Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

The pyramids also which are from the square, are generated
by the addition of squares to each other, as will be evident
from an inspection of the following schemes.
Squares.
1 4 9 16 25 36 49 64 81 100
Pyramids from Squares.
1 5 14 30 55 91 140 204 285 385
And after the same manner all the forms that proceed from
the other rnultangles are produced. For every multangular
form proceeds ad infinitum from unity, by the addition of
unity to a figure of its own kind. Hence it necessarily appean,
that triangular forms are the principles of the other figures;
because every pyramid from whatever base it may proceed;
whether from a square, or pentagon, or hexagon, or heptagon,
etc. is contained by triangles alone, as far as to the vertex.
CHAPTER XII.
On defective pyramids.
As the pyramid is perfect which proceeding from a certain
base arrives as far as to unity, which is the first pyramid in
power and capacity; so the pyramid whose altitude does not
reach to unity, is called defective. Thus, if to the square 16,
the square 9 is added, and to this the square 4, but unity is
omitted to be added, the figure indeed is that of a pyramid, but
because it does not arrive as far as to the summit, it is called
defective, and has not for its summit unity, which is analogous
to a point, but a superficies. Hence if the base is a square, the
ultimate superficies will be a square. And if the base is a pentagon,
or hexagon, or heptagon, etc. the ultimate superficies
will be of the same form as the base. If however, the pyramid