Dsplicate ratio (8orareafier AS-ren) has been defined by Euclid in the manner herelnbefore given. But it is in fact the ratio arising from the composition of two equal ratios. Suppose we want to compound the ratio of P to Q with the ratio of P to Q. Take a magnitude to begirt with, which may as well be r itself ; alter it in the ratio of r to Q; It then becomes Q. Alter Q into a in tho ratio of r to that Is, let n be a third proportional to P and Q. Then r is changed into It at these two steps, each Involving an alteration in the ratio of r to Q : hence Euclid's duplicate ratio is the ratio compounded of two equal ratios; and, similarly, triplicate ratio (rporAacricov) is that compounded of three equal ratios, and so on.
The sulaluplicate, subtriplinate, sesquiplicate, &c. ratios, which later geometers used, completed that Language of multiplication and division applied to operations of powers and roots which finally suggested the Idea of logarithms. [See also ADDITION OF RATIOS.] The propositions requisite for the establishment of the direct use of compound ratio are contained in the fifth book. But in the inverse use there is a manifest hiatus in the converse part of vi. 22. It Is supplied by n lemma added at the end of the proposition ; which is found in almost all the manuscripts (even in the Vatican manuscript, and Peyrard admits it accordingly). This is a pretty sure sign that Euclid did not give the lemma ; for he never refers to anything which is to come after what lie has in hand. Robert Simson omits this lemma, and so leaves the proposition undemonstrated. What is wanted is the following :—lt is impossible that the same ratio should have two sub-duplicate ratios, or should be the duplicate ratio of two different ratios ; or, if A be to 13 in the duplicate ratio of A to x, and also in the duplicate ratio of A to Y, then a and I' must be equal. If possible, let them be unequal ; say that x is the greater : A X B A Y B Then because x is greater than I', the ratio of A to a is less than that of A to T. But the ratio of A to x is that of x to n ; and the ratio of A to Y
is that of Y to B ; therefore the ratio of x to n is less than that of v to n. Therefore x is less than Y; but it is also greater, which is absurd. Consequently x and Y cannot be unequal, &c. Bye continuation of this process it may easily he established that a given ratio can only be the triplicate of one ratio, only the quadruplicate of one, and so on.
It is unnecessary to say anything on the decomposition of ratios. Clear as it becomes in arithmetic, after a while, that every multiplica tion is a division and every division a multiplication, it is much clearer from the beginning, in this subject, that every composition is a decomposition, and every decomposition a composition. Suppose that P to q is the ratio compounded of A to n and c to n, and wo wish to return back again to the ratio of A to B. We must compound the ratio of r to Q with that of D to c; for it is easily made obvious that the ratios of c to n and D to c compounded give the ratio of a magni tude to itself, the ratio of equality, the use of which effects no alteration.
It is now easy to see that all the operations of algebra which spring from multiplication inclusive, must be represented in geometry by operations of composition, &e. Robert Simson, who, an we have seen, has left a demonstration of the sixth book absolutely unfinished, though " Theon or some unskilful commentator" had provided a lemma which supplied what was wanting, has thought it necessary to add some very complicated propositions on compound ratio at the end of the fifth book. If they were intended as illustrations of the great difficulty of rendering the most common propositions of algebra into geometrical language (and what else could have been meant it is bard to imagine) the algebraical equivalents should have been intro duced. Take the proposition s, for instance, which it may safely be asserted no beginner ever fathomed. The following is an arithmetical case of it. If