Other famous geometers could also be named, but such men as Euclid, Archimedes, and Apollonius are enough to show that geometry was cultivated to a great extent by the philosophers of antiquity. It progressively advanced, like philosophy itself, from the time of Thales until it had reached the perfection of which it was capable, when it became merged into astronomical science. It was cultivated more particularly by the disciples of Plato, who placed over his school this inscription: "Let no one ignorant of geometry enter here." He believed that the laws by which the universe is governed are in accordance with the doctrines of mathematics. The same opinion was shared by Pythagoras, the great founder of the science, whose main formula was that _number_ is the essence or first principle of all things. No thinkers ever surpassed the Greeks in originality and profundity; and mathematics, being highly prized by them, were carried to the greatest perfection their method would allow. They did not understand algebra, by the application of which to geometry modern mathematicians have climbed to greater heights than the ancients; but then it is all the more remarkable that without the aid of algebraic analysis they were able to solve such difficult problems as occupied the minds of Archimedes and Apollonius. No positive science can boast of such rapid development as geometry for two or three hundred years before Christ, and never was the intellect of man more severely tasked than by the ancient mathematicians.