TRAINS OF REASONING. 293 



cible from the known laws of the propagation of motion 

 through an elastic medium ; while facts already empi- 

 rically known respecting sound, became an indication 

 of corresponding properties of vibrating bodies, pre- 

 viously undiscovered. 



But the grand agent for transforming experimental 

 into deductive sciences, is the science of number. The 

 properties of numbers, alone among all known phe- 

 nomena, are, in the most rigorous sense, properties 

 of all things whatever. All things are not coloured, 

 or ponderable, or even extended ; but all things are 

 numerable. And if we consider this science in its whole 

 extent, from common arithmetic up to the calculus of 

 variations, the truths already ascertained seem all but 

 infinite, and admit of indefinite extension. 



These truths, although aflirmable of all things 

 whatever, of course apply to them only in respect of 

 their quantity. But if it comes to be discovered that 

 variations of quality in any class of phenomena, corre- 

 spond regularly to variations of quantity either in 

 those same or in some other phenomena; every formula 

 of mathematics applicable to quantities which vary in 

 that particular manner, becomes a mark of a corre- 

 sponding general truth respecting the variations in 

 quality which accompany them : and the science of 

 quantity being (as far as any science can be) altogether 

 deductive, the theory of that particular kind of 

 qualities becomes, to this extent, deductive likewise. 



The most striking instance in point which history 

 affords (though not an example of an experimental 

 science rendered deductive, but of an unparalleled 

 extension given to the deductive process in a science 

 which was deductive already), is the revolution in 

 geometry which originated with the illustrious Des- 

 cartes, and was completed by Clairaut. These phi- 



