Page:Encyclopædia Britannica, Ninth Edition, v. 20.djvu/101

PTOLEMY 89 established. Ptolemy does not give them, hut in each case when required applies the theorem of Menelaus for spherics directly. This greatly increases the length of his demonstrations, which the modern reader finds still more cumbrous, inasmuch as in each case it was necessary to express the relation in terms of chords the equivalents of sines only, cosines and tangents being of later invention. Such, then, was the trigonometry of the Greeks. Mathe- matics, indeed, has ever been, as it were, the handmaid of astronomy, and many important methods of the former arose from the needs of the latter. Moreover, by the found- ation of trigonometry, astronomy attained its final general constitution, in which calculations took the place of dia- grams, as these latter had been at an earlier period sub- stituted for mechanical apparatus in solving the ordinary problems. 1 Further, we find in the application of trigon- ometry to astronomy frequent examples and even a sys^ tematic use of the method of approximations, the basis, in fact, of all application of mathematics to practical questions. There was a disinclination on the part of the Greek geometer to be satisfied with a mere approximation, were it ever so close ; and the unscientific agrimensor shirked the labour involved in acquiring the knowledge which was indispensable for learning trigonometrical cal- culations. Thus the development of the calculus of approximations fell to the lot of the astronomer, who was both scientific and practical. 2 We now proceed to notice briefly the contents of the Almagest. It is divided into thirteen books. The first book, which may be regarded as introductory to the whole work, opens with a short preface, in which Ptolemy, after some observations on the distinc- tion between theory and practice, gives Aristotle's division of the sciences and remarks on the certainty of mathematical knowledge, " inasmuch as the demonstrations in it proceed by the incontrovert- ible ways of arithmetic and geometry." He concludes his preface with the statement that he will make use of the discoveries of his predecessors, and relate briefly all that has been sufficiently explained by the ancients, but that he will treat with more care and develop- ment whatever has not been well understood or fully treated. Ptolemy unfortunately does not always bear this in mind, and it is sometimes difficult to distinguish what is due to him from that which he has borrowed from his predecessors. Ptolemy then, in the first chapter, presupposing some preliminary notions on the part of the reader, announces that he will treat in order what is the relation of the earth to the heavens, what is the position of the oblique circle (the ecliptic), and the situation of the inhabited parts of the earth ; that he will point out the differences of climates ; that he will then pass on to the consideration of the motion of the sun and moon, without which one cannot have a just theory of the stars ; lastly, that he will consider the sphere of the fixed stars and then the theory of the five stars called " planets." All these things i.e., the phenomena of the heavenly bodies he says he will endeavour to explain in taking for principle that which is evident, real, and certain, in resting everywhere on the surest observations and applying geometrical methods. He then enters on a summary exposition of the general principles on which his Syntaxis is based, and adduces arguments to show that the heaven is of a spherical form and that it moves after the manner of a sphere, that the earth also is of a form which is sensibly spherical, that the earth is in the centre of the heavens, that it is but a point in comparison with the distances of the stars, and that it has not any motion of translation. With respect to the revolution of the earth round its axis, which he says some have held, Ptolemy, while admitting that this supposition renders the explanation of the phenomena of the heavens much more simple, yet regards it as altogether ridiculous. Lastly, he lays down that there are two principal and different motions in the heavens one by which all the stars are carried from east to west uniformly about the poles of the equator ; the other, which is peculiar to some of the stars, is in a contrary direction to the former motion and takes place round different poles. These preliminary notions, which are all older than Ptolemy, form the subjects of the second and following chapters. He next proceeds to the construction of his table of chords, of which we have given an account, and which is indispensable to practical astronomy. The employment of this table presupposes the evaluation of the obliquity of the ecliptic, the knowledge of which is indeed the foundation of all astronomical science. Ptolemy in the next chapter indicates two means of determining this angle by observation, describes the instruments he employed for that purpose, and finds the same value which had already been found 1 Comte, Systeme de Politique Positive, iii. 324. 2 Cantor, Vorlesungen iiber Geschichte der Mathematik, p. 356. by Eratosthenes and used by Hipparchus. This "is followed by spherical geometry and trigonometry enough for the determination of the connexion between the sun's right ascension, declination, and longitude, and for the formation of a table of declinations to each degree of longitude. Delambre says he found both this and the table of chords very exact." 3 In book ii. , after some remarks on the situation of the habitable parts of the earth, Ptolemy proceeds to make deductions from the principles established in the preceding book, which he does by means of the theorem of Menelaus. The length of the longest day being given, he shows how to determine the arcs of the horizon intercepted between the equator and the ecliptic the amplitude of the eastern point of the ecliptic at the solstice for different 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 con- siders 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 observa- tions ; 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 par- ticulars 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 4 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 inter- sections 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 longi- tudes ; 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 especi- ally those of Hipparchus, whom he praises "as a lover of labour and a lover of truth" (avopi <pioir6v<f> re ofiov Kal <f>ia-r)0ei). He begins by telling us how Hipparchus was led to discover the pre- cession of the equinoxes ; he relates the observations which led Hipparchus to his second great discovery, that of the eccentricity of the solar orbit, 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 circum- stances on which the equation of time depends. All this the reader will find in the article ASTRONOMY (vol. ii. p. 750). 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. 5 This fine principle, 3 De Morgan, in Smith's Dictionary of Greek and Roman Biography, s.v. "Ptolemacus, Claudius." 4 KaipiKal, temporal or variable. These hours varied in length with the seasons ; they were used in ancient times and arose from the division of the natural day (from sunrise to sunset) into twelve parts. 6 Aim., ed. Halma, i. 159. XX. 12