296 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[OcTOBEB, 



call this very book of Playfair's edition, an algeliraiml treatise .' 

 The confusion of idea we so frequently meet with amongst mathe- 

 maticians of no mean fame, with respect to the relations or analo- 

 fj;ies of arithmetic and geometry, is most extraordinary. Who that 

 really understood the grn\indx of his science, could for one moment 

 confound an abl)reviated mode of writing (which is all the essen- 

 tial cliaiige made by Playfair of Simson's edition of the fifth 

 book) with the abstractions of arithmetic ? 



It will he generally conceded that nothing could come from the 

 ]>en of Professor De Morgan which did not bear the impress of 

 profound and independent thinking. He is no imitator. We may 

 differ from him : but we can never fail to admire tlie earnest spirit 

 in which he writes, nor to wonder at the extraordinary resources 

 of his intellect. Of course, then, when (a dozen years ago) he 

 l)ublished his tract on the 'Connection of Number and Magni- 

 tude,' we were prepared to expect that the doctrine of ratio would 

 l>e placed in a new and more philosopliical light than before. The 

 great object of the work, however, turned out to be — an attempt 

 to bring intelligent pupils into a position to philosophise on this 

 subject for tliemselves. In this the author has been eminently 

 successful; and his illustrations are admirably adapted to prevent 

 the student from resting with those vague and misty notions, with 

 which too many even of matured geometers are so supinely satis- 

 fied. 



Mr. De Morgan starts with "an extension of the arithmetical 

 notion of ratio, to magnitudes in general and especially to 

 space-magnitudes." Perhaps, au fond, his view is not diiferent 

 from that which we have taken some pains in the earlier part of 

 this review to enforce; although we view the arithmetical idea as 

 a mere suggestion of the general one, whilst Mr. De Morgan con- 

 siders the general one as an extension of the arithmetical one. 

 'l"he difference may, possibly, be only verbal ; but we think we see 

 sonietliing more in it. Our views, as well as his, are now before 

 the reader — whether they wholly agree or partially differ, we sliall 

 not here further stop to inquire, for it is time to say a word or two 

 respecting Mr. Browning's work itself. 



Mr. Browning "takes the bull by the horns," and at once starts 

 with the consideration of "concrete quantities."^ The theorems 

 respecting the limits of variable concrete quantities are both neat 

 in form, and we think, witli the author, tliat they are new in man- 

 ner. These are brought in for ulterior purposes. 



We have quoted Euclid's fifth definition litei-ally translated from 

 the Greek : we now give Mr. Browning's. "The ratio of A to B 

 is a relation of magnitude, which is determined by comparison of 

 A with the several fractions of B in regard to equality, excess, or 

 defect : so that C has to D the same ratio which A has to B, when 

 C is equal to, or greater than, or less than any fraction of D, ac- 

 cording as A is equal to, greater than, or less than the same frac- 

 tion of B." (p. 11.) The neatness of the Euclidean definition is 

 here replaced by a too close imitation of the manner of Simson's 

 version. Setting this aside, the marked difference between it and 

 Euclid's is, — that the first and third terms are here compared re- 

 spectively with fraet ions of the second and fourth; whereas, Eu- 

 clid compares eqni-multiples lyi the first and third with etjui-niii/ti]i!es 

 of the second and fourth. This difference is not an essential one, 

 except it shall prove that an essentially different mode of subse- 

 quent demonstration can be built u])on it. 



Mr. Browning asstanes as an axiom that when three terms of a 

 proportion are fixed upon, a fourth exists. There is nothing in the 

 details or the spirit of the ancient geometry analogous to this 

 assumption. Euclid never assumes the existence of anything 

 which he does not first show how to actually find; and most 

 (though not all) modern geometers of any authority have fol- 

 lowed his example in this respect. Still, we will not quarrel with 

 tlie assumption, though we could wish the author had been able to 

 dispense with it. 



If the proportional quantities be commensurable, Mr. Brown- 

 ing's fractions will amongst their varieties express the ratio of any 

 <iuantity whatever to its fellovv. The test "equal-equal" amongst 

 Euclid's eijui-multiples are there formed by the multiplication of 

 the terms by tlie denominator of the fraction. So far, then, we 

 run nearly parallel with Euclid's argument, however different our 

 routes may seem to be. Nothing lost and nothing gained, then, 

 by the change, thus far. 



There is one point of view under which exception will be taken 



9 We wish the term " quantity," or Homething analogous, were by common consent 

 adopted instead ut " magnitude." The " magnitude 01 a force " is indeed a common but 

 an aivltvvard expression , and it has led to much quibbling and inconsecutive reasoning in 

 I tie elements of mechanical science. But the " magnitude of a period of duration" in- 

 volves an absurdity loo gross eveu for tlie '* magnates " of sclentilic license in language 

 and demonstration. Yet we tvant the phrase " magnitude of the time," in order to per. 

 tect generality, if we must be compelled to use the term magnitude iu these researches. 

 Evea Mr. Broivnlng is not uuiformly consistent on this point. 



by many to Mr. Browning's definition: — that the cases are very 

 few in which we can exhiint the arbitrary fractions of any one of the 

 quantities concerned; either in the same form as the original, 

 which we hold to be a superfluous condition, — or in any form 

 whatever, which is a more weighty consideration. We only indi- 

 cate it here — not discuss it. 



If algebra be considered, as many writers of the present day 

 consider it, the "act of combining symbols according to given 

 laws," there will be as many algebras as there can be formed 

 laws of combination — that is, innumerable ones. Mr. Browninjy 

 says of such algebras, tliat each "should have Rules limited by its 

 Definitions; and that an extension of the Definitions is the only 

 license for an extension of the Rules." Considering the "Defini- 

 tions" to signify the laws of combination, and the "Rules" to sig- 

 nify the resulting formulas, we quite concur in his statement. We 

 must guard ourselves, however, by saying that the interpretation of 

 those resulting formuliB, and of everything obtained by means of 

 them, must also be consistent with the ideas upon which the laws 

 of operation were founded. W^e can introduce into the interpre- 

 tation nothing different from, nor even more special than, what we 

 introduced amongst the original conditions. Nothing can be e.x- 

 plicitly got out of an equation that was not implicitly put into it. 

 The extrication is all that we can do. 



This is a view which, obvious though it be, is often forgotten in 

 the course of a general system of reasoning. We so often find 

 assumptions which are not contained in the "definitions" in such 

 reasonings, that we habitually look out for them ; and though, by 

 some modification or other, some of these may be removed or other- 

 wise deduced, yet it veiy often happens that the whole force of the 

 reasoning as reasoning is rendered nugatory. It appears to us that 

 Mr. Browning has a little to do in the way of amendment here, — 

 though we cannot suggest h->w it is to be done. 



His definition of proportion (p. 11) is tantamount to this: that if 



A = I . B, then C = | . D; 

 A > ? . B, then C > ^ . D; 



A Z 



B, then C Z 



D; 



for all values of y and z expressible in finite terms. In this if y and 

 z be not numerical symbols, we apprehend that the term "fraction" 

 will be deemed inappropriate ; and not only so, but that the defini- 

 tion itself is without distinct meaning. We have viewed them, 

 then, as "numerical symbols " On the next page, however, they are 

 for the present deprived of tlielr arithmetical character, and are 

 directed to be understood as " symbols of ratio" only. This seems 

 to us to invalidate the definition itself; and we apprehend that, to 

 render this consistent, a new definition of ratio which does not 

 involve any numerical considerations whatever, ought to be given. 

 We remit this to Mr. Browning's consideration. 



Again, what idea can we form of adding, substracting, multiply- 

 ing, and dividing ratios, when the ratio itself is deprived of all 

 defined meaning.? Symbols of number they are not allowed to be, 

 — and symbols of magnitude (or of any kind of quantity) it would 

 be preposterous to suppose Mr. Browning meant them to be. This 

 however is the pivot upon which Mr. Browning's escape from the 

 difficulty of incommensurables entirely turns. How it happened 

 that the very terms " factor" and " quotient" occurring in his in- 

 vestigations respecting his symbols of ratio, did not enforce upon 

 his mind that he was really conducting an arithmetical inquiry, he 

 himself, by looking back at the history of his own mind during the 

 research, will be best able to tell. Such lapses are, however, to be 

 expected in all attempts at logical generalization founded on mere 

 generalizations or changes of definition. Definition is a two-edged 

 sword ; and few persons in wielding it escape a cut or two from the 

 back-edge. 



Till this fundamental diiBculty is removed it would be useless to 

 pursue the mere consequences. It is sufficient to say generally 

 that the work itself manifests great ingenuity and much real skill 

 in dealing with very abstract and difficult topics. There are many 

 instances of consummate address in dealing with the details of his 

 reasoning ; and we tliiiik tliat when a more intelligible basis is laid 

 for the superstructure, the greater part of his materials will be 

 found to be of a- long-enduring character. It is by these that we 

 were led in an earlier page to say that Mr. Browning has " made a 

 step in advance;" and with the caution we have given, we strongly 

 recommend bis book to the careful reading of all who take interest 

 in this recondite inquiry. 



