During this period the great philosophic school of Plato (429-348 B.C.) flourished at Athens, and to this school is due the first systematic attempt to create exact definitions, axioms, and postulates, and to distinguish between elementary and higher geometry. It was at this time that elementary geometry became limited to the use of the compasses and the unmarked straightedge, which took from this domain the possibility of constructing a square equivalent to a given circle ("squaring the circle"), of trisecting any given angle, and of constructing a cube that should have twice the volume of a given cube ("duplicating the cube"), these being the three famous problems of antiquity. Plato and his school interested themselves with the so-called Pythagorean numbers, that is, with numbers that would represent the three sides of a right triangle and hence fulfill the condition that _a_^2 + _b_^2 = _c_^2. Pythagoras had already given a rule that would be expressed in modern form, as 1/4(_m_^2 + 1)^2 = _m_^2 + 1/4(_m_^2 - 1)^2. The school of Plato found that [(1/2_m_)^2 + 1]^2 = _m_^2 + [(1/2_m_)^2 - 1]^2. By giving various values to _m_, different Pythagorean numbers may be found. Plato's nephew, Speusippus (about 350 B.C.), wrote upon this subject. Such numbers were known, however, both in India and in Egypt, long before this time. One of Plato's pupils was Philippus of Mende, in Egypt, who flourished about 380 B.C. It is said that he discovered the proposition relating to the exterior angle of a triangle. His interest, however, was chiefly in astronomy. Another of Plato's pupils was Eudoxus of Cnidus (408-355 B.C.). He elaborated the theory of proportion, placing it upon a thoroughly scientific foundation. It is probable that Book V of Euclid, which is devoted to proportion, is essentially the work of Eudoxus. By means of the method of exhaustions of Antiphon and Bryson he proved that the pyramid is one third of a prism, and the cone is one third of a cylinder, each of the same base and the same altitude. He wrote the first textbook known on solid geometry. The subject of conic sections starts with another pupil of Plato's, Menaechmus, who lived about 350 B.C. He cut the three forms of conics (the ellipse, parabola, and hyperbola) out of three different forms of cone,--the acute-angled, right-angled, and obtuse-angled,--not noticing that he could have obtained all three from any form of right circular cone. It is interesting to see the far-reaching influence of Plato. While primarily interested in philosophy, he laid the first scientific foundations for a system of mathematics, and his pupils were the leaders in this science in the generation following his greatest activity.