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He was, so far as is known, the first compiler of a book of Elements; and he was the first to prove the important theorem of Eucl. XII. 2 that circles are to one another as the squares on their diameters, from which he further deduced that similar segments of circles are to one another as the squares on their bases.
The rule is stated in general terms, but the above representation of its effect shows that it is a piece of pure algebra. The Pythagorean contributions to geometry were even more remarkable. The most famous proposition attributed to Pythagoras himself is of course the theorem of Eucl.
Warned by the unanswerable arguments of Zeno against infinitesimals, mathematicians substituted for this the statement that, by continuing the construction, we can inscribe a polygon approaching equality with the circle as nearly as we please. The method of exhaustion used, for the purpose of proof by reductio ad absurdum, the lemma proved in Eucl.
Another problem solved by the Pythagoreans was that of drawing a rectilineal figure which shall be equal in area to one given rectilineal figure and similar to another. Plutarch mentions a doubt whether it was this problem or the theorem of Eucl. I. 47 on the strength of which Pythagoras was said to have sacrificed an ox.
Four such numbers, namely 6, 28, 496, 8128, were known to Nicomachus. The law of formation for such numbers is first found in Eucl. Secondly, Theon of Smyrna gives the law of formation of the series of 'side-' and 'diameter-' numbers which satisfy the equations 2x²-y²=±1. The law depends on the proposition proved in Eucl.
The most elementary case appears in Eucl. I. 44, 45, where it is shown how to apply to a given straight line as base a parallelogram with one angle equal to a given angle and equal in area to any given rectilineal figure; this construction is the geometrical equivalent of arithmetical division.
The method of exhaustion is seen in operation in Eucl. XII. 1-2, 3-7 Cor., 10, 16-18. Props. 3-7 Cor. and Prop. 10 prove that the volumes of a pyramid and a cone are one-third of the prism and cylinder respectively on the same base and of equal height; and Archimedes expressly says that these facts were first proved by Eudoxus.
This would be in accord with the Greek instinct for generalization and their wish to be able to account for everything on rational principles. Eucl. Elementary as these things are, they represent a new departure of a momentous kind, being the first steps towards a theory of geometry.
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. 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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