196 On the Annhjlkul and Sijnthd'uul Modes of Reasoning 



l^clicvc that ihrouch its means the geometers of the two last 

 centuries found t)ut the nunierous cli^coveries which made 

 thciti so illustrious, and \^ hieh have served as a foundation for 

 tlie lahour.i of their successors. But either to conceal their 

 proceedings, or probal)lv not being sufficientlv used to the 

 method, thcv dared not entirelv trust to it; but when thev 

 had discovered a proposition, they always demonstrated it 

 svntlieticallv. It appears from the posthumous \vorks of 

 Pascal and Robcrval, that they llrst made use of the method 

 of indivisibles to resolve problems, and afterwards demon- 

 strated them according to the manner of the autients. Ther 

 generally concealed the proceedings which thev made use 

 of; because, their ways of invention not being reduced to 

 general rules and methods, they had the greatest interest in 

 keeping them secret, for to assure themselves of arms 

 proper to make them superior in the attacks which their 

 rivals made b-v their defiances, m hich increased every 

 dav*. 



Certain authors, in very different scicTices, supposing that 

 the evidence of which geometry exclusively boasted wa;^ 

 owing to the method of geometers, thought, by applying 

 this method to the object of their researches, they should be 

 enabled to protect it from opposition; but it is easy to per- 

 ceive that tills imitation of method is imperfect, and that 

 tliere will also be some ditrerence owing to the nature of the 

 subject. 



It is in chemistry that the application of the two methods 

 appears most evident, and ct)nformable to the etymology of 

 their names. We combine together certain simple sub- 

 slanees, or regarded as such, and thus operate bv synthesis. 

 VV'e take a compovind body, and separate it into its com- 



* 



* This could not be Newton's moti'.'L* ; for he a[)[icars to have supposed 

 that a mathematic.il proposition was not tit to be pubiibhed but with a 

 syntheiical demonstration. In his Treatise upon Fluxion;., he exprtises 

 himst-if as follows upon this subject: 



" Po-itiiuaiii area curve alitujus ita (analytice) repcrtn est et con- 

 structs, indagnnda est dcmonstnuio coni-tructionisj ut oini->so, quatenus 

 lieri pctjsf, calculo algcbraicu, theorema liat concinnum et elcgans, ac 



LLMtN I't'EI.IClJM SLSTINKIU VAl.KAT.'' 



Laplace thinks likewise that Newton •' had discovered by analysis the 

 i^reatest pur: of his theorems ; but his predilection for synthesis, and great 

 cstimarion for the t^eometry of tlie antieins, induced hinr.ro give a synthe- 

 lifal fon-ii to his tiieorcniN, and likewise his method of Jluxions." (£;f- 

 f'ostt. ({■> Syst. cln Mondr.. 2d edit. p. 33;-;.) 



We lind in^he luminous reflections upon the character and respective 

 advanta^^s of svnthc.is and analysis which follow tliis quotation, ail that 

 precision and clearness w Inch the autiior has jtiade use of in the rest of hU 

 excellent vvork, upon the niost abstract principles of mechanics, 



5 ponent 



