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BOOK III.

PROPORTIONAL LINES AND SIMILAR POLYGONS.

ON THE THEORY OF PROPORTION.

245. DEF. The Terms of a ratio are the quantities compared.

246. DEF. The Antecedent of a ratio is its first term. 247. DEF. The Consequent of a ratio is its second term. 248. DEF. A Proportion is an expression of equality between two equal ratios.

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

:

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Form 1 is read, a is to b as c is to d.

Form 2 is read, the ratio of a to b equals the ratio of c to d. Form 3 is read, a divided by b equals c divided by d. The Terms of a proportion are the four quantities compared.

The first and third terms in a proportion are the antecedents, the second and fourth terms are the consequents.

terms.

terms.

249. The Extremes in a proportion are the first and fourth

250. The Means in a proportion are the second and third

251. DEF. In the proportion a:b:: cd; d is a Fourth Proportional to a, b, and c.

252. DEF. In the proportion a: b

Proportional to a and b.

253. DEF. In the proportion ab Proportional between a and c.

bc; c is a Third

bc; b is a Mean

254. DEF. Four quantities are Reciprocally Proportional when the first is to the second as the reciprocal of the third is to the reciprocal of the fourth.

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If we have two quantities a and b, and the reciprocals of

these quantities

1

a

and; these four quantities form a recipro

cal proportion, the first being to the second as the reciprocal of the second is to the reciprocal of the first.

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255. DEF. A proportion is taken by Alternation, when the means, or the extremes, are made to exchange places.

Thus in the proportion

a b c d,

we have either

a:cb: d, or,

dbc: a.

256. DEF. A proportion is taken by Inversion, when the means and extremes are made to exchange places.

Thus in the proportion

abcd,

by inversion we have

bad: c.

257. DEF. A proportion is taken by Composition, when the sum of the first and second is to the second as the sum of

the third and fourth is to the fourth; or when the sum of the first and second is to the first as the sum of the third and fourth is to the third.

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258. DEF. A proportion is taken by Division, when the difference of the first and second is to the second as the difference of the third and fourth is to the fourth; or when the difference of the first and second is to the first as the difference of the third and fourth is to the third.

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

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It is to be observed that the product of two quantities implies that at least one of the quantities is an abstract number. We cannot multiply dollars by dollars, or lines by lines. Still we may speak of the product of two lines, if we understand by the expression the product of the numbers which represent the lines when they are measured by a common unit.

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Thus in the rectangle of the lines A B and A C, suppose AL to be a common measure of A B and A C, and to be contained in A B seven times, and in A C four times.

At the several points of division on AC draw lines perpendicular to A C,

and at the several points of division on AB draw lines perpendicular to A B.

Then the rectangle will be divided into squares, all equal to each other. There will be seven in a row, and four rows;

that is four times seven squares.

PROPOSITION II.

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

In the proportion a : b:: b : C,

b2 = a c,

(the product of the extremes is equal to the product of the means).

Whence, extracting the square root,

b = √ac.

$ 259

Q. E. D.

PROPOSITION III.

261. If the product of two quantities be 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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Divide both members of the given equation by bd.

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262. If four quantities of the same kind be in proportion, they will be in proportion by alternation.

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(the product of the extremes is equal to the product of the means).

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§ 259

Q. E. D.

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