The word sreauderas has been translated "quantity," by many editors, which makes nonsense of the whole ; for magnitude has hardly a different meaning from quantity, and a relation of magnitudes with respect to quantity may give clear ideas to those who want a word to convey a notion of architecture with respect to building, or of battles with respect to fighting ; and to no others. Wallis, we believe, restored the true meaning of the word, and was followed by Gregory, as seen above : and Euclid himself, or some very old commentator, in another place, shows in what sense he used it. In the fifth definition of the sixth book (omitted by many editors), he says that a ratio is com pounded of two other ratios when the wnAmortrrf r of the latter multiplied (weAMorNacrialreficrai) together, make the former. Now, this would be unmeaning if the Greek word meant simply quantities, unless they were quantities represented by numbers (though Gregory has here forgotten his own previous correction, and writes quantitaa instead of quantuplicitas). The lexicographers generally give " quan titas ;" but they are not adepts in the mathematical use of terms implying relations of magnitude.
The first and rough notion of ratio being thus given, we may find a synonymo for the word in the more intelligible term relative magnitude. Six feet, though greater than three feetos, relatively to four feet, a less magnitude than three feet is, relatively to one foot : the number of times which six feet contains four feet is less than the number of times which three feet contains one foot. The relative magnitude of six to four is less than the relative magnitude of three to one; or the ratio of six to four is less than the ratio of three to one.
Given two magnitudes, how are we to find the means of expressing the first in terms of the second I Euclid answers this question, when it can be answered, in the tenth book, by giving the rule for finding the greatest common measure of two magnitudes, in which he employs a process exactly the same as that of the arithmetical rule in common use. Of two magnitudes which are multiples of the same third magnitude, each is a definite arithmetical fraction of the other.
But it may happen that the magnitudes have no common measure [Iscomarmisunamx], in which case the means of expression would fail. We can describe the diagonal of a square as a part of a certain figure, and the description is perfect ; but if we attempted a description secundum quantuplicitatem, we should never succeed ; for no possible line exists of which it can be said that the diagonal of a square and its side both contain that line exactly. Such quantities are called by Euclid eatrya, irrational, or leaving no ratio; and in the primitive meaning of the term this is correct, for there is no quantuplicitative mode of expressing one by the other. lint the term ratio, both in Euclid and all other writers, immediately acquires another sense ; and it is this new sense in which we proceed to speak of ratio. Since the
relative magnitude of two quantities is always shown by the quantu plicitative mode of expression, when that is possible, and since propor tional quantities (pairs which have the same relative magnitude) are pairs which have the same mode (if possible) of expression by means of each other ; in all such cases sameness of relative magnitude leads to sameness of mode of expression ; or proportion is sameness of ratios (in the primitive sense). But sameness of relative magnitude may exist where quantuplicitative expression is impossible ; thus the diagonal of a larger square is the same compared with its side as the diagonal of a smaller square compared with its side. It is an easy transition to speak of sameness of ratio even in this ease ; that is, to use the term ratio in the sense of relative magnitude, that word having originally only a reference to the mode of expressing relative magnitude, in cases which allow of a particular mode of expression. The word irrational does not snake any corresponding change, but continues to have its primitive meaning, namely, incapable of quantuplieitative expression. And it is worth noting that this of itself shows that the original meaning of Ahos referred to expression, not to the thing expressed; for liktrya (not having a ratio) would have been absurd as applied to incommensurable quantities, if the primitive mathematical meaning of the first word had coincided with its modern one.
The idea of relative magnitude is one which strikes us in all cases in which wo compare the parts of an original with the corresponding parts of any model or imitation. It does not closely connect itself with any mode of expression or measurement : if a part of the model were only in a slight degree too large or too small, the detection of the error might require a formal measurement, but anything which is very much out would be rejected by ono glance of the eye. Let us suppose now that the formal measurement is attempted. The first and simplest notions of relative magnitude are gained from repetition ; and the ideas of two, three, four, &c., originally used in their simple cumulative sense, soon become the representatives of those simple relative magni tudes which are suggested by pairs in which one is quantuple of the other. The next step is to those magnitudes in which neither is quantuplo of the other, but both are quantuple of a third : from which wo learn how, admitting aliquot parts, to extend the mode of expression. Thus, of the magnitudes 10 It and 7 n, wo see that every relation of quantuplieity can be derived from the simple numbers 0 and 7 : the first number is If of the Becloud, a mode of expression which equally applies to the magnitudes 10 it and 7 is.