Book I

, containing the necessary preliminaries to the study of the Ptolemaic system, gives a Table of Chords in a circle subtended by angles at the centre of ½° increasing by half-degrees to 180°. The circle is divided into 360 μοιραι {moirai}, parts or degrees, and the diameter into 120 parts (τμηματα {tmêmata}); the chords are given in terms of the latter with sexagesimal fractions (e. g. the chord subtended by an angle of 120° is 103^{p} 53′ 23″). The Table of Chords is equivalent to a table of the _sines_ of the halves of the angles in the table, for, if (crd. 2 α {a}) represents the chord subtended by an angle of 2 α {a} (crd. 2 α {a})/120 = sin α {a}. Ptolemy first gives the minimum number of geometrical propositions required for the calculation of the chords. The first of these finds (crd. 36°) and (crd. 72°) from the geometry of the inscribed pentagon and decagon; the second ('Ptolemy's Theorem' about a quadrilateral in a circle) is equivalent to the formula for sin (θ-φ) {th-ph}, the third to that for sin ½ θ {th}. From (crd. 72°) and (crd. 60°) Ptolemy, by using these propositions successively, deduces (crd. 1½°) and (crd. ¾°), from which he obtains (crd. 1°) by a clever interpolation. To complete the table he only needs his fourth proposition, which is equivalent to the formula for cos (θ+φ) {th+ph}.

Ptolemy wrote other minor astronomical works, most of which survive in Greek or Arabic, an _Optics_ in five Books (four Books almost complete were translated into Latin in the twelfth century), and an attempted proof of the parallel-postulate which is reproduced by Proclus.

Heron of Alexandria (date uncertain; he may have lived as late as the third century A. D.) was an almost encyclopaedic writer on mathematical and physical subjects. He aimed at practical utility rather than theoretical completeness; hence, apart from the interesting collection of _Definitions_ which has come down under his name, and his commentary on Euclid which is represented only by extracts in Proclus and an-Nairīzī, his geometry is mostly mensuration in the shape of numerical examples worked out. As these could be indefinitely multiplied, there was a temptation to add to them and to use Heron's name. However much of the separate works edited by Hultsch (the _Geometrica_, _Geodaesia_, _Stereometrica_, _Mensurae_, _Liber geëponicus_) is genuine, we must now regard as more authoritative the genuine _Metrica_ discovered at Constantinople in 1896 and edited by H. Schöne in 1903 (Teubner).