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BOOK III.

PROPORTIONAL LINES AND SIMILAR
POLYGONS.

THE THEORY OF PROPORTION.

292. A proportion is an expression of equality between two equal ratios.

A proportion may be expressed in any one of the following forms:

α C

b=2; a:b=c:d; a:b::c:d;

and is read, "the ratio of a to b equals the ratio of c to d."

293. The terms of a proportion are the four quantities compared; the first and third terms are called the antecedents, the second and fourth terms, the consequents; the first and fourth terms are called the extremes, the second and third terms, the

means.

294. In the proportion a: b = c: d, d is a fourth proportional to a, b, and c.

In the proportion a: bbc, c is a third proportional to a and b.

In the proportion a:bb: c, b is a mean proportional between a and c.

PROPOSITION I.

295. In every proportion the product of the extremes is equal to the product of the means.

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296. A mean proportional between two quantities is equal to the square root of their product.

In the proportion a: b = b : c,

b2 = ac,

§ 295

(the product of the extremes is equal to the product of the means). Whence, extracting the square root,

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297. If the product of two quantities is equal to the product of two others, either two may be made the extremes of a proportion in which the other two are made the means.

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or,

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Divide both members of the given equation by bd.

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PROPOSITION IV.

298. If four quantities of the same kind are in proportion, they will be in proportion by alternation; that is, the first term will be to the third as the second to the fourth.

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299. If four quantities are in proportion, they will be in proportion by inversion; that is, the second term will be to the first as the fourth to the third.

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PROPOSITION VI.

300. If four quantities are in proportion, they will be in proportion by composition; that is, the sum of the first two terms will be to the second term as the sum of the last two terms to the fourth term.

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301. If four quantities are in proportion, they will be in proportion by division; that is, the difference of the first two terms will be to the second term as the difference of the last two terms to the third term.

Let a:bc: d.

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PROPOSITION VIII.

302. In any proportion the terms are in proportion by composition and division; that is, the sum of the first two terms is to their difference as the sum of the last two terms to their difference.

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303. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

Let a:bc:d=e:f=g: h.

To prove a+c+e+g:b+d+f+h=a: b.

Denote each ratio by r.

or,

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Whence, a=br, c=dr, e=fr, g=hr.

Add these equations.

Then a+c+e+g= (b+d+f+h)r.

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a+c+e+g:b+d+f+ha: b.

Q. E. D.

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