GEOMETRY

This article appears in Volume V11, Page 675 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: GAG-GEO

GEOMETRY , the general

term

TERM

for the branch of

mathematics

MATHEMATICS (Gr. /saBnľar1Kil, Sc. vOcvn or E7rlQTt'ľn; from p iN.La, "learning" or "science ")

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

SYSTEM

may be based upon inductions from experience, in which case the deduced geometry may be regarded as a branch of

physical

PHYSICAL

science; or, on the other hand, the

system

SYSTEM

may be formed by purely logical methods, in which case the geometry is a phase of pure

mathematics

MATHEMATICS (Gr. /saBnľar1Kil, Sc. vOcvn or E7rlQTt'ľn; from p iN.La, "learning" or "science ")

. Obviously the geometry with which we are most

familiar

FAMILIAR (through the Fr. familier, from Lat. familiaris, of or belonging to the familia, family)

is that of existent spacethe 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

FRAME

systems of axioms which definitely characterize some other

kind

KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

of space, and from these axioms to deduce a

series

SERIES (a Latin word from serere, to join)

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

ELEMENT (Lat. elementum)

.
VI. Non-Euclidean Geometry: a discussion of geometries other than that of the space of experience.
of geometry.

Special

SPECIAL

subjects are treated under their own. headings: e.g.

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