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, parts or degrees, and the diameter into 120 parts (τμηματα); the chords are given in terms of the latter with sexagesimal fractions (e. g. the chord subtended by an angle of 120° is 103p 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) represents the chord subtended by an angle of 2 α (crd. 2 α)/120 = sin α. 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), the third to that for sin ½ θ. 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).
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 = √{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 found by Archimedes and more exact than the 3-1/7 and 3-10/71 obtained in the Measurement of a Circle.
Book I of the Metrica calculates the areas of triangles, quadrilaterals, the regular polygons up to the dodecagon (the areas even of the heptagon, enneagon, and hendecagon are approximately evaluated), the circle and a segment of it, the ellipse, a parabolic segment, and the surfaces of a cylinder, a right cone, a sphere and a segment thereof. Book II deals with the measurement of solids, the cylinder, prisms, pyramids and cones and frusta thereof, the sphere and a segment of it, the anchor-ring or tore, the five regular solids, and finally the two special solids of Archimedes’s Method; full use is made of all Archimedes’s results. Book III is on the division of figures. The plane portion is much on the lines of Euclid’s Divisions (of figures). The solids divided in given ratios are the sphere, the pyramid, the cone and a frustum thereof. Incidentally Heron shows how he obtained an approximation to the cube root of a non-cube number (100). Quadratic equations are solved by Heron by a regular rule not unlike our method, and the Geometrica contains two interesting indeterminate problems.
Heron also wrote Pneumatica (where the reader will find such things as siphons, Heron’s Fountain, penny-in-the-slot machines, a fire-engine, a water-organ, and many arrangements employing the force of steam), Automaton-making, Belopoeïca (on engines of war), Catoptrica, and Mechanics. The Mechanics has been edited from the Arabic; it is (except for considerable fragments) lost in Greek. It deals with the puzzle of ‘Aristotle’s Wheel’, the parallelogram of velocities, definitions of, and problems on, the centre of gravity, the distribution of weights between several supports, the five mechanical powers, mechanics in daily life (queries and answers). Pappus covers much the same ground in Book VIII of his Collection.
We come, lastly, to Algebra. Problems involving simple equations are found in the Papyrus Rhind, in the Epanthema of Thymaridas already referred to, and in the arithmetical epigrams in the Greek Anthology (Plato alludes to this class of problem in the Laws, 819 B, C); the Anthology even includes two cases of indeterminate equations of the first degree. The Pythagoreans gave general solutions in rational numbers of the equations x²+y²=z² and 2x²-y²=±1, which are indeterminate equations of the second degree.
The first to make systematic use of symbols in algebraical work was Diophantus of Alexandria (fl. about A. D. 250). He used (1) a sign for the unknown quantity, which he calls αριθμος, and compendia for its powers up to the sixth; (2) a sign (
) with the effect of our minus. The latter sign probably represents ΛΙ, an abbreviation for the root of the word λειπειν (to be wanting); the sign for αριθμος (
