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 degrees of obliquity of the sphere; hence he finds the height of the pole and reciprocally. From the same data he shows how to find at what places and times the sun becomes vertical and how to calculate the ratios of gnomons to their equinoctial and solstitial shadows at noon and conversely, pointing out, however, that the latter method is wanting in precision. All these matters he considers fully and works out in detail for the parallel of Rhodes. Theon gives us three reasons for the selection of that parallel by Ptolemy: the first is that the height of the pole at Rhodes is 36°, a whole number, whereas at Alexandria he believed it to be 30° 58′; the second is that Hipparchus had made at Rhodes many observations; the third is that the climate of Rhodes holds the mean place of the seven climates subsequently described. Delambre suspects a fourth reason, which he thinks is the true one, that Ptolemy had taken his examples from the works of Hipparchus, who observed at Rhodes and had made these calculations for the place where he lived. In chapter vi. Ptolemy gives an exposition of the most important properties of each parallel, commencing with the equator, which he considers as the southern limit of the habitable quarter of the earth. For each parallel or climate, which is determined by the length of the longest day, he gives the latitude, a principal place on the parallel, and the lengths of the shadows of the gnomon at the solstices and equinox. In the next chapter he enters into particulars and inquires what are the arcs of the equator which cross the horizon at the same time as given arcs of the ecliptic, or, which comes to the same thing, the time which a given arc of the ecliptic takes to cross the horizon of a given place. He arrives at a formula for calculating ascensional differences and gives tables of ascensions arranged by 10° of longitude for the different climates from the equator to that where the longest day is seventeen hours. He then shows the use of these tables in the investigation of the length of the day for a given climate, of the manner of reducing temporal to equinoctial hours and vice versa, and of the nonagesimal point and the point of orientation of the ecliptic. In the following chapters of this book he determines the angles formed by the intersections of the ecliptic—first with the meridian, then with the horizon, and lastly with the vertical circle—and concludes by giving tables of the angles and arcs formed by the intersection of these circles, for the seven climates, from the parallel of Meroe (thirteen hours) to that of the mouth of the Borysthenes (sixteen hours). These tables, he adds, should be completed by the situation of the chief towns in all countries according to their latitudes and longitudes; this he promises to do in a separate treatise and has in fact done in his Geography.

Book iii. treats of the motion of the sun and of the length of the year. In order to understand the difficulties of this question Ptolemy says one should read the books of the ancients, and especially those of Hipparchus, whom he praises “as a lover of labour and a lover of truth” (undefined). He begins by telling us how Hipparchus was led to discover the precession of the equinoxes; he relates the observations by which Hipparchus verified the eccentricity of the solar orbit imperfectly known to his Chaldaean predecessors, and gives the hypothesis of the eccentric by which he explained the inequality of the sun’s motion. Ptolemy concludes this book by giving a clear exposition of the circumstances on which the equation of time depends. Ptolemy, moreover, applies Apollonius’s hypothesis of the epicycle to explain the inequality of the sun’s motion, and shows that it leads to the same results as the hypothesis of the eccentric. He prefers the latter hypothesis as more simple, requiring only one and not two motions, and as equally fit to clear up the difficulties. In the second chapter there are some general remarks to which attention should be directed. We find the principle laid down that for the explanation of phenomena one should adopt the simplest hypothesis that it is possible to establish, provided that it is not contradicted by the observations in any important respect. This fine principle, which is of universal application, may, we think—regard being paid to its place in the Almagest—be justly attributed to Hipparchus. It is the first law of the “philosophic prima” of Comte. We find in the same page another principle, or rather practical injunction, that in investigations founded on observations where great delicacy is required we should select those made at considerable intervals of time in order that the errors arising from the imperfection which is inherent in all observations, even in those made with the greatest care, may be lessened by being distributed over a large number of years. In the same chapter we find also the principle laid down that the object of mathematicians ought to be to represent all the celestial phenomena by uniform and circular motions. This principle is stated by Ptolemy in the manner which is unfortunately too common with him—that is to say, he does not give the least indication whence he derived it. We know, however, from Simplicius, on the authority of Sosigenes that Plato is said to have proposed the following problem to astronomers: “What regular and determined motions being assumed would full account for the phenomena of the motions of the planetary bodies?” We know, too, from the same source that Eudemus says in the second book of his History of Astronomy that “Eudoxus of Cnidus was the first of the Greeks to take in hand hypothesis of this kind,” that he was in fact the first Greek astronomer who proposed a geometrical hypothesis for explaining the periodic motions of the planets—the famous system of concentric spheres. It thus appears that the principle laid down here by Ptolemy can be traced to Eudoxus and Plato; and it is probable that they derived it from the same source, namely, Archytas and the Pythagoreans. We have indeed the direct testimony of Geminus of Rhodes that the Pythagoreans endeavoured to explain the phenomena of the heavens by uniform and circular motions.

Books iv., v. are devoted to the motions of the moon, which are very complicated; the moon in fact, though the nearest to us of all the heavenly bodies, has always been the one which has given the greatest trouble to astronomers. Book iv., in which Ptolemy follows Hipparchus, treats of the first and principal inequality of the moon, which quite corresponds to the inequality of the sun treated of in the third book. As to the observations which should be employed for the investigation of the motion of the moon, Ptolemy tells us that lunar eclipses should be preferred, inasmuch as they give the moon’s place without any error on the score of parallax. The first thing to be determined is the time of the moon’s revolution; Hipparchus, by comparing the observations of the Chaldaeans with his own, discovered that the shortest period in which the lunar eclipses return in the same order was 126,007 days and 1 hour. In this period he finds 4267 lunations, 4573 restitutions of anomaly and 4612 tropical revolutions of the moon less 7° q.p.; this quantity (7°) is also wanting to complete the 345 revolutions which the sun makes in the same time with respect to the fixed stars. He concluded from this that the lunar month contains 29 days and 31′ 50″ 8‴ 20⁗ of a day, very nearly, or 29 days 12 hours 44′ 3″ 20‴. These results are of the highest importance. In order to explain this inequality, or the equation of the centre, Ptolemy makes use of the hypothesis of an epicycle, which he prefers to that of the eccentric. The fifth book commences with the description of the astrolabe of Hipparchus, which Ptolemy made use of in following up the observations of that astronomer, and by means of which he made his most important discovery, that of the second inequality in the moon’s motion, now known by the name of the “evection.” In order to explain this inequality he supposed the moon to move on an epicycle, which was carried by an eccentric whose centre turned about the earth in a direction contrary to that of the motion of the epicycle. This is the first instance in which we find the two hypotheses of eccentric and epicycle combined. The fifth book treats also of the parallaxes of the sun and moon, and gives a description of an instrument—called later by Theon the “parallactic rods” devised by Ptolemy for observing meridian altitudes with greater accuracy.

The subject of parallaxes is continued in the sixth book of the Almagest, and the method of calculating eclipses is there given. The author says nothing in it which was not known before his time.

Books vii., viii. treat of the fixed stars. Ptolemy verified the fixity of their relative positions and confirmed the observations of Hipparchus with regard to their motion in longitude, or the precession of the equinoxes. The seventh book concludes with the catalogue of the stars of the northern hemisphere, in which are entered their longitudes, latitudes and magnitudes, arranged according to their constellations; and the eighth book commences with a similar catalogue of the stars in the constellations of the southern hemisphere. This catalogue has been the subject of keen controversy amongst modern astronomers. Some, as Flamsteed and Lalande, maintain that it was the same catalogue which Hipparchus had drawn up 265 years before Ptolemy, whereas others, of whom Laplace is one, think that it is the work of Ptolemy himself. The probability is that in the main the catalogue is really that of Hipparchus altered to suit Ptolemy’s own time, but that in making the changes which were necessary a wrong precession was assumed. This is Delambre’s opinion; he says, “Whoever may have been the true author, the catalogue is unique, and does not suit the age when Ptolemy lived; by subtracting 2° 40′ from all the longitudes it would suit the age of Hipparchus; this is all that is certain.” It has been remarked that Ptolemy, living at Alexandria, at which city the altitude of the pole is 5° less than at Rhodes, where Hipparchus observed, could have seen stars which are not visible at Rhodes; none of these stars, however, are in Ptolemy’s catalogue. The eighth book contains, moreover, a description of the milky way and the manner