Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

which 4 is, is compared with the row immediately under it,
sesquialter ratio will be unfolded. Thus 3 to 2, or 6 to 4, or
9 to 6, or 12 to 8, is a sesquialter ratio. And in a similar manner
in the rest. Nevertheless, one number here exceeds another
by the same quantity as in the naturally double species. For
the first surpasses the first, ix. 3 surpasses 2 by 1. The second
surpasses the second by 2. The third surpasses the third
by 3, and 60 on. But if the fourth order is compared to the
third, as 4 to 3, 8 to 6, 12 to 9, etc. in all these the sesquitertian
ratio will present itself to the view.
This too is divine in the table, that all the angular numbers
are squares. But a square number in short, is that which is
produced by the multiplication of a number into itself. Thus
in this table, one multiplied by itself is one, and this is in
power or capacity a square. Also, twice 2 is 4, thrice 3 is 9,
four times 4 is 16; and so of the rest. But the numbers which
are placed about the angular numbers, are longilateral. These,
however, are such as are produced by the multiplication of two
numbers that differ from each other by unity. Thus 2 and 6
are situated about 4; but 2 is produced by the multiplication
of 1 by 2, and 2 differs by 1 from 1. But 6 is produced by the
multiplication of 2 by 3; and these differ from each other by
unity. Again, 6 and 12 are situated about 9: and 12 is formed
from 3 multiplied by 4, and 6 from 2 by 3. All which are produced
from sides that differ from each other by unity. And in
a similar manner in the other angular numbers, the numbers
which are situated about them will be found to be longilateral.
Again, from the aggregate of two longilateral numbers, and
twice the square which they surround, a square number will
be formed. Thus 2+6 added to twice 4 is equal to 16. Thus,
too, 6+12 added to twice 9 is equal to 36. And so of the rest.
A square number also will be formed from the aggregate of
two proximate square numbers together with the double of the