GREEK SCIENCE IN ALEXANDRIA 101 



white, gray, dun, or piebald; the number of piebald bulls was 

 less than the number of white bulls by (^ + |) of the number of 

 gray bulls, it was less than the number of gray bulls by (| + ) 

 of the number of dun bulls, and it was less than the number of 

 dun bulls by ( -f- y) of the number of white bulls. The number 

 of white cows was (| + j) of the number of gray cattle (bulls and 

 cows), the number of gray cows was (j + -3-) of the number of 

 dun cattle, the number of dun cows was (-5- + ) of the number of 

 piebald cattle, and the number of piebald cows was (^ + y) of the 

 number of white cattle." The seven equations are insufficient to 

 determine the eight unknown quantities. The solution attributed 

 to Archimedes consists of numbers of nine figures each. 



Again he succeeds in summing the series of squares : 1, 4, 9, 15, 

 25, 36, etc., to n terms, expressing the result in geometrical form. 

 Both proof and formulation are of course much more complicated 

 by reason of the entire lack of an algebraic symbolism, the same 

 remark naturally applying also to the preceding cattle problem 

 and to the cubic equation referred to above. This last was in- 

 deed to Archimedes not primarily an equation at all, but a pro- 

 portion 



a x : b : : % a 2 : x 2 . 



In his Circle Measurement already outlined, he showed mastery 

 of square root, and the comparison of irrational numbers with 

 fractions, showing for example that 



^W->V3>Mf. 



How these fractions were obtained cannot be certainly deter- 

 mined, but it was presumably by a process analogous at least to 

 the modern method of continued fractions, though such fractions 

 themselves could not have been known to him. 



In the Sand Counting, Archimedes undertakes to give a number 

 which shall exceed the number of grains of sand in a sphere with a 

 radius equal to the distance from the earth to the starry firma- 

 ment. The treatise begins : " Many people believe, King Gelon, 

 that the number of sand grains is infinite. I mean not the sand 



