VI.

When the first of four magnitudes has the same ratio to the second which the third has to the fourth (such sameness of ratios being defined by the preceding def"), the four magnitudes are said to constitute a proportion, or to be proportional.

OBS. 1. When four magnitudes, A, B, C, D, are proportional, the proportion which they constitute is usually expressed by saying that A is to B as C to D, and may be written

A : B :: C: D.

But whenever either of these forms is used to denote a proportion, we must bear in mind that it is but a brief way of stating that A has the same ratio to B which C has to D, according to the defn of sameness of ratio given in Def. 5.

OBS. 2. When four magnitudes are proportional :

:

(1) they are called the first, second, third, and fourth terms of the
proportion;

(2) the fourth term is called a fourth proportional to the three
magnitudes which form the first, second, and third terms;
(3) the first and fourth terms are called the two extremes, and
the second and third the two means.

VII.

The first of four magnitudes is defined to have a greater ratio to the second than the third has to the fourth (or, what is the same thing, the third to have a less ratio to the fourth than the first has to the second) when there can be taken such equimultiples of the first and third, and such equimultiples of the second and fourth, that the multiple of the first shall be greater than the multiple of the second, but the multiple of the third not greater than the multiple of the fourth.

OBS. When this def" is referred to, it is cited as the "definition of one ratio being greater (or less) than another."

VIII.

By proportion is meant the sameness of ratios.

OBS. This is nothing more than a different way of expressing Def. 6.

P

IX.

In order that four magnitudes may constitute a proportion, no more than two of them are allowed to be equal to the same magnitude.

OBS. 1. Hereby Euclid excludes from consideration all such proportions as A is to A as A to B, A is to B as B to B, &c. But we shall meet with such proportions as A is to B as A to B, A is to A as B to B (Def. 4), where two pairs of the four magnitudes are equal to the same magnitude; and with such as A is to B as C to A, A is to B as B to C, where one pair of the four are equal to the same magnitude. Since proportions like the last frequently occur, we shall speak of them in the next observation.

OBS. 2. When of three different magnitudes of the same kind, one can form the first term of a proportion, another each of the second and third terms, and the remaining magnitude the fourth term; i. e. if the first of the three magnitudes be to the second as the second is to the third: then

(1) the three magnitudes are said to be proportional;

(2) the third magnitude is said to be a third proportional to the first and second;

(3) the second magnitude is said to be a mean proportional

between the first and third.

X.

The duplicate ratio of the ratio which one magnitude has to another is defined to be the ratio which the first magnitude has to a third magnitude such that the first is to the second as the second is to the third.

OBS. I. Hence when three magnitudes are proportional (Def. 9. Obs. 2), the ratio which the first bears to the third is the duplicate ratio of the ratio which it bears to the second.

OBS. 2. This definition will be made clearer by the following example:

Ex. It is required to find the duplicate ratio of the ratio of A to B.

Take a magnitude C of the same kind as A and B, such that A is to B as B to C. Then the ratio of A to C will be the duplicate ratio of the ratio of A to B.

XI.

When there are any number of magnitudes of the same kind, the ratio of the first of them to the last

is defined to be the ratio compounded of the ratios of the first to the second, of the second to the third, of the third to the fourth, &c., and of the last but one to the last.

OBS. 1. In the particular case when the ratios of the first magnitude to the second, of the second to the third, &c., are all the same one to another, the ratio compounded of them as above defined is called a multiplicate ratio of either of the component ratios, the order of multiplicity being equal to their number.

The defn and the observation will be made clearer by the following examples :

:

Ex. 1. If there are three magnitudes of the same kind, A, B, C; the ratio of A to C is the ratio compounded of the two ratios of A to B, and of B to C. And in the particular case when the ratio of A to B is the same as that of B to C, the ratio of A to C is called the duplicate ratio of that of A to B, or of B to C.

Ex. 2. If there are four magnitudes of the same kind, A, B, C, D; the ratio of A to D is the ratio compounded of the three ratios of A to B, and of B to C, and of C to D. And in the particular case when the ratios of A to B, of B to C, and of C to D are the same to one another, the ratio of A to D is called the triplicate ratio of that of A to B, or of B to C, or of C to D. OBS. 2. The ratios which this defn, as it stands, enables us to compound, must consist of magnitudes of the same kind, and the second term of each ratio must be the same as the first of the succeeding one. The manner of modifying the defn to compound any ratios whatever will be seen from the annexed example.

Ex. Let there be any three ratios whatever; viz. the ratio of A to B, the ratio of C to D, and the ratio of E to F. It is required to find the ratio compounded of these three ratios.

Take two magnitudes of any kind (usually straight lines) K, L, such that A is to B as K to L; take a magnitude M of the same kind as K and L, such that C is to D as L to M; and take a magnitude N of the same kind as K, L, M, such that E is to F as M to N. Then the ratio compounded of the three ratios of A to B, of C to D, and of E to F, is the same as the ratio compounded of the three ratios of K to L, of L to M, and of M to N; but the ratio compounded of the three ratios of K to L, of L to M, and of M to N is by defn the ratio of K to M. Hence the ratio of K to M is the ratio compounded of the three ratios of A to B, of C to D, and of E to F. And a similar method must be employed whatever be the number of ratios to be compounded.

XII.

Of two magnitudes, having a ratio to one another, the first is called the antecedent, and the second the

consequent; and of four magnitudes constituting a proportion, the first and third are called the antecedents, and said to be homologous to one another, and the second and fourth are called the consequents, and are likewise said to be homologous to one another.

POSTULATES.

I.

Let it be granted that of a given magnitude there may be taken any multiple required.

II.

Let it be granted that any given multiple of a magnitude may be divided into parts, each of which is equal to the magnitude of which it is the multiple.

AXIOMS.
I.

Equimultiples of the same magnitude, or of equal magnitudes, are equal to one another.

II.

Magnitudes, of which the same magnitude is an equimultiple, or of which equal magnitudes are equimultiples are equal to one another.

III.

A multiple of a greater magnitude is greater than the same multiple of a less magnitude.

IV.

Of two magnitudes, the first is greater or less than the second according as the multiple of the first is greater or less than the same multiple of the second.

PROPOSITIONS.

PROP. I. THEOR.

If a set of magnitudes be equimultiples of the same number of other magnitudes: then, whatever multiple either one in the first set is of the corresponding one in the second, the same multiple shall all the first set of magnitudes taken together be of all the second set of magnitudes taken together.

I. Let the number of the magnitudes in each set be two. Let the two magnitudes AB, CD be respectively equimultiples of the two E, F. Then whatever multiple AB is of E, the same multiple shall AB and CD together be of E and F together.

Since by hyp AB, CD are equimultiples of E, F, there are as many magnitudes in AB, each equal to E, as there are in CD, each equal to F. Divide (v. Post. I) AB into magnitudes, each equal

to E, viz. AG, GB, and

GB being equal to that of CH, HD.

F

H

By const AG is equal to E, and CH to F; hence, adding equals to equals, AG and CH together are equal (Ax. 3) to E and F together. Similarly GB and HD together are equal to E and F together: and so on, if there were more magnitudes in AB and CD. Hence there are as many