A History of Greek Mathematics and Astronomy

Page 15

Lastly, Archimedes tells us that Democritus was the first to state, though he could not give a rigorous proof, that the volume of a cone or a pyramid is one-third of that of the cylinder or prism respectively on the same base and having equal height, theorems first proved by Eudoxus.

We come now to the time of Plato, and here the great names are Archytas, Theodoras of Cyrene, Theaetetus, and Eudoxus.

Archytas (about 430-360 B. C.) wrote on music and the numerical ratios corresponding to the intervals of the tetrachord. He is said to have been the first to write a treatise on mechanics based on mathematical principles; on the practical side he invented a mechanical dove which would fly. In geometry he gave the first solution of the problem of the two mean proportionals, using a wonderful construction in three dimensions which determined a certain point as the intersection of three surfaces, (1) a certain cone, (2) a half-cylinder, (3) an anchor-ring or tore with inner diameter nil.

Theodorus, Plato's teacher in mathematics, extended the theory of the irrational by proving incommensurability in certain particular cases other than that of the diagonal of a square in relation to its side, which was already known. He proved that the side of a square containing 3 square feet, or 5 square feet, or any non-square number of square feet up to 17 is incommensurable with one foot, in other words that √3, √5 ... √17 are all incommensurable with 1. Theodorus's proof was evidently not general; and it was reserved for Theaetetus to comprehend all these irrationals in one definition, and to prove the property generally as it is proved in Eucl. X. 9. Much of the content of the rest of Euclid's Book X (dealing with compound irrationals), as also of Book XIII on the five regular solids, was due to Theaetetus, who is even said to have discovered two of those solids (the octahedron and icosahedron).

Plato (427-347 B. C.) was probably not an original mathematician, but he 'caused mathematics in general and geometry in particular to make a great advance by reason of his enthusiasm for them'. He encouraged the members of his school to specialize in mathematics and astronomy; e. g. we are told that in astronomy he set it as a problem to all earnest students to find 'what are the uniform and ordered movements by the assumption of which the apparent motions of the planets may be accounted for'. In Plato's own writings are found certain definitions, e. g. that of a straight line as 'that of which the middle covers the ends', and some interesting mathematical illustrations, especially that in the second geometrical passage in the Meno (86E-87C). To Plato himself are attributed (1) a formula (n²-1)²+(2n)²=(n²+1)² for finding two square numbers the sum of which is a square number, (2) the invention of the method of analysis, which he is said to have explained to Leodamas of Thasos (mathematical analysis was, however, certainly, in practice, employed long before). The solution, attributed to Plato, of the problem of the two mean proportionals by means of a frame resembling that which a shoemaker uses to measure a foot, can hardly be his.

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

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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