The _Sphaerica_ of Theodosius of Bithynia (written, say, 20 B. C.) contains no trigonometry. It is otherwise with the _Sphaerica_ of Menelaus (fl. A. D. 100) extant in Arabic; Book I of this work contains propositions about spherical triangles corresponding to the main propositions of Euclid about plane triangles (e.g. congruence theorems and the proposition that in a spherical triangle the three angles are together greater than two right angles), while Book III contains genuine spherical trigonometry, consisting of 'Menelaus's Theorem' with reference to the sphere and deductions therefrom. Ptolemy's great work, the _Syntaxis_, written about A. D. 150 and originally called Μαθηματικη συνταξις {Mathêmatikê syntaxis}, came to be known as Μεγαλη συνταξις {Megalê syntaxis}; the Arabs made up from the superlative μεγιστος {megistos} the word al-Majisti which became _Almagest_. 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.