Chapter IX: Introduction (4)

He was, however, very famous as a mathematician, and also as an observer. Ptolemy quotes his determination of the summer solstice of the year b.c. 281, and this tells us the date at which he flourished. He was renowned for a very ingenious method by which he tried to discover how much further from us the sun is than the moon. When the moon is half full the angle sun-moon-earth is a right angle, and if the angle sun-earth-moon be measured, by pointing the astrolabe first to sun and then to moon, the third angle, at the sun, may be computed, and then the ratio sun-earth to moon-earth will be known. The method is perfect theoretically, and if the sun were comparatively near, say about ten times the moon’s distance, it would be practicable; but the distance is really so much greater that the angle at the sun almost vanishes, and a very small error in estimating it causes an error equal to many millions of miles in the result. It is also impossible to determine from looking at the moon the exact time when the division between light and dark is a straight line. Aristarchus made the angle at Earth 87° instead of 89° 50′, and this gave the sun a distance of about 19 times as far as the moon, instead of 400 times, which is the true value.

The foolish fad (as they thought it) about Earth’s motion, held by this otherwise great man, is described quite clearly by the two writers above-mentioned. He suggested that the stars might be immoveable, and Earth be turning on her axis at the same time that she moves in a circle round the sun. Moreover realizing all that this implies with regard to the immense distance of the stars, he said that the circle in which Earth revolves round the sun, compared with the sphere of the stars, is as the centre of a sphere compared with its circumference. That is, not only Earth, but Earth’s whole orbit, shrinks to a point when compared with the infinite distance of the stars.

We have unfortunately absolutely no information as to the way in which Aristarchus was led to these remarkable truths, and can only make conjectures from what we know of his times. Evidently the Central Fire theory was a suggestive step, and so was another theory which was afloat about this time, and has been called the “Egyptian system” on the authority of Macrobius in his commentary on Cicero’s _Dream of Scipio_. No one really knows where it arose, but it is ascribed with much probability to Heracleides of Pontus. According to this, the two planets Mercury and Venus circled round the sun, and all three together circled round Earth, which still remained the centre of the Universe and of the other celestial motions. It was an idea which might have occurred to any unprejudiced observer, since the oscillations of Mercury and Venus from side to side of the sun are more striking than their movements through the stars. They never go far from him in the sky, like the other planets, and seem to belong to him.

Further, the clear understanding of the periodic motions of Mars, Jupiter, and Saturn, and the accurate observation which had been introduced by Eudoxus, must have revealed the fact that the loops in the orbits of these planets are connected with the apparent movement of the sun; and the varying brightness (especially noticeable with Mars) is inconsistent with the assumption of unvarying distance from Earth.

The great importance of the sun, above all the other planets, had of course always been recognised, and it is possible that Aristarchus was struck by an unconscious suggestion in Aristotle’s advice to the Pythagoreans to examine the nature and place of the sun, “that other centre of the Universe,” for that was also a point of origin, and a noble one.[43] His own work, too, would lead him to attribute a commanding position to the sun, for as it was nineteen times as distant as the moon, it must also be nineteen times as large (since they appear equal), so he must have felt certain that it was a very great size, probably much the largest of all bodies in the Universe.

------------------------------------------------------------------------ [43] See page 101. ------------------------------------------------------------------------

It was the grandest and truest of all the Greek astronomical theories. But it was not accepted, it was hardly even discussed, and we can scarcely be surprised at this. Such an improbable theory needed many more convincing proofs than Aristarchus could bring forward. Observation, calculation, comparison of theory with facts, this was what was needed before safe ground could be won for belief, and in this Eudoxus was a true pioneer. But freedom of thought and courage in imagination is needed also in science, and Aristarchus seems to have been almost the last to possess this.

[Sidenote: Seleucus B.C. 160.]

[Sidenote: Aryabhata born A.D. 467.]

A certain Babylonian, named Seleucus, who lived about B.C. 160, perhaps on the Tigris, and an Indian astronomer, Aryabhata, of the fifth century A.D., both taught that Earth turns on her axis, but Aristarchus stood alone in suggesting that she also has a movement of revolution round the sun. The possibility of Earth’s motion was alluded to but seldom by classical and mediæval writers, and then almost always as a foolish fancy hardly worth discussion;[44] until at last Copernicus, hardly daring to publish his bold idea in sixteenth century Europe, for fear of persecution, sought and found among the followers of Pythagoras, and in Aristarchus of Samos, kindred spirits with his own.

------------------------------------------------------------------------ [44] Indian astronomers also refused to accept the doctrine of Aryabhata. Varâha Mihira (sixth century A.D.) says:—“Others maintain that the earth revolves and not the sphere: if that were the case, falcons and other birds could not return from the ether to their nests.” ------------------------------------------------------------------------

6. THE SCHOOL OF ALEXANDRIA.

The spheres of Eudoxus would not work. The system was already overladen, and more variations in velocity were becoming known. Observation shows, too, that the planets, especially Mars and Venus, vary greatly in brightness, in regular periods which correspond with their movements, and this could not be accounted for by any possible number or arrangement of spheres if all were to remain concentric to Earth, so that every planet remained always at a uniform distance. Yet Aristotle had said that Earth must be at the centre of the Universe. The new philosophers of the Stoic school agreed with him, and among mathematicians only Aristarchus dared to disagree.

But they were not long at a loss. The homocentric spheres[45] were thrown aside, and during the third century two new hypotheses were suggested (beside that of Aristarchus), and although we do not certainly know by whom, there is little doubt as to the place in which they originated. By a strange fate, Egypt, the home of astronomy many centuries before, became the seat of the latest, most brilliant, and most successful astronomical school of ancient times, and the knowledge won there in the five centuries between B.C. 300 and A.D. 200 spread, following in the wake of Alexander’s conquests, over the whole of the civilized world.

------------------------------------------------------------------------ [45] Spheres all centring in one point. ------------------------------------------------------------------------

It was a purely Greek school, however. Greece lost her independence under Alexander, and was finally crushed by Rome in 146 B.C., but never was Greek learning and culture so much honoured and sought after as in this age. In Egyptian Alexandria, Greek men of science found a welcome, and opportunities of research which did not exist in any other place in the world. In the Museum, founded and liberally endowed by the royal Ptolemies, was a great library whose custodians were bidden to obtain every book that had ever been written, and it is said that when any stranger arrived with a new book it was taken from him and copied for the Museum, and the copy returned to the owner. Within the great marble colonnaded building were lecture-halls, and reading rooms, and laboratories; there were gardens for botanists and zoologists, and observatories for astronomers. These astronomers were all Greeks, and though now living in Egypt they do not seem to have learned anything more from the Egyptians. Perhaps the priests resented the intrusion, and kept their secrets jealously to themselves; perhaps Eudoxus and his immediate followers had learned all they had to teach. This seems the more probable because, although the Greeks of this age did use Babylonian records of eclipses to form a lunar theory, they complained of the insufficient accuracy of all available planetary records. Moreover, they were not hindered from learning astrology: the sacred books of the Egyptians which taught its principles were translated into Greek about 300 B.C.; but it was always treated by them as quite a separate branch of study. Their geometrical methods were entirely their own. They introduced a system of notation which greatly simplified calculation[46]; their discovery of the principles of spherical trigonometry inaugurated a new era in astronomy; and they invented a new class of astronomical instrument.

------------------------------------------------------------------------ [46] Mahaffy, _The Progress of Hellenism in Alexander’s Empire_, p. 119. ------------------------------------------------------------------------

We know what these instruments were like, and it is even possible to give an illustration; for, from descriptions in Ptolemy’s _Almagest_, we find that the “well-made copper circles,” the gnomons, and the celestial globes, which were set up in the Square Portico of the Museum, were of the same pattern as instruments which existed in Pekin, in the ancient observatory on the ramparts, until they were looted by the Germans during the late Chinese war. The Alexandrian instruments were not supported on their stands by beautiful bronze dragons, but on the other hand the circles were more accurately divided, which after all was of more importance from the astronomers’ point of view. Fig. 21 gives a general view of the Pekin Observatory, and Fig. 22 one of their astrolabes dating from the 13th century A.D.

At first glance there seems to be here absolutely nothing like our modern observatories. Ancient and mediæval astronomers had indeed no telescopes, being ignorant of the properties of lenses: therefore they were unable to study the features of any heavenly body except the moon, and they had no way of finding out anything about their physical constitution; but they had many ways of measuring their distances and motions, and even the angular sizes of sun and moon, and their instruments were the forerunners of our sextants, micrometers, and transit instruments, our chronometers and sidereal clocks.

_From a photograph taken in 1888, and published in the “Bulletin de la Société belge d’Astronomie”._]

The gnomon has been already described[47], and it was one of the most valuable instruments used by the Greeks. The Pekin gnomon at the right of figure 21 was more than 40 feet in height, and on the top had a little plate of copper which was pierced by a hole as fine as the eye of a needle: the observations made with this were much more exact than observations of the end of the shadow, which must always be vague, and the Chinese records of the sun’s movements made with this instrument between 1270 and 1280 A.D. are of great value in modern research. Ptolemy explains that his gnomon was made accurately vertical by the use of a plumb-line, and that one way of testing the level of the surface on which the shadow fell was to flood it with water.

------------------------------------------------------------------------ [47] See p. 26. ------------------------------------------------------------------------

Clepsydras, or water-clocks, were used by the Greeks, and many kinds of sundials for telling the time by day. Tables were also made of the risings of bright stars which served for clocks by night.

The instrument in the middle of the platform is a quadrant, and beyond this on the left is a large celestial globe, which, however, only dates from the Jesuit missionaries of the seventeenth century. Ptolemy says that his globe was made the colour of the night sky; Sirius being marked in his proper place, all other stars were placed relatively to him, and in their own colours as nearly as might be; the Galaxy was drawn, and the figures of the constellations outlined. The globe was arranged to turn on either the poles of the ecliptic or of the equator; circles of wood represented the horizon and meridian, and the pole could be arranged at any altitude according to the latitude of the place.[48]

------------------------------------------------------------------------ [48] _Almagest,_ Bk. viii. ------------------------------------------------------------------------

But perhaps the most interesting are the astrolabes, and these owe their origin to the Greeks. The essential part of any instrument for determining angular distances is a divided circle and a pointer: the pointer is directed first to one object then to another, and the angle between them is then read off on the circle. In the astrolabe, the pointers themselves were also circles, provided with little perforated rods for “sights.” (These are not visible on the instrument in Fig. 22, and have probably been broken off.) There were two fixed circles, set in the plane of the ecliptic and perpendicular to it. Three other circles could be rotated round the poles of the ecliptic. One of these was directed (by means of the sights), to some body whose position was already known, another to the body whose position was to be ascertained, and the angle between them was read off on the ecliptic circle; on the third the angular distance north or south of the ecliptic circle could be read. This last and the ecliptic circle were both divided into 360 degrees, and as many fractions of a degree as space and skill would allow.

The equinoctial astrolabe was similar, but the fixed circle was in the plane of the equator, instead of the ecliptic. One of each of these is seen in the view of the Pekin Observatory.

But how did the old astronomers know how to find the ecliptic and the equator in the sky, and set their circles in those planes? This they did by means of the sun’s motion. The gnomon told them the day of the equinox (see p. 25), and on that day the sun was in the equator: therefore, if a circle was set up so that the shadow of the upper part fell symmetrically upon the lower, with a little line of light each side, it must be exactly in the plane of the equator. In the Square Porch such a circle was erected, a large one of copper, and when once correctly adjusted it was a standard plane, and also showed the date of the equinoxes, as accurately as the gnomon itself. Since the ecliptic is the path of the sun as seen in the sky, it is obvious that it could be determined from a number of different observations of his position at different times of the year.

_From a photograph taken in 1888, and published in the “Bulletin de la Société belge d’Astronomie”._]

Finally, accurate solar tables were drawn up, showing the sun’s position in the sky in degrees for different dates, and then from these it was possible to find the places of planets and stars. They could not of course be compared directly, but the position of sun and moon were compared during the day, when both were in the sky, and then after dark the planets and stars were compared with the moon, allowing for her motion among the stars in the meantime. Or secondly, when the moon was eclipsed, and therefore known to be in the ecliptic and exactly opposite the sun, the places of stars could be found directly.

This very brief description will give some idea of the chief instruments and methods used, and when we see how very rough and elementary they were, and remember that the Greeks had to work out their observations without algebra, or decimal notation, we are amazed at their results, and their far-reaching ambitions.

[Sidenote: Eratosthenes B.C. 276-194.]

Already in the very early days of the Museum, Eratosthenes, a celebrated geographer, made a bold attempt to utilize observations of the sun measuring the size of the earth. It was known that in Syene (the modern Assuan) on the day of the summer solstice at noon no shadows were thrown, and the bottoms of wells could be seen: evidently therefore the sun was in the zenith. Eratosthenes found that the sun’s distance from the zenith in Alexandria at noon on the same day was 7° 12′, or one-fiftieth of the circumference of the heavenly sphere, consequently the two towns must be distant from one another (assuming them to be nearly in the same meridian) one fiftieth of the circumference of the earth. The distance from Alexandria to Meroe was known, and from Meroe to Syene had been paced by the king’s professional pacers; the whole was 5000 stadia. 50 times 5000 = 250,000. The figure always quoted by the ancients is however 252,000. If the stadium used by Eratosthenes was the measure generally used for long distances which have been paced, this estimate is equal to 24,662 miles, only about 200 miles less than the modern value. It was partly by luck that Eratosthenes got such a good result, for he was evidently only working with round numbers, and the extra 2000 stadia seem to have been added in order to make one degree equal to exactly 700 stadia. But in any case it was a highly creditable performance.

[Sidenote: Euclid _c._ B.C. 300.]

[Sidenote: Apollonius _c._ B.C. 270.]

There were celebrated mathematicians and geometers at Alexandria, whose work was most useful to astronomy, such as Euclid, and Apollonius of Perge. The latter is specially mentioned by Ptolemy in connection with the new theory of “moveable eccentrics,” which was invented to account for the varying brightness of the planets, as well as their peculiar movements.

Fig. 23 explains this theory. Let P A be a great revolving circle upon which Mars is fixed. (In the hands of the Alexandrian mathematicians the spheres almost disappear, and they deal practically only with circles.) If the earth were at its centre, as Eudoxus demanded, Mars must always be at the same distance, but if we make the circle eccentric to Earth, by putting its centre at C while Earth is at E, then the distance and consequently the brightness will constantly vary, and Mars will be brightest when at perigee P (point nearest Earth), and faintest when in apogee A (point furthest from Earth).[49]

------------------------------------------------------------------------ [49] Greek _peri_ near, _apo_ away from, _ge_ Earth. ------------------------------------------------------------------------

But, as the Greeks had discovered, Mars attains his greatest brilliance at different points of the zodiac, so P must be made moveable, and it always happens when he is opposite the sun, therefore P must keep pace with the sun’s apparent motion in the zodiac and P E always point towards him. This was accomplished by making P C A turn round upon the fixed point E, so that for instance when the sun had moved through a quarter of his circle (in three months) P A had moved to P′ A′, and the whole eccentric had moved into the new position shown in the diagram, its centre C being now at C′. In other words, the centre of the eccentric moves round Earth in the same time and in the same direction as the sun, that is in one year, and “with the signs” (from west to east).

At the same time, Mars is moving in an opposite direction on the eccentric, and without entering into all the details of the problem, we may add that the Greek geometers found that by determining the proper relative sizes of the large and the small circle they could make the two motions neutralize one another when the planet reached its stationary points, and the retrograde motion prevail over the direct when it retrograded. A similar arrangement was made for Jupiter and Saturn.

In this very ingenious way the varying brightness as well as the varying motions of these three planets were accounted for, without violating the principle of uniform circular motion, and without removing Earth from the centre of the Universe. She was also still the centre of planetary motion, in a certain sense; but to place the true centres of these planets’ spheres _outside Earth and in the direction of the sun_ was a very suggestive step, and may well have helped Aristarchus to his bold hypothesis. For he had only to put the sun, not at some indefinite point along the line E A, but exactly at the point C, and it became the centre of motion for Mars, Jupiter, and Saturn, just as in the “Egyptian theory” it was the centre of motion for Venus and Mercury. In this way he would arrive at the conception of the sun circling round Earth and carrying all the planets with him, (a theory which was held by the great astronomer Tycho Brahé in the sixteenth century A.D.). Then a flash of insight may have revealed to him the fact that this motion of the sun is apparent only, being but the reflection of Earth’s own motion; for she is circling round the sun like all the other planets.

It is, however, only a guess that the Moveable Eccentrics played this part in the theory of Aristarchus. They did not long hold the field, because they were not applicable to Venus and Mercury, which are never seen in opposition to the sun. So they were thrown aside for another system, the Epicycles, which illustrates much more simply the stations and retrogressions of the planets, and can be used for them all.

Later on, when more irregularities of motion were discovered, it was found necessary to combine eccentrics and epicycles, and by means of this joint system it became possible at last to represent completely, and as accurately as they could be observed, all the apparent movements of the heavens. First, however, an immense amount of work had to be done, and new methods devised, both in observation and mathematics. The man who contributed most, in both ways, to make it possible, was Hipparchus.

7. HIPPARCHUS.

[Sidenote: Hipparchus _c._ 140 B.C.]

Of this great man we know scarcely anything but what can be gathered from the work he did, and this corroborates Ptolemy’s description of him: “Hipparchus, lover of toil and truth φιλοπονον και φιλαληθεα.” He lived about B.C. 140, since this is the date of the only book of his still extant, and his work was not done in Alexandria, though he may have studied there in his youth, and he used the Museum records. We count him among the Alexandrians, as he belongs to this era, but he seems to have been a private astronomer, who set up an observatory of his own in Rhodes, his native place. Here we seem to see him, surrounded by his primitive instruments and his papyrus books, patient, eager, modest, seeking no fame and no reward but the joy of his work. By day he would keep watch over the sliding shadow of his gnomon, would write up his observations, make long calculations, and devise new methods in mathematics, improve and modify his astrolabes and his clepsydras; at night he would spend long hours with moon, planets, and stars, making up for the defects and shortcomings of his instruments by the skill and care with which he applied them to measure positions in the sky. Nothing but the most loving and conscientious care could have raised his work to such a pitch of accuracy, and made such rude means suffice for such splendid achievements.

The book we possess, apparently an early one, is chiefly concerned with the positions, the risings and the settings, of stars, and at the end is a list of sixteen which came to the meridian at intervals of an hour: from this list and the knowledge of spherical trigonometry which he possessed, it would be possible to calculate the time at night to within about a minute.

Hipparchus was able to construct a satisfactory theory of the sun and to some extent of the moon, but he found more irregularities in the planetary motions than Eudoxus had suspected. The records of his predecessors were not accurate enough for him to construct a theory for the planets, and he soon realized that one life-time would not be long enough to collect all the data necessary, so, as Ptolemy tells us, “Hipparchus, who loved truth above all things,” quietly set to work to make as good and as many observations as possible, leaving it to his successors to complete and explain them.

In the same spirit he undertook the laborious task, of which Pliny speaks with awe as a presumptuous scheme, even for a god, “_rem etiam Deo improbam_,” of numbering the stars. Pliny says he was led to do this by the appearance of a New Star, which blazed out suddenly in the constellation of Scorpio in b.c. 136, just as Nova Persei did in Perseus in February 1901. He saw that even in the upper regions of the eternal heavens, which Aristotle had supposed absolutely changeless, changes may occur, and in order that even the least of these should not pass unnoticed, he set to work to note the number, brightness, and position of all he could see. This great catalogue of 1080 stars, copied by Ptolemy in his _Almagest_, was the basis for all succeeding catalogues, from Spain to Turkestan, until quite modern times. In it, for the first time, the places of the stars were not merely described according to their position in the constellation figures, but were noted in degrees on the sphere, as is done to-day.

[Sidenote: Timocharis _c._ B.C. 280.]

One day, when comparing his notes with those of Timocharis, who had worked at Alexandria about a century and a half earlier, he found that the brilliant star Spica, the Ear-of-Corn which the Virgin carries in her hand, had apparently moved nearer to the autumnal equinox by about 2°. (Two degrees is about four times the angular diameter of the sun). Of course he or Timocharis might have made a mistake, or Spica might really be moving among the stars, or she might be carried along with the rest by a slow movement of the whole star sphere. Apparently Hipparchus satisfied himself that he could rely upon Timocharis’ observation, and took pains to verify his own; the second hypothesis could be disproved by the fact that Spica does not change her place perceptibly among her neighbours; and finally it became clear that her motion is part of a slow apparent movement of the whole heavens.

Here was a discovery of first importance, an unexpected reward of patient accuracy, of which the white Spica, flashing down from summer skies, may always remind us. Hipparchus had discovered the grand cycle which we call the “Precession of the Equinoxes,” and before Spica returns to the same position in which he saw her then, when she led him to his great discovery, she will have been watched by generations of astronomers for another twenty-four thousand years. No notice of the cycle has been found as yet among the records of any other nation, although it seems as if the astronomers of Babylon and Egypt, and other countries where observations had been carried on for many centuries, must have been aware of it. We can only imagine that at long intervals of time they found that the stars had somehow changed, and made corrections accordingly, but without understanding the nature of the change. What Hipparchus thought about its cause we cannot tell: probably he left all speculations to future astronomers, and confined himself to noting the fact.

The displacement of Spica which he observed is shown in the diagram.

Both Timocharis and Hipparchus evidently measured her position indirectly by comparing it with that of the moon, which was eclipsed at the time,[50] and therefore known to be in the ecliptic and opposite the sun. To find the sun’s distance from the equinox was an easy matter, since his yearly course had long been carefully studied, and the days on which he passed the equinoxes were regularly observed with the gnomon. Spica, then, had moved eastward along an arc parallel to the ecliptic, and since celestial latitude and longitude are referred to the ecliptic, we may define her apparent movement in astronomical language by saying that while her latitude had remained constant, her longitude had increased by about two degrees; and further, as the celestial equator is oblique to the ecliptic, this implied that her declination (position north or south of the equator) had also varied. The diagram shows that she had a less northerly declination than before.

------------------------------------------------------------------------ [50] _Syntaxis_, book VII.; Delambre, _Histoire de l’Astronomie Ancienne_ Vol. II. page 247 (1817 edition). ------------------------------------------------------------------------

At this rate, Spica, which was now only 6° from the autumnal equinox, would reach it in less than five hundred years, and thereafter would lie east instead of west of it; and that she has in fact done so, may be seen by consulting a modern star atlas. She is now 22½° of longitude east of the autumnal equinox, and nearly 11° south of the equator. Her south declination will continue to increase for about five thousand years, after which she will come north again.

Ptolemy says that Hipparchus examined other stars, and found that they also were increasing their longitude at what appeared to be the same rate as Spica. The yearly amount of the movement, derived from the Spica observations, is within a few seconds of arc of the true value, which is 50¼ seconds; but Hipparchus would not fix any value until it had been tested by further observation, and merely stated that it could not be _less_ than one degree in a century _i.e._ 36 seconds per annum.

This is a very uncomfortable phenomenon for astronomers, since every star is for ever changing its measured position on the celestial sphere. Take three stars, one at the north pole, another on the equator, and a third in the southern hemisphere. After some years, the first will no longer be a pole star, the second no longer an equatorial star, and the third may have so far increased its south declination that it will be invisible at latitudes in our northern hemisphere where formerly it used to rise above the horizon. One compensation for this inconvenience is that if we know what star was near the pole, or which stars lay along the equator, on any given occasion, we can calculate the date. Thus it is believed that the Great Pyramid was built when Alpha Draconis was the Pole Star, that is, nearly 3000 years B.C.; and by a similar method Mr Maunder determines the epoch at which the ancient southern constellations were invented, as we have already seen.

The greatest inconvenience, and also the greatest historical interest, attaches to stars like Spica which belong to constellations of the zodiac, for if they are not stationary with regard to the equinoxes and solstices they are not such simple guides to the length of the solar year as the ancients supposed them to be. The scheme of the Babylonians for beginning their month Nisan when the stars of Dilgan rose just before the sun was an excellent one for a time, but if they had continued it for many centuries they would have found that their year was too long, and the months were all falling in the wrong seasons. This has actually happened with Hindus and Parsis, who now keep their New Year in the middle of our April, although when their calendar was fixed, about thirteen hundred years ago, the years began at the spring equinox. For the sun is like a runner in a circular race-course who thinks he has completed a lap when he returns opposite a group of spectators originally standing at the starting-point, but after several laps he finds that the spectators and the goal no longer coincide; either they, and also all the others surrounding the course are walking away from it, or an unseen hand has been moving the flag towards him, and so shortening the lap.

It is the flag which must count, in any case, not the spectators, and with the sun it is the equinox which must count, and not the stars, for this is the point at which he crosses the equator, making day and night equal, and from this we count the beginning and ending of our seasons. So our year is counted from equinox to equinox, and is twenty minutes shorter than the “sidereal,” or star year, of the ancient Babylonians. Hipparchus, from observations of equinoxes and solstices, made the year 365 days 5 hours and nearly 55 minutes, which is only 6 minutes longer than the correct value.

After Hipparchus had made his discovery, astronomers agreed upon a somewhat clumsy and very confusing device, by which the zodiac was divided into twelve equal “signs” of 30 degrees, which bear the same names as the zodiacal constellations, but whose beginning is always reckoned from the vernal equinox. These twelve “signs of the zodiac,” therefore, do not now agree with the twelve constellations of the zodiac, and our present “first point of Aries,” which marks the vernal equinox, is in the _constellation_ of Pisces.

What is the true cause of this strange phenomenon? Are the stars really all in motion, or is it the equinox which moves?

The successors of Hipparchus, who believed that the stars were fixed on a sphere, found no great difficulty in conceiving that this sphere had a very slow easterly motion, round the poles of the ecliptic, completing a revolution once in 36,000 years (_i.e._ one degree in century). To us, however, it is impossible to believe that the stars, which we have found to be at enormous and varied distances, are all revolving at one rate, parallel to the ecliptic. The ecliptic, to us, is simply the plane of Earth’s own orbit, and as she moves in it she has a very slow “wobbling” motion on her axis, as well as the rapid spinning of the diurnal motion, like the “wobble” of a spinning top. The top has this motion because gravity is trying to pull it down from its upright position; the earth because the sun is trying to drag her slightly protuberant equator into the plane of her orbit.

The resulting motion is not a revolution of the earth, nor an apparent revolution of the star sphere round Earth: what really happens may be illustrated with the traditional orange and knitting needle.

Ignoring all motions but the one we are speaking of, let the points of the knitting needle (Earth’s axis of rotation) trace out small circles in space, and the equator of the orange will be seen to alter the direction of its tilt, but without turning round (Fig. 26). Stick a pin in the equator, and others in north or south latitudes, between equator and pole; these will always remain facing you, but while the pole makes its small circle, the equatorial pin will be seen to move up and down, while the tropical and temperate pins trace out ellipses. These are the movements which we see reflected in the stars; and if Earth’s diurnal rotation were suddenly to cease, while her revolution in her orbit and the movement of “precession” continued, we should see Spica, for instance, sink slowly lower in the southern sky and after ages rise again northwards, but there would be very little preceptible movement east or west.

The movement observed by Hipparchus, then, was not a movement of Spica and other stars, but a movement of the equinox. For the celestial equator is simply a reflection of Earth’s equator in the skies, and as it keeps changing the direction of its tilt in the way described, it changes the point at which it cuts the ecliptic. This may best be seen by taking two rings or hoops (two large curtain rings, for instance), one of which just fits inside the other. Tilt the inner ring, so that half of it is above and half below the other ring, and they touch at two points, 1 and 2 (Fig. 27). The outer ring is the Ecliptic, the inner the Equator, and where they touch each other are the Equinoxes. Now move the inner ring, not sliding it round, nor making any difference in the angle between the two, but simply so that they touch at fresh points, 1′ and 2′. In this way you may make the points of contact revolve entirely round. This is what the real equinoxes are doing: while the equator opposite the group of stars in figure 25 rises and falls, the equinox travels on, and finally returns to the same place.

_V_ is the vernal equinox, at the intersection
of the equator and plane of the ecliptic: _APX_
the earth’s axis, which always preserves the same
inclination (23½%) to the plane of the ecliptic. As
_APX_ slowly revolves round _T_ in the direction of
the arrow, the vernal equinox is gradually shifted
to _V_′, and so on.

(_From Young’s “Manual of Astronomy,” 1902._)]

The phenomenon is called “precession of the equinoxes,” because they thus move on to meet the sun in his yearly course.

The discovery of precession is what has chiefly made Hipparchus famous, but the invention of the astrolabe and of spherical trigonometry, both believed to be due to him, his star catalogue, and his many observations, more accurate than any made before, were so valuable as pioneer work that Ptolemy justly called him the Father of Astronomy. If Hipparchus could visit one of our observatories to-day, and see the clock-driven equatorials, the transit instruments, the beautifully divided circles read with microscopes, and the sidereal clocks, one wonders whether he would be more astonished at the advance on his astrolabes and clepsydras or at the homage paid to him as one in whose footsteps all astronomers are proud to tread.

8. PTOLEMY.

_Claudius Ptolemœus._ I know that I am mortal
and ephemeral, but when I scan the multitudinous
circling spirals of the stars, no longer do I touch
Earth with my feet, but sit with Zeus himself, and
take my fill of the ambrosial food of the gods.

For more than two centuries after Hipparchus very little original work was done in astronomy, and no one seems to have had the courage to take up his unfinished task and study seriously the difficult problem of planetary motions.

[Sidenote: Posidonius _c._ 135 B.C. to _c._ 50 B.C.]

Posidonius the Stoic, who lived for some years in Rhodes, made a fresh determination of the earth’s circumference, basing it not on observations of the sun, like Eratosthenes, but of the star Canopus, which in his time was just visible at Rhodes while in Alexandria it rose “a quarter of a sign” (_i.e._ 7½ degrees) above the horizon. By his method, the earth was a little smaller, (240,000 stadia instead of 250,000), but it must have been difficult to measure the distance between Rhodes and Alexandria over the sea, and it is impossible to say when a star is exactly on the horizon. Posidonius also observed the tides in the Mediterranean, and showed that “Ocean follows the movements of the heavens,” and especially of the moon, having daily and monthly periods.

[Sidenote: Geminus _c._ 70 B.C.]

A little later Geminus wrote an Introduction to Astronomy, which was an excellent little book as far as it went, but although he was apparently a native of Rhodes, and speaks of Hipparchus, he seems to know nothing of his work, for he does not quote his careful determination of the length of the year, nor his discovery of precession.

[Sidenote: Cleomedes _c._ 20 B.C.]

[Sidenote: Theon of Smyrna _c._ 100 A.D.]

Nor were other writers, such as Cleomedes and Theon of Smyrna, better informed, and they added nothing new to the advance of astronomical science.

[Sidenote: Ptolemy _c._ 140 A.D.]

But at last a worthy successor arose at Alexandria, the immortal Ptolemy, whom Dante met in that Limbo of antique spirits which was almost Elysium, although on the brink of the Inferno.

We do not know when Ptolemy was born nor when he died, nor where was his native town: we only know that his first recorded observation was made in the eleventh year of the Emperor Hadrian, that is A.D. 127, and his latest in A.D. 150, and that he lived and studied in Alexandria. He had splendid opportunities for carrying on the work of Hipparchus, for besides the use of the instruments in the Museum Observatory, he had at hand all the Museum records, which included the writings of Hipparchus. Ptolemy was not so painstaking and accurate an observer as Hipparchus, but he was a very able mathematician to whom it was evidently a joy to handle figures and work out problems. By examining a number of observations spread over several centuries, and combining them with his own, he was able to accomplish the task in which so many others had failed, and to frame a system which embraced all the celestial motions then observed. The monumental summary in which he set forth this system contains a great deal of interesting information about his methods and instruments, and about the work of Hipparchus, for whom he always expresses the most generous admiration. The original name of his book was the “Mathematical System of Astronomy,” but his admirers having called it the “Great System,” _Megiste Syntaxis_, the Arabs affixed their article _al_ and gave it the name it preserves to this day, of _Almagest_. It remained the standard treatise on astronomy until the _De Revolutionibus Orbium Celestium_ of Copernicus appeared in 1543.

The name of his book indicates the scope of Ptolemy’s work. It was to represent all the observed motions of the heavenly bodies by means of a mathematical system, so that they became amenable to calculation; but to explain the causes of these motions was thought to lie quite outside an astronomer’s province. It was not for a mere observer and calculator to determine which motions were real and which apparent, else Ptolemy must have decided in favour of Earth’s rotation, for he says that it would be much easier to account for celestial motions on this assumption. Nor was it his business to investigate the substance of the stars. Little did he dream that astronomers would one day solve such problems, and uphold their conclusions in the face of all the world: for him, as for his contemporaries, the decisions of the philosophers, and especially of Aristotle, were final, and his task was to describe what he saw in the light of their teaching.

He brings forward, in the introduction to his book, a few arguments against the absurd notion, taught by some, that Earth is in motion, turning on her axis or moving through space; but all that he proves is the immense difficulty, even to a trained mind, of accepting these theories, and the great authority of the philosophers who denied them on abstract principles. Educated Greeks might still discuss the nature of the heavenly bodies, as in Plutarch’s delightful dialogue _On the Face of the Moon_, (written about half a century before the days of Ptolemy), and might ridicule the law of gravity, laughing at the absurdity of supposing that if the middle of a man’s body were at the middle of the earth, his feet as well as his head would be “up,” and that falling weights if they reached this point would stop short, or oscillate to and fro. Yet even they all agreed that the fixed stars do most probably move “in a circle of eternal and never-ending revolution,” and that they are of a pure and eternal substance unlike Earth; and for the professional astronomers of Alexandria these axioms were assumed as the basis of all their work. Earth must be immoveable at the centre of the Universe because the heavy stuff of which she is made sinks necessarily to the centre and there remains in globular form without need of support; the heavenly bodies must be in motion, because, being of ethereal substance, it is their nature to revolve eternally in circles.

This being granted, we can feel nothing but admiration for the extent of Ptolemy’s knowledge, the comprehensiveness of his scheme, and the skill and patience with which he overcame its difficulties.

Earth, according to Ptolemy, is but a point compared with the immense surrounding sphere in which the stars are set, and this turns always round us, communicating its motion (he does not inquire how) to sun, moon and planets, so that day follows night, and the heavenly bodies daily rise and set. For the slow movement of precession, which also affects all heavenly bodies, Ptolemy accepted the least value of Hipparchus, one degree in a century, only testing it in rather a perfunctory way, which was a great pity, for he might have determined it much more closely after an interval of some 250 years. But one man cannot do everything, and he doubtless thought it best to spend more time on the planets, whose intricacies had baffled Hipparchus and gave him also a great deal of trouble.

He retained the great spheres which were supposed to carry them round Earth, inside the star sphere, but the chief feature of his system is the use of small spheres, which were fixed on the larger, and therefore called “epicycles,” while the large were known as “deferents” or carriers. The general principle of epicycles is very simple, as may be seen by comparing the two diagrams.

Fig. 28 shows the path of Mars as we saw it among the stars of Pisces in the year 1909. Throughout July the planet was travelling in its usual direction, “with the signs,” but on August 22nd it came to a stop, then turned and travelled backwards “against the signs” until October 26th, when it stopped again, reversed its direction once more, and during the rest of the year moved rapidly forward.

Fig. 29 shows the principle on which Ptolemy would have explained this curious track. Each planet was supposed to be fixed on a small circle, the epicycle, and this was fixed upon a large circle, or deferent, upon which it travels in the direction shown by the arrow, at a uniform speed, returning to the same place in the sidereal period of the planet. Thus Mars, as seen from Earth, which is near C the centre of the deferent, makes a great circle through all the zodiac in two years, Jupiter in twelve, and so on. But meanwhile the epicycle is rotating round its own centre, C1, and when the planet reaches the point marked S, the two motions neutralize one another, so that it appears stationary, as Mars did on August 22, 1909. After this, the motion of the epicycle more than counter-balances the motion of the deferent, and the planet seems to reverse its direction until it reaches the point on the epicycle marked S′. After this the two motions are once more in the same direction, so the planet is seen to move rapidly forward, as Mars did after October 26.

Ptolemy’s method of accounting for movements such as those shown in Fig. 28.]

In this extremely ingenious way the strange planetary oscillations were accounted for, without violating the law of uniform circular motion, and in a more convenient and satisfactory way than by the concentric spheres of Eudoxus or the moveable eccentrics of Apollonius. Each of the five planets was provided with an epicycle and a deferent, and these were made of the proper relative size and given the right speed, so that the motions should correspond with what we see in the sky.

Ptolemy calls the oscillation a planet’s “anomaly with regard to the sun,” because (as we have seen when discussing the moveable eccentrics) it was known to be connected in every case with a planet’s angular distance from the Sun. On September 24, when Mars was in the middle of his retrograde arc in Pisces, the sun was exactly opposite, in the constellation of Virgo. This is found to be always the case, not only with Mars, but with Saturn and Jupiter too. Whenever one of these planets has the position O on its epicycle, and therefore is retrograding, the sun will be found to be exactly opposite in the sky. Mars comes into this position, and is opposite the sun, once in 780 days; this, therefore, Ptolemy called the period of his epicycle, while a little less than two years was the period of the deferent. The two periods of Saturn are 378 days and 29½ years nearly; of Jupiter 399 days and nearly 12 years.

Venus and Mercury betray their dependence upon the sun in more striking fashion, for since these planets simply oscillated from one side to the other of the sun, their epicycles must be supposed to be keeping pace with him all the way round the zodiac. Fig. 31 shows their relation to one another. On September 25, 1911, Mercury was seen from Earth as a morning star as far west of the sun as it is possible for him to travel, while Venus, after shining as an evening star all the summer, had come into line with the sun and become invisible.[51] On December 7 following, the rotation of the epicycles (Ptolemy would say) had brought both planets to new positions, Mercury now being an evening star at his “greatest elongation east,” and Venus a morning star. But the centres of the two epicycles always remain in a line with one another and the sun, and so their periods on the deferents are the same as his, viz. one year. The epicyclic periods, or intervals between two “greatest elongations” west or east, are 116 days for Mercury, 584 days for Venus.

------------------------------------------------------------------------ [51] Venus had passed her “inferior conjunction with the sun” on Sept. 15. ------------------------------------------------------------------------

We still explain the complicated course of the planets by resolving it into two approximately circular motions, but we know now that only one belongs to the planet itself, the other is Earth’s own motion. The reason why the sun’s position affects the position of every planet is simply that the epicycles of Mars, Jupiter, and Saturn, and the deferents of Venus and Mercury are reflections of Earth’s yearly journey round the sun.

Ptolemy had by no means finished with the planets when he had provided each one with an epicycle to represent the “anomaly with regard to the sun.” Hipparchus had noticed that there was another lesser irregularity, which seemed to be periodical likewise, although no former system had taken it into account; and this he called the “anomaly with regard to the zodiac,” because the speed of each planet and the amplitude of its loop varies slightly according to the part of the zodiac it happens to be in. He suggested that this might be dealt with by combining the two theories of epicycles and eccentrics, and this suggestion Ptolemy adopted with success. He placed each deferent with its centre not exactly at the earth, but at a certain small distance which was different for each planet. (This is not shown on our small-scale diagrams, therefore Earth appears at the exact centre.) The true explanation of this irregularity is that each planet’s path is not strictly circular, but elliptical.

Besides this, Ptolemy had to represent the planet’s movements north and south (note how this varies in fig. 28.). This was partly managed by the aid of small wheels, rotating in such a way that they lifted and lowered the epicycle as required.

Although Ptolemy quotes Babylonian observations of lunar eclipses dating back as far as the eighth century B.C., the oldest planetary observations that he uses were made only four hundred years before his time, and they were probably Greek. Even these were generally very rough. For instance:—

In the 496th year of Nabonassar, on the 17th day
of Choeac, in the morning, Mercury was three
moon-breadths north of the tail of Capricorn.

In the same year, Phamenoth the 30th, Mercury was
three moon-breadths south of the horn of Taurus
which is also the foot of the Charioteer.

The first year of Nabonassar (a Babylonian epoch) corresponds with B.C. 747, so the 496th year is B.C. 251. The months used by Ptolemy are usually Egyptian. The later observations, made with an astrolabe, were much more precise; Ptolemy quotes one from Theon of Smyrna, which states that Mercury was 3° 50′ in advance of the Heart of the Lion (Regulus), and for his own observations he also usually gives the sign, degree, and minute. Sometimes the planets had been observed so near stars that their positions could be very accurately determined by the aid of Hipparchus’ star catalogue. Timocharis, on a certain morning in the 13th year of Ptolemy Philadelphus (B.C. 273), saw Venus beside the last star in the wing of the Virgin (Beta Virginis); Ptolemy himself saw her so close behind a certain star in Aquarius that she seemed to touch it with her rays; and in the 83rd year after the death of Alexander, Jupiter had been observed to eclipse the Southern Ass, that is the southernmost of the pair of stars on either side of the little cluster in Cancer which the ancients called the Manger.