12. The power of the diagony is twise asmuch, as is the power of the side, and it is unto it also incommensurable.
Or thus: The diagonall line is in power double to the side, and is incommensurable unto it, H.
As here thou seest, let the first quadrate bee aeio: Of whose diagony ai, let there be made the quadrate aiuy: This, I say, shall be the double of that: seeing that the diagonies power is equall to the power of both the equall shankes. Therefore it is double to the power of one of them.
This is the way of doubling of a square taught by Plato, as Vitruvius telleth us: Which notwithstanding may be also doubled, trebled, or according to any reason assigned increased, by the [25 e iiij], as there was foretold.
But that the Diagony is incommensurable unto the side it is the 116 p x. The reason is, because otherwise there might be given one quadrate number, double to another quadrate number: Which as Theon and Campanus teach us, is impossible to be found. But that reason which Aristotle bringeth is more cleare which is this; Because otherwise an even number should be odde. For if the Diagony be 4, and the side 3: The square of the Diagony 16, shall be double to the square of the side: And so the square of the side shall be 8. and the same square shall be 9, to wit, the square of 3. And so even shall be odde, which is most absurd.
Hither may be added that at the 42 p x. That the segments of a right line diversly cut; the more unequall they are the greater is their power.
13. If the base of a right angled triangle be cut by a
perpendicular from the right angle in a doubled reason, the power of it shall be halfe as much more, as is the power of the greater shanke: But thrise so much as is the power of the lesser. If in a quadrupled reason, it shall be foure times and one fourth so much as is the greater: But five times so much as is the lesser, At the 13, 15, 16 p xiij.
