RATIO. 



RATIO. 



94!i 



Let D and s be two incommensurable magnitudes : how are we to 

 describe their relative magnitudes ? That they have a definite relation 

 is certain ; suppose, for precision, that s is the side of a square, and D 

 its diagonal ; any alteration of D, or any error in D, s being given, would 

 make the figure cease to be a square. There are many mathematical 

 notions in which accuracy is not attainable in finite terms, but is the 

 limit towards which we approach when number or magnitude, as the 

 case may be, is increased or diminished without limit. In the present 

 case the expression of ratio or relative magnitude, which is not accu- 

 rately attainable by one or more relations, can be continually amended 

 by adding one or more relations, until the inaccuracy of the mode of 

 expression is rendered as small as we please ; in such a case, accuracy 

 must be imagined to reside in the supposition of an infinite number of 

 given, or at least of attainable, relations. 



To explain our meaning, suppose that the person whom we address 

 is altogether ignorant of the relative magnitude of the diagonal D and 

 the side s. He asks for a relation, and knowing the mode of dealing 

 with the ratios of comrnensurables, naturally desires to know how many 

 diagonals make an exact number of sides. If we could answer this 

 question, if for instance we could say that 100 diagonals make 142 sides 

 exactly, the question would be settled : for an arithmetical rule would 

 always deduce the diagonal when the side is given. But we are obliged 

 to reply, that no number of diagonals whatsoever will make an exact 

 number of sides. He then asks how he is to form a perfect conception 

 of the diagonal ; we answer by placing two equal sides at right angles 

 to one another, and joining the extremities. This, he replies, and 

 properly, is not a mode of finding the relative magnitude, which is 

 something connected with magnitudes only, and that the permission 

 given by Euclid to join two given or determined points is not any real 

 determination of the included length. We then tell him that it ia at 

 his pleasure to name a fraction of the side, and we can express the 

 diagonal with an error not so great as that fraction ; he names, say one- 

 millionth of the side, and we give him the promised information in 

 tolling him that 1,000,000 of diagonals exceed 1,414,213 sides, but fall 

 short of 1,414,214 sides. The consequence is, that the diagonal lies 

 between V414213 and 1-414214 of the side : these differ from one 

 another by one-millionth of the side, and the error of the diagonal is 

 of course less. If he should ask how he is to carry this process yet 

 further for himself, we give him the arithmetical symbol \/ 2, and 

 instruct him how to perform the arithmetical operation of approxima- 

 ting to its value. In this we show him how to find between what 

 number of sides any number of diagonals lies ; and hi so doing we give 

 the ratio of the diagonal to the side, so far as the nature of the case 

 will admit of its expression. 



The relative magnitude, then, of two magnitudes is given, when the 



place of any multiple whatsoever of the one among the multiples of 



the other can be found from the data. For example, we carry on the 



'< icale of the aide and diagonal of a square, in the power of 



extending which ad injinitum lies that of expressing the ratio, so far 



as expression is possible, and of absolutely comparing the ratio with 



. ", in as accurate a manner as if expression had been perfect. 



, D, 2s, 2n, 3s, 4s, 3D, 5s, 4D, 6s, 7s, So, 8s, 60, 9s, ID, 10s, 11s, 

 80, 12s, 9o, 13s, 14s, 10o, 15s, llo, 16s, 12D, 17s, 18s, iyi>, 

 19s, 14o, 20s, 21s, 16o, &c. 



In this table we see, for instance, that 10 diagonals are more than 

 1 4 and less than 15 sides, and so on. The only doubt that can possibly 

 rrni.iin may be thus expressed :--Is the preceding scale a property of 

 tin- diagonal and of nothing else > May there not be a length so near 

 to the diagonal that its multiples shall never fall out of the same 

 intervals as those of the diagonal ? Let K be a given quantity, no 

 matter how small ; we say that it is impossible that all the multiples 

 of D K can lie in the same places among the multiples of s as the 

 multiples of D. Take m times both ; then m (D + K) and BID differ by 

 mK. Now however small K may be, it is possible to take m so great 

 that mK shall exceed s, or any multiple of s previously named : whence 

 the thing asserted is evident. The definition of the ratio of s to D 

 lies, then, in this scale ; or rather, whatever the definition may be, the 

 mode of finding all relations between s and D lies in the formation of 

 this scale so far as may be necessary for the purpose in hand. The 

 definition r>f proportion is then contained in sameness of multiple 

 scale* ; that is, i> is to 8 as A to B, when any multiple whatever of D is 

 contained between the same two multiples of s, that the same multiple 

 of A is cont ained between those of B. We here come to the subject of 

 PROPORTION, which the reader should now consult as a continuation 

 of the pres ;nt one. 



It is wei 1 known that the word \6\os came to mean fraction, the 

 expression of commensurable ratio. Fractional arithmetic was called 

 loffittic arithmetic even down to the 17th century. It is not so well 

 known th it this use of the word is as old as the time of Euclid. 

 Aristotle ;ives, as his instances of discontinuous quantity, ipiQuis KO! 

 \iyot, nu; nber and fraction. The logicians have always taken this to 

 mean nun iber and tjieech, some transcriber having interpolated a sub- 

 sequent passage to this effect in the text of Aristotle. Plato tells us 

 that the Egyptian Theuth invented ipiOjuiirf icol \oytantis, meaning, 

 no doubt, integer and fractional reckoning. The idea implied in com- 

 position of ratio is very imperfectly treated in Euclid : and yet upon 

 AXD SCI. DIV. VOL. VI. 



the correct understanding of it depends whether the boasted victory 

 over the difficulties of incommensurables which the fifth book gives is 

 real or imaginary. 



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 confusion 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 (cri/yKevVos) of ratios in the 

 fifth book ; the word translated composition (amBis iris) refers to such 

 a process as the formation of the ratio of A + B to B from that of A to B. 

 But the definitions of duplicate, triplicate, &c. ratio, are laid down ; which, 

 as we shall see, are particular cases of compound ratio. These defini- 

 tions are as follows : if A, B, c, D, &c., be in continued proportion, so 

 that as A to B, so B to c, o to D, and so on, then the ratio of A to c is 

 called the duplicate ratio of that of A to B, 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, L, M, we are to say the ratio of K 

 to M is compounded of the ratios of K to L and of L to M. And further 

 that if A be to B as K to L, and v to w as L to M, then the ratio K to M 

 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 

 TT7jAiK<jT7rr of the ratios multiplied together make a certain [ratio]." 

 On the word left untranslated (which, we believe, must be translated by 

 'f'licitifs), 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 

 larentheses 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 7x6 to 4x11; or possibly, that, 

 expressing the above ratios as those of J to 1, and T r to 1, the com- 

 pounded ratio ia that of J x ft 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 / to h is produced from those 

 of / to y and </ 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 ir>jAiic<iT7|s as \ve have done. And we find that Wallis was the 

 person who suggested to Gregory cjitanlitplicitas instead of quantitas 

 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 (Project, 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 

 agree entirely with Savile. The word TTJAII!T?)S 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 



4ai/T TOXAoirXao-iofffleio-ai iroiuffl Tim. TUe Scholium presently cited K ivi s 

 it with \frjov for T iW in one place, and TTT;\IK((T)TO \6yov in another. 



3 I- 



