Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 81
parity. Hence, in consequence of its sameness and immutable
nature, when it mutiplies itself, either in breadth or depth, it
still retains its own form, or if it multiplies any other number
by itself, the number which it multiplies does not recede from
its own quantity ;-a property which cannot be found in any
other number. But the duad is the leader of the even series,
and is likewise the principle of all difference. For if it multiplies
itself either in breadth or depth, or any other number, a
number different from itself immediately produced. Squares
however are produced by adding the terms in the above series
of odd numbers to each other. Thus 1+3=4, (1 being the
first square in power), 1+3+5=9, 1+3+5+7=16, 1+3
+5+7+9=25, and so of the rest. But from the addition of
the terms in the series of even numbers, the heteromekeis are
produced. For the first term of the series 2 is produced from
twice one. But 2+4=6, and 6 is produced from 3 multiplied
by 2. Again 2+4+6=12, and 12 is produced from 4 multiplied
by 3. And after a similar manner all the rest are produced.
CHAPTER XVI.
On the generation of the numbers called LATERCULI or tyles,
and of those denominated usseres or planks.-Their definition;
and, On circular or spherical numbers.
THE numbers called laterculi, which are also themselves
solids, are generated after the following manner. When spaces
are equally extended in length and breadth, but have a less
depth, the solid produced from these is a laterculus. Of this
kind are 3X3X2=18, or 4X4X2=32, or any other of 3
similar formation. But they are defined to be solids produced
by the multiplication of equal numbers equally into a less
number. The solids however called acseres or planks, are pro-