THE MATHEMATICAL SCIENCES 157 



3. The apagogee (ajiaycoyrj) } which changes the ques- 

 tion propounded into another more simple ; 



4. The solution, which shows the possibility of solv- 

 ing this simpler question by means of the data of the 

 enunciation in defining by division the conditions of 

 possibility ; 



5. The construction, which completes the ecthesis by 

 defining the various accessory lines which it is necessary 

 to consider in order to make the demonstration ; 



6. The demonstration properly so called, which 

 deduces from the construction the figure required ; 



7. The conclusion, which affirms that this figure 

 satisfies the required conditions. 2 



As M. Zeuthen remarks, " whilst the analysis con- 

 tained in Nos. 3 and 4, i.e. in the transformation and 

 the solution, is methodically important for finding the 

 solution, it is no longer necessary when it is merely a 

 question of expounding in an unassailable manner 

 what has been found, which was always the chief aim 

 of Greek writers. It is therefore very often omitted, 

 so that the exposition consists only of the use of 

 operations numbered 1, 2, 5, 6, 7 ; thus the form which 

 we call synthetic is obtained." 3 By their very nature 

 theorems assume the form of a synthetical rather than 

 an analytical exposition. They are capable, however, 

 of an antithetical demonstration, the procedure of 

 which is analytical. One supposes that the proposed 

 theorem be true or false, then one considers whether 

 the consequences deduced from this supposition be 

 apparently right ; according to the conclusion reached, 

 the theorem will be judged true or false. One supposes, 

 for example, that two triangles, having one side and 



1 G. Friedlein, In primum Euclidis Elementorum librum 

 Prodi Commentarii, Teubner, Leipzig, 1873, p. 212. 



a 4 Boutroux, Ideal, p. 55. — 29 Zeuthen, Histoire des mathe- 

 matiques, p. 80. — 26 Tannery, Geo. grecque, p. 148. 



3 29 Zeuthen, Histoire des mathematiaues, p. 83. 



