used In Mathematics and other Sckrices. 195 



The demonstration of theorems in the manner called re- 

 ductio ad absurdum is, properly speaking, an analytical 

 proceeding; tor we suppose the converse of the proposition 

 is true, and bv seeking consequences which are absurd 

 make it appear that the hypothesis is so likewise. 



I believe the characteristics of synthesis and analysis \h 

 the mathematics are rendered tolerably clear by the pre- 

 ceding descriptions. In the first method, the proposition 

 enunciated is always the last consequence of the chain of 

 reasonings which form the demonstration: it is a composi- 

 tion; for we add principle continually to principle, until we 

 come to this consequence. 



In analysis, on the contrary, by supposing the question 

 resolved, we take in the whole of the subject ; and making it 

 pass with different forms, or making diverse traductions of 

 the enunciations, we come to the solutions sought. 



Condillac, in the fourth volume of his Course of Studies, 

 makes it appear that all the art of reasoning merely consists 

 in discovering the identity of diverse propositions; it is the. 

 order in which we connect the propositions that constitutes 

 the method: moreover, when we reason synthetically, all 

 the propositions that we make use of are identic until the 

 last, which is itself a consequence of the preceding ones ; 

 and, by containing the subject of the enunciation, shows 

 that the proposition advanced is true. When we reason 

 analytically, we proceed from the emuiciation, which is i:iot 

 identic bv itself; and all the traductions which we pass by 

 are merely hypothetical, until we arrive at the last, which 

 always ought to render it identic; and from that results the 

 determination of the quantity sought : likewise by the con- 

 nection of the anterior ideas all the intLrmcdiate proposi- 

 tions become identic, and consequently the proposed ques- 

 tion is resolved. 



Those who understand algebra will readily perceive that I 

 have traced the order made use of in resolving equations; 

 they will see that at the last operation, when we have ob- 

 tained the value of the unknown quantity, liie final equa- 

 tion will become identic by its substitution ; which will also 

 be the case with all liu)se which precede it. 



Analysis is in general the mclhud of invention; and we 



inverse order, the difFiTcnt p^rts of the sn:i'yis The irrpossibility of 

 tlie lyit result of ihf analysis will prove tv dcntly, in this case us in tiie 

 prccetlinj; one, that of tlic thing deiv.aiiflid. 



Tlicrc is liktwise, in the solution of every prgblem, what is called the 

 deterniinatiuni ih^t is to say, the reasoning uy ^siucli ue show when, in 

 wliat manner, and how miny difF- rent vv,iy», Ux j.ioi> tm can be 'ohi'-". 



N 2 ' beheve 



