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GEOMETRY

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Originally appearing in Volume V11, Page 675 of the 1911 Encyclopedia Britannica.
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GEOMETRY , the See also:

general See also:term for the See also:branch of See also:mathematics which has for its See also: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 See also:proof; and these axioms, in See also:conjunction with the mathematical entities of the point, straight See also:line, See also:curve, See also:surface and solid, appropriately defined, are the premises from which the geometer draws conclusions. The geometrical axioms are merely conventions; on the one See also:hand, the See also:system may be based upon inductions from experience, in which See also:case the deduced geometry may be regarded as a branch of See also:physical See also: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 See also: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 See also:frame systems of axioms which definitely characterize some other See also:kind of space, and from these axioms to deduce a See also:series of non-contradictory propositions; such geometries are called non-Euclidean. It is convenient to discuss the subject-See also: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 See also:account of See also: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. See also:Analytical Geometry: the See also:representation of geometrical figures and their relations by algebraic equations. V.

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

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

End of Article: GEOMETRY

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