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

Emerson.

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

“Yet you will not mark your days

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

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

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

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

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

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

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

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

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

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

23. While their oriental predecessors had confined themselves chiefly to astronomical observations, the earlier Greek philosophers appear to have made next to no observations of importance, and to have been far more interested in inquiring into causes of phenomena. Thales, the founder of the Ionian school, was credited by later writers with the introduction of Egyptian astronomy into Greece, at about the end of the 7th century B.C.; but both Thales and the majority of his immediate successors appear to have added little or nothing to astronomy, except some rather vague speculations as to the form of the earth and its relation to the rest of the world. On the other hand, some real progress seems to have been made by Pythagoras11 and his followers. Pythagoras taught that the earth, in common with the heavenly bodies, is a sphere, and that it rests without requiring support in the middle of the universe. Whether he had any real evidence in support of these views is doubtful, but it is at any rate a reasonable conjecture that he knew the moon to be bright because the sun shines on it, and the phases to be caused by the greater or less amount of the illuminated half turned towards us; and the curved form of the boundary between the bright and dark portions of the moon was correctly interpreted by him as evidence that the moon was spherical, and not a flat disc, as it appears at first sight. Analogy would then probably suggest that the earth also was spherical. However this may be, the belief in the spherical form of the earth never disappeared from Greek thought, and was in later times an established part of Greek systems, whence it has been handed down, almost unchanged, to modern times. This belief is thus 2,000 years older than the belief in the rotation of the earth and its revolution round the sun (chapter IV.), doctrines which we are sometimes inclined to couple with it as the foundations of modern astronomy.

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

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

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

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

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

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

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

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

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

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

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

Fig. 8.—The phases of the moon.

Fig. 9.—The phases of the moon.

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

Fig. 10.—The phases of the moon.

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

Fig. 11.—The phases of the moon.

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

Fig. 12.—The curvature of the earth.

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

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

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

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

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

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

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

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

Fig. 14.—The equator and the ecliptic.

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

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

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

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

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

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

Fig. 16.—The measurement of the earth.

36. To Eratosthenes (276 B.C. to 195 or 196 B.C.), another member of the Alexandrine school, we owe one of the first scientific estimates of the size of the earth. He found that at the summer solstice the angular distance of the sun from the zenith at Alexandria was at midday 1∕50th of a complete circumference, or about 7°, whereas at Syene in Upper Egypt the sun was known to be vertical at the same time. From this he inferred, assuming Syene to be due south of Alexandria, that the distance from Syene to Alexandria was also 1∕50th of the circumference of the earth. Thus if in the figure S denotes the sun, A and B Alexandria and Syene respectively, C the centre of the earth, and A Z the direction of the zenith at Alexandria, Eratosthenes estimated the angle S A Z, which, owing to the great distance of S, is sensibly equal to the angle S C A, to be 7°, and hence inferred that the arc A B was to the circumference of the earth in the proportion of 7° to 360° or 1 to 50. The distance between Alexandria and Syene being known to be 5,000 stadia, Eratosthenes thus arrived at 250,000 stadia as an estimate of the circumference of the earth, a number altered into 252,000 in order to give an exact number of stadia (700) for each degree on the earth. It is evident that the data employed were rough, though the principle of the method is perfectly sound; it is, however, difficult to estimate the correctness of the result on account of the uncertainty as to the value of the stadium used. If, as seems probable, it was the common Olympic stadium, the result is about 20 per cent. too great, but according to another interpretation20 the result is less than 1 per cent. in error (cf. chapter X., §221).

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

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

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

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

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

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

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

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

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

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

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