Making x ny, n being unknown, we have, from (1),

Whence, taking the positive value of n, only,

5. Required two numbers, whose product is equal tc the square of two thirds the first, and the difference of whose squares is greater, by 1, than the square of twice the second.

6. Find two numbers, whose sum, greater, is 209, and whose difference, less, is 24.

7. Find two numbers, such that squares is equal to 181, and the squares equal to 19.

Ans. 9 and 4.

multiplied by the multiplied by the

Ans. 11 and 8. the sum of their difference of their Ans. 9 and 10,

RATIO,

CHAPTER X.

PROPORTION, AND SERIES.

RATIO AND PROPORTION.

150. THE MEASUREMENT of a quantity is the opera tion of finding how many times it contains some other quantity of the same kind, taken as a standard. The latter quantity is called the UNIT OF MEASURE, and may be any quantity which is of the same kind as the quantity measured. The unit of measure is supposed to be known beforehand, and is therefore called the ANTECEDENT. The value of the quantity measured becomes known, in terms of the antecedent, by the operation of measurement, and is therefore called the CONSEQUENT. Mathematically, the measurement is performed by dividing the consequent by the antecedent; the result is an abstract number, which, being prefixed to the unit employed, expresses the value of the thing measured. Hence, the following definition :

151. The RATIO of one quantity to another of the same kind, is the result obtained by dividing the second quantity by the first. The first quantity is called the ANTECEDENT, and the second the CONSEQUENT.

Thus, the ratio of a to b, is

b

or a)b; or, it may

be written, a: b. Here, a is the antecedent, or stand ard, and the consequent, and both are called terms.

152. A PROPORTION is the expression of equality between two equal ratios. Thus,

a

is a proportion, expressing the fact that the a to b is equal to the ratio of c to d. portion may also be written thus,

abc: d.

ratio of This pro

In either case, the proportion is read, a is to bas c is to d. In most algebraical operations, the former method of writing a proportion is the better; in geometrical operations, the latter method has been most frequently employed.

153. There are four terms in every proportion which have received different names with respect to each other The first and third are antecedents; the second and fourth are consequents. The first and fourth are extremes; the second and third are means. The first and second form the first couplet; the third and fourth form the second couplet. The fourth term, is called a fourth proportional to the other three. When the second term is equal to the third, it is said to be a mean proportional between the other two. In this case, there are but three different terms in the proportion, and the last term is saki to be a third proportional to the other two. Thus, in the

a and c are antecedents, b and d consequents, a and

b

d extremes, b and c means; or ab, is the first

couplet,

d

a'

or cd, is the second couplet, and d is

a fourth proportional to a, b, and c. Also, in the proportion,

b is a mean proportional between a and c, and c is a third proportional to a and b.

154. Quantities are in proportion, by alternation, when antecedent is compared with antecedent, and consequent with consequent.

155. Quantities are in proportion, by inversion, when antecedents are made consequents, and consequents are made antecedents.

156. Quantities are in proportion, by composition, when the sum of antecedent and consequent is compared with either antecedent or consequent.

157. Quantities are in proportion, by division, when the difference of antecedent and consequent is compared with either antecedent or consequent.

158. Two varying quantities are reciprocally, or inversely proportional, when one is increased as many times as the other is diminished. In this case, their product is a fixed quantity, as xy = m.

159. Equimultiples of two quantities, are the results. obtained by multiplying both by the same quantity. Thus, ma, and mb, are equimultiples of a and b, whatever may be the value of m.

160. We shall proceed to demonstrate some of the most important properties of proportions, adopting both methods of writing proportions.

1. If four quantities are in proportion, the product of the means is equal to the product of the extremes. Conversely, if we divide both members of (2) by ca, we shall have,

b

a

=

d

; or, a: b::c: d. Hence,

If the product of two quantities is equal to the product of two other quantities, the first two may be made the means, and the second two the extremes, of a proportion.

It follows, that of three proportional quantities, the square of the mean is equal to the product of the

extremes.

If we multiply both members of (1) by, and reduce the result to its simplest form, we shall have,

2. If four quantities are in proportion, they will be in proportion by alternation.