In Geometry, where there is no independent definition of ratio, it is the enunciation of this proposition which forms the definition of duplicate ratios. 415. "If there be four magnitudes in continued pro- Triplicate portion, the first shall have to the fourth the duplicate ratio in ratio of the first to the second." Geometry. then therefore X = a3 : b3 :: a : d. or 813 818 It is the enunciation of this proposition which forms the definition of triplicate ratio in Geometry. resolution 416. "If there be four magnitudes, a, b, c, d, which Composiare proportionals, and four others, a', b', c', d', which tion and are proportionals also, then their corresponding products or of proporquotients, tions. or aa', bb', cc', dd', 417. "If four quantities, a, b, c, d, be proportionals, then a", b", c", d", shall also be proportionals." 418. "If there be any number of ratios which are equal to each other, then as one antecedent is to its consequent, so shall all the antecedents together be to all the consequents together." Let a b c d e f, then also, a : b :: a + c + e b + d + f. The same demonstration may easily be extended to any number of ratios which are equal to each other. consequen definition of ways self of the geo 419. There are many propositions in the fifth book of Self evident Euclid, where hypotheses are made concerning ratios as ces of the greater and less than, as well as equal to, each other, and algebraical the circumstances under which they are so, which are either proportion self evident, or so nearly self evident, consequences of the are not alalgebraical definition of ratio, that it will not be neces- evident consary to notice them in this place, as distinct propositions sequences to be demonstrated: the case however is very different in a metrical desystem of Geometry, where the want of a definition of ratio, finition. as disconnected with the definition of proportion, makes it necessary to consider such properties of ratios, as much the objects of demonstration, as any of the properties of proportions or proportionals: thus when it is said that "of unequal magnitudes, the greater has a greater ratio to the same than the less has," it is required to bring this proposition within the operation of the definition of quantities which are not proportionals, and to shew that the conditions are such as coincide with the circumstances under which the first of four magnitudes is defined to have to the second a greater ratio than the third to the fourth, which is in this case identical with the second: if however we represent algebraically the same three magnitudes by a, b, and c, where a is greater than c, then a the first ratio is represented by and the second by с and it is a necessary consequence of the meaning attached to the term fraction, which is synonymous with that of ratio, that the first fraction under such circumstances is greater than the second: no formal demonstration could add to the evidence of such a proposition. The same observation would apply to the following proposition: "if there be six magnitudes, and if the first bears to the second a greater ratio than the third to the fourth, but the third does not bear to the fourth a less ratio than the fifth to the sixth: then the first bears to the second a greater ratio than the fifth to the sixth." If the six magnitudes be severally denoted by a, b, c, d, e, f, then the three ratios which are the objects of ately that is greater than وتر and no demonstration can b add to the evidence of such a conclusion: in the absence however of any geometrical mode of defining, independently of each other, the values of these ratios, it becomes requisite to shew that there are some equimultiples of the first and fifth, and also of the second and sixth, where the multiple of the first is greater than that of the second, but the multiple of the fifth not greater than that of the sixth a conclusion by no means self evident, but requiring the authority of a formal demonstration. CHAP. XII. General ThEORY OF SIMPLE ROOTS, WITH THE PRINCIPLES the terms 420. WE have before explained the meaning of the General term root in its ordinary acceptation, as the inverse of the meaning of term power, (Art. 13.): and have also investigated and root and exemplified, at considerable length, the rules for the deter-power. mination of the roots from their corresponding powers, both for numbers and for simple and compound algebraical expressions (Chap. VIII.); there are other consequences of those operations, which are due to the fundamental assumptions of Algebra, which are amongst the most important in the whole science of symbols, which we shall now proceed to consider: it will form however a useful, and in some respects a necessary, introduction to this subject, to consider and determine the extent of the meaning which can be properly given to the term root in Arithmetic, and in a system of arithmetical algebra, properly so called. the term root in 421. The term root, when applied to numbers or Meaning of numerical quantities, denotes as we have seen, the number or numerical quantity, whether admitting of finite Arithmetic. determination or not, which used as a factor in multiplication as often as its denomination indicates, produces the required number, whether fractional or not thus the square or quadratic root of 25 is 5: for 5 x 5 = 25: the square root of 5329 73 361 19 73 73 5329 ; X = the cubic or cube root of 1.030301 is 1.01; for 1.01 x 1.01 x 1.01 produces 1.030301; the square root of 2 is denoted symbolically by 2, there being no finite arithmetical |