GEOMETRY, the science of space, discusses and investigates the properties of definite portions of space under the fourfold division of lines, angles, surfaces, and volumes, without regard to any physical properties which they may have. It has various divisions, e.g., plane and solid geometry, analytical or algebraical geometry, descriptive geom etry, and the higher geometry. Plane and solid geometry are occupied with the consideration of right lines and plane surfaces, and with the solids generated by them, as well as with the properties of the circle, and, it may be said, the sphere; while the higher geometry considers the conic sections and curved lines generally, and the bodies generated by them. In the higher geometry, immense advances have recently been made through improved methods, the application of modern analysis, and the various calculi in algebraical geometry, the nature of which is explained in the article Co-or dinates (q.v.). Descriptive geometry, a division of the science so named by Monge (q.v.), is properly an extension or general application of the principle of projections (q.v.), its object being to represent on two plane-surfaces the elements and character of any solid figure. It has many practical applications. When one surface penetrates another, for instance, there often result from their intersection curves of double curva ture, the description of which is necessary in some of the arts, as in groined vault-work, and in cutting arch-stones, etc. and this is supplied by descriptive geometry.
The history of geometry is etc., of interest, but no more can be given here than a very bare sketch of it. The name of the science (Gr. and Lat. geometria) originally signified the art of measuring land. Herodotus, the earliest authority on the subject, assigns the origin of the art to the necessity of measuring lands in Egypt for the purposes of taxa tion, in the reign of Sesostris, about 1416-1357 n.e (Hero, book ii., chap. 109). This is probable, not only as resting on such authority, but also because, a priori, we should expect the necessity. of measuring lands to arise with property in land, and to give birth to the art. Of the of the science, however, among the Chaldeans and Egyptians, we have no record.
The story of Herodotus is further confirmed by tradition. Proclus, in his commes tary on Euclid's Elements (b. ii. c. 4), says that the art was brought to Greece from
Egypt by Thales,- who was himself a great discoverer in geometry. The Greeks at once took keenly to the study; various disciples of Thales excelled in it, chief among them Pythagoras, who, according to Proclus, first gave geometry the form of a deductive science, besides discovering some of its most important elementary propositions, among others, it is said, the 47th Prop. 'Enc. b. I. See article PYTHAGORAS for a notice of his other contributions to the science. Pythagoras had illustrious successors: Anaxagoras of Clazomenm; 2Enopidis, the reputed discoverer of Enc. b. I. 12,23; Briso and Autipho; Hippocrates of Chios, who "doubled the cube," and quadrated the lunula, which bear his name, and is said to have written a treatise on geometry; Zenodorns; Democritus of Abdera; and Theodortts of Cyrene, who is said to have been one of the instructors of Plato, whose name marks an epoch in the history of the science. his academy at Athens, Plato placed the celebrated inscription, ifedeis agcometretos eisito (` Let no one ignorant of geometry enter here"), thus recognizing it as the first of the sciences, and as the proper introduction to the higher philosophy. He is the reputed inventor of the method of geometrical analysis, and of geometrical loci and the conic sections, called in his time the higher geometry. From his academy proceeded many who advanced the science, of whom Proclus mentions thirteen, and more than one of them as having written treatises on the subject,.that have been lost. We shall mention but two of these: Eudoxus, who is said to have brought into form and order in a treatise the results of the studies at the academy, and to have invented the doctrine of proportion, as treated in the 5th book of Euclid's Elements; and the great-Aristotle, who assigned geometry as high a place as Plato did, and who wrote a treatise on the subject, as did at least two of his pupils, Theophrastus and Eudemus, from the latter of whom Proclus took most of his facts. Autolycus, a disciple of this Theophrastus, wrote a treatise on the movable sphere, yet extant; while Aristmus, the reputed instructor of Euclid in geometry, is said to have written five books on the conic sections, and five ou solid loci, all of which are lost.