Excerpt from Miscellanies, Vol. 5: Chiefly Academic
This was the first Science cultivated by Demonstration, which means, Complete Proof. Plato may have hoped that it would supersede the mere dogma of Pythagoreans. Much sound pro gress had been made in Geometry earlier than Socrates; but this sturdy thinker was alienated rom all superhuman science (72; ampo’ma) by the mutual refutations of the too rapid speculators. The practical value of Geometry was well known in the Athens of Pericles; indeed the grossly false satire of Socrates in the Clouds of Aristophanes may indicate that the elements of Astronomy also were becoming a staple of education for aristo cratic youth. Certainly in the next generation Plato took up Geometry with enthusiasm, and with a panegyric which passed all bounds. He is said to have written up over his gate Let no one enter, who is untaught in Geometry, as though it were indispensable to all who might seek instruction in his philosophy. To his followers accordingly the Science was indebted for some notable additions. Among them are named the celebrated problems of Trisecting an Angle, and what we now call Ex tracting the Cube Root, geometrically. To solve these by rule and compass certain new curves were imagined. The geometrical method of Analysis is ascribed to Plato himself.
It may seem paradoxical to assert, that the successes of the Greeks in Geometry were a false light, damaging to their philosophy: at least we think it so acted on Plato, who seems always to import the sphere and the triangle into moral thought. The sphere was his perfect form.
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