PROPORTION. 



PROPORTION. 



H 



at but give a different distribution : that is, the correct distance, and 

 the correct distance only, satisfies all the conditions required by 

 Euclid's definition. 



Let us now, by the dittrilmtum of one set of magnitudes among 

 than of another set, agree to mean the placing of the first magnitude 

 among those of the second set, the latter having been previously 

 arranged in ascending order of magnitude. Thus, in the following 

 instance, we distribute the multiples of 3 among those of 8, the latte 

 being in Roman numerals : 



3 6 riii 9 12 15 xvi 18 21 xxiv 27 30 mii Ac. 

 24 



This use of the word distribution having been well learnt, the 

 fallowing way of stating the definition will be easier than that o 

 Euclid : " tour magnitudes A and B of one kind, and c and D of the 

 ame or the same other kind are proportional when all the multipl 

 of A are distributed among the multiples of B in the same intervals as 

 the corresponding multiples of c among those of D." Or, whatever 

 numbers maud n may be, if m A lie between >IB and (n + 1) B,m c lies 

 between * D and (a + 1) D. 



If the preceding test be always satisfied from and after any given 

 multiples of A and c, it must be true before those multiples. For 

 bMtam, let the test be always satisfied from and after 100 A and lOOo 

 and let 5 A and 5 c be instances for examination, falling before 100 i 

 I and 100 c. Take some multiple of 5 which will exceed 100, say 6( 

 times, and let it be found on examination that 250 A lies betwean 

 678 B and 679 B ; then 250 o lies between 678 D and 679 D. Divide 

 them by 60,and it follows that 5 A lies between 13MB and ISffiB, am 

 still more between 18s and 14 B. Similarly, 60 Ue between 18 ttD 

 and ISgD, and still more between 13 D and 14 D. Or 5 A lies in the 

 me interval among the multiples of B in which 5 o lies among the 

 multiples of D; and the same demonstration applies to any other 

 instance. 



Again, the test is also satisfied if the multiples of any multiple (m) 

 of A are distributed among the multiples of any multiple (M) of B in 

 the same manner as the multiples of m c among the multiples of n : 

 for instance, if the multiples of 3 A be distributed among those of 5 B 

 in the same manner as the multiples of 3 c among those of 5 D. Let 

 !1 A and 11 c be given for examination ; take any multiple of 3 greater 



th T JSL " , 3 ' L 9 'o and , cxarnino n ( A) and 11 (9 o), or 33 (3 A) 

 and 33 (3 c). Let 83 (3 A) he between 27 (5 B) and 28 (5 B) ; then by 

 hypothesis 33 (3 c) lies between 27 (5 D) and 28 (5 D). Divide all by 9 

 and we find that HA lies between 15 B and 15 JB, while lie lies 

 between ISpand 15D. Hence 11 A lies between 15s and 16s, while 

 3 in the same interval among the multiples of D ; and in the 

 same manner any other instance may be proved. 



The principles of the fifth book of Euclid are by many supposed to 

 be inevitably connected with the apparatus of straight lineYdrawn 

 parallel to one another, by which Euclid represents his magnitudes and 

 their multiples. This is not the case; and the simple expression of 

 magnitudes by large letters, and of numerical multiplier* by small 

 ones, will very much facilitate the demonstrations, without altering 

 anything but mere modes of expression. 



The next point to be considered is the infinite character of the 

 definition of proportion; four magnitudes are not to be called pro- 

 portional, until it is shown that every multiple of A foils in the same 

 al among the multiples of B, in which the same ^multiple of c is 

 Mind among the multiples of D. So that this definition is a negative 

 .ne, like that of parallel lines, which may be thus stated : two lines 

 el when every point of one of them, however far produced, is 

 on one side of the other. We might expect then to find that the test 

 is simple and positive, and an examination of the 

 istration already produced will confirm this. 



.Suppose that the distribution of the railings among the columns 

 should be found to agree in the model and the original as far aTthe 

 millionth railing. TlS proves, as we have seen, onlythat the raTling 

 dutance of the model does not err by the millionth part of the^orre 

 .ponding column-distance; for if it did err so much, the lultipuSn 

 >ramillion.fold would have placed the millionth railing (if 

 none before it) wrong by at least one interval It is then obvious that 

 .dividual cases, however extensive, will enable an 

 t the construction and its model to affirm the proportion or 

 " * : a" that it can do is to enable him to fix LitsVwhich 





point "f vi<<w, to ill-fine disproportion, and to make proportion consist 

 in the absence of all disproportion. Similarly, obvious as is the notion 

 of parallelism, and the connection of the non-intersection of two straight 

 lines with that of their always keeping the same distance with each 

 otlier, it is more easy to define this relation by the absence of all inter- 

 section than by any of its positive properties. 



The negative character of the definition of parallels does not prevent 

 it in. in being very easily proved that such lines exist ; and an exami- 

 nation of the first or last propositions of the sixth li-..k of Ku. lid will 

 show that the existence of proportional quantities is as easily esta- 

 blished on the definition given as on any other. To take an iii^- 

 however, in which nothing but lines shall be the objects of con 



shall here prove, In a different manner, the second proposition 

 of Kuclid's sixth book, or one to the same effect. 



Let o A B be a triangle, to one side of which a b is drawn parall. 

 in o A produced set off A A,, A, A,, Ac., equal to OA, and oo,, 0,0., Ac., 

 equal to oa. Through every one of the point* so obtained draw 

 parallels to A B, meeting o B produced in 6, b., B t , Ac. Then it it, easily 

 proved that 6 b v i,4,, Ac., are severally equal to o 6, and B B-, B.B Ac. 

 to OB. Consequently a distribution of the multiples of OAalnong 

 those of o a is made on one line, and of o B among those of o 6 on the 

 other. The examination of this distribution in all its extent (which 

 is impossible, and hence the apparent difficulty of using the dcfin 

 is rendered unnecessary by the known property of parallel line*, 

 since A a lies between <, and a,, B. must lie between k, and o ; for if 

 not, the line A.,BJ would cut either a s i, or 0.6 . Hence, without 

 inquiring where A does fall, we know that if it fall K-tue,n a. ;,,,d 

 a.+i, B. must fall between b, and 6. +1 : or if m x o A fall in i 

 tude between n x o a and (n + 1) x o a, then m x o B must fall between 

 n x o 6 and (n + 1) x o b. Thus it is established that OAistooaasoB 

 to 06. 



The propositions of the fifth book become very simple when the. 

 definition is fully elucidated, and symbolic expression is substitut 

 the words at length of Euclid. They will be found thus treat 

 Playfair's or Lardner's editions of Euclid, and in the ' Connect 

 Number and Magnitude ' (London, 1836). 



When quantities are commensurable, a multiple of one may be found 

 which is exactly equal to a multiple of the other : thus if A=3\B, 

 13A = 43B. In this case the arithmetical definition of proportion is 

 sufficient, and the other may be shown to follow from it Let A= 3An, 

 and o = 8AD, so that, arithmetically speaking, A is to B as c to D. Let 

 TO A lie between n B and (n + 1) B, or (3& m) x B lies between n B and 

 ( + 1)B. Then the number or fraction 3^ m must lie between n and 

 n + 1 ; whence (3^m)D lies betweennDand(n + l)D,ormc lies between 

 n D and (n + 1) D. 



It is however perfectly allowable to leave out of sight the. possible 

 case in which a multiple of A is exactly equal to a multiple of B ; since 

 if the test be true in all other cases, it is therefore true in this, 

 if possible, let 4A = ?B, and 4c be (say) greater than 7D. Then 

 m (4 c) exceeds m (7 D) by m times this difference, which may be made 

 as great as we please, or 4 m o, and multiples succeeding it, may bo 

 made to fall in an interval as many intervals removed from that of 

 7mD and (7m + I)D as we please. But 4 m A ia eqiml to 7 m B, whence 

 (4 m + 1) A, Ac., must fall among the multiples of B in intervals of given 

 nearness * to the interval of 1ms and (7 m+ 1) B. Consequently the 

 multiples of A following 4mA cannot always fall among the multiples 

 of B in the same intervals as the same multiples of c among those of D ; 

 and the rest of the test cannot be true, unless 4 c = 7u ; that is, if the rest 

 of the test bo true, then 4 = 7 D. 



The following question will enable the reader to see for himself how 

 :ar he is able to apply the method of Euclid : Returning to the 

 ;ration, suppose that the columns, instead of being mathematical lines 

 are of a given thickness, and that the columns in the model ai 

 >roportionate thickness ; let it also be supposed that when a railing is 

 >rojected upon the column, there are no means of determining on 

 vhich side of the axis of the column it falls. It is to be shown that 

 f the distribution be according to the definition as to every railing 

 which is not so projected, and about which there is therefore no 

 doubt, it must also be true as to every case in which the doubt exists. 



The advantages of the study of proportion in the manner laid down 

 by Euclid, or some other equivalent in extent and strictness, are pre- 

 cisely the advantages of accuracy over inaccuracy, and of real demon- 

 stration over a false and slovenly appearance of it, which, though a 

 :loso resemblance, is therefore all the more dangerous. And it - 

 M remembered what wo mean by demonstration, namely, the process 

 of obtaining a conclusion by sound logic from premises known to bo 

 If the prolixity of Euclid's method could be avoided l>y an 

 juumption, we should not object, provided the assumption wore both 

 true and easily seen to bo true. For instance, if the theory of PAK.VL- 

 LKL8 could Iw established without its axiom by means of a htn. 



We lean the render to put this demontrtion into more exact form. 



