AND ITS SELF- CONSERVATION. 99 



application of mathematics to them, must keep within the 

 limits of measure, seeing that they have constantly to do 

 with the quantitative phases of the extended world, and 

 must, therefore, bear the mark of "relativity" inherent 

 in all things within the realm of measure. 



In sober truth, that the application of mathematics to 

 the actual extended world may be brought within the 

 range of finite powers of thinking, it is necessary to con- 

 fine the calculations to a simple set of relations more or 

 less arbitrarily chosen, and to regard this set of relations 

 as if completely isolated from the rest of the universe. 

 For example, Thomson and Tait, in their " Treatise on 

 Natural Philosophy," call special attention to the fact 

 that even in so simple a case as that of the investigation 

 of the lever it is necessary to assume that a lever is a 

 bar, perfectly rigid, inflexible, and without weight an 

 assumption which, of course, can never be realized. 



In short, the assumption made in every single instance 

 in the application of mathematics to the concrete sciences 

 is more or less in direct contradiction to the actual facts. 

 Or, if not exactly this, at least all except certain more or 

 less arbitrarily chosen aspects of those facts are of neces- 

 sity ignored in each and every problem proposed. 



It is true that the very purpose of the mathematical 

 phase of the sciences is to discover the exact measure of 

 things. And yet the really exact is not the approximately 

 exact. The former is, no doubt, that which is desired, 

 though the latter is the utmost that is ever actually 

 attained. The " exact sciences" propose an ideal which 

 they can never hope to realize ; and this is inevitable from 

 the very nature of the case. The so-called exact sciences 

 are necessarily restricted to the realm of measure that is, 



