NUMBEK AND MAGNITUDE 643 



pages. Further, there are infinitesimals which are infinitely small 

 with reference to those of the first order, and infinitely large with 

 respect to the order 1 + e, however small e may be. Here, then, are 

 new terms intercalated in our series; and if I may be permitted to 

 revert to the terminology used in the preceding pages, a terminology 

 which is very convenient, although it has not been consecrated by 

 usage, I shall say that we have created a kind of continuum of the 

 third order. 



It is an easy matter to go further, but it is idle to do so, for we 

 would only be imagining symbols without any possible application, and 

 no one will dream of doing that. This continuum of the third order, 

 to which we are led by the consideration of the different orders of 

 infinitesimals, is in itself of but little use and hardly worth quoting. 

 Geometers look on it as a mere curiosity. The mind only uses its 

 creative faculty when experiment requires it. 



(2) When we are once in possession of the conception of the math- 

 ematical continuum, are we protected from contradictions analogous to 

 those which gave it birth ? No, and the following is an instance : 



He is a savant indeed who will not take it as evident that every 

 curve has a tangent ; and, in fact, if we think of a curve and a straight 

 line as two narrow bands, we can always arrange them in such a way 

 that they have a common part without intersecting. Suppose now 

 that the breadth of the bands diminishes indefinitely: the common 

 part will still remain, and in the limit, so to speak, the two lines will 

 have a common point, although they do not intersect i.e., they will 

 touch. The geometer who reasons in this way is only doing what we 

 have done when we proved that two lines which intersect have a com- 

 mon point, and his intuition might also seem to be quite legitimate. 

 But this is not the case. We can show that there are curves which 

 have no tangent, if we define such a curve as an analytical continuum 

 of the second order. No doubt some artifice analogous to those we 

 have discussed above would enable us to get rid of this contradiction, 

 but as the latter is only met with in very exceptional cases, we need 

 not trouble to do so. Instead of endeavoring to reconcile intuition 

 and analysis, we are content to sacrifice one of them, and as analysis 

 must be flawless, intuition must go to the wall. 



Tlie Physical Continuum of Several Dimensions. We have dis- 

 cussed above the physical continuum as it is derived from the imme- 

 diate evidence of our senses , or, if the reader prefers, from the 

 rough results of Feclmer's experiments; I have shown that these 

 results are summed up in the contradictory f ormula? : A= B, B = 

 C, A > C. 



Let us now see how this notion is generalized, and how from it may 

 be derived the concept of continuums of several dimensions. Consider 

 any two aggregates of sensations. We can either distinguish between 



