er and mistress, of whom the body was to be the subject. And he
made her out of the following elements and on this wise: Out of the
indivisible and unchangeable, and also out of that which is divisible
and has to do with material bodies, he compounded a third and
intermediate kind of essence, partaking of the nature of the same and of
the other, and this compound he placed accordingly in a mean between the
indivisible, and the divisible and material. He took the three elements
of the same, the other, and the essence, and mingled them into one form,
compressing by force the reluctant and unsociable nature of the other
into the same. When he had mingled them with the essence and out of
three made one, he again divided this whole into as many portions as was
fitting, each portion being a compound of the same, the other, and the
essence. And he proceeded to divide after this manner:--First of all, he
took away one part of the whole (1), and then he separated a second part
which was double the first (2), and then he took away a third part which
was half as much again as the second and three times as much as the
first (3), and then he took a fourth part which was twice as much as the
second (4), and a fifth part which was three times the third (9), and a
sixth part which was eight times the first (8), and a seventh part
which was twenty-seven times the first (27). After this he filled up the
double intervals (i.e. between 1, 2, 4, 8) and the triple (i.e. between
1, 3, 9, 27) cutting off yet other portions from the mixture and placing
them in the intervals, so that in each interval there were two kinds of
means, the one exceeding and exceeded by equal parts of its extremes (as
for example 1, 4/3, 2, in which the mean 4/3 is one-third of 1 more than
1, and one-third of 2 less than 2), the other being that kind of mean
which exceeds and is exceeded by an equal number (e.g.

- over 1, 4/3, 3/2, - over 2, 8/3, 3, - over 4, 16/3, 6, - over 8: and
- over 1, 3/2, 2, - over 3, 9/2, 6, - over 9, 27/2, 18, - over 27.

Where there were intervals of 3/2 and of 4/3 and of 9/8, made by the
connecting terms in the former intervals, he filled up all the intervals
of 4/3 with the interval of 9/8, leaving a fraction over; and the
interval which this fraction expressed was in the ratio of 256 to 243
(e.g.

243:256::81/64:4/3::243/128:2::81/32:8/3::243/64:4::81/16:16/3::242/32:8.

And thus the whole mixture out of which he cut these