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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 com

pared.

246. DEF. The Antecedent of a ratio is its first tern. 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:

1. a : 6 ::c:d
2. a : b = c :d

3.

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.

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

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

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

252. DEF. In the proportion a : b:: 6:0; c is a Third Proportional to a and b.

253. DEF. In the proportion a :b::b:c; b is a Mean Proportional between a and c.

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.

1 1 1 Thus

a:6::-:

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If we have two quantities a and b, and the reciprocals of these quantities and ; these four quantities form a reciprocal proportion, the first being to the second as the reciprocal of the second is to the reciprocal of the first.

a : 6 :: 1.1

As

b a 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 :: 0 :d, we have either

a:c::6:d, or, d:6:: : a. 256. DEF. A proportion is taken by Inversion, when the means and extremes are made to exchange places. Thus in the proportion

a : 6 :: 0 :d, by inversion we have

b: a ::d : 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. Thus if

a : 6 :: 0 :d,

or,

we have by composition,

a + b : 6 ::c+d : d,

a+b:a ::c+d : c. 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 proportion the product of the extremes is equal to the product of the means.

Let a : 6::c:d.

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In the treatment of proportion, it is assumed that fractions may be found which will represent the ratios. It is evident that a ratio may be represented by a fraction when the two quantities compared can be expressed in integers in terms of any common unit. Thus the ratio of a line 24 inches long to a line 34 inches long may be represented by the fraction 38 when both lines are expressed in terms of a unit II of an inch long.

But it often happens that no unit exists in terms of which both the quantities can be expressed in integers. In such cases, however, it is possible to find a fraction that will represent the ratio to any required degree of accuracy.

Thus, if a and b denote two incommensurable lines, and 6 be divided into any integral number (n) of equal parts, if one of these parts be contained in a more than m times, but less than m+1 times, then > but <m+1; so that the error in taking either of these values for is < Since n can be increased at pleasure, - can be made less than any assigned

6

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n

n

value whatever. Propositions, therefore, that are true for ", and m + 1

e , however little these fractions differ from each other, are

n

true for ; and " may be taken to represent the value of

PROPOSITION II.

260. A mean proportional between two quantities is equal to the square root of their product. In the proportion a :b :: 6 : C, 12 = ac,

§ 259 (the product of the extremes is equal to the product of the means). Whence, extracting the square root, b=Vac.

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.

Then

or,

a : 6 ::6:d.

Q. E, D.

PROPOSITION IV.

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

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