Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

CHAPTER X.
On solid numbers.-On the pyramid, that it is the principle of
all solid figures, in the same manner as the triangle, of plane
figures; and on its species.
HENCE, however, the way becomes easier to the contemplation
of solid figures. For by foreknowing what the power of
quantity naturally effects in the plane figures of numbers,
there will be no impediment in the transition to solid numbers.
For as we add to the length of numbers another interval, i.e.
superficies, in order that breadth may be exhibited, so now by
adding to breadth that which is called depth, the solid body of
number will receive its completion.
It appears, however, that as in plane figures the triangular
number is the first, so in solids that which is called the pyramid,
is the principle of depth. For it is necessary to find the
primordials of all the established figures in numbers. But one
kind of pyramid is that which raises itself into altitude from a
triangular base. Another kind is that which raises itself from
a square base; another, from a pentagonal base; and others,
from other multangular bases; as may be seen in the following
diagrams :
These pyramids also are denominated from their bases; so
that the first is called a triangular; the second, a square; the