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For, by the supposition, we have n A= n. m E = n m E, and mB = m. n Enm E; hence n A = m B.*
Secondly, Suppose n A =m B, and n CmD; we are to prove that if P and q are any other two numbers which give pA=q B, they will likewise give p C = q D.
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 C = A + 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 Am 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 B. 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 B-B', 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.
For, by hypothesis, we have
nA = mB
qBpA; and multiply
ing together the terms which stand above each other, we obtain
nq AB=mp AB; hence nqmp; hence nqCmp C. But by hypothesis we have
n Cm D; hence nq Cmq D.
Now we have already shewn
nq Cmp C; hence mp Cmq D, hence p C = qD.
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 be illustrated by observing, that when we have A: B:: C: D, 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 or 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, Bis a mean proportional between A and C, C between B and D, D between C and E; and the magnitudes A, B, C, D, E are said to be in continued proportion, or sometimes, in geometrical progres
* 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. What is understood by it, the Author has explained at pp. 48. 49.
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;
nAxmD=m Bxn C, or
nm AD=nm BC, or
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.
* If A and B are incommensurable, still we shall have AD-BC. For, if not, one of them must be greater: suppose AD greater, 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 Ã=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
)—n m AD'=n m CB―n m CB'; or dividing, 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 shew in the same manner that it is not greater: hence it is equal to BC.
If B and C are incommensurable, the reasoning is exactly analogous to that employed in the note to the foregoing case, and need not be repeated here.
Cor. 1. Thus we have obtained a new test of proportionality; and henceforth, whenever we find four factors capable of forming two equal products, we are at liberty to constitute an analogy of these factors, making those of the one product means, those of the other extremes. For this reason, if we have
A: B:: C:D,
then also we shall have A: C:: B: D, which is termed alternando, B: A:: D: C, which is termed invertendo: because in both cases the product of the extremes is still equal to that of the means.
Cor. 2. Hence also supposing A: B: C: D, we shall have A:B:.pC:pD, p being any number whole or fractional; because if we have AD=BC, then also p AD=p BC whatever be the value of p. Hence a ratio is not affected by multiplying or dividing its terms by the same number.
Cor. 3. If we have A: B::C:D, and A:E::F:D; then from the first of these AD-BC, from the second AD=EF; hence BC EF, hence E: B: C:: F; which inference is said to be drawn ex equali perturbate, in allusion to the position of the terms.
Cor. 4. Also if we have A: B:: C: D, and B: E::D: F; then from the first of these (Cor. 1.) we have B: D::A: C, and from the second B: D:: E: F; hence A: C::E: F; which is said to be ex equali directe, for a similar reason.
Cor. 5. If we have A: B:: B:C; then B2= AC, and BAC; hence a mean proportional is equal to the square-root of the product formed by multiplying the two extremes.
Scholium. From this Proposition is derived the mode of operating in the common arithmetical Rule of Three, where three terms of a proportion being given, it is required to find the fourth. We have A: B:: C: x; hence A =BC, hence x=
which is the rule adverted to. The right arrangement of
Α the three given terms, or the stating of the question, as it is called, does not properly form an arithmetical problem: it depends on a knowledge of the objects treated of by the question; which objects may be geometrical, mechanical, commercial, or of any conceivable kind.
The ratio of two magitudes is not affected, when they are respectively increased or diminished by any pair or pairs of magnitudes having the same ratio.
Thus having A: B::C: D::E: F, we shall likewise have A:B::A±C÷E:B±D±F.
For by the last Theorem we have
Adding or subtracting which, we have
ABAD + AF BA± BC± BE
or A (R± D±F)=B (A± C±E).
Hence by the last Theorem
A: B:: A+ C+E:B+D±F.
And the same may be shewn of any number of magnitudes having the same ratio.
Cor. 1. If we have A: B:: C: D, then alternando we shall have A: C:: B: D, and by the Proposition A: C:: A+B: C+D; hence, alternando once more,
A: A+B:: C:C+D;
which inference is said to be drawn convertendo.. Sometimes also it is written
the reasons for which are exactly similar.
Cor. 2. By the very same process we deduce
A: A-B:: C: C-D,
or A-B: B:: C-D: D;
which is said to be dividendo.
Cor. 3. And combining these two Corollaries with Cor. 4. of the last Theorem, we have
A+B: A-B:: C+D: C-D;
which is said to be miscendo.
The products of the corresponding terms of two analogies are proportional.
Suppose we have
(A: B::C: D, and
AE: BF:: CG: DH.
}; then we shall like