2:m 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[OcTOBEn, 



'Ilio iil(jebraist vievrs tlie PxprossiDn rnlin ns a fraclioii,' and to 

 liiiri, tlicreforp, it has a certain (l('f!;ree of intellitfiliility, tlioiiifli it 

 ninst of necostiity be somewhat confused and very iti(oni|dete. 

 He seldom cares, however, for this: )iis iiiacliine is in f;('ar, and )ie 

 can (rririd out results at liiw own jdeasure — tliouj,"^!! his intellect 

 have as little to do with the process as the heart of a Tartar has 

 »ith the pravers and clianns wliirh he puts into his little windmill, 

 to perform alike his <levotions and his jiliysical cure. 



'I'he ireometer takes a different course. A ratio, like a fraction, 

 in itself and alone, is a mere conception of the mind, and, taken 

 «/«;ic, without the least use. His ohject is not to (ijid or express 

 the relation of one magnitude to anotlier, hut to ilc/cniiiiic the. lawn 

 liy which sevenil mapnitudefi, havin;? a s|iecified set of relations, 

 flre connected tofjether in respect to some other set of relations. 

 To effect this, as far as propcjrtion is concerned in the task, he 

 t'oni|)ares the ratio of two magnitudes with the ratio of two 

 others; and the e.\|)ression of tlie sameness of these ratios it is, 

 which III' calls ^^iinipiirtimi." Uiidiu' tliis aspect, projporti(ui becomes 

 intellifjihle. Still this assumption furnishes no criterion which 

 determines wljether f(mr inafjnitudes, ijirrii hi/ lunj ntlirr miiditinii.s^- 

 are proportionals or not; and we xccm to have made little way — 

 thouf^h, indeed, the step was a. wiile one. 



In coni|parin;^ any ratios capalde of numerical exjiression in 

 finite terms (that is, comrnensurahh^ (piantities), it would readily 

 be discovered "such eipii-multiples of the first and third and such 

 t'qui-multiples of tlie second ami fourth" coulil 1)0 taken, that the 

 multiple so taken of the first shiinld lir ecpial to that of the second, 

 and that the multiple of the thinl wintlil hi; thru eipial to that of 

 the fourth. 'I'he result is obtained by cross-multiples. It would 

 naturally occur to iiupiire what would take nlace between the 

 niiiltiplea of the third and fourth if the multiple of the first 

 should be taken bo as to be /Greater or to be less than that of tliu 

 Hecoiid: and the third and fourth liavinjjf in these cases universally 

 the same relation as to f!;reater, equal, less, this relation was proved 

 to be a priipcrli/ iif pnijiurliona/s, as long as they were coinnicnsur- 

 ahle. 



"But are those properties confined to commensurable propor- 

 tionals alone?" would be a natural (|uestion. 'I'he primary one, 

 respecting the e(piality of the multiples, it would ho clear at once 

 could never he more than approximately fulfilled by finite cross- 

 multipliers:'' but during that a]iproximation, however closely 

 pushed, the other two iriteria were invariably fiil/lUed — as, for 

 instance, in the case of the siiles and diagonals of two S(|uares. 

 It would also be seen and rea<lily proved that in all coinmen- 

 surable magnitudes, those other two criteria being made the hy- 

 pothesis (or the definition of propiirti(uiality), the princi])al con- 

 dition of "ecpial-eipial" could he shown to follow as a conse- 

 (jiience; — so that, in reality, those two conditions ("greater- 

 greater" and "less-less") were ade(iuateto form a ilefining criterion 

 of propiu'tional magnitudes, as long as they were c(unineiisui'able. 

 The "etpial-equar' criterion was ;i conseipience of conimensura- 

 liility only; and the other two lieitig capable of existence in- 

 dependently of commensurability, were not of necessity bound 

 by that peculiar relation. 



Kucliil, then, hail good reason for choosing the "complicated" 

 criterion of |)roportion which constitutes his fifth definition, and 

 (if iioii-proportionality which forms his seventh. The tests of 

 greater and less, completely and without other aid defined the 

 ])roi)orti(Miality of commensurahles; and they were eipially appli- 

 cable and definitive when the magnitudes were iiicoiiiiueiisurable, 



I Not ratius null/. It liiti, ulti-r tin- t'niicy of Kniii^nU, Am|l^rc, iiiiil lliifuiiicy, 

 becDiiii^ u fniiliton iit (^iinibrldKe lu fUiiHliIi-r an intiflu an a rutin, ami a ratio a» a Irac. 

 tlon. One conlil alinoMt think that the merry p'renchlnen meant to "hoax (he jill- 

 ferliiR .lohntunii," by ijlvhig them a bait that wunld render them rldlculuuii tu all men and 

 throu^'h all time, 



'i When four mnpnltude* nr« fflven by^flctual «ihlbUlon, witbout any speclQed connec- 

 tion or dependence, of efmrse It li only hy a nerles of cxperlnienta upon the multiple! 

 thtit w»t can brln^ the dellnlthni to hear upiin the proof of propurthiinility or diHi)ropor- 

 tionidlty. Hut tn Kenmetrleal liiij.lrleif are ihev ever so «lven > —There U nUvayn given a 

 relation amoai/itt t/ii'm : and ntibjcct to thin relation It la that they are to be proved to 

 be proportional or not. 



8uub a remark would he alinont too trivial to make, were It not that we bave observed 

 It to escape the notice ol sturlentu Mo often aH to create yrcat va^ueuesa In their mlinla, of 

 the olijecta and coridUlonH of proportionality. " How am 1 to aasure tnyHelf that the 

 four lines drawn here are proportionala } Must I make experhnents with (he inuUipleH 

 tin I am satlNlled > And bow, liuleed, can 1 be satltilied idiiudntely and loKicidly, froiii ii 

 limited nundler of expi-rlMieuts, such as I can iimke, that the conditions will be unlvera 

 nlly fulUlled ?" — QocHtloiis sncli as these from InteiilKent'stodents show that this note Is 

 not superflnoiit. We have many a time heard them asked— and In no captious spirit. 



I* The approximation may be made by narrowUiK the limit of diOerencu between tbe 

 eipilniultlpleH to luiy ylvun Unite extent. 'I'hu problem is, manifestly — "Two magnitudes 

 ol the same knid bciiiK given to lind multiples of them that shall <lltrer hy a nuk^nltnde 

 less than nily Klven ina)^iiVLude." The process would be that Kivcn iiy ICuelid tor rnullii< 

 the common measure of tvvo inaitnitudeH lu book\x. prop. 7, till we obtained a remulndwr 

 less than tbe (tlven miV({nUude. These operations are aiitlimelically represented )ty a 

 continued fractitm, which, reduced to u comtuon fraction, baa Us nuiucrutor uud denoiui- 

 nutor lur the required uiultipllcrH. 



where the "equal-ctiual" test could not be applied, hut, indeed, 

 where it was not at all required. This definition, then, beini< 

 alike applicable to commensurable and to incommensurable magni- 

 tudes, is fully justified in principle, and contains all the ipialitiea 

 of a philosophically constructed definition. The only olijectioa 

 urg(!d against it is its "(WH/y/cr;///"; but there is to be urged 

 against all others which essentially differ from it — their incomplete 

 iii'xx, (imltiijidtii, or iniiiiiilii'iihili/i/. 



'I'hose who are accustomed to look into Simson's "notes" on his 

 edition of Kuclid, will see how often he has made changes in the 

 text; and, in fa(^t, of all editions, as I'eyrard has remarked, this is 

 the farthest removed from the whole of the existing texts, or of 

 books jirinted from those texts. We can hardly, therefore, refer 

 to Simson to ascertain the views of Euclid upon disputed points, 

 and especially upon points of great delicacy of thinking. Simson's 

 labours on this book ;ire indeed exceedingly valuable; and he him- 

 self in (uie of his notes expresses that to be his own opinion. His 

 very expanded expression of the fifth definition, however, has more 

 of the verbosity of the lawyer th.in of the neat discrimination of 

 the philosophical geometer. Williamson translates the Greek 

 literally as follows : — 



"Magnitudes are said to be in the satne ratio; the first to the second, and 

 the tliird tu the fourth; when tlic equi-inultiplea of the first and third, ac- 

 cordini; to any iimlti[)lication, are at the same time less, or at the same time 

 equal, or at the same time greater, than each af the equi-iuultiples of tbe 

 second and fourth, compared with one anotlier." 



If, therefore, the texts of Euclid could be depended on (which 

 they certainly cannot) as genuine and uncorrui)te(l, wo should 

 think that Euclid had committed a grave oversight by introducing 

 as a ])art of a general definition a circumstance that was only 

 accidental and confined to a limited class of cases — the introduc- 

 tion of the "e(|ual-e(piar' test as an essential one. The two are 

 universal and sufficient: the other is accidental and casual — in- 

 capable of being generally made to exist, — and even when so made 

 in any particular case, adding no force to the ])roof.' 



'I'his we hold to be a ([uestion of paramount importance in the 

 doctrine of proportion; and more esjiecially as it has done much 

 towards not only creating confused views of proportion, hut given 

 rise to numberless visionary books and the loss of much valuable 

 time both of writers and readers. Men of mediocre minds do not 

 select such topics as this for their crude speculations; and it is 

 unfortunate that able men should be led the dance of years after 

 an ii/iiii.1 /'atitiix.'' There has indeetl been a race of jiersons who 

 have "reformed" proportion, — so also has there been who manu- 

 f.ictiired "geometry without axioms," — and another still, who 

 "s(juared the circle ' and "trisected an angle." It is not, however, 

 of these cognate races that we speak: but of men of truly ])hilo- 

 sophic minds, who have been led by the admission as genuine of a 

 phrase occurring in a probably corrupted text, or by an accident- 

 ally suiierfluous jphraso of Euclid himself, to establish the necessary 

 co-existence of the three tests instead of two. 



That one great obstacle to the reception of Euclid's method of 

 proportion arises from the cumbrous language in which it is 

 delivered, there can be no doubt. The tautology in the expression 

 of the conditions of the text, and enunciation of the conclusions 

 deduced by them, is so wearisome to the eye and ear, as to operate 

 like an opium-]iill or the "passes" of the mesmerist — to prodnt^e a 

 catalepsy of mind, if not a idiysical catalepsy. The (ireeks, how- 

 ever, had nothing that could bo called "pure arithmetic " (their 

 nearest ai)iiroach to it being the doctrine of numbers as delivered 

 in Kucliil's 7th, Hth, !)th, and partly the 10th books), and of course 

 their language respecting its most elementary truths is necessarily 

 im|)erf'e(^t. This is one cause, indeed the chief one, of thi'.funii of 

 the fifth book. For instance (to use the language of De .Morgan), 

 "V. ] , 2, 3, .5, (i. These are simple propositions of c(uicrete arith- 

 metic covered in langu.ago which makes them unintelligible to 

 modern ears. The first, for instance, states no more than that 



^ It Is very probable that Ruclld gave (supposlnt; he gave it at all) tbe ** e(iual-equol" 

 test as an Incidentally occurring variation of form of the test depending on the Other 

 two conditlonsi and which. In the cases where It might occur, would of itself be suf- 

 bcleiit. We caimot, however, bring ourselves to think that he gave It as u part and 

 parcel of the fundamental drliidlion. We do not need the testimony of Simson to the 

 corniptlon of Kuclid by "unskilful commentators;" for almost every page ctmtalns In- 

 ternal evidence of the ' Klements' not being lu liielr original condition. The present 

 hioica li ke a scboliuiu or corollary concentrated into the text. Poor Tbeou I be bus much 

 to answer for. 



* We may Instance here o circumstance that has been often, and Indeed generally, 

 misappreliendetl— vti. the Props. 7, S, 11, In. of Kuclld's lifth book. It is very natural to 

 say that " if ratio be the relatlcm lietwcen two maftnitndes with respect to ijii'intiti/, then 

 equal magnitudes have tlie sjiine relatUm as to i/nantity to any third magnitude of the 

 same species, and fience the same ratio." Ail this is very true, and so are corresponding 

 objections to the other three: but it was incumbent on Kuclid to show that this result 

 also flowud as a cmaetpience/ro/tt his dejiltition. These propositions formed u practical 

 test of the correspondence ot bis system with our common notions, wliere they could be 

 brought into comparison. Kxtreme cases are the severest tests of every system, fiu- 

 clld'a bears it riguruusly. 



