Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

Two
does not only not arrive at unity and the extremity, but not
even at the first multangle in energy, which is of the same
kind as the base, it is called twice defective. Thus, if a pyramid
proceeding from the square 16, should terminate in 9, and
not arrive at 4, it will be twice defective; and in short, as many
squares as are wanting, so many times is the pyramid said to be
deficient. And all pyramids will be denominated in a similar
manner, from whatever multangular base they may proceed.
CHAPTER XIII.
On the n tr rn bers called cubes, wedges, and parallelo pipedons
AND thus much concerning the solid numbers which have
the form of a pyramid, equally increasing, and proceeding
from a proper multangular figure as from a root. There is
however another orderly composition of solid bodies, such as
cubes, wedges, and parallelopipedons, the superficies of which
are opposite to each other, and though extended to infinity
will never meet. Squares therefore, being disposed in an orderly
series, viz. 1. 4. 9. 16. 25, etc. because these have alone
length and breadth, but are without depth; if each is multiplied
by its side, it will have a depth equal to its breadth or
length. For the square 4 has 2 for its side, and is produced
from twice two. From the multiplication therefore of the
square 4 by its side 2, the form of the cube 8 is generated. And
this is the first cube in energy. Thus also the square 9 multiplied
by its side 3, produces the cube 27. And the square 16
multiplied by its side 4 generates the cube 64. And after a
similar manner the other cubes are formed. Every cube also,
which is formed by the multiplication of a square into its side,
will have six superficies, the plane of each of which is equal to
its forming square. It will likewise have twelve sides, each of
them equal to the side of the square by which it is produced;