Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

intermediate longilateral number. Thus in the following series,
1. 2. 4. 6. 9. 12. 16. 20. 25. 30. 36. 42. 49.
in which between two proximate squares there is a longilateral
number, if 4 is added to 1 and to twice 2 the sum will be 9.
Thus, too, 4 added to 9, added to twice 6, is equal to 25. And
thus also, 9 added to 16, added to twice 12, is equal to 49. And
so of the rest. From this aggregation likewise, it is evident that
the squares thus formed are odd numbers; but those arising
from the former aggregation are even numbers. For 9,25, 49,
etc. are odd, and 16, 36, 64, etc. are even numbers.
Moreover, of the monads or unities at the four corners of
the table, the first and the third are squarcs, viz. 1 and 100;
but the other corners have the two other monads 10 and 10.
Farther still, the product arising from the multiplication of the
two squares 1 and 10, is equal to that of the two other me
nads multiplied into each other, viz. is equal to 10 X 10. 'I'hese
squares too, cut the table into two equal triangles. They also
become, as it were, the diameter of the figure; and hence they
are called diametral.
CHAPTER XX.
On the third species of inequality, which is called superpartient
:-Its species, and the generation of them.
AFTER the two first habitudes therefore, the multiple and
the superparticular, and those habitudes which are under
them, viz. the submultiple and the subsuperparticular, the third
species of inequality presents itself to the view, which we have
already denominated superpartient. But this takes place when
one number on being compared to another contains the whole
of it in itself, and certain parts of it besides, such as two, three,
or four parts, or any other that may arise from the comparison.
This habitude also begins from two third parts: for if one