April 8, 240 b.c.e.

Eratosthenes of Cyrene

Chief Librarian

Great Library of Alexandria

Alexandria, Egypt

Dear Eratosthenes,

Greetings my dear friend. I congratulate you on your recent measurements of the globe, though I suspect your eyes will suffer from so much gazing at the sun to achieve it. If you are hungry for it I require your assistance in the computation of the number of Helios’ cattle. They are horned, though that may make little difference as all are worth the same money horns or no horns. They live in four herds of different colors, white, black, yellow, and dappled. Each herd has bulls in these proportions: the white bulls are equal to a half and a third of the black together with all of the yellow. The black are equal to the fourth part of the dappled, and a fifth, together with again, all of the yellow. The remaining bulls, the dappled ones, are equal to a sixth part of the white and a seventh, together with all of the yellow. And, you’re not done. These are just the boys. There’s more.

Here are the cows: the white are equal to the the third part and a fourth of the whole herd of the black. The black are equal to the fourth part again of the dappled, and with if a fifth part, when all, including the bulls, don’t forget the bulls, are together. With me? This is the property of a god; he’s got masses of cows, all reproducing, so you must keep up: the dappled in four parts are equal in number to a fifth part and a sixth of the yellow herd. The yellow are equal to a sixth part and a seventh of the white herd.

Keep in mind, when the white ones mingle with the black, and if you can get them to stand still, they don’t like to stand still, they make a shape equal in depth and breath. Helios likes to line them up like this, takes forever but he has the time. He likes to arrange the yellow and the dappled bulls as one herd into a triangle shape, starting with one and then two in the next row, then three, and so on. There are exactly enough of them to achieve this, so try it in your calculations but don’t let the other colors in there! The others like to triangulate too, but you must exile them from the equation. Just the yellow and dappled bulls invited.

If you can do this Eratosthenes, I’ll come there myself, call you perfect, and crown you in glory. At minimum I will buy you lunch, you were looking quite starved last we met.

Yours etc.,

Archimedes

A:vt