664 POPULAR SCIENCE MONTHLY. 



It would be thought that, after this investigation by Amthor, the 

 question of solving the cattle problem would have been finally dropped, 

 but such was not the case. The certainty that numbers could be 

 found satisfying all nine conditions existed, and until they had been 

 actually computed the challenge of the author of the problem still 

 remained open. The way to solve it was well understood from the 

 theory of indeterminate analysis. Let the preceding equation be mul- 

 tiplied by 8 and unity added to each member, and let 2 ?i -f- 1 be called 



y; then it reduces to 



y- — 410,280,423,278,424 x" = 1 



which is of the form y 1 — Ax 1 = 1, and it is known that when A is 

 an integer there can always be found integral values of y and x which 

 satisfy the equation. The method of obtaining such values of x and 

 y can not well be explained here, but such a method was devised many 

 years ago by Pell and by Fermat, and it is well known to those skilled 

 in higher arithmetic. For example, take the simple case where 

 A = 19, or y- — 19 x- = 1, then the smallest integral values of y and 

 x which satisfy this equation may be found to be 170 and 39, so that 

 the square of 170 minus 19 times the square of 39 equals unity. 



In 1889 A. H. Bell, a surveyor and civil engineer of Hillsboro, 

 Illinois, began the work of solution. He formed the Hillsboro Mathe- 

 matical Club, consisting of Edmund Fish, George H. Eichards and 

 himself, and nearly four years were spent on the work. They com- 

 puted thirty of the left-hand figures and twelve of the right-hand 

 figures of the value of x~ without finding the intermediate ones. This 



value is 



x- = 34,555,906,354,559,370 252,058,980,100 



in which the dots indicate fifteen computed figures which it is here 

 unnecessary to give and 206,487 uncomputed ones; the total number 

 of figures in this number is 206,531. The final step is to multiply 

 each of the numbers of the first solution by 4,456,749 and by this 

 value of x 2 , and thus are obtained 



White bulls = 1,596,510 341.800 



Black bulls = 1,148,971 178,600 



Dappled bulls = 1,133.192 894,000 



Yellow bulls = 639,034 026,300 



White cows = 1,109,829 564,000 



Black cows = 735,594 645,400 



Dappled cows = 541,460 318,000 



Yellow cows = 837.676 113,700 



Total cattle 7,760,27 1 08 1,800 



in which the dots represent 206,532 figures, the total number of figures 

 in each line being either 206,545 or 206,544. In each of these lines 

 there are omitted twenty-four figures at the left-end and six at the 



