Definition of Hyperspace of Points. What, then, is a hyperspace of points! How is the notion arrived at? And what is its utility? The values of a single continuous variable x are familiarly representable by the points of a right line; the ordered pairs of values of two independent variables x, and x,, by the points of a plane; and the ordered trip lets of values of three independents x,, xi, by the points of ordinary space. To the analyst with geometric bias or predilection, the sug gestion immediately and forcibly presents itself and there ought to be a space whose points would serve to represent, as in the preceding cases, all ordered sets of values of n independ ent variables xi, xi, . . . xn. Such a space is not present to intuition, vision, or visual imagination. The mathematician is not in the least concerned, however, whether his two or three dimensional geometry appeals to any visi ble space or not. His geometry is simply the theory of a certain structure which is exempli fied by the system of pairs or triads of numbers, and after an imperfect fashion by the space of our experience. The transition to the theory of the systems exemplified by number tetrads, pentads or n-ads is simple and natural.
Co-ordinates, etc.In point space of n dimensions the simplest co-ordinates of the point are the distances xi, x2, . . . xn of the point from n mutually perpendicular point spaces of n 1 dimensions. These co-ordinate spaces, taken n 1 at a time, determine n co ordinate axes. A linear equation elm em+ en,+ 1 = 0 defines or represents an n 1 dimensional space of order one, the analogue of the plane in ordinary space. The $'s are the negative reciprocals of the axal intercepts of the n 1-space. Holding the x's fixed and letting the 's vary, the foregoing equation will repre sent a point as envelope of its generating 21-1-spaces. Two such equations together de fine an n 2-space as their intersection or a straight line as their envelope. Similarly, three such equations serve to represent an n 3-space as locus of points or a plane as envelope of n 1-spaces, and so on. A space that is n-fold in points is also n-fold in spaces of n 1-di mensions. Its dimensionality is 2(n 1) alike in lines and in spaces of n 2 dimensions. In general, its dimensionality is p + 1) if the point space either of p or of n dimen sions be taken as generating element. Not only, however, do the two last-mentioned elements furnish the same dimensionality, but they are indeed reciprocal elements of n-fold point space, for the same system of equations which on proper interpretation defines one of the ele ments admits of a second (dual) interpreta tion defining the other. It thus appears that by taking as elements the various simple spaces of less than n dimensions for generating ele ments of n-fold point space, there arise n geometries of this space; or, if we regard two reciprocal theories as but two aspects of i one geometry, the elements in question yield n : 2 or 1 n 1) :2 geometries according as n is even or odd, the element having (n 1) :2 dimensions being, in case of n odd, its own reciprocal, or self-reciprocal, like the line in ordinary space. See LINE GEOMETRY AND
ALLIED THEORIES.
Remarks on Four-space.Thus point space of four dimensions is also 4-dimensional in ordi nary spaces (say lineoids), the point and the lineoid being reciprocal elements. It is 6-di mensional in lines and also in planes, which are also reciprocal elements of this space. It appears that this space, unlike ordinary space, does not admit of self-reciprocal construction. An equation of degree n in point (lineoid) co-ordinates xi, x,, xs, x3, ($3, $3, E. represents a locus (envelope) of order (class) n. If n1,the locus (envelope) is a lineoid (point). Two linear equations define a plane as locus or a line as envelope; three, if independent, repre sent a line as locus or a plane as envelope; and four give a point or a lineoid. In general, two planes have, not a line, but only a point in common; reciprocally, two planes are not in general in a same lineoid. A lineoid being de termined by four independent points, it appears that two arbitrary lines determine a lineoid. In 4-space a point can pass from the inside to the outside of a (2-dimensional) closed surface, such as an ordinary sphere, without going through the surface, just as in ordinary space a point can pass from the inside to the outside of a circle without crossing the circumference. Accordingly, in 4-space a 3-fold solid like the human body could be literally seen through, and no ordinary prison-house could confine.
Do hyperspaces exist? Undoubtedly they have logical existence, the concept of hyper space being interiorly consistent and available for thought. More mathematics does not de mand. The hypothesis of their ex istence, science may yet be compelled to employ. Indeed it has been conjectured that certain chemical phenomena (as of the carbon compounds) may be due to greater freedom of motion than ordinary space affords. How ever, except in so far as time and space are part of one 4-dimensional system, as is re quired by the theory of relativity (q.v.) the ob servations of physics do not find any clear evi dence of a motion in the fourth dimension.
Bibliography. The literature of the geom etry (both pure and analytical) of hyperspaces is very extensive. It is, however, chiefly con tained in the mathematical journals. All scien tific nations have contributed to the subject, the Italians probably more than any other. The best work for the beginner is P. H. Schoute's (Mehrdimensionale Geometric' (1902). An excellent explanation, addressed to the non mathematician, of the concept of 4-space is found in Hermann Schubert's Essays and Recreations.'