Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 79
to have four angles and four sides. They are not however all
equal, but are always less by unity. For neither are all the
sides equal to all, nor is the breadth equal to the length; but
as we have said, one part being greater, it precedes and surpasses
the less part by unity alone. If therefore the series of
natural numbers is disposed in order, and the second is multiplied
by the first, or the second by the third, or the third by the
fourth, or the fourth by the fifth, and so on, the numbers called
heteromekeis will be produced:
Hence 2x1, 2x3, 3x4, 4x5, etc. i.e. 2, 6, 12, 20, etc. will
be numbers longer in one part.
If therefore the numbers that are multiplied differ only from
each other by unity, the above mentioned numbers will be produced.
But if they differ by any other number, as thrice seven,
or thrice five, etc. and their sides do not differ by unity alone,
these numbers are not called heteromekeis, or longer in the
other part, but oblong. For Pythagoras, and the heirs of his
wisdom, ascribed other or the different to the binary or duad
alone. And this they asserted to be the principle of difference.
But they said that the primaeval and unbegotten monad was
the principle of a nature always the same, and which is similar
to, and in concord with itself. The duad however, is primarily
dissimilar to the monad, because it is first disjoined from it ;
and therefore is the principle of a certain difference, because it
is only dissimilar by unity to that primary essence which yossesses
an invariable sameness of subsistence. Thcse numbers
therefore, are deservedly called longer in the other part, because
their sides precede each other by unity alone.
This however is an argument that difference is justly constituted
in the binary number, that the term other or different, is
only asserted of two things, by those who are not negligent in