Page:A History of Mathematics (1893).djvu/76

 The foundation of this science was laid by the illustrious Hipparchus.

The Almagest is in 13 books. Chapter 9 of the first book shows how to calculate tables of chords. The circle is divided into 360 degrees, each of which is halved. The diameter is divided into 120 divisions; each of these into 60 parts, which are again subdivided into 60 smaller parts. In Latin, these parts were called partes minutœ primœ and partes minutœ secundœ. Hence our names, 'minutes' and 'seconds.'[3] The sexagesimal method of dividing the circle is of Babylonian origin, and was known to Geminus and Hipparchus. But Ptolemy's method of calculating chords seems original with him. He first proved the proposition, now appended to Euclid VI. (D), that "the rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides." He then shows how to find from the chords of two arcs the chords of their sum and difference, and from the chord of any arc that of its half. These theorems he applied to the calculation of his tables of chords. The proofs of these theorems are very pretty.

Another chapter of the first book in the Almagest is devoted to trigonometry, and to spherical trigonometry in particular. Ptolemy proved the 'lemma of Menelaus,' and also the 'regula sex quantitatum.' Upon these propositions he built up his trigonometry. The fundamental theorem of plane trigonometry, that two sides of a triangle are to each other as the chords of double the arcs measuring the angles opposite the two sides, was not stated explicitly by him, but was contained implicitly in other theorems. More complete are the propositions in spherical trigonometry.

The fact that trigonometry was cultivated not for its own sake, but to aid astronomical inquiry, explains the rather