Comparing (4) with (1), we see that their first mem bers are equal; hence, their second members must also be equal; that is,

Hence, the following principle:

3. If the first couplets of two proportions are the same, the second couplets will form a proportion.

Consequently, by alternation, if the antecedents of two proportions are the same in both, the consequents will be in proportion.

If we take the reciprocals of both members of (1), we shall have,

Hence, the following principle:

4. If four quantities are in proportion, they will be in proportion by inversion.

If we add 1 to both members of (1), and also subtract 1 from both members, we shall have,

Whence, by reducing to a common denominator, we have,

5. If four quantities are in proportion, they will be in proportion by composition or division.

If we multiply both terms of the ratio

b

a'

by m, its

value will not be changed (Art. 61), and we shall have,

Hence, the following principle:

6. Equimultiples of two quantities are proportional to the quantities themselves.

If we multiply both terms of the first member of (1), by m, and both terms of the second member by n, the equality will not be destroyed, and we shall have,

mb

ma

nd

= -; or, ma: mb :: nc: nd (9.)

пс

Hence, the following principle:

7. If four quantities are in proportion, any equi multiples of the first couplet will be proportional to any equimultiples of the second couplet.

If, in Equation (8), we suppose m = 1±

any fraction whatever, we shall have the following

principle:

8. If two quantities be increased or diminished by like parts of each, the results will be proportional to the quantities themselves.

If, in Equation (9), we suppose, m = 1±

p'

we shall have the following principle.

9. If both terms of the first couplet of a proportion be increased or diminished by like parts of each, and if both terms of the second couplet be increased or diminished by any other like parts of each, the results will be in proportion.

A continued proportion, is one in which several ratios are successively equal to each other, as,

h

= &c.; or, a:b: : c : d : : e: ƒ :: g : h,

(10.)

From the preceding continued proportion, we evi dently have the following equations:

Changing this equation into a proportion (Principle 1), we have,

a + c + e +g+ &c. : b+d+f+h+ &c. :: a : b; or,

10. In any continued proportion, the sum of all the antecedents is to the sum of all the consequents, as any antecedent is to the corresponding consequent.

Let us assume the two equations,

bf dh

= ; or, ae: bf :: cg: dh (12.)

ae cg

Hence, the following principle:

11. If two proportions be multiplied together, term by term, the products will be proportional.

This principle may be extended to the multiplication of any number of proportions, term by term.

SERIES.

161. A SERIES is a succession of terms, each of which is derived from one or more of the preceding ones, by a fixed law. This law is called the law of the series.

When a certain number of terms are given, and the law of the series is known, any number of terms may be found. There are, in general, an infinite number of terms in every series.

The simplest series are, Arithmetical and Geometrical Progressions.

162. An ARITHMETICAL PROGRESSION is a series in which each term is derived from the preceding term, by adding to it a constant quantity. This quantity is called the common difference.

When the common difference is positive, each term is greater than the preceding, and the progression is said to be increasing. When the common difference is nega tive, each term is less than the preceding, and the progression is said to be decreasing.

Thus, 2, 4, 6, 8, &c., is an increasing arithmetical progression, in which the common difference is +2.

The series, 18, 16, 14, 12, &c., is a decreasing arithmetical progression, in which the common difference

Although there are an infinite number of terms in every progression, it is customary to speak of any finite number of consecutive ones, as constituting a progression. Thus, we call the succession of terms,

3, 5, 7, 9, 11,

a progression of 5 terms.

If the terms of any increasing progression be taken in a reverse order, beginning at the last, there will result g decreasing progression. Thus, the progression,

4, 8, 12, 16,

becomes, when reversed,

16, 12, 8, 4.

If a decreasing progression, be, in like manner ro versed, there will result an increasing progression.