For a ratio is not altered, by adding to it, or subtracting from it, the terms of another equal ratio. (Art. 312.). If a:b::c:d, and a:b::m:n,

Then by adding to, or subtracting from a and b, the terms of the equal ratio m:n, we have,

a+m:b+n::c:d, and a-m: b-n::c:d.

And by adding and subtracting m and n, to and from c and d, we have,

a:b::c+m:d+n, and a:b::c―m: d―n.

Here the addition and subtraction are to and from analogous terms. But by alternation, (Art. 344,) these terms will

become homologous, and we shall have,

a+m:c::b+n:d, and a-mic::b-n: d.

Cor. 1. This addition may evidently be extended to any number of equal ratios. (Euclid 2. 5. cor.)

(c:d
h:1

Thus if a:b::

m:n

XC : :y

then a:b::c+h+m+x:d+l+n+y.

Cor. 2. If a:b::c:d then a+m:b::c+n:d. (Eu. 24.5.)

And m:b::ndS

For by alternation a:c::b:d min::bid

And

350. Hence, if two analogous or homologous terms be added to, or subtracted from the two others, the proportion will be preserved.

Thus, if a:b::c:d, and 12:4::6:2, then,

1. Adding the last two terms, to the first two.

a+c:b+d::a:b

and a+c:b+d::c:d

or atc:a::b+d:b

and a+c:c::b+d: d

12+6: 4+2::12:4
12+6: 4+2:: 6:2
12+6:12::4+ 2:4
12+6: 6:4+ 2:2.

2. Adding the two antecedents to the two consequents.

a+bb::c+d:d

a+b:a::c+d: c, &c.

12+4: 4::6+2:2
12+4:12:6+2:6, &c.

This is called composition. (Euclid 18. 5.)

3. Subtracting the first two terms from the last two. c-a: a::d-b:b, or c-a:c::d-b: d, &c.

4. Subtracting the last two terms from the first two. a-c: b―d::a: b, or a―c: b—d::c:d, &c.

5. Subtracting the consequents from the antecedents. a-b:b::c-d:d, or a: a-b::c:c-d, &c.

The alteration expressed by the last of these forms is called conversion.

6. Subtracting the antecedents from the consequents. b-a:a::d-c: c, or b:b-a::d: d-c, &c.

7. Adding and subtracting, a+b:a—b::c+d:c-d. That is, the sum of the first two terms, is to their difference, as the sum of the last two, to their difference.

Cor. If any compound quantities, arranged as in the preceding examples, are proportional, the simple quantities of which they are compounded are proportional also.

Thus, if a+b:b::c+d: d, then a:b::c:d. division. (Euclid 17. 5.)

This is called

CASE V. Compounding Proportions.

351. If the corresponding terms of two or more ranks of proportional quantities be multiplied together, the products will be proportional.

If the cor

QUEST. What is composition? Conversion? Division? responding terms of two or more ranks of proportionals are multiplied together, how will the product be?

This process is called compounding proportions. It is the same as compounding ratios. It should be distinguished from what is called composition, which is an addition of the terms of a ratio. (Art. 350, 2.)

If a:b::c:d

And h:l::m:n

Then ah: bl:: cm: dn

12:4::6:2

10:5::8:4

120:20:48:8

For from the nature of proportion, the two ratios in the first rank are equal, and also the ratios in the second rank. And multiplying the corresponding terms is multiplying the ratios, (Art. 311,) that is, multiplying equals by equals, (Ax. 3;) so that the ratios will still be equal, and therefore the four products must be proportional.

The same proof is applicable to any number of proportions. a:b:: c :d

From this it is evident that if the terms of a proportion be multiplied, each into itself, that is, if they be raised to any power, they will still be proportional. (Art. 308.)

If a:b::c:d
a:b::c:d

Then a2:62::c2: d2

2:4::6:12
2:4:: :6:12

4:16:36: 144

Proportionals will also be obtained, by reversing this process, that is, by extracting the roots of the terms.

If a:b::c:d, then a/b/c:√d.

For taking the products of the extremes and means, ad-bc.

And extracting the root of both sides,

That is, (Arts. 210.a, 339,)

Vad-√bc

QUEST.-What is meant by compounding proportions? What is the difference between compounding proportions and composition? If several quantities are proportional, how are like powers or roots of them?

CASE VI. Involution and Evolution of the terms.

352. If several quantities are proportional, their like powers or like roots are proportional.

And man: m/b :: m ch: m/d", that is, an; b::c: dr. 353. If the terms in one rank of proportionals be divided by the corresponding terms in another rank, the quotients will be proportional.

This is sometimes called the resolution of ratios.'

This is merely reversing the process in Art. 351, and may be demonstrated in a similar manner.

N. B. This should be distinguished from what geometers call division, which is a subtraction of the terms of a ratio. (Art. 350, 7.)

354. When proportions are compounded by multiplication, it will often be the case that the same factor will be found in two analogous or two homologous terms.

Here a is in the first two terms, and c in the last two. Dividing by these, (Art. 345,) the proportion becomes m:b::n:d. Hence,

QUEST.-What is meant by the resolution of ratios?

355. In compounding proportions, equal factors or divisors in two analogous or homologous terms, may be rejected.

This rule may be applied to the cases, to which the terms "ex æquo" and "ex æquo perturbate" refer. (Arts. 346.a, 348.) One of the methods may serve to verify the other.

356. When four quantities are proportional, if the first be greater than the second, the third will be greater than the fourth; if equal, equal; if less, less.

a=b, cd-LA

a>b, c>d

a<b, c<d.

proportional, their reciprocals.

For in each of these proportions, we have, by reduction, ad-bc.

PROBLEMS IN GEOMETRICAL PROPORTION.

Prob. 1. Divide the number 49 into two such parts, that the greater increased by 6, may be to the less diminished by 11 as 9 to 2.

QUEST.-In Compounding proportions, what may be done with equal factors or divisors? When four quantities are proportional, if the first is greater than the second, how is the third? If equal? If less? If four quantities are proportional, how are their reciprocals?