1819.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



331 



of one easy problem; and the analysis of this consists of a^in^/e 

 step only. Nor is the language and manner of this one analysis and 

 synthesis so very precise as to preclude the possibility of miscon- 

 ception of the analytical principle. At any rate, tliis example is 

 preposterous as a specimen of the method itself, or even as a ijuide 

 for those wizards who "may be desirous of obtaining a Govern- 

 ment certificate." Mr. Tate is insulting the scientific public, and 

 hood-winking the jeHera/ public, by such arrant trifling as this. 



One word more about this work. The stuilent who stops short 

 at the third book might as well not begin Euclid at all, whatever 

 might be the case with the aspirants for a Government diploma. 

 He cannot get the remaining books of Euclid separately from 

 these ; and hence he is obliged to buy over again what Mr. Tate 

 had sold him, in order to his getting what Mr. Tate professed to 

 give him — an "edition of Euclid." This mode of imjxising upon 

 the poor youth of humble life who is desirous to study, by ab- 

 stracting from his pocket his hard-earned sliillings (all under the 

 ])retence of serving him too!) is very much like some of the bene- 

 volence which occasionally comes before the metropolitan police 

 magistrates. 



Differences of detail, indeed, there are ; and, according to the 

 trader's code of ethics, the difference of detail between an ambi- 

 guous transaction and a masterly piece of enterprise need not be 

 very great. The Christian moralist may see more to condemn 

 than to admire in such a code; but what matters this, when that 

 code is framed by the great city lawgivers— the magnates who 

 count the day's gains by hundreds, and sometimes by thousands .'' 

 " Why not then allow Mr. Tate, in his small way of business, the 

 privilege of other traders.''" Aye — why, indeed.'' We do not 

 wish to deprive him of it ; but we ask consistency even in trade. 

 We only object to his one day deluging the public mind with a 

 subtle poison at a profit, in order that he may the next make a 

 fortune by its antidote, or perhaps by a semi-antidote. In the 

 case of physical poisons the law would step in ; but public opinion 

 can alone arrest a moral or intellectual pestilence. The press is 

 the only tribunal before which such cases as these can be tried : 

 and we believe that we "judge righteous judgment," in an unhesi- 

 tating condemnation of the same man writing the two works in 

 question ; and still more so, of that man putting in any claims to a 

 consistent scientific character. We consider him to have damaged 

 the cause of science itself, — to have held geometry up to ridicule 

 in public estimation, as a thing having one character last year, 

 and another this, — as destroying public confidence, in the earnest 

 enforcement of the value of science by its sincere cultivators, — and 

 as placing reason on a par with mere sentiment, variable as the 

 digestion, and uncertain as the megrims of the morrow. We yet 

 hope that the public may be led, on forming its judgment of the 

 sincerity of the convictions of scientific men, to select sjiecimens 

 to judge from, very much contrasted to that which Mr. Tate will 

 afford them. 



REPLY TO REVIEW OF 'ALGEBRA OF RATIOS.' 



Sir — Having observed in the review of my 'Algebra of Ratios, 

 in your last number, a very serious misconception on some import- 

 ant points, I will, with your permission, offer a few remarks, ))rin- 

 cipally by way of reference to certain portions of the work which 

 the nature of the objections clearly shows have either escaped 

 perusal, or been greatly misunderstood. 



1st. It is said, that "Mr. Browning assumes 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 geo- 

 metry analogous to this assumption. Still, we will not quarrel 

 with the assumption, though we could wish the author had been 

 able to dispense with it." 



That this assumption has been dispensed with, a reference to the 

 index will at once show; and on turning to the pages therein re- 

 ferred to, it will be seen that the mode of proof is as follows: — "If 

 )h'B is a variable fraction of B increasing to the limit A, then m'D 

 is a variable fraction of D increasing to some limit C." — IX. Art. 1. 



This limit is proved in I. Art. 2, to be "a quantity wliich has the 

 same ratio to D which A has to B; and therefore when the ratio of A 

 to B is given, and D is any quantity whatever, we thus prove the ex- 

 istence of another quantity C which has the same given I'atio to D." 



By means of these propositions, in the proof of which the wliole 

 difficulty lies, it is shown in I. Art. 5'2, that when three terms of a 

 proportion are given, a fourth exists. 



2ndly. It is objected "that the cases are very few in which rce 

 can exhibit the arbitrary fractions of any one of the quantities con- 

 cerned.' 



This clearly is not required, and were it possible, would be of 

 little use. It is only necessary tliat we should be able to conceive 

 the existence of such fractions, not to exhibit them in a iiainericat 

 form J and for all purposes of reasoning on tlie nature of ratio in 

 general, this mental conception, witli tlie use of general symbols to 

 denote what we know to exist but cannot otherwise express, will 

 be found sufficient. 



3rdly. It is said, "His definition of proportion is tantamount to 

 this: that if 



_ y 



.•^' 



V 



then C = — 



z 



A > 



Azi^. 



B, then C > 



D; 



D: 



B, then C / — . D 



.'/ 



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

 and c be not numerical symliols, we apprehend that tlie terra 'frac- 

 tion' will be deemed inappropriate; and not only so, but that the 

 definition itself is without distinct meaning. We have viewed 

 them, then, as 'numerical symbols.' On tlie next page, however, 

 they are deprived of tlieir arithmetical character, and are directed 

 to he 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 

 anv numerical cnnsiilerutions whatever, ought to be given." 



here I am constrained to complain somewhat of unfairness (un- 

 intentional I am sure), in exhibiting my definition in a symbolical 

 form very different from my own — w Inch, for the most part, is ver- 

 bally expressed — and then framing objections from the form which 

 is tlius introduced. I have no where said Xhni fractions and whole 

 numbers verbally expressed, as in my definition, are to be deprived 

 of arithmetical meanings, and regarded as ratios; but tliat symbols 

 which hitherto have been used to represent fractions and whole 

 numbers, will, with special exceptions, he regarded as symbols of ratio 

 in the sense previously defined. 



The distinction between tlie arithmetical and the extended 

 meanings of these symbols has, in all cases, been carefully ob- 

 served. Thus, the symbol m is used in mU to denote an arithme- 

 tical multiple or fraction of B; but as a subscript symbol, it is 

 used in B,,, to denote the quantity whose ratio to B is m. This 

 notation has been retained until, in the course of investigation, it 

 could be shown that j»B and B,„, having the same value, could 

 allowably be interchanged. And in page 55, will be found this 

 remark: — "Consistently with this, inB may in future be used to 

 denote either a concrete number of which B is the unit, or a quan- 

 tity whose ratio to B is m. The notation of Art. 3, which was 

 adopted only to avoid confusion with arithmetical symbols, will there- 

 fore no longer be required." 



The foregoing remarks will, I think, tend to show that the 

 logical niceties which I am rejiresented to have disregarded, were 

 invariably present to my mind during the investigation of the 

 subject, and have in reality been carefully observed in every por- 

 tion of the work. 



I am, &c. 



Henry B. Browning. 



Stamford, October 20, lSi9. 



[\Ve have inserted Mr. Browning's letter entire, although a large 

 portion of it is taken up with quotations from our review of 

 his book. Mr. Browning is of opinion that we have misrepre- 

 sented his meaning; and we deemed it the best way to allow him 

 to explain away, as he best could, the passages and steps of his 

 processes to which we had expressed objections. We make this 

 exception to our general rule (of not allowing our pages to be 

 made the vehicle of reply from authors dissatisfied with our stric- 

 tures,) entirely on account of the abstract character of the subject 

 under discussion, and the almost insurmountable difficulty of 

 expressing verbally the conceptions of the mind respecting it. 

 We consider that Mr. Browning, under these circumstances, is 

 entitled to that indulgence : but were we to open our pages to 

 replies in general, we should find it necessary to only notice those 

 works upon which we could bestow unqualified praise — for what 

 author, but those fortunate few, would not otherwise deluge us 

 with ''reams of reply?" 



1. AVhetlier Mr. Browning's proof of the existence of the fourth 

 proportional be, as tie says" it is, legitimately made out from the 

 passages referred to, we shall not liere stop to inquire. We appre- 

 hend the existence in. posse of the fourth proportional has not been 

 questioned— only the exhibition of it in esse. No one doubts the 



43* 



