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). Book I on the measurement of areas is specially interesting for (1) its statement of the formula used by Heron for finding approximations to surds, (2) the elegant geometrical proof of the formula for the area of a triangle Δ {D} = √{_s (s-a) (s-b) (s-c)}, a formula now known to be due to Archimedes, (3) an allusion to limits to the value of π {p} found by Archimedes and more exact than the 3-1/7 and 3-10/71 obtained in the _Measurement of a Circle_.