Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

etc. the greater term will always be found to differ from the
less by the quantity of the ratio less by unity.
In continued geometrical proportionality also, the rectangle
under the extremes is equal to the square of the mean. Thus
in the three terms 2.4. 8. which are in geometrical proportion,
2 X 8=4X4=16. The same thing will also take place though
there should be a greater number of terms than three, if the
number of terms is odd. Thus in the five terms 16. 8. 4. 2. 1,
the rectangle under the extremes, i.e. 16x1 is equal to the
square of the middle term 4. Thus too in the seven terms 64.
32. 16. 8. 4. 2. I., 64X1=8)(8. But where the terms are not
continued, the rectangle under the extremes is equal to the rectangle
under the means; as is evident in the terms 2. 4. 8. 16;
for 2X 16=4X8=32. And if the terms are more than four,
if only they are even, the same thing will take place. Thus in
the six terms 2. 4. 8. 16. 32. 64, 2 X64=128=4X32=8 X 16.
Another property of this middle is, that there is always an
equal ratio both in the greater and the less terms. Thus in all
the terms 2. 4. 8. 16. 32. 64, there is a duple ratio. This is
likewise the case in the terms, 3. 9. 27. 81. 243. 729. And in
a similar manner in others.
And in the last place, it is peculiar to this middle, squares
and numbers longer in the other part, being alternately arranged,
to proceed from the first multiple into all the habitudes
of superparticular ratios, as will be evident from the following
scheme.
, , , ,
Squares and numbers longer in the other part, aIternately arranged.
1 2 4 6 9 12 16 20 25 30 37
Puple, Duple slqui- . Sesqui- Sesqui- Suqui- Sesqui- Sesqui- Stsqui- Suquiquintan
alter I alter ttrtian I tertian quartan ( quatan qintan 1
Hence the arithmetical middle is compared to an oligarchy,
or the republic which is governed by a few, who pursue their
own good, and not that of the community; because in its less