6 SCIENCE IN GRECO-ROMAN ANTIQUITY 



they desire to have as a fundamental number, the num- 

 ber 2 2.3.5 which is divisible by the majority of small 

 numbers in constant use ? Or again, is it because the 

 hexagon, inscribed in a circle, divides it into six equal 

 parts ? 1 



It is very difficult to decide between these various 

 hypotheses. 



It will be seen that our knowledge of the arithmetic 

 of the Oriental nations is very small ; the same is true 

 concerning their geometry. According to the accepted 

 tradition of Greek writers, 2 this science owed its birth 

 to purely practical needs. It was the overflowing of 

 the Nile which led the Egyptians to think of geometry, 

 for, as soon as the inundations were over, they 

 endeavoured to restore to each cultivator the 

 boundaries of his fields. Hence the necessity for an 

 exact survey. The formulae used were, however, 

 empirical, and were far from being always accurate. 

 For example, to estimate the surface of a quadrilateral, 

 the Egyptians did not attempt more than finding the 

 product of half the sum of the opposite sides ; in order 

 to calculate the area of a circle they used a value of n 

 equal to 3-1604 instead of 3-1415.... They knew, 

 however, that if the sides of a triangle are respectively 

 5, 4, 3, it is a right-angled triangle, and they made use 

 of this property to erect in the field a perpendicular 

 to a straight line. For this purpose, they used a cord 

 divided by two knots into lengths equal to 5, 4, and 3 ; 

 by means of pegs they made the length 4 coincide 

 with the straight line at the extremity of which the 

 perpendicular had to be erected (Fig. 2), then keeping 

 taut the lengths 5 and 3, they brought them together 

 in such a way as to join the ends. 3 



It is for this reason that the Egyptian geometricians 



1 29 Zeuthen, Histoire des mathematiques, p. 7. 



2 Proclus, Com. Euclid, 1, p. 64, 18. 



3 22 Milhaud, Nouvelles Etudes, p. 66. 



