KATIO. 



ItATIONALISM. 



Euclid (died from Dasypodius by Heibomius Mid Wallis) is of 

 .pmiou tliat n|Ai*4nt is used raW than the more natural word 

 mtrw, precisely that it may understood in a wider MOM, go as to 

 include fractional and incommensurable ratio*. That U, as WallU 

 expreMs it. kormirhfold is used instead of *MM*-*U. that * 

 may suggest the klea of a part of a time (commensurable or not) where 

 M.I' would only suggest that of an integer. We cannot much admire 

 thin refinement; nor doe* it give any help : for the introduction of 

 the idea of nrommnMrabUity numerically expressed, o aa to be tit for 

 arithmetical multiplication, would vitiate, or at least would trauimute, 

 Kudid's whole system of proportion. 



But the oldMt testimony, both to the existence of the definition, 

 and the meaning of the disputed word, U Eutocius, in hia commentary 

 on book ii. prop. 6 (of Torelli. 4 of preceding editor*). He here cites, 

 expressly from the elements, the definition as given ; and odds, as the 

 explanation of WTAIK>TIH, that it is the number which, by multiplica- 

 tion, turns the consequent into the antecedent This number, he says, 

 Kiv.w name to the ratio, and he cites Nicoinaclnu and Huron as under- 

 standing it in the same way. But. he goes on to my, the word is more 

 pr|>erly taken when this number is an integer. 



Leaving now out of view what Kuclid really did write, we shall 

 proceed to consider the subject'of composition of ratios, so as to supply 

 what, on any supposition, must be acknowledged to be wanted in the 

 elements. The notion of a ratio is easily and almost necessarily con- 

 nected with the idea of alteration I'M that ratio. We cannot express a 

 ratio without two magnitudes, the first of which, altered in the 

 ratio given, becomes the second. If we want to alter in the i 

 t to o,, this is easily done when the quantity to be altered is r ; fur 

 tli.-n the process is only writing Q instead of p. But if the quantity lie 

 A, then B must be found, so that A and B shall have the same ratio aa 

 p and >,-. 



If it be a numerical ratio which we consider, say that of 3 to S, 

 alteration of any number in that ratio implies that we change all it* 

 tltrrrt into fva, and any remaining fraction of three into the same 

 fraction of 'five. Alteration of any magnitude, say a length, in that 

 ratio implies that, choosing any length as a measure, we alter every 

 three such lengths which the given magnitude contains into five, and 

 every fraction ... three into the same fraction of five. This amounts 

 to changing the number or magnitude into five-thirds of what it was ; 

 and generaUy,alteration in the ratio of a to 6 (numbers) is nothing but 



multiplication by 

 a 



if x 



Take a magnitude A, alter it in the ratio of p to q : say that it then 

 becomes B ; that is, A U to B as p to q. Take the magnitude we left 

 off with, B, alter it in the ratio of E to s, making it c. Take c, alter it 

 in the ratio of V to w, making it D. Then at three processes, by three 

 successive alterations dictated by given ratios, we have changed A into 

 D, or have altered A in the ratio of A to D. Say that the ratio of A to 

 D is more simply expressed by that of H to >. Then, if we begin with 

 A, and alter it at once in the ratio of M to N, we change it into D, pro- 

 ducing tho same effect as if we hail successively altered in the ratios 

 of p to q, R to s, and v to w. Hence the ratio of M to N is properly 

 said to be compounded of the ratios of P to q, K to s, ami v to w.: 

 it dictates the alteration which will produce at once the effect of the 

 three alterations prescribed by the three other ratios. In like manner, 

 ,- t.. 'I'l'litim, that 10 is compounded of G ami 4 ; for addition 



, 



of 1 is equivalent to the addition of 6 and of 4. In multiplication 

 we nay that 24 is con>i>ounded of 6 and 4. And generally, the com- 

 pound should be defined as that which produces the united effect of all 

 the components, when both components and compound are used in the 

 tm way. Kuclid, vi. 23, is now more than a mure addition t to the 

 phraseology of geometry. The parallelograms A 11 c D and E F <; H [the 

 rradur may draw the figure for himself 1 being mutually equiangular, 

 it tell* us that if we take any magnitude and alter it in the ratio of 

 AC to iti, and then alter the result in tho ratio of AB to KF, 

 the change thus made at two steps might be made in one by 

 altering tie original magnitude in the ratio of the area A U C D to 

 thf area Kf H. 



This process applies equally to commensurable and iucuiun. 

 able ratios ; but in the former case of course the arithmetical substitute 

 {or composition of ratios is easy. We want to compound the ratios of 

 M to M and of a to 6, all four being integer numbers : it being known 

 that every commensurable ratio is expressible by the ratio of two 

 integer numbers. Take any magnitude p, and alter it in the ratio of 

 w to" H : it becomes n wtha of p. Alter this in the ratio of o to It : we have 

 tlifii Votus of M-mths of p, or Irtt-ouithx of r, which would also be obtained 

 by altering Pin the ratio of axMto6x n. Hence composition of nume- 

 rical ratios is performed by multiplication of the antecedent* for an 

 antecedent and of the consequents for a consequent. Tho process than is 



merely equivalent to that of the multiplication of fractions. If 



and were called the quautuplicities (irqAunfnrrcf ) of the ration, then 



a 



tin- quimuplicity of the compound ratio i, the product of the quantu- 

 - of the components, as in the definition (be it Euclid's or no) 

 which is found in the manuscripts of tin- dements. 



l>n/ilimtr ntli'j (ta\ufflur \iyot) him l>een diMiiied by Kuelid in the 

 m.inner bantnbeCoire given. Hut it is in faet the ratio an'sini; from 

 the composition of two equal ratios. Suppose i |,ound 



the ratio of P to q with the ratio of p to ij. Tak, ,,lc t,> 



begin with, which may as well be P itself : alter it in the ratio of p to 

 q ; it then becomes q. Alter q into R in the ratio of p to q ; t ! 

 let R be a third proportional to p and Q. Then r i.s changed into n at 

 these two steps, each involving an alteration in the ratio of p to q : 

 hence Kudid'a duplicate ratio ii the ratio compounded of 

 ratios ; aud, similarly, triplicate ratio (rpirXaaluy) is that compounded 

 of three equal ratios, and to 



The subduplieate, subtriplicat, sesqniplicate, to. ratio-, which 

 geometers used, completed that Language of multiplication and division % 

 applied to operations of power.- and roots which finally suggested the 

 idea of logarithms. [See also ADDITION OF RATIOS.] 



The pro)x>sitions requisite for the establishment of ' 

 compound ratio are contained in the fifth book. But in the r 

 use there is a manifest hiatus in the c ' of vi. 2J. It H 



supplied by a lemma added at the end of the proposition; which is 

 found in almost all the manuscripts (even in the Vatican i. 

 and Peyrard admits it accordingly). This is a pretty sure sign 

 Euclid did not give the lemma ; for he never refers to anything which 

 is to come after what he has in hand. Robert Simson omits thi< 

 lemma, and so leaves the proposition undemonstrated. What i- 

 wanted is the following : It is impossible that the same ratio should 

 have two Hub-duplicate ratios, or should be the duplicate ratio of two 

 different ratios ; or, if A be to B in the duplii 



also in the duplicate ratio of A to Y, then x and Y must be equal. If 

 possible, let them be unequal ; say that x is the greater : 



A X B 



A V B 



Then because x is greater than Y, the ratio of A to x is less than that 

 of A to Y. But the ratio of A to x is that of x to Ii ; and the ratio of A to v 

 is that of Y to B ; therefore the ratio of x to B is less than that of Y to 

 B. Therefore x is less than Y ; but it is also greater, which is absurd. 

 Consequently x and Y cannot be unequal, tc. By a continuation of i his 

 process it may easily be established that a given ratio can only ! 

 triplicate of one ratio, only the quadrnpUoaM of one, and so on. 



It is unnecessary to say anything on the decomposition of i 

 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 compositi 

 decomposition, and every decomposition a composition. Suppose that 

 P to q is the ratio compounded of A to B aud c to i\ and we v 

 return back again to the ratio of A to B. We must compound the 

 ratio of p to q with that of D to c ; for it is easily made obvious that 

 the ratios of c to I) 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 .-prim; 

 from multiplication inclusive, must be < 

 operations of composition, tc. Rob n, who, as we havi 



has left a demonstration of the sixth book absolutely unfinished, 

 though "Theon or gome unskilful commentator" had provided a 

 lemma which supplied what was wanting, has thought it n 

 add gome very complicated propositions on compound ratio ;.t, tin- 

 end of the fifth book. If they were intended as illustrations of the 

 great difficulty of rendering the m 



into geometrical language (and what else could have been meant it is 

 hard to imagine) the algebraical equivalents should ha>e b, , n intro- 

 duced. Take the proposition K, for instance, which it may safely be 

 ; ever fathomed. The following is an arithmetical 

 case of it. If 



_ 



j 



The, 



LI 



i L 



n 



o 



UATIOXAU A quantity, algebraic or arithmetical, U rational 

 when it can be expressed without the use of the sigu-ot e\,,liit,i,,ii, 

 such as those of tin- s pure root, cube root, &o. [ lim.vTio: At,.] 



KATIi >N AI.ISM i i a system of theology, which M n to be 



.lev, .loped iii Germany during the latter half of the la.-i century. It 

 arose in a great degree as a re-action against tin- prineipl * and the 

 method of biblical criticism promulgated by Voltaire, Volney, and the 

 r'rench Kncyelopa dists. Kant led the way, an.l hi .-\steni . 

 voured to preserve a medium between infidelity and superiutm 

 He was followed, more or less adopting his system, by S,cmlc.r, J. J>. 



