303. Ratios that are equal to the same ratio are equal to each

304." If four quantities are proportional, they will be proportional by composition.

Add unity to each of these equals, and we have

305. If four quantities are proportional, they will be proportional by division.

Subtract unity from each of these equals, and we have

306." If four quantities are proportional, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference.

307." If four quantities are in proportion, any equimultiples of the first couplet will be proportional to any equimultiples of the second couplet.

Multiply both terms of the first fraction by m, and both terms of the second fraction by n, and we have

308." If four quantities are in proportion, any equimultiples of the antecedents will be proportional to any equimultiples of the consequents.

Multiply each of these equals by m, and we have

309. If any number of quantities are proportional, any one antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents.

310. If there are two sets of proportional quantities, the prod ucts of the corresponding terms will be proportional.

311. If four quantities are in proportion, like powers or roots of these quantities will also be in proportion.

Raising each of these equals to the nth power, we obtain

312. If three quantities are in continued proportion, the product of the extremes is equal to the square of the mean.

313. If three quantities are in continued proportion, the first is to the third in the duplicate ratio of the first to the second.

Multiply each of these equals by, and we have

314. If four quantities are in continued proportion, the first is to the fourth in the triplicate ratio of the first to the second.

Multiplying together (1), (2), and (3), we have

315. Proportions are often expressed in an abridged form. Thus, if A and B represent two sums of money put out for one year at the same rate of interest, then

A: B:: interest of A: interest of B.

This is briefly expressed by saying that the interest varies as the principal. A peculiar character (o) is used to denote this relation. Thus interest ∞ principal

denotes that the interest varies as the principal.

316. One quantity is said to vary directly as another when the two quantities increase or decrease together in the same ratio. Thus, in the above example, A varies directly as the interest of A. In such a case, either quantity is equal to the other multiplied by some constant number.

Thus, if the interest varies as the principal, then the interest equals the product of the principal by some constant number, which is the rate of interest.

If the space (S) described by a falling body varies as the square of the time (T), then

S = mT2,

where m represents a constant multiplier.

317. One quantity may vary directly as the product of several others.

Thus, if a body moves with uniform velocity, the space described is measured by the product of the time by the velocity. If we put S to represent the space described, T the time of motion, and V the uniform velocity, then we shall have

S∞ Tx V.