THE CATTLE PROBLEM OF ARCHIMEDES. 66 1 



The Wolfenbiittel manuscript has an appendix, also in Greek, 

 giving numbers in answer to the problem, the total number of cattle 

 being stated as 4,031,126,560, but the results satisfy only the first 

 seven conditions. Lessing also gives results computed by Leiste, a 

 clergyman of Wolfenbiittel, whose solution satisfies these seven condi- 

 tions and likewise the eighth one. In 1821 J. and K. L. Struve pub- 

 lished a critical and mathematical discussion, which was followed in 

 1828 by another from G. Hermann. The latter makes the interesting 

 remark that Gauss had arrived at a complete solution ; but, if so, no 

 further information regarding it has been obtained. Many other 

 attempts at solution were made, but the large numbers required to 

 satisfv the nine conditions discouraged many investigators. Some 

 critics thought that the original problem of Archimedes included only 

 the first seven conditions and that the two others had been added by a 

 later writer. It is, of course, clearly seen that the exercise as stated 

 includes two problems, the first to find integral numbers that satisfy 

 the first seven conditions, and the second to find integral numbers that 

 satisfy all the nine conditions. As the poem says, the first problem 

 may be solved by those of moderate proficiency in numbers, while the 

 second can only be done by those of the highest skill. 



The first problem is an easy one for a boy in the high school. Let 

 W, B, D, Y represent the number of white, black, dappled and yellow 

 bulls and let w, b, d, y represent the number of white, black, dappled 

 and yellow cows. The seven conditions then give the seven equations: 



and these contain eight unknown quantities. The problem, therefore, 

 is of the kind called indeterminate, for many sets of numbers may be 

 found to satisfy the seven equations. That set having the smallest 

 numbers is the one required, for any other set may be found by multi- 

 plying these numbers by the same integer. If B and W are eliminated 

 from equations (1), (2), (3), there will be found the single equation 

 891 D = 1,580 Y, and hence T = 891 and D = 1,580 are the 

 smallest integral numbers satisfying it; from these are found 

 5 = 1,602 and W = 2,226. These values, before insertion in equa- 

 tions (4) to (7), are to be multiplied by a factor m, the value of 

 which is later to be determined, so that the number of cows in each 

 herd shall be an integer. Proceeding with the elimination, the values 

 of iv, b, d, y are deduced in terms of m, and it is then seen that 4,657 

 is the least value of m which will make the results integers. It is thus 

 easily found that 



