to get an accurate value of the moon's mean motion. [Halley, in 1693, compared this value with his own. measurements, and so discovered the acceleration of the moon's mean motion. This was conclusively estab lished, but could not be explained by the Newtonian theory for quite a long time.] He determined the plane of the moon's orbit and its inclination to the ecliptic. The motion of this plane round the pole of the ecliptic once in eighteen years complicated the problem. He located the moon's excentric as he had done the sun's. He also discovered some of the minor irregularities of the moon's motion, due, as Newton's theory proves, to the disturbing action of the sun's attraction. In the year 134 B.C. Hipparchus observed a new star. This upset every notion about the permanence of the fixed stars. He then set to work to catalogue all the principal stars so as to know if any others appeared or disappeared. Here his experiences resembled those of several later astronomers, who, when in search of some special object, have been rewarded by a discovery in a totally different direction. On comparing his star positions with those of Timocharis and Aristillus he found no stars that had appeared or disappeared in the interval of 150 years; but he found that all the stars seemed to have changed their places with reference to that point in the heavens where the ecliptic is 90° from the poles of the earth-i.e., the equinox. He found that this could be explained by a motion of the equinox in the direction of the apparent diurnal motion of the stars. This discovery of precession of the equinoxes, which takes place at the rate of 52′′.1 every year, was necessary for the progress of accurate astronomical observations. It is due to a steady revolution of the earth's pole round the pole of the ecliptic once in 26,000 years in the opposite direction to the planetary revolutions. Hipparchus was also the inventor of trigonometry, both plane and spherical. He explained the method of using eclipses for determining the longitude. In connection with Hipparchus' great discovery it may be mentioned that modern astronomers have often attempted to fix dates in history by the effects of precession of the equinoxes. (1) At about the date when the Great Pyramid may have been built y Draconis was near to the pole, and must have been used as the pole-star. In the north face of the Great Pyramid is the entrance to an inclined passage, and six of the nine pyramids at Gizeh possess the same feature; all the passages being inclined at an angle between 26° and 27 to the horizon and in the plane of the meridian. It also appears that 4,000 years ago—i.e., about 2100 B. C.-an observer at the lower end of the passage would be able to see yDraconis, the then polestar, at its lower culmination.1 It has been suggested that the passage was made for this purpose. On other grounds the date assigned to the Great Pyramid is 2123 B.C. (2) The Chaldæans gave names to constellations now invisible from Babylon which would have been visible in 2000 B.C., at which date it is claimed that these people were studying astronomy. (3) In the Odyssey, Calypso directs Odysseus, in accordance with Phoenician rules for navigating the Mediterranean, to keep the Great Bear ever on the left as he traversed the deep" when sailing from the pillars of Hercules (Gibraltar) to Corfu. Yet such a Phil. Mag., vol. xxiv., pp. 481-4. course taken now would land the traveller in Africa. Odysseus is said in his voyage in springtime to have seen the Pleiades and Arcturus setting late, which seemed to early commentators a proof of Homer's inaccuracy. Likewise Homer, both in the Odyssey1 (v. 272–5) and in the Iliad (xviii. 489), asserts that the Great Bear never set in those latitudes. Now it has been found that the precession of the equinoxes explains all these puzzles ; shows that in springtime on the Mediterranean the Bear was just above the horizon, near the sea but not touching it, between 750 B.C. and 1000 B.C.; and fixes the date of the poems, thus confirming other evidence, and establishing Homer's character for accuracy.2 (4) The orientation of Egyptian temples and Druidical stones is such that possibly they were so placed as to assist in the observation of the heliacal rising3 of certain stars. If the star were known, this would give an approximate date. Up to the present the results of these investigations are far from being conclusive. Ptolemy (130 A.D.) wrote the Suntaxis, or Almagest, which includes a cyclopædia of astronomy, containing a summary of knowledge at that date. We have no evidence beyond his own statement that he was a practical observer. He theorised on the planetary motions, and held that the earth is fixed in the centre * Πληιάδας τ' ἐσορῶντε καὶ ὀψὲ δύοντα βοώτην "The Pleiades and Boötes that setteth late, and the Bear, which they likewise call the Wain, which turneth ever in one place, and keepeth watch upon Orion, and alone hath no part in the baths of the ocean." 2 See Pearson in the Camb. Phil. Soc. Proc., vol. iv., pt. ii., p. 93, on whose authority the above statements are made. 3 See p. 6 for definition. of the universe. He adopted the excentric and equant of Hipparchus to explain the unequal motions of the sun and moon. He adopted the epicycles and deferents which had been used by Apollonius and others to explain the retrograde motions of the planets. We, who know that the earth revolves round the sun once in a year, can understand that the apparent motion of a planet is only its motion relative to the earth. If, then, we suppose the earth fixed and the sun to revolve round it once a year, and the planets each in its own period, it is only necessary to impose upon each of these an additional annual motion to enable us to represent truly the apparent motions. This way of looking at the apparent motions shows why each planet, when nearest to the earth, seems to move for a time in a retrograde direction. The attempts of Ptolemy and others of his time to explain the retrograde motion in this way were only approximate. Let us suppose each planet to have a bar with one end centred at the earth. If at the other end of the bar one end of a shorter bar is pivotted, having the planet at its other end, then the planet is given an annual motion in the secondary circle (the epicycle), whose centre revolves round the earth on the primary circle (the deferent), at a uniform rate round the excentric. Ptolemy supposed the centres of the epicycles of Mercury and Venus to be on a bar passing through the sun, and to be between the earth and the sun. The centres of the epicycles of Mars, Jupiter, and Saturn were supposed to be further away than the sun. Mercury and Venus were supposed to revolve in their epicycles in their own periodic times and in the deferent round the earth in a year. The major planets were supposed to revolve in the deferent round the earth in their own periodic times, and in their epicycles once in a year. It did not occur to Ptolemy to place the centres of the epicycles of Mercury and Venus at the sun, and to extend the same system to the major planets. Something of this sort had been proposed by the Egyptians (we are told by Cicero and others), and was accepted by Tycho Brahe; and was as true a representation of the relative motions in the solar system as when we suppose the sun to be fixed and the earth to revolve. The cumbrous system advocated by Ptolemy answered its purpose, enabling him to predict astronomical events approximately. He improved the lunar theory considerably, and discovered minor inequalities which could be allowed for by the addition of new epicycles. We may look upon these epicycles of Apollonius, and the excentric of Hipparchus, as the responses of these astronomers to the demand of Plato for uniform circular motions. Their use became more and more confirmed, until_the_seventeenth century, when the accurate observations of Tycho Brahe enabled Kepler to abolish these purely geometrical makeshifts, and to substitute a system in which the sun became physically its controller. 4. The Reign of Epicycles-From Ptolemy to Copernicus. After Ptolemy had published his book there seemed to be nothing more to do for the solar system except to go on observing and finding more and more accurate values for the constants involved-viz., the periods of revolution, the diameter of the deferent,1 and its ratio to that of the epicycle, the distance of the excentric3 from the centre of the deferent, and the position of the 2 |