are obviously necessary in all
works, as the finger, palm, foot, and cubit. These they apportioned so
as to form the "perfect number," called in Greek [Greek: teleion], and
as the perfect number the ancients fixed upon ten. For it is from the
number of the fingers of the hand that the palm is found, and the foot
from the palm. Again, while ten is naturally perfect, as being made up
by the fingers of the two palms, Plato also held that this number was
perfect because ten is composed of the individual units, called by the
Greeks [Greek: monades]. But as soon as eleven or twelve is reached, the
numbers, being excessive, cannot be perfect until they come to ten for
the second time; for the component parts of that number are the
individual units.

6. The mathematicians, however, maintaining a different view, have said
that the perfect number is six, because this number is composed of
integral parts which are suited numerically to their method of
reckoning: thus, one is one sixth; two is one third; three is one half;
four is two thirds, or [Greek: dimoiros] as they call it; five is five
sixths, called [Greek: pentamoiros]; and six is the perfect number. As
the number goes on growing larger, the addition of a unit above six is
the [Greek: ephektos]; eight, formed by the addition of a third part of
six, is the integer and a third, called [Greek: epitritos]; the addition
of one half makes nine, the integer and a half, termed [Greek:
hemiolios]; the addition of two thirds, making the number ten, is the
integer and two thirds, which they call [Greek: epidimoiros]; in the
number eleven, where five are added, we have the five sixths, called
[Greek: epipemptos]; finally, twelve, being composed of the two simple
integers, is called [Greek: diplasios].

7. And further, as the foot is one sixth of a man's height, the height
of the body as expressed in number of feet being limited to six, they
held that this was the perfect number, and observed that the cubit
consisted of six palms or of twenty-four fingers. This principle seems
to have been followed by the states of Greece. As the cubit consisted of
six palms, they made the drachma, which they used as their unit, consist
in the same way of six bronze coins, like our _asses_, which they call
obols; and, to correspond to the fingers, divided the drachma into
twenty-four quarter-obols, which some call dichalca others trichalca.

8. But our countrymen at first fixed upon the ancient numbe