TRANSCENDENTAL GEOMETRY. 507 



TRANSCENDENTAL GEOMETRY. 



Br ALFEED C. LANE. 



TRANSCENDENTAL geometry is the geometry of solids and sur- 

 faces in n-dimensional or in curved space. Exactly what surfaces 

 and what solids is a hard question to answer, and the answer is still 

 harder to understand. Let us, then, first find out in what way or ways 

 the science of transcendental geometry arose. 



Descartes invented a method of applying algebra to geometry by 

 the well-known Cartesian co-ordinates. As you remember, a point in 

 a plane is determined by two co-ordinates, x and y, for example; a 

 point in space by three, x, y, and 2. Now, the question is not unnatu- 

 ral, " What would x, y, z, v, determine ? " The natural answer is, " A 

 point in space of four dimensions." 



Moreover, we see that, although we have no experience of space of 

 four dimensions, we could form equations between four variables, and 

 transform and combine them as we do in analytic geometry of three 

 dimensions. By adopting a code of interpretation as like to our ordi- 

 nary code as circumstances would permit, we could interpret the rela- 

 tions of our equations as geometrical relations. 



But, as the idea of a fourth dimension to space is almost if not 

 quite inconceivable, let us endeavor to render it less so if possible. 

 Imagine a man deprived of everything but vision, in the way of sen- 

 sible experience. The world to him would be two dimensional. If, 

 then, he were taken out to drive, he would see continual changes in his 

 plane of vision, but he would ascribe them merely to the effects of 

 time. Eor example, were he to go through a covered bridge, his sensa- 

 tions might be as follows : A small dark spot, gradually enlarging till 

 it covers the field of vision ; then a small bright spot in the middle of 

 it, which would similarly enlarge. 



Now, suppose our universe sliced in two by a plane which moved 

 along through it. Suppose sentient beings inhabited this plane. They 

 would perceive at once two dimensions of our universe and the third as 

 a succession in time. So we might suppose ourselves conscious of 

 three dimensions of our universe, and of the fourth as the succession 

 of things in time. Thus we might consider time as a dimension. It 

 is so considered in the mechanical curves of position. Yet we should 

 then have to bring in time relative to time. 



We will illustrate still further by considering the theory ^ ^ 

 of knots. It is evident that, so long as the line represented f C~W 

 in the adjoined figure is kept in the plane, the knot or kink I ^J 

 can not be got out of it. But, by turning the loop up, it 

 can be removed at once. 



The annexed knot the type of all knots in ordinary space can 



