TRAINS OF REASONING. 249 



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, de- 

 ductive 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 de- 

 ductive process in a science which was deductive already), is 

 the revolution in geometry which originated with Descartes, 

 and was completed by Clairaut. 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 pecu- 

 liar relation of quantity between either two or three rectilineal 

 co-ordinates ; insomuch that if the law were known according 

 to which those co-ordinates vary relatively to one another, 

 every other geometrical property of the line or surface in 

 question, 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 

 algebraical one could ; and geometry received an accession 

 (actual or potential) 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 commonly so called, have 

 been made algebraical. The varieties of physical phenomena 

 with which those sciences are conversant, have been found to 

 answer to determinable varieties in the quantity of some 

 circumstance or other ; or at least to varieties of form or 

 position, for which corresponding equations of quantity had 

 already been, or were susceptible of being, discovered by 

 geometers. 



In these various transformations, the propositions of the 

 science of number do but fulfil the function proper to all pro- 



