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tion A + B B :: C+D: D; and from (20) we have A - B : B:: C-D: D; hence, by (17), since the consequents of these proportions are equal, the antecedents will constitute a proportion, and we shall have A+B: A-B::C+D: C - D. In like manner, when B is greater than A, since for the equation we may by (12) put

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

B

A

=

C

Ꭰ '

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we shall get A+

Hence (mixedly, or uniting composition and division), we say that the sum of the first two terms of any proportion is to their difference as the sum of the last two terms is to their difference.

(22.) From the equation

C

A
B D'

=

or proportion A: B ::

C: D, when A is greater than B, we have the proportion A: A-B:: C: C-D; and when B is greater than A, we shall have B: B-A::D:D-C. Since this proposition has in fact been proved in (20), it will not be necessary to repeat the proof in this place.

Hence, we say (by conversion) that the greater of the first two terms of any proportion is to the difference of the first two terms as the greater of the last two terms is to the difference of the last two terms. And it is easy to see that the less of the first two terms is to their difference as the less of the last two terms is to their difference.

(23.) We shall now suppose that A, B, C, D are quantities

of one kind, and that we have the equation

A C

=

B D'

or the

proportion A: B:: C: D; then if A is greater than C, it is easy to see that B will be greater than D, and we shall have the proportion A-C: B-D:: A: B. For by alternating the terms of the given proportion by (13), we get A : C :: B: D, hence, by (20), we shall get AC: A :: B – D : B, and alternating this proportion by (13) we get A - C : BD: A: B, as required.

Hence, if a whole quantity is to a whole quantity as a quantity taken from the first quantity is to a quantity taken from the second quantity; then the first remainder is to the second remainder as the first whole quantity is to the second whole quantity.

(24.) Suppose that we have the proportion A: B:: C: D, such that its terms are all of one kind; then if A is the greatest term, it is easy to show that D is the least term, and we shall have A+D greater than B+ C. For, by (22), A: A-B:: C: C-D; consequently, since A is greater than C, we must have A - B greater than C - D; and adding B + D to these unequals, we have A + D greater than BC. Again, if we suppose that B is the greatest term of the given proportion, then inverting its terms by (12), we shall have B: A: D: C, and we shall have (as before) C for the least term, and B + C will be greater than A + D.

Hence, in any proportion whose terms are of one kind, the sum of the greatest and least terms is always greater than the sum of the other two.

(25.) If A, B, C are three quantities of one kind, and if

A = B, then we shall clearly have

Also, if we have the equation

that we must have A = B.

A B

=

T or by (12)
C

A B C C

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C C

A B

it is clear

Hence (since the equations used are virtually proportions), we say that equal quantities have the same ratio to the same quantity; and that the same quantity has the same ratio to equal quantities.

And, reciprocally, quantities which have the same ratio to the same quantity are equal to each other; and those quantities to which the same quantity has the same ratio, are equal to each other.

(26.) If A, B, C are three quantities of one kind, and if A is greater than B, it is clear that we shall have the ratio

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it is plain that A is greater than B; and if

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Hence, of unequal quantities, the greater has a greater ratio to the same quantity than the less has; and the same quantity has a greater ratio to the less quantity than it has to the greater.

Reciprocally, if one of two quantities has a greater ratio to another quantity than the remaining quantity has, then the remaining quantity is the less of the two; and if the same quantity has a greater ratio to one of two quantities than it has to the other, then the quantity to which it has the greater ratio is the less of the two.

G

H

(27.) We shall now suppose that we have

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= etc., where we shall suppose that A, B, C, D, etc., are (either numbers or) quantities of one sort, then we shall have A+C+E+ G + etc. A C E

=

=

= = etc.

B+D+F+ H+ etc. B Ꭰ F

For if m and n are any two positive integers, we shall get mA mc ME

from the supposed equations the equations

=

mG nH

=

=

=

nB nD nF

= etc., which show that when mA is greater than nB, we shall have mC greater than nD, mE greater than nF, and so on; also, when mA equals nB, we have mC equal to nD, mE = nF, and so on; and when mA is less than nB, mC is less than nD, mE is less than nF, and so on. Hence, when mA is greater, equal to, or less than nB, we shall have mA + mC+mE + etc. m(A + C + E + etc.), greater, equal to, or less than n(B + D + F + etc.); hence, by (14), A+ C+E+ G + etc. A C E we shall have = = etc. B+D+F+ I + etc. B D F Hence, when (numbers or) quantities of the same kind are proportionals, we say that the sum of all the antecedents is to the sum of all the consequents, as any antecedent is to its consequent.

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=

=

A

C

E F

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B

B

then by adding these

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Consequently, if we call A, B, C, D, etc., the first, second,

third, etc., quantities, successively, we may say that if the first has to the second the same ratio that the third has to the fourth, and the fifth to the second the same ratio that the sixth has to the fourth, then the sum of the first and fifth is to the second as the sum of the third and sixth is to the fourth. In like manner, if the seventh is to the second as the eighth is to the fourth, we shall have the sum of the first, fifth, and seventh is to the second as the sum of the third, sixth, and eighth is to the fourth; and so on for any number of quantities that are connected with the second and fourth by equations, as above.

(29.) OF THE COMPOSITION AND RESOLUTION OF RATIOS.

Because ratios are numbers, it is evident that they may be added or subtracted, multiplied or divided, like numbers. The product of any number of ratios is said to be a ratio that is compounded of all the ratios that enter as factors into A C E G the product. Thus, X X X x etc., is a ratio that

B D F H

is compounded of the ratio of A to B, and of the ratio of C to D, and of the ratio of E to F, and so on.

A ratio which is compounded of two equal ratios is called a duplicate ratio, which is clearly equal to the square of either of the equal ratios.

Thus, if we have

A C
B D'

then

tio, which is evidently the same as

A C

X is a duplicate ra

B D

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A ratio that is compounded of three equal ratios is called a triplicate ratio, which is manifestly the same as the cube or third power of either of the equal ratios. Thus, if we

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A C E
B=D F

X X is a triplicate ratio, and B D F

3

it is clearly the same as either of the expressions (4), (C);

In like manner, a ratio that is compounded of four equal ratios is called a quadruplicate ratio, and a ratio that is com

pounded of five equal ratios is termed a quintuplicate ratio, and a ratio that is compounded of six equal ratios is termed a sextuplicate ratio; and universally, if n denotes any positive integer, then a ratio that is compounded of n equal ratios may be said to be of the nth order, if we consider either of the equal ratios as being of the first order; and it is clear that the ratio of the nth order is the same as the nth power of either of the equal ratios.

Thus, if we have the n equal ratios

A с E

A с E

=

=

B D F

= etc.,

then X X x etc., to n factors, is a ratio of the nth

B D F

order, which is evidently the same as either of the expres

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Reciprocally, if a ratio is resolved into two equal factors or ratios, then either of the equal ratios is named a subduplicate ratio, which is manifestly the same as the square root of the given ratio.

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which is clearly the same as the second or square root of the

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Also, if a ratio is resolved into three equal ratios, then either of the equal ratios is called a subtriplicate ratio, which is clearly the same as the third or cube root of the given ratio. Thus, if we find that

A

equals the product of

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B then either of these equal

ratios is a subtriplicate ratio, which is plainly the same as

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the third or cube root of the given ratio, or V

B

also called the subtriplicate ratio of the given ratio.

And when a ratio is resolved into four equal ratios, either

VA, which is

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