SEQUEL TO THE EPOCH OF H1PFARCHUS. 167 



struction even in modern times. Another writer on the same subject 

 is Menelaus, who lived somewhat later, and whose Three Books on 

 Spherics still remain. 



One of the most important kinds of deduction from a geometrical 

 theory, such as that of the doctrine of the sphere, or that of epicycles, 

 is the calculation of its numerical results in particular cases. With 

 regard to the latter theory, this was done in the construction of Solar 

 and Lunar Tables, as we have already seen ; and this process required 

 the formation of a Trigonometry, or system of rules for calculating the 

 relations between the sides and angles of triangles. Such a science 

 had been formed by Hipparchus, who appears to be the author of 

 every great step in ancient astronomy. 21 He wrote a work in twelve 

 books, " On the Construction of the Tables of Chords of Arcs ;" such a 

 table being the means by which the Greeks solved their triangles. 

 The Doctrine of the Sphere required, in like manner, a Spherical 

 Trigonometry, in order to enable mathematicians to calculate its re- 

 sults ; and this branch of science also appears to have been formed by 

 Hipparchus, 22 who gives results that imply the possession of such a 

 method. Hypsicles, who was a contemporary of Ptolemy, also made 

 some attempts at the solution of such problems : but it is extraor- 

 dinary that the writers whom we have mentioned as coming after 

 Hipparchus, namely, Theodosius, Cleomedes, and Menelaus, do not 

 even mention the calculation of triaugles, 23 either plain or spherical ; 

 though the latter writer 24 is said to have written on "the Table of 

 Chords," a work which is now lost. 



We shall see, hereafter, how prevalent a disposition in literary ages 

 is that which induces authors to become commentators. This tendency 

 showed itself at an early period iu the school of Alexandria. Aratus, ss 

 who lived 270 b. c. at the court of Antigonus, king of Macedonia, de- 

 scribed the celestial constellations in two poems, entitled " Phenome- 

 na," and "Prognostics." These poems were little more than a versifi- 

 cation of the treatise of Eudoxus on the acronycal and heliacal risings 

 and settings of the stars. The work was the subject of a comment by 

 Hipparchus, who perhaps found this the easiest way of giving connec- 

 tion and circulation to his knowledge. Three Latin translations of this 

 poem gave the Romans the means of becoming acquainted with it : 

 the first is by Cicero, of which we have numerous fragments ex- 



« Delamb. A. A. ii. 87. « A. A. i. 117. 23 A. A. i. 249. 



« A. A. ii. 37. 25 A. A. i. 74. 



