ANALYTIC GEOMETRY - THREE OR MORE DIMENSIONS In three dimensions, a point requires three co-ordinates; a single equation between them represents a surface. For a curve, two equations at least are necessary; since each by itself gives a surface, the points whose co-ordinates satisfy both equations are those lying on both surfaces, and the curve represented is their complete intersection. If this total intersection breaks up into distinct curves, to represent one curve k apart from the other, we use the equation of another surface which passes through k but not through the residual curve. There may or may not exist two surfaces whose total intersection is k; if not, k cannot be represented by less than three equations. Thus the intersection of two quadrics can consist of a line and a twisted cubic curve. We can represent the line alone by the equations of two planes passing through it, for these have no other common curve; but to express the cubic we use the equation of a third quadric through it, meeting the first two in different residual lines.
In space, the general bilinear equation in two sets of three variables ax-1-13y+ yz = I, according to the way we interpret it, represents either a point or a plane, and these elements are dual to each other. A line may be regarded either as the intersection of two planes or as the join of two points, and, in either point or plane co-ordinates, can be represented by two linear equations. The whole set of lines in space is a fourfold infinity, and a line requires four independent or five homogeneous co-ordinates. It is more. symmetrical and convenient to use a set of six homo geneous co-ordinates, which are connected by one quadratic identity, and whose ratios involve four independent quantities. If the line is defined as the intersection of two planes whose point equations are alx+a2y+aaz+a4 = o = 0 , the co-ordinates of the line are the six quantities P23, p31, p12, P14, p34, where Pap = aabl — a p ba, connected by the identity : p23P14+P31P24+ p12p34 — o.
As far as the algebra is concerned, there is no essential dif ference, except in complexity, between three dimensions and any higher number. Most people's imagination can deal with line, plane and space, and stops short there; but analysis is more than imagination. With proper extensions, the ideas and language suitable to the lower dimensions, and partly supplied by everyday life, can be carried on. We agree to call a set of n numbers the co-ordinates of a point in a space of n dimensions, the series of sets satisfying one linear equation a hyperplane, and so on. This may seem to be geometry in name only; but properties of lower space suggest valid and interesting extensions in higher space, that retain their geometrical tinge, and, conversely, many re sults in ordinary space have been suggested, or their mutual relations illuminated, by work in higher dimensions, which, to a specialist imagination, is as geometrical as the lower ranges are to everybody. Moreover, though it is not easy to imagine a four fold infinity of points, we are familiar with the higher orders of infinity in other objects, for example, all the lines of ordinary space, or all the conics in a plane. Thus the six homogeneous co-ordinates pap of a line in three dimensions may be interpreted as homogeneous co-ordinates of a point in space of five dimen sions, lying on the quadric in that space whose equation is given by the quadratic identity satisfied by the p's.
BIBLIOGRAPHY.-G. Salmon, A Treatise on Conic Sections, 6th ed. Bibliography.-G. Salmon, A Treatise on Conic Sections, 6th ed. (1879) ; A Treatise on Higher Plane Curves, 3rd ed. (Dublin, 5879) A Treatise on Analytic Geometry of Three Dimensions, 5th rev. ed. (1912-15) ; R. F. A. Clebsch, V orlesungen fiber Geometrie, ed. F. Lindemann (1906-10. Full references both to the classical authorities and to modern research are given in the Encyklopddie der mathe matischen Wissenschaften, Bd. iii., ed. W. F. Meyer and M. Mohrmann (Leipzig, 1903-15, in progress) . (H. P. Hu.)