smooth in Lobachevski's space, but if transported into Euclidean space, it can only lie crinkly and fluted around its edges, for this environment, though boundless, is less roomy and the expanse will be too niggardly to accommodate the carpef s ample proportions. Suppose, in addition, that the carpet, while at the non-Euclidean factory, receives on its surface a checker-board pattern of black and white squares separated by criss-cross parallel lines, truly straight lines in the sense of being the shortest distances between points. The squares can not be perfect rectangles because, as already observed, such figures are not among the non-Euclidean possibilities. In Euclidean space, they would look like the black and white areas in Fig. 4. The figure is not, however, a perfect representation, because the carpet could not be made to lie flat in Euclidean surroundings without violent stretching, while to distend it would be to destroy the spatial relations of the lines of the pattern, after which, for geometrical purposes, it would no longer be the same carpet.
Spread out the carpet, nevertheless, as evenly as Euclidean space allows. Xo part will lie perfectly flat, of course; and only a limited portion can be made to lie smooth; the outlying portions will refuse to be spread out and must remain in folds. The smooth portion will then be slightly curved into the shape of a saddle, trending upward at front and back, and rolling off downward on either side, the whole forming a surface of constant negative curvature, called by mathematicians a pseudosphere, and being simply Lobachevski's plane surface after its transportation into the Euclidean environment. Upon such a surface we can draw diagrams suited to illustrate any problem in Lobachevski's plane geometry just as for Euclid's plane geometry we make use of the flat surface of a blackboard. Lines drawn on the pseudosphere can not be straight; they can only be the straightest lines that the surface will allow; but, limiting our discussion to lines lying wholly within the surface, these straightest lines will still be the shortest distances between points in the surface and would remain so, even if the surface were crushed into a wrinkled heap.
We do not know upon what kind of a surface Euclid drew his diagrams, perhaps upon sand, but it is reasonable to presume that it was approximately flat. Had he used a pseudospherical surface, he might have developed a different conception of space. Had he, on the contrary, chosen a sphere, he might have arrived at the geometry of Riemann, for the plane surface of Eiemannian space becomes simply a sphere under Euclidean conditions. The opportimity is so favorable just now that I may be permitted briefly to set down some of the results derived from this third type of geometry. Obviously, the straightest line that one could draw upon a sphere, as, for instance, by stretching a string between two points on the surface, would, if extended, go completely round and form a great circle. Certain conclusions follow:
