THE MATHEMATICAL SCIENCES 121 



what must be understood by this science, as distinct 

 from pure arithmetic. Inspired by this scholium, P. 

 Boutroux justly points out that " Far from likening 

 magnitudes to numbers, according to Greek tradition 

 it was not permissible to consider as true numbers the 

 numbers resulting from measurements of magnitudes, 

 such as phialitic numbers, or relating to phials, melitic 

 numbers, or relating to apples (or flocks). And this 

 is why problems dealing with magnitudes were enun- 

 ciated in concrete and not theoretical terms ; what for 

 us is the * solution of an equation of such or such type ' 

 was formerly the solution of the problem of the oxen, 

 the problem of the trees, the problem of the rabbits, 

 etc." x Even in our own times schoolboys speak of 

 the problem of the runners, the problem of the foun- 

 tains, etc. 



At first, however, the distinction between logistics 

 and pure arithmetic was not clearly defined. It is 

 certain that though Euclid surpassed the knowledge 

 of the Pythagorean school, he left aside many of the 

 questions studied by it. 2 The Pythagorean arithmetic 

 was certainly more varied in its researches and, up to 

 a certain point, in its conceptions, than the arithmetic 

 of its successors. The fact is easily explained. 



Although the Pythagoreans had the indisputable 

 credit of laying the foundations of mathematical science 

 in Greece, they were not able to free them from all 

 metaphysical considerations. This fact is especially 

 striking in regard to arithmetic, which was in a sense 

 the corner stone of the Pythagorean philosophy, in 

 whose eyes number and its properties constituted the 

 basis of reality. In truth, sensible phenomena which 

 are most diverse from a qualitative point of view, can 

 show identical numerical relations. There is, for 

 example, from the standpoint of the impression received, 



1 3 Boutroux, Analyse, I, p. 121. 



2 25 Tannery, Science Hellene, p. 370. 



