THE CATTLE PROBLEM OF ARCHIMEDES. 663 



equating it to -J- n(v -f- 1) and computing the value of n by the solu- 

 tion of the quadratic equation; this value of n is found to be not an 

 integer. 



Some mathematicians who desired to solve the cattle problem have 

 claimed that W + B is not required to be a perfect square, because 

 the statement of the eighth condition in the Greek manuscript does 

 not use the term square number, but mentions ' a square figure.' Since 

 the length of a bovine animal is greater than its breadth, it was main- 

 tained by Wurm, about 1830, that W + B is required to be a rect- 

 angular number, that is, a number having two factors. On this 

 hypothesis he made a solution which gave the total number of cattle 

 as 5,916,837,175,686, and the number of white and black bulls as 

 2,093,299,351,328, which has the factors 704,538 X 2,971,166, as well 

 as many others, while the number of dappled and yellow bulls is 

 1,351,238,949,081, which is a triangular number, so that these bulls 

 could be arranged in a triangle with 3,287,843 rows. 



The consensus of opinion regarding the eighth and ninth condi- 

 tions is expressed, however, in the statement of the problem as given 

 above, namely, that the terms e square figure ' and ' triangular figure ' 

 should be understood to mean square number and triangular number. 

 Since 51,285,802,909,803 is the number of dappled and yellow bulls 

 which results from a solution that satisfies conditions (1) to (8) 

 inclusive, it is plain that the ninth condition may be expressed by 



51,285,802,909,803, ar = % n (n + 1 ) , 



in which x and n are to be integers. When x~ has been found, each 

 of the numbers of the first solution is to be multiplied by 4,456,749 x 2 , 

 in order to give the number of bulls and cows in each herd, satisfying 

 the nine imposed conditions. 



These numbers were readily seen to be so great that the island of 

 Sicily could not contain all the cattle, as the problem seems to demand. 

 This requirement, however, was understood to be only figurative, and 

 mathematicians agreed that the numbers, though very large, could be 

 found, but that no useful purpose would be attained by computing 

 them. Thus the question rested until 1880, when Amthor undertook 

 to determine how many figures were required to express one of the 

 numbers. His lengthy investigation demonstrates that 206,545 figures 

 are needed to express the total number of cattle. He further computed 

 that 766 are the first three figures of this number, so that 766 X 

 10206,542 j g ^ e a pp rox } ma te number of cattle. This is an enormous 

 number, and it is easy to show that a sphere having the diameter of 

 the milky way, across which light takes ten thousand years to travel, 

 could contain only a part of this great number of animals, even if the 

 size of each is that of the smallest bacterium. 



