294 REASONING. 



losophers remarked, that to every variety of position 

 in points, direction in lines, or form in curves or 

 surfaces, (all of which are Qualities,) there corre- 

 sponds 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 relating 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 (actual or potential) of new truths, cor- 

 responding 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 cor- 

 responding 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 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 cannot directly ascertain (or not so 

 conveniently) by experiment. We travel from a given 

 visible or tangible fact, through the truths of numbers, 

 to the fact sought. The given fact is a mark that a 

 certain relation subsists between the quantities of 



