a

For being compounded with the ratio of minority, is

And being compounded with a ratio of majority, is

(312.) If the homologous terms of two ratios be added together, the ratio of the sums is of intermediate magnitude.'

a

с

Let and be the ratios, the former being the greater,

but as adbc, therefore the numerator of the former frac

tion exceeds that of the latter; and hence

с

a

a+c

b

Again reducing and to the same denominator,

they become

a+c b+d

and since adbc, the numerator of the former fraction

exceeds that of the latter, and hence

a+c

с

d

7

d

b+d

(313.) 'The duplicate ratio of two quantities is equal to

the ratio of these quantities compounded with itself." Let a, b, be two quantities, then a2: 62 is the ratio of a:b

compounded with itself, or with a:b (301); or (a: b, a: b) = a2: b2.

(314.) The triplicate ratio of two quantities is equal to a ratio compounded of three ratios, each of which is equal to the ratio of the quantities."

Let a, b, be two quantities; then the ratio compounded of the three identical ratios a: b, a:b, and a:b, is a3: 63 (301), or (a: b, a: b, a: b) a3: 13.

=

(315.) The ratio of two quantities is equal to their subduplicate ratio compounded with itself."

Let ab be the subduplicate ratio of two quantities a and b; then a:b, compounded with a:b, gives a: b, or (a: √b, Ja: √b) = a:b.

(316.) The ratio of two quantities is equal to the ratio which is compounded of three ratios, each of which is equal to their subtriplicate ratio.'

Let ab be the subtriplicate ratio of a and b; then the three identical ratios a: /b, a: /b, and a:b, being compounded (301), give the ratio a: b, or (a: 3/b, /ab/a: /b) = a:b.

(317.) The ratio of the first of any number of quantities to the last, is equal to the ratio which is compounded of that of the first to the second, of the second to the third, of the third to the fourth, and so on to the last."

For let a, b, c, d, e, be five quantities, then (a: b, b: c, a b с d

c:d, d:e) =

a

The preceding theorems may all be illustrated by numerical examples.

Thus, to exemplify the theorem in art. (312), let the two

5 3

ratios be and then a 5, b = 6, c=3, and d = 4; 6 4'

therefore a + c = 8, and b+d=10, or

Now, reducing and to a common

a + c 8 4 b+d 10 5

=

denominator, they

5 3

6

4

5 4

4 3

6 5.

hence 7 Again, reducing and to the same deno

5 4

other theorems may be similarly illustrated, by giving the letters particular values.

OF PROPORTION.

DEFINITIONS.

(318.) A proportion or an analogy consists of two equal ratios, and therefore of four terms."

Let the ratio a: b be equal to that of c:d, then a:b=c:d is a proportion, which is sometimes expressed thus, a: b::c:d, and is enunciated thus: the ratio of a to b is equal to that of c to d; or a is to b as c is to d.

(319.) 'The terms of equal ratios, taken in order, are said to be proportional, and they are called proportionals, or proportional quantities."

Thus, if a:bc:d, the terms a, b, c, d, are said to be proportional, and are called proportionals. So if a:bc:d =e:f, a, b, c, d, e, and f, are said to be proportional quantities, or proportionals.

(320.) The first and last terms of a proportion are called extremes, and the second and third means.

Thus, if a:bc:d, a and d are called extremes, and b and c means.

(321.) When of any number of quantities, the first has to the second the same ratio as the second to the third, and the second to the third the same ratio as the third to the fourth, and so on, the quantities are said to be in continued proportion, and are called continual proportionals."

Thus, if a:bb:cc:d, a, b, c, and d, are in continued proportion. So are 2, 4, and 8, for 2:44:8.

(322.) When three quantities are in continued proportion, the first and third are called extremes, and the second a mean proportional."

Let a:bb:c, then a and c are the extremes, and b is a mean proportional between them.

(323.) 'Four quantities are said to be directly proportional, when the first is to the second as the third to the fourth."

Thus, A, B, C, and D, are directly proportional when A: B C: D. So 3, 4, 6, and 8, are directly proportional, for 3:46:8.

'Four quantities are said to be indirectly or inversely proportional, when the first is to the second as the fourth to the third."

Thus, A, B, D, and C, are indirectly proportional when A:B= = C: D. So are 3, 12, 32, and 8, for 3: 128:32.

(324.) Two quantities are said to be reciprocally proportional to other two, when one of the former is to one of the latter, as the remaining one of the latter to that of the former; that is, when the former are taken for extremes, and the latter for means, or conversely."

Thus, A, D, are reciprocally proportional to B and C, when A: BC: D, or when A: CB: D. So 2 and 15 are reciprocally proportional to 10 and 3, for 2:10= 3:15, or 2:3=10:15.

(325.) 'Two or more analogies are said to be compounded when their corresponding ratios are compounded; that is, by taking the products of their corresponding terms."

(326.) The terms of an analogy may undergo various changes in respect to their order or magnitude, and still a proportion will exist; and to denote these changes, the following technical terms are employed :

-

(327.) By inversion, when the second is to the first as the fourth to the third;

(328.) By alternation, when the first is to the third as the second to the fourth;

(329.) By composition, when the sum of the first and second is to the second, as the sum of the third and fourth to the fourth;

(330.) By addition, when the first is to the sum of the first and second, as the third is to the sum of the third and fourth;

(331.) By division, when the excess of the first above the second is to the second, as the excess of the third above the fourth to the fourth;

(332.) By mixing, when the sum of the first and second is to their difference, as the sum of the third and fourth to their difference;

(333.) By direct equality, when there are several analogies, and two homologous terms in each are equal to two homologous terms in the following, and it is inferred that the remaining terms of the first analogy are directly proportional to the remaining terms of the last; and

(334.) By indirect equality, when there are several analogies, and two terms not homologous in each are equal to two terms not homologous in the following, and it is inferred that the remaining terms of the first are reciprocally proportional to the remaining terms of the last."

AXIOMS.

(335.) If the antecedents and consequents of two ratios are respectively equal, so are the ratios.

(336.) If two ratios are equal, and also their antecedents, so are their consequents.

(337.) If two ratios are equal, and also their consequents, so are the antecedents."

These three axioms may be more concisely stated thus: of these three conditions-the equality of two ratios, of their antecedents, and of their consequents-if any two be given, the third also exists.

(338.) 'If one ratio exceeds another, and their consequents are equal, the antecedent of the former exceeds that of the latter.

(339.) If one ratio exceeds another, and their antecedents are equal, the consequent of the former is less than that of the latter.

(340.) If the consequents of two unequal ratios are equal, the greater ratio is that which has the greater antecedent.