MATHEMATICAL SCIENCES SEVENTEENTH CENT. 153 



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In this case three co-ordinates are required for each point*, ajfjd theii 

 relations are expressed by two equations, each involving a pair of! the, 

 variables. By another extension of Descartes' idea, oujytd surface* 

 may be represented, and thus the geometry df solids also'is brought 

 within the grasp of algebraical analysis. To explairi to/ the non-mathe- 

 matical reader the manner in which the geometrical properties of curves, 

 etc., are deducible from their equations, would be to write a treatise on 

 analytical geometry. It must suffice to say, that a great part of such 

 investigation consists in determining the directions, etc., of tangents, 

 and that the geometry of co-ordinates easily furnishes general methods 

 for such problems. 



Descartes not only reconstructed the science of geometry and made 

 important improvements in algebraical analysis, but he was also the 

 author of a famous " System of the World," and he published besides 

 treatises on optics, on anatomy, and various metaphysical subjects. 

 His theory of the world was an attempt to explain the causes of all 

 phenomena by principles which were deduced from a few axioms first 

 assumed as the simplest and most obvious propositions. From the 

 fundamental axioms we have already mentioned (page 148), and certain 

 ideas as to the attributes of the Deity, Descartes supposed that he could 

 reason down to the law of nature. He does not, however, wholly re- 

 ject experiment and observation, to which, of course, inductive methods 

 must be applied. The problem he proposes is the converse of Bacon's, 

 who from effects sought to infer the cause. Descartes expressly says : 

 " W T e employ experiment, but not as a reason by which anything is 

 proved ; for we wish to deduce effects from their causes, and not causes 

 from their effects. We appeal to experience only in order that we may 

 be able, out of the innumerable effects which may be produced by the 

 same cause, to direct our attention to one rather than the other." 

 Descartes appears, nevertheless, to have admitted the value of Bacon's 

 method, for in some of his writings he distinctly refers to Bacon in 

 terms of approval. Possibly, in experimenting, he applied the induc- 

 tive method so far as the object he had in view admitted ; but this 

 kind of research was, with Descartes, merely an occasional and sub- 

 ordinate means of supporting his a priori theories. He introduced 

 some original views into his. speculations on mechanical science as, 

 for example, the idea of inertia regarded not as mere passive resistance 

 to motion, but as a really acting power. In .this way he first pointed 

 out with distinctness, that in every case of motion in a circle there must 

 be some deflecting force, since, without the operation of such a force, 

 the moving body would move in a straight line ; that is, if the deflect- 

 ing force ceased at any instant to act, the body would, from that instant, 

 move in the direction of the tangent. His notions of inertia, etc., are 

 deduced from a theoretical principle of the permanence of such quali- 

 ties, and by regarding motion as in some cases a kind of latent quality, 

 he lays it down that the quantity of motion in the universe is always 



