Page:Popular Science Monthly Volume 67.djvu/669

equating it to 1/2 n (n $+$ 1) and computing the value of n by the solution 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 maintained by Wurm, about 1830, that W $+$ B is required to be a rectangular 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 $\times$ 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 conditions is expressed, however, in the statement of the problem as given above, namely, that the terms '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, x² $=$ 1/2 n (n $+$ 1),

in which $x$ and n are to be integers. When $x^{2}$ 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 $\times$ 10^206,542 is the approximate 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.