The elementary plane geometry ordinarily studied in our schools is based directly, or in -directly through the work of Legendre, upon Euclid's Elements. Of this classic work, the first four and the sixth 'books' are devoted to plane geometry, that is, geometry in which the figures can all be imagined in one plane, even though, for purposes of superposition, they may be imagined as taken out of that plane in the course of the discussion. Euclid's treatment of solid geometry, in which the figures are imagined as occupying three dimensions, was so meagre that the elementary treatment of the sub ject to-day differs quite radically from that in the Elements. One of the principles of Eu clid's work now most often violated is the at tempt to avoid hypothetical constructions. For Euclid seeks to show how to construct each of the figures needed before he makes use of it. Thus, since it is impossible to trisect a general angle by the use of the compasses and the un marked straight edge, Euclid would have been estopped from asking such a question as, Do the arms of an angle, and the two lines which trisect the angle, trisect a transversal of these lines? At present it is more common to assume that the necessary figures can be constructed, and see what propositions can be proved from certain assumed postulates and axioms. Later, the question of the figures admitting of construc tion by the compasses and straight edge is con sidered by itself. Euclid's work is still used as a text-book in the schools of England and her colonies; but it has long since given way to a more modern treatment in most other countries.
The basis of ancient geometry as set forth in the Elements went practically unchallenged until the nineteenth century. The renewed interest in the science, growing out of the Renaissance, in spired the investigation of Euclid's assumptions, and led mathematicians to seek to demonstrate the fifth postulate or twelfth axiom (given by Brill as the eleventh), viz. that two unlimited
straight lines intersect on that side -of a trans versal on which the sum of the interior angles is less than a straight angle. Among the eminent mathematicians who sought to show the de pendence of this proposition upon those preceding it were Legendre and Gauss. Lobatehevsky and Bolyai were the first to construct a geometry in dependent of Euclid's assumption, and thus to found the so-called non-Euclidean geometry. Then at once followed a great advance toward exploring the new field, and from the researches of Riemann, Helmholtz, and Beltrami, it is con cluded that ten of the Euclidean assumptions are valid for all geometry, but that the one just mentioned and "two straight lines (or, more generally, two geodetic lines) include no space," are limited to the properties of particular space. Riemann and IIelmholtz formulated assumptions for a geometry in space of n-ply manifoldness and with constant curvature, and observed that on the sphere, whose curvature is constant and posi tive, the sum of the angles of a triangle is less than a straight angle, this characterizing the space of the geometry of Bolyai and Lobntehevsky. Klein has designated these three geometries re spectively, the elliptic, parabolic, and hyperbolic Starting with this broader view, many of the leading mathematicians of the last quarter of a century, including Cayley, Lie, Klein, Pasch, Killing, Fiedler, and Mansion, have given much attention and made valuable contributions to the subject of geometry.
Without questioning the validity of Euclidean geometry, there have grown out of it in modern times two great systems—an analytic, or coor dinate (see ANALYTIC GEOMETRY), and a syn thetic, or 'modern' geometry. The latter em braces descriptive and projective geometry, al though systems of coordinates have been intro duced also in the latter.