we must consider what astronomy was before his time. I doubt not that if a full account of the laborious work of Chaldæan, Egyptian. Indian, Chinese, and other ancient astronomers had reached our time, we should find many among them who well deserved to rank among great discoverers; for an immense amount of work had to be accomplished to place astronomy in the position which it occupied when Hipparchus began his work. Yet if we rightly apprehend what that position was, and consider what astronomy became when the labours of Hipparchus had produced their full effect, or rather their first fruits, we shall appreciate to some degree the importance of his researches.

I will not here discuss the history of astronomy before the time of Hipparchus. It would occupy too much space, and would be outside of my subject. It would also lead us to the discussion of many doubtful and difficult questions. But the position of the science before the days of Hipparchus can be fairly well ascertained from the account which Ptolemy has given of the labours of his great predecessor.

Astronomers had ascertained the general motions of the sun and moon, and of all the heavenly bodies visible to the naked eye. They knew that the earth is surrounded on all sides by the stellar sphere, on the concave surface of which, in appearance, the stars. are set in apparently unchanging groups,-the constellations. They had learned that this hollow sphere is seemingly carried round once a day, as if turning on an axis. This motion of rotation they had found to be absolutely uniform.

Further, by long-continued and careful observations they had found that the sun appears to circuit the stellar sphere on an unchanging path once a year. I speak of a year as though this measure of time and that occupied by a revolution of the sun around the stellar sphere were not necessarily identical,-for this reason, that the year in common acceptance means, and ever has meant, the cycle of the seasons. This cycle we now know indeed, to be brought about by the sun's motion round the stellar sphere on an inclined path, which brings him in midsummer nearer than at any other time to the visible pole of the heavens, and in winter nearest to the unseen pole. But the coincidence, or rather the exceedingly close approach to coincidence between the year of seasons and the period of the sun's circuit of the stellar sphere, was in reality one among those earlier discoveries by astronomers, of whose history ve know so little.

The moon had been found in like manner to circuit the stellar sphere the same direction as the sun, moving on a path somewhat inclined o his, in a period (variable somewhat in length) of about 27 days, called a sidereal lunar month. The ancient astronomers had also in determining the general laws according to which eclipses of the sun and moon recur, ascertained the general laws of the moon's motions, and had found that her path among the stars is not unchanging like the sun's. Within certain limits of inclination this path undergoes constant changes, the points where it crosses the sun's path shifting constantly,

hot always in one direction, yet always with a balance of motion in one direction. If these points were fixed, the sun would of course pass them at intervals of half-a-year. When he was passing them, or within certain distance from them, eclipses of the sun and moon would tccur, so that at intervals of six months eclipse seasons, so to call them, would recur. But the ancients found that the average interval separatag eclipse months is only about 1733 days, instead of half-a-year, or 182ğ days; which shows that these points where the moon's track crosses the sun's are (on the whole) constantly moving to meet the advancing sun, or are constantly moving backwards.

The earlier astronomers had also learned much about the motions of the five planets or wandering stars known to them. I ought perhaps to say the five other planets, for they called the sun and moon planets, because these bodies moved among the stars. They found that the planets, though on the whole advancing, are moving the same way round as the sun and moon, yet at regularly recurring epochs cease thus to advance, travel backwards for awhile, and ceasing to travel backwards, begin again to advance,—making always a much longer journey forwards than backwards, as they advance on a path showing a series of loops and twistings of a most complicated nature.

Not to occupy more space than can be spared with the account of what the astronomers before Hipparchus had discovered, let it suffice to say, that they had in a general way determined the periods of the sun's and moon's motions and of the planetary revolutions, and had recognized the regular recurrence of certain changes in the distances of sun, moon, and planets, indicating peculiarities in their paths which might (as they judged) admit of being explained, but which certainly none among them had succeeded in interpreting.

It is, however, necessary to notice that more than a century before the time of Hipparchus the Alexandrian School of Astronomy had been founded by Ptolemy Soter, one of Alexander's generals, who reigned over Egypt after the death of Alexander. His son, Ptolemy Philadelphus, gave to the astronomers of this school a large edifice containing an observatory and the celebrated library formed by Dimetrius of Phalos. Here Aristillus and Timocharis made their observations, and to this school also belonged the well-known astronomers Aristarchus of Samos and Eratosthenes. The latter was the first successful measurer of our earth's globe, and has been called the Father of Chronology.

According to Strabo, Hipparchus was born at Nicæa, in Bithynia. Although we know neither the year of his birth nor of his death, it is certainly known that his labours were in progress during the thirty-five years following 160 B.C. Probably his first observations were made in Bithynia. But it is certain that he afterwards continued his work at Rhodes. It has been supposed by some that he also observed for some time at Alexandria; but although Ptolemy refers to the views of Hipparchus respecting observations made at Alexandria, he nowhere says that Hipparchus himself observed there.

From among the many services rendered to astronomy and to mathematics by Hipparchus, I propose here to consider three only: first, his determination of the length of the year; secondly, his discovery of that mighty motion of the rotational axis of the star sphere which gives rise to what is technically called the procession of the equinoxes; and thirdly, his investigation of the motions of the sun and moon. All three were noble achievements; all three were based on exact observation; but they were exceedingly diverse in character. The first was a triumph of mensurational astronomy; the second revealed the existence of constant mutation where everything had seemed fixed and unchanging; the third revealed order and regularity really existing among movements apparently most complicated and perplexing.

The year had been supposed in the time of Hipparchus to last exactly 365 days 6 hours. It is indeed probable that the ancient Chaldæan astronomers had made more exact determination of this important time-measure. But it is certain, that the astronomers of the Alexandrian school had regarded 365 days as the true length of the year of seasons. Hipparchus was the first to recognize from direct observation of the sun that the year is somewhat less than 365 days in length. Aristarchus of Samos, in the year 281 B.C., had observed as closely as he could the time when the sun reached his greatest range north of the celestial equator, or made his nearest approach towards the visible pole. In other words, Aristarchus had timed to the best of his ability the summer solstice of the year 281 B.C. Hipparchus, in the year 134 B.C. or nearly a century and a half later, made a similar observation. By dividing the time between the two epochs into as many parts as there were years in the interval, he inferred that three hundred years contain 109,574 days, instead of 109,575 days, as they would if a year lasted exactly 365 days. This made the length of the year 365 days 5 hours 55 minutes 12 seconds. The result is not strictly correct. Three hundred years contain in reality about 109,5723 days. But the correction made by Hipparchus was important in itself, and still more as showing the necessity for further observation.

Hipparchus himself recognized the probability that his determination of the length of the year would require correction; and the way in which he showed this involved the recognition of two most important principles.

In the first place Hipparchus observed that the correctness of his estimate depended mainly on the length of time which had elapsed between his own observation and that made by Aristarchus. The errors, whatever they might be which Aristarchus and he himself might have made in determining the true epochs of the solstices they respectively observed, combined to produce a certain error in the total estimated interval between the two solstices. This error might be large in itself. If one observation had been made at the summer solstice of one year, and the second at the summer solstice of the next year, the interval, instead of being a true year of 365 days 5 hours 48 minutes 49 seconds, might

be a day or so too long, or a day or so too short. Hipparchus himself acknowledged that his error in determining the time of the solstice might amount to a quarter of a day. Aristarchus might have made a similar error, or a larger one. Suppose, however, the errors in the two observations to have been each half a day in length, and such that either both seemed to shorten the interval (the first being too late, the second' too early), or both to lengthen it (the first being too early, the second too late). Then if the interval had been but a single year, the measure of the year would have been a day too short or a day too long. But the interval between the two observations being in reality 147 years, an error of a day in determining this interval would correspond only to an error of 1-147 of a day for each year, or not quite ten minutes. As the error actually amounted to little more than six minutes, we see that the actual error as between the observations of Aristarchus and Hipparchus amounted to not much more than half a day. The result is creditable to both astronomers. We must be careful not to assume, as some have done, that the error lay in the observation of Hipparchus. It arose from the comparison between his observation and that made by Aristarchus, and most probably the largest part of the error was in the work of the earlier observer. Hipparchus, in measuring the year in this manner, recognized the important principle (one of the fundamental principles of modern astronomy), that the wider apart two observations are for determining such a period, the smaller is the resulting error in the determination of the period itself. But he clearly perceived also that the observation of solstice is not a satisfactory method for determining the length of the year. He only selected this method at first because he had no other way of dealing with a long period. If we consider what the summer solstice of the sun really means we shall perceive why the moment of its occurrence cannot be accurately determined. As midsummer approaches, the sun passes farther and farther north from the celestial equator, but with a constantly diminishing motion,-just as towards noon the end of the hour hand of a clock passes higher and higher, but more and more slowly, (so far as height is concerned). At the true midsummer solstice the sun is at his farthest rorth, just as the end of the hour hand of a clock is at its highest at true noon. Now it is clear that if we had no other way of telling the moment of true noon than by noting when the end of the hour hand was highest, we should be apt to make a mistake of several seconds at least, because of the small change which takes place in this respect near the hour of noon. Or suppose a horizontal line traced on a clock face at exactly the level reached by the end of the hour hand at twelve, and that we were to determine the hour by noting when the end of the hour hand exactly reached this level, it is clear that the slow change of height would cause our estimate to be, very probably, erroneous. On the other hand, suppose a horizontal line across a clock face at the exact level of the end of the hour hand at three and at nine; then we should be able very readily to de

termine when it was three or nine, by noting when the end of the hour hand passed this level,-for the end crosses this level at right angles. In other words, because the height of the hour hand is changing most quickly near three and at nine, and least quickly near twelve and six, we could much more exactly time the hours three and nine by noting the height of the hour hand's end, than we could time six or twelve. Now in much the same way spring and autumn are the seasons when the sun's midday height is changing most rapidly from day to day. It is then much easier to determine exactly when he is on the celestial equator, or the true epoch of either the vernal or the autumnal equinox, than to determine either when he is farthest north or farthest south of the equator, or the true epoch of the winter or the summer solstice.

The

Hipparchus clearly perceived this point. In other words, he clearly recognized the principle, a most important one in observation, that the best opportunity for a time observation is obtained when the body observed is most rapidly changing as respects that particular circumstance which is to be noted (in this case the distance of the observed body from the equator or from the visible pole of the heavens.) principle is simple enough, and seems obvious enough when explained; but it is certain that Hipparchus was the first definitely to indicate its nature and to apply it in astronomical observation. He timed the sun's passages of the celestial equator during a period of thirty-three years. From these observations he deducted the same value for the length of the year as from the solstitial observations, though, as already mentioned, these covered a period of one hundred and forty-seven years, or nearly five times as long.

It should be added that although Hipparchus himself (through no fault of his own) was prevented from determining the year exactly, yet the modern estimate owes its accuracy to his observations. It is from a comparison of more observations of equinoxes by him with recent observations that we have been able to infer the length of the year within a second or so of its true length.

The second great discovery of Hipparchus was a very remarkable one. The first related to a period which has elapsed more than two thousand times since Hipparchus dealt with it; the second relates to a period of which not one-twelfth part has elapsed since his time,—the tremendous precessional period of nearly 25,900 years.

Although we cannot see the stars around the sun in the daytime, we know that his course carries him along a definite track among the stars. Careful observations enable the astronomer to determine the exact position of the various stages of the sun's course, his equinoxes, his solstices, and the limits of the twelve equal divisions called signs. Until the time of Hipparchus, it was supposed, at least by the astronomers of the Greek school, and certainly at the beginning of his career it was believed by Hipparchus himself, that the positions of the equinoxes and solstices are unchanging. In other words, it was believed that when the sun reaches a particular point in his track among the stars,