Page:Popular Science Monthly Volume 67.djvu/666

THE CATTLE PROBLEM OF ARCHIMEDES.

By Professor MANSFIELD MERRIMAN,

LEHIGH UNIVERSITY.

THE sleepy town of Wolfenbüttel 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 arithmetical 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 mathematicians.

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:

${\begin{aligned}&{\text{White bulls}}=({\tfrac {1}{2}}+{\tfrac {1}{2}})\ {\text{black bulls}}+{\text{yellow bulls}},\\&{\text{Black bulls}}=({\tfrac {1}{4}}+{\tfrac {1}{5}})\ {\text{dappled bulls}}+{\text{yellow bulls}},\\&{\text{Dappled bulls}}=({\tfrac {1}{6}}+{\tfrac {1}{7}})\ {\text{white bulls}}+{\text{yellow bulls}},\\&{\text{White cows}}=({\tfrac {1}{3}}+{\tfrac {1}{4}})\ {\text{black herd}},\\&{\text{Black cows}}=({\tfrac {1}{4}}+{\tfrac {1}{5}})\ {\text{dappled heard}},\\&{\text{Dappled cows}}=({\tfrac {1}{3}}+{\tfrac {1}{6}})\ {\text{yellow herd}},\\&{\text{Yellow cows}}=({\tfrac {1}{6}}+{\tfrac {1}{7}})\ {\text{white heard}}.\end{aligned}}$

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. Consider, however, the following additional relations between the bulls of the sun:

${\begin{aligned}&{\text{White bulls}}+{\text{black bulls}}={\text{a square number}},\\&{\text{Dappled bulls}}+{\text{yellow bulls}}={\text{a triangular number}}.\end{aligned}}$

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.