iii.i GEOMETRY AND INDUCTION 215 



me yesterday to be the only thing wanting to complete 

 the system, will complete it to-morrow, no matter when 

 to-morrow may be. What is there at the base of this 

 belief? Notice that the belief is more or less assured, 

 according as the case may be, but that it is forced upon the 

 mind as an absolute necessity when the microcosm con- 

 sidered contains only magnitudes. If two numbers be 

 given, I am not free to choose their difference. If two 

 sides of a triangle and the contained angle are given, the 

 third side arises of itself and the triangle completes itself 

 automatically. I can, it matters not where and it matters 

 not when, trace the same two sides containing the same 

 angle: it is evident that the new triangles so formed can be 

 superposed on the first, and that consequently the same 

 third side will come to complete the system. Now, if 

 my certitude is perfect in the case in which I reason on 

 pure space determinations, must I not suppose that, in 

 the other cases, the certitude is greater the nearer it ap- 

 proaches this extreme case? Indeed, may it not be 

 the limiting case which is seen through all the others 

 and which colors them, accordingly as they are more or 

 less transparent, with a more or less pronounced tinge 

 of geometrical necessity? 1 In fact, when I say that 

 the water on the fire will boil to-day as it did yesterday, 

 and that this is an absolute necessity, I feel vaguely that 

 my imagination is placing the stove of yesterday on that 

 of to-day, kettle on kettle, water on water, duration on 

 duration, and it seems then that the rest must coincide 

 also, for the same reason that, when two triangles are 

 superposed and two of their sides coincide, their third 

 sides coincide also. But my imagination acts thus only 

 because it shuts its eyes to two essential points. For the 



1 We have dwelt on this point in a former work. See the Essai sur les 

 donnies immHiates de la conscience, Paris, 1889, pp. 155-160. 



