146 



REASONING. 



phenomenon better known. Thus the 

 science of sound, which previously 

 stood in the lowest rank of merely 

 experimental science, became deduc- 

 tive when it was proved by experiment 

 that every variety of sound was con- 

 sequent on, and therefore a mark of, 

 a distinct and definable variety of 

 oscillatory motion among the particles 

 of the transmitting medium. When 

 this was ascertained it followed that 

 every relation of succession or co-ex- 

 istence which obtained between phe- 

 nomena of the more known class, 

 obtained also between the phenomena 

 which correspond to them in the 

 other class. Every sound, being a 

 mark of a particular oscillatory mo- 

 tion, became a mark of everything 

 which, by the laws of dynamics, was 

 known to be inferrible from that 

 motion ; and everything which by 

 those same laws was a mark of any 

 oscillatory motion among the particles 

 of an elastic medium became a mark 

 of the corresponding sound. And thus 

 many truths, not before suspected, 

 concerning sound become deducible 

 from the known laws of the propaga- 

 tion of motion through an elastic 

 medium ; while facts already empiri- 

 cally known respecting sound become 

 an indication of corresponding pro- 

 perties of vibrating bodies, previously 

 undiscovered. 



But the grand agent for trans- 

 forming experimental into deductive 

 sciences is the science of number. 

 The properties of number, alone 

 among all known phenomena, 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 ascer- 

 tained seem all but infinite, and 

 admit of indefinite extension. 



These truths, though affirmable of 

 all things whatever, of course apply 

 to them only in respect of their 

 quantity. But if it comes to be dis- 



covered that variations of quality in 

 any class of phenomena correspond 

 regularly to variations of quantity 

 either in those same or in some other 

 phenomena ; every formula of mathe- 

 matics applicable to quantities which 

 vary in that particular manner be- 

 comes a mark of a corresponding 

 general truth respecting the varia- 

 tions 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 be- 

 comes, to this extent, deductive like- 

 wise. 



The most striking instance in point 

 which history affords (though not an 

 example of an experimental science 

 rendered deductive, but of an unparal- 

 leled extension given to the deductive 

 process in a science which was deduc- 

 tive already) is the revolution in 

 geometry which originated with Des- 

 cartes and was completed by Clair- 

 aut. These great mathematicians 

 pointed out the importance of the 

 fact, that to every variety of position 

 in points, direction in lines, or form 

 in curves or surfaces, (all of which 

 are Qualities,) there corresponds a 

 peculiar relation of quantity between 

 either two or three rectilineal co-ordi- 

 nates ; insomuch that if the law were 

 known according to which those co- 

 ordinates vary relatively to one an- 

 other, every other geometrical pro- 

 perty of the line or surface in ques- 

 tion, whether relating to quantity or 

 quality, would be capable of being 

 inferred. Hence it followed that 

 every geometrical question could be 

 solved, if the corresponding alge- 

 braical one could ; and geometry 

 received an accession (actual or poten- 

 tial) of new truths, corresponding to 

 every property of numbers which the 

 progress of the calculus had brought, 

 or might in future bring, to light. In 

 the same general manner, mechanics, 

 astronomy, and in a less degree every 

 branch of natural philosophy com- 

 monly so called, have been made 

 algebraical. The varieties of physi- 



