66o POPULAR SCIENCE MONTHLY 



THE CATTLE PROBLEM OP ARCHIMEDES. 



By Professor MANSFIELD MERRIMAN, 



LEHIGH UNIVERSITY. 



r I MiE sleepy town of Wolfenbiittel in northern Germany is the proud 

 -*- possessor of a library containing about 240,000 books and 10,000 

 manuscripts, many of the latter being Greek and Latin writings of 

 interest and value. Lessing, the poet and philosopher, was appointed 

 its librarian in 1769, and a few years later he published translations 

 of some of the unique manuscripts with commentaries thereon. One 

 of these was a Greek poem of forty-four lines which states an arith- 

 metical problem that has since attracted much attention on account of 

 the difficulty of its solution and the enormous numbers required to 

 fulfil its conditions. The name of Archimedes appears in the title of 

 the poem, it being said that he sent it in a letter to Eratosthenes, the 

 Cyrean, to be investigated by the mathematicians of Alexandria. 

 Opinions differ as to the truth of this statement, and it may well be 

 doubted if Archimedes was the real author, particularly as no mention 

 of the problem has been found in the writings of the Greek mathe- 

 maticians. 



The following statement of the cattle problem has been abridged 

 from the German translations published by Nesselmann in 1842, and 

 by Krumbiegel in 1880 : 



Compute, O friend, the number of the cattle of the sun which once grazed 

 upon the plains of Sicily, divided according to color into four herds, one milk- 

 white, one black, one dappled and one yellow. The number of bulls is greater 

 than the number of cows, and the relations between them are as follows: 

 White bulls = (i + i) black bulls + yellow bulls, 

 Black bulls - - (J -f- i) dappled bulls + yellow bulls, 

 Dappled bulls = (J + y) white bulls + yellow bulls, 

 White cows = (A + \) black herd, 

 Black cows = ( \ + J) dappled herd, 

 Dappled cows = (* + &) yellow herd. 

 Yellow cows = (J + \) white herd. 

 If thou canst give, O friend, the number of each kind of bulls and cows, 

 thou art no novice in numbers, yet can not be regarded as of high skill. Con- 

 sider, however, the following additional relations between the bulls of the sun: 

 White bulls -\- black bulls = a square number, 

 Dappled bulls + yellow bulls = a triangular number. 

 If thou hast, computed these also, O friend, and found the total number of 

 cattle, then exult as a conqueror, for thou hast proved thyself most skilled in 

 numbers. 



