A Short History of Astronomy (1898)/Chapter 2

by Arthur Berry

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 291/2 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 3651/4 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 3651/4 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^[1] 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 (ἁντιχθών), 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.^[2]

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.^[3] 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.^[4] 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.

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

Fig.—8. The phases of the moon.

(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
Fig. 9.—The phases of the moon. 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.^[5]

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
Fig. 12.—The curvature of the earth. 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.

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 occupations 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.^[6]

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.

32. Among the earlier members of the Alexandrine school were Aristarchus of Samos, Aristyllus, and Timocharis, three nearly contemporary astronomers belonging

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

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 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), he also found that the diameter of the moon was about 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.

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,

Fig. 14.—The equator and the ecliptic.

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,^[7] the zenith,^[8] 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.^[9] 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 obhque, 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.

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
Fig. 15.—The equator, the horizon, and the meridian. 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 231/2°. Consequently at places at which the height of the pole is less than 231/2° 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 231/2°, its distance from the pole is 661/2°; therefore in regions so far north that the height p k of the north pole is more than 661/2°, 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.

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

Fig. 16.—The measurement of the earth.

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^[10] 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,^[11] 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.^[12]

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.

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

Fig. 17.—The eccentric.

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

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

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.

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 921/2 days, the length of the year being 3651/4 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.

Hipparchus saw that the motion of the sun could equally well be represented by the other device suggested
Fig. 19.—The epicycle and the deferent. 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,^[13] 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 291/2 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,^[14] 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 inchned 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.

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
Fig. 20.—The eclipse method of connecting the distances of the sun and moon.it with the known angular diameters of the sun and moon, he obtained, by a simple calculation,^[15] 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

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

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.

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'

Fig. 22.—The movement of the equator.

was slightly to the east of m. This change m m' being nearly the same for all stars, it was simpler to attribute it Fig. 23.—The precession of the equinoxes. 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. 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".

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

Fig. 24.—The precession of the equinoxes.

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 3651/4 X 50/1296000 days or 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 3651/4 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 11/2 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 3651/4 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.

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

Fig. 25.—The earth's shadow.

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

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

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,

Fig. 27.—The sun and moon.

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

Fig. 28.—Partial eclipse of Fig. 29.—Total eclipse of
the moon. the moon.

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.

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

Fig. 31.—Parallax.

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.

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."^[16]

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^[17], 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

Fig. 32.—Refraction by the atmosphere.

refraction more than the upper, so that a contraction of the vertical diameter results, the horizontal contraction being much less.^[18]

47. The Almagest is avowedly based largely on the world 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,^[19] and a mathematical table^[20] 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.^[21] 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.^[22] 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.^[23]

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

Fig. 33.—Parallax.

about 1/20 of the true number, as indicated by modern work (chapter xiii., § 284).

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 astroriomy. 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

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

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.

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

Fig. 35.—The equant.

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.

By means of a system of this kind, worked out with great care, and evidently at the cost of enormous labour, Ptolemy was able to represent with very fair exactitude the motions of the planets, as given by the observations in his possession.

It has been pointed out by modern critics, as well as by some mediaeval writers, that the use of the equant (which played also a small part in Ptolemy's lunar theory) was a violation of the principle of employing only uniform circular motions, on which the systems of Hipparchus and Ptolemy were supposed to be based, and that Ptolemy himself appeared unconscious of his inconsistency. It may, however, fairly be doubted whether Hipparchus or Ptolemy ever had an abstract belief in the exclusive virtue of such motions, except as a convenient and easily intelligible way of representing certain more complicated motions, and it is difficult to conceive that Hipparchus would have scrupled any more than his great follower, in using an equant to represent an irregular motion, if he had found that the motion was thereby represented with accuracy. The criticism appears to me in fact to be an anachronism. The earlier Greeks, whose astronomy was speculative rather than scientific, and again many astronomers of the Middle Ages, felt that it was on a priori grounds necessary to represent the "perfection" of the heavenly motions by the most "perfect" or regular of geometrical schemes; so that it is highly probable that Pythagoras or Plato, or even Aristotle, would have objected, and certain that the astronomers of the 14th and 15th centuries ought to have objected (as some of them actually did), to this innovation of Ptolemy's. But there seems no good reason for attributing this a priori attitude to the later scientific Greek astronomers (cf. also §§ 38, 47).^[24]

It will be noticed that nothing has been said as to the actual distances of the planets, and in fact the apparent motions are unaffected by any alteration in the scale on which deferent and epicycle are constructed, provided that both are altered proportionally. Ptolemy expressly states that he had no means of estimating numerically the distances of the planets, or even of knowing the order of the distance of the several planets. He followed tradition in accepting conjecturally rapidity of motion as a test of nearness, and placed Mars, Jupiter, Saturn (which perform the circuit of the celestial sphere in about 2, 12, and 29 years respectively) beyond the sun in that order. As Venus and Mercury accompany the sun, and may therefore be regarded as on the average performing their revolutions in a year, the test to some extent failed in their case, but Ptolemy again accepted the opinion of the "ancient mathematicians" (i.e probably the Chaldaeans) that Mercury and Venus lie between the sun and moon. Mercury being the nearer to us. (Cf. chapter i., § 15.)

52. There has been much difference of opinion among astronomers as to the merits of Ptolemy. Throughout the Middle Ages his authority was regarded as almost final on astronomical matters, except where it was outweighed by the even greater authority assigned to Aristotle. Modern criticism has made clear, a fact which indeed he never conceals, that his work is to a large extent based on that of Hipparchus; and that his observations, if not actually fictitious, were at any rate in most cases poor. On the other hand his work shews clearly that he was an accomplished and original mathematician.^[25] The most important of his positive contributions to astronomy were the discovery of evection and his planetary theory, but we ought probably to rank above these, important as they are, the services which he rendered by preserving and developing the great ideas of Hipparchus—ideas which the other astronomers of the time were probably incapable of appreciating, and which might easily have been lost to us if they had not been embodied in the Almagest.

53. The history of Greek astronomy practically ceases with Ptolemy. The practice of observation died out so completely that only eight observations are known to have been made during the eight and a half centuries which separate him from Albategnius (chapter iii., § 59). The only Greek writers after Ptolemy's time are compilers and commentators, such as Theon (fl. A.D. 365), to none of whom original ideas of any importance can be attributed. The murder of his daughter Hypatia (A.D. 415), herself also a writer on astronomy, marks an epoch in the decay of the Alexandrine school; and the end came in A.D. 640, when Alexandria was captured by the Arabs.^[26]

54. It remains to attempt to estimate briefly the value of the contributions to astronomy made by the Greeks and of their method of investigation. It is obviously unreasonable to expect to find a brief formula which will characterise the scientific attitude of a series of astronomers whose lives extend over a period of eight centuries; and it is futile to explain the inferiority of Greek astronomy to our own on some such ground as that they had not discovered the method of induction, that they were not careful enough to obtain facts, or even that their ideas were not clear. In habits of thought and scientific aims the contrast between Pythagoras and Hipparchus is probably greater than that between Hipparchus on the one hand and Coppernicus or even Newton on the other, while it is not unfair to say that the fanciful ideas which pervade the work of even so great a discoverer as Kepler (chapter vii., §§ 144, § 151) place his scientific method in some respects behind that of his great Greek predecessor.

The Greeks inherited from their predecessors a number of observations, many of them executed with considerable accuracy, which were nearly sufficient for the requirements of practical life, but in the matter of astronomical theory and speculation, in which their best thinkers were very much more interested than in the detailed facts, they received virtually a blank sheet on which they had to write (at first with indifferent success) their speculative ideas. A considerable interval of time was obviously necessary to bridge over the gulf separating such data as the eclipse observations of the Chaldaeans from such ideas as the harmonical spheres of Pythagoras; and the necessary theoretical structure could not be erected without the use of mathematical methods which had gradually to be invented. That the Greeks, particularly in early times, paid little attention to making observations, is true enough, but it may fairly be doubted whether the collection of fresh material for observations would really have carried astronomy much beyond the point reached by the Chaldaean observers. When once speculative ideas, made

definite by the aid of geometry, had been sufficiently developed to be capable of comparison with observation, rapid progress was made. The Greek astronomers of the scientific period, such as Aristarchus, Eratosthenes, and above all Hipparchus, appear moreover to have followed in their researches the method which has always been fruitful in physical science—namely, to frame provisional hypotheses, to deduce their mathematical consequences, and to compare these with the results of observation. There are few better illustrations of genuine scientific caution than the way in which Hipparchus, having tested the planetary theories handed down to him and having discovered their insufficiency, deliberately abstained from building up a new theory on data which he knew to be insufficient, and patiently collected fresh material, never to be used by himself, that some future astronomer might thereby be able to arrive at an improved theory.

Of positive additions to our astronomical knowledge made by the Greeks the most striking in some ways is the discovery of the approximately spherical form of the earth, a result which later work has only slightly modified. But their explanation of the chief motions of the solar system and their resolution of them into a comparatively small number of simpler motions was, in reality, a far more important contribution, though the Greek epicyclic scheme has been so remodelled, that at first sight it is difficult to recognise the relation between it and our modern views. The subsequent history will, however, show how completely each stage in the progress of astronomical science has depended on those that preceded.

When we study the great conflict in the time of Coppernicus between the ancient and modern ideas, our sympathies naturally go out towards those who supported the latter, which are now known to be more accurate, and we are apt to forget that those who then spoke in the name of the ancient astronomy and quoted Ptolemy were indeed believers in the doctrines which they had derived from the Greeks, but that their methods of thought, their frequent refusal to face facts, and their appeals to authority, were all entirely foreign to the spirit of the great men whose disciples they believed themselves to be.

1. We have little definite knowledge of his life. He was born in the earlier part of the 6th century B.C., and died at the end of the same century or beginning of the next.
2. Theophrastus was born about half a century, Plutarch nearly five centuries, later than Plato.
3. Republic, VII. 529, 530.
4. Confused, because the mechanical knowledge of the time was quite unequal to giving any explanation of the way in which these spheres acted on one another.
5. I have introduced here the familiar explanation of the phases of the moon, and the argument based on it for the spherical shape of the moon, because, although probably known before Aristotle, there is, as far as I know, no clear and definite statement of the matter in any earlier writer, and after his time it becomes an accepted part of Greek elementary astronomy. It may be noticed that the explanation is unaffected either by the question of the rotation of the earth or by that of its motion round the sun.
6. See, for example, the account of Galilei's controversies, in chapter vi.
7. The poles of a great circle on a sphere are the ends of a diameter perpendicular to the plane of the great circle. Every point on the great circle is at the same distance, 90°, from each pole.
8. The word "zenith" is Arabic, not Greek: cf. chapter iii., § 64.
9. Most of these names are not Greek, but of later origin.
10. That of M. Paul Tannery: Recherches sur l'Histoire de l'Astronomie Ancienne, chap. v.
11. Trigonometry.
12. The process may be worth illustrating by means of a simpler problem. A heavy body, falling freely under gravity, is found (the resistance of the air being allowed for) to fall about 16 feet in 1 second, 64 feet in 2 seconds, 144 feet in 3 seconds, 256 feet in 4 seconds, 400 feet in 5 seconds, and so on. This series of figures carried on as far as may be required would satisfy practical requirements, supplemented if desired by the corresponding figures for fractions of seconds; but the mathematician represents the same facts more simply and in a way more satisfactory to the mind by the formula s = 16 t², where s denotes the number of feet fallen, and t the number of seconds. By giving t any assigned value, the corresponding space fallen through is at once obtained. Similarly the motion of the sun can be represented approximately by the more complicated formula l = nt + 2 e sin nt, where l is the distance from a fixed point in the orbit, t the time, and n, e certain numerical quantities.
13. At the present time there is still a small discrepancy between the observed and calculated places of the moon. See chapter xiii., § 290.
14. The name is interesting as a remnant of a very early superstition. Eclipses, which always occur near the nodes, were at one time supposed to be caused by a dragon which devoured the sun or moon. The symbols ☊ ☋ still used to denote the two nodes are supposed to represent the head and tail of the dragon.
15. In the figure, which is taken from the De Revolutionibus of Coppernicus (chapter iv., § 85), let d, k, m represent respectively the centres of the sun, earth, and moon, at the time of an eclipse of the moon, and let s q g, s r e denote the boundaries of the shadow-cone cast by the earth; then q r, drawn at right angles to the axis of the cone, is the breadth of the shadow at the distance of the moon. We have then at once from similar triangles

    g k—q m : a d—g k :: m k : k d.

    Hence if k d = n . m k and ∴ also a d = n. (radius of moon), n being 19 according to Aristarchus,

    g k—q m : n. (radius of moon)—g k :: I : n

    n. (radius of moon)—g k = n g k—n q m

    ∴ radius of moon + radius of shadow

    = (i + i/n) (radius of earth).

    By observation the angular radius of the shadow was found to be about 40' and that of the moon to be 15', so that

    radius of shadow = 8/3 radius of moon;

    ∴ radius of moon = 3/11 (I + I/n) (radius of earth).

    But the angular radius of the moon being 15', its distance is necessarily about 220 times its radius,

    and ∴ distance of the moon

    = 60 (I + I/n) (radius of the earth),

    which is roughly Hipparchus's result, if n be any fairly large number.

16. Histoire de l'Astronomie Ancienne, Vol. I., p. 185.
17. The chief MS. bears the title μεγάλη σύνταξις, or great composition though the author refers to his book elsewhere as μαθηματιὴ σύνταξις (mathematical composition). The Arabian translators, either through admiration or carelessness, converted μεγάλη, great, into μεγἰστη, greatest, and hence it became known by the Arabs as Al Magisti, whence the Latin Almagestum and our Almagest.
18. The better known apparent enlargement of the sun or moon when rising or setting has nothing to do with refraction. It is an optical illusion not very satisfactorily explained, but probably due to the lesser brilliancy of the sun at the time.
19. In spherical trigonometry.
20. A table of chords (or double sines of half-angles) for every 1/2° from 0° to 180°.
21. His procedure may be compared with that of a political economist of the school of Ricardo, who, in order to establish some rough explanation of economic phenomena, starts with certain simple assumptions as to human nature, which at any rate are more plausible than any other equally simple set, and deduces from them a number of abstract conclusions, the applicability of which to real life has to be considered in individual cases. But the perfunctory discussion which such a writer gives of the qualities of the "economic man" cannot of course be regarded as his deliberate and final estimate of human nature.
22. The equation of the centre and the evection may be expressed trigonometrically by two terms in the expression for the moon's longitude, a sin θ + b sin (2Φ — θ), where a, b are two numerical quantities, in round numbers 6° and 1°, θ is the angular distance of the moon from perigee, and Φ is the angular distance from the sun. At conjunction and opposition Φ is 0° or 180°, and the two terms reduce to (a—b) sin θ. This would be the form in which the equation of the centre would have presented itself to Hipparchus. Ptolemy's correction is therefore equivalent to adding on

    b [sin θ + sin (2 Φ — θ)], or 2 b sin Φ cos (Φ — θ),

    which vanishes at conjunction or opposition, but reduces at the quadratures to 2 b sin θ, which again vanishes if the moon is at apogee or perigee (θ = 0° or 180°), but has its greatest value half-way between, when θ = 90°. Ptolemy's construction gave rise also to a still smaller term of the type,

    c sin 2 Φ [cos (2 Φ + θ) + 2 cos (2 Φ — θ)],

    which, it will be observed, vanishes at quadratures as well as at conjunction and opposition.

23. Here, as elsewhere, I have given no detailed account of astronomical instruments, believing such descriptions to be in general neither interesting nor intelligible to those who have not the actual instruments before them, and to be of little use to those who have.
24. The advantage derived from the use of the equant can be made clearer by a mathematical comparison with the elliptic motion introduced by Kepler. In elliptic motion the angular motion and distance are represented approximately by the formulæ nt + 2e sin nt, a (1—e cos nt) respectively; the corresponding formulæ given by the use of the simple eccentric are nt + e' sin nt, a {1—e' cos nt). To make the angular motions agree we must therefore take e' = 2e, but to make the distances agree we must take c' = c; the two conditions are therefore inconsistent. But by the introduction of an equant the formulæ become nt + 2e' sin nt, a (1—e' cos nt), and both agree if we take e' = e. Ptolemy's lunar theory could have been nearly freed from the serious difficulty already noticed (§ 48) if he had used an equant to represent the chief inequality of the moon; and his planetary theory would have been made accurate to the first order of small quantities by the use of an equant both for the deferent and the epicycle.
25. De Morgan classes him as a geometer with Archimedes, Euclid, and Apollonius, the three great geometers of antiquity.
26. The legend that the books in the library served for six months as fuel for the furnaces of the public baths is rejected by Gibbon and others. One good reason for not accepting it is that by this time there were probably very few books left to burn.