CHAP. XI. The term Ratio. Importance of ascertaining its ordinary meaning. In what manner represented. Geometrical repre ratio. ON RATIOS AND PROPORTIONS. 349. THE term Ratio, in ordinary language, is used to express the relation which exists between two quantities of the same kind with respect to magnitude: thus we speak of the ratio of two numbers, of two lines, of two areas, of two forces, of two periods of time and of any other concrete quantities, the relation of whose magnitudes to each other admits of being estimated. 350. The definition of ratio in Algebra, like all other definitions in that science, must be an assumption adapted as much as possible to its popular usage, in Arithmetic or Geometry or in both, in order that the conclusions which are founded upon it or represented by means of it, may be transferred to those subordinate sciences: in order properly to effect this object, it will be necessary to commence by endeavouring to ascertain what the meaning and usage of this term in those sciences really is. 351. A ratio, (the word is here used absolutely) consists of two terms or members, which are denominated the antecedent and the consequent: it is denoted in arithmetic as well as in geometry, by writing the antecedent before the consequent, with two dots, one above the other, between them: thus the ratio of 3 to 5 is written as fol lows; 3: 5. In a similar manner, if a and b denoted any two other numbers, lines or other magnitudes of the same kind, their ratio, whatever it may mean, would be denoted by a b. 352. Such a mode of representing a ratio, merely sentation of exhibits its terms to the eye as objects of comparison, and consequently conveys to the mind no idea of absolute magnitude: it may be called the geometrical representation of ratio, being the only one which is used in that science: for trical defi though in geometry we may agree upon various modes of representing ratios, yet they must all of them be equally arbitrary or independent of each other: for there is no No geomegeometrical definition of ratio, by which the equivalence nition of of different modes of representation may be ascertained ratio. as necessary consequences of it: for ratio is said to be the relation of quantities, of the same kind with respect to magnitude, a description of its meaning much too vague to be considered as a definition, and therefore not capable of being made the foundation of other propositions respecting it: it is for this reason that ratios in geometry are only considered in connection with each other, as constituting or not constituting a proportion. 353. A little examination however of some of the con- Popular ditions which ratios, taken according to the popular usage the term meaning of of the term, must satisfy, will lead to an arithmetical mode ratio. of representing them, by which their absolute magnitude may be ascertained, and which will thus conduct us to an arithmetical and also to an algebraical definition of ratio, which will be independent of the connection of ratios with each other for it is perfectly conformable to our common idea of ratios, to consider them in the first place, as necessarily the same for the same magnitudes, in whatever manner they may be represented; and in the second place, as independent of the specific affections or properties (of the same kind) of the magnitudes themselves. terms of the may un dergo. 354. Thus, if two lines admitted of resolution into 3 Changes and 5 parts respectively, which were equal to each other, which the the lines themselves might be correctly represented by the same ratio numbers 3 and 5, and their ratio by 3: 5. But the common unit of their primary division is itself divisible into 2, 3 or m equal parts, and the numbers of these successive parts which the original lines would contain, would be severally 6 and 10, 9 and 15, 3m and 5m, which might denote them equally with the original numbers 3 and 5; their ratio therefore, which remains the same, in conformity with the principle referred to, would be equally represented by 6: 10, 9: 15, 3m 5m. Conclu sions thence deduced. Again, this mode of representing lines and their ratio, which possess this particular relation to each other, is equally applicable to any other magnitudes of the same kind which possess the same relation to each other: thus two areas, two solids, two forces, two periods of time, may be so related to each other, as to admit of resolution. into 3 and 5 parts or units respectively, which are equal to each other: under such circumstances they must admit likewise of resolution into numbers of parts or units equal to each other, which are any equimultiples of 3 and 5: such pairs of numbers therefore, will equally represent those magnitudes, and will likewise equally form the terms of the ratio which expresses their relation to each other. 355. The preceding observations will conduct us naturally to the following conclusions: (1) Magnitudes of the same kind, which admit of resolution into any numbers of parts or units, which are equal to each other, may be properly represented by such numbers, or by any equimultiples of them. (2) The numbers which represent two magnitudes of the same kind will form the terms of the ratio, which expresses their relation to each other: and this ratio remains unaltered, when its terms are replaced by any equimultiples of them. (3) Such ratios are dependent upon the numbers which form their terms only, and are the same, whatever be the nature and magnitude of the concrete unit of which those numbers may be respectively composed. A ratio 356. All these conditions will be fully satisfied, if we may be represented agree to denote a ratio by means of a fraction, of which by means of the antecedent is the numerator, and the consequent the denominator: for the value of this fraction is deter a fraction. * We shall generally give an enlarged signification to the term multiple, as denoting the result of multiplication by fractions as well as by whole numbers. mined solely by the numbers which form its numerator and denominator, and is entirely independent of the specific value or nature of the units of the same kind, of which they are respectively composed: and it remains unaltered, when its numerator and denominator are multiplied or divided by the same number, that is, when the terms of the ratio corresponding, are replaced by any equimultiples of them. cal definition of In arithmetic, therefore, a ratio may be defined, as Arithmetithe fraction whose numerator is the antecedent, and denominator is the consequent of the ratio. ratio. definition 357. In algebra likewise we adopt the same definition, Algebraical considering the ratio of two quantities expressed by symbols of ratio. a and b, as a phrase synonymous with the fraction a : in this science, therefore, there is a meaning to be attached to the ratio of a to b, whenever there is a a meaning to be attached to the fraction whether the b' quantities which they denote are of the same or a different kind, or possess the same or different algebraical signs: in other words, there is no limit to the interpretation of the term ratio, when thus applied, which is different from that which belongs to the interpretation of the equivalent fraction. It appears, therefore, that both in Arithmetic and Algebra, the theory of ratios becomes identified with the theory of fractions. other mag nitudes. 358. The symbols of algebra represent geometrical as Includes geometrical well as other quantities, and the lines, areas and solids of as well as geometry, are thus brought within the range of this definition: it must be kept in mind however, that it is only by considering geometry as a science subordinate to algebra, that such quantities admit of the mode of representation which that definition renders necessary: for there is no geometrical mode of representing the division of one line by Reason why there in geome try. another, or the result of such a division: for this result is no defini- can bear no analogy to the quantities which produce it, tion of ratio being essentially numerical and consequently not capable of being represented by a line, unless in a symbolical sense, which under all circumstances must be different from that in which the other lines are used. It is of great importance to attend to this distinction, as it serves not only to explain the reason why there is no independent definition of ratio in geometry, but also why in comparing different ratios of geometrical lines or areas with each other, with reference to their identity or diversity, we are not at liberty to avail ourselves of the algebraical definition of ratio, unless we first change the mode of representing the quantities which are the objects of the investigation, and resort to the use of symbolical language. It is indif ther the 359. The identity of the definitions of ratio in arithferent whe- metic and algebra, makes it indifferent whether the quantisame ratio ties which compose the terms of a ratio, present themselves presents it- under an arithmetical or symbolical form: for the tranan arithme- sition from one to the other is immediate, so long as it tical or a can be effected by an operation, which in one is indicated and not performed: thus the ratio of 4 to the 9 or self under symbolical form. √4 is identical with, though one ratio is symbolical √9 or algebraical, and the other arithmetical: the same transition may be likewise made, whenever there is a symbolical common measure of the terms of a ratio, which leaves, when removed, an arithmetical result: thus |