RATIO, in law, an account; a cause, or the giving judgment therein. In mathematics: (1) The measure of the relation which one quantity bears to an other of the same kind; that is, it is the number of times that one quantity con tains another regarded as a standard. This is found by dividing the one by the other. The quotient or ratio thus ob tained is the proper measure of the rela tion of the two quantities. Some writers define the ratio of one quantity to an other as the quotient of the first quantity divided by the second, while others define it as the quotient of the second divided by the first. Thus, the ratio of 2 to 4, or of a to b, may be taken either as 2±4 or 4-=2, and a÷b or In every ratio there are two quantities compared, one of which is supposed known, and is assumed as a standard; the other is to be determined in terms of this standard. These quantities are called terms of the ratio; the first one, or that which is antecedently known, is called the ante cedent, and that whose value is to be measured by the antecedent, is called the consequent. Ratios are compared by comparing the fractions; thus, the ratio of 8:5 is compared with the ratio of 9:6, by comparing the frac tions % and %; these fractions are re spectively equal to and and since 4% is greater than the ratio of 8:5 is greater than that of 9:6. Ratios are compounded together by multiplying their antecedents together for a new an tecedent, and their consequents together for a new consequent; thus the ratio of a:b, compounded with that of c:d, is ac: bd. Proportion is the relation of equal ity subsisting between two ratios. See PROPORTION. (2) A name sometimes given to the rule of three in arithmetic.
Compound ratio: (a) The ratio of the product of the antecedents of two or more ratios to the product of the conse quents: thus if 3 : 6 : : 4 : 12, then 12:72 is the compound ratio. (b) When one quantity is connected with two others in such a manner that if the first is in creased or diminished, the product of the other two is increased or diminished in the same proportion, then the first quantity is said to be in the compound ratio of the other two.
Direct ratio, two quantities are said to be in direct ratio when they both in crease or decrease together, and in such a manner that their ratio is constant.
Duplicate ratio, when three quantities are in continued proportion, the first is said to have to the third the duplicate ratio of that which it has to the second, or the first is to the third as the square of the first to the square of the second.
Inverse ratio, two quantities or mag nitudes are said to be in inverse ratio, when if the one increases the other neces sarily decreases, and, vice versa, when the one decreases the other increases.
Mixed ratio or proportion: a ratio or proportion in which the sum of the ante cedent and consequent is compared with the difference of the antecedent and con sequent. Thus, if a:b::c:d, then a+ b: (1-1)::e-i-d:c—d is the mixed ratio or proportion.
Prime and ultimate ratios, a method of analysis, devised and first successfully, employed by Newton in his "Principia." It is an extension and simplification of the method known among the ancients as the method of exhaustions. To conceive the idea of this method, let us suppose two variable quantities constantly ap proaching each other in value, so that their ratio continually approaches 1, and at last differs from 1 by less than any assignable quantity; then is the ultimate ratio of the two quantities equal to 1. In general when two variable quantities, simultaneously approach two other quantities, which, under the same cir cumstances, remain fixed in value, the ultimate ratio of the variable quanti ties is the same as the ratio of the quantities whose values remain fixed. They are called prime or ulti mate ratios, according as the ratio of the variable quantities is receding from or approaching to the ratio of the limits. This method of analysis is generally called the method of limits.
Extreme and mean ratio, in geometry, the ratio where a line is divided in such a manner that the greater segment is a mean proportional between the whole line and the lesser segment: that is, that the whole line is to the greater segment as that greater segment is to the less.
Composition of ratios, the act of com pounding ratios.
Ratio of a geometrical progression, the constant quantity by which each term is multiplied to produce the succeeding one. To find the ratio of a given pro-. gression, divide any term by the preced ing one.
Ratio of exchange, a phrase used in political economy to denote the propor tion in which a quantity of one com modity exchanges for a given quantity of another. The expression can never be used with any degree of accuracy, ex cept in those cases where the commodi ties are homogeneous in quality, and sus ceptible of weight or measurement, as in the exchange of gold for silver, copper, iron, etc., or that of wheat for barley, oats, etc.