nitudes are said to be commensurable, when there is some magnitude which is contained in each of them a certain number of times exactly.

Magnitudes which have no common measure, are said to be incommensurable.

Magnitudes A, B, which have one common measure M, have also many others, indeed, an unlimited number of common measures, for (as will be shown in Prop. 1.) every magnitude which is contained an exact number of times in M, is contained also an exact number of times in A and B; and whether M be divided into two, or three, or any other number of equal parts, one of these parts will be contained an exact number of times in M.

Among the common measures of the same two magnitudes, there is, however, always one which is greater than any of the others, and which (as will be shown in Prop. 6.) is measured by every other. This greatest common measure is always to be understood when "the common measure" is spoken of without further specification.

5. The numerical ratio of one magnitude to another with which it is commensurable, is a certain number, whole or fractional, which expresses how many, and what parts of the second are contained in the first: for example, if the common measure of A and B be contained in Å five times, and in B six times, or, which is the same thing, if A contain ths of B, then A is said to have to B the numerical ratio "5 to 6" which is thus written 5; 6, or, in the fractional form, 5.

In fact, the particular ratio of two given magnitudes, whether commensurable or otherwise, can be conceived only by means of the numbers which denote how often the same magnitude is contained, or nearly contained, in each: without these, no idea can be formed of their relative magnitude; they constitute its measure, true or approximate.*

To these numbers, therefore, when speaking of commensurable magnitudes, the term "ratio" alone, i. e. without the addition of "numerical," will be found commonly applied in what follows.

* The numerical_ratio is accordingly designated, by some writers, "The measure of the ratio" of one magnitude to another. This term has, however, been applied in a different sense, to which deference is more particularly due, as it has given rise to the word "logarithm," of which it is the literal interpretation.

6. The terms of the ratio of two magnitudes, are the numbers which denote how often a common measure of the two is contained in each of them. They are distinguished by the names of antecedent and consequent, according to the corresponding magnitudes. In the foregoing example 5 and 6 are the terms of the ratio of A to B, 5, the antecedent, and 6, the consequent.

The terms of the same ratio of A to B, will be different according to the common measure by which they are determined; the lowest terms being in all cases those which are determined by the greatest common measure. It must be observed, however, that no other terms can express the same ratio, but such as are either the lowest, or equimultiples of the lowest terms; for the magnitudes compared, can have no common measure, which is not either the greatest, or contained a certain number of times in the greatest common measure. the other hand, any terms whatever which are equimultiples of the lowest terms will express the same ratio: thus, if A contain ths of B, it will contain also ths of B, ths, and, generally,

3 x n

5x n

On

For example, let A B C D, and EFGH be two rectangles, having equal altitudes, and let their bases A B, E F, 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, A b c D, &c. Efg H, &c. all equal to one another (I. 25.). Then, 5; 6 is the ratio of the rectangle ABCD to the rectangle E F G H, because A B CD and EFGH contain the same rectangle 5 and 6 times respectively; and the same 56 is also the ratio of the base AB 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, A: B: C, i. 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.

Ꭰ

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 X 6 times.

Cor. 3. Hence, also, reversely, the ratio of A to B being expressed by any terms, as 10×5: 10×6, 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 times B and three times B' respectively, proposition, A and A' being equal to three contain C and C' respectively 3x4 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

magnitudes A and B,measures also their sum A + B, and their difference A~B.

For, if M be contained in A any number of times, as 7, and in B any number of times, as 4; it will evidently be contained in the sum of A and B, 7 + 4 or 11 times, and in their difference 7-4 or 3 times, and therefore will measure their sum and difference.

The same may be said, if, instead of 7 and 4, any other numbers be taken.*

Therefore, &c.

PROP. 4.

If there be two magnitudes A, B, and if one of them be contained in the other a certain number of times with a remainder; any common measure of the two magnitudes shall measure the remainder, and any common measure of the remainder and the lesser magnitude, shall measure the greater also.

Let B be contained in A twice with a remainder R, and let M be A any common measure of A and B; then, since M is contained a certain number of times in B, it is also contained a certain number of times in twice B (1.)† and measures twice B: but it also measures A; therefore (3.) it measures the difference of A and twice B, that is R.

Next, let N be any common measure of R and B: then, as before, N measures twice B; therefore (3.) it measures the sum of R and twice B, that is, A.

And the reasoning, in either case, is independent of the particular numbers

assumed.

Therefore, &c.

Cor. The greatest common measure of the remainder and lesser magnitude is also the greatest common measure of the two magnitudes. For, since every common measure of A and B is also a common measure of B and R; the greatest common measure of A and B will be found among the common measures of B and R; and it has been

The example of Euclid has been followed in annexing straight lines to illustrate this and many subsequent propositions, which are, however, not the less to be understood as applicable to and demonstrated of magnitudes generally, as is evident from the language of the enunciation and demonstration.

The reference is here to the first proposition of the present Book; and generally, in such references as have no Roman numeral to indicate the Book, the current Book is always to be understood.

shown that every one of the latter measures both A and B; therefore the greatest among them is the greatest common measure of A and B.

PROP. 5,

By repeating the process indicated in the last proposition, with the remainder and the lesser magnitude, and again with the new remainder (if there be one) and the preceding, and so on, the greatest common measure of two given commensurable magnitudes A, B may be found.

Let B, for instance, be contained in A twice (as in the last proposition), with a remainder R; let R be contained in B three times, with a second remainder R.; let R, be contained in R four times, with a third remainder R., and let R, be contained in R, five times exactly. Then, because (by 4. Cor.) the greatest common measure of A and B is the greatest common measure of B and R, that is (by the same Cor.) of R and R,, that is, again, of R, and R,, and because R,, being contained in itself once and a certain number of times in R,, is the greatest common measure of R, and R., it is likewise the greatest common measure of A and B.

2

2

The same may be said, if instead of 2, 3, 4, 5, any other numbers, supposed to arise from a similar examination of any two given commensurable magnitudes, be taken. At every step of the process, the remainder, as R, is diminished by the following remainder R, or by as many times R, as are contained in it, that is, in either case, by a magnitude greater than the supposed greatest common measure, to procure the new remainder R,. In all cases therefore, after a number of steps, which is less than the number of times the lesser magnitude contains the supposed greatest common measure, a remainder will be found which is equal to the greatest common measure. Therefore, &c.

Cor. 1. If the process admit of being continued through an unlimited number of steps, without arriving at a remainder which measures the next preceding, the magnitudes which are subjected to it, have no common measure, i. e. they are incommensurable.

Cor. 2. By proceeding in a similar manner, the greatest common measure of three magnitudes A, B and C may be found; for if M'be taken, the greatest common measure of A and B, and M, the greatest common measure of M and C, then because the greatest common