12. If there are any number of ratios, and a set of magnitudes is taken such that the ratio of the first to the second is equal to the first ratio, and the ratio of the second to the third is equal to the second ratio, and so on, then the first of the set of magnitudes is said to have to the last the ratio compounded of the given ratios.

:

Thus, if A: B, C: D, E F be given ratios, and if P, Q, R, S be magnitudes taken so that

13.

When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.

then A is said to have to C the duplicate ratio of that which it has to B.

it is clear that the ratio compounded of two equal ratios is the duplicate ratio of either of them.

14. When four magnitudes are in continued proportion, the first is said to have to the fourth the triplicate ratio of that which it has to the second.

It may be shewn as above that the ratio compounded of three equal ratios is the triplicate ratio of any one of them.

Although an algebraical treatment of ratio and proportion when applied to geometrical magnitudes cannot be considered exact, it will perhaps be useful here to summarise in algebraical form the principal theorems of proportion contained in Book V. The student will then perceive that its leading propositions do not introduce new ideas, but merely supply rigorous proofs, based on the geometrical definition of proportion, of results already familiar in the study of Algebra.

We shall only here give those propositions which are afterwards referred to in Book VI. It will be seen that in their algebraical form many of them are so simple that they hardly require proof,

SUMMARY OF PRINCIPAL THEOREMS OF BOOK V.

PROPOSITION 1.

Ratios which are equal to the same ratio are equal to one another. That is, if Y and C : D=X : Y; then A B C D.

A: B X

PROPOSITION 3.

If four magnitudes are proportionals, they are also proportionals when taken inversely.

This inference is referred to as invertendo or inversely.

PROPOSITION 4.

(i) Equal magnitudes have the same ratio to the same magnitude.. For if

then

A=B,

AC BC.

(ii) The same magnitude has the same ratio to equal magnitudes.

PROPOSITION 6.

(i) Magnitudes which have the same ratio to the same magnitude are equal to one another.

(ii) Those magnitudes to which the same magnitude has the same ratio are equal to one another.

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

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

This inference is referred to as alternando or alternately.

PROPOSITION 12.

If any number of magnitudes of the same kind are proportionals, then as one of the antecedents is to its consequent, so is the sum of the antecedents to the sum of the consequents.

.. A B A+C+E+ : B+D+F+....

This inference is sometimes referred to as addendo.

PROPOSITION 13.

(i) If four magnitudes are proportionals, the sum of the first and second is to the second as the sum of the third and fourth is to the fourth.

This inference is referred to as componendo.

(ii) If four magnitudes are proportionals, the difference of the first and second is to the second as the difference of the third and fourth is to the fourth.

That is, if

then

A: B= C: D,

A~B: B C~D: D.

The proof is similar to that of the former case.

This inference is referred to as dividendo.

PROPOSITION 14.

If there are two sets of magnitudes, such that the first is to the second of the first set as the first to the second of the other set, and the second to the third of the first set as the second to the third of the other, and so on to the last magnitude: then the first is to the last of the first set as the first to the last of the other.

First let there be three magnitudes, A, B, C, of one set, and three, P, Q, R, of another set,