INTRODUCTION.

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ON PROPORTION.

THE doctrine of Proportion belongs properly to Arithmetic, and ought to be explained in works which treat of that science. Its object being to point out the relations which subsist among magnitudes in general, when viewed as measured, or represented by numbers, the counexion it has with Geometry is not more immediate than with many other branches of knowledge, except indeed as Geometry affords the largest class of magnitudes capable of being so measured or represented, and thus offers the widest field for reducing it to practice. Owing, however, to our general and long-continued employment of Euclid's Elements, the fifth Book of which is devoted to Proportion, our common systems of Arithmetic, and even of Algebra, pass over the subject in silence, or allude to it so slightly as to afford no adequate information. For the sake of the British student, therefore, it will be requisite to prefix a brief outline of the fundamental truths connected with this department of Mathematics; at least, in so far as a knowledge of them is essential for understanding the work which follows.

The proper mode of treating Proportion has given rise to much controversy among mathematicians; chiefly originating from the difficulties which occur in the application of its theorems to that class of magnitudes denominated incommensurable, or having no common measure, Euclid evades this obstacle; but his method is cumbrous, and, to a learner, difficult of comprehension. All other methods have the disadvantage of frequently employing the principle of reductio ad absurdum, a species of reasoning, which, though perfectly conclusive, the mathematician wishes to employ as seldom as possible. The opposite advantages, however, have generally overcome this reluctance; and Euclid's method is now almost entirely abandoned in elementary treatises. On this matter, we are happily delivered from the necessity of making any selection; the author having himself provided for the application of proportion to incommensurable quantities, and demonstrated every case of this kind as it occurred, by means

of the reductio ad absurdum. He has also in various parts of these Elements interspersed explanations of the sense in which geometrical magnitudes may be viewed, as coming under the dominion of numbers, and bearing a proportion to each other. So that our duty, on the present occasion, is reduced to little more than defining a few terms, and, in the briefest manner, exhibiting the leading truths of the subject, when referred to mere numbers, or to magnitudes capable of being completely represented by numbers.

DEFINITIONS.

I. One magnitude is a multiple of another, when the former contains the latter an exact number of times: thus, 6 is a multiple of 2, 10 of 5, &c. Like, or equimultiples, are such as contain the magnitudes they refer to, the same number of times thus, 8 and 10 are like multiples of 4 and 5.

One magnitude is a submultiple, or measure of another, when the former is contained by the latter an exact number of times thus, 2 and 3 are submultiples of 6. Like submultiples, or equal submultiples, are such as are contained by the magnitudes they refer to, the same number of times thus, 5 and 4 are like submultiples of 10 and 8.

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II. Four magnitudes are proportional, if when the first and second are multiplied by two such numbers as make the products equal, the third and fourth being respectively multiplied by the same numbers, likewise make equal products.

Thus, 6, 15, 8, 20, are proportional; because, multiplying the first and second by 5 and 2 respectively, so as to make 6x5=15×2, we have likewise 8×5=20×2; and generally, the magnitudes A, B, C, D, are proportional, if m and n being any two numbers such that n A=m B, we have likewise n C=m D. The magnitude A is said to be to B, as C is to D; the four together are named a proportion or analogy, and are written thus, A: B:: C: D.

Note. To make this definition complete, two things are required: first, that such a pair of numbers n, and m, be always discoverable as shall make n A=m B; secondly, we must prove that when one such pair of numbers m and n is discovered, and found likewise to make n C=m D, every other pair of numbers p and q, making p A=q B, will also make p C=q D.

First. With a view to the former of these conditions, we must find a common measure of A and B ; the mode of doing which is explained at large in Article 157, Book II. of these

Elements. Suppose this common measure to be E, and that A=m E, B=n E: the numbers required will be n and m. For, by the supposition, we have n An.m E=n m E, and m B=m.n E=nm F; hence n A= =m B.*

Secondly. Suppose n A=m B, aud n C=m D; we are to

* It is obvious, however, that when the magnitudes A and B are incommensurable, or have no common measure, this method will not serve. If, for example, the first term A were the side of a square, B the second term being its diagonal, and the third term CA + B the sum or the difference of the former two, there could exist no common measure between any of the terms, and no such pair of numbers n and m could be found as would make n A = = m B. Hence our Definition, not being applicable to the magnitudes in question, could, strictly speaking, form no criterion for distinguishing their proportionality, or any other property possessed by them. Nevertheless it is certain that, if C were made the side of a new square, and the diagonal were named D, the two lines C and D would stand related to each other in regard to their length, exactly as the lines A and B stand related to each other in regard to theirs and though a line, measuring any one of the four, must of necessity be incapable of measuring any of the remaining three; though when expressed by numbers, each of them, except one, must form an infinite series; yet these four lines are undoubtedly proportional, as truly as if they admitted any given number of common measures: and consequently they, and all other magnitudes, exhibiting similar properties, ought to be included, directly or indirectly, in every definition of proportion.

It is likewise certain, that if the Definition given above could be applied to such magnitudes, it would correctly indicate their proportionality: that if in the example just alluded to, a pair of numbers n and m could be found giving n Am B, they would also give n C = m D. We have now, therefore, to inquire in what manner our Definition can be brought to bear on this class of magnitudes, with regard to which, it appears to be, as it were, potentially true, though never actually applicable.

If B is divided by any measure of A, it will leave a certain remainder less than that measure: if B is then divided by a half, a third, a ninth, a sixteenth, or any submultiple of that measure, the remainder will evidently in each case be less than that submultiple; and as the submultiple, which of course will still measure A, may be made as little as we please, the remainder may also be made as little as we please; and thus a magnitude be found which shall correctly measure A and B — B', B' being less than any assigned magnitude. Suppose E were a measure of A, such that A = m E, B — B'n E; we shall have n A= m (B — B'); and if at the same time we have n C = m (D— D′), the magnitudes A, B—B', C, D-D', are proportional by the Definition. Now, if it is granted that, as by using a sufficiently small submultiple of E we can diminish the remainder B', so also we can diminish the remainder D', and at length reduce them both below any assigned magnitude; then it is evident that BB', D-D', may approximate to B and D, as near as we please; and since the proportion still continues accurate at every successive approximation, we infer that it will, in like manner, continue accurate at the limit which we can approach indefinitely, though never actually reach. In this sense our Definition includes incommensurable as well as commensurable quantities; and whatever is found to be true of proportions among the latter, may also, by the method of reductio ad absurdum, be shewn to hold good when applied to the latter.

prove that if Ρ and 9 are any other two numbers which give p A=q B, they will likewise give p C=q D.

For, by hypothesis, we have

SnA=mB) {q B=p A } ; and multiplying* together the terms which stand above each other, we obtain

nq A B=mp AB; hence nq=mp; hence ng C=mp C. But by hypothesis we have

n C=m D; hence ng C=mq D.

Now we have already shewn

nq C=mp C; hence mp C=mq D; hence p C=q D.

III. The ratio or relation which is perceived to exist between two magnitudes of the same kind, when considered as mere magnitudes, appears to be a simple idea, and therefore unsusceptible of any good definition. It may he illustrated by observing, that when we have A: B:: CD, the ratio of A to B is said to be the same as that of C to D. In Arithmetic, the ratio of two numbers is usually represented by their quotient, or by the fraction which results from making the first of them numerator, and the second denominator. It is in this sense that one ratio is said to be equal to, less, or greater, than another ratio.

IV. The first and third terms of a proportion are called the antecedents; the second and fourth, the consequents. The first and fourth are likewise called the extreme terms, or the extremes; the second and third, mean terms, or means. When both the means are the same, either of them is called a mean proportional between the two extremes; and if in a series of proportional magnitudes each consequent is the same as the next antecedent, those magnitudes are said to be in continued proportion.

Thus, if we have A: B:: C: D, A and C are antecedents, B and D are consequents; A and D are extremes, B and C are means. If we have A:B::B: C :: C : D :: D : E, B is a mean proportional between A and C, C between B and D, D between C and E; and the magnitudes A, B, C,

* Unless one of the magnitudes A and B is a number, they evidently cannot, in a literal sense, be multiplied together. The expression product of A and B must therefore, in all such cases, be regarded as elliptical, or employed merely for the sake of brevity.

D, E, are said to be in continued proportion, or sometimes, in geometrical progression.

THEOREM I.

If four magnitudes are proportional, the product of the extremes will be equal to that of the means; and conversely, if two products are equal, any two factors composing the first will form the extremes of a proportion, in which any two factors composing the second, form the means.

First. Suppose A: B::C: D; then is AD=BC.

Find a common measure of A and B, if they have one: suppose it to be E; and that A=m E, B=n E. Then we shall have n A=m B; and therefore (Def. 2.) n C=m D. Equal quantities multiplied by equal quantities yield equal products; hence

n AxmD=m Bxn C, or

nm AD=nm BC, or
AD=BC.*

Secondly. If we have AD=BC; then we are to prove that A:B::C: D.

Find the common measure of A and B ; and suppose we have n Am B.

Now we have AD=BC,

Also

mB=nA

}

; hence by multiplying,

m DAB=n CAB, hence by dividing
m D=n C; and therefore (Def. 2.)
A: B::C: D.

* If A and B are incommensurable, still we shall have AD BC. For, if not, one of them must be greater: suppose AD less, and that we have AD=BC-CF. By the method explained in Def. 2, find a measure of A which shall be less than F; and suppose n A=m (B-B'), B' being of course less than the measure, and therefore still less than F. By the same Definition, we shall have n C=m (D—D′), D′ being a positive quantity. Hence we have

n A×m(D—D')=n CXm (B—B'); that is

nm AD-nm AD'= n m CB-nm CB'; or dividing,

AD
AD

AD' CB-CB'; but by the supposition, we had
CB-CF.

Hence AD-AD' being less than AD, CB-CB' must also be less than CB-CF; hence CB' is greater than CF, and B' is greater than F. On the contrary, however, it is less than F by hypothesis: hence that hypothesis was false; hence AD is not less than BC. We could show, in the same manner, that it is not greater: hence it is equal to BC.