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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 see verally the extreme and the mean terms.
term of this analogy being equal to the fourth, therefore (W. 4.) AD= BC.
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.
For the extreme and mean terms are thus only mutual
ly 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.
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.
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.
The terms of an analogy are proportional by miring ; or the sum of the first and second is to the difference, as the sum of the third and fourth to their difference.
PROP. XIII. THEOR.
A proportion will subsist, if the homologous terms be multiplied by the same numbers.
Let A : B : : C : D ; then pa : q B : ; p( : qL).
For, since A : B : : C ; D, alternately A : C : : B : D; but the ratio of A to C is the same as pa : po (V. 3.), and the ratio of B to D is the same as qB : qI). 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 (W. 4.) A–B is greater than C–D ; to each add B+D, and A+D is greater than B+C.