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To express that the ratio of A to B is equal to the ratio of C to D, we write the quantities thus,

A : B :: C : D.

and read, A is to B, as C to D.

The quantities which are compared together are called the terms of the proportion. The first and last terms are called the two extremes, and the second and third terms the two means.

170. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order.

171. Three quantities are in proportion when the first has the same ratio to the second that the second has to the third; and then the middle term is said to be a mean proportional between the other two.

172. Quantities are said to be in proportion by inversion, or mversely, when the consequents are made the antecedents and the antecedents the consequents.

173. Quantities are said to be in proportion by alternation, or alternately, when antecedent is compared with antecedent and consequent with consequent.

174. Quantities are said to be in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent.

175. Quantities are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent.

176. Equi-multiples of two or more quantities are the products which arise from multiplying the quantities by the same numbeT. Thus, m x A and m x B are equi-multiples of A and B, the common multiplier being m.

177. Two quantities, A and B, are said to be reciprocally proportional, or inversely proportional, when one increases in the same ratio as the other diminishes. When this relation exists, either of them is equal to a constant quantity divided by the other.

178. If we have the proportion

A : B :: C : D,

we have (Art. 169);

A L,

and by clearing the equation of fractions, we have

BC=AD;

that is, of four proportional quantities, the product of the two extremes is equal to the product of the two means.

179. If four quantities, A, B, C, and D, are so related to each other that

AxD=BxC,

we shall also have ?-=^,

A O

and hence, A : B :: C : D;

that is, if the product of two quantities is equal to the product of two other quantities, two of them may be made the extremes, and the other two the means of a proportion.

180. If we have three proportional quantities

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hence, IP^AC;

that is, the square of the middle term is equal to the product of the two extremes.

181. If we have

B I)

A : B : C : D and consequently, ———,

A O

Q

multiply both numbers of the last equation by we obtain

B

C D

A~b:

and hence, A : C : : B : D;

that is, if four quantities are proportional, they will be in proportion by alternation.

182. If we have

A : B :: C : D and A : B : : E : F, we shall also have

B D , B F

A=C and T=£;

hence, ^=4- and C : D : : E : F."

That is, if there are two sets of proportions having an antecedent and consequent in the one equal to an antecedent and consequent of the other, the remaining terms will be proportional.

183. If we have

B D

A : B :: C : D and consequently

A O

we have, by dividing 1 by each member of the equation, A C

~jj=~jj, al)d consequently B : A : : D : C.

That is, Four proportional quantities will be in proportion, when taken inversely (Art. 172).

184. The proportion

A : B :: C : D gives AxD=BxC.

To each member of the last equation add BxD. We shall then have

(A+B)xDz=(C+D)xB;

and by separating the factors, we obtain

A+B : B :: C+D : D.

If, instead of adding, we subtract BxD from both memhers, we have

(A-B)xD=(C-D)xB;

which gives A—B : B :: C—D : D.

That is, If four quantities are proportional, they will be in propor^ tion by composition or division.

185. If we have

B D

A = C'

and multiply the numerator and denominator of the first member by any number m, we obtain

mB D

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that is, Equal multiples of two quantities have the same ratio as the quantities themselves.

186. The proportions

A : B :: C : D and A : B : : E : F, give AxD-BxC and AxF=BxE;

adding and subtracting these equations, we obtain

A{D±F)=B(C±E), or A : B : C±E : D±F.

That is, If C end D, the antecedent and consequent be augmented or diminished by quantities E and F, which have the same ratio as C to D, the resulting quantities will also have the same ratio.

187. If we have several proportions

A : B :: C : D, which gives AxD=BxC,

A : B : : E : F, AxF=Bj<E,

A : B :: G : H, AxH=BxG,
&c, &c,

we shall have by addition

A(D+F+H)=B(C+E+G); and by separating the factors

A : B : C+E+G : D+F+H

That is, In any number of proportions having the same ratio, any antecedent will be to its consequent, as the sum of the antecedents to the sum of the consequents.

188. If we have four proportional quantities

B D

A : B :: C : D, we have —=-^;

A O

and raising both members to any power, as n, we have

B" -- D"

~af=~ot'

Mid consequently A" : B" :: C" : D*.

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