5 o8 THE POPULAR SCIENCE MONTHLY. 



not be undone without severing the ends. In four-dimensional space it 

 could. By this means Zollner 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 to- 

 gether. 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 rela- 

 tions by themselves prove nothing in regard to real or possible relations 

 between external facts. Moreover, the algebraic theory of dimen- 

 sionality will be interpreted fully by nothing less than a space of infi- 

 nite 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 differ- 

 ence in direction of the initial and terminal tangents. Its differential 



expression is D t s or rnr ^xyyjr ^ S et tne 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 investi- 

 gated. 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 simi- 

 larly 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 



