4 SCIENCE IN GRECO-ROMAN ANTIQUITY 



express as a simple fraction, A for example, appeared 

 to the Egyptians as a problem, viz., to what sum of 

 fractions of the unit is the division of 2 by 29 equal ? 

 They showed that the sum was equal to A + ^ + t4t 



+ ITS-?. 



When the number to divide was greater than 2, e.g. 

 / ¥ , the Egyptians resolved it in the following manner : 



srV = "2"V ~f~ TB + "2"? + ITS'* 



Replacing 7 % by the value found above, one obtains 

 finally after simplification : 



irV = J "f" ttV 4~ IT'S" + "sV + ?!;?• 

 The manual of the scribe Ahmes gives a table of reduc- 

 tion for all fractions having 2 as numerator, and the 



odd numbers from 3 to 99 as denominator : i.e. 



2« + I 



(where n may have any value from 1 to 49). x By what 

 process has it been possible to compile this table ? This 

 is difficult to say, owing to want of information on this 

 point. According to M. Zeuthen the operation was 

 originally purely empirical, as follows : 2 Given the 

 fraction § , we represent the numerator by the length 

 a b (Fig. 1), and the denominator by the length a c. 



€/' 6 



I 1 — I ~~-H 



1 



Fig. 1. 



Now, let us take a cord, equal in length to a c, which we 

 can fold in such a way as to get one-half, one-third, etc., 

 of its length. If we mark off on a c half of this cord, we 

 reach a point beyond b, if we take one-third, we fall short 

 of b, at the point d. There still remains a length d b, 

 which, marked off 15 times, is equal to the whole length 



1 9 Cantor, Geschichte der Math., 1, p. 25. 



2 30 Zeuthen, Math. Wissensch., p. b 19. 



