82 THE FOURTH DIMENSION. 



There is no necessity of our entering in any detail into the charac- 

 ter and structure of the last-mentioned surfaces. 



Intimately related with the plane, however, are all those sur- 

 faces, which, like the plane, have the curvature zero ; in this cate- 

 gory belong especially cylindrical surfaces and conical surfaces. A 

 sheet of paper of the form of the sector of a circle may, for exam- 

 ple, be readily bent into the shape of a conical surface. If two con- 

 gruent triangles, now, be drawn on the sheet of paper, which may 

 by displacement be translated the one into the other, these triangles 

 will, it is plain, also remain congruent on the conical surface ; that 

 is, on the conical surface also we may displace the one into the 

 other; for though a bending of the figures will take place, there 

 will be no distension or contraction. Similarly, there are surfaces 

 which, like the sphere, have everywhere a constant positive curva- 

 ture. On such surfaces also every figure can be transferred into 

 some other position without distension or contraction of its parts. 

 Accordingly, on all surfaces thus related to the plane or sphere, 

 the assumption which underlies the eighth axiom of Euclid, that it 

 is possible to transfer into any new position any figure drawn on 

 such surfaces without distortion, holds good. 



The eleventh axiom in its turn also holds good on all surfaces 

 of constant curvature, whether the curvature be zero or positive ; 

 only in such instances instead of "straight" line we must say 

 " shortest "line. On the surface of a sphere, namely, two shortest 

 lines, that is, arcs of two great circles, always intersect, no matter 

 whether they are produced in the direction of the side at which the 

 third arc of a great circle makes with them angles less than two 

 right angles, or, in the direction of the oth^r side, where this arc 

 makes with them angles of more than two right angles. On the 

 plane, however, two straight lines intersect only on the side where 

 a third straight line that meets them makes with them interior an- 

 gles less than two right angles. 



The twelfth axiom of Euclid, finally, only holds good on the 

 plane and on the surfaces related to it, but not on the-sphere or other 

 surfaces which, like the sphere, have a constant positive curvature. 

 This also accounts for the fact that one of the three postulates 



