GEOMETRY, History of the Elements of. The. history of the science of geometry begins in Greece. It is true that mensuration was developed to a considerable extent at an early period in Egypt, Babylonia and India, and that this work involved the measurement of angles in the astronomical observations of the people of these countries, but the abstract science of form never attained any prominence before the Greek period. In Egypt, for exam ple, the harpedonapue (rope stretchers) knew the right-angled triangle whose sides are 3, 4, 5 and stretched their ropes to lay out a right angle in much the same way that a modern sur veyor erects a perpendicular by the help of his chain; but there is no evidence that the Egyp tians thought of proving the Pythagorean The orem. Herodotus testifies to the fact that the Egyptians divided the land that was subject to the overflow of the Nile into quadrilaterals, and therefore they must have had some knowl edge of elementary surveying. Indeed, long before this time Ahmes (see ALGEBRA, HISTORY OF THE ELEMENTS OF) gave certain rules, partly incorrect, for measuring areas and vol umes, in particular an interesting one for the area of the circle, d)'. All of this work was, however, very elementary, and the rules were merely the result of unscientific ob servation.
In Greece the science of geometry may be said to have begun with Thales (q.v.), who was born at Miletus, c. 640 a.c., and who died at Athens in 548. Brought up in contact with the learning that drifted from the East to the shores of the Mediterranean, in his younger days devoted to those commercial pursuits that made Miletus a centre of wealth, he traveled extensively and devoted his later years to losophy. From Egypt he seems to have taken back to Ionia whatever of primitive geometry was known, and his school at Miletus was voted to the study of philosophy, astronomy, and the science of form. Thales is supposed to have proved the propositions concerning the equality of vertical angles, the sides opposite the equal angles of a triangle, the tion of a triangle by one side and two angles, the bisection of a circle by a diameter, and the nature of an angle inscribed in a semicircle. His most famous pupil was Pythagoras (q.v.), i who was born at Samos c. 580 and died in southern Italy c. 501. A man of great personal
magnetism, a mystic, and versed in the lore of the Orient, Pythagoras made his school at Cro tona the mathematical centre of his time. Al though none of his works is extant, if, in deed, he wrote any, it is known that he proved the famous theorem which bears his name (Euclid I, 47), a proposition already known empirically to the Egyptians and Chinese, and probably to the Hindus, at least for special cases. Pythagoras also gave much attention to the study of proportions and irrational quanti ties, always from the standpoint of geometry. He also knew the size of the angle of a regular n-gon, and the stellar pentagon was made the badge of his order.
The century following Pythagoras was one of discovery. Among the most noted geometers was Hippocrates of Chios, c. 440 B.C., who must not be confounded with the great physician who may have been the author of a treatise on the mystic number 7. Hippocrates, who had come in contact with the Pythagoreans, wrote the first Greek text-book on mathematics and designated the geometric figures by letters placed at the angles. To him is due the first example of the quadrature of a curvilinear fig ure, a proposition known as the lunes of Hip pocrates. The theorem asserts that if semi circles be described on the three sides of an isosceles right-angled triangle in such way as to form lunes on the two shorter sides, the area of the lunes equals that of the triangle. In his attempt to duplicate the cube he showed that the problem can be solved if two mean propor tionals can be found between e and 2e, where e is the edge. This problem was one of the three famous problems of antiquity, the others being to square the circle and to trisect any given angle. It is now known that these problems, easily solved if the necessary instruments are allowed, cannot be solved merely by the use of an unmarked ruler and a pair of compasses. Contemporary with Hippocrates lived Hippias of Elis, to whom is probably due the quadra trix which Dinostratus afterward studied and named. About the same time Antiphon and Bryson sought the quadrature of the circle by means of inscribed and circumscribed polygons, the number of sides being successively doubled, and with them begin the theories of limits and of exhaustions.