Surds of whose bases natural 254. If we write down a series of the nth roots of the na- the same tural numbers, all of them will be surds except those whose order, bases are perfect nth powers: and all those surds will be irre- are the ducible to a common surd, unless their bases have a common numbers, factor, multiplied by perfect nth powers. In a similar manner, or surds of the same if we write down the series of successive roots of a number a base, the from the quadratic downwards, all those descending roots will denominabe surds unless a be resolvible into two or more equal factors, whose in which case there will be as many whole numbers in the series, the natural as there are different ways in which is a resolvible into equal numbers. factors. tion of orders are of surds indices denomina 255. Again all powers of a surd will be surds except those All powers whose denominations are equal to or multiples of, the denomination whose of the root: thus the successive powers of "/a, will form a series are less Ja, Ja3, "\/a3, "a"-1 a, a*/a, a "fa", ......, the first n-1 of than the which are surds irreducible with each other, and all the other tion of the terms of the series are whole numbers or multiples of the first root are (n−1) terms: and if we could conceive the existence of a root surds. whose denomination was an incommensurable number, such as we shall afterwards be called upon to consider, then all its powers, however far continued, will be surds irreducible with each other. irreducible tions in the theory. 256. The theory of surds, under the form in which we have Imperfecjust presented it, is necessarily very imperfect, in consequence preceding of our not being able to avail ourselves of many propositions in algebra, which are essential to its more complete developement: those properties of them however which we have noticed in the preceding articles will be sufficient to point out their general relation to each other, and also the extent to which they are capable of being compared with each other by means of the ordinary processes of arithmetic. U CHAPTER IV. The term ratio. In what manner re ON RATIOS AND PROPORTIONS. 257. 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 forces, of two periods of time, and of any other concrete quantities of the same kind, the relation of whose magnitude to each other admits of being estimated or conceived. 258. A ratio (the term is here used absolutely) consists of presented. two terms or members, which are denominated the antecedent and the consequent, from the order of their position: it may be 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 3: 5. Geometrical representation of ratio. No geometrical definition of ratio properly so called. In a similar manner, if a and b denoted any other two numbers, lines or other magnitudes of the same kind, their ratio, whatever meaning it may possess or receive, would be denoted by a b. 259. Such a mode of representing a ratio, merely exhibits its terms to the eye, in a certain order, 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. Whatever modes, however, we may adopt in geometry for the representation of ratios, they must all of them be equally arbitrary and independent of each other: for there is, properly speaking, no definition of ratio in geometry, by which the equivalence of different modes of representation may be ascertained as necessary consequences of it: for ratio is said to be (Euclid, Book V. Def. 3.) the mutual relation of two magnitudes of the same kind to one another, with respect to quantity, a description of its meaning much too vague and general to be considered as a proper definition, inasmuch as it cannot be made the foundation of any propositions respecting it. It is for this reason that ratios in geometry are only considered when placed in connection with each other, as constituting or not constituting a proportion. 260. A little examination however of some of the conditions Popular which ratios, taken according to the popular usage of the term, the term meaning of must satisfy, will lead to an arithmetical mode of representing ratio. them, by which their absolute magnitude may be ascertained, and which will thus conduct us to an arithmetical 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. may un 261. Thus, if two lines admitted of resolution into 3 and Changes which the 5 parts respectively, which were equal to each other, the lines terms of the themselves might be correctly represented by the numbers 3 and same ratio 5, and their ratio therefore by 3 5. But their common primary dergo. unit is itself divisible into 2, 3 or m equal parts, and the numbers of these successive parts which the original lines, under such circumstances would contain, would be severally 6 and 10, 9 and 15, 3 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, and 3m : 5m. 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 subordinate 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. Conclusions thence deduced. A ratio Arithmetical definition of ratio. Includes geometrical as well as other magnitudes. 262. 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 only, which form their terms and are the same, whatever be the nature and magnitude of the concrete unit of which those numbers may be respectively composed. (4) The ratios of two magnitudes of the same kind, which have no common measure with each other, and which are therefore incommensurable with each other, may be approximately represented by such numbers of common units of those magnitudes as approximate to them in value. 263. All these conditions will be fully satisfied, if we agree to denote a ratio by means of a fraction, of which the antecedent is the numerator, and the consequent the denominator: for the value of this fraction is determined 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. ratio In arithmetic, therefore, and also in arithmetical algebra, a may be defined, as the fraction whose numerator is the antecedent, and denominator is the consequent of the ratio. It will follow, therefore, that both in Arithmetic and Arithmetical Algebra, the theory of ratios will be identified with the theory of fractions. 264. The symbols of arithmetic represent geometrical as well as other quantities, and the lines, areas and solids of geo • 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. in geome metry, are thus brought within the range of this definition: it must be kept in mind however, that it is only by considering geometry as thus connected with arithmetic, 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 another, or the result of such a division: Reason why there for this result can bear no analogy to the quantities which pro- is no definiduce it, being essentially numerical and consequently not capable tion of ratio of being represented by a line, unless in a symbolical sense, which try. under such 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 the symbols of arithmetic or algebra. 265. We shall now proceed to the statement of some of the more common propositions concerning ratios, which, though merely properties of arithmetical and other fractions, require, from custom, the use of a new and peculiar phraseology, and are connected with the formation of some important theories. "Ratios are compared with each other, by comparing the Ratios: fractions by which they are denoted." Thus, the ratios of 3 to 5 and of 5 to 8, are denoted by the 5 fractions and : these are identical with the fractions 25 3 5 24 40 and (Art. 127.): it is the second of these ratios, therefore, which is the greater of the two. how com pared. greater 266. A ratio of greater inequality is one, whose antecedent Ratios of is greater than its consequent: a ratio of less inequality is one, or less whose antecedent is less than its consequent: a ratio of equality inequality. is one, whose antecedent is equal to its consequent: the first corresponds to an improper fraction, the second to a proper fraction, and the third to unity. The following proposition, connected with ratios which are thus denominated, is frequently used. |