In every matter connected with elementary geometry, confusion may and often does arise from mixing together criticisms of two different kinds ; on Euclid as a writer, and on the subject as a matter of thought. To avoid such coufusion in reference to composition 'of ratio, we shall begin with the consideration of what we find in Euclid, —not in Simson's Euclid, nor Playfair's Euclid—but in Euclid of Alexandria.
There is nothing on ratio compounded (crvyseadvos) of ratios in the fifth book; the word translated composition (ativOcats) refers to such a process as the formation of the ratio of A +n to a from that of A to n.
But the definitions of duplicate,triplieate,&c. ratio, are laid down ; which, as we shall see, are particular cases of compound ratio. These defini tions are as follows :—if a, n, c, &c., be in continued proportion, so that as A to B, so B to c, o to n, and so on, then the ratio of A to o is called the duplicate ratio of that of A to n, the ratio of A to D is called the triplicate ratio of A to B, and so. on.
In one proposition, and in one only, is the, phrase composition of ratios used : in the 23rd of the sixth book, where it is said " Equi angular parallelograms have to one another the ratio compounded of the sides." There is no definition (at least, it is now so supposed ; here however we must refer to our next paragraph) given of the words in italics, and on looking into the demonstration of the proposition, we find we must assume, as a matter of phraseology merely, that of any three quantities of the same kind, K, a, ar, we are to say the ratio of K to sr is compounded of the ratios of K to a and of L to at. And further that is said to be the ratio compounded of the ratios of A to B and of v to w. If there be anything more than mere phraseology in this, it must be because Euclid makes a tacit reference to some arithmetical system current in his time.
It is true that there is found in a great preponderance of manu scripts (in all, we believe) a definition of compound ratio. It is among the definitions of the sixth book, and literally translated* is as follows :—" A ratio is said to be compounded of ratios, when the nniticornTES of the ratios multiplied together make a certain [ratio]." On the word left untranslated (which, we believe, must be translated by gemntoplicitks), we refer to what precedes and to what follows. This de finition is admitted into the editions of Basel and Oxford, and into Briggs's edition of the six books. Peyrard has omitted it in the Paris edition, because, in his celebrated Vatican manuscript, it is not in the text, but has been added at the side. The Berlin editor admits it in parentheses as a disputable passage. Set a scholar to make the text of Euclid from the ordinary mode of weighing the evidence of manu scripts, and there is no doubt this definition must appear as a part of the elements. Set a geometrical reasoner to settle the question by the internal evidence of the passage and its keeping with the rest of the book, and there is as little doubt that it would be rejected. The
meaning of the passage is, apparently, that if two ratios be expressed numerically, as those of 7 to 4 and 6 to 11, the ratio compounded of those ratios is to be the ratio of 7 x 6 to 4 x 11; or possibly, that, expressing the above ratios as those of 1 to 1, and to 1, the com pounded ratio is that of x to 1.
In the early translations from the Arabic, the definition is omitted, and reference is made, in demonstrating vi. 23, to a note inserted among the definitions of the fifth book, which is very insufficient. But the phrase there is that the ratio of f to h is produced from those of f to g and g to h: and to the definitions of the seventh book several are added, one of which is, that in a series of numbers the ratio of the first to the last is produced from the successive ratios of each to the one following.
In many manuscripts there is a scholium preceding the sixth book, which August, the Berlin editor, though not admitting it into Euclid, thinks must be of high antiquity; in which we fully agree with him.
It is to be found in the Basel edition, and in the notes to the Berlin. This scholium, 'while it gives confirmation to the preceding view (which hardly wants it), takes the same side on the meaning of the word wsioolTar as we have done. And we find that Wallis was the person who suggested to Gregory guantuplicitcts instead of guantitas as the translation. See his discussion of this point at length in his English Algebra (1684), ch. 19 and 20; revised in his Latin Algebra (Works, vol. ii. ch. 19, 20), and again at p. 665 of the same volume, where there is a defence of this definition against Henry Savile, who (Prwlect. in Eucl.) had considered it as a great defect. To the text of Euclid we have only further to say that this consent of Savile, Wallis, and Gregory, as to the genuineness of the definition in question, is of great weight. But with regard to the matter of the definition we I agree entirely with Savile. The word ssioaSvas needs definition quite as much as the phrase composition of ratios itself. This definition, it will be observed, either restricts the composition to ratios which are of commensurable magnitudes, or implies and assumes the multiplica tion of two interminable decimal fractions. An old scholiast on Euclid (cited from Da.sypodium by Meibernins and Wallis) is of opinion that ingustkijr IS used rather than the more natural word TOChlif, precisely that it may be understood in a wider sense, so as to Include fractional and incommensurable ratios. That is, as Wallis expresses it, is used instead of hoicsntan,-foll, that snuck may suggest the Idea of a part of a time (commensurable or not) where easy would only suggest that of an integer. We cannot much admire this refinement; nor does It giro any help : for the introduction of the idea of incommensurability numerically expressed, no as to be fit fur arithmetical multiplication, would vitiate, or at least would transmute, Euclid"s whole system of proportion.