used in Mathematics and other Scmices. 1 99 



fbat in the Elements of Pluclid there are to be found a grea 

 number of places in which it is very necessary that all the 

 intermediate truths should be exposed; theywere not known 

 to the inventors, and it would be difficult to produce them 

 although their existence is evident. 



Analysis produces these intermediates, and makes them 

 pass under tlie eves of the operator, although in an inverse 

 order; and when they become so numerous that it is im- 

 possible to express them otherwise than by algebraical for- 

 nuihe, it then becomes necessary to employ calculation ; 

 and this makes known truths to which otherwise the rea- 

 soning faculty could never attain. 



It will sometin)cs happen, that synthesis, by representing 

 things much simpler than analysis, will lead to a conclusion 

 in a much less complex manner. The researches upon the 

 attraction of spheroids by Maclaurin, are a remarkable in- 

 stance of this circumstance ; but in the hands of Lagrange, 

 Laplace, and Lcgendre, analysis re-assumes, m these cases, 

 those advantages which it possesses in all others. 



I think that to all those who have clear notions of the me- 

 thod emploved bv geometers, itwill appear proved from what 

 precedes, that the true method of analysis has never beea 

 applied to metaphysics, which does not appear susceptible 

 of this application, at least in the present state of science. 



It is not because Locke and Condillac made use of the 

 analytical mclhod, that motaphvsics made such great pro- 

 gress in their hands, but because they sought their first no- 

 tions from nature, and not from their imaginations ; it is 

 b(X'ause they ascended to the true origin of all our know- 

 ledge, rather than create a system according to Uieir fancy. 

 If the first o-eometers had been willing, or perhaps able, to 

 form to themselves other notions of the right line and circle 

 than those they received from nature, doubtless they would 

 have formed a geometry wiiich had no resemblance to that of 

 nature, but would have been entirely imaginary. The 

 metiiod of aeumeters is not the sole cause of the certainty 

 of ihcir results; this certainty is principally owing to the 

 nature of the notions which they have to combine. It is 

 possible for a mathematical demonstration to be obscure, 

 embarrassed, and incomplete, and yet at the same time to 

 conduct to the truth of the enunciated proposition any per- 

 son who shall have the patience and sagacity necessary to 

 follow and rectify this demonstration, 'i'his takes place 

 from mathematicians eniploving only complete notions, or 

 such that the property whicn forms their principal charac- 

 ter excludes all others. When they wish to reason upon 

 N 4 ' notions 



