Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

tervals. For there is one interval from the first to the second,
another from the second to the third, and another from the
third to the fourth, which is the ultimate distance. Plane figures,
therefore, are very properly said to be contained by two,
and solid, by three intervals. Let there be two squares, therefore,
viz. 4 and 9. Of these, therefore, there can only be one medium
constitgted in the same ratio; and this medium is 6.
For 6 is sesquialter to 4, and 9 is sesquialter to 6. This, however,
takes place, because the side of the one square, multiplied
by the side of the other, produces the middle term 6. For the
side of 4 is 2, and of 9 is 3, and these multiplied together produce
6. Hence, in order to find the medium of two squares,
their sides must be multiplied together, and the product will be
the medium that is sought. But if two cubes are given, as 8
and 27, two media only can be constituted between these in the
same ratio, viz. 12 and 18. For 12 to 8, and 18 to 27 are conjoined
by a sesquialter ratio alone.
These two media also are found as follows: If the side of
the greater cube is multiplied by the side of the less, and afterwards
the product is multiplied by the less, the less of the twa
media will be obtained. But if the square of the side of the
greater cube, is multiplied by the side of the less cube, the
product will be the greater medium. Thus in the two cubes 8,
and 27. The side of the greater cube is 3, and the side of the
less is 2, and 3X 2 X 2=12, the less medium. And 3X3X 2
-18, the greater medium. Thus too in the cubes 27 and 64.
The side of the greater cube is 4, and the side of the less is 3,
and 4x3 )(3=36, the less medium. And 4X4X3=48, the
greater medium. Hence the less medium derives two sides
from the less cube, and one side from the greater cube. And
after the same manner the greater medium derives two sides
from the greater cube, and one side from the less cube. Thus
12 the less medium between the cubes 8 and 27, has two sides