Two geometers, Bryson and .Antiphon, appear to have lived about the time of Hippocrates, and a little before Aristotle. These are only known by some animadversions of this last philosopher on their attempts to square the circle. It appears that before this time, geometers knew that the area of a circle was equal to a triangle whose base was equal to the circumference, and perpendicular equal to the radius.
Flaying briefly traced the progress of geometry during the two first ages after its introduction into Greece, we come now to the origin of the Platonic school, which may be considered as an xra in the history of the science. Its celebrated founder had been the disciple of a philosopher (Socrates) who set little value on geomen v ; but Plato en tertained a very different opinion of its utility. After the example of Thales and Pythagoras, he travelled into Egypt, to study under the priests. He also went into Italy, to consult the famous Pythagoreans, Philolaus, Ti mxus of Locris, and Archytas, and to Cyrene to hear the mathematician Theodorus. On his return to Greece, he made mathematics, and especially geometry, the basis of his instructions. He put an inscription over his school, forbidding any one to enter, that did riot understand gco• metry ; and when questioned concernin.; the probable em ployment of the Deity, he answered, that he geometrized continually, meaning no doubt that he governed the uni verse by geometrical laws.
It does not appear that Plato composed any work him self on mathematics, but he is reputed to have itiventid the Geometrical Analysis : (Sec ANALYSIS.) The theory of the Conic Sections originated iu this school ; some have even supposed that Plato himself invented it, but there does not seem to be any sufficient ground for this opinion. See CONIC SECTIONS.
A third discovery due to the Platonic school was that of the geometrical loci ; when the conditions which determine the position of a point are such as to admit of its being any where in Zi lime Of a particular kind, but do not admit of its being out of that line, then the line is called the locus of the point: Thus, if one end of a straight line of a given length be at a given point, the locus of the other end will be the circumference of a given circle : Again, if the base of a triangle of a given area be given in position and mag nitude, the locus of its vertex will be a given straight line, which will be parallel to the base ; also, if the base of a triangle be given in position and magnitude, and its verti cal angle be given in magnitude, the locus of its vertex will be the circumference of a given circle: all this is evi dent from the elements of geometry. Geometrical loci,
considered mere], as speculative truths, are interesting; but their chief value arises from their utility in the resolu tion of problems, of which, in general, they suggest the most elegant solutions. See Locus.
The celebrated problem concerning the duplication of the cube, acquired its celebrity about the time of Plato. Its origin, however, was earlier; for it appears, that Hip pocrates had reduced it to the determination of two mean proportionals between two given lines ; but it had not then excited much attention among geometers. \Ve have al ready given its history in the introduction to CONIC SEC TIONS. Plato himself gave a solution, and it was also resolv ed by Archytas, Eudoxus, Eratosthenes, and Alenzechmus. The solutions of eleven of the ancient geometers, are pre served in Eutocius' commentary on Archimedes, de SM. et Cyl.
It is probable that tire trisection of an angle, a problem of the same difficulty as the duplication of the cube, was likewise considered in the Platonic school. There is no absolute testimony of its being so ancient ; but, according to the natural progress of the human mind, it must have occurred as soon as geometry assumed the form of a sci ence ; for the transition from the bisection of an arc to its division into three, or any number of equal parts, or into parts which have a given ratio to one another, is easy. The quadrat•ix, a curve almost as old as the time of Plato, appears to have been invented with a view to the solution of the problem in its most general form. One difficulty in the problems of doubling a cube, and trisecting an an gle, must have arisen from the impossibility of resolving them by straight lines and circles alone; and of this the ancient geometrical analysis gave no certain indication. The modern analysis teaches how to resolve every such pro blem, and also spews by what fines it may be effected.