DIMENSION, a term used in geometry to denote a magni tude measured in a specified direction, as, for instance, along a diameter or a principal axis or an edge. A point is said to be with out dimension; a line has the one dimension of length, a surface has the two dimensions of length and breadth, while a solid has the three dimensions of length, breadth and thickness. Since the lengths of lines, the areas of surfaces and the volumes of solids are represented respectively by linear, quadratic and cubic alge braic expressions, the term dimension has been carried over into algebra. Thus quadratic, cubic, biquadratic algebraic expressions or equations are said to be respectively of two, three, four dimen sions. Similarly, the term dimensions is used in mechanics with reference to the units of time, length and mass and various derived units (see UNITS, DIMENSIONS OF), and it occurs likewise in many ofher parts of physics, notably in the theory of electricity and magnetism (see PHYSICAL UNITS).
The fundamental descriptive proposition concerning space, as we are accustomed to it in experience, is that space is a continuum (a continuous or unbroken distribution of points) having three dimensions. The intuitive basis of this proposition may be eluci dated as follows. If on a curve (or line) we mark certain points (elements without dimension) we separate the curve into parts bounded by the points in such a way that we cannot pass along the curve from one part to another without encountenng and passing over one of these marked points. Since the curve .may be separated into parts by elements without dimensions it is itself said to be a figure having one dimension. But a surface cannot be thus separated into parts by marking isolated points on it; for, in going from one place to another on the surface, we can always avoid passing over these marked points by going around them. If we draw in the surface a suitable closed curve (a figure of one dimension) then the surface is separated into parts in such a way that we cannot move over the surface from one part to another without encountering and passing over a point on the curve. Since a surface can not be separated into parts by points (figures without dimension) but can be separated into parts by a suitable figure of one dimension, we say that a sur face itself has two dimensions. Similarly space cannot be sepa rated into parts by isolated points or curves or both taken to gether, while it can be so separated into parts by means of a closed surface (a figure of two dimensions). For this reason we say that space has three dimensions. This, according to Henri Poincare (Dernieres Pensees, Flammarion, Paris, 1917, pp. 6I ff.), is the fundamental qualitative ground for ascribing three dimen sions to the usual space of experience.
The mathematician introduces three co-ordinates to represent the points of ordinary space and much of his analysis of its prop erties is carried out algebraically. Now the algebraic analysis is competent to deal with sets of any number n of co-ordinates. Thus algebraic geometry leads readily to the conception of spaces of any number n of dimensions; and these have been extensively treated, though no one has a lively mental picture of spaces having more than three dimensions. Popular interest in these higher spaces (as opposed to the interest of mathematicians in them) has been centered principally around the concept of the "fourth dimension." But the number of dimensions of these higher spaces is unlimited; and, in fact, several kinds of space with an infinite number of dimensions have been investigated.
That the higher spaces may be given a concrete representa tion in terms of experience is shown by the fact that the totality of straight lines in our usual space of three dimensions constitutes a veritable space of four dimensions (see PROJECTIVE GEOMETRY). This arises from the fact that four independent co-ordinates are necessary to define completely the position in space of a line of unlimited extent. Therefore a geometry in which the elements are the lines of ordinary space is a geometry of four dimensions.
(R. D. CA.)