156 SCIENCE IN GRECO-ROMAN ANTIQUITY 



inscribed angles are right angles ; on the contrary, 

 to inscribe an equilateral triangle in a circle is really a 

 problem, since it is possible to inscribe in it a triangle 

 which is not equilateral." x 



The disagreement is deeper in appearance than in 

 reality, and arises, as Proclus explains, from a difference 

 of point of view. The distinction between ideal science 

 and didactic science is itself sufficient to show that both 

 Geminus and Carpus may be right, " for if it is accord- 

 ing to the order that Carpus gives the pre-eminence to 

 problems, it is according to the degree of perfection 

 that Geminus gives it to theorems." 2 In as far as it 

 is ideally conceived of, mathematical truth only contains 

 theorems, but to the mind that conquers it by degrees 

 it appears in the form of problems. However, whether 

 it is a question of problems to solve or theorems to 

 demonstrate, it is necessary to have recourse to methods 

 of which the Greeks, starting from Plato, had carefully 

 fixed the stages. By analysis they decomposed a 

 complex whole into simpler propositions, already 

 admitted or demonstrated. For example, to draw a 

 tangent to two circles, they supposed the problem 

 solved, and showed that in order to find this solution, 

 it is necessary to start from the known construction of 

 a tangent drawn to a circle through an external point. 

 Synthesis, on the contrary, enables the complex 

 geometrical relation, of which the demonstration is 

 needed, to be reconstructed by means of primitive 

 propositions. 



For the Greeks the typical question consists of seven 

 parts : 



1. The protasis, or enunciation indicating the data of 

 the problem and what is required ; 



2. The ecthesis, or repetition of the enunciation in 

 relation to a particular figure ; 



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



2 Quoted according to 4 Boutroux, Ideal, p. 63. 



