Plooij1.pdf

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EUCLID'S CONCEPTION OF RATICAND HIS DEFINITION OF PROPORTIONAL MAGNITUDES

AS CRITICIZED BY ARABIAN COMMENTATORS(including the text in focsimile with trqnslo,tion of thecommentory on rotio ofAbO (Abd Allh Muhqm'mod ibn Mu'dh ol-Dioiin)

PROEFSCHRIFTter verkriiging von de grood von Doctor in de Wis- enNotuurkunde oon de Riiksuniversi'teit te Leiden, op gezog vonde Rector-Mogniicus Dr B. A. von Groningen, hoogleroorin de Foculteit der Lettren en Wiisbegeerte, publiek teverdedigen op Dinsdog 20 Juni 1950 e 14 uur

doorEDWARD BERNARD PLOOIJgeboren te Heeze (N. Br.)

Uitgeverii W. J. von HengelRotterdo'm


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Promotor: Prof. Dr J. H. KRAMERS

CONTENTS

L Mothemotics in the scientific civilisotion of lslom.ll. he commentory o ol-Dioiin.lll. Other commentories on rqtio.lv. Duscussion of the dierent commentories ond evoluotion o{ their purpor.Chopter ll. Notes on the text.Chopter ll. Noies on the ronslotion.C,hopter lll. Notes.Chopiter lV. No'tes.

chddhrzssigdlzn w(u) h lfitN.B. ln the tronslite.rotion of Arobic words the following chorocters in order rpthe Arobic o1phobet'

)btthdishfqkl hm


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I. MATHEMATICS IN THE SCIENTIFIC CIVILISATION OF IsLAM.l. ln the dork oges, when in our regions the light wos token from the condlestick,o new ond vigorous civiljsotjon orose ond spreod olong the opposite shores of theMediierroneon ond inlond. Towords the end of the 8th century it begon to ossimilotethe scientiic ocquisitions o the Greeks, ond in o much lesser degree of the lndions.Syrion Christions, living in Mesopotomio, lronsloted philosophicol ond.scieniific reotisesfrom Greek originols into Arobic. Exploiting the rich possibilities of this longuoge theysucceeded in moking jt on odmiroble vehicle for thoughts ond doctrines hitherio olienond stronge to it. Sverol o{ these tronslotors were eminent scholors {hemselves. At thistime, whel the lslomic civilisotion wos still in the woy of formotion, religious oppositio.nsdid not yet ploy o dominont pori. ln ihe course of the 9th century, however, thecultivotion o,f science in the Arobic longuoge become more ond more exclusively fhework of Mohommedons, who for the gieoter port were not of the Arobic roce, butdescendonts of the inhobitonts of the ierritories in Syrio, Mesopotomio, Egyp ondPersio conquered by the Arobs in the 7th ond 8lh centuries. For this reoson it seemsrecommen,doble to speok of the "lslomic civilisotion", olthough the term "Arobiccjvilisotion" would be iustiied to o cerioin exten by the common use of fhe Arobiclonguoge os o medium of scientiic ociiviiy ond inercourse'2. Sorokin ond Merton composed o slotisticol survey of "the course of Arobionintellectuol development (700 - l3O0 A.D.)" (l). Their results do not cloim ony obsolufevolue but ore striking enough {or giving us on imoge o{ the rise ond oll othe scientific tide os ell oi o the muluol proportion of the different bronchesof reseorch. The morked predominonce, however, o mothemotics is portly couse.dby the oc thot osironomy ond even osirology ore. included. lt is procticollyimpossible to seporote these sciences, ot ony rote for the centuries under considerotion. For more detoils oboui the meihod used by the outhors I refer to the originolConcerning the development o mothe'motics the outhors ound, with the exception otwo discontinuities, o ropid increose beginning in the second holf o{ the 8th cenfuryreoching its climox o century loter ond {ollowed by o slow decline. The discontinuiiieore: o notoble bockwordness in lhe irst holf of the 1Oth century ond o suddenfoll obout he middle o the l lth century. The totol imoge connot help suggesiingconspicuous correlotion wifh the odventures of the stote ond its rulers. Sufice it to poinio the rise ond efflorescence of the eoslern Coliphote in the 8th ond 9th ceniury; thperiod of decoy ond disinegrotion o{ter thot; the estoblishment o the Buwoyhibultonote in the middle o the lOth century ond its downfoll in 1058; ond somewhoi lotethe rise of the Goznovid Sultonote, which begon to disintegrote ot obout the some time


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lf we did not know it olreody we could gother from oll this thot the protection of owell-estoblished court wos o viol condition {or the florescence of science, ot leost inthose doys.3. By woy of survey the ollowing condensed exposition moy be useful.The second holf o the 8th century sow the beginning of Muslim mothemotics. wo(Abbsid coliphs, ol-MonsOr (the 2nd, 754-775), he ounder o Bogdod, ond Hr0nol-Rosid (the Sth, 786-809) ore porticulorly notorious os potrons of science. Hindumofhemotics ond osronomy (the siddhnto) ployed for some time o prominent port ondihe tronslotion of Greek moihemoticol treotises wos initioted.ln the first hol of the 9ih ceniury rhe cAbbsid coliph ol-Mo)mOn (the zth, Bl3-g33)founded on ocodemy ot Bogdod, the Boit ol-hikmo, which become the centre o ociivityfor rhe ironslolors. Trigonometry went through o heydoy; some problems o puregeomery were studied; Euclid wos lronsloted ond commented on; in orithmetic ondolgebro Hindu ond Greek knowledge were combined. ln 'ihe second holf of the 9thcentury the work wos continued on o higher level. Greek mothemotics were tronslofeclond studied oll olong the line in o brilliont woy.The first holf of the lOth century wos o time of foll, not so much in quoli'ry os in inlensity.ln the second holf of the lOth century, however, morhemoficol octivity *or run"we,not only in Bogdod, where it wos siimuloted by the Buwoyhid sultons (Adud ol-Doulo(949-982) ond his son Siorof ol-Doulo (982-989), but now olso in Cordovo, porticulorlyunder the potronoge of its 9rh Umoiiod coliph ol-Hokom ll (9l-976). The number ooriginol contributions increosed in oll bronches.The first holf o the I lth century sow o continuotion of the eflorescence of mothemotics.while spoin's conlribution wos of minor importonce, the Dr ol-hikmo, on ocodemyin Coiro like rhe two cenuries older one in Bogdod ond founded by ihe th Fiimii,ol-Hkim, bore its first ruits. The works of mosl eosiern scientists, olthough considerqble,were olmost eclipsed by those of ol-Bir0n, who worked ot the court of Mohm0d ofGhozno ond his son Most 0d. Two distinct trends of mothemoicol thought wereperceivoble, o more lheoreticol one qnd o more procticol one.From the second holf of the llth century onwords Muslim mothemoticol productivitywos percepibly reduced. lt is true, o few mothemolicions supplied contributions of highvolue, but in the long run no reol push wos left. At lost ihe moin interest seemed toconine itself to procticol problems omong which the division of inheritonces ployedo dominont port.4" A clossiicotion of Muslim mothemolics. Muslim clossificotion of the different bronchesof mothemotics divides them in elemenls (pure mothemotics) ond derivotives. The {irstgroup is throughout invorioble ond contoins the some subdivisions os the medioevol2

quodrivium, viz. geomery, orithmetic, ,ostronomy. ond music' ,lt goes boci".ff.t"^iU eorpyry qni'not to Aristotle (2). According to modern views theofostronomyinthisgroupisro.therstronge'However,weshouldbeorinminin those doys the motions'of heovenly bod]es were consldered to be perfectroil*t, circulor indeed, ond invoriobi" o, oppnr"d to physicol mognitudes'V"i, fr+"rrlng o division more odequote to our present purpose twould roth"n plr" r"if,moticol ol*iuity inlo o productive pori in norrower sense ond oone, ihot focusses ottention io the oundotions of molhemoticol knowledge ond rMuslims would soy tnor the firur occupies itsel with the bronches ({ur0() ond th*iit nn" roots (,us0l) o moihemotics. the subiect motter of the lotter is moforot"d in the deiinitinr, it'," postulotes, ondlthe common notjons (oxioms) oELEMENTS.5.ThefollowingsurveyoflheodventuresofEuclid,sELEMENTSintheMus{rom the Bth to the 14th lentury is os complete os possible. Moreover i reveols to Muslim scientists in speclol opics. ln ihe next chopter the commentory oon rotio, one of tn"re"topi.r. is reproduced ot fuli length, together with otronslotion o the text. ln the 3rd chopter the originol Greek text is proddiscussed, while other Arobic commeniories on th some iopic ore quot+,n-.nopi"t the different opinions ore compored ond d jscussed'EUCLID'S ELEMENTS IN THE MUSLIM WORLD' N)'1. AbCt MOs Dibir ibn Hoiin ol-Azdi'The olchemist GEBER oi tn" Vtiaate Ages; {lourished mostly in K6o, c' 77hove written o commentory on Euclid'Brockelmonn G(2). l, 278; S' l, 42'Sqrton l, 532.Suler, Arabet, Arl- 3.Kopp lI, 7.Meli, Por. 8, 9.2. Al-Hodididi ibn J0suf ibn Motor'Flourished some time between 786 and 833, probobly in Bogdod. He twiceEucljd,s ELEMENTS, fir.i ,nd", Hr0n ol-Rosiid, then'ogoin under ol-Mo'mClotter tronslotlon the Books l- ond I l-'l3 ore exionl'Brockelmqnn G(2). l' 221; S' ', 33'Sonon I, 562.Suler, Araber, Art. 1,Mieli, Por. 15.

") The notes refer to rhe bibliogrophy on poge 14'


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3. Al-(Abbs ibn So(d ol-Diouhor.Flourished under ol-Mo)mOn ond-iook port in the osironomicol observotions ot Bogdodin 829-30 ond ot Domoscus in 832-33. He wrote o commeniory on Eucrid,s ELEMENTS,odditions to Book I (some propositions); ond odditions to Bookt. ih" lott", ore extont.Brockelmonn S. l, 382.Sorlon I, 52,Suler, Arober, Arl. 21.Kapp ll,7l.Krouse, Hondsch riten, 44"4. AbO 'l-Toiiib Sonod ibn (,Al.Flourished under ol-Mo)mOn, died oter B4. A Muslim osironomer ond mqthemoiicionof Jewish birrh. He wfoie on the subiect morfer o Book 5 0f Eucrid,s ELEMENTS.Sorton I, 5,Sver, Arober, Art. 24.Kapp ll, 91.5. lbn Rhiwoih ol-Arrodini.From Arrodjn, o town in Chuzistn beiween Bosro ond Foris. He wroe o commenioryon Book l0 of Euclid,s ELEMENTS. His dotes qre uncertoin.Suter, Araber, Art. 33.Kopp Il, 51.!: Abr) J0suf Jo(q0b ibn tshq ibn ol_Sobbh ol-Kind.he philosopher of the Arobs (i.e. of the Arob roce). Born in Bosro ot the beginningof the 9th cenrury. Frourished in Bogdod ""a Jii ,." ars. iJt'riroru on improvededition o Euclid's work ond o reotise on its oims (oghrd).Erocke/monn c(2). t, 230; S. I, 372.Sorlon l, 559.Suler, Arober, ,Arf. 45.Kopp lll, 11.7. AbO (Abd Allh Muhommod ibn (ls ol_Mhni.Moihemoticio'n ond ostronomer from Mhdn, Kirmdn, persio. Flourished c. B0, died c.874-884. He wrore obout rhe 2 propositions of Book r of the ELEMENTS rhot conbe proved without o reductio od obsurdum. xtqnt ore three (different?) treotises onroiio (Book 5), o porr of o commentory on Book r0, o.d ;; u"ftlnotion o obscureploces in Book 13.Brockelmonn S. ,, 383.Sqrion l, 597,Suler, Arober, Arl. 17.Kopp lll, 0.Krouse, Hondschriten, 450,4

8. AbO 'l-Hoson Thbit ibn Qurro ibn Morwn ol-Horrn.Physicion, mothemolicion, ond ostronomer from Horrn, Mesopotomiq; {louBogdod ond died in 90'l,75 (or 64) yeors of oge. He improved the tronslotioELE ENTS by lshq ibn Hunoin ond Qustd ibn Loq (Books 14 ond 15 by thOf this improved edition some copies ore exlont. He wrote on introductioELEMENTS ond treotises obout the premisses ond propositions o Euclid. Aimproved edition of Euclid wos even better thon the irst. He wrote o comto the Books l4 ond ,l5. Extqnt ore severol treoises on the "postuloie of pBrockelmann G(2). l, 241; S. t, 384.Sorton l, 599.Suler, Arober, Art. .Kopp ll, 58.Krouse, Hondschriten, .453.9. AbO Jo


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12' 'Ab0 Muhommod ol-Hqsqn ibn (uboid Ailh ibn suroimn ibn wohb.lion of o wozir of ol-Mu'rodid, who died in 901 H;-;;;il"o'-.orr"niory on rhe,ltrbious ploces in Euclid,s wort obout rotio in one reotise tff. " -Suler, Arober, Art. 94 ond Ann. 23 p. 2ll.Kopp ll, 77.13. AbO cUrhmn So(d ibn Jo(q0b ol-Dimosiqi.Muslim physicion ond mohemoticion, frourished oi bogdod under or-Muqtodir (gog-g32).llo ronsloied some Books of Euclid's ELEMENTS incluiing t" rtr,,,ra the commentoryo[ Poppos on it. The lotter lronslotion is exlont.Sorlon I,631.Svler, Arober. Art. 98-Kopp ll, 81.Mioli, Por. 2l .l4' Ab. Nosr Muhommod ibn Muhommod ibn Torchn ibn uzrogh sr-Frb.t-)r cr Turkish omity; srudicd i. B.sJ;;; ii.rn.i.i.;;;i;;; ^"*.", died in Domos-rrrs' oged c 80, in 950. He wrote o commenlory on the difficulies in Euclid,s premissesof the Books I ond S, exiqnt in He,brew.Brockelmqnn G(2). I, 234-3; S. l, 375_27.Sorion l, 28.Suler, Araber, A. It6.Suler, Nochtrdge, p, l5.Kopp ll, 97.15. ,AbCl Do)0d Suloimon ibn cUqbo.A- corrtemporory of ol-Chzin (no. lz). H" wrote c conrmentory on ihe subiect nrcriter.[ I'he second port of Euclid,s ELEMENTS, Book 10, which i, u*t'"ni.Suter, rober. Art. lt7.1. Ahmod ibn ol-Husoin, ol-Ahwz, ol_Kiib.llourished c 940. He wrote o commentory on Book i0 of the ELEMENS, which is extont.l-lr: is not sofely identified.Brockelmonn S. l, 387.Sutcr, Arober, Art. 123.Kopp ll, 57.Krouse, Hondscriten, 42.17. AbCl Dio(for ol-Chzin.llotrr itr Khursn, died between 9l oncj 92l. 4othenroticion ond ostronomer. He wrote(t

o commenlory on the irst port of Book l0 of the ELEMENTS. Severol copies ore exlqnBrockelmonn S. t, 387.Soron l, 64,Suler, Arsber, Arr. 124.Kopp ll,77.Krouse, Hondsch riflen, 42,18. J0honn ibn J0suf ibn ql-Hrith ibn ol-Botrq, ol-Qoss.Died c.980. He tronsloted Euclid from the Greek. Of his hond o treotise on rotionoond irrotionol mognitudes is exlont.Brockelmonn S. ,, 389.Suer, Arober, Arl. l3l.Kopp lll, 37.19. AbO 'l-Qsim (Al ibn Ahmod ol-Antk, ol-Muditob.Flourished in Bogdod ond died in 987. Of hjs commentory on Euclid's ELEMENTS ihsecond port is extoni.Suler, Arober, Atr. 140.Kopp II, 54.20. Ahmod ibn (Umor ol-Korbs.Mothemoticiqn. He wrole o commentory on Euclid. Exlont is o commentory on tbeginnings (:ud0r) of Euclid's work. His dotes nre uncertoin'Brockelmonn G(2). I, 27; S. l, 390.Suter, Arober, Ad. 144,Kopp lll, 37.21 . AbO J0suf Jo(qOb ibn Muhommod ol-Rfizi'He wrote o commentory on Book 10 o Euclid's ELEMENTS ot the instoncelbn ol-'Amd.Suler, Arober, At. 147.Kopp ll, 9.Mieli, Psr. 17.22. Nozf ibn Jumn ol-Mutotqbbib.A christion psysicion of Greelk origin. Flourished under (Adud ol-Doulo; died c.99He tronsloted Book l0 (?) of Euclid's ELEMENTS. Extont is his tronslotion of soodditions in Greek to l0,l ond ']0,, slight modificotions of 'l0,7 ond 10,8 ond o pof the corroiorium of 10,9 (Oxford edition).Brockelmonn S, l, 387.Sorion I, 4.Suler, A,rober, Arl. 158,Kopp lll' 8'


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23. 'b0 lshq lbrhim ibn Hilr ibn rbrhm ibn Zohr0n ol-Horrn.A sobion; died in Bogdod 9g4. He wrote o commenrory on EucLid,s trurNrs.Sorlon I, 59.Svler, Arober, Arl, l1.24- AbO 'l-wof Muhommod ibn Muhommod ibn Johi ibn lsm(l ibn ol-


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32' AbO (Abd Allh Muhommod ibn J0suf ibn Ahmod ibn Mu(dh ol-Dioiin.From sevillo; flourished- c. 1080; wozir, qd, ond foqh. He wrote o commentory onrotio (the firsi seven definifions of Book 5) which is extqnt.Brockelmonn S. ,, 80.Sorlon ll, 342.Suter, Arober, A,il. 213.Suer, Nochrcige, p. 170.Kopp 11,77; ibidem note I7B.33. AbO 'l-Foth (Umor ibn lbrhm ol-Choiim Ghiirh ol_Din.Persion mothemoticion, ostronomer ond poet;'born in or neor Nsib0r c. 1038_'l04g;died there in I'l23. He wrote (in 'l078) o commentory on the difficuities in the premissesof Euclid, which is exront.Brockelmqnn G(2). I, 620; S. ,, 85j.Sqrlon l, 759-Suler, Araber. Art. 2.Kopp 11,79.34. Muhommqd ibn Muhommod ol-Boghdd.Flourished c. ll00 or loter. Proboble oulhor of on extont commentory on Book l0 ofhe ELEMENTS, "Liber iu.dei super decimum Euclidis,,, by Gerord of r"rono. Accordingfo suter we could rend "iudicis" i.s.o. "iudei". one bo Bokr Muhommod ibn cAbdol-Bq. wos qd 'l -mrostn, iudge of the hospitol, in Bogdod-.. I tzg. ln rhe ms.the oulhor is colled Abbocus, which moy come from ob(dol)bJcus. lt contoins numericolexomples, iust whot ol-Qift mentions obou't the treotise of Abd ol-Bdq.Brockelmonn S, l!, 1023.Suler, Arober, A,rl. 517.Suer, Nochtrdge, p. 181.Kopp lll, .Biiirnbo, Ueber zwei moth. Hondschr. o.d. 14. Johrh.. Bib, Moth. 3(3),71, lg)2.sufer, ueber den Kommentor des Muh. b. (Abdathoqi zum r0. Buch v.'Euktid. g;b. Moft,(g),7,2g4,1906_7.Mieli, Por. 21.35. Al-Muzoqr ibn lsm(l ol-Asizr.Muslim mothemoticion ond physicisi, coiloborotor of (Umor ol-Choilm; died beore1122.He wrote o summory of Eucrid's ELEMENTS, of which Book r4 s exront.Brockelmqnn S. t, 85.Sorton Il, 204.Suler, roer, Ar. 268.Krouse, Hondschriflen, p, 483.l0

36. AbO Muhommod Dibir ibn Afloh.igfn, Hispono-Muslim stronomer ond mothemoticion; lived in Sevillo (or wothere) 'ond died probobly obout ihe middle o fhe thjrteenth century. He wrexplonotion o{ Euclid, o Hebrew copy of whjch is exlon'Sarlon ll,206.Suier, Arober, Art' 284.Kopp ll, 69.Mieli, Por. 43.37. AbO'l-Fui0h Ahmqd ibn Muhommod ibn ol-Sur Noim ol-Din (lbn ol-SPersion physicion in Bogdod, loter in Domoscus; died in I'l53. He wrote lreotdiferent ploces of the ELEMENTS, which ore extont'Brockelmqnn S. I,857,Suler, Arober, A'rl. 287.Krouse, Hondscfi ri[Ien, 485.3g. Abo (Abd Allh Muhommod J0suf ibn Muhommod Muwoffoq ol-Dn olLived in Sjohruz0r ond Domoscus, died in I189 in Arbelo' He wrote explonotEuclid's ELEMENTS.Suler, Arober, Art' 305.39. AbO 'l-Wqld Muhommod ibn Ahmod ibn Muhommod ibn Rusid'AVERROES; born jn cordovo I12, djed in Morokko 'l]98. He is soid to hoveotreotiseonwhotisindispensobleofEuclid{orthesiudyo{theALMAGEST.Erockelmonn G(2). l, 604; S' ', 833'Sorton l, 355.Suler, Arober, Art. 315.40, Abo (Abd Allh Muhommqd ibn (Umor ibn ol-Husoin ibn ql.Chotb Fool-Din ol-Rzi'persion; lived from l150-.l2,l0; wrote obout the musdort of Euclid ond pendeeply into the theory of the ELEMENTS (?)'Brockelmann 611). I, 6; S' t' 920'Sarlon ll, 34'Suler, Araber, Ail' 328.Kopp ll, 94.41. Qoisor ibn Ab 'l-Qsim ibn (Abd ol-Ghon ibn Musfr (Alom ql-Dn olEgyption mothemoticion, rironot"r ond engineer; born o Asf0n in 1178;


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Domoscus in 'l25,l. He wrote to Nosr ol-Dn ol-T0si obout Euclid's premisses. The letteris extont.Brockalmonn S. l, 87.Ssrlon ll, 23.Suler, Arober, At, 3SB.Krouse, Hondscriften, 491.42' Ab0 Zokori Johi (or Ahmod) ibn Muhommod ibn (Abdn, ol-shib Nodimol-Dn, lbn ol-Lub0d.Syrion physicion, morhemcrticion, osrronomer ond philosopher. Born in Holob, 1210/11;clied oer 1267; lived mony yeors in Egypt. He wrote on exrroct irom Euclid, onexplonotion of Euclid's postulotes (?) ond ) o .r,nnrory of his premisses.Sorlon ll. 24.Suler, Arober, Arl. 35.Kopp lll, 7.43. AbO Dio(for Muhommod ibn Muhommod ibn ol-Hoson, Nosr ol-Dn ol-T0s,ol-Muhoqqiq.Persion philosopher, mothemoticion, ostronomer, physicion ond scientist, who wrote inArobic ond Persion. Born in i20l; died in )274. He wrote two redoctions of theELEMENTS, o lorger ond o shorter one. The first is extont in Florence ond wos printedin Rome, 1594. There ore two versions, wh 13 ond with 12 Books, with ond withouto Lotin tifle. Of the shorter edition in l5 Books mony copies ore extont. lt wos printedin Constontinople in 180], he Books j- olso ot Colcutio in 1824. ll wos commentedon by AbO lsh6q. He wrote on Euclid's posiurotes (?) (ol-uq0l ol- moud0co), retters toQoisor ibn Ab'l-Qsim obout ihe sth postuloie ond on 105 problems bosed on theELEMENTS. Of oll these treotises copies ore extont ond the contents moy be even morevoried lhon is indicoted.Brockelmonn G(2). t, 7Oi S. I, 921.Sorlon Il, l0OI.Suler, A,raber, Art. 38.Krouse, Hondschrilen, 199.Mieli, Por. 29.44. So


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BIBLIOGRAPHY.(l) soRoKlN, Pitirim A.; MERToN, Robert K.. The course of Arobion intelleciuoldevelopment, T00-1300 A.D.. A study in melhod. lSlS 22, 516-24, j935.(2) STEPHNSON, J.. The Clossiicotion of lhe Sciences occording to Nosiruddin Tusi.lSlS 5,329-338. The outhor cires Zellner, Die philosophie der Griechen.(3) BROCKELMANN, Corl. Geschichte der Arobischen Liiierotur. Zweiie den Supplement-bonden ongeposste Aufloge. Leiden, E. J. Brill, 1943-44. Supplemenibond l, 1937;Supple'mentbond ll, i938; Supplemenrbond lll, 1942.(4) SUTER, Heinrich. Die Mothemotiker und Asronomen cier Arober und ihre Werke.Abh. z. Gesch. d. Moih. Wissensch. mit Einschluss ihrer Anwenclungen. X. Heft. Leipzig,r 900.(5) suER, Heinrich. Nochiroge und Berichtigungen zu,,Die Mothemotiker ....,,.Abh. z. Gesch. d. Moth. Wiss. erc. XIV. Her, p. ISS-85, Leipzig, 1902.() KAPP, A.G. . Arobische Uebersetzer und Kommentotoren Euclids, sowie deren moth.-noturw. werke ouf Grund des To'rikh ol-Hukom' des lbn ol-eift. lsls 22, 23, 24,1934-36.(7) KRAUSE, Mox. . Stombuler Hondschriten islomischer Mothemqtiker. euellen undstudien zur Geschichte der Mcrfhemoik, Astronomie und physik. Abt. B, Siudien; Bond 3;p.437-532;1936.(8) MlELl, Aldo. Lo science Arobe et son rle dons l'volution scientifique mondiole.Leden, F. J. Brill, .l938.

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II.

CommenioryMuhommod onibn

AATHE COMMENARY OF AL-DJAJJANI.

rotio occording to the reosoning o the qdi AbO 'Abd AMu ( dh ol-0111oni, God, who is lofty, hove mercy upon

Note: The English translation {ollows upon the Arabic text as closely as seemconrpatible with the demands o{ intelligibility. It is printed against the correspondinoriginal. The beginning o a new page in the Arabic text is indicaled in the translatiouv (g), (c), etc. The {igures are designed in accordance with mediaeval practice, butnot claim any authenticitY.


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ln the nome of God, grocious ond merciful. God bless our mosier Muhommod.The qodi Abti (Abd AllOh Muhommod ibn Mu(dh ol-Diolin, God be pleosed withhim, soys, We intend to exploin whot moy not be cleor in the fifrh book o Euclid'swriting to such os ore not solisied with it, though I hove the conviclion os to Euclid'sreosoning, his inlentions ond his oims ihoi hey ore more evident ond cleorer insubdivision to ony true ond oir exominer thon much by meons of which one hos triedio moke it cleor ond hoped to illustrote it; except some things which he left becousehis reosoning ond the orrongement of his writing were sufficient to moke themunderstood ond reolised. Concerning thot which mef with opposition ond people thoughtnot o be complefe or cleor, so thot they modo it comple'ie or cleor occording to theirown thinking, well, their reosoning on this point is more it for being exploined ondis more in wont of proof thon Euclid's writing. So, or instonce, is our opinion on thewriting oscribed to Syndus (l); for he who considers his proos of thot which Euclid giveswithout proof, os"'lf two stroighi lines ore drown over less thon two right ongles, heywill inevitobly meet" (2) ond "The diometer bisects he circle" (3) ond "Two stroight linesdo not enclose o surfoce" (4) will see weokness ond debility in ihem ond the necessityof ossuming things which ore more extensive thon those Euclid cloims to ossume osself-evident. Now somebody moy remork' "l Euclid's reosoning in his writing is os youmeniion, ond nobody mokes it cleorer thon Euclid did, why then do you desire to exploinhis reosoning in rhe fifth book ond elucidote whot is obscure in it?" Then I soy, By thiswriting we only meon to exploin thot Euclid's reosoning in this book is ihe very soundone ond his method the very monifest one. For mony think thot Euclid opprooches theexplonotion of rolio rom o door other thon its proper door, ond introduces it in o wrongwoy by his definition o it by toking multiples, ond in his seporoting from its deinitionconcerning its essence thot which is undersiood by the very conception of rotio; ond theyiudge thot there is no obvious conneclion between roio ond toking multiples. But uponmy life, wiih nothing rotio is more closely connecied thon with toking multiples of thetwo compored mognitudes. Upon the whole I soy thot rhe fith book cor.rtoins someobscuriy, ond thot ihe study of it moy be tiring; however oll strivings ore meosured bythe nobleness of the obiect to be ochieved, ond he who ospires to o beouty's hond mustpoy the price. For ihe writing rom this point onwords is of o spirituol ncriure ond whotprecedes is of o corporol one. We will soy in this motter thol which I hope God will mokeuseful to the ottentive reoder, ond which moy help him to remove the difficulties in it.To Him, who is lofty ond is nol equolled in groce, I pro{fer my humble proyer {or rightconduct ond preservotion from error ond conceit; for He is mighty ond there is no lordbeside Him. I soy, ond God be my help, thot the things thot come under the quontity usedin the ort of geometry ore five, viz. ihe nurnber (5), (B) which is the irst ond simplesto fiem, the line, the surfoce, the ongle o'nd the solid. Let us use the term mognitude ingenerol to denote ony o these things, to which it is o nome or the species comprisingoll of them. Wherr the mognitudes ore of one of hese five species let them be colledhomogeneous. These hove the property thot by {requent multiplicotion the less csnI


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equol the greoier or exceed it. Also con by requent division he greoter becomeequol to ihe less or foll short of it. When there ore two homogeneous mognifudesond the less meosures the greoter, ihen the less is colled q port of the greoter ondthe greoter is colled he multiple. lt is olso colled the superior () to thotport. When the less o the two homogeneous mognitudes meosures the greoter,or when lhere is o third mognitude o the some species thol meosures eochof both, then they ore colled commensuroble; ond when the less does not meosurelhe greoter, ond no third mognitude is found thot meosures them, then they orecolled incommensuroble. Rotio is size of o mognitude os compored with oothermognilude of the some species, viz. o comporison is mode beiween the twomogniludes or the purpose ihot the size moy be known of one of them oscompored with the other (7). Proportion is equolity of rotios (8). When o mognitudeis reloted to onother mognitude the loter is colled consequeni ond the -for11-1"1"ontecedent, ond either of ontecedent ond consequeni is colled the componion o theother. The whole is greoter thqn the port ond likewise it is usuol to odmit thot theport is less thon the whole. Now from the whole only such ports ore token thot donot reoch up to the whole; or if it exceeds the whole it is colled greoter thon it, or omultiple of it, ond the term "oking ports" is not opplicoble to it. we however willinlroduce in this writing o ours lhe proctice thot when we ioke ports of some mognitudewe will poy no ottention lo the question whether such ports do equol the mognltudethey ore token from or foll short o it or exceed it, the efect of which *o-rld buthrowing the dificulty on repetition of our reosoning ond prolixity in whoi we iniendto exploin hereofter, if God is willing. For ihis will neither domoge the proof in onywoy, nor does it deviote from ony ossumed bose or is ot vorionce with it. lt onlyserves to focilitote ihe reosoning ond to shorten it. So we obsolve ourselves from theobligotion o indicote in every cose which mognitude is the greoier in order to rokeports rom it.lf the mognitudes ore commensuroble, then one of them contoins exoct ports completelyilling up the other. For i the third mogniude, which meosures both, is produce:d, thisthird is o port of eoch of them, ond when it is oken o number of imes from one of thetwo mognitudes ond exhousts thot mognitude in numbering ihen evidently the meosuredmognitude is porls of lhe other. lf however the mognitudes ore incommensuroble noneof ihem conoins exoct ports of the other. As o the terms o ihe proportion the firsto them contoins os mony ports of lhe second os lhe fhird contoins such porls ofihe fourth (9).This is evident, no veil is over it ond it needs no proof. Euclid deols (c) wirh it in thesome woy in lhe seventh book, but it is only eective if the first mognitude is commen-suroble wiih the second ond the third is commensuroble with rhe urth, so thot eocht8


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lcontoins o whole number. of pcrrts of is componion. when however the componionsore incommensuroble ond it is impossible to express the size o eoch ontecedent oscompored with its consequent ond to reduce it to o portition ond to ""pr"* il rvmeons of porfs, then it is,evident, olthough the wellknown view does not exist, nomelyexpressing by ports, ond is improcicoble ond no ports of the consequent fill upcompletely eoch o he ontecedents wifhout remoinder, fhot it is by no ,un, porriui!or the rotio of the irst to lhe second to be like the rotio o thJ third to the ourthwhile ihe first contoins more ports of ihe second then the third o th" forrth on'noless either. This needs no proof, becouse ii oppeols immediotely to the mind. For wheno mognitude contoins more ports of o second mcrgnitude ihon o third mognitude suchports o o {ourth mognitude, then the size o thJ irst os compored with the secondis not like the size of the third os compared with the fourth. Over this there is no veil,nor is it mode cleorer by prolixity.in the reosoning, since likewise things thot orecleor ond evident to the mind without need o proo r" not mode .l.or", uv prixrvin the explonotion, becouse there is no method to moke cleor whot is olreody cleorin it' This preomble however detrocts nothing from the foci thot in this cose lhere is noexpressing by meons o ports thot cover eoch of the two ontedents ond thot it cononly be expressed by less ond more.However whot is less thon o thing is not determined, nor whot is more either; equolityonly is determined. But the like we oten ind in this ort; for instonce, when we desireto know the length of the circumference o o circle ond its size os compored with thediometer, then we leorn this by less ond more; for similority o^a eqrotrty oil in it,becouse equoliy never occurs berween o curved line orrd o rtriglt one, or they orenot of the some kind (.l0). And so too ore the commensuroble ond-the incommensurqbleomong the mognitudes. these being not o the some kind. For expressing by portsis impossibe omong incommensurobles, iust us ii is impossible wiih one speciol meosureomong circumerence ond diometer. About the circumference it is only soid thot ii isgreoter thon the sides of 'every figure inscribed in the circle ond less thon the sidesof every igure circumscribed oboui the circle. And likewise, riun *" relote somemognitude to o mognitude incommensuroble with it ond ore obliged to express thisor to opproximote it io some expressible rorio for some design- or other, we soythot it is greoter thon such ond, such ports ond ress thon such ond such ports.so too it is with irrotio'nol roots. lf we ore obligod to reolise tnu, in ports we soythot the root. of o given number which is not o squore is greoter thon the root ofeoch squore less ihon the given number ond less hon the rooi oi eoch squore greoterthon the given number; ond the omission ond impossibiliry o expressing do not intrudeupon the purport of it ond do not (D) ofect the sound bo.e noi tronsform it.When now it is oll rue whot we hove mentioned obout this, we only expotiote on it

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ond bring it forword ond emphosise it in order thot it moy be o bosis to rely uponin whot follows hereoter, i{ God is.willing. This is thor *"-h" rotio o the irst tothe second is os he rotio of the thirc{ tJ the fourth it connot hppe. thot of onyports of the second the,.irst should lonioin onything more thon the third o suchports oken rom he fourth ond no less either. Howevei no indeiniteness must be seenin our words "ony ports of the second . . . . onyihing,,, becouse i tr,* nonn", of the portsof rhe second conroined in rhe firsr differ from rh" ;.;;;;ii" p;';, of rhe {ourrh con_toined in the third, rhe rorios ore differenr ond nor "qrol, ir;;r;lctlve whor porrs oreconcerned. on this ooi.nr.we wiil give some erucidorion by';";,i-;i o porticuror cose,becouse there is some herp.in ,r.h" on "l"rpr", which nobody wirr refuse, orthough wehove fostened down o,lrecrdy the generor oi *ior.fpri", t.'oil 'porti.rro, coses in qsuficient woy. We will give it hoiever for ihe soke'of gruor; .i;rness. so we soy:when somebody supposes rhe rorio o o mogniiude AB;"-; ;;;ude cD ro be osrhe rotio o EF to GH, then Jhe reosoning obout this is creor*if AB ond cD orecommensuroble; lmeon thot AB conroins J,,ony ports o CD os EF contoins suchports of GH. Bur if rhey ore incommensurobre, then, othorg ;"; unqbre to expresshow mony ports AB contoins o CD, onyon" who supposes these mogniiudes to beproportionol is nor ollowed to soy, or instonce, thot AB is more thon wo thirds ofD, ond EF less rhon fwo thirds oi GH or equor io two r.,i. .i H. etuo h" moy norsoy thof AB is two thirds of CD, ond EF more hon two thirds oi Cu no,. ress either;nor thot AB is less hon wo thirds of cD, ond EF more thqn -two thirds of GH orquol to two thirds. These hings ore essentioily ob.r. rlr"rry'n"" we exominethe other ports ond cire them, fr whot is on qbsurd ui"* in ri"i, or ,n* ports weentioned is olso obsurd in respect of the other poris, be they ever-ro smoll. Thereforei somebody thinks these mog'nifudes to be proportionor ond soys hot AB is, fornstonce, more rhon three hundred ond sixiy seven ten thousondth ports of CD ondhot EF is less thon such ports token from G or equor to them, there is no differencebeiween this,onf the precding. This we odduced qitl.l*gh-;.'rloroning needs norore o the kind in order trrot it migrrt be creor thot ihe ,"or*, is quite generor, sohot..the condition opplies to oll portslsmoll of lorge, no ports beino dis. A B ptinguished oove orhers. rni, ii tn"t --* ? f,---------**t Hwhen of proportionol mogniludes the t------rrotio o the firs't to the second is os the roiio o the third to the ourth, it is impossibleto ind ports of the second ond ourih equor in number ond denominoiion, whoieverports rhey moy be, so ihot_the ports of the second ore found to exceed fie irstmognitude, unless the pors of the fourh too exceed he third irl ognirra"; or thot theports of he second ore found o be equor io the irst..g"irrJJ, unress the pors22


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o the ourt'h mognitude too ore equol to the ihird mognitude; or thot the ports ofthe second ore found to oll short of the first mognitude, unless the ports of thefourth ioo foll short of the third mognitude. For if ports of the second ore foundexceeding the first ond ports o he fourth being equol to the ihird or folling shortof it, the irst contoins less porfs of the second thon the third of the fourth, nd othis we proved olreody the obsurdity. And likewise, when ports of the second ore foundbeing equol to $e first mogniiude ond poris of the fourih exeeding or folling shortof the third mognitude, the irst contoins less or more ports of the second thn thethird of lhe ourth; ond likewise, when ports o the second. ore found folling short ofthe first mognitude ond pors of the ourih being equor to the hird,.nojnitrdu o..exceeding it, the irst.contoins more ports of the second thon the third of ihe fourth.All this is obsurd, oni whot remoins for us in fhis preomble is to show whether it isconvertible or not.We then soy thot it is convertible, viz. when there ore our mognitudes while of oll.ports equol in number ond denominoion token rom the ,".onJ ond the fourth theports of the second ore not ound to exceed the first mognitude, unless the ports of thefourth too exceed the third mognitude, nor ore the poits of t'he second found to beequol to the first mognitude, unless the ports of the fourth too be equol to the thirdmognitude, nor ore the ports of the second ound to foll shori of the irst mognitude,unless the ports of the fourh too foll short of the third mognitude, rhen th1 rotioof the first to the second is os the rotio o the third to the fouith. l, for insfonce, weioke for the {our mognitudes AB, c, DE, F, ond suppose thot of oll some ports foundof C ond F the ports of C ore not found to exceed Ab, unless the ports of F too exceedDE, ond the ports of c ore noi found ot be equol to AB, unless ihe ports of F too beequol to DE, ond ihot the ports of C ore noi ound to foll short of AB, unless heporis of F too oll short of DE, then lsoy ihot the rotio of AB to c is os the rotioof DE to F.he proof of this is thot it con not be otherwise. For i thot could be so, let the rotioo AB to c be os the rotio of DG to F (lr). lf GE is less thon F, we soy: we divide Fogoin ond ogoin, first toking the holve, then which ollows in the division, viz. the thirdond so on till we come to the first port of F thot is less thon GE ond iolt tnot por n. ihi,is less thon GE. From C we toke the some port; be it K. hen we subtroct from DEpieces equ.ol to H, beginning to subtroct from D, until it comes 1o the first mork necrE. not reoching up to G, or GE is greoter thon H; ond ret these pieces b" DL, iM, MN.Then we subtroct rom AB pieces equol to K; it will contoin K os mony times os DEcontoins H. The result posses B ond exceeds AB, iust os DN exceeds DE, becouse wehove supposed so obout he mognitudes (F) AB, c, DE, F. Let theseporis be Ao, op, pe.consequently AQ is ports ot c ond DN is ihe some kind of polt, of F, in the some

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number. Bu AQ is greoter hon AB ond DN is less thon DG; thereore AB contoinsless ports of C thon DG of F. And yet the rotio o AB to C should be o,s the rotioo DG to F. This is o controdiction; it is impossible ond consequently the rotio of ABio c is unlike the rotio of something greoter thon DE io F. Likewise it is proved to beunlike the rotio o{ something less thon DE to F. Consequently the rotio of AB to Cis only os the ,rotio of DE to F ond lhis is whot intended to demonstrote.AOPBot----+i D L M, E N G##k CFAs whot we hove previously soid. ond emphosized is cleor now ond we hove got osound ideo o proportion in the cbse o commensuroble mognitudes, where expressingis possible in on exoct woy by meons o ports, ond in he cose of incommensuroblemognitudes, olthough expressing by meons of ports in the strict sense is not otoinoblehere, so moy suffice whot we hove mentioned, if God, who is lofty, is willing, ond letus soy whot is our intention in this writing. This is the view which brought Euclid to hisdeinition of rotio by meons o toking multiples with restriction lo thot; without comple-ting oll thot is occidentol ond complementory to rotio ond its increosing ond decreosing;ond to the neglect of oll this being content with toking multiples. Therefore lsoy,ond God be my help for right conduct, thol, os proportion orises, os we mentioned,from the considerotion of ports ond their comporison, oll thot occurs with the portswe toke occurs likewise with the multiples, os to exceeding, folling short or beingequol, ond whot is unottoinoble with ports is olso unoltoinoble with multiples, becousemultiplicotion ond portilion ore of one cotegory. For they both originoie from unity,but thot muliplicofion originotes rom ii by increose ond enlorgement ond portitioninversely by decreose ond diminution. The some principle opplies to eoch ofboth. The two mogniudes indeed, when it is possible to find ports of one ofihem thot ore equol to some ports of the other, then it is olso possible totoke o multiple of one of them thoi is equol to some multiple o the other; whichwill occur in the cose o commensuroble mogniiudes. Bui when it is entirelyimpossible to find ports of one of them thot ore equol to ports of he other,then it is olso impossible thot there is some multiple of one of them equol to somemultiple of the other; which will occur in the cose of incommensuroble mognitudes.And likewise, when to two unequol mognitudes o third mognitude is found less thonthese two, meosuring them ond being o port of eoch of them, then olso o commonmultiple of hem will be found, so ihot the conformlty of the conditions of the multiplesin respec of the con'ditions of the ports is yet more evideni.I will now present something of which on exomple is required, viz. thol if A is some

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multiple of B, while the some multiple is token o A ond B, viz. CD ond EF, then I soy,(G) thot CD is the some multiple of EF os A is o B. Proof: We divide CD in its portsequol to A, which moy be CG ond GD, ond divide EF in its ports equol to B, whichmoy be EH, HF, then CG is equol to A ond EH equol to B, ond likewise CG is hesome multiple of EH os A is of B, ond likewise GD o HF. Thereore the whoie CDis the some multiple of EF os A is of B,ond the proo is complete. Now we soy thot ,, o I cF__f___jwhen lhere ore loken two diferent mognitu- I E H Fdes, ond ports ore token of one of them, ond }* |+|these ports ore compored with the other mognitude, hen their condition os toexceeding, folling short ond being equol is lhe some os the condition of o certoinmultiple loken of he ported mognitude in respect of o certoin multiple token of theofier mognitude, if (the number of) the multiple of ihe ported mognitude is equol tolhe number o the ports, ond likewise (the number of) the multiple of the othermogniiude is equol to the denominoor of the ports. An exomple of this is, thot themognitudes A ond B ore different ond thot of A o number of poris is token, whichmoy be CD, one of which pors being CE, ond thot of ihe mognitude A olso o multipleis token corresponding to the number of ports cE contoined in CD, whlch moy be F; ondthct of B o multiple is token corresponding io fhe multiple thot A is of its port cE,which moy be G; then I soy thot the condition of CD in respect o B, os to being equol,exceeding ond folling short, is the some os ihe condition of F in respect of G, sothot F, which is o multiple of A, corresponds in corrdition with cD, which is portsof A, ond G corresponds in condition with B, o which it is o multiple, if CD is equolto B, then F is equol to G; ond if cD is greoter thon B, then F is greoter thon G; ond,if CD is less thqn B, then F is less then G. The proof of this is, thot G is the somemultiple of B os A is of cE; ond olso, thot F is the some multiple of A os CD is of cE.Therefore A is o multiple of cE. Now the some multiple is token of A ond cE, whichore F ond CD, so thot F is the some multiple of CD os A is of cE. And A is the somemultiple of CE os G is of B, therefore F is fhe some multiple o{ cD os G is of B.Consequently if cD is equol to B, then olso its mulriple, which is F, is equol to G,which is the some multiple of B; ond if CD is less thon B, then olso F is less thon G;ond if CD is greoter thon B, then olso F is greoter ihon G (12). This is whot we inten_dedtodemonstrote. A cE DNow thot this is cleor we soy: When we F+-+-'+{ +4{intend to compore some mognitude with on F--9--{ # Gother mognitude by meons of its ports, ond #substitute o multiple of eoch of them for the ports in the woy we hove illustroted inthis diogrom, then oll is cleor to us obout the multiples whot wos cleor to us obout28


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the,ports, os to being equol ond exceeding ond olling short. And likewise, when weconvert-this proposition, (H) we soy: when two mognitu--des ore diferent, ond when ofeqch of them o different multiple 'is token, ihen th-e condiion of th" ,rttiple o,f onemognitude in respecf of .the murtiple of lhe other mognitude is (the condition) o beingequol, exceeding or olling short in occordonc" *h the condition of certoin porof one of both mognitudes 'in respect of the other mognitude lt..il, *n"n the denomi-notor of the ports is the number of the multiple toke"n o the mognitude thot is notported, ond the number of ,the -ports is equol io the number ol'the multiple tokenof -the ported mognitude. Therefore it is sound whot we hove soid ond mentionedbeore, viz. if. multiples ore substituted or ports they do not give up onything ofthem nor do they closh with it.Euclid now, insteod of toking poris of the second ond the ourth ond comporing eochof them wirh its componion, r meon comporing the ports of rhu ,.ond with the firslmognitude ond comporing the ports of the ouith mognitude with the third mognituJe,only tokes o some multiple of the second ond the fo"urth ond ,ubrtitut", the ulriplesof eoch o them or iis pors, ond tokes o some multiple of rhe first ond the thirdond subsiifutes the multiple o eoch of them for lsel; ond oll thol is cleor in the coseof the multiples ond hoppens with them in respect o the multiple oi the other is cleorlr.th".:g:" of the porrs. ond hoppens wirh them in respecr ; ; other mognitudls.We will illustrqte this further with. on exomple ond soy, W'hen there or" fou1. ,onitrJur,ond when, whotever ports 'equol in number ond denominqtion be token o the secondcrnd the ourth, the ports of the second ore nor foun'd to exceed the first mognituJe,unless the ports of the fourth ore found to exceed,the third mognitude, ona t. port.of the second ore not ound to oil short of ihe first mognitud, unless the pori, otthe fourth ore found o foll short of the third mognitude, nd the ports of the secondore not ound to be equol .to .ihe irst mognitude, unless the ports of the fourih oreound io.be equol fo the third mognitude; then oll ,oru,rtiifl", tok"n of the firsrond the third ond oll some muJtiples token of ihe second ond the fourth ore not found,unless, i{ the multiple of the irst exceeds the multiple o thu ,..ond, the multiple oithe third too will exceed. the murtipre o{ the fourth, ond if the murtipre of the iirst isequol to the multiple of the second, the multiple.of the third is "qrol io the multiple ofthe fourth, ond if the multiple of the irst fqlis shorr of the multiiie of the second, rhemultiple of the third folls,short of the.multiple of the fourth. t--t soy: when fhemultiples ore in ihis condition, then the poris ore in the first-mentioned condition.And whot of one of these two conditions is found to be sotisfied is found to U" *t.rrJof the other too, ond the proportionolity of the mognitudes in order is ossured, ond therotio of the first ro the second is os the rotio of the third o the fourth.For insonce, let ihe mognitudes be A, B, C, D, ond let the oforesoid condition os to30


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the ports be ound; I meon thot, whoiever equivolent ports ore token of B ond D,the ports of B ore not fo_und (K) to exceed A, unless the ports of -o ioo exceed c, ondthe ports of B ore not ound to be equol to.A, unless he ports of D too uu "quoio.c,. ond the ports of B ore not ound to foll short o A, unless the porrs of D toofoll short of C; then I soy ihot whotever some multiple be token of he irs ond thethird, ond whotever some multiple be token of the second ond ihe urth, the multipleo the first will not be found to exceed the multiple of the seconJ, unless rhe multi,fleof he third too be found to ,exceed the multiple of the fourth, onJ th" rnrttipte or in"first will not be found to foll shori of the multiple of ihe ,".ona, unless the ,rHrfio ththird foo be found to foll short o{ the multiple of the ourrh.The proo is thot it con nof be otherwise. For ii such were possible, let be qkeno A ond c eq_uimultiples, which moy be E (ond) F, ond of d ond D equimultiples,which moy be G ond H, whire rhe condirion of E in' ,"rpu.i .i u unrike r,he con-dition of F in respect of H os to being equor, exceeding or foiling short; (e.g.) E greoterthon G o'd F not greoter hon H,-if such *"r" poriible. Let us now toke of B theport denominoted ofter the number of the multiple ihor E is o A ond F is o C, {orthese ore the some. This moy be KL. And of D the some port, wiich ,oy bu MN.hbn.we moke KO, which muit be the some murtipre o KL os C i, o B ond H is of D,for these ore ihe some. Likewise we moke Mp, which must be rhe some multiple oYN o: these multiples were. fg-* Ko is ports of B, ond Mp is rhe some ports o D.Therefore A ond B ore.two different mognitudes, ond of one o them, viz. B, portsore oken, viz. Ko, ond of B olso o muliiple is token, viz. G, rhe number o whichis. the number o the ports in Ko; ond of A q multiple is tqken, viz. E, fhe numberof which is the denominotor of ihe ports, in respect of which it hos been demonstrotedin. the foregoing proposition fhot the condition o E in respect ol C iu like the conditionof A in respect of Ko, os to being equol, exceed,ing ond olling short.-Now we supposedE to be greoter thon G, ond thereore A is greoter tr,o ro. The some conductis hot. of the mognirudes C (ond) D ond iheir ,rlrifl", ond their ports, sothot the condirion of F in respect of H is like the condition of C in ,u.pJ?f Mf, os to being equol or exceeding or foiling short. But F wos not greoter thon H,thereore c is not greoter thon Mp; ona R is gieoter thon KO, ond KO ond Mp oreports equol in number ond denominotion o B nd D. Therefore pqris ore found of B,viz. KO, less thon A, ond ports ore ound of D, viz. Mp, not ress thon c. This is ocontrqdiction, it is im.possible, becouse we hove supposed obout ihe mognitudes thoino ports o them could be ound but in the condiiion we hove mentioneios to beingequol or exceeding or folling short; hence it is cleor thot when the soid conditionof the ports is found, fhe condition.o the multiples is found too ond the proportionolityis found. And o this we wonted he proof.32


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+lB

col-{

f.H

MNPF-+-l

(L) By the some Procedureit is mode cleor to us, iwe suppose obout thesemognitudes thot the con-dition of the multiPles isfound, thot the condition othe ports is ound too, ondthot the proPortionolitY isfound (13).LOAs it is cleor now conclusively thoi the deinition o rcrtio by meons of multiplesir-rorna,-ond thot oll thot is ound with ports is ound likewise with multiples'.ondthot these two methods to reoch the truth ond to oscertoin the soundness ore equol' osone o them is not opposd to ihe other nor does one of them moke necessory whot the"tt,"r a".iorus impossible or'declore impossible whot the other clolms os necessory, so wesoy thot the seoichlng o{ proofs obout rotio ond whot is odditionolly or occidentollyconnected with it ond te investigotion o this with multiples is more elegont thon withports ond is more convenient for mony reosons. One of these is thot to toke multiplesof eoch of the two mognitudes is not dificulr, be ii o low or o high multiple; but .toioke ports of some mog"nitucle in such.o woy thoi these ports ex1e9d thot mognitude,crnd to coll whot is grter ihon the thing poris of it, thot is difficult os it orises fromon unusuol concePtion.o,. in fort, of o thing i, only expressed whot is less thon ii. This we mentioned olreodyin the 'prefoce of this writing. lf thereore we ovoid this view ond strive to toke porsf th" 'gr"of"r o the two (mponions) only, be it the ontecedent or the consequent'then we ore not ollowed nor oble to do so beore we know the greoter o ontecedeniond consequent in.uery.oru of o rotio. with multiples we d.o not wont onything o{ thef.i.a. in" ocility o thil l .ompored with the difiiculty only he knows who hos riedthe exposition of the Jeinition o{ rotio by meons of ports os well os by meons ofrrltlplur, or there is between them o lorge difference in diiculty o reseorch ondowkwordness of exposition. Moreover i wL would corry out the de{inition of rotioby meons of ports, *u *orta hove to olter the necessory propositions thot ore odvoncedi the fith book, ond we would hove io tronspose them {rom multiples into ports. Buttoking ports gives more intork ond di{iculty in reseorch, becouse in he cose of multiplesrn",i"g.ituj" is only multiplied by some number, whereos in the cose o poris oneneeds the number of the ports ond of the denominoiors of the pors crnd besides (oneneeds) the whole. truif i" kno*t"age of rhe lorge in orithmetic is eosier thon thekrrowledge o tf," rn.''o;. The oncieni scholors in oll kind of sciences we hove {ound

crcting in the some *oy. ror hose omong them who hod to reckon with froctions34

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