Return to Vignettes of Ancient Mathematics

The text used is the edition of Tannery (1893), but I have also consulted the translation of ver Eecke (1959) and the paraphrase of Heath (1910).

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For notes on translation, go to the introduction to Book I.

In fact, let it be prescribed to divide 16 into two squares.
And let the 1st be assigned Δ^{Υ} 1, therefore the other will be 16 Δ^{Υ}. It will be required, therefore, for 16 Δ^{Υ} to be equal to a ⬜.

I form the ⬜ from however many ’s Μ’s that are the side of 16 . Let it be 2 4. Therefore, the ⬜ will be Δ^{Υ} 4 16 16. These are equal to 16 Δ^{Υ}. Let a common be added, the deficiency and similars from similars.

Therefore, Δ^{Υ} 5 are equal to 16, and becomes 16 of fifths.

One will be 264^{25´}, another 144^{25´}, and the two added make 400^{25´}, or 15, and each is a square.

In fact let it again be required to divide square 16 into two squares. Let again the side of the 1st be assigned 1, that of the other however many ’s of as many as the side of the divided. In fact, let it be 2 4.

Therefore, the ⬜’s will be, one Δ^{Υ} 1 and the other Δ^{Υ} 4 16 16. Remaining, I want the two added equal to 16.

Therefore, Δ^{Υ} 5 16 16 are equal to 16. And becomes 16^{5´}.

The sd. of the 1st will be 16^{5´}; therefore, it will be 256^{25´}. An the sd. of the 2nd 12^{5´}; therefore, it will be 144^{25}. And the demonstration is obvious.

Let it be required with 13, which is composed from squares 4 and 9, to divide it again into two other squares.

Let there be taken of the mentioned squares the sds, 2, 3, and let there be assigned the sds of the prescribed squares, one 1 2, the other of as many it may be of as many as is the side of the remaining. Let it be 2 3. And the squares become, one Δ^{Υ} 1 4 4, the other Δ^{Υ} 4 9 12.

Remaining is to make the two added 13. But the two added make
Δ^{Υ} 5 13 8. These are equal to 13. and becomes 8^{5}.
For the actualities: I assigned the sd of the 1st, 1 2. It will be 18Μ^{5}.
The sd of the 2nd I assigned 2 3. It will be the side of one of them. And the ⬜’s themselves will be, one 324^{25´}, which is of one. And the two added make 325^{25´}, which brings together the prescribed 13.

In fact, let it be prescribed that the excess of them is 60.

Let it be assigned one whose side is 1 and the other whose side is 1 and however many ’s you want, merely so that the ⬜ from the ’s does not exceed the excess that’s given, nor, truly, that it be equal. For, when one form is left equal to one form, the problem will be established.

Let it be 1 3. Therefore the squares will be Δ^{Υ} 1 and Δ^{Υ} 1 6 9. And the excess of them is 6 9. These are equal to 60. And becomes 8 𐅵.
The side of the 1st will be 8 𐅵, and that of the 2nd 11 𐅵. The ⬜’s themselves will be, one 72 4, the other 132 4, and things of the proposition are obvious.