Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

side less. And in the first monadic side and diameter indeed,
the square in power of the diametrical monad, will be less by
one monad, than the double of the square of the side produced
by the monad. For the monads or units are in equality. But
1 is less than 2 by 1. Let us therefore add one to the side in
power, i.e. to the monad. Hence the side will be two monads.
And let us add to the diameter two sides, i.e. to the monad two
monads. The diameter therefore will consist of three monads.
And the square indeed, which is made from the side, i.e. from
2 is 4. But the square which is made from 3 is 9. Hence 9
exceeds the double of the square, which is made from 2 by I.
Again, let us add 3 in power to the side 2. The side thercfore
will consist of 5 monads. But to the triad in power, let
us add two sides, i.e. twice two, and the aggregate will be 7
monads. Hence, the square from the side will be 25; but the
square from 7 will be 49. The square 49 therefore is less by
the monad than the double of the square 25. Farther still, if to
the side you add 7, the sum will be 12. And if to the diameter
7 you add twice the side 5, the square from 17, viz. 289, will
exceed the double of the square from 12, i.e. the double of 144,
by 1. And thus successively, an addition being similarly made.
there will be an alternate analogy, the square from the diameter
being sometimes less by l, and sometimes more by 1,
than the double of the square from the side. And such sides
and diameters are effable." But the diameters alternately sometimes
exceed by 1. the double in power of the sides, and are
sometimes less by 1 than the double of them. All the diameters
therefore will become double in power of all the sides,
by an alternate excess and defect, the same monad in all of
them producing equality, so that the double in all of them
neither exceeds, nor is deficient. For what was deficient in the
* Effable quanities are such as can be expressed either in whole numbers
or fractions.