IS EUCLID'S GEOMETRY MERELY A THEORY? 561 



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 rect- 

 angles 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 mathe- 

 maticians a pseudosphere, and being simply Lobachevski's plane sur- 

 face 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 pseu- 

 dosphere 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 wiU still be the shortest dis- 

 tances 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 con- 

 trary, chosen a sphere, he might have arrived at the geometry of Rie- 

 mann, 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 com- 

 pletely round and form a great circle. Certain conclusions follow : 



