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and b, supposing h contained in a, m times, and in b, n times; then man therefore ; =m; but is the nth part of b; therefore the nth part of b is con

tained in a, m times.

Again, let c=mi, and we shall have made man; wherefore, by this equation, d=ni; therefore å; and, consequently, á=m: wherefore, also, c contains the nth part of d, m times.

115. COROLLARY.—Hence, if two fractions are equal, the numerator of the one will be to its denominator, as the numerator of the other is to its denominator.

To indicate that four quantities, a, b,c,d, are proportionals, the sign : is placed between the first two and between the last two; and the sign :: is placed between the two terms that stand in the middle; thus, a : 6 :: 0 :d, is read, as a is to b, so is c to d,

116. Four proportionals are termed a proportion.

Of four proportional quantities, the last term is called the fourth proportional to the other three,

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

SCHOLIUM.

117. Since, from ad = be, we may choose four different ways of taking out the first term, in order to make a proportion; and since there are two ways of making choice of the second term after the first, there are eight ways in which the positions of the terms will be different: these are-

a : 0 :: 0 :d a :c :: b : d b: a :: d:c b :d :: a :c C: a :: d: 6 c:d :: a : 6 d:b :: C: a d:c :: b: a

118. So that, in every two sets of proportionals, where the same quantity stands first, the terms of the one set are placed alternately to those of the other; and we may also observe that, four of these sets of proportionals have their terms inverted with regard to the other four, and are therefore the same when read in contrary order.

THEOREM 41.

119. If the corresponding terms of any number of proportions are multiplied together, the products, taken in the same order, will be proportionals.

Thus, as a : 6::C:d

e:f::g:h

i :k :: 1 :m
then, as aei : bfk :: cgl : dhm

For ad=bc

eh=fg

imkl

therefore, adehim=befgkl. Consequently, aei : bfk :: cgl: dhm; therefore the proposition is manifest.

120. COROLLARY.—Hence, if the proportions are the same as the first, we shall have a : 63 :: C3: .

121. The proportion which is formed by the multiplication of the corresponding terms of two or more proportions is said to be compounded of these proportions.

When two of the proportions to be compounded are the same, the proportion compounded of the two is said to be the duplicate of either : thus, a:b?::co: d’ is in duplicate proportion to that of a : b::c:d.

When three proportionals, which are the same, are compounded, the compound is said to be triplicate to either of the simple ones: thus, a': 63 :: c': do, is the triplicate proportion of a : b::c:d.

THEOREM 42.

122. If four quantities, a, b, c, d, be proportionals, the first will be to the sum of the first and second as the third is to the sum of the third and fourth. For, since a, b, c, d, are proportionals, ad=bc. To each side of this

equation add the product ac, and we have ad+ac=bc+ac; that is, a(c+d)=c(a+b).

Therefore, a : a+b :: C: c+d,

123. COROLLARY.—Since out of the two equal products ad, bc, four combinations in twos may be chosen, viz.

ab, ac, bd, cd, we may therefore have the four following equations all different, by adding each combination to each side of the equation ad=bc, viz.

No. 1....a(c+d)=c(a+b)

2....a(b+d) = bla+c)
3....d(a+b) = b[c+d)
4....d(a+c) = c(b+d)

Since each of these equations will give eight sets of proportionals; therefore the whole four will give thirty-two.

THEOREM 43.

124. If four quantities, a, b, c, d, be proportionals, as the first is to the difference of the first and second, so is the third to the difference of the third and fourth.

For, since a, b, c, d, are proportionals, ad=bc. Subtract each side of this equation from the product ac, and we have ac- ad = ac— bc; that is, a(c-d)=c(a - b), whence a: a-6::c:c-d.

125. COROLLARY.—Since out of the two equal products ad, bc, we may choose the four combinations in twos, viz. ab, ac, bd, cd; and may therefore have the four following equations, by subtracting each side of the original equation, ad=bc, from each combination.

No. 1....a(c-d)=c(a-6)

2....a(b-d)=b(a-c)
3....d(b-- a) =bd-c)

4....d(c-a)=c(d-b)
126. Again, by subtracting each of the products, ab, ac, bd, cd, from each
side of the original equation, ad=bc, we have

No. 1....a(d-c)=c(b-a)

2....a(d5)=b(c-a)
3....d(a-6)=b(c-d)

4....da-c)=c(b-d)
Now, since every one of these eight different equations will give eight sets
of proportionals, in each set of which the situations of the terms will be
varied ; therefore the whole will give sixty-four sets of proportionals, in which
the terms will have different situations in every two sets.

THEOREM 44.

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127. If four quantities, a, b, c, d, be proportionals, it will be, as the sum of the first and second is to their difference, so is the sum of the third and fourth to their difference.

For, dividing the equation, No. 3, pr. 123, by the equation, No. 3, pr. 126, we shall have

atb ctd
a-b

c-d
And, by dividing the equation, No. 3, in 123, by the equation, No. 3, in
pr. 125, we have,

atb ctd

b-ad-c Wherefore a +b:ab::c+d:cnd* 128. Three quantities are said to be proportionals when the first is to the middle quantity as the middle quantity is to the third ; thus, let a, b, c, be three proportionals; then a :b::b:c.

* The character ~ between two quantities signifies the difference, as 579=4.

THEOREM 45. 129. If three quantities be proportionals, the product of the two extremes is equal to the square of the mean.

. Let a, b, c, be the three proportionals ; then, ac = b’, for, since = , multiply both sides of this equation by ab, and « * = abc, that is, bo=ac.

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130. If the first and second terms of two proportions are alike, the third and fourth terms of both, placed in the order of their antecedents and consequents, will be proportionals.

Let a : 6::c:d, and a :b::e:f; then will c:d::e:f. From the first proportion we have ad = bc, and from the second we have be = af ; therefore, multiplying the corresponding sides of these equations, we have adbe=bcaf; and, throwing out the common factors on each side of this equation, there will remain de=cf; wherefore, c:d::e : f.

THEOREM 47.

131. In any number of proportionals, of which all the ratios are equal, it will be, as the antecedent of any ratio is to its consequent, so is the sum of all the antecedents of the other ratios to the sum of all the consequents.

For, let = å, å=j, =, then will ; = = ; =%.

Therefore, ad = bc

af = be

ah =

bg
Whence ald+f+h) = b(c+e+g.)
Wherefore, a :b:: c+e+g:d+f+h, as was to be shown.

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