1 94 On the Analytical and Synthetical Modes of Reasoning 



attributed to Plato. By this method we suppose the pro- 

 posed problem is already resolved ; from which it results 

 that one condilioii i^ tultilled, or, what comes to the same, 

 that an equality takes place amongst certain magnitudes 

 civen and others soiight. It is by finding out the conse- 

 quences of the condition which we suppose fulfilled, or of 

 the equality which is the consequence of it, that in the 

 end we discover the unknown quantity, or trace the pro- 

 ceeding which it is necessary to follow in order to perform 

 what is demanded. 



I cannot do better than show here the definitions which 

 Vieta has given of syntliesis and analysis, after Theon, a 

 geometrician of Alexandria, who, from his living so nmch 

 nearer the times of the anlients, was better able to judge of 

 their opinions than we are. 



'* Est veritatis inquirendae via qiisedam in mathematici?, 

 qnam Plato prinuis invenisse dicitiir, a Theone nominata 

 analysis, et ab eodem detinita, ads^umptio qutesiti tamquam 

 concessi per consequentiaad verum concessum. Ut contra 

 synthesis, adsumptio concessi per consequentia ad qiiassiti 

 finem«t comprehensionem." {Fictoe opera, pag. 1.)* 



The 



* There is likewise a definition n{ analysis and synthesis given in the 

 preface to tbe seventh book of the JVTathematical Collections of Pappus, 

 which, for its curiosity, I shall insert. 



Analysis is the way wliich, procecdir.g from the thing demanded, ar- 

 rives, by means of certain established consequences, to somtwhit knowtt 

 before, or placed among the number of principles acknowledged For true. 

 This method makes Msgo from a truth or proposition througli all its an- 

 tecedents i and we call It ;malysis. or resolution, or an inverted solution. 

 In synthesis, on ihe contrary, we begin from the proposition last found in 

 analysis, ordering properly the above antecedents, wlii«h now present 

 themselves as consequents; and by combining them amongi,t themselves 

 w; arrive at the conclusion sought, from which we piocetdcd in the first 

 case. 



We distinguish two sorts of analysis :— in one, which may be named 

 contemplative, it is proposed to discover the truth or falsity of an ad- 

 vanced proposki'On ; the other is reUtciiT ro the soiiition of problems, or 

 the research of unknown truths. In tli^- tirst, In- ussuming f.,>r ti ue, or as 

 tormerly known, the iubjtcc of the advanced proposition, v.c proceed by 

 the conseipitnces of the hypothftsis to something known ; and if the result 

 k) true, the projwsition sdvancul is likowiie true. 1 he dirett dcmonstra- 

 ti>n is Ustly found by taking, in an inverse crtk-r, the difercnt parts of 

 theanalvsisi ; if the cunsfcpicnce to which w« iiirive;ii last is found to bie 

 f.ilse. we conclude that the p^opusitioa analysed is likewise f^ilse. When 

 hrtvc to prove a problem, we suppose it already known, and we proceed 

 upon this supposition till at arrive at something known. If the last re- 

 ,ult which we can obtaiii is compiised in the nundjcr of whdt geometri- 

 cians call known truths, the question proposed can hi- resolved. jThe de- 

 monstration (or properly the construction ) ii again formed, by taking, in aa 



inverse 



