The Cattle Problem of Archimedes

Archimedes

A problem that Archimedes solved and, in a letter to Eratosthenes of Cyrene, sent to those who were working on such problems in Alexandria.

Compute the number of cattle of the Sun, O stranger, and if you are wise apply your wisdom and tell me bow many once grazed on the plains of the island of Sicilian Thrinacia, divided into four herds by differences in the color of their skin—one milk-white, the second sleek and dark skinned, the third tawny-colored, and the fourth dappled.

In each herd there was a great multitude of bulls, and there were these ratios. The number of white bulls, mark well, O stranger, was equal to one half plus one-third the number of dark skinned, in addition to all the tawny-colored; the dark skinned bulls were equal to one-fourth plus one-fifth the number of dappled, in addition to all the tawny-colored. The number of dappled bulls, observe, was equal to one-sixth plus one seventh the white, in addition to all the tawny-colored.

Now for the cows there were these conditions: the number of white cows was exactly equal to one third plus one-fourth of the whole dark skinned herd; the number of dark-skinned cows, again, was equal to one-fourth plus one-fifth of the whole dappled herd, bulls included; the number of dappled cows was exactly equal to one fifth plus one-sixth of the whole tawny-colored herd as it went to pasture; and the number of tawny-colored cows was equal to one sixth plus one-seventh of the whole white herd.

Now if you could tell me, O stranger, exactly how many were the cattle of the Sun, not only the number of well-fed bulls, but the number of cows as well, of each color, you would be known as one neither ignorant nor unskilled in numbers, but still you would not be reckoned among the wise. But come now, consider these other facts, too, about the cattle of the Sun.

When the white bulls were mingled with the dark-skinned, their measure in length and depth was equal as they stood unmoved,1 and the broad plains of Thrinacia were all covered with their number. And, again, when the tawny-colored bulls were joined with the dappled ones they stood in perfect triangular form beginning with one and widening out, without the addition or need of any of the bulls of other colors.

Now if you really comprehend this problem and solve it giving the number in all the herds, go forth a proud victor, O stranger, adjudged, mark you, all powerful in this field of wisdom.