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 threedimensional 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 noncontradictory propositions; such geometries are called nonEuclidean.
It is convenient to discuss the subjectmatter 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. NonEuclidean 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.
End of Article: GEOMETRY 

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