32 SCIENCE AND METHOD. 



with a calculation which has taught us apparently all 

 that we wished to know? The reason is that, in 

 analogous cases, the lengthy calculation might not be 

 able to be used again, while this is not true of the 

 reasoning, often semi-intuitive, which might have 

 enabled us to foresee the result. This reasoning 

 being short, we can see all the parts at a single glance, 

 so that we perceive immediately what must be changed 

 to adapt it to all the problems of a similar nature 

 that may be presented. And since it enables us to 

 foresee whether the solution of these problems will 

 be simple, it shows us at least whether the calculation 

 is worth undertaking. 



What I have just said is sufficient to show how vain 

 it would be to attempt to replace the mathematician's 

 free initiative by a mechanical process of any kind. 

 In order to obtain a result having any real value, it 

 is not enough to grind out calculations, or to have 

 a machine for putting things in order : it is not order 

 only, but unexpected order, that has a value. A 

 machine can take hold of the bare fact, but the soul 

 of the fact will always escape it. 



Since the middle of last century, mathematicians 

 have become more and more anxious to attain to 

 absolute exactness. They are quite right, and this 

 tendency will become more and more marked. In 

 mathematics, exactness is not everything, but without 

 it there is nothing : a demonstration which lacks 

 exactness is nothing at all. This is a truth that I 

 think no one will dispute, but if it is taken too 

 literally it leads us to the conclusion that before 1820, 

 for instance, there was no such thing as mathematics, 

 and this is clearly an exaggeration. The geomctri- 



