*THE POPULAR SCIENCE MONTHLY.*

not be undone without severing the ends. In four-dimensional space it could. By this means Zöllner interpreted some of the knot-untying performances of Slade, the American spiritualist. Let us, for example, interpret these facts, using time as fourth dimension, bringing in, of course, time relative to time. If time were a fourth dimension, parts of the state of things at different instants might be visible together. Thus we could have A, C, B, after being tied, joined to A, D, B, and then A, C, B, before being tied.

But we must remember, in passing on, that algebraic equations are capable of other than geometrical interpretations, and that their relations by themselves prove nothing in regard to real or possible relations between external facts. Moreover, the algebraic theory of dimensionality will be interpreted fully by nothing less than a space of infinite dimensionality.

We come now to the most difficult branch of the subject, that of curved surfaces and of curved space. The curvature of a plane curve at any point is the limit of the ratio of the length of the curve to the difference in direction of the initial and terminal tangents. Its differential expression is or . To get the curvature of a curved surface at any point, we slice it up by planes normal to it at that point. On each of these planes it will describe a curve. These curves will have different curvatures at the original point. The reciprocal of the product of the greatest and least of these is called by Gauss the measure of curvature. This name he also applied to an analogous function of the co-ordinates of a point in space. The expression, for a plane curve, of the curvature is the reciprocal of the radius of the circle of closest possible contact at the point investigated. Hence, some have argued that transcendental geometry was inconsistent, in that it talked about the curvature of a space where there were not Euclidean straight lines, hence no radii, and nothing to refer the curvature to. This argument is open to other answers, but it is enough to say that the measure of curvature has no necessary connection with radii.

To return, the condition that a rigid figure can be moved about on a surface without changing its shape, or that a rigid body can be similarly moved in space, is that the measure of curvature of the surface or space is constant in value. Some one might say that, if a body is rigid, no motion can change its shape. This, however, is not true of the mathematically rigid body except under the above conditions, taking the most general definition of a rigid body.

It is assumed in Euclid that motion of a figure does not alter it. That is, if an angle, A B C, is equal to an angle B C D, it will be equal to it however it is as a whole moved or rotated. This is an assumption that the measure of curvature of the plane or space is constant. Moreover, if we assume it constantly equal to naught, the