Plooij2.pdf

7/27/2019 Plooij2.pdf

1/20

ovoided the operotion withports in odding them qnd subtrocting them ond convertingthem into whole numbers. They contrived this by odministering numbers to the porwhich they derived from the denominoiors of the ports ond by ultiplicotion they modethese numbers the denominolors of the ports in such o woy thoi these were oboutwhole numbers or the beginner. The mosters omong the oble scholors we find doingthe some. (M) For those who occupied the'mselves -with ostronomicol colculotions, o-sPtolemoeus ond others, did not siick to the ports, but possed rom them ond the,irkind into newly devised denominotors. They divided unity into sixty ports in order toreploce whoi reloies io one hol ond one ihird ond whot is yet more owkword thonthese.ond the operotion of its multiplicotion ond oddition ond thl like by whole numbers.And likewise, when ihey needed such ports they divided them olso into o whole numbero ports, every unit inio sixty ports ond ihey creoted nomes os minutoe, secundoe,lertioe, etc..After mentioning this, lei us now remind the introduclion contoining the conditions bymeons of multiples which Euclid used ond let us sum them up. Often rie will olter Euclidisorder in them becouse of the occordonce we think there is in it with the obiecr weintend to goin. So we soy: When the rotio of ihe irst io the second is like the rotioof the third o the ourth, then ony equimultiples found of the irst ond the thirdond ony equimultiples found of the second ond iourth ore only ound with the propertfthot when the multiple of the irst exceeds the multiple of te second the multip'le o'fthe third too exceeds the multiple of the fourth, ond when rhe multiple o rhe irstis equol to the multiple o the second the multiple of the rhird too is equol to themultiple of the fourth, ond when' the multiple oi the first olls short .f th; ,rft'fiuof the second the multiple of the third too olls short of the multiple of the fourrh.And olso when we convert this we soy: when there ore four mognitudes ond ony equi-multiples found of the irst ond the third ond ony equimultiplei found o the secondond the ourth ore only.found with the property thoi when' the multiple of the firstexceeds the multiple of the second, the multiple of the hird too ur."jd, the multipleof ihe fourh, ond when the multiple of the fiist is equol to the multiple of the ,".ond,the multiple of the third too is equol to the multiple o the ourih, ond when ihemultiple of the irst folls short of the multiple of the second, the multiple of the thirdtoo folls shor of the multiple o the fourth, then the rotio of the firsi to the secondis like the roiio of the third to the fourrh,Whot remoins for us in this writing is to mention the rotio which is greoler hononoiher rotio..So lsoy: When there ore four mogniludes, the {irst conioining more portsof the second thon the third contoins o the some ports o the fourth, ihen the sizeof the.first os compored with the second is greoier thon the size of rhe third or.o,-npor.Jwith the ourth. This wonts no proo, becuse whot contoins more- ports is o greoter36


2/20

roiio thon whot contoins less ports.Also I soy thot this sentence is convertible 1oo. viz. ihot when the size o the firstos compored with the second is greoler thon ihe size of the ihird os compored withthe ourh, some ports moy be ound of the second ond ourth so thctt the irst(N) contoins more ports o the second thon the hird of the fourth.Some things o this kind thot we con very well do without ore presentecl omong whotis necessory. So lsoy lhot when there ore our mognitudes ond ony ports {ound ofthe second ond the fourth ore not {ound, unless if the ports o the second exceedthe irst mogniude, the ports of the fourth exceed the hird mognitude, ond if theporfs o the second oll short o the first mognitude, the ports of rhe fourth oll shoro the third mogniiude, then the mognitudes oie proportionol even if we clo not menionlhe conditiorr o equolity. For to mention equolity or to leove it out comes to lhesome thing. For when they ore ixed (enclosed) by exceeding ond olling short otcr time, the condition of equolity is necessorily existont by force o logic. oncl theproportionolity in the mognitudes exists. For instonce; When we cjssume foui rnqgnitudesAB, c, DE ond F, ond suppose thot ony ports ogreeing in number ond denoinotiontoken rom C ond F ore only found with the property thot when the'ports o C exceedAB, then the ports of F exceed DE too. ond when the ports o C oll short o AB,then the poris o F fcrll short of DE too, equolity not being mentioned, rhen lsoy thorports o C equol to AB ore only found if olso the pors of F crre equol to DE. The proofof this is ihot i con not be otherwise. For i rhcri coulcl be, let AB be exoctporis of c, ond let DE contoin less or more of such ports o F; let DE be more.Let DG contoin os mony ports of F os AB contoins porfs o C. Now you must no stopdlviding F until you reoch the. first port of it thot is less thon EG. Let thot port be .This is less thon EG. Let us toke o similor pori o c, soy K. Then we subtroct rom DEthe ports equol to H thot it contoins ond no doubt their remoinder will oll betweenG ond E, becouse GE is greoter thon H. Let ihose pieces be DL, LM, MN. Then wesubtroct from AB the pieces like K thor it contoins. From it con be subirocted os mony.pieces os from DG con be subtrocted pieces like H, becouse we hove supposed thoithe rotio of AB to C is os the rotio of DG to F, for either contoins the some numberof ports of its componion. Let these ports be AO, op, pe. Now Ae is greoter thonAB, iust os DN is greoter thon DG. Therefore Ae is ports of c, ond DN is iimiloir portsof F in the some number. However AQ is greoter thon AB ond DN is less thqn D,so thot some ports o C ond F ore ound ond the poris of C ore found to exceed ABond the ports of F to oll short of DE. And we hod supposed thot no some ports of cond F would be found so thot lhe ports o c would be ound to exceed AB, unlessthe ports of F would be found to exceed DE. This is o conirodiction, il is impossible.Therefore if ports of C ore not found equol to AB, unless similor ports of F oie {ound.t


3/20

equol to DE, then the three condilions o being equol, exceeding ond olling short,.r"afy ound, ond therefore the proportion () is'sound, ond thot is whot we wishedto demonstrote (14).Alsolsoythotwhentherotio A o p I o L MG N Eof the first to the second ;5 -l!j- F--{-{-.-|_{greoter thon the rotio of the # *--t-tlhird to the fourth, some Ports cof he second ond the LrJn ot" only {ound with the property thot the ports of thesecond ore iound to exceed the {irst mognitude if ihe ports ol l[" fourth too orefound to exceed the first mognitucle. lf or instonce the rotio of AB to c is greoterthon the rorio of DE 1; f, I ioy thot oll equol ports token rom C ond F hove the;;;;til tnoi *f,"" the ic,*s of C ""te"d' AB, the ports of F too will exceed DE't "' proot is thot it con not be otherwise thon we soid. For, if thot could be so, viz' if,o*u ,or" ports of c ond F hod the property thot the ports of C were found to exceedS ond the ports o{ F were not {ound to exceed DE, iet the ports of C exceeding ABbe AG ond such ports of F thq do not exceed DE, DH' Now DH ond AG ore someport, or F o,nd C, ond DE conioins more ports of F thon AB contoins ports of c, ondso the rotio o DE to F is greoter thon the rotio of AB to C; bui the rotio of AB o C*r' *" g*oter. This is o controdiciion, it is impossible. Jherefore no ports of C orefouncl exeeding AB, unless the some ports of F exceed DE; ond this is whot we wontedto demonslrote. A BG D H EAfter mentioning oll this, I soy hot when -t-+f -r'#oi iow mognitJdes the rotio o the irst to l c I hr----lihe second is greoier thon the rotio of hethird to the {ourih, some ports con be found o the second ond the ourth with ther"riy rhor rhe port, oi the second foll short of the first mognitude ond the portsof the fourth do not foll short o rhe third mognitude. lf, or instonce, the rotio of Ato B is greoter thon the roio of C to D, then I soy 'thot -it is- possible io find some.ru pJrt, of B ond -*ii nf.' property thot the ports o_B foll short of A ond thep"rt, ii D do no, olirho* of c. the proo is thot ihis is ound o some ports tokenr"* S ond D, {o,- othur*ir" ports of {olling short o A ore not found, unless theoorts of D oll short o C. And olso, becous we hove supposed thot the rotio o{-i. A 'r gr";ter thon the rotio o C to D, ii is certoin thot whotever equol ports weind of g ;na p, the ports of B ore only ound to exceed A i he ports of D too exceedC, os we hove demonstroted in the preceding proposition. Therefore A, B, C, D ore-our mognltudes ond oll ports token irom B ond D hove lhe property hot if the portsof B exed A the ports of D exceed C, ond when the ports o B foll short of A the"Ur "t D oll shori of C. Thereore the rotio o A to B is os the rotio of C to D' We


4/20

supposed however thot the rotio of A to B wos greoter thon the rotio of C to D.This is o controdiction, it is impossible. Thereore it is not (true) ihot whofever portstoken from B ore ound less (P) thon A the poris token form D ore necessorily less 'honc too. lt is rother possible to {ind ports of B olling short of A, while the ports o Ddo not foll short of C, ond this is whot we wished to demonstrote (.l5).From he point of view of the multiples on theother hond it is now sound too thot when t--1---{the rotio of the first to the second is unlike Ithe roiio of the fhird to the fourth, it is F--------._{possible thot equimultiples ore ound of the first ond the third ond equimultiples of theof the second ond the fourth with the property thot the multiple of the secondis found to foll short of the multiple o{ the irst ond the multiple of the fourth is notound to foll short of the multiple o{ the third, in the sqme woy os we found portsof the second folling short of the irst ond ihe ports of the ourth not folling shortof the third. And likewise os we soy thot the multiple of the second folls short of themultiple of the first ond the multiple of the ourth does not oll short of the multipleof the third, we soy too thot the multiple of the first exceeds the multiple o the secondond the multiple of the third does not exceed the multiple of he fourth.Let us now mention ihings seporotedly os Euclid mentions ihem. so we soy, There orefour mognitudes ond ii is possible to find equimultiples of the first ond the thirdond equimultiples of the second ond the fourth so thot the multiple o the first isfound to exceed the multiple of lhe second ond the multiple of the third not to exceedthe multiple of the fourth, then the rotio of the first to the second is greoter thon therotio of the ihird to the fourth. And os to its converse, o{ which we demonstroted thewoy of its finding, it is this: When the rotio of the first to he second is greoter thonthe rotio of the third to the ourth, then it is possible to ind equimultiples of thefirst ond the third ond equimultiples of the second ond the ourth so thot the multipleof t'he first is found to exceed the multiple of the second ond the multiple of ihethird not io exceed the multiple of the fourth.We thus demonstroted in his writing the soundness of whot Euclid soys oboui themultiples ond thot he does not choose the explonotion of rotio by meons of themultiples withour the reoson of their sound connection with rotio. lt is even impossibleto ind onylhing thot is more soundly connected with rotio ond stronger in rigour ondexoctness os to its sections ond properties lhon the multiples, becouse nobody condoubt thot rotio is not but o comporison of o mognitude with onother mognilude inorder to know the size of one of the two os compored with ihe other. And the com-porison o the less os compored with the greoter is only known by porls of ihe greoter,ond the multiples ore only on enlorgement of fhe ports, ond the multiples inclide the

42

c# D


5/20

comporison of the less with the greoter os well os the comporison of the greoerwith the less, ond with ports it is only usuol lo compore the less with the greoter,olthough in ihis writing, while mentioning the ports, we hove poid no ottention to whotploce we oiloch io ihem becouse of the opinion (Q) we hove given obout ocilitoiing thereosoning ond the veriicotion of it in beholf of our need of its frequent repetitionond use. For the more it is used in the reosoning, the more there is occosion for reliefond obbrevioiion.And os io {inding in the cose of the greoier rotio ond the equlmultiples token of thefirst ond he hird ond the equimultiples token of the second ond the fourth with theproperty thot the multiple of the first exceeds the multiple of the second ond themultiple of the third does not exceed the multiple o the fourth, how is the woy toproduce these multiples, os this condition is not found in oll multiples necessorily, butonly in some multiples, os we hove mentioned, well, we know o meons by which it isproduced, but only ofier we know how to produce o ourth mognitude hoving o rotioto lhree given ond known proportionol mognitudes. We wont to f ind onother,ourth, mognitude which will be the fourth of them ond fitting them in such owoy thot the roiio of the irst given one to the second given one is os the rotioof the third given one o the fourth sought one. For if we do noi know how toproduce this {ourth mognitude to the three given mognitudes then we will notknow the meons to get ot inding the multiple in the oforesoid condition either. ln thesixth book (is wriiien) how to produce to two mognitudes o third proportionol ond t isolso within the scope of thot book how to produce o fourth (proporionol) to three.However this only opplies when the mognitudes ore oll lines, but os to ongles, or suroces,or solids we do not see this. We hove seen people trying to produce his oforesoidmultlple ond exerling themselves in finding this fourth mog,nitude without o propermeihod of working which might enoble hem to find it. For me there is no diferencebetween him who is exerting himsel in inding this fourth mognitude ond (him who is)exerting himself in finding the multiple with the oforesoid property. And on this groundwe think,ond ihe Lord knows beiter, thot Euclid uses this multiple in the twelfth propo-sition of the {ith book (l) wihout it bein'g menlioned before,how is the woy to find it.Something of the kind does not weoken the proof ond does not detroct onything fromit, becouse, if the thing is necessorily exislent, it con do no horm thot its exisienceis used in the orgumentoiion ond ihot it is used in proving other things with it. For ifthis were no ollowed in o proof we were no more permitted, or instonce, to believethot the side of the sepogon inscribed in he circle is less thon the side of the hexogonond greoter thon the side of ihe octogon both inscribed in one circle, nor thot heside of the plone hendecogon is less thon the side of the decogon ond greofer thon theside of the plone dodecogon both in one circle. For the construction of the septogon-

r44


6/20

chord is not possible in the geometricol woy. Bu when we wont to show thot it isgreoter thon the side of the ociogon (R) we drow o chord ond suppose it to subtend theseventh port o the circle, by ossumption only, not in reolity, ond then we properlyorronge the proof, ond the lock of procticobility does not horm in this theoreiicol motter.It only horms in the performonce of the drowing of it, e.g. when we intend to constructthe side o 'the plone regulor tetrocoidecogon inscribed in the circle, we ore not ollowedo drow o chord ond lo suppose, ot vorionce wi,th the truth, thot it subtends the sevenfhport of the circle, then'to divide the orc into two equol ports ond to finish the {igure.The like is not ollowed ond ihe distinction is cleor. For thot the like is ollowed inestoblishing o proo ond is not {it or procticol perormonce of o drowing is evi.dent,i God, who is lofty, is willing.And os our words hove conried us so for ond we hove mentioned whot we hopedto mention, it is the momeni for us to breok off our reosoning ond to conclude it tothe proise of the Lord, Whose glory is greot, hollowed be His nomes. We proy to Himfor orgiveness ond to excuse our error ond stumbling ond o guide (us) to whotpleoses Him of truth in word ond deed.Finished is the commentory on rotio occord'ing to the reosoning of the okhAbO (Abd Allh Muhqmmod ibn Mu(dh ol-Dioiin,God hove mercy upon him.


7/20

ru. OHER COMMENTARIES ON RATIO.'I . Al-Dioiin.wrote his commentory (l) in deence of Euclid's work becouse somepeople were not sotisfied with it ond tried io moke it complele or cleor occording to theirown thinking' ln porticulor in view of Euclid's definiiion of proportionol mogniiudes heremorks' For mony think thot Euclid opprooches the explonotion of rotio from o doorolher thon ils proper door, ond introduces it in o wrong woy by his definiiion o itby ioking multiples, ond in his seporoting rom its deinition concerning its essencethot which is understood by the very conception of rofio; ond they iudge ihot thereis no obvious conneclion between rotio ond ioking multiples.2. Beore veriying os to how or oiher monuscripts beor oui ihis slolement ofol-Dioiini's, lom bound to discuss the Greek text thot wos subiected to the soidcriticism, confining myself os much os possible to Euclid's conception of roio qnd hisdefinition of proportionol mognitudes.For o better understonding of whot could be soid oboui the outherrticity of ihe Greektext one must toke into occount the port thot Theon of Alexondrio octed in is trodition.About ihis Diiksterhuis (2) writes the ollowing,"The woy in which the text of the Elemenls hos reoched us hos been greotly inluencedby the version o it written by Theon of Alexondrio (end of the 4th century A.D.).Theon's obiect seems to hove been no so much to render the text os pure os possibleos to moke the confenls os cleor os posslble; he rectified ,n-ors, or whcrt he thoughtto be errors, modernised the mode of expression, inserled propositions, corrolloriesond coses thot Euclid opporently hod thought superfluous, ond elucidoted the courseof proos where he thought it lioble for possible dificulties. The consequence of thisopening up the work wos thot loter Greek outhors used Theon's iext olmost exclusively,ond perhops one never wculd hove ound out how much he oltered in ihe otherwordings, if Peyro.rd hod not discovered in I 8l 0 o monuscript the ext of whichwos older thon Theon's. This so-ccrlled monuscript P wos used by Heiberg in composingihe fomous text-edition thot by this time generolly underlies the study of Euclid."3. The firsi seven definitions in Book 5 occording to Heiberg ore,oc'. Mpoq oci pye0oe g.ey0ouq c l. A mogniiude is o port o o mogni-orooov roU pe(ovoq, 6rav xxru-perp r pe(ov.p'. IIo),ocn).otov 8 c pre(ov coU&rcovoq, (nav x*ragerpicar, rroU ),rtovoq.y'. yoq oti o pr.eye0v pr.oyevdv\ xa-r n"r1)l.N6r7r now otr,or,q.48

tude. the less of the greoter, whenit meosures the greoter.2. The greoter is o multiple of the lesswhen it is meosured by the less.3. Rotio is some stote of two mognitudesin conneciion with size.

'. yov 7.er,.r rpq ,ii).l1).oc peyOr1yerocr,, & 8vorrar no),),ocrcocor,a(-pr,eva ritr1cov rcepXer,v.e'. 'Ev rQ arlcQ ).yqo g.e101 ),ye'catev*r, rp6rov npg ecepov xaitprov Yrpq crapcov, 6tav td, coUnptou xocl cptou iod.xr.6 no),),ot-r\ctovtx rv roU eurpou v"d.i 'ce-rdpcou ioxr,q no),ocnao


8/20

c. or ofter 6': d,vaoyor 8 otrv { rv y


9/20

is lhe stoe of pors (hlu'l-'odiz)i), whirst ihot remoinder is one of the ports othe greoter, or the ofoir dces noi orrive ot o remoinder thot meosures the foregoingenirely, but the process continues od ininitum. This losi stote of one mognitude wilhregord fo the other is the stote of incommensurobility.,,After this the outhor remorks thot Euclid soys: ,,Being proporiionol is similority ofroos", ond thcrt this similority implies "o comporison -(muqlosotun) of the stote oftwo mognitudes (with regord o eoch other) ond ihe sfote of two (other) msgn;1gjs5(wit'h regord io eoch other) thot ore in ihe some rotio ((ol nisboti,hi),,.,,Wen fhissloie o the first two mogniiudes is olso he stote of the other two mognitudes, thenit is soid thoi this stote is the siote of similority of rotios. When however motters stondolherwise, then this siole is no the siote of similority ond there is no being proporionol.He (i.e. Euclid) soys similority, becouse this stote is o quolity (koifiiotun) ond no oquo.ntity (kommiotun)' For when the stote of ihe irst two mognitudes (with regord toeoch other) is the stote of equolity, then the stote of the othr two mognitudes (withregord to eoch other) is the stote of equolity too; ond when the stote of he irst twomogniudes is the stote of multiple, or the siote of port, or the stote of poris, or oneof the other slotes of rotio thot ore found in the cose of commensuroble mognitudes(ow s)iru ohwli 'l-nisbofi 'ili hiio li'l-moqdri ,l-musiiorikoii), rhen the lrote ofihe two olher mognifudes with reqord to eoch other is the some stote too. This is oquolity ond not o quontity. And in occordonce with this is the stote o incommensurobles(with regord to eoch other)." The outhor then eloborotes this point giving due ottention1o the numbers of imes (quotients) thot eoch remoinder meosure-s thJ foregoing, oIess remoinder olwoys being left ot the end. Thereupon he soys,,,When thi. qrnce(ol-todwulu) in meosuring the remoinders never ceoses to lurn out equolly od infinitum.then it is soid thot the mogniiudes, their stote being os mentioned, ore proporlionol,ond this stote is he stote of similority o roios.,,After this he outhor first gives on exomple ond extends his reosoning to the stoteof being greoter in rotio, whereupon he soys thot Euclid gives ihe deinition,,,Fourmogniludes ore soid to'be in one roiio (f nisboiin whidoiin), the first to ihe secondond the third 1o the fourth, when equinumerol (ol-mutoswiiotu 'l-morrri) multiplesof the first ond third, eiiher olike exceed, or olike ore equol to, or olike foll shortof equinumerol multiples of the second ond fourih, when compored consecutively (idhqisot col'l-wil)i) one with the other. The outhor expresses his opinion,,thot tLis is omcrler lhot does not need o proo, becouse it belongs to the principles for him who hosodvonced so {or. For every book (10) hos its principles occording to its ploce in theronge of books (inno hdh sioi)un loiso luhtdiu (oloihi il burnin li-onnohu mino'l-ow'ili (indo mon inroh il hdh'l-mowdi(i idh kno li-kulli moqlotin ow)ilu

J1

bihosbi moroboii tilko 'l-moqdloli)." Then follow Euclid's wordore in one rotio be colled proportionql (muionsibotun)", ond "multiples ore such thot the multiple of the first of them excesecond, ond the multlple of the third does not exceed the mulit is soid thot the rotio of the first tot ihe second is greoter thoto he fourth."So for ol-Noiriz.6. Ibn ol-Hoithom (l l) in his commentory on Euclid's premisseis concerned, writes the following, "Rotio is o certoin relotioncertoin relotion it is only found with two things." Then he dividerotio (ol-nisbotu 'l-mudjmolotu) ond the deinje rotio (ol-nisbfirst contoins ossertions obout greoier ond less, or somethingdefinite roiio however is subdivided into three species; l the double, or o certoin multiple, or the holf, or o certoin port, ois composed o these; 3. when we soy: the rotio o this mognis the rotjo o this other mognitude to this other mognitude.combined in two species: "the numerol rotio (nisbotun (ododiiois not numerol (ghoiru .ododlotin)." Numerol roiios ore themultiple ond the ports, o non-numerol rotio is the roiio hot cbe expr,essed (hiio md iumkinu on i0diodo wo Id iumkinurotjo is found wiih the continuous quoniities (i'l-kommioti 'l-mube ound with numbers, or inslonce where irrotionol rootsrotio ound with continuous mognitudes he outhor soys thotinto two species, the numerol rotio ond the non-numerol one. "Tcose of continuous mognitudes, one of hem to the other, is likto o number; ond this is cleor. For omong mognitudes (c(mogniude) of whlch is o multiple of the other, or a portequol to it. All these rotios ore found with numbers too."As to the mogniiudes with o non-numerol rolio the outhor remistic of eoch poir o mognitudes of thjs kind (min chs,sohdhihi 'l-sifoti) hot when one is meosured by the other, etc., tdo not orrive ot o remoinder meosuring the oregoing one.After hoving expotioied on the numerol rotio the outhor soys:found in oll specles o he continuous quontiiy, which compriseoncients, is thot rotio is the idenlicotion of the meosure (oijiomognitudes with regord to the other. A porophrose (tofsrunquontity of mognltudes is in question, rotio is the very notion


10/20

'l-mo(n 'lldhi f kommijori 'l-moqdiri 'lldh jus)olu .onhu bi-oilin). More recenroulhors defined rcrtio with other worclings (bi-olzin ghoiri hdhihi) ond this is whothey soid: lt is o cerloin relotion (idfotun m) of t*o"ho,nog"neous rnognitudes os iomognitude (fi 'l-miqdri). Euclid mentions the definirion of iotio in Book V; in somemonuscripts ( bo(di 'l-nusochi) it is found in the {irst wording, irr other lronus.ripts;in the second one." lbn ol-Hoithom's opinion is ihot both qreorrect.Then he enters into detoils obout ports ond multiples, bringing up- ,or" geometricolquestions ond one of linguisiic r.roture too when soying th*ot-,,di'.un,, ,"tty rrl"on,"equol" (ol-di(fu huwo,l-motholu).Then ollows the definition: "The mognitucJes thot hove o rotio to one onother ore thosethot mulriplied (idh d0(ifot) ore topoble of exceecling on, noth"r.,, The outhordemonsiroles thot it points to being of ihe some species. The next de{inition runs osollows"'Mognitudes ore soid to be in one rotio, the first o the second qnd the thirdto the ourth, when equimultiples token of the irst ond the ihird either olike exceeclequimultiples of the second ond the ourth, or olike ore equol to them, or fqll shortof the,m, when token in (due) order, whotever the mulripies moy be.,, The outhoreloborotes on this point,. ofter which he quotes Euclid,s words, ,,Le mognitudes whichhove one rotio be colled proporlionol." He then recopitulotes the foregoing by soyingthot .proportionol mognitudes ore hose hoving the soid chorocteriiiic l.h*orni"And", soys ibn ol-Hoithom, "upon my lie, it is on indispensoble chorocleristic(chssoun lzimotun) o. proportionol mognitudes. However, when only stoted ii isnot cleorly underslood (ill onnoh loisot zhiroton li 'l-fohmi bi- mudiolrodi ,l-qouli),but needs o proof." To the proof he gives lwill return loter on (13).So for ibn ol-Hoithom.7.. Of lhe commentory (14) of cUmor ol- Choiimi (lS) rhe second chopter deols wirh"roiio ond being proportionol ond the true essence of either.,, lt begins os follows,"he element-writer soys obout the essence of rotio thot it is the ideniificotion o themeosu.re*.(oij1otu qodri) of two homogeneous mognitudes, one wiih regord to theother." Then the ouhor exploins the meoning of htmogeneous by meons o Euclid,snext deinition (the 4th), ofter which he returns to the foregoing one. He developsthe succession of possibilities in the usuol woy, viz. orguing- thJt the less is eithero port of the greoter ond tokes it up entirely when they ors reloted to one onother,or it is ports o it, or yet something else ((olo wodihin choro).,, After this he soysthot "it is chorocteristic o quontity to be submittobie to the considerotion of eqrolor unequol. Now rotio is ihe essence of this considerotion when two homogeneousmognitudes ore reloted lo one onolher ond of the consideroiion of someth'g elseconnected with it, ond this is the mogniude (miqdru) of thot rotio when it is J rotioLt^JT

of mognitudes (nisboun miqddriiotun). This is more cleor in th(ol-(ododldtu). The irst thing found os to roiios is oundthot numbers ore reloted to one onother, ond it is ound thor unequol. Then unequols ore compored ond it moy be foun(icr(uddu) the greoter, os three (meosures) nine. Then the nuis looked for thot three meosures nine, ond three is found: nine three imes. Of this the nome is deduced in occordoond it is soid to be the thi,rd port; hence the roiio is one ihthis considerofion to ihe cose of poris, ofter which he conor this notion with mognitudes too ond finds wih them,o third cose in consequence of the fqct thot mogniudes ore nports, ond thot their division hos no definite end os is thedhliko onno'l-moqdiro ghoiru murokkobotin mino'l-odizloiso li-'nqismihim nih6jotun mohd0dotun kom li'l-'ododposed of indivisible ports, eoch of which being the unit."Ater this the outhor discusses how the less is subtroctedremoinder is left less thon the less. ond how this process leoo numbers. For ihis he refers to Book 7. With mognitudes hobound te leod to of end, which Euclid discusses in Book .l0."However", tUmor declores, "we do not need this in ourhow motters stond it does noi {ollow os o motter of courmognitudes the less is either o port of the greoter, r porpossibilily is to be observed, not in the woy of numbers, but ilf now somebody should soy thot these ihree coses do notbui only the two coses thot occur with numbers, then we replgive us crny trouble to view he rules o rotio ond proporiionShould this subdivision prove io be obsurd in the course owould not be to blome or ii. Should it prove however no tomentioned it ot leost ond hove exhousted oll possible cosrom which other very deep logicol mysteries ore to be lgrosp it l""Then he (Euclid) comes to being proportionol qnd sols: TFrom the linguistic point of view this is o nice phrose, butof it he deviotes excessively from the rue essence (hoqqotunAfter this the outhor quotes the sth definition ot ull lengremcrrking however thot it does not infringe upon tlre importFor when, for instonce, ihe first is the holf of the second heholf o the fourth ioo. Then he quotes the definition of who


11/20

continues os follows: "These ore his (Euclid's) words obout being proportionol, ondwe will coll it the common woy of being proportionol (ol-tonsubu 'l-mosjh0ru). As forourselves we will deol with the true woy of being proportionol (ol-tonsubu'l-hoqiqiu),whilst the whole Sth book deols with the common woy."hen the outhor develops his own ideo, the first port of which he recopitulotes osollows, "When there ore four mognitudes ond the irt is o port o the second, ondfhe third the some port of the fourth; or the irst is ports of the second ond the thirdthe some ports of the fourth, ihen the rotio o the first to the second will not oil tobe like the rotio of the third io the ourth. This is the numerol rotio (ol-nisboiu'l-(ododilotu)." Ater this follows on exposiiion of whot is found when the mognitudesore incommensuroble ond the successjon of remoinders in the we]lknown process doesnot come io on end. lf under lhese circumsionces the number of imes (quotient)thot o remoinder rneosures the foregoing (remoinder) in the relotion of the irst(mognitude) to the second mognitude is equol o the number o{ times (quotlent)tho the corresponding remoinder meosures he foregoing (remoinder) in therelotion of the third (mognitude) to the fourth mognitude, ihen he soys thot therotio of the irst to he second is like the roiio of ihe third to the ourth, "and", he soys,"this is the true woy o being proportioncrl in the geometricol kind."So for cUmor ol-Choiinri.

IV. o,taussroN oF THF DTFFERENT ..MMENTARTES AOF THEIR PURPORT.l The Arobion commenlories quoted in the foregoing similority. As opposed to Euclid's woy o{ deoling with the sthey propose onother method, which is in oll four coses essenquestion orises whether we ore entilled to toke this foct orof thinking typicol of the scientists o the Moslim period. Aoctivity however wos so lorgely dependent on the Greek trodbeing very suspicious of conclusions obout essentiolly new ideiri recent yeors the birth ond growh of Greek mothemoilcsgoted (1). The notion we ore now oble to form of ils evo Dijksterhuis o.o., shows o twoold crisis cri the end ofLeoving oside the difficulties with regcrrd io continuity ond ininto the other stumbling-block' the discovery thot irrotionolity iviz. thot "the implied ossumption thot eoch poir of homogroiio o o number to o number" is incorrect. ln this connecithot the Greek notion o{ number comprised only the iniegerssolution of the dificulties orising from the soid discovery woEuclid deolt with proportions in Book 5 of the Elements. Howto be seen whether the Greeks could hove cieored fheir retheir notion of proportionoliiy on the oforesoid error in o woriginol conception of rotio.2. lt is more thon likely thot oitempts of the kind hove bpresumobly olmost crt ihe some time in which Eudoxus mouldeln the concrete orm known to us rom the prefoce of BoThe following short exposition moy support the conlecture {olf two homogeneous mognitudes ore commensuroble they hviz. o mognitude exists going into both precisely when they osuppose for o moment thoi ihe common meosure of two m35 times in o ond l times in b, which numbers however oremeosure lhe greoter by the less ond find,thot is goes 2 timeless thon b. This remoinder is 3 times the common meosure, buThen we meosure the less by this first remoinder ond findleoving o second remoinder 12, less thon 11 . This secondmeosure itself, but we do not know this beforehond. We thremoinder by the second remoinder ond ind thot it goeHence we hove, 11 : 3r23=ilr :; =l;Ji,=f


12/20

ln his woy we find the rotio o two mognitudes together wiih iheir common meosure.Euclid proctised the method in the Books 7 ond 10, using the term d,v()ugar,pevto denote the olternote meosuremenl o the successive remolnders.Now we toke two different poirs of mognitudes ond try to find the roio o either poirin the qforesoid woy. Eoch poir for itself mus be homogeneous but no more is required.The relotion of the mognitudes of one poir "in respec o size" is effectively chorocterizedby the series of numbers (quotients) indicoting how mony times eoch mognitude orrernoinder goes into the {oregoing (i.e. next greoter) one. Therefore it is not for-fetchedto soy ihot it would hove been in perfect occordonce with the originol (numerol) Greeknotion of rolio to coll the rqtio of the {irst poir equol to the rotio of ihe second poiri the soid series o quolients in both coses is the some.However, of the end o the Sth century B.C. the Greeks were well owore of the foctthot or from oll poirs o mogniiudes hove o common meosure. Hence they olso knewperfectly well thot the series o{ quotients in mony coses does not orrive ot o definiie end.For if it does o common meosure is found ond the rotio will be numerol.l now, finolly, iwo pcrirs of mognitudes produce the so,me in{inite series of quotients,neither hos o rotio in the originol (numerol) sense. Nevertheless it would not hove beeno lorge step for the Greeks to declore: Two poirs o mognitudes hove the some rotioif the series of quolients in one poir is the some os the series of quotients in the otherpoir, irrespeciive of whether these series ore finite or infinite."Thot this method hos ployed o port in Greek mothemotics onyhow", soys Diiksterhuisin 1929 (4), "is confirmed by the monifoid occurrence o opproximofions of irrotionolvolues thot upon exominoiion oppeor io be opproximoting froctions of developmentsinto continued froctions". ln our exomple of this porogroph the rotio o o to b con beexpressed by' d^lb_r It-;' -1which in o corresponding woy lecrds to on infinitely continued froction in cose the rotlofor lock of o common meosure is "irrotionol", tippq'coq.Moreover we hove evidence yet more closely connected with the heory of rolio.3. ln Topics Vlll.3, 158b, 29-35 Aristoile refers to o definition of "hoving the somerotio" ihot comes o "hovlng lhe some d,vravaipeor,q ". Heoth (in ,l908) reolisedthot it wos not identicol with the Euclideon one, but {oiled to give o sotisfoctoryexplonotion. This wos, probobly for the irst time, given by Zeuthen (in lglZ), but itseems to hove remoined unobserved until 192 when Junge independently of the formerpointed to lhe some obvious solulion. A more eloborote study of it wos then producedby Hosse ond Scholz (in 'l928) (5) ond in porticulor by Becker (in 'l93) ().

Zeuthen's solution is the following.ln his commentory Alexonder soys upon the possoge mentioof proportion thot the oncients used is this: proportionol ore mwhich 'hove the so,me d,v0ugcrpE6tq . Aristotle however c&vravaLpectq ."It is evldent ihot for Alexonder the meoning of dv0ugaperfectly cleor. From where then did he get the ierm? An obEuclid, whom he cites more thon once ond whose work heknowledge. Now Euclid does no use lhe noun, but in 7; 1,2verb civ0ugaupev in the sence of subtrocting in turn onor o remoinder of ii from the other or its remoinder in order to hqndled o proportion by meons of on d,v0ugotpeor,6 , i.emeosure. At this point of the reosoning he conclusion leops to continued roction. They were perectly owore of the foct thorotios, "the relotion in respec of size" being the sqme, the imthis someness wos the oppeoronce of two identicol "choins" oor infinite. Their definition there{ore covered rotionol os welond in this respect wos not inferior to the Euclideon one.Why then, we might osk, did it foll into the bockground? A plobecouse o its improcticobility. This solution however wos ddeveloped o generol heory of proportion bosed on the oond exten.ded so or os to cover the needs of Euclid l0 ()'4. Hoving thus found by o lucky chonce o single possoge iin o mothemoticol work) pointing to the existence in themothemotics of o current conception of rotio devioting esseone, we find in the Moslim period on ovowed preference foGreeks seem to hove kept dork so coreully thot it oppeintentionolly. ln olmost every writing on the subiect fhe Euche greotest common meosure of two mognitudes is ossumedoll ideos obout rotio ond proportionolity. Nevertheless we m{oct thot qn eloborote Arobion theory of proportions, boseconception of rotio hos no been found u,p to the present.this is moinly o consequnce o the greot perfection of ucthe subiect motter, o woy thot took the wind out of the soilsThot the Greeks did not so much qs mention the whole thoter oll. At thot time their ospirotion for beoufy on'd virtue, fmet with he temptotion of octuol finolity. Perfection seem

58


13/20

fheir reoch, thence they wei"e impotient wiih oll thot could noi stond its est. For Sportonworrjors to dispose of weok infonts wos os noturol os the proctice of Greek mothe-moticions in Euclid's oge never to publish onything in o siote o deeciiveness oruninished development; ond crs long os the onthyphoireiicol woy of deoling withproporiions could not beor comporison wiih the Euclideon method the)r were rglhrinclined to ignore il. They invented the synthetic woy of enuncioting fhe outcome offhe)r invesligotions, ihot up io the present is used in most mothemoticol textbooks. Theexception thot proved the rule wos ound in 'l90 by Heiberg, the so-colled Ephodosof Archlmedes, o writing by the greot mothemoiicion deoling with he woy in which hehod deduced some theorems thot he hod proved elsewhere in o rigorous woy b,yexhoustion (7).5. A posiive evoluotion of Moslim mothemoticol proc{uctivity must oke occount of odiference in ottitude belween the Greeks ond the Arobion scientists o the middle o'ges.The Iotier enterloined o deeprooted venerotion for the Greek clossics, toking them forperfect olmost by oxiom. As to their own work, however, they hod no pretensions ofthe kjnd, ond they reveoled iheir troin of thoughts without ony difidence, beinginerested ot the some time in tho of olhers. This mode hem view the study ofmothemotics from the psychologicol ongle more thon ihe Greeks did, who confinedthemselves olmost exclusively to logic. The Euclideon doctrine of proportions indeedmel the relentless requirements o contemporory logic, when previous woys of deolingwith roiio hod become disc'redited crs o result of the discovery o irrotionolity os ogenerol phenomenon. From the form in which it wos presented, however, liitle or norhingcould be deduced regording lhe woy in which it hod come into being.Now obou rolio ond proportionolity, unlike most other dificult notions in mothemotics,o certoin omount of knowledge will originote more or less intuitively ond spontoneouslyin the mind of ony person occupying himself with them in on oitentive woy. lf he is oiroined mothemoticion he will no doubt be disposed to odmi thot o logicol iustificctiono his views is required. When, however, o iustiicotion of the kind ossumes the chorqcterof o theory not reminding him in ony woy of his own conceptions, then his, inner selfwill rebel ogoinst ii, irrespeciive of his possible odmission thot the soid theory connotbe opposed on logicol grounds. The chonces ore thot he will either try to obtoin onequivolent resul in o woy more in line with his own thoughts, or try to build o bridgebetween them ond the unsotisfying theory.6. The quoted commentories show tho both otlempts hove been mode in the Moslimworld, olthough the irst does not oppeor crnywhere to be corried through to qn ext'enflike Becker's effort in .l933. The different methods exhibit o morked similority. ln oll0

coses the reosoning storts from the numerol rotio, whilst the Euhe greotest common meosure underlies oll ottempts o opFuclid's conception of rotio ond his definition of proportionoI will troce this briefly in the quoied references.Al-Mohn (respectively Thbit ibn Qurro) brings out cleorlymutuol be,hoviour of two mogniiudes when compored with ohe Euclideon process o finding the greotest common meosumognitudes ore proportionol when the two series o quotientsore identicol.Al-Noirz develops esseniiolly the some iheory. He builds no bo proportionol mognitudes, but expresses his opinion thot thof o principle.thot must be ossumed ond connol be proved.lbn ol-Hoithqm disiinguishes between numerol ond non-numerit is chorocteristic of the lotter thoi the Euclidesn process of inmeosure does not orrive ot on end with mogniudes hoving suAs opposed to ol-Noirizi's his opinion is thot Euclid's definproo functions os o bridge between the two diferent ideThrough the intermediory of i lbn ol-Hoithom hopes to preservconception, viz. psychologicol sotisfoction os well qs logicol iusto the following.Three premisses precede:l. When there ore four mognitudes ond the rotio of the firthe rotio f the third to the whole fourth, then the rotio oof ihe second is the rotio of the third io o portion of ih2. Eoch mogniiude is copoble of being holved ond eoc,hholved ond e'och hol of o holf is copoble of being hoports becomes groter fhon o prescribed number.3. When there ore two di{ernt mognitudes the less is countil the multiple becomes greoter thon the greoter mognThen the outhor tokes our mognitudes, supposes them to bequimultiples o the first ond third ond of the second ond outhoi the property of olike exceeding, folling short, or beingthem. He reminds fhe reoder of the oct thot the rotio ofnumerol or not numerql ond begins by supposing thot it isis oble to prove hot the mognitudes hove the property in qWhen however the rotio of the first to the second is not thnumber the outhor begins by supposing the multiple of themultiple of the second. Using his premisses he now tokes o m


14/20

thon the second, hoving o numerol roiio to the irst. lt follows thot olso o mognitude M2con be ound, less thon the fourth, ,hoving the some numerol r.otio to the third. He isoble to moke the difference beween the mognitude M1 ond the second mognitudeos smoll os he desires, so smoll indeed thot the multiple of the first mogniude is stillless thon the multiple of M1 . The lotter mognitudes hqd o numerol rotio fo one onother,whence it follows thoi the multiple of the third mognitude oo is less thon the multipleof M2, which is less thon the fourth. But then ihis multiple is less thon the multiple o{the {ourth mognitude o fortiori.In other possible coses the reosoning runs in o corresponding woy.(umor ol-choiim cleorly tokes his stond right owoy. He too boses his reosoningon the numerol rolio ond from there deols with non-numerol roios by meons of Euclidtprocess of finding the greotest common meosure. His definition is: proportionolity issimilorify of rotios. He tokes Euclid's definition for on interpretotion o lhe former,-buthe is not ot oll content with it. lnsteod he is of the opinion thot Euclid deviqtes exces-sively {rom he true essence of being proporlionol, embodied in the identity of theoforesoid series of quotients. his true woy of being proporfionol is the subiect motterof his treotise.7. Coming bock to ol-Dioiln lwish to begin by stoting thot the criticism of Euclidwhich he so violently obiected to reolly existed. Even the rother strong ossertions os1o the lock of connection between rotio ond the muliiples we find bock-in the treotiseof (Umor ol-Choiimi, who must hove been o coniemporory o his,As for h'imself ol-Dioiin is full o{ odmiroiion for Euclid. Thot is why he undertook todefend him. He is convinced of the correciness of Euclid's definitions ond subsequentreosoning; hence he desires io convince others too. ln order lo ensure o reosonobleomount of succes he irst needs o common bose occeptoble for both porties. He tokesthe line thot o primitive conception o rotio ond proportionoiity is {ound in the mindof every right-thinking person. From this he derives o number of truihs chorocreristicof proportionol mognitudes, whic,h he tokes or perfectly evident (9). The irst is thot ino proportion he first 'er'm contoins os mony ports of the second qs the third contoinsports of the fourih. Expressly he soys thot this is so evideni thot o closer proof issuperfluous. ln view o the phenomenon of irrotionolity, however, this truth constitutesno fruitful ideo to be used os o storiing-point for the theory of proportions in the coseof mognitudes. lf however the irst is o whole number of poris of he second ond thethird the some number of porls of the fourth, then proporionolity is ossured. But yetsornething else is not. open to ony doubt' if the first contoins more or less ports of ihesecond ond the third the some.number of ports of the fourth, then proportionolity isoltogether out of the question. This too needs no proof "becouse it oppeols immedioely62

fo the mind", or in this cose "the size of the irst os comporelike the size of the third os compored with the fourth". Thisof the cited critics too when they speok o the meosure o oto onother mognitude, whilst Euclid conines himself to "somein conneclion wiih size", o view not unknown to the soid criticdistinguished from ihe former. The subtle diference is thotleods to meosuring o rotio, whilst Euclid only meont to introsimilority o fwo rotios. Al-Dioiini is convinced thot his stotenot need o proo, "since hings ihot ore cleor ond evident to tproo ore not mode cleorer by prolixity in the explonotion, becto moke cleor who is olreody cleor in it".Ater this the outhor opporently hos ocussed his otientioto prove, viz. thot the Euclideon condilion of multiples impproportion. Hoving exploined olreody whot he meons by "procee'ds to moke the connection. For thot purpose he conveporls, whot comes to porting by the number o the irst muit is evident thot mognitudes ruly proportionol occording Euclid's condition too, For otherwise, if some ports of the seconthe first mognitude while the some ports of the fourth did not ewould contoin fewer ports of the second thon the third o the oolreody proved the impossibility.Now only the converse wos yet io be proved, viz. thot mogniEuclideon woy were proportionol in virtue o his own considhe produces is on indirect one showing much resembloncol-Hoithom. The existence of o {ourth proportionol ond themognitudes underlie his reosoning. The demonstrotion tokes thelf our mogniudes AB, C, DE ond F ore o such o kind thn->- F =- DE ond conversely, then AB : C : DE : F. Fom


15/20

inconsistent with the supposilion hot AB ' C : DG ' F. This supposition thereoremus be reiected.ln the some woy it con be proved thot the supposition AB : C : DG , F leods to ocontrodiction i{ DG < DE Therefore, soys ol-Dioiin, it is only true thotAB:C-DE:F.Afier hoving demonstroted in this woy thot Euclid's chorocteristic of proportionol mogni-udes ogrees with the conception of proportionolity thot will orise spontoneously in omon's mind, ol Dioiin inlroduces the Euclideon multiples. This would come to o simplemultiplicotion by m in our, more modern. notoiion, but the ouihor, not hoving hedisposol o our surveyoble meons of expression, needs our poges.Then he roises in o rother unconvincing woy the odvontoges of Euclid's multiples.He closes the second port recopituloting Euclid's "introduction", only chonging hisorder. ln occordonce with his previous orgumentotion, however, this meons thot hereploces Euclid's s,totement, which is o mere definition. by o convertible proposiion.The third port of ol-Dioiidn's treotise deols with unequol rotios. The storting-point,once more not needing o proof, is thot the first rotio is greoter tho,n the seco-nd, if theirsi mogniiude contoins more ports of the second mognitude thon the third o thesome ports o the fourth. When, on lhe other hond, the first roiio is greoter thon the'second, then "some ports moy be ound of the second ond fourth so thot the firstcontoins more ports of the second thon ihe third of ihe fourth."Before possing on to the proof fhe outhor first introduces the following simpliicotion.ln order to be sure o pro'portionolity, he soys, it is sufficient to know thot ; B ? Agoes with a O ? C, without equolity being mentioned. ln point of foct he demonstrotes-m\thot the relotion of the inequoliiies entoils the relotion o the equolities. When nowthe irst rotio is greoter thon the second the ports of the second mognitude will onlyexceed he ports of the irst mognitude, if the ports of the fourth too exceed the thirdmognitude. For otherwise the third would contoin more ports o the fourth thqn thefirst of the second, so thot the second roiio would be the greoter.Then follows the proof tho it must be possible to ind ports of the second less thonthe first, whilst the some ports of the'fourth exceed the third, os soon os it is estoblishedthot the irst rotio is greoier thon the second. For i, for instonce, this would not opplyto the mognitudes A, B, C ond D, then t a A would entoil p D < C withoutcxception,while? B > Awould entoil I O > C, becouseiheroiioof AtoBisgreoterq-9'64

thon the rotio o C to D. Then, however, we would hove pwith the supposition.All this is convered into rnultiples, oter which the outhorof o greoter rotio by o convertible proposition.8. By woy of summory I now wish io remork thoi the siudymothemoiiciens betroys o mentolity diferent from ihe Greirst, o consequence o the foct thot they ore not the creotototors of the molhemoticcil ocquisitions under considerotion.loy on convincing olongside of proving points to their opconnecied with ihe tronser of science, which con not be mo(or in their iime "copying"). hey kepi the fire burning. lnbecome, os o noturol philosop'her, ihe predecessor ond leono doubt deserves o slmllor honour os o mothemoticion. Athe spirit gives woy to erudition, the productiviy ef 11.'u Arobicvolue, dries up in o desert of drob sterility. To conclude mythis view by the following quototion from ol-TOs (,l0).9. ln his introduction o Book 5 ql-T0s declores thot meothe other is the essence of rotio. He distinguishes four comultiple, pors, ond the cose of incommensurobility.When mognitudes ore copoble o exceeding one onoiher ihno doubt hove o roiio to one onother. lf two oiher (or pohove o rotio o one onother ihot in no respect is dihe four (or three) mognitudes ore colled proportionol. Thewoy of inding (or not finding) the common meosure ond ihe othot, consciously or unconsciously, his conception of proporiokind o relotion to one onother in finding the common meoo nto no iresis."Concerning proporlionol mogniludes the or,thor slotes thol wony some multiple is token ond likewise of the second crndthird will exceed the multiple of the ourth if the multiplmultiple o the second; ond likewise in the cose of equolityOf this "definition" of Euclid's the outhor gives on ingenious bwhich I reproduce ot full length. lt runs os follows."Let the rotio of A to B be os the rotio of C to D, ond let someC, viz. E ond F; ond o B ond D some muliiple, viz. H ondH, then F exceeds K, ond if it is equol it is equol, ond i


16/20

Proof' As he rotio of A to B is os the rotio o C to D, C exceeds D if A exceeds B,ond is equol i it is equol to it, ond qlls short i il olls short. But E ond F oreequinumer.o'l multiples of A ond C, therefore F exceeds D i E exceeds B, ond is equolif it is equol, ond folls short if it olls short. Bul H ond K ore equinumerol multiplesof B ond D; lherefore F exceeds K if E exceeds H, ond is equol i it is equol, ond follsshor if it folls short; ond this is whot we wonted to demonstrote.And when there ore our mognitudes ond the rqtio o theirs o the second is unlike the roiio of the third to the ourth,then it is impossible when some equinumerol multiples o thefirst ond third ore token ond likewise of the second ond{ourth, thot the multiple o{ the f irst does not exceed themultiple of the second unless the multiple of the third exceeds Hthe mul,tiple o the fourth, ond is not equol to it unless it isequol to it, ond does not foll short of it unless it folls shortof it. For if otherwise, let the roiio of A to B be unlike therotio of C to D, ond let be token some equinumerol multipleof A ond C, viz. E ond F, ond let be token some equinumerolmultiple of B ond D, viz. H ond K. Becouse E does not exceedH unless F exceeds K, ond is not equol to it unless it is equolto it, ond does not foll short o it unless it folls short of it, ond Kthese ore equinumerol multiples of A ond C, iherefore A doesnot exceed H unless C exceeds K, ond is not equol to it unlessit is equol to it, ond does not foll short of it unless it olls s,hortof it. And H,ond K ore equinumerol multiples of the mognitudes

"B ond D, therefore A does not exceed B unless C too exceeds D, ond is noi equolo it unless il is,equol io it, ond does not foll short of it unless it folls shorr of it.But A exceeds B ond C does not exceed D, or is equol to B ond C is not equol to D,or folls short of B ond C does'not oll short o D by ossumption. This is q controdiction;herefore the sentence is well-founded ond hot is whot we wonted to demonstrote."

'I I^

I.

66

8.38.9b. zC. 1BD.5D. l48,4E. 12,13E. 17,18E. r9E. 23,24F.3F. 14r. 21t. 22H.ilH. 17H. 23K.4K,BK. lrN.2N. I4N 19,20N. 2rN. 22o. 19P, IP. t8Q.4

CHAPTER II. NOTES ON THE TEXT.I reod , bi-kithroii tod(finI reod ' iuqlu lo-hum 'l-mutobo(inniI reod , ol-miqdrniI reod ' li-onnohumI reod , sioi)unI reod , wo-l on ioqOloond ot the end : wo-l onnoI reod ' nqisoton (on-huond hol-woy : oqollu mim-mI reod , idh knor odlz'u 'l-r6bi(i iuswi'l-miqI reod , ill wo-odjz)u z nqisotun tr:n dh _ond further, o-oq0lu onno nisboto 'obI reod , o-in kdno h-h osghoro min zI reod , wol-tokun titko 'l-fus0lu dh lrn mnI reod , nisbotu 'U lta a1I suppress the second , whidinI reod ' dhO od'finI reod , wo-uqoddomu m iuhfddju iloihi mithdluhI reod , mo(o 'l-od'fi li'l-dchoro iozhoru f 'l-odI reod (ot the end) ' li 'l-thnI reod ' li-ihdhumoI reod , l t0diodu qd({u 'l-owwoliI reod, ol-ju)chodh li-wo Ji od'fun mutosond urther , wo-lijb wo I reod , wo-min-d dkliko 'l-diuz)oI reod ot the end : ollqtI reod , osghoru min hhI reod, fo-jok0nu to okboro min )ob kom oI reod , 'o okboru min EbI reod , odlz'u i nqisoionI insert ot the beginning : o-in lom lr)diodI reod ' ol-mo'ch0dholu min dI reod ' hdh 'l-kitAbiI reod ' li 'l-thn wo 'l-rdbi(i


17/20

CHAPTER I. NOTES ON THE TRANSLATION.(l) I do not know ony commentotor of this nome.The word remotely recolls'ovo8o too.(2) The 2nd Postuloe.(3) The oddition to the ISth Definition of Book l.(4) This ossertion used to be known os the th Postulote or os the 9th or the l2tlrAxiom.Proc'los olreody considered il on unnesessory oddition.(5) Thot numbers ore mognitudes does not ogree with the Aristteleon (ond no doubt'Euclidesn view thot mognitudes ore divisible od infinilum, while numbers oscomposed of lndivisible units ore not.() I do not know o Greek oouivolent.(7) This deinition ossure6 the possibility o the ollowing one.(8) A striking deviolion from fhe Eucl,ideon view.(9) This implies the view thot o rotio is o froction.(.l0) This reosoning must hove met with opposition even in ol-Dioiin's rime.(ll) The existence of such o mognitude DG is not colled in question.(12\ lf I A S g.ihennA S mB.m't- ">"-'(13) lf 1B 5Ainvoluurl p5C thennB -< mAinvolvesnD 5 mCm>ond vice verso.(14) lf a C: ne involves I r DE.then I c-ng involves-1 F:DE.m ! P.qq

Now we toke D G = L, F.qLetbeH: + F C' D,thereore, I B > A involves lA, B : C, D, which leods to o controd,iction.(l) Euclid 5; l2 hos nothing o the kind, but ; l2 gives theproportionol. he possoge is not quite cleor.


18/20

CHAPTER III. NOTES., , .: . i I .., I -( l ) Algiers 1446, 30.(2) Diiksterhuis, Dr. E. J., De Elemenen von Euclides. Historische Bibliotheek voor deExocte Wetenschoppen, deel I (.l929)l en deel lll (1930), 1P. Noordhff N.V.,(3)(4)(5)()

Groningen. Deel l, blz. 109. : :l);lo-. 8 of he suryey in chop,tgr 1.,No. 7 of idem.Poris 2467, 160, 197 Vo (-207). Probobly (portly) the some, Berlin 009, I o,34 b - 38 o; Corulloh 1502, 50, 25 o - 26 b.Krokou 59. Anoritii in decem Iibros priores elementorum Euclidis commentorii.Tronslotion by Gerord of Cremono. Books I - 10.ldem edidit Mox. Curtze, Teubner, Leipzig, 'l899 (Euclidis opero omnio, Supple-mentum).Codex Leidensis 399, l. Euclid,is Elemento ex interpreiotione ol-Hodschdschodschiicum commenoriis ol-Norizii, Arobice el Lotine ediderunt notisque instruxerunR.O. Besthorn ef J. L. Heiberg. (Books I - 4; Ihe Books 5 ond were loterpublished by G. Junge, J. Roeder ond W. T,homson.) lB93 ond loter.(7) No. I I of the survey in chopter L(8) No. 2 of idem.(9) lhove been in doubt obout this tronslotion, but lthink it is right ofter oll. Theword did not occur in t'he dictionories I consulted.(]0) Add' o fhe Elements.(l l) No. 3l o the survey in chopter L(12) Algiers 1446, 10. Probobly (portly) the some: Oxord I 908, I 0. Feyzulloh 1359,2o,150-237o. Seroy 3454,20, (Books 5 ond only). Brusso, Horroccizode,Heyet 20, I o. Steinschneider, Hebr. Uebers, 314, 24.Poge 1.Paris 4946,40. he some: Leyden, Cod. or. 'l99 (8).No. 33 of the survey in chopter l.From this lunderstond th,ot (Umor ol-Choii6mi tokes Euclid's 5h definition for oninerpretotion or commentory (siorhun) of the 4th.

(r 3)(14)(r5)fi)

v0

CHAPTER IV. NOTES.(l) Diiksterhuis, De Elementen von Euclides, deel l, ofd. l.(2) ldem, l.c. hoofdstuk V.(3) ldem, l.c. hoofdstuk V, 4.(4) ldem, l.c. blz.73.(5) Hosse, Helmut und Scholz, Heinrich.Die Grundlogenkrisis der Griechischen Mothemotik. Ch1928.() Becker, oscor. Eudoxos-studien l. Quellen und studien z2, H. 4,31 I - 333, 1933.(7) Dijksierhuis, Lc. blz. 58.(8) The oufior here uses bo(dun, not diuz)un.(9) Cf. Korl Duncker, Zur Psychologie des Produktiven DenkeI 935.

(10) No. 43 of the survey in chopter l.


19/20

STELLINGENt.

Ten onrechte schriift Klomroth (Ueber den Arobischen Euklid, Z.D". . . . und ijr metrein wird qodoro oder (oddo geselzt, iecontinuirliche oder discrete Grssen hondelt./'il.

De mening von Oscor Becker (Eudoxos-Studien l, Quellen undMoth. Abt. 8,2, H.4, 3l l-333, 1939) ". . dosz von der onthyous keine direkte Beweismglichkeit von V, fijr olgemeine Green redenheorie geboseerd op de onwikkeling von verhoudingde stelling, dot in een evenredigheid de binnentermen verwisseldvoor olgemene grootheden bewezen kon worden) is ongegrond.1il.De bedoeling von W. de Geus om in zijn dissertotie (ConAmserdom, 'l943) een syslemoiische opbouw te geven von de couitsluilend op de grondslogen von de infinitesimoolrekening is doniuiste opzet niet voldoende verwezenliiki.tv.

De evolutietheorie voor slerren von Hoyle en Lyttleton (ProceedinPhilosophicol Socieiy, i939 en i940) vindt een onvoldoende quohierbij in oonmerking genomen occretie-mechonisme.V.

Dot voor het volgen von wiskundeonderwiis op de middelboreoonleg nodig is, is nief bewezen.VI

Met de woorden: "souvenez-vous que le hosord ne fovorise que(in .]854 gesproken lot de studenten te Rijssel) formuleerde Posthef produciieve denken.


20/20

vll.Bii her Voorbereidend Hoger Onderwiis kon en moet de wiskunsto voor het productieve denken.vlll.De leer von otto selz (versuche zur Hebung des !ntelligenzniveous'l935), die uitgoot von hel grondbeginsel, dot intelligentie nietgove is, moor gedefinieerd kon worden ols een structuur vongedrogingen, ingesteld op het verwerven en qcnwenden von igrondslog voor de opbouw von een didoctiek der wiskunde.tx.

Het door Korl Duncker (Zur Psychologie des Produktiven Denkensuitgesproken beginsel, ". . . . dosz ein sochverholt sich in derMomenle (Aspekte) oufbouen lciszt, ols nochher - vermge neuer- von ihm obgelesen werden knnen", kon een ruirne loepossing vder wiskunde.X,Uit de betreffende literotuur (o.o. Hfler, Psychische Arbei, in Zen ll e.v.; Binet et Henri, Lo fotigue intellectuelle, lB98; LehmAijszerungen psychop'hysicher Zustcin,de ll ('l901) I l8 e.v.; Foucgeneroles de l'octivjt mentole, in An. Ps. l9 (.l913) z5 e.v.; E. L. TPsychology lll, 1914; F. G. en C. G. Benedict, Mentol efort inexchonge, heort rote, ond mechcrnics of respiroiion ('l933); J. Jvon geesteliike orbeid op de toole sfofwisseling, in het Ned. T. v.blz. 2012 e.v.) bliikr, dot vermoeidheid no geesreliike orbeigeschreven qon de insponning der willekeurige (d.i. door de wilconcenlrotie.xt.Dot de bloedinoxicotie von Mosso en de stofwisselingsverhogidirec verbond stoon met de eigenliike geesteliike orbeid, is nieixt1.Voor de opleiding von toekomstige leroren kon de besoonde opleortsen ioi voorbeeld slrekken.

Documents

Plooij2.pdf