THE MATHEMATICAL SCIENCES 151 



cases only renders knowledge more difficult of attain- 

 ment. 



" From all these points of view, it will be found that 

 the elementary treatise of Euclid surpasses any other : 

 if its utility be considered, it leads to the theory of 

 primordial figures ; x its lucidity and regular chain of 

 reasoning are ensured by its progression from the most 

 simple to the most complex, and by basing the theory 

 on common ideas ; the generality of the demonstra- 

 tions, by the choice of the starting-point in the problems 

 to be dealt with, in the theorems which set forth the 

 principles " (Proclus, Comm. Eucl. I, p. 73, 15 et 

 seq.). 2 



The elementary treatise of Euclid is indeed a model 

 of truly rational science. It begins by a collection of 

 primary propositions which are enunciated in such a 

 way as to make them universally acceptable and which, 

 although as limited in number as possible, are sufficient 

 to secure the construction of the whole mathematical 

 edifice. This construction proceeds from the simple to 

 the complex by way of demonstration and resolution 

 of problems. It begins by establishing the properties 

 of the most elementary figures, then by their means 

 it demonstrates the properties of more and more com- 

 plex figures. In this way the work of synthetic 

 geometry is accomplished, and this work must be 

 logically unassailable. 



In dealing with the primary propositions, the 

 Elements, as they have come down to us, distinguish 

 between definitions and hypotheses (postulates and 

 axioms) . 



The definitions (oqol) define the meaning and limits 

 of the concepts used. The postulates (cdrrj/iara) 

 demand that certain constructions (for example, to 

 draw a straight line between two points) shall be 



1 Polyhedra composed of material elements. 

 "Quoted from 26 Tannery, Geo. grecque, p. 142. 



11 



