PROP. XIV. THEO.

If the first has the same ratio to the second which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less.

PROP. XV. THEO.

Magnitudes have the same ratio to one another which their equimultiples have.

0:40 4. (12 B. v.).

And as the same reasoning is generally applicable, we have

OMO: M'

... Magnitudes have the same ratio, &c.

Let A and B be two numbers, and C and D equimultiples of

them,

A B C D.

Let Cn A and Dn B;

then it is evident that A: B:: n A: n B,

DEFINITION XIII.

The technical term permutando, or alternando, by permutation or alternately, is used when there are four proportionals, and it is inferred that the first has the same ratio to the third which the second has to the fourth; or that the first is to the third as the second is to the fourth: as is shown in the following proposition:

Let : Δ O: A,

by "permuntando" or "alternando" it is inferred

: Δ.

It may be necessary here to remark that the magnitudes O, ▲, must be homogeneous, that is, of the same nature or similitude of kind; we must therefore, in such cases, compare lines with lines, surfaces with surfaces, solids with solids, &c. Hence the student will readily perceive that a line and a surface, a surface and a solid, or other heterogenous magnitudes, can never stand in the relation of antecedent and consequent.

PROP. XVI. THEO.

If four magnitudes of the same kind be proportionals, they are also proportionals when taken alternately.

.. If four magnitudes of the same kind, &c.