so-called geometrical axioms that two straight lines can not inclose a surface, etc., are true. For example, a spherical surface has a constant measure of curvature not equal to zero, and positive. Since the shortest distance between two points is a straight line, let us, extending the analogy, call the shortest distance between two points of a spherical surface, lying wholly in that surface, a straight line of that surface. Now, as the measure of curvature of a spherical surface is constant, we can slide a figure about over the surface without altering it, as is evident at once. On a sphere, however, more than one perpendicular can be drawn on the surface from a point to a straight line, and two straight lines can inclose a surface.
In a surface whose curvature is negative, an infinite number of straight lines of the surface can be drawn through a given point which will never meet a given straight line. Such a surface would be like a spool. Some of its sections would be concave and others convex to the same point. We have analogous results in what is called curved space. These results were first suggested by Riemann, who was a pupil of Gauss.
For this mathematical treatment all that is needed is, first, algebra and the differential calculus; secondly, a method of interpreting them geometrically. We have found a code of interpretation for some algebraic equations which give geometrical results, and we apply it so far as we can to all.
So far the mathematicians might have gone without let or hindrance, and there some of them, as Boole and Grassman, stopped. But others thought they had settled whether the geometrical axioms were a priori truths or not. We have just worked out a system of geometry, said they, which is not, as we think, impossible, where these axioms do not hold. Therefore these axioms are the results of an experience of things as they are. If we had had a different order of things, as is possible, these axioms would not have been true nor thought of. I shall, however, try to prove that, although not thought of, they are true.
The geometrical axioms express relations; relations between what?
Geometry is a branch of mathematics. Therefore the geometrical axioms express mathematical relations. What, then, is mathematics, and with what does it deal?
Mathematics, in its widest sense, I will define as the science which treats of logical—that is necessary—relations. Between outside things there are no necessary relations. The relation of cause and effect is sometimes called necessary; but, if so, it is not usually handled mathematically. The relations must, then, be of mental things.
They are not relations between images or imaginations of outside things, for two reasons: First, the relations between imaginations can be no more necessary than the things they image; second, the imaginations of men's minds are different. One may imagine a line as a