126 SCIENCE IN GRECO-ROMAN ANTIQUITY 



properties from those of geometrical figures. He 

 studied the theory of rational numbers, indicated the 

 rules for finding the greatest common factor and the 

 least common multiple ; he also studied fractions and 

 geometrical progressions and demonstrated that the 

 number of prime numbers is unlimited. 



Is it to the system of numeration in use amongst 

 the Greeks that their lack of progress in arithmetic 

 should be attributed ? Certainly this system was not 

 as practical as our own, but this was not an insur- 

 mountable barrier, as is shown by the Arenarius of 

 Archimedes. 



However this may be, arithmetical speculations were 

 only revived in Greece by Diophantus and then in an 

 algebraical form. The originality of Diophantus con- 

 sists in the first place in having entitled his work 

 aQid/urjTLxd (Arithmetic) , and then in treating of matters 

 which are logistical. This innovation was more than 

 a matter of words, it brought into abstract science 

 that which had formerly been considered to belong to 

 concrete science ; it announced a change in form and 

 method. With one exception (Opera I, p. 385) the 

 numbers of Diophantus are abstract and do not relate 

 to oxen or rabbits ; the problems also are treated 

 methodically, their solution is not merely enunciated 

 without demonstration, as had been the case with the 

 logisticians. 



Although Diophantus had eclipsed all his predeces- 

 sors, his aim was not understood in the way he desired. 

 Nicomachus, in his treatise on arithmetic, still considers 

 the numbers of Diophantus as concrete. The tradi- 

 tional distinction between arithmetic and calculation 

 remained, although the deep abyss which separates 

 them is henceforward filled up. 1 



As a matter of fact, the Arabs did not translate 

 Diophantus until the tenth century, and it was only 

 1 26 Tannery, Geo. grecque, p. 52. 



