Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

and eight angles, each of which is contained under three such
angles as those of its producing square.
Because also every cube proceeds from equilateral squares,
it is equal in all its parts. For the length is equal to the
breadth, and the depth, to each of these. It is likewise necessarily
equal to itself according to six parts, viz. upward, downward,
on the right hand, on the left hand, behind and before.
But it is requisite that the solid which is opposite and contrary
to this, should be that which has neither the length equal
to the breadth, nor the depth to either of these, but has all these
unequal. Solids of this kind are such as thrice four multiplied
by 2, or thrice four multiplied by 5, and the like, which
unequally proceed through unequal degrees of spaces. Solids
of this form are called in Greek scalenoi scalene, but in Latin
gradati, because they increase like steps from the less to the
greater. Certain Greeks also called them spheniscoi; i.e. little
wedges. By some of the Greeks likewise they were called
bomiscoi, i.e. certain little altars, which in the Ionic region of
Greece, as Nicomachus says, were fashioned after this manner,
so that neither the depth corresponded to the breadth, nor the
breadth to the length. Between cubes therefore, extending
themselves in equal spaces, and those solids which increase
gradually from the less to the greater, those subsist as media,
which are neither equal in all their parts, nor unequal in all;
and these are called parallelopipedons.
CHAPTER XIV,
On the numbers cded HETEROMEKEIS, or, LONGER IN THE
OTHER PART;-^^ on oblong numbers, and the generation of
them
THE numbers which are longer in one part than in the other
when they are considered according to breadth, will be found