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PROP. VI. THEOR.

If four numbers be proportional, the product of the extremes is equal to that of the means; and of two equal products, the factors are convertible into an analogy, of which these form severally the extreme and the mean terms.

Let A B C: D; then AD=BC.

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For (V. 3.) A.D: B.D:: B.C: B.D; and the second term of this analogy being equal to the fourth, therefore (V. 4.) AD=BC.

Again, let AD=BC; then A : B :: C: D.

For, by identity of ratios, AD: BD:: BC: BD, and hence (V. 3.) A: B:: C: D.

Cor. 1. Hence the greatest and least terms of a proportion, are either extremes or means.

Cor. 2. Hence also a proportion is not affected, by transposing or interchanging its extreme and mean terms.--- On this principle are founded the two following theorems.

PROP. VII. THEOR.

The terms of an analogy are proportional by. inversion, or the second is to the first, as the fourth to the third.

Let A B C D; then inversely B: A:: D: C.

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For the extreme and mean terms are thus only mutually interchanged, and consequently the same equality of products AD and BC still obtains.

PROP. VIII. THEOR.

Numbers are proportional by alternation, or the first is to the third, as the second to the fourth.

Let A: B:: C: D; then alternately A: C:: B: D. For the extreme terms being still retained, the mean terms are merely transposed with respect to each other; the same equality of products, therefore, also here subsists.

PROP. IX. THEOR.

The terms of an analogy are proportional by composition; or the sum of the first and second is to the second, as the sum of the third and fourth to the fourth.

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Let A B C: D; then by composition A+B: B:: C+D: D.

Because A : B :: C: D, the product AD=BC (V. 6.) ; add to each of these the product BD, and AD+BD= BC+BD. But (V. 1.) AD+BD=D(A+B), and BC+BD=B(C+D); wherefore (V. 6.) assuming the factors of these equal products for the extreme and mean terms, A+B: B:: C+D: D.

PROP. X. THEOR.

The terms of an analogy are proportional by division; or the difference of the first and second is to the second, as the difference of the third and fourth to the fourth.

Let A B C : D; suppose A to be greater than B, then will C be greater than D (V. 4.): It is to be proved that A-B: B:: C-D: D.

For, since A B :: C: D, the product AD=BC (V. 6.), and, taking BD from both, the compound product AD—BD is equal to BC—BD; wherefore, by resolution, (A-B)D=B(C-D), and consequently A-B: B:: C-D: D.

If B be greater than A, then BD-AD=BD-BC, and, by resolution, (B-A) D=B (D-C); whence B-A : B :: D-C: D.

PROP. XI. THEOR.

The terms of an analogy are proportional by conversion; that is, the first is to the sum or difference of the first and second, as the third to the sum or difference of the third and fourth.

Let A: B::C:D, and suppose AB; then A:AB ::C:C=D.

For, since (V. 6.) the product AD=BC, add or sub

stract these to or from the product AC; and ACAD =AC BC. Wherefore, by resolution, A(C=D)= C(A+B), and consequently A: A±B:: C: C÷D.

If AB, then AD-AC-BC-AC, and, by resolution, A(D-C)=C(B-A), whence A: B-A :: C: D-C. Cor. Hence, by inversion, AB : A :: C±D: C, or B-A: A:: DLC: C.

PROP. XII. THEOR.

The terms of an analogy are proportional by mixing; or the sum of the first and second is to the difference, as the sum of the third and fourth to their difference.

Let A: B::C: D, and suppose AB; then A+B: A-B:: C+D: C-D.

For, by conversion, A: A+B:: C: C+D, and alternately AC: A+B: C+D.

Again, by conversion, A: A-B :: C: C-D, and alternately A: C:: A—B : C—D. Whence, by identity of ratios, A+B : C+D :: A−B: C—D, and alternately A+B: A-B::C+D: C-D.

The same reasoning will hold if A be less than B, the order of these terms being only changed.

PROP. XIII. THEOR.

A proportion will subsist, if the homologous terms be multiplied by the same numbers.

Let A B C D; then pA: qB:: pC : qD.

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For, since A: B:: C: D, alternately A: C:: B : D; but the ratio of A to C is the same as pA: pC (V. 3.), and the ratio of B to D is the same as qB: qD. Wherefore pA: pC :: qB: qD, and, by alternation, pA: qB : PC: qD.

Cor. The Proposition may be extended likewise to the division of homologous terms, by employing submultiples.

PROP. XIV. THEOR.

The greatest and least terms of a proportion, are together greater than the intermediate ones.

Let A: B:: C: D; and A being supposed to be the greatest term, the other extreme D is the least (V. 6. cor. 1.): The sum of A and D is greater than the sum of B and C.

Because A: B:: C: D, by conversion A: A-B :: C: C-D, and alternately A: C ; : A—B : C—D; but A, being the greatest term, is therefore greater than C, and consequently (V. 4.) A-B is greater than C-D; to each add B+D, and A+D is greater than B+C.