INTRODUCTION 5 



of the cord. Then f == $ -f A- It is, however, open to 

 question whether this process always leads to the exact 

 results given us by the table of reduction. However 

 that may be, the practice of expressing fractional 

 quantities by a sum of fractions all having unity as 

 numerator, persisted amongst the Greeks until the 

 sixth century of our era. This practice, besides, 

 facilitated the treatment of certain problems which, 

 for us, lead to the solution of a numerical equation. 

 Such is the following, propounded by Ahmes : To find 

 a number, which, increased by its seventh, is equal to 

 19. The answer given : 16 + \ + J is accurate. 1 



The tablets of Senkereh, discovered in 1854 in the 

 library of Sardanapalus IV, give undeniable proof that 

 the Chaldeans, besides the decimal system, used an 

 advanced sexagesimal system based on the principle of 

 the position value of figures. 2 



These tablets, taking sixty as unit base, give us a list 

 of squares and cubes of which the following is an 

 example : 



1 -4 (i.e. 60 + 4) is the square of 8. 

 1 -2i (i.e. 60 + 21) is the square of 9. 

 More recent inscriptions even show an empty space and 

 sometimes a special sign representing zero, when that is 

 necessary. 3 The positional notation which charac- 

 terizes our arithmetic was thus clearly known by the 

 Chaldeans, and it is very curious, seeing its practical 

 advantages, that it did not pass into Greco-Roman 

 science. 



How were the Chaldeans led to choose the sexagesimal 

 division as well as the decimal system ? 4 Is it because 

 they originally divided the year into 360 days ? Or did 



1 29 Zeuthen, Histoire des mathdmatiques, p. 8. — 6 Boyer, 

 Histoire des mathematiques, p. 4. 



2 22 Milhaud, Nouvelles Etudes, p. 54. 



8 30 Zeuthen, Math. Wissensch., p. b 12. 

 4 Ibid., p. b 13. 



