258 MATHEMATICAL CLASSIFICATION 



relations as a certain other system in an old science, which has already been 

 reduced to a mathematical form, and its problems solved by mathematicians. 



Thus, when Mossotti observed that certain quantities relating to electro- 

 static induction in dielectrics had been shewn by Faraday to be analogous to 

 certain quantities relating to magnetic induction in iron and other bodies, he 

 was enabled to make use of the mathematical investigation of Poisson relative 

 to magnetic induction, merely translating it from the magnetic language into 

 the electric, and from French into Italian. 



Another example, by no means so obvious, is that which was originally 

 pointed out by Sir William Thomson, of the analogy between problems in 

 attractions and problems in the steady conduction of heat, by the use of vhic-h 

 we are able to make use of many of the results of Fourier for heat in explaining 

 electrical distributions, and of all the results of Poisson in electricity in explaining 

 problems in heat. 



But it is evident that all analogies of this kind depend on principles of a 

 more fundamental nature ; and that, if we had a true mathematical classification 

 of quantities, we should be able at once to detect the analogy between any 

 system of quantities presented to us and other systems of quantities in known 

 sciences, so that we should lose no time in availing ourselves of the mathe- 

 matical labours of those who had already solved problems essentially the same. 



All quantities may be classed together in one respect, that they may be 

 defined by means of two factors, the first of which is a numerical quantity, and 

 the second is a standard quantity of the same kind with that to be defined. 



Thus number may be said to rule the whole world of quantity, and the 

 four rules of arithmetic may be regarded as the complete equipment of the 

 mathematician. 



Position and form, which were formerly supposed to be in the exclusive 

 possession of geometers, were reduced by Descartes to submit to the rules of 

 arithmetic by means of that ingenious scaffolding of co-ordinate axes which he 

 made the basis of his operations. 



Since this great step was taken in mathematics, all quantities have been 

 treated in the same way, and presented to the mind by means of numbers, or 

 symbols which denote numbers, so that as soon as any science has been 

 thoroughly reduced to the mathematical form, the solution of problems in that 

 science, as a mental process, is supposed (at least by the outer world) to be 

 carried on without the aid of any of the physical ideas of the science. 



