1819. 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



295 



Um acres and ten roods mnke ten times ns much lis one acre and one 

 rood." — VV^e want more of iiritlimetical aii<l liMf,'-iial reform in tlie 

 iiftli book of Kuclid, than reform (or reconstruction, or novelty) 

 »kf iirf::ument. Tlie essence of Euclid's reasoriiiiir is pure. 



In all the editions of Euclid, the ma^^iiitudes are reiiresented hy 

 lini'x — never by any other species of lif^ure, as anj^les, circles, poly- 

 fliins, polyhedra, cones, or spheres. Vet tlie reiisiuiiii^ itself, as 

 well as the lani;iiaf;e in whicli it is couched, is perfectly general — 

 is appliciihle to all classes of mai;nitudes (as niinihers, time, or 

 force), as well lis to (j;eometrical entities. A C(Mitined view of the 

 Application of proportion is thus .i!ii/(/exteil; and as no vvariiinf.f is 

 ^{iven, the suffnesti(ui too often taUes the pliice of a positive and 

 wiforced limitation of the doctrine to linear maijiiitmles. That 

 tliis is no imafjinary case is plain from this — that many able and 

 Hceomplislied mathematicians, who were no mean thiiiUers neither, 

 have «.v.v»»/(:(Z this limitation in their variations of Euclid's proofs. 

 For instance, in the demonstration of r. IK, it has been proposed 

 over and over apiin by Eucli<rs coniinentatwrs, even down to the 

 ])resent day, to interchaiifie the secoiul and third miifjiiitiides as 

 the ])reliminary to an abridfjed proof! If, then, ttie musters in 

 gt>ometry can thus delil)erately enunitiate such a liiiiitation, wo 

 CJinnot wonder that the ine.\i)erieuced student should be led to 

 form it from the sufrfjestions of the )i!.;urcs. Of course the cure is 

 obvious: make, in all cases where the hypothesis does not imply 

 all beinfi; of the same species, the third and fourth matrnitudes (d" 

 different species from the first and seconil. Let, for instance, the 

 first pair be lines, and the second angles— the Hrst be aiifjles and 

 the second rectangles, circles, pidybedra, or cones, 'i'he only ciui- 

 ditiou that is essential is that the two members of each [lair shall 

 bo of the siiiiie sjieeies — no matter what that species be. It is not 

 even necessary that the members of either pair shall bo of the 

 mime form; for it is uot form but magnitude which is the relation 

 under discussion. 



This last remark is an answer to objections wliich have been 

 made to the use of any other figures but lines; namely, tliat as far 

 8S we know at this stage of our learniiiij, wo cannot make multi- 

 jiles «( any other figures of the siime/orm (or, at least, of all otlier 

 figures), so as to fulfil the conditions. Docs Euclid prescribe 

 nutiieiiexs (if form as a condition amongst the eiiui-multiples? or 

 even that the multiple shall constitute one figure (in the ordinary 

 sense of the word figure) similar to that of which it is the multi- 

 ple.'' Every child knows that the twenty equal bits of clay which 

 he has rolled up into "marbles," if made into a single one, would 

 be twenty times as large as one of the small (a suli- multiple) mar- 

 bles; and this child would admit, the terms being made intelligible 

 to him, that the twenty small marbles formed as truly (or even 

 more truly) a multiple of one of them as the large ball' into which 

 they were all combined, did. It is left as a discovery to be made 

 by fastidious geometers, that the reverse is the case! — or at least 

 to found an objection to any argument upon such an assumption! 



We have spoken of iiroportion mainly in reference to geometry; 

 but neither the fundamental idea nor the laws (d" proporlimi are 

 jiiH-uliar to magnitude, jiroperly so called — though doubtless ori- 

 ginally suggested by them. It is even rendered familiar to early 

 hoyhood in its cxtensioUfhy the (piestions which we solve in arith- 

 metic, under the heads of the "rule-of-three, direct and inverse," 

 tlie "rule-of-five," etc. We have, indeed, brmiglit (rudely and 

 im])erfectly laid down, it is true, hut still brought) under our 

 notice four temis, which are, generally, in pairs of different spe- 

 cie.s, even as to form;" as money and the goods the money will 

 purchase: ami always different in reajiect to concrete signillcatiim; 

 H^ principal and interest, length and breadth, etc. Wa have, 

 tliexefore, much elementary training to introduce the conception 

 of sameness of ratio, where the pairi* of magnitudes or entities of 

 aiiy kind are of different species, the first pair from the second. 

 Yet, withal this, writers enforco the total forgetfulness or abstrac- 

 tion (if all concrete considerations when we come to general in- 

 quiries concerning proportion; and teach us to view it in reference 

 to numbers or the symbols of number only, and to consider the 

 ratio as only a numeriual fraction arising from the division of the 

 first term by the second or the second by the first. This carries 

 the research back to its first rude and imperfect conception: but 

 the dilficulties of that conception are got over by considering all 



• It U »lm.ige what on Influtncs curly.formed hoblts have upon ns throuKh llff.-n 

 niuulfrstiitlon ot which Is .Imost universal In respect to "statlnK a rule-of.tlirec sum." 

 . "^ "'7'<"V*'' "f'*'; l'"^ celebrnu-d Krancls Wulklnghuine, to make the llrat and third 

 Urnis of the same kind. In occorduucs ivllh the vencrnl.le precepts irodltlunally descended 

 from Iho early Italian merchants, and as rellt;lou»ly iidhcrcd to by our " school-arithmetic " 

 mannfacturers as ihough It were " a aavluR article ol their Inlth." No elementary author, 

 wo bellev.. belore Uounycastle, ventured to depart Irom this mode of statement • and. as 

 far OS we know, his heresy has been almost left to die with him. We most confess that 

 for ourselves, even alter the llflh decade of our life has been completed, we have some- 

 times detected ourselves unconnclously returning to tlie old worship, and grsvely perform- 

 ing the old rllM In solving a queatluii In the '• rulj.of.thrcc I " 



functions of all numbers, whether expressible in finite terms or 

 not, as niinibcrs — tliiis including (not very logically, indeed, but 

 with a logic that suits and satisfies most algeliraists) the iiicum- 

 mensurablcs as well as cominciisiiraldes.' I'roiiortion is then 

 expressed as an eipiation botwi'en two siiidi ratios; and all the 

 properties of projioitiouals are then obtained liy ordinary algebraic 

 transformation. We do not say that such a process is altogether 

 inconclusive with resjicct to iiruiuirtidniil nnnilnrs: but we do say 

 that the "otl'-liand way" in which it is usually developed, is so far 

 wanting in precision ami completeness, as to render the logic of it 

 very dilficult to discover. It is slwrt enough; and the facts ar« 

 visibly tabulated — which, to too many minds, are the great deside- 

 rata of mathematical learning. The indolence of niaiikiud will 

 always render "shtu't cuts" in si'iencc matters of high estimation; 

 and we fear, too, that the system of t'oUegiate and I'niversity 

 examination, involving so much "book-work," and enforcing so 

 much "writing out," has a tendency to periictuate this stenogra- 

 ]diic system of de\elo[)ing science. Nevertheless, we feel con- 

 fident of this — that an algebraic system of proportion, (•onijilcfe 

 in all its jiarts and written out intelligibly, would be but little 

 less expanded a treatise than one founded on the most general 

 views and carried out to the same extent. Nay, more, — we think 

 it would be extremely dilUciilt to devise a cinirso of reasoning 

 upon algebraic ratio, Hhicb can be considered perfectly legitimate, 

 but which does not iuvoh e principles that are far more geiieriil than 

 those of simple algebra — indeed, the most general jiriiiciples <d' 

 ratio. This is taiitaiiiount to saying (and we mean it to be) that 

 all denionstrati<uis of proportion which confine the idea of ratio 

 to arithmetic in its most generalised forms, is little less than a 

 mathematical farce; — either on the one hand giving us mere le- 

 gerdemain under the title of demonstration, — or, on the other, 

 representing the principles which are oonimou to all ratio as pecu- 

 liar to mere aritliiuctic. 



The structure of Eiiclids 'Elements' (including the books omit- 

 ted by Simson), leads us to the ciniclusion, that the illustrious 

 (ireek had in view to include arithmetic and geometry in his 

 demonstrations — the abstract and concrete. In truth, the idea 

 of yijrcc, as we view it, does not appear to have ever been mooted 

 as a conceivable a]ii>lication of tlii" exact methods of research in 

 the Schoid of I'lalo; and lime is no otherwise an clement of the 

 ap|died mathematics, even now, than in connection with force. 

 Contrary, indeed, to modern practice, the (jreeks treated number 

 in subordination to, and by means id', geometry; instead of treat- 

 ing geometry as subordinate to, and by means of, number, as is 

 the modern fashion. His reasoning, too, reaches both views; and 

 as he did not conceive any other entities cmild come under the 

 shelter of exact science, he straineil after no greater generaliza- 

 tion either of idea or language. He saw arithmetic (the pro- 

 ])erties of numbers) only as a subordinate branch of geometry — or 

 at least as entirely dependent upon geometry for all its evidence; 

 and he gave, thcrcl'ore, to his doctrine of ratio a form which ren- 

 ders this de|ieiidence obvious and (as he thought) real. We can 

 not lieri>, however, enter into the instructive iinpiiry which the 

 circumstance just ijuoted suggests: we may perhaps find an 

 op))ort unity hereafter. 



Many writers have attemjited to give the fifth book of Euclid 

 in general terms and symbids, unrestricted by the references to 

 figures:" but, in general, both their language and their jirofessed 

 intention have besptdien their treating it as a branch of geometry 

 only. Of all these writers, I'layfair is perbajis the most successful; 

 and did we know less of the details of his work than we do, we 

 shtuild I'eel great confidence in this editor's quick logical [lercep- 

 tioiis having prevented him from making any change that would 

 vitiate the reasoning. Yet (our readers must believe us, unlikely 

 as the statement may appear), we have heard not only good geo- 

 metrical investigators and teachers of mathematics of high repute 



T It may be worth the rendur's while to deulODBtrat«, after Euclid's manner, the tniili 

 of such a proportion as thlK: — 



2 ; .v/2 :: ^/ri ; 2, 

 or Indeed any other Involvlne radicals or tratiHCeinlcntals. It eqnullynpplles to all forms. 

 lint Itft him keep In mind that only /'/ift'^tTinimhers sre admlttsalile as multipliers, \\ he- 

 tUt}r /unctions o/ Htiinlirrft can fwr be rendered uvallable a» muUipiitrs, ainl the leidlU 

 uia( y of the procuss rendered unassailable, we offer no optniuu : It bus uot yet been done 

 at all events. 



8 The assumption of n tetter of the atp/tatiet to dcHlgnate nny magnllude or (luantUy, 

 has been considered by many persons to constitute the research rlependant on Unit as- 

 Stmiptl.in ay an algcbralinl process. They would not say that Ihe process was alKcbraical 

 (even Ihonyh it may be) If the whole were written out In " Kn^-llsb. Dictionary words." 

 The mode of writing has no more to do with the ()nestloii than Tentcrden steeple with 

 Goodwin Sands. 'I he fact Is, the magnitude or tpninlity Is labelled with tlnit letter ss a 

 condition for more brief cxpreHsion, — It becomes the name of the (luanllty, not a iiMioeii. 

 cai represenlnlion of its t'tttue. In algebra it Is solely the name of a nnmtier, not of a 

 concrete (/uantitt/. Kven in tliose very Imaginative places, a government olllce oi a nier. 

 chant's counltUK-honse. he most he n " dull fellow" who supposed that "Form A" or 

 " Ledger B " designated the number of llgures, words, or letters In the one, or the actual 

 amount of the "Cr. balance," In the other. Yet the ciuies are analogous. 



