A History of Greek Mathematics and Astronomy

Page 30

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.

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T. L. Heath
A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath
A History of Greek Mathematics and Astronomy