TRAINS OF REASONING. 167 



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 corresponding general truth re- 

 specting the variations in quality which accompany them: and the sci- 

 ence 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 Avhich history affords (though not 

 an example of an experimental science rendered deductive, but of an un- 

 paralleled extension given to the deductive 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 peculiar 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 re- 

 lating to quantity or quality, would be capable of being inferred. Hence 

 it followed that every geometrical question could be solved, if the corre- 

 sponding algebraical one could ; and geometry received an accession (act- 

 ual or potential) of new truths, corresponding to every property of num- 

 bers 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 va- 

 rieties 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 num- 

 ber do but fulfill the function proper to all propositions forming a train of 

 reasoning, viz., that of enabling us to arrive in an indirect method, by 

 marks of marks, at such of the properties of objects as we can not direct- 

 ly ascertain (or not so conveniently) by experiment. We travel from a 

 given visible or tangible fact, through the truths of numbers, to the facts 

 sought. The given fact is a mark that a certain relation subsists between 

 the quantities of some of the elements concerned ; while the fact sought 

 presupposes a certain relation between the quantities of some other ele- 

 ments : now, if these last quantities are dependent in some known manner 

 upon the former, or vic^ versa, we can argue from the numerical relation 

 between the one set of quantities, to determine that which subsists be- 

 tween the other set; the theorems of the calculus affording the intermedi- 

 ate links. And thus one of the two physical facts becomes a mark of the 

 other, by being a mark of a mark of a mark of it. 



