The influence of Plato (429-348 a.c.) on elementary geometry was greater than is usually supposed. He found the science in a disordered state, a mass of unrelated propositions, very likely covering much of plane geometry as found in Euclid. His philosophic mind led him to the attempt to put the science on a more satisfactory foundation by insisting upon accu racy of definition, upon a limited number of postulates (including axioms), and upon definite bounds to plane geometry. As a result, only those figures capable of construction by the help of an unmarked ruler and the compasses are recognized as belonging to the field of ele mentary geometry. To the school of Plato is also due the analytic method of attack in geom etry, including the reductio ad absurdum. Although not himself a great discoverer in mathematics, two of Plato's pupils reached high eminence in geometry. Of these the first was Eudoxus, who extended the theory of propor tion, founded the doctrine of similar figures, gave much attention to the problem of the golden section, applied the method of exhaus tions to the mensuration of solids, and wrote the first text-hook upon stereosnetry. The sec ond was Menmchmus, who, in his attempts to solve the duplication (or Delian) problem, dis covered the conic sections. The study of the five regular polyhedra also occupied the at tention of Plato's pupils, so much so that they received the name of Platonic bodies.
The influence of Aristotle was directed to the encouragement of the study of the history of geometry and the applications of mathe matics. As a result, his followers began to collate the work of the earlier Greeks and to consider its relation to physical problems. Ele mentary geometry now enters the text-book period and several attempts at works of this character appear in the 4th century, a.c. This
movement culminated in the works of Euclid (q.v.), a man of whose personal life we know practically nothing save that he taught and wrote in Alexandria c. 300 B.C. Probably of little originality in the way of mathematical discovery, Euclid had a genius for compilation, and this showed itself in the 2reigela (con nected series), or Elements, as it was called in later times. This famous work is devoted prin cipally to plane geometry, and it has formed the basis of practically all elementary treatises up to the present time. The natural effect of Euclid's work was to give the impression that the field of elementary plane geometry was ex hausted. Mathematicians therefore directed their energies to the applications of geometry, to stereometry and to conics. Archimedes (q.v.), writing at Syracuse c. 240 B.C., opened the great field of mathematical physics and car ried the study of elementary geometric solids to its greatest height among the Greeks. To him is also due the limits 3i and 3,14 for 7r, the study of the spiral that bears his name, and the quadrature of the parabola. Apollonius of Perga (qv.), ((the great wrote eight books on conics c. 225 a.c., and set a standard which still influences the text-books in analytic geometry. Of the minor geometers who fol lowed Apollonius, two may be mentioned. Nicomedes (c. 180 a.c.), who invented the con choid, a curve which easily solves the trisection problem, and his contemporary, Diodes, whose cissoid furnishes an easy means for duplicating the cube. Of the later Greek geometers the most noteworthy is Hero of Alexandria (see