GEOMETRY, Cartesian. Between num ber and the properties of number, on the one hand, and space and the properties of space, on the other, there is, strictly speaking, no re semblance; and the science of number, i.e., algebra or analysis, and the science of space, i.e., geometry, are essentially, psychologically, i and logically independent doctrines. But de spite their fundamental unlikeness and inde pendence, there is between the two, broadly speaking, a fact-to-fact correspondence. For example, there subsists, or may be established, a unique and reciprocal, or one-to-one, corre spondence between the real numbers and the points of a straight line or other curve; between the real numbers and the lines of a flat pencil (see Paojecrtvt GEOMETRY) or the tangents to a curve; between the pails of real numbers and the points or the lines of a plane; between the triplets of real numbers and the circles of a plane or the points or planes of space; between the quartets (permutations four at a time) of the real numbers and the lines or the spheres of space. (See LINE GEOMETRY AND ALLIED Tnrnaias). The theory of the correspondence thus simply exemplified, the logically organic body of propositions setting it forth, is the science called analytic or algebraic geometry. It is often called co-ordinate geometry from the fact that the set of numbers determining or corresponding to a geometric element are called the co-ordinates of the element. By virtue of the correlation between analytic facts and geometric facts, it is frequently possible, when facts of the one type are known, to infer the corresponding facts of the other,• and so to investigate space analytically (algebraically, arithmetically) and to investigate number geometrically. Under either of these aspects, analytical geometry appears as a method: ana lytic investigation of geometry, geometric in vestigation of analysis. Usually it is the former aspect under which the doctrine is regarded, geometry being the subject-matter, and analysis the means or instrument of re search.
The science presents numerous branches or varieties. These differ among themselves in various ways. Two varieties may differ in respect to what is often called their %paces,* i.e., in respect to the domains, fields, regions, or extents (as curve, surface, space) containing the configurations with which they deal. Thus arise such distinctive designations as (on, in) a plane, or plane geometry, geometry on a surface or a curve, geometry of space. Again, a given or domain may be con ceived in countless ways. It may be con ceived as the assemblage of its points or of its lines or of its circles, and so on. Accordingly two geometries relating to a same ((space or domain may yet differ in respect to their primary elements, in respect, i.e., to the ele
ments of which the configurations investigated are regarded as composed. So arise, for ex ample, such distinct theories as the point, line, circle, . . . , geometries of the plane, and the point, line, plane, circle, sphere, . . . , geome tries of space. Once more, as will appear in this and related articles herein cited, chosen element in any given domain may be referred to different kinds of configurations of refer ence; it may, in other words, be determined by, made to correspond or be associated with, different kinds of co-ordinate systems. Upon the choice of co-ordinate system depends, ceteris paribus, the analytic form of a given geometric theory. ,Accordingly, two geome tries that are identical in content may differ in form, in algebraic guise or garb.
The primitive, by far the oldest, variety of analytic geometry, the parent of all other varieties, is the Cartesian, so called from its founder, Rene Descartes (1596-1650). Though originally a plane geometry, its procedure is equally adapted, and has been extended, to spaces of every dimensionality in points. (See HYPERSPACES). It is characterized partly by its primary element, the point, and partly by its co-ordinate system, which will be explained raison et chercher la viriti, dans les sciences, published in 1637, is to be regarded, on account of its influence on mathematics and upon knowl edge in general, as one of the very greatest contributions ever made to science. Descartes was not indeed the first to apply algebra to geometry. That had been done by the great geometer,' Apollonius of Perga (about 260 200 a.c.), who had referred the conic sections to their tangents and diameters, expressing the relations by linear equations between areas. In the 14th century Oresme and others had applied numbers (tlatitudo" and precursors of the modern ordinate and ab scissa) to refer a point to two chosen rectan gular lines or axes. The point was confined, however, to the first quadrant. In this way the straight line, the circle and the parabola were studied. Other predecessors of Descartes were Vieta (1540-1603), Cavalieri (159£1-1647), Roberval (1602-75), and the brilliant Fermat (1601-65), who more nearly than any other approaches Descartes in his understanding of the analytic method. Even Fermat, however, had apparently not seen what Descartes saw, the possibility of referring at once to a single co-ordinate configuration different curves of different orders.
The following paragraphs give a very brief account of the elements of Cartesian, or ordinary analytical, geometry with special reference to the straight line and the conic section and the simplest configurations of space.