Or thus: If the square of an odde number given for the first
foote, have an unity taken from it, the halfe of the remainder shall be the other foote, and the same halfe increased by an unitie, shall be the base: H.
As in the example: The sides are 3, 4, and 5. And 25. the square of 5. the base, is equall to 16. and 9. the squares of the shanks 4. and 3.
Againe, the quadrate or square of 3. the first shanke is 9. and 9 - 1. is 8, whose halfe 4, is the other shanke. And 9 + 1, is 10. whose halfe 5. is the base. Plato's way is thus by an even number.
11 If the halfe of an even number given for the first shanke be squared, the square number diminished by an unity shall be the other shanke, and increased by an unitie it shall be the base.
As in this example where the sides are 6, 8. and 10. For 100. the square of 10. the base is equall to 36. and 64. the squares of the shankes 6. and 8.
Againe, the quadrate or square of 3. the halfe of 6, the first shanke, is 9. and 9 - 1, is 8, for the second shanke. And out of this rate of rationall powers (as Vitruvius, in the 2. Chapter of his IX. booke) saith Pythagoras taught how to make a most exact and true squire, by joyning of three rulers together in the forme of a triangle, which are one unto another as 3, 4. and 5. are one to another.
From hence Architecture learned an Arithmeticall proportion in the parts of ladders and stayres. For that rate or proportion, as in many businesses and measures is very commodious; so also in buildings, and making of ladders or staires, that they may have moderate rises of the steps, it is very speedy. For 9 + 1. is, 10, base.