equal to the sum of two, three, or any number of given squares, viz. the squares of A, B, C, &c.

Take D E equal to A; from the point D, draw DF (44.) at right angles to DE, and equal to B: join EF: from F, draw FG at right angles to EF, and equal to C; and so proceed for the rest of the given squares. Join E G. Then, because E D F is a right angled triangle, the square of EF is equal to the squares of E D, D F (36.), that is, to the squares of A, B: again, because EFG is a right angled triangle, the square of EG is equal to the squares of EF, FG, that is, to the squares of A, B, C; and

magnitude is greater or less than a fourth.

The ratio of one magnitude to another is independent of the kind of magnitudes compared; for it is obvious that one may contain the other, or the sixth, or twelfth, or hundredth part of the other the same number of times, whether they be lines, or surfaces, or solids, or again, weights, or parts of duration.

It is required only for the comparison we speak of, that the magnitudes be of the same kind, containing the same magnitude, each of them, a certain number of times, or a certain number of times nearly. Upon these numbers, and upon these only, the ratio depends.

Hence it appears that this theory pertains in truth to Arithmetic. The use of Proportion is, however, so indispensable in Geometry, that it has been usual, either to introduce its theorems into the body of the science, after the example of Euclid, or to premise a few of the most important of them as a manual for reference. In the present, and two following sections, the subject will be discussed at a length commensurate with its importance.

Def. 1. When one magnitude is compared with another of the same kind, the first is called the antecedent, and the second the consequent.

2. One magnitude is said to be a multiple of another, when it contains that other a certain number of times exactly and the other magnitude, which is contained in the first a certain number of times exactly, is said to be a submultiple, or measure, or part of the first.

Hence, also, one magnitude is said to measure another when it is contained in the other a certain number of times exactly.

3. Two magnitudes are said to be equimultiples of two others, when they contain those others the same number of times exactly: and the other magnitudes which are contained in the first the same number of times exactly, are said to be like parts of the two first.

Thus, 7 A, 7 B, are equimultiples of A, B; and A, B, are like parts of 7 A,

7 B.

4. Two magnitudes are said to be commensurable with one another, when a common measure of the two may be found, i. e. a magnitude which is contained in each of them a certain number of times exactly.

In like manner, any number of mag

For example, let A B C D, and EFGH be two rectangles, having equal altitudes, and let their bases A B, EF, contain the same straight line M 5 and 6 times respectively: divide A B, E F, into the parts A b, &c. Ef, &c. each equal to M, and through the points of division draw right lines bc, &c. fg, &c. parallel to AD, and EH, thereby dividing the rectangles A B CD, EFGH, into 5 and 6 smaller rectangles respectively, Abc D, &c. Efg H, &c. all equal to one another (I. 25.). Then, 5; 6 is the ratio of the rectangle A B C D to the rectangle EFGH, because ABCD and EFGH contain the same rectangle 5 and 6 times respectively; and the same 56 is also the ratio of the base A B to the base EF, because A B and EF contain the same straight line 5 and 6 times respectively. Therefore, the ratios of the first to the second, and of the third to the fourth, are expressed by the same terms, and the two rectangles, and their two bases, are proportionals.

In the preceding, and in every other instance of commensurable proportionals, the first two, and the second two, have a common numerical ratio: and, in every case, if two magnitudes, and other two, have a common numerical ratio, the four magnitudes are, according to this definition, proportionals.

It is evident from the observations on def. 6, that if four magnitudes be proportionals, any other terms expressing the ratio of the first to the second, must likewise express the ratio of the third to the fourth. For the terms which determine the proportion, are either the lowest terms, or equimultiples of them (see Prop. 6. Cor. 1.): and in either case, the lowest terms which express the ratio of the first to the second, and of the third to the fourth, must be the same; therefore, because any other terms expressing the ratio of the first to the second must be equimultiples of the lowest terms, that is, of the lowest terms of the ratio of the third to the fourth, such terms express also the ratio of the third to the fourth (see observations on Def. 6.)

The same will be demonstrated more at large in Prop. [9].

Def. [8]. If there be two magnitudes of the same kind, and other two, the first is said to have to the second a greater ratio than the third has to the fourth, when the first contains some measure of the second a greater number of times than the third contains a like measure

of the fourth: also, when this is the case, the third is said to have to the fourth a less ratio than the first has to the second.

This definition can, in no case, apply to the same magnitudes which come under def. [7], i. e. one magnitude cannot have to another the same ratio as a third to a fourth by def. [7], and at the same time a greater or a less ratio than the third has to the fourth, by this definition. (See Prop. 9. Cor. 1 and 2.)

Much less can one magnitude have to another a greater ratio than a third has to a fourth, and at the same time a less ratio than the third has to the fourth, by this definition.

9. When four magnitudes A, B, C, D, are proportionals, they are said to constitute a proportion, which is thus written, A:B::C: D.

i. e. "A is to B as C is to D."

Of a proportion, the first and last terms are called the extremes, and the second and third the means; thus A, D are the extremes, and B, C the means, of the proportion A: B::C: D. The terms A and C are said to be homologous, as also B and D; the former being antecedents, and the latter consequents, in the proportion.

It is indifferent in the statement of a proportion which of the ratios precedes the other, for it is evident, that if A has to B the same ratio as C has to D, C has to D the same ratio as A to B. Hence, A: B::C: D, and C: D::A: B, signify the same proportion-the only difference being, that the extremes of one expression are the means of the other.

Since magnitudes cannot be compared, except they be of the same kind, it is manifest that the first and second terms of a proportion must be of the same kind, as also the third and fourth; yet the first and third may be of different kinds : e. g. 10lbs 6lbs::15 ft.: 9ft. is a true proportion; for the ratio of the first to the second, as well as of the third to the fourth, is 5: 3, and yet 10lbs. and 15ft. are not magnitudes of the same kind.

10. Three magnitudes of the same kind are said to be proportionals or in continued proportion, when the first has to the second the same ratio which the second has to the third. Magnitudes A, B, C, which are in continued proportion, may be written thus, ABC,

.e. "A is to B, as B to C." In this case. B is called a mean proportional or geometrical mean between A and C, and C a third proportional to A and B.

D

11. Any number of magnitudes of the same kind are said to be in continued proportion, when the first is to the second, as the second to the third, as the third to the fourth, and so on. Magnitudes A, B, C, D, &c. which are in continued proportion, may be thus written, A: B:C:D: &c.

The magnitudes of such a series are said to be in geometrical progression · and B, C, are called two geometrical means between A and B ; again, B, C, D, three geometrical means between A and E; and so on.

Also, in this case, the first A is said to have to the third C, the duplicate ratio of that which it has to the second B-to the fourth D, the triplicate ratio of that which it has to B, and so on: and reciprocally, A is said to have to B the subduplicate ratio of that which it has to C, the subtriplicate ratio of that which it has to D, and so on.

12. If there be any number of magnitudes of the same kind, A, B, C, D, the first A is said to have to the last Da ratio which is compounded of the ratios of A to B, B to C, and C to D.

Also, if K and L, M and N, P and Q, be any other magnitudes, and if the ratios of K to L, of M to N, and of P to Q, be the same respectively with the ratios of A to B, of B to C, and of C to D, A is said to have to D a ratio which is compounded of the ratios of K to L, M to N, and P to Q.

AXIOMS. (EUc. v. Ax. 1, 2, 3, 4.)

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

2. Those magnitudes of which the same, or equal magnitudes are equimultiples, are equal to one another.

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

4. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

PROP. 1.

If one magnitude be a multiple of another, and if this be likewise a multiple of a third magnitude, the first shall be a multiple of the third.

Let A contain B three times, and let B contain C four times; then because B contains C four times, three times B or A must contain C thrice as often, or twelve times, i. e. a certain number of times exactly; therefore A is a multiple of C. The

same may be said, if, instead of three and four, any other numbers whatever be taken, i. e. if A be any multiple of B, and B any multiple of C. Therefore, &c.

Cor. 1. If one magnitude measure another, it will measure any multiple of that other.

Cor. 2. Hence, the ratio of A to B being expressed by certain given terms, as 5:6, the same ratio may be expressed by any terms which are equimultiples of the given terms, as 10 × 5:10×6. For, if M be the common measure which is contained in A five times and in B six times, any measure of M, as the tenth part, will also measure A and B (Cor. 1.) and will be contained in A 10 × 5 times, and in B 10 × 6 times.

Cor. 3. Hence, also, reversely, the ratio of A to B being expressed by any terms, as 10×5: 10x6, which have a common factor, the same ratio may be expressed by any terms, as 5: 6, which are like parts of the given terms.

For, if M be the common measure which is contained in A 10×5 times, and in B 10x 6 times, it is evident that 10 M will be contained in A 5 times, and in B 6 times.

PROP. 2. (EUC. v. 3.)

If two magnitudes be equimultiples of two others, and if these be likewise equimultiples of two third magnitudes, the two first shall be equimultiples of the two third.

Let A and A' contain B and B' respectively three times, and let B and B' contain C and C' respectively four times; then, as in the demonstration of the preceding proposition, A and A' being equal to three times B and three times B' respectively, contain C and C' respectively 3 x 4 times, i.e. the same number of times exactly; therefore A and A' are equimultiples of C and C'.

The same may be said, if, instead of 3 and 4, any other numbers be taken, i. e. if A, A' be any equimultiples of B, B', and B, B' any equimultiples of C, C'. Therefore, &c.

Cor. If two magnitudes A, A' be equimultiples of two others B, B', and likewise of two third magnitudes C, C', and if one of the second, B, be a multiple of the corresponding one, C, of the third, the other second, B', shall be the same multiple of the other third, C'.

PROP. 3.

A common measure M of any two