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Let A contain M 4 times, and N 5 times: then, if A be divided into 4 × 5, or 20 equal parts, M will contain 5 and N 4 of those parts; and, therefore, one of those parts being contained an exact number of times in M, and an exact number of times in N, is a common measure of M and N.

Next, let M contain a magnitude a 5 times, and let N contain the same a 4 times: then, if there be taken a magnitude A, which contains a 5x4, or 20 times, A will contain M 4 times, and N 5 times, and, therefore, will be measured by each of the magnitudes M and N.

And the demonstration will be the same whatever numbers are taken. Therefore, &c.

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mensurable with the second, the second with the third, the third with the fourth, and so on to the last, all the magnitudes shall be commensurable.

Cor. 2. If two magnitudes be commensurable with one another, and if one of them be incommensurable with a third magnitude, the other shall likewise be incommensurable with the third.

SECTION 2.-Proportion of Commen surable Magnitudes.

In the foregoing Section it has been seen how the greatest common measure, and hence the lowest terms of the ratio, of two magnitudes may be determined (5. and 5. Cor. 3.). It has been seen, also, that every other common measure of the same two magnitudes is contained in the greatest a certain number of times exactly, and hence that no other terms but such as are equimultiples of the lowest can express the same ratio (6. and 6. Cor. 1.); which last simple relation given terms to the lowest terms, and affords an easy Rule for reducing any hence determining whether two given numerical ratios which are differently expressed be the same or not (6. Cor.

2 and 3).

We now proceed to the theorems of proportion, the end of which is for the most part to show, that if two magnitudes have the same ratio with other two, magnitudes which are related after a certain manner to the two former, will have the same ratio to one another with magnitudes which are similarly related to the two latter.

In the demonstration of these theorems, an illustration by particular numbers will be preferred, for the sake of the less practised reader, to the use of general symbols for numbers. In following this plan hitherto, we have had occasion to observe at the conclusion of each demonstration, that the steps are not upon this account the less general, but apply equally to any other numbers which may be substituted in place of the particular numbers assumed. The same observation will be found equally true, and is accordingly here premised with regard to particular numbers, wherever they are introduced in the following demonstrations. The demonstrations are exactly similar, whatever numbers be substituted in the place of them; and, accordingly, the general propositions are not less evident

than if the reasoning had been conducted by general symbols.

The present Section treats only of commensurable proportionals as described in def. [7]. The theorems, however, (that is, all of them, the first excepted which is expressly enunciated of commensurables) are likewise true with regard to the more general description of proportionals to be considered in Section 3. Hence the brackets which the reader will have already noticed in "def. [7]" and "def. [8]," and in which the numbers 9, 10, &c. of the following propositions are inclosed. Hence, too, the absence of all reference to Euclid, whose theorems on the subject of Proportion are stated not of commensurable magnitudes only, but of such as, whether commensurable or otherwise, come under a more general definition of proportionals. In the next Section, these propositions, (prop. [9] excepted) will be repeated with reference to a similar general definition, the brackets will be removed from the numbers 9, 10, &c., and the references to Euclid will be annexed as usual.

PROP. [9].

If there be four magnitudes A, B, C, D, which are commensurable proportionals, the second and fourth, and any like parts of the second and fourth, shall be contained in the first and third the same number of times exactly, or the same number of times with corresponding remainders less than the parts.

Let the common ratio of A to B and C to D be any whatever, for example, 845; and first, let the second and fourth, viz. B and D, be taken: B and D shall be contained in A and C respectively, the same number of times exactly, or the same number of times with corresponding less remainders.

Let M and N be the common measures of A and B and of C and D, by which the ratio 84: 5 is determined; and which are therefore contained in B and D respectively 5 times, and in A and C respectively 84 times. Then, because M and N are contained in A and C respectively 84 times, and that B and D contain M and N respectively 5 times; B and D are contained in A and C respectively, as often as the number 5 is contained in 84, that is, the same number of times exactly, if 5 be exactly contained in 84, or the same number of times with corresponding remainders, if 5 be contained in 84 with a remainder.

Next, let any like parts, for example the 6ths of B and D, be taken: these shall be contained in A and B the same number of times, either exactly, or with corresponding less remainders.

Let M and N be the common measures of A and B and of C and D, as before, by which the ratio 84: 5 is determined; and which are therefore contained in B and D respectively 5 times, and in A and C respectively 84 times. And let M be divided into 6 parts each equal to m, and N also into 6 parts each equal to n. Then it is evident that m

and n must be contained in A and C respectively 84×6 times, and in B and D respectively 3 x 6 times (1.). And, because the sixth-parts in question, i. e. the sixth-parts of B and D, are contained in B and D respectively 6 times, they must contain m and n respectively 5 times. Therefore, the parts in question are contained in A and C respectively as often as the number 5 is contained in 84 × 6, that is, the same number of times exactly, if 5 be exactly contained in 84 x 6, or the same number of times with corresponding remainders, if 5 be contained in 84 × 6 with a remainder.

Therefore, &c.

Cor. 1. Hence it appears that, if two magnitudes and other two have a common numerical ratio, any other terms expressing the ratio of the first to the second must also express the ratio of the third to the fourth; as was observed in the remarks upon def. [7].

Cor. 2. Hence it appears, also, that A cannot be said to have to B the same ratio which C has to D, according to def. [7], and at the same time a greater or a less ratio than C has to D, according to def. [8]; as was observed at def. [8].

Cor. 3. And much less can A be said to have to B a greater ratio than C has to D, according to def. [8], and at the same time a less ratio than C has to D, according to the same definition.*

* As the General Theory of Proportion in Section 3. contains the remaining propositions of this Section with reference to the new definitions 7 and 8 there given, the reader may, if he pleases, pass on to that Section, or rather, to the concluding Scholium of the present one; by which he will omit nothing that will be cited in the future pages of this treatise. On the other hand, it is recommended to beginners, and such as are not curious about the general and the remainder of the present Section, with the opencomplete theory of proportion, to peruse with care ing paragraphs of Section 3., and then pass on to Section 4.; which they may do with the assurance of no difficulty being presented to them upon that account in the remainder of the treatise.

PROP. [10].

Equal magnitudes have the same ratio to the same magnitude: and the same has the same ratio to equal magnitudes.

For if any part, as a 5th, of C be contained in A any number of times, as 4, and if B be equal to A, the same part of C will evidently be contained in B the same number of times, 4; and therefore, 4: 5 will be the common ratio of A to C, and of B to C.

Again, if any part of A, as a 4th, be contained in C any number of times, as 5, and if B be equal to A, the like part of B (ax. 2.) will evidently be contained in C the same number of times, 5; and, therefore, 54 will be the common ratio of C to A, and of C to B.

Therefore, &c.

Cor. If a ratio which is compounded of two ratios be a ratio of equality, one of these must be the reciprocal of the other. For if there be three magnitudes of the same kind A, B, and C, and if A be equal to C, the ratio of B to C must be the same with the ratio of B to A in other words, if the ratio of A to C, which is compounded of the ratios of A to B and of B to C, be a ratio of equality, the ratio of A to B must be the reciprocal of the ratio of B to C, (def. 6. and 12.).

PROP. [11].

Of two unequal magnitudes the greater has a greater ratio to the same magni tude; and the same magnitude has a greater ratio to the lesser of the two.

B

For, the magnitudes A, B, and C, being supposed to be commensurable, if A be greater than B, it must contain the common measure of A, B, and C, that is, a measure of C, a greater number of times than B contains the same measure of C, and therefore (def. [8]) A has to C a greater ratio than B has to C.

Again, if A be greater than B, and if C be any magnitude commensurable with each of them, like measures of A and B may be found, which are each of them contained a certain number of times in C. For, since A, B, and C (3.) contain the same magnitude M, each of them a certain number of times, let them be equal to 7 M, 5 M, and 12 M,

respectively. Then, if a and b be taken, the 35th parts (7×5) of A and B, a will be a 5th part of M, and therefore will be contained an exact number of times in C, and b a 7th part of M, and therefore also contained an exact number of times in C (1. Cor. 1.). But any measure of A is greater than the like measure of B (ax. 4.); therefore, of the like measures in question, viz. a and b, C will contain that of A a less number of times than it contains that of B. There

fore (def. [8]) C has to A a less ratio than it has to B, or a greater ratio to B than it has to A.

Therefore, &c.

Cor. 1. Magnitudes which have the same ratio to the same magnitudes those to which the same magnitude has are equal to one another: as likewise the same ratio ([9] Cor. 2.)

Cor. 2. A ratio which is compounded of two ratios, one of which is the reci

procal of the other, is a ratio of equality. For, if there be three magnitudes of the same kind, A, B, and C, and if the ratio of B to C be the same with the ratio of B to A, A must be equal to C: in other words, the ratio of A to C, which is compounded of the ratios of A to B and of B to C, one of which is the reciprocal of the other, is a ratio of equality, (def. 6. and 12.).

and

Cor. 3. If one of two magnitudes have a greater ratio to the same magnitude than the other has, the first must be greater than the other if the same magnitude have a greater ratio to one of two-magnitudes than it has to the other, the first must be less than the other ([9] Cor. 2, 3.).

PROP. [12].

Magnitudes A, B and C, D, which have the same ratio with the same magnitudes P, Q, have the same ratio with one another.

For, since A, B have the same ratio with P, Q, some part of B is contained in A as often as a like part of Q is contained in P; therefore, if any other part of Q be contained a certain number of times in P, a like part of B will be contained as often in A ([9]).

But, because C, D have the same ratio with P, Q, some part of D is contained in C as often as a like part of Q is contained in P. Therefore, whatever part this be, which is taken of D, and contained in C as often as the like part of Q in P, the like part of B is con

tained as often in A; that is, A: B:: C: D.

Therefore, &c.

Cor. 1. If A have to B the same ratio as C to D, and C to D a greater or a less ratio than E to F, A shall have to B a greater or a less ratio than E to F. For, from what we have seen ([9]) it follows, that whatever part of D it be that is contained in C a greater or less number of times than the like part of F is contained in E, the like part of B must be contained in A the same greater or less number of times.

Cor. 2. And in the same manner it may be shown, that if A have to B a greater or a less ratio than C to D, and C to D the same ratio as E to F, A shall have to B a greater or a less ratio

than E has to F.

PROP. [13].

If four magnitudes be proportionals, and if the first be any multiple or part of the second, the third shall be the same multiple or part of the fourth: and conversely, if one magnitude be the same multiple, or part, of another, that a third magnitude is of a fourth, the four magnitudes shall be proportionals.

For, if A, B, C, D be proportionals, and if the second, or any part of the second, be contained a certain number of times in the first, the fourth, or a like part of the fourth, will be contained the same number of times in the third ([9]). Therefore, if the second be contained in the first seven times, the fourth will also be contained in the third seven times or again, if a seventh part of the second be contained in the first once, a seventh part of the fourth will also be contained in the third once.

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Again, if A contain B seven times, and C also contain D seven times, A, B and C, D have a common ratio 7 1; and, in like manner, if A be contained in B seven times, and C in D also seven times, A, B, and C, D, have a common ratio 17. Therefore, in either case, A,B, C, D are proportionals. Therefore, &c.

PROP. [14].

If four magnitudes A, B, C, D, be proportionals, and if the first be greater than the second, the third shall be greater than the fourth, if equal, equal, and if less, less.

For if A be greater than B, any common measure M of A and B will be con

tained a greater number of times in A than it is in B, and therefore ([9]) the like measure, N of D, will be contained in C a greater number of times than it is in D, that is, C will be greater than D. And in like manner it may be shown, that if A be equal to B, C will be equal to D, and if less, less. Therefore, &c.

PROP. [15].

If four magnitudes A, B, C, D, be proportionals, they shall also be proportionals when taken inversely; that is, invertendo* B: A::D: C.

For if 7 5 be the common ratio of A to B and of C to D, there will be common measures M, N, the first of A and B, contained in B and D respectively five and the other of C and D, which are times, and in A and C respectively seven times; and therefore 5: 7 will be a common ratio of B to A and of D to C.

This may be stated as follows— "the reciprocals of equal ratios are equal to one another." Therefore, &c.

PROP. [16].

If four magnitudes, A, B, C, D be proportionals, and if there be taken any equimultiples of the first and third, and also any equimultiples of the second and fourth; these equimultiples shall likewise be proportionals.

Let A, 4 C be any equimultiples of A, C, and 6 B, 6 D, any equimultiples of B, D: and let 7:5 be the common ratio of A to B, and of C to D, as in the last proposition. Then, if the measures M and N by which it is determined be taken, because M and N are contained in A and C respectively 7 times, they are contained in 4 A and 4 C respectively 4×7 times; and in like manner it may be shown that they are contained in 6 B and 6 D respectively 6 × 5 times. Therefore 4×7 6x5 is at once the ratio of 4 A to 6 B, and of 4 C to 6 D. Therefore, &c.

*The Latin words "invertendo," "alternando," "dividendo," "convertendo," "componendo," "ex æquali in proportione directâ," "miscendo," and "ex æquali in proportione perturbatâ," ([15], [19], [20], [20] Cor. 1. [21], [24], [24], Cor. 2. and [26]) carry with them, particularly to such as are strangers to the language, an air of mystery we should rather have dispensed with. They are in such constant use, however, that we cannot well do without them. They mean no more than " by inverting," "by alternating," "by separating,"" by exchanging," " by comportion," "by mixing," and "by reason of equal inbining," ," "by reason of equal intervals in direct pro

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tervals in cross proportion:" and they serve as so many titles to their respective theorems, which are those most frequently cited out of the whole theory of proportion.

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If four magnitudes of the same kind be proportionals, and if the first be greater than the third, the second shall be greater than the fourth; if equal, equal; and if less, less.

Let A, B, C, D be proportionals, and

let M and N be the common measures

of A and B and of C and D, by which their common ratio is determined; that is, let M and N be like parts of B and D, which are contained the same number of times in A and C respectively, (def. [7]). Then, because M and N are like parts of A and C, it is evident that if A be greater than C, M must be greater than N; and therefore also B, which is a multiple of M, greater than D, which is the same multiple of N (ax. 3.).

In the same manner it may be shown, that if A be equal to C, B will be equal to D; and if less, less.

Therefore, &c.

Cor. Hence also, if four magnitudes of the same kind be proportionals, and if the second be greater than the fourth, the first will be greater than the third; if equal, equal; and if less, less.

PROP. [19].

If four magnitudes A, B, C, D of the same kind be proportionals, they shall also be proportionals when taken alternately; that is, alternando A: C ::B:D.

For, if M and N be the common measures of A and B and of C and D, by which their common ratio is determined; then, because A and C are equimultiples of M and N ([17]), A: C::M : N‚— and, for the like reason, B: D::M: N. Therefore ([12].) A: C::B: D. Therefore, &c.

PROP. [20].

If four magnitudes A, B, C, D be proportionals, they shall also be proportionals when taken dividedly; that is, the difference of the first and second shall be to the second as the difference of the third and fourth to the fourth;or dividendo, A-B: B::C-D: D.*

For, M and N being taken as in the preceding propositions, if they be contained in A and C 7 times, and in B and D 4 times, they will be contained in A~B and CD, 7~4 or 3 times; and therefore 34 will be the ratio of A~ B to B, and also of CD to D. Therefore A B:B::C~D: D.

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And the same may be said, when A and C are less than B and D respectively.

Therefore, &c.

Cor. 1. If four magnitudes A, B, C, D be proportionals, they shall also be proportionals by conversion; that is, the first shall be to the difference of the first and second, as the third to the difference of the third and fourth; or

convertendo A: A~B::C: C~D.

For invertendo, B: A::D: C, dividendo, A-B: A::C~D: C, and invertendo A: A-B::C; C~D.

Cor. 2. If four magnitudes A, B, C, D of the same kind be proportionals, the greatest and least of them together shall be greater than the other two together.

The sign placed between two letters denotes the difference of the magnitudes which are represented by them, without supposing the first to be the greater, as is the case when we write A--B by itself.

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