THE MATHEMATICAL SCIENCES 155 



or principal proposition ; then the lemma, a secondary 

 proposition intended to facilitate the demonstration of 

 a theorem to follow ; and the corollary, a direct conse- 

 quence of a theorem which has just been established. 



But how are these propositions to be demonstrated ? 

 Although all agree as to the method to be followed, there 

 is a divergence of views as to the interpretation to be 

 given to the demonstration. At the time of Plato 

 and probably of Euclid also 1 there were subtle discus- 

 sions on the question whether mathematical pro- 

 positions must be considered as problems to be solved, 

 or on the contrary as theorems to be demonstrated. 

 Proclus (Comm. Eucl., I, p. 77, 15 et seq.) sums up the 

 discussions on this subject in the following way. The 

 Platonists such as Speusippus and Geminus held that 

 figures and their properties exist in the eternal world 

 of ideas independently of the construction the mathe- 

 matician can make of them ; the latter can only make 

 manifest to the understanding what already existed. 

 For example, equilateral triangles are such by 

 definition, that is to say, by an eternal relation of 

 ideas, and the fact of constructing them cannot add to 

 or take away anything from their existence. There- 

 fore it is not correct to speak of problems, but only 

 of theorems (objects of contemplation). Some philo- 

 sophers, such as the mathematicians of the school of 

 Menaechmus, were of the opinion that all should be 

 regarded as problems ; others said with Carpus that 

 problems as a class precede theorems, because it is by 

 the former that the subjects are found to which belong 

 the properties to be studied. 



Finally, many considered as a theorem that which 

 contained only one possibility, and as a problem 

 that which was capable of several possibilities. For 

 example, " to propose to inscribe a right angle in a 

 semicircle is not to speak geometrically, since all the 

 1 26 Tannery, Geo. grecque, p. 145. 



