Geometrical Proportion

370. But if four quantities are in geometrical proportion, the product of the extremes is equal to the product of the means.
      If a:b = c:d, ad = bc
      For by supposition, (Arts. 341, 359.) $\frac{a}{b } =\frac{c}{d } $
      Multiplying by bd, (Ax. 3.) $\frac{abd}{b } =\frac{cbd}{d } $
      Reducing the fractions, ad = bc.
      Thus 12:8 = 15:10, therefore 12.10 = 8.15.

Cor. Any factor may be transferred from one mean to the other, or from one extreme to the other, without affecting the proportion. If a:mb = x:y, then a:b = mx:y. For the product of the means is, in both cases the same. And if na:b = x:y, then a:b = x:ny.

371. On the other hand, if the product of two quantities is equal to the product of two others, the four quantities will form a proportion, when they are so arranged, that those on one side of the equation shall constitute the means, and those on the other side, the extremes.
      If my = nh, then m:n = h:y, that is $\frac{m}{n } =\frac{h}{y } $
      For by dividing my = nh by ny, we have $\frac{my}{ny} =\frac{nh}{ny } $
      And reducing the fractions, $\frac{m}{n } =\frac{h}{y } $.

Cor. The same must be true of any factors which form the two sides of an equation.
If (a + b).c = (d - m).y, then a + b:d - m = y:c.

372. If three quantities are proportional, the product of the extremes is equal to the square of the mean. For this mean proportional is, at the same time, the consequent of the first couplet, and the antecedent of the last. (Art. 366.) It is therefore to be multiplied into itself, that is, it is to be squared.
If a:b = b:c, then mult, extremes and means, ac = b2.
Hence, a mean proportional between two quantities may be found, by extracting the square root of their product.
If a:x = x:c, then x2 = ac, and x√ac.

373. It follows, from Art. 370, that in a proportion, either extreme is equal to the product of the means, divided by the other extreme; and either of the means is equal to the product of the extremes, divided by the other mean.
      1. If a:b = c:d, then ad = bc
      2. Dividing by d, $a=\frac{bc}{d} $
      3. Dividing the first by c, $b=\frac{ad}{c} $
      4. Dividing it by b, $c=\frac{ad}{b} $
      5. Dividing it by a, $d=\frac{bc}{a} $ ; that is, the
fourth term is equal to the product of the second and third drnded by the first.

On this principle is founded the rule of simple proportion in arithmetic, commonly called the Rule of Three. Three numbers are given to find a fourth, which is obtained by multiplying together the second and third, and dividing by the first.

374. The propositions respecting the products of the means, and of the extremes, furnish a very simple and convenient criterion for determining whether any four quantities are proportional. We have only to multiply the means together, and also the extremes. If the products are equal, the quantities are proportional. If the products are not equal, the quantities are not proportional.

375. In mathematical investigations, when the relations of several quantities are given, they are frequently stated in the form of a proportion. But it is commonly necessary that this first proportion should pass through a number of transformations before it brings out distinctly the unknown quantity, or the proposition which we wish to demonstrate. It may undergo any change which will not affect the equality of the ratios; or which will leave the product of the means equal to the product of the extremes.

It is evident, in the first place, that any alteration in the arrangement, which will not affect the equality of these two products, will not destroy the proportion. Thus, if a:b = c:d, the order of these four quantities may be varied, in any way which will leave ad = bc. Hence,

376. If four quantities are proportional, the order of the means, or of the extremes, or of the terms of the both couplets, may be inverted without destroying the proportion.
      If a:b = c:d,
      And 12:8 = 6:4
then
1. Inverting the means,
      a:c = b:d
      12:6 = 8:4
that is
      The first is to the third
      As the second to the fourth.
In other words, the ratio of the antecedents is equal to the ratio of the consequents.

This inversion of the means is frequently referred to by geometers, under the name of Alternation.

2. Inverting the extremes,
      d:b = c:a
      4:8 = 6:12
that is,
      The fourth is to the second,
      As the third to the first.

3. Inverting the terms of each couplet,
      b:a = d:c
      8:12 = 4:6
that is,
      The second is to the first,
      As the fourth to the third.
This is technically called Inversion.
Each of these may also be varied, by changing the order of the two couplets. (Art. 365.)

Cor. The order of the whole proportion may be inverted.
      If a:b = c:d, then d:c = b:a.
In each of these cases, it will be at once seen that, by taking the products of the means, and of the extremes, we have ad = bc, and 12.4 = 8.6.
If the terms of only one of the couplets are inverted, the proportion becomes reciprocal. (Art 367.)
If a:b = c:d, then a is to b, reciprocally, as d to c.

377. A difference of arrangement is not the only alteration which we have occasion to produce, in the terms of a proportion. It is frequently necessary to multiply, divide, involve, etc. In all cases, the art of conducting the investigation consists in so ordering the several changes, as to maintain a constant equality, between the ratio of the two first terms, and that of the two last. As in resolving an equation, we must see that the sides remain equal; so in varying a proportion, the equality of the ratios must be preserved. And this is effected either by keeping the ratios the same, while the terms are altered; or by increasing or diminishing one of the ratios as much as the other. Most of the succeeding proofs are intended to bring this principle distinctly into view, and to make it familiar. Some of the propositions might be demonstrated, in a more simple manner, perhaps, by multiplying the extremes and means. But this would not give so clear a view of the nature of the several changes in the proportions.

It has been shown that, if both the terms of a couplet be multiplied or divided by the same quantity, the ratio will remain the same; (Art. 355.) that multiplying the antecedent is, in effect, multiplying the ratio, and dividing the antecedent, is dividing the ratio; (Art. 352.) and farther, that multiplying the consequent, is, in effect, dividing the ratio, and dividing the consequent is multiplying the ratio. (Art. 353.) As the ratios in a proportion are equal, if they are both multiplied, or both divided, by the same quantity, they will still be equal. (Ax. 3.) One will be increased or diminished as much as the other. Hence,

378. If four quantities are proportional, two analogous or two homologous terms may be multiplied or divided by the same quantity, without destroying the proportion.

If analogous terms be multiplied or divided, the ratios will not be altered. (Art, 355.) If homologous terms be multiplied or divided, both ratios will be equally increased or diminished. (Arts. 352, 353.)
If a:b = c:d, then,
1. Multiplying the two first terms, ma:mb = c:d
2. Multiplying the two last terms, a:b = mc:md
3. Multiplying the two antecedents, ma:b = mc:d
4. Multiplying the two consequents, a:mb = c:md
5. Dividing the two first terms, $\frac{a}{m}:\frac{b}{m}=c:d$
6. Dividing the two last terms, $a:b=\frac{c}{m}:\frac{d}{m }$
7. Dividing the two antecedents, $\frac{a}{m}:b=\frac{c}{m}:d$
8. Dividing the two consequents, $a:\frac{b}{m}=c:\frac{d}{m}$.

Cor. 1. All the terms may be multiplied or divided by the same quantity.
            ma:mb = mc:md, $\frac{a}{m}:\frac{b}{m}=\frac{c}{m}:\frac{d}{m} $.

Cor. 2. In any of the cases in this article, multiplication of the consequent may be substituted for division of the antecedent in the same couplet, and division of the consequent, for multiplication of the antecedent. (Art. 354, cor.)

379. It is often necessary not only to alter the terms of a proportion, and to vary the arrangement, but to compare one proportion with another. From this comparison will frequently arise a new proportion, which may be requisite in solving a problem, or in carrying forward a demonstration. One of the most important cases is that in which two of the terms in one of the proportions compared, are the same with two in the other. The similar terms may be made to disappear, and a new proportion may be formed of the four remaining terms. For,

380. If two ratios are respectively equal to a third, they are equal to each other.
This is nothing more than the 11th axiom applied to ratios.
1. If a:b = m:n
And c:d = m:n
then a:b = c:d,or a:c = b:d. (Art.376.)
2. If a:b = m:n
And m:n = c:d
then a:b = c:d,or a:c = b:d.

Cor. If a:b = m:n
m:n > c:d
then a:b > c:d.
For if the ratio of m:n is greater than that of c:d, it is manifest that the ratio of a:b, which is equal to that of m:n, is also greater than that of c:d.

381. In these instances, the terms which are alike in the two proportions are the two first and the two last. But this arrangement is not essential. The order of the terms may be changed, in various ways, without affecting the equality of the ratios.

1. The similar terms may be the two antecedents, or the two consequents, in each proportion. Thus,
If m:a = n:b
And m:c = n:d
then
By alternation, m:n = a:b
And m:n = c:d
Therefore a:b = c:d, or a:c = b:d, by the last article.

2. The antecedents in one of the proportions, may be the same as the consequents in the other.
If m:a = n:b
And c:m = d:n
By inver. and altern. a:b = m:n
By alternation c:d = m:n:
Therefore a:b, etc. as before.

3. Two homologous terms, in one of the proportions, may be the same, as two analogous terms in the other.
If a:m = b:n
And c:d = m:n
By alternation, a:b = m:n
And c:d = m:n
Therefore, a:b, etc.

All these are instances of an equality, between the ratios in one proportion, and those in another. In geometry, the proposition to which they belong is usually cited by the words "ex aequo" or "ex aequali". The second case is this article is that which in its form, most obviously answers to the explanation in Euclid. But they are all upon the same principle, and are frequently referred to, without discrimination.

382. Any number of proportions may be compared, in the same manner, if the two first or the two last terms in each preceding proportion, are the same with the two first or the two last in the following one.
      Thus if a:b = c:d
      And c:d = h:l
      And h:l = m:n
      And m:n = x:y
            then a:b = x:y.
That is, the two first terms of the first proportion have the same ratio, as the two last terms of the last proportion. For it is manifest that the ratio of all the couplets is the same.

And if the terms do not stand in the same order as here, yet if they can be reduced to this form, the same principle is applicable.
      Thus if a:c = b:d
      And c:h = d:l
      And h:m = l:n
      And m:x = n:y
            then by alternation
      a:b = c:d
      c:d = h:l
      h:l = m:n
      m:n = x:y.
      Therefore a:b = x:y, as before.

In all the examples in this, and the preceding articles, the two terms in one proportion which have equals in another, are neither the two means, nor the two extremes, but one of the means, and one of the extremes; and the resulting proportion is uniformly direct.

383. But if the two means, or the two extremes, in one proportion, be the same with the means, or the extremes, in another, the four remaining terms will be reciprocally proportional.
If a:m = n:b
And c:m = n:d
then a:c = $\frac{1}{b}:\frac{1}{d} $, or a:c = d:b

For ab = mn
And cd = mn
(Art. 370) Therefore ab = cd, and a:c = d:b.

In this example, the two means in one proportion, are like those in the other. But the principle will be the same, if the extremes are alike, or if the extremes in one proportion arei like the means in the other.
      If m:a = b:n
      And m:c = d:n
      then a:c = d:b.

      Or if a:m = n:b
      And m:c = d:n
      then a:c = d:b.
The proposition in geometry which applies to this case, is usually cited by the words "ex aequo perturbate"

384. Another way in which the terms of a proportion may be varied, is by addition or subtraction.

If to or from two analogous or two homologous terms of a proportion, two other quantities having the same ratio be added or subtracted, the proportion will be perserved.

For a ratio is not altered, by adding to it, or subtracting from it, the terms of another equal ratio. (Art. 357.)
            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. 376,) these terms will me homologous, and we shall have,
      a+m:c = b+n:d,      and a-m:c = b-n:d.

Cor. 1. This addition may, evidently, be extended to any number of equal ratios.
Thus if
      a:b = c:d
      a:b = h:l
      a:b = m:n
      a:b = x:y
Then a:b = c+h+m+x:d+l+n+y.

Cor. 2. If a:b = c:d
      And m:b = n:d
then a+m:b = c+n:d.

For by alternation a:c = b:d
And m:n = b:d
therefore
      a+m:c+n = b:d
      or a+m:b = c+n:d.

385. From the last article it is evident that if, in any proportion, the terms be added to, or subtracted from each other, that is,

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 two last terms, to the two first.
      a+c:b+d = a:b      12+6:4+2 = 12:4
    and a+c:b+d = c:d      12+6:4+2 = 6:2
      or a+c:a = b+d:b      12+6:12 = 4+2:4
    and a+c:c = b+d:d      12+6:6 = 4+2:2.

2. Adding the two antecedents, to the two consequents.
      a+b:b = c+d:d      12+4:4 = 6+2:2
      a+b:a = c+d:c, etc.      12+4:12 = 6+2:6, etc.
            This is called Composition.

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

4. Subtracting the two last terms from the two first.
            a-c:b-d = a:b
            a-c:b-d = c:d, etc.

5. Subtracting the consequents from the antecedents.
            a-b:b = c-d:d
            a:a-b = c:c-d, etc.
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
            b:b-a = d:d-c, etc.

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

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

This is compounding ratios, (Art. 347,) or compounding proportions. It should be distinguished from what is called composition, which is an addition of the terms of a ratio. (Art 385. 2.)
      If a:b = c:d         12:4 = 6:2
      And h:l = m:n         10:5 = 8:4
Then ah:bl = cm:dn         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. 352. cor.) 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.
            If
              a:b = c:d
              h:l = m:n
              p:q = x:y
            Then ahp:blq = cmx:dny.
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.
      If a:b = c:d         2:4 = 6:12
       a:b = c:d         2:4 = 6:12
Then a2:b2 = c2:d2         4:16 = 36:144
Proportionals will also be obtained, by reversing this process, tnat is, by extracting the roots of the terms.
      If a : b:: c : d,      then √a:√b = √c:√d.
For taking the product of extr. and means, ad = bc
And extracting both sides, √ad = √bc
That is, (Arts. 254, 371,) √a:√b = √c:√d.
Hence,

387. If several quantities are proportional, their like powers or like roots are proportional.
              If a:b = c:d
Then an:bn = cn:dn,      and ma:mb = mc:md.
And man:mbn = mcn:√dn, that is, am/n:bm/n = cm/n:dm/n.

388. 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.
      If a:b = c:d      12:6 = 18:9
      And h:l = m:n      6:2 = 9:3
      Then $\frac{a}{h}:\frac{b}{l }=\frac{c}{m}:\frac{d}{n} $      $\frac{a}{h}:\frac{b}{l }=\frac{c}{m}:\frac{d}{n} $.
This is merely reversing the process in Art. 386, and may be demonstrated in a similar manner.
This should be distinguished from what geometers call division, which is a subtraction of the terms of a ratio. (Art. 385. cor.)

389. In compounding proportions, equal factors or divisors in two analogous or homologous terms, may be rejected.
If
      a:b = c:d         12:4 = 9:3
      b:h = d:l         4:8 = 3:6
      h:m = l:n         8:4 = 6:15
Then a:m = c:n          12:20 = 9:15.
This rule may be applied to the cases, to which the terms "ex aequo" and "ex aequo perturbate" refer. See Arts. 381 and 383. One of the methods may serve to verify the other.

394. The changes which may be made in proportions, without disturbing the equality of the ratios, are so numerous, that they would become burdensome to the memory, if they were not reducible to a few general principles. They are mostly produced,
1. By inverting the order of the terms, Art. 376.
2. By multiplying or dividing by the same quantity, Art. 378.
3. By comparing proportions which have like terms. Art. 380, 381, 382, 383.
4. By adding or subtracting the terms of equal ratios, Art. 384, 385.
5. By multiplying or dividing one proportion by another, Art. 386, 387, 388.
6. By involving or extracting the roots of the terms, Art. 387.

391. 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.

For, the ratios of the two couplets being the same, if one is a ratio of equality, the other is also, and therefore the antecedent in each is equal to its consequent; (Art. 345,) if one is a ratio of greater inequality, the other is also, and therefore the antecedent in each is greater than its consequent; and if one is a ratio of lesser inequality, the other is also, and therefore the antecedent in each is less than its consequent.
Let a:b = c:d; then if
            a = b, c = d
            a > b, c > d
            a < b, c < d.

Cor. 1. If the first be greater than the third, the second will be greater than the fourth; if equal, equal; if less, less.
For by alternation, a:b = c:d becomes a:c = b:d, without any alteration of the quantities. Therefore, if a = b, c = d, etc. as before.

Cor. 2. If a:m = c:n
and m:b = n:d
then if a = b, c = d, etc.
For, by equality of ratios, (Art. 381. 2.) or compounding ratios, (Arts. 386, 389.)
a:b = c:d. Therefore, if a = b, c = d, etc. as before.

Cor. 3. If a:m = n:d
and m:b = c:n
then if a = b, c = d.

391. b. If four quantities are proportional, their reciprocals are proportional; and v. v.
      If a:b = c:d, then $\frac{a}{h}:\frac{b}{l }=\frac{c}{m}:\frac{d}{n} $.
For in each of these proportions, we have, by reduction, ad = bc.


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