GEOMETRY  , the

general term for the branch of mathematics which has for its province the study of the properties of space . From experience, or possibly intuitively, we characterize existent space by certain fundamental qualities, termed axioms, which are insusceptible of proof; and these axioms, in conjunction with the mathematical entities of the point, straight line, curve, surface and solid, appropriately defined, are the premises from which the geometer draws conclusions . The geometrical axioms are merely conventions; on the one hand, the system may be based upon inductions from experience, in which case the deduced geometry may be regarded as a branch of physical science; or, on the other hand, the system may be formed by purely logical methods, in which case the geometry is a phase of pure mathematics . Obviously the geometry with which we are most familiar is that of existent space—the three-dimensional space of experience; this geometry may be termed Euclidean, after its most famous expositor . But other geometries exist, for it is possible to frame systems of axioms which definitely characterize some other kind of space, and from these axioms to deduce a series of non-contradictory propositions; such geometries are called non-Euclidean . It is convenient to discuss the subject-matter of geometry under the following headings: I . Euclidean Geometry: a discussion of the axioms of existent space and of the geometrical entities, followed by a synoptical account of Euclid's Elements . II . Projective Geometry: primarily Euclidean, but differing from I. in employing the notion of geometrical continuity (q.v.)—points and lines at infinity . IV . Analytical Geometry: the representation of geometrical figures and their relations by algebraic equations . V .

Line Geometry: an analytical treatment of the line regarded as the space

element . VI . Non-Euclidean Geometry: a discussion of geometries other than that of the space of experience . of geometry . Special subjects are treated under their own. headings: e.g .