentric being inclined to the ecliptic at an angle of 5°, and sliding back, so as to represent the motion of the nodes already described.

The result cannot, however, have been as satisfactory as in the case of the sun. The variation in the rate at which the moon moves is not only greater than in the case of the sun, but follows a less simple law, and cannot be adequately represented by means of a single eccentric; so that though Hipparchus’ work would have represented the motion of the moon in certain parts of her orbit with fair accuracy, there must necessarily have been elsewhere discrepancies between the calculated and observed places. There is some indication that Hipparchus was aware of these, but was not able to reconstruct his theory so as to account for them.

41. In the case of the planets Hipparchus found so small a supply of satisfactory observations by his predecessors, that he made no attempt to construct a system of epicycles or eccentrics to represent their motion, but collected fresh observations for the use of his successors. He also made use of these observations to determine with more accuracy than before the average times of revolution of the several planets.

Fig. 20.—The eclipse method of connecting the distances of the sun and moon.

He also made a satisfactory estimate of the size and distance of the moon, by an eclipse method, the leading idea of which was due to Aristarchus ([§ 32]); by observing the angular diameter of the earth’s shadow (Q R) at the distance of the moon at the time of an eclipse, and comparing it with the known angular diameters of the sun and moon, he obtained, by a simple calculation,[25] a relation between the distances of the sun and moon, which gives either when the other is known. Hipparchus knew that the sun was very much more distant than the moon, and appears to have tried more than one distance, that of Aristarchus among them, and the result obtained in each case shewed that the distance of the moon was nearly 59 times the radius of the earth. Combining the estimates of Hipparchus and Aristarchus, we find the distance of the sun to be about 1,200 times the radius of the earth—a number which remained substantially unchanged for many centuries (chapter VIII., [§ 161]).

42. The appearance in 134 B.C. of a new star in the Scorpion is said to have suggested to Hipparchus the construction of a new catalogue of the stars. He included 1,080 stars, and not only gave the (celestial) latitude and longitude of each star, but divided them according to their brightness into six magnitudes. The constellations to which he refers are nearly identical with those of Eudoxus ([§ 26]), and the list has undergone few alterations up to the present day, except for the addition of a number of southern constellations, invisible in the civilised countries of the ancient world. Hipparchus recorded also a number of cases in which three or more stars appeared to be in line with one another, or, more exactly, lay on the same great circle, his object being to enable subsequent observers to detect more easily possible changes in the positions of the stars. The catalogue remained, with slight alterations, the standard one for nearly sixteen centuries (cf. chapter III., [§ 63]).

The construction of this catalogue led to a notable discovery, the best known probably of all those which Hipparchus made. In comparing his observations of certain stars with those of Timocharis and Aristyllus ([§ 33]), made about a century and a half earlier, Hipparchus found that their distances from the equinoctial points had changed. Thus, in the case of the bright star Spica, the distance from the equinoctial points (measured eastwards) had increased by about 2° in 150 years, or at the rate of 48″ per annum. Further inquiry showed that, though the roughness of the observations produced considerable variations in the case of different stars, there was evidence of a general increase in the longitude of the stars (measured from west to east), unaccompanied by any change of latitude, the amount of the change being estimated by Hipparchus as at least 36″ annually, and possibly more. The agreement between the motions of different stars was enough to justify him in concluding that the change could be accounted for, not as a motion of individual stars, but rather as a change in the position of the equinoctial points, from which longitudes were measured. Now these points are the intersection of the equator and the ecliptic: consequently one or another of these two circles must have changed. But the fact that the latitudes of the stars had undergone no change shewed that the ecliptic must have retained its position and that the change had been caused by a motion of the equator. Again, Hipparchus measured the obliquity of the ecliptic as several of his predecessors had done, and the results indicated no appreciable change. Hipparchus accordingly inferred that the equator was, as it were, slowly sliding backwards (i.e. from east to west), keeping a constant inclination to the ecliptic.

Fig. 21.—The increase of the longitude of a star.

The argument may be made clearer by figures. In fig. 21 let ♈ M denote the ecliptic, ♈ N the equator, S a star as seen by Timocharis, S M a great circle drawn perpendicular to the ecliptic. Then S M is the latitude, ♈ M the longitude. Let S′ denote the star as seen by Hipparchus; then he found, that S′ M was equal to the former S M, but that ♈ M′ was greater than the former ♈ M, or that M′ was slightly to the east of M. This change M M′ being nearly the same for all stars, it was simpler to attribute it to an equal motion in the opposite direction of the point ♈, say from ♈ to ♈′ (fig. 22), i.e. by a motion of the equator from ♈ N to ♈′ N′, its inclination N′ ♈′ M remaining equal to its former amount N ♈ M. The general effect of this change is shewn in a different way in fig. 23, where ♈ ♈′ ♎ ♎′ being the ecliptic, A B C D represents the equator as it appeared in the time of Timocharis, A′ B′ C′ D′ (printed in red) the same in the time of Hipparchus, ♈, ♎ being the earlier positions of the two equinoctial points, and ♈′, ♎′ the later positions.

Fig. 22.—The movement of the equator.

Fig. 33.—The precession of the equinoxes.

The annual motion ♈ ♈′ was, as has been stated, estimated by Hipparchus as being at least 36″ (equivalent to one degree in a century), and probably more. Its true value is considerably more, namely about 50″.

Fig. 24.—The precession of the equinoxes.

An important consequence of the motion of the equator thus discovered is that the sun in its annual journey round the ecliptic, after starting from the equinoctial point, returns to the new position of the equinoctial point a little before returning to its original position with respect to the stars, and the successive equinoxes occur slightly earlier than they otherwise would. From this fact is derived the name precession of the equinoxes, or more shortly, precession, which is applied to the motion that we have been considering. Hence it becomes necessary to recognise, as Hipparchus did, two different kinds of year, the tropical year or period required by the sun to return to the same position with respect to the equinoctial points, and the sidereal year or period of return to the same position with respect to the stars. If ♈ ♈′ denote the motion of the equinoctial point during a tropical year, then the sun after starting from the equinoctial point at ♈ arrives—at the end of a tropical year—at the new equinoctial point at ♈′; but the sidereal year is only complete when the sun has further described the arc ♈′ ♈ and returned to its original starting-point ♈. Hence, taking the modern estimate 50″ of the arc ♈ ♈′, the sun, in the sidereal year, describes an arc of 360°, in the tropical year an arc less by 50″, or 359° 59′ 10″; the lengths of the two years are therefore in this proportion, and the amount by which the sidereal year exceeds the tropical year bears to either the same ratio as 50″ to 360° (or 1,296,000″), and is therefore (365-1∕4 × 50)∕1296000 days of about 20 minutes.

Another way of expressing the amount of the precession is to say that the equinoctial point will describe the complete circuit of the ecliptic and return to the same position after about 26,000 years.

The length of each kind of year was also fixed by Hipparchus with considerable accuracy. That of the tropical year was obtained by comparing the times of solstices and equinoxes observed by earlier astronomers with those observed by himself. He found, for example, by comparison of the date of the summer solstice of 280 B.C., observed by Aristarchus of Samos, with that of the year 135 B.C., that the current estimate of 365-1∕4 days for the length of the year had to be diminished by 1∕300th of a day or about five minutes, an estimate confirmed roughly by other cases. It is interesting to note as an illustration of his scientific method that he discusses with some care the possible error of the observations, and concludes that the time of a solstice may be erroneous to the extent of about 3∕4 day, while that of an equinox may be expected to be within 1∕4 day of the truth. In the illustration given, this would indicate a possible error of 1-1∕2 days in a period of 145 years, or about 15 minutes in a year. Actually his estimate of the length of the year is about six minutes too great, and the error is thus much less than that which he indicated as possible. In the course of this work he considered also the possibility of a change in the length of the year, and arrived at the conclusion that, although his observations were not precise enough to show definitely the invariability of the year, there was no evidence to suppose that it had changed.

The length of the tropical year being thus evaluated at 365 days 5 hours 55 minutes, and the difference between the two kinds of year being given by the observations of precession, the sidereal year was ascertained to exceed 365-1∕4 days by about 10 minutes, a result agreeing almost exactly with modern estimates. That the addition of two erroneous quantities, the length of the tropical year and the amount of the precession, gave such an accurate result was not, as at first sight appears, a mere accident. The chief source of error in each case being the erroneous times of the several equinoxes and solstices employed, the errors in them would tend to produce errors of opposite kinds in the tropical year and in precession, so that they would in part compensate one another. This estimate of the length of the sidereal year was probably also to some extent verified by Hipparchus by comparing eclipse observations made at different epochs.

43. The great improvements which Hipparchus effected in the theories of the sun and moon naturally enabled him to deal more successfully than any of his predecessors with a problem which in all ages has been of the greatest interest, the prediction of eclipses of the sun and moon.

That eclipses of the moon were caused by the passage of the moon through the shadow of the earth thrown by the sun, or, in other words, by the interposition of the earth between the sun and moon, and eclipses of the sun by the passage of the moon between the sun and the observer, was perfectly well known to Greek astronomers in the time of Aristotle ([§ 29]), and probably much earlier (chapter I., [§ 17]), though the knowledge was probably confined to comparatively few people and superstitious terrors were long associated with eclipses.

The chief difficulty in dealing with eclipses depends on the fact that the moon’s path does not coincide with the ecliptic. If the moon’s path on the celestial sphere were identical with the ecliptic, then, once every month, at new moon, the moon (M) would pass exactly between the earth and the sun, and the latter would be eclipsed, and once every month also, at full moon, the moon (M′) would be in the opposite direction to the sun as seen from the earth, and would consequently be obscured by the shadow of the earth.

Fig. 25.—The earth’s shadow.

Fig. 26.—The ecliptic and the moon’s path.

As, however, the moon’s path is inclined to the ecliptic ([§ 40]), the latitudes of the sun and moon may differ by as much as 5°, either when they are in conjunction, i.e. when they have the same longitudes, or when they are in opposition, i.e. when their longitudes differ by 180°, and they will then in either case be too far apart for an eclipse to occur. Whether then at any full or new moon an eclipse will occur or not, will depend primarily on the latitude of the moon at the time, and hence upon her position with respect to the nodes of her orbit ([§ 40]). If conjunction takes place when the sun and moon happen to be near one of the nodes (N), as at S M in fig. 26, the sun and moon will be so close together that an eclipse will occur; but if it occurs at a considerable distance from a node, as at S′ M′, their centres are so far apart that no eclipse takes place.

Now the apparent diameter of either sun or moon is, as we have seen ([§ 32]), about 1∕2°; consequently when their discs just touch, as in fig. 27, the distance between their centres is also about 1∕2°. If then at conjunction the distance between their centres is less than this amount, an eclipse of the sun will take place; if not, there will be no eclipse. It is an easy calculation to determine (in fig. 26) the length of the side N S or N M of the triangle N M S, when S M has this value, and hence to determine the greatest distance from the node at which conjunction can take place if an eclipse is to occur. An eclipse of the moon can be treated in the same way, except that we there have to deal with the moon and the shadow of the earth at the distance of the moon. The apparent size of the shadow is, however, considerably greater than the apparent size of the moon, and an eclipse of the moon takes place if the distance between the centre of the moon and the centre of the shadow is less than about 1°. As before, it is easy to compute the distance of the moon or of the centre of the shadow from the node when opposition occurs, if an eclipse just takes place. As, however, the apparent sizes of both sun and moon, and consequently also that of the earth’s shadow, vary according to the distances of the sun and moon, a variation of which Hipparchus had no accurate knowledge, the calculation becomes really a good deal more complicated than at first sight appears, and was only dealt with imperfectly by him.

Fig. 27.—The sun and moon.

Fig. 28.—Partial eclipse
of the moon.

Fig. 29.—Total eclipse of
the moon.

Eclipses of the moon are divided into partial or total, the former occurring when the moon and the earth’s shadow only overlap partially (as in fig. 28), the latter when the moon’s disc is completely immersed in the shadow (fig. 29). In the same way an eclipse of the sun may be partial or total; but as the sun’s disc may be at times slightly larger than that of the moon, it sometimes happens also that the whole disc of the sun is hidden by the moon, except a narrow ring round the edge (as in fig. 30): such an eclipse is called annular. As the earth’s shadow at the distance of the moon is always larger than the moon’s disc, annular eclipses of the moon cannot occur.

Fig. 30.—Annular eclipse of the sun.

Thus eclipses take place if, and only if, the distance of the moon from a node at the time of conjunction or opposition lies within certain limits approximately known; and the problem of predicting eclipses could be roughly solved by such knowledge of the motion of the moon and of the nodes as Hipparchus possessed. Moreover, the length of the synodic and draconitic months ([§ 40]) being once ascertained, it became merely a matter of arithmetic to compute one or more periods after which eclipses would recur nearly in the same manner. For if any period of time contains an exact number of each kind of month, and if at any time an eclipse occurs, then after the lapse of the period, conjunction (or opposition) again takes place, and the moon is at the same distance as before from the node and the eclipse recurs very much as before. The saros, for example (chapter I., [§ 17]), contained very nearly 223 synodic or 242 draconitic months, differing from either by less than an hour. Hipparchus saw that this period was not completely reliable as a means of predicting eclipses, and showed how to allow for the irregularities in the moon’s and sun’s motion (§§ 39, 40) which were ignored by it, but was unable to deal fully with the difficulties arising from the variations in the apparent diameters of the sun or moon.

An important complication, however, arises in the case of eclipses of the sun, which had been noticed by earlier writers, but which Hipparchus was the first to deal with. Since an eclipse of the moon is an actual darkening of the moon, it is visible to anybody, wherever situated, who can see the moon at all; for example, to possible inhabitants of other planets, just as we on the earth can see precisely similar eclipses of Jupiter’s moons. An eclipse of the sun is, however, merely the screening off of the sun’s light from a particular observer, and the sun may therefore be eclipsed to one observer while to another elsewhere it is visible as usual. Hence in computing an eclipse of the sun it is necessary to take into account the position of the observer on the earth. The simplest way of doing this is to make allowance for the difference of direction of the moon as seen by an observer at the place in question, and by an observer in some standard position on the earth, preferably an ideal observer at the centre of the earth. If, in fig. 31, M denote the moon, C the centre of the earth, A a point on the earth between C and M (at which therefore the moon is overhead), and B any other point on the earth, then observers at C (or A) and B see the moon in slightly different directions, C M, B M, the difference between which is an angle known as the parallax, which is equal to the angle B M C and depends on the distance of the moon, the size of the earth, and the position of the observer at B. In the case of the sun, owing to its great distance, even as estimated by the Greeks, the parallax was in all cases too small to be taken into account, but in the case of the moon the parallax might be as much as 1° and could not be neglected.

Fig. 31.—Parallax.

If then the path of the moon, as seen from the centre of the earth, were known, then the path of the moon as seen from any particular station on the earth could be deduced by allowing for parallax, and the conditions of an eclipse of the sun visible there could be computed accordingly.

From the time of Hipparchus onwards lunar eclipses could easily be predicted to within an hour or two by any ordinary astronomer; solar eclipses probably with less accuracy; and in both cases the prediction of the extent of the eclipse, i.e. of what portion of the sun or moon would be obscured, probably left very much to be desired.

44. The great services rendered to astronomy by Hipparchus can hardly be better expressed than in the words of the great French historian of astronomy, Delambre, who is in general no lenient critic of the work of his predecessors:—

“When we consider all that Hipparchus invented or perfected, and reflect upon the number of his works and the mass of calculations which they imply, we must regard him as one of the most astonishing men of antiquity, and as the greatest of all in the sciences which are not purely speculative, and which require a combination of geometrical knowledge with a knowledge of phenomena, to be observed only by diligent attention and refined instruments.”[26]

45. For nearly three centuries after the death of Hipparchus, the history of astronomy is almost a blank. Several textbooks written during this period are extant, shewing the gradual popularisation of his great discoveries. Among the few things of interest in these books may be noticed a statement that the stars are not necessarily on the surface of a sphere, but may be at different distances from us, which, however, there are no means of estimating; a conjecture that the sun and stars are so far off that the earth would be a mere point seen from the sun and invisible from the stars; and a re-statement of an old opinion traditionally attributed to the Egyptians (whether of the Alexandrine period or earlier is uncertain), that Venus and Mercury revolve round the sun. It seems also that in this period some attempts were made to explain the planetary motions by means of epicycles, but whether these attempts marked any advance on what had been done by Apollonius and Hipparchus is uncertain.

It is interesting also to find in Pliny (A.D. 23-79) the well-known modern argument for the spherical form of the earth, that when a ship sails away the masts, etc., remain visible after the hull has disappeared from view.

A new measurement of the circumference of the earth by Posidonius (born about the end of Hipparchus’s life) may also be noticed; he adopted a method similar to that of Eratosthenes ([§ 36]), and arrived at two different results. The later estimate, to which he seems to have attached most weight, was 180,000 stadia, a result which was about as much below the truth as that of Eratosthenes was above it.

46. The last great name in Greek astronomy is that of Claudius Ptolemaeus, commonly known as Ptolemy, of whose life nothing is known except that he lived in Alexandria about the middle of the 2nd century A.D. His reputation rests chiefly on his great astronomical treatise, known as the Almagest,[27] which is the source from which by far the greater part of our knowledge of Greek astronomy is derived, and which may be fairly regarded as the astronomical Bible of the Middle Ages. Several other minor astronomical and astrological treatises are attributed to him, some of which are probably not genuine, and he was also the author of an important work on geography, and possibly of a treatise on Optics, which is, however, not certainly authentic and maybe of Arabian origin. The Optics discusses, among other topics, the refraction or bending of light, by the atmosphere on the earth: it is pointed out that the light of a star or other heavenly body S, on entering our atmosphere (at A) and on penetrating to the lower and denser portions of it, must be gradually bent or refracted, the result being that the star appears to the observer at B nearer to the zenith Z than it actually is, i.e. the light appears to come from S′ instead of from S; it is shewn further that this effect must be greater for bodies near the horizon than for those near the zenith, the light from the former travelling through a greater extent of atmosphere; and these results are shewn to account for certain observed deviations in the daily paths of the stars, by which they appear unduly raised up when near the horizon. Refraction also explains the well-known flattened appearance of the sun or moon when rising or setting, the lower edge being raised by refraction more than the upper, so that a contraction of the vertical diameter results, the horizontal contraction being much less.[28]

Fig. 32.—Refraction by the atmosphere.

47. The Almagest is avowedly based largely on the work of earlier astronomers, and in particular on that of Hipparchus, for whom Ptolemy continually expresses the greatest admiration and respect. Many of its contents have therefore already been dealt with by anticipation, and need not be discussed again in detail. The book plays, however, such an important part in astronomical history, that it may be worth while to give a short outline of its contents, in addition to dealing more fully with the parts in which Ptolemy made important advances.

The Almagest consists altogether of 13 books. The first two deal with the simpler observed facts, such as the daily motion of the celestial sphere, and the general motions of the sun, moon, and planets, and also with a number of topics connected with the celestial sphere and its motion, such as the length of the day and the times of rising and setting of the stars in different zones of the earth; there are also given the solutions of some important mathematical problems,[29] and a mathematical table[30] of considerable accuracy and extent. But the most interesting parts of these introductory books deal with what may be called the postulates of Ptolemy’s astronomy (Book I., chap. ii.). The first of these is that the earth is spherical; Ptolemy discusses and rejects various alternative views, and gives several of the usual positive arguments for a spherical form, omitting, however, one of the strongest, the eclipse argument found in Aristotle ([§ 29]), possibly as being too recondite and difficult, and adding the argument based on the increase in the area of the earth visible when the observer ascends to a height. In his geography he accepts the estimate given by Posidonius that the circumference of the earth is 180,000 stadia. The other postulates which he enunciates and for which he argues are, that the heavens are spherical and revolve like a sphere; that the earth is in the centre of the heavens, and is merely a point in comparison with the distance of the fixed stars, and that it has no motion. The position of these postulates in the treatise and Ptolemy’s general method of procedure suggest that he was treating them, not so much as important results to be established by the best possible evidence, but rather as assumptions, more probable than any others with which the author was acquainted, on which to base mathematical calculations which should explain observed phenomena.[31] His attitude is thus essentially different from that either of the early Greeks, such as Pythagoras, or of the controversialists of the 16th and early 17th centuries, such as Galilei (chapter VI.), for whom the truth or falsity of postulates analogous to those of Ptolemy was of the very essence of astronomy and was among the final objects of inquiry. The arguments which Ptolemy produces in support of his postulates, arguments which were probably the commonplaces of the astronomical writing of his time, appear to us, except in the case of the shape of the earth, loose and of no great value. The other postulates were, in fact, scarcely, capable of either proof or disproof with the evidence which Ptolemy had at command. His argument in favour of the immobility of the earth is interesting, as it shews his clear perception that the more obvious appearances can be explained equally well by a motion of the stars or by a motion of the earth; he concludes, however, that it is easier to attribute motion to bodies like the stars which seem to be of the nature of fire than to the solid earth, and points out also the difficulty of conceiving the earth to have a rapid motion of which we are entirely unconscious. He does not, however, discuss seriously the possibility that the earth or even Venus and Mercury may revolve round the sun.

The third book of the Almagest deals with the length of the year and theory of the sun, but adds nothing of importance to the work of Hipparchus.

48. The fourth book of the Almagest, which treats of the length of the month and of the theory of the moon, contains one of Ptolemy’s most important discoveries. We have seen that, apart from the motion of the moon’s orbit as a whole, and the revolution of the line of apses, the chief irregularity or inequality was the so-called equation of the centre (§§ 39, 40), represented fairly accurately by means of an eccentric, and depending only on the position of the moon with respect to its apogee. Ptolemy, however, discovered, what Hipparchus only suspected, that there was a further inequality in the moon’s motion—to which the name evection was afterwards given—and that this depended partly on its position with respect to the sun. Ptolemy compared the observed positions of the moon with those calculated by Hipparchus in various positions relative to the sun and apogee, and found that, although there was a satisfactory agreement at new and full moon, there was a considerable error when the moon was half-full, provided it was also not very near perigee or apogee. Hipparchus based his theory of the moon chiefly on observations of eclipses, i.e. on observations taken necessarily at full or new moon ([§ 43]), and Ptolemy’s discovery is due to the fact that he checked Hipparchus’s theory by observations taken at other times. To represent this new inequality, it was found necessary to use an epicycle and a deferent, the latter being itself a moving eccentric circle, the centre of which revolved round the earth. To account, to some extent, for certain remaining discrepancies between theory and observation, which occurred neither at new and full moon, nor at the quadratures (half-moon), Ptolemy introduced further a certain small to-and-fro oscillation of the epicycle, an oscillation to which he gave the name of prosneusis.[32] Ptolemy thus succeeded in fitting his theory on to his observations so well that the error seldom exceeded 10′, a small quantity in the astronomy of the time, and on the basis of this construction he calculated tables from which the position of the moon at any required time could be easily deduced.

One of the inherent weaknesses of the system of epicycles occurred in this theory in an aggravated form. It has already been noticed in connection with the theory of the sun ([§ 39]), that the eccentric or epicycle produced an erroneous variation in the distance of the sun, which was, however, imperceptible in Greek times. Ptolemy’s system, however, represented the moon as being sometimes nearly twice as far off as at others, and consequently the apparent diameter ought at some times to have been not much more than half as great as at others—a conclusion obviously inconsistent with observation. It seems probable that Ptolemy noticed this difficulty, but was unable to deal with it; it is at any rate a significant fact that when he is dealing with eclipses, for which the apparent diameters of the sun and moon are of importance, he entirely rejects the estimates that might have been obtained from his lunar theory and appeals to direct observation (cf. also § 51, note).

49. The fifth book of the Almagest contains an account of the construction and use of Ptolemy’s chief astronomical instrument, a combination of graduated circles known as the astrolabe.[33]

Then follows a detailed discussion of the moon’s parallax ([§ 43]), and of the distances of the sun and moon. Ptolemy obtains the distance of the moon by a parallax method which is substantially identical with that still in use. If we know the direction of the line C M (fig. 33) joining the centres of the earth and moon, or the direction of the moon as seen by an observer at A; and also the direction of the line B M, that is the direction of the moon as seen by an observer at B, then the angles of the triangle C B M are known, and the ratio of the sides C B, C M is known. Ptolemy obtained the two directions required by means of observations of the moon, and hence found that C M was 59 times C B, or that the distance of the moon was equal to 59 times the radius of the earth. He then uses Hipparchus’s eclipse method to deduce the distance of the sun from that of the moon thus ascertained, and finds the distance of the sun to be 1,210 times the radius of the earth. This number, which is substantially the same as that obtained by Hipparchus ([§ 41]), is, however, only about 1∕20 of the true number, as indicated by modern work (chapter XIII., [§ 284]).

Fig. 33.—Parallax.

The sixth book is devoted to eclipses, and contains no substantial additions to the work of Hipparchus.

50. The seventh and eighth books contain a catalogue of stars, and a discussion of precession ([§ 42]). The catalogue, which contains 1,028 stars (three of which are duplicates), appears to be nearly identical with that of Hipparchus, It contains none of the stars which were visible to Ptolemy at Alexandria, but not to Hipparchus at Rhodes. Moreover, Ptolemy professes to deduce from a comparison of his observations with those of Hipparchus and others the (erroneous) value 36″ for the precession, which Hipparchus had given as the least possible value, and which Ptolemy regards as his final estimate. But an examination of the positions assigned to the stars in Ptolemy’s catalogue agrees better with their actual positions in the time of Hipparchus, corrected for precession at the supposed rate of 36″ annually, than with their actual positions in Ptolemy’s time. It is therefore probable that the catalogue as a whole does not represent genuine observations made by Ptolemy, but is substantially the catalogue of Hipparchus corrected for precession and only occasionally modified by new observations by Ptolemy or others.

51. The last five books deal with the theory of the planets, the most important of Ptolemy’s original contributions to astronomy. The problem of giving a satisfactory explanation of the motions of the planets was, on account of their far greater irregularity, a much more difficult one than the corresponding problem for the sun or moon. The motions of the latter are so nearly uniform that their irregularities may usually be regarded as of the nature of small corrections, and for many purposes may be ignored. The planets, however, as we have seen (chapter I., [§ 14]), do not even always move from west to east, but stop at intervals, move in the reverse direction for a time, stop again, and then move again in the original direction. It was probably recognised in early times, at latest by Eudoxus ([§ 26]), that in the case of three of the planets, Mars, Jupiter, and Saturn, these motions could be represented roughly by supposing each planet to oscillate to and fro on each side of a fictitious planet, moving uniformly round the celestial sphere in or near the ecliptic, and that Venus and Mercury could similarly be regarded as oscillating to and fro on each side of the sun. These rough motions could easily be interpreted by means of revolving spheres or of epicycles, as was done by Eudoxus and probably again with more precision by Apollonius. In the case of Jupiter, for example, we may regard the planet as moving on an epicycle, the centre of which, j, describes uniformly a deferent, the centre of which is the earth. The planet will then as seen from the earth appear alternately to the east (as at J₁) and to the west (as at J₂) of the fictitious planet j; and the extent of the oscillation on each side, and the interval between successive appearances in the extreme positions (J₁, J₂) on either side, can be made right by choosing appropriately the size and rapidity of motion of the epicycle. It is moreover evident that with this arrangement the apparent motion of Jupiter will vary considerably, as the two motions—that on the epicycle and that of the centre of the epicycle on the deferent—are sometimes in the same direction, so as to increase one another’s effect, and at other times in opposite directions. Thus, when Jupiter is most distant from the earth, that is at J₃, the motion is most rapid, at J₁ and J₂ the motion as seen from the earth is nearly the same as that of j; while at J₄ the two motions are in opposite directions, and the size and motion of the epicycle having been chosen in the way indicated above, it is found in fact that the motion of the planet in the epicycle is the greater of the two motions, and that therefore the planet when in this position appears to be moving from east to west (from left to right in the figure), as is actually the case. As then at J₁ and J₂ the planet appears to be moving from west to east, and at J₄ in the opposite direction, and sudden changes of motion do not occur in astronomy, there must be a position between J₁ and J₄, and another between J₄ and J₂, at which the planet is just reversing its direction of motion, and therefore appears for the instant at rest. We thus arrive at an explanation of the stationary points (chapter I., [§ 14]). An exactly similar scheme explains roughly the motion of Mercury and Venus, except that the centre of the epicycle must always be in the direction of the sun.

Fig. 34.—Jupiter’s epicycle and deferent.

Hipparchus, as we have seen ([§ 41]), found the current representations of the planetary motions inaccurate, and collected a number of fresh observations. These, with fresh observations of his own, Ptolemy now employed in order to construct an improved planetary system.

As in the case of the moon, he used as deferent an eccentric circle (centre C), but instead of making the centre j of the epicycle move uniformly in the deferent, he introduced a new point called an equant (E′), situated at the same distance from the centre of the deferent as the earth but on the opposite side, and regulated the motion of j by the condition that the apparent motion as seen from the equant should be uniform; in other words, the angle A E′ j was made to increase uniformly. In the case of Mercury (the motions of which have been found troublesome by astronomers of all periods), the relation of the equant to the centre of the epicycle was different, and the latter was made to move in a small circle. The deviations of the planets from the ecliptic (chapter I., [§§ 13], 14) were accounted for by tilting up the planes of the several deferents and epicycles so that they were inclined to the ecliptic at various small angles.

Fig. 35.—The equant.