805 



PROPORTION. 



PROPORTION. 



600 



minutest falsehood ; whereas, on examining it with a powerful micro- 

 scope, we find the so-called sharp edge capable of being magnified into 

 a plane of any dimensions, though it may appear a sharp edge to the 

 Unassisted sensfs. 



The imperfection of the definite arithmetical definition of proportion 

 is universally admitted, while the complexity of the rigorous definition 

 by which Euclid supplied its place is almost as universally felt to be a 

 grievance. Many attempts have been made to avoid the trouble 

 without incurring the reproach of inaccuracy. One or two of these 

 we shall notice. 



Legeudre, in his otherwise excellent work on geometry, refers the 

 student to works on arithmetic for the theory of proportion ; and, 

 having stated that when A is to B as o to E, it is known that 

 A x D= B x 0, adds (twelfth edition, page 61), " This is certainly true 

 for numbers ; it is true also for any magnitudes, provided they can 

 be expressed, or that we imagine them expressed by numbers, and 

 this we may always suppose." A system of geometry which tells the 

 learner in so many words that he may always suppose that which is 

 not true, needs no further comment, even though Legendre were its 

 author. It is true that in subsequent parts of the work we find 

 demonstrations adapted to the case of incommensurable quantities, 

 but they want that moat important element of a proposition involving 

 proportions, namely, a definition of what the term means; these 

 demonstrations turn upon the theorem that when four quantities are 

 proportional, the first is greater than, equal to, or less than the second, 

 according as the third is greater than, equal to, or less than the fourth ; 

 but it has not been previously stated what the author means by four 

 quantities being proportional. In the English translation of the pre- 

 ceding work, a preliminary chapter is added on proportion, in which 

 the definition given as to incommensurable magnitudes amounts to the 

 following : when A and B are incommensurable, and also and D, the 

 four are said to be proportional when A' and o' can be found, as near 

 as we please to A and c, and which, being commensurable with B and 

 D, are proportional to them, in the arithmetical sense of the term. 

 This is a sufficient definition ; but it really amounts to that of Euclid 

 (as do all sufficient definitions which we have seen), and is not so 

 easily used. 



M. Lacroix (' Siemens de Ge'ome'trie,' p. 6) makes the approximate 

 finding of a common measure stand in place of an exact process, and, 

 fairly stating that the error of the process may be made too small to be 

 visible, rests the exactness of his geometry on its not being sensibly 

 erroneous. 



The author of the ' Elements of Geometry,' in the ' Library of 

 Useful Knowledge,' states the proportion of incommensurable mag- 

 nitudes to consist in " their ratios admitting of being approximately 

 represented by the same numbers, to how great an extent toner the 

 degree of approximation may be carried." In virtue of the words in 

 italics, this definition may be considered as being, when properly used, 

 capable of forming the basis of an exact theory ; and that it is properly 

 used in the work cited we fully admit, since its first application is to 

 the establishment of the definition of Euclid. The only objection we 

 should make to the work in question is that its expressions (page 48) 

 would lead the student to imply that cotnmensurabUity is the general 

 rule, and incommensurability the exception ; an extended theory is 

 given, because magnitudes are not always commensurable. Now it is 

 important the student should know, and should always bear in mind, 

 that of two magnitudes of the same kind taken at hazard, or one being 

 given and the other deduced by a geometrical construction, it is very 

 much more likely that the two should be incommensurable than that 

 they should be commensurable. 80 that the apparently cumbrous 

 theory of proportion is not introduced to meet a few cases which 

 sometimes occur, but to prevent the majority of instances from being 

 treated incorrectly. 



The definition* of proportional quantities given by Euclid is as 

 follows : " Magnitudes are said to have the same ratio to one another 

 the first to the second, and the third to the fourth, when equimultiples 

 of the first and third, and equimultiples of the second and fourth 

 whatever the multiplications may be, yield a multiple of the first 

 greater than, equal to, or less than, that of the second, according as the 

 multiple of the third is greater than, equal to, or less than, that of the 

 fourth." That is, if A, B, C, and D be the four magnitudes, and m anc 

 n any two whole numbers whatsoever, m A must be greater than, equa 

 to, or less than n B, according OB m o is greater than, equal to, or less 

 than n D. Otherwise thus : whatever whole numbers m and may be 

 A must exceed, equal, or fall short of n-mths of B, according as c 

 exceeds, equals, or falls short of n-Mths of D. A person practised in 

 algebra would comprehend the definition most easily when stated thus 

 m A n B must have the same sign as m o n o, for all whole values o 

 m and n. 



This definition equally applies whether A and B be commensurable 

 or incommensurable, since no attempt is made to measure B by an 



The Greek of Euclid's definition ia very short, and can only be intelllgibl 

 rendered Into English by circumlocution. 'Ev T$ ai/r$ \6ycp nfytQij Kiyrrai 

 tlmi, ffttnav n-pot ttfatfoti Kol Tflrov irpis TfTaprov, STO.V ri TOV ir 

 (cal TpiTou icrkxit woAAairAciffHi TO>I> roi Sfuripau rtoi Ttrdprov l 

 oAAairAair(<i>, Kttff tmoiovom iroAAairAa<7ia<rjitoi', iKUTfpov tl fi/xa 



aliquot part of A. The two questions which must be asked, and satis- 

 actorily answered, previously to its reception, are as follows : 



1. What right had Euclid, or any one else, to expect that the 

 jrecediug most prolix and unwieldy statement should be received by 

 he beginner as the definition of a relation the perception of which is 

 me of the most common acts of his mind, since it is performed on 



very occasion where similarity or dissimilarity of figure is looked for 

 ir presents itself ? 



2. If the preceding question should be clearly answered, how can 

 he definition of proportion ever be used ; or how is it possible to com- 

 lare every one of the infinite number of multiples of A with every one 

 if the multiples of B ? 



To the first question we reply, that not only is the test proposed by 

 Cuclid tolerably simple, when more closely examined, but that it is, or 



might be made to appear, an easy and natural consequence of those 

 undamental perceptions with which it may at first seem difficult to 

 icmpare it. To elucidate this, suppose the following case : 



There is a straight colonnade composed of columns at equal distances 

 rom each other, the first being distant from a bounding wall by a 

 ength equal to the distance between any two successive columns. In 

 ront of the colonnade let there be a row of railings * equidistant from 



each other, the first being at the same distance from the wall at which 

 he railings are from each other. Let the columns be numbered from 

 -he wall, and also the railings ; remember also that it is not supposed 



that there goes any exact number of railings to the interval of two 



columns, but that the interval of the columns may be to the interval 

 if the railings in any ratio, commensurable or incommensurable. 



12 a 4 io 7 8 a 10 11 12 13 H 15 10 



If we may suppose this construction carried on to any extent, it is 

 easily shown that a spectator, by mere inspection, without measure- 

 ment, may compare the column-distance (o) with the railing-distance 

 [R), to any degree of accuracy. For example, since the tenth railing 

 falls between the fourth and fifth columns, it follows that 10 K is 

 jreater than 4 c and less than 5 o, or that B lies between ^ of c and A 

 of c. To get a more accurate notion, he may examine the 10,000th 

 railing : if it fall between the 4674th and 4U75th columns, it follows 

 that 10,000 B lies between 4674 o and 4675 o, or B lies between ;& 

 and jjjgJSj of o. There is no limit to the degree of accuracy thus obtain- 

 able ; and it can also be shown that the ratio of c and R is determined 

 when the order of distribution of the railings among the columns is 

 assigned ad injinitum ; or, which is the same thing, when the position 

 of any given railing can be found, as to the numbers' of the columns 

 between which it lies. Any alteration, however small, in the place of 

 the first railing, must at last affect the order of distribution. Suppose, 

 for instance, that the first railing is moved farther from the wall by 

 one part in a thousand of the distance between the columns, the 

 second railing must then be pushed forward twice as much, the third 

 three times as much, and so on : those after the thousandth are 

 pushed forward more than a thousand times as much, that is, bv 

 more than the interval between the columns, or the order with 

 respect to the columns is disarranged. 



Let it now be proposed to make a model of thtf preceding con- 

 struction, in which c shall be the distance between the columns, and r 

 that between the railings. It needs no definition of proportion, nor any- 

 thing more than the conception which we have of that term prior to 

 definition (and with which we must show the agreement of any 

 definition we may adopt), to assure us that o must be to R in the same 

 proportion as c to r, if tie model be truly formed. Nor is it drawing 

 too largely on that conception of proportion if we assert that the 

 distribution of the railings among the columns in the model must be 

 everywhere the same as in the original : for example, that the model 

 would be out of proportion if its S6th railing fell between the 18th and 

 19th columns, while the 56th railing of the original fell between the 

 17th and 18th columns. Here, then, the question as to the dependence 

 of Euclid's definition upon common notions is settled ; for the obvious 

 relation between the construction and its model which has just been 

 described contains the collection of conditions, the fulfilment of which, 

 according to Euclid, constitutes proportion. According to Euclid, 

 whenever m o exceeds, equals, or falls short of n B, then m c must 

 exceed, equal, or fall short of r : by the most obvious property of 

 the preceding constructions, according as the mth column comes after, 

 opposite to, or before the nth railing, in the original, th mth column 

 must come after, opposite to, or before the nth railing, in the correct 

 model. 



That the test proposed by Euclid is necessary, appears from the 

 preceding ; and also that it is sufficient. For, admitting that, to a 

 given original, with a given column-distance in the model, there is one 

 correct model railing-distance (which must therefore be the one which 

 distributes the railings among the columns as in the original), we have 

 seen that any other railing-distance, however slightly different, would 



* By the terms column and railing wo mean throughout to designate vertical 

 lines which are the axes of the columns or railings, 



