TRANSCENDENTAL geometry is the geometry of solids and surfaces 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 z. Now, the question is not unnatural, "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 ordinary code as circumstances would permit, we could interpret the relations 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 sensible 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. For example, were he to go through a covered bridge, his sensations 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 in the adjoined figure is kept in the plane, the knot or kink 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