PAOLO MANCOSV ARISTOTELIAN LOGIC AND EUCLIDEAN MATHEMATICS: SEVENTEENTH-CENTURY DEVELOPMENTS QUAESTZO DE CERTZTUDZNE OF THE A4ATHEA4ATZCARUA4 noitcudortnI GNOMA the factors hcihw deyalp a elor ni the htrib of naelilaG ecneics si the process of lacitirc noisiver of nailetotsirA yhposolihp hcihw took ecalp gnirud the htneetfif dna the htneetxis .seirutnec sA part of siht process tsum eb dedulcni the yrutnec-htneetxis noitcelfer no the ygolometsipe‘ of .’scitamehtam nihtiW the dnuorgkcab of nailetotsirA yhposolihp a rebmun of seussi were desiar tuoba the erutan of scitamehtam hcihw del emos srohtua (e.g. -olocciP ,inim ,anetaC )aryereP ot the lacixodarap siseht that scitamehtam si ton a .ecneics esehT ,snoitisop ,ylbadnatsrednu detareneg the snoitcaer of other srohtua (e.g. ,izzoraB ,inacnaiB )onatimoT who deirt ot etatsnier scitamehtam otni the krowemarf of nailetotsirA‘ ’.’ecneics sihT etabed si netfo denoitnem as tnemtrapeD* fo ,yhposolihP elaY ,ytisrevinU .O.P xoB 0563 elaY ,noitatS weN ,nevaH TC ,02560 .A.S.U Received 21 February ;1991 ni revised form 14 June 1991, yB‘ nailetotsirA‘ ’ecneics si tnaem ereh eht snoitidnoc hcihw a ydob fo egdelwonk tsum yfsitas ni redro ot teem s’eltotsirA noitinifed fo cifitneics .egdelwonk roF ,eltotsirA ot wonk yllacifitneics ,si gnoma rehto ,sgniht ot wonk eht esuac no hcihw eht tcaf sdneped dna cifitneics snoitartsnomed cra esoht hcihw ecudorp cifitneics .egdelwonk See rof ,elpmaxe Posterior .scit&nA kooB .1 noitceS .2 .J senraB ni s’eltotsirA Poslerior scirylnnA ,drofxO( ,5791 .p ,)69 sesu ’noitanalpxe‘ redner’ot eht keerG aitia. eH snialpxe sih eciohc sa .swollof ‘ ”noitanalpxE“ dna sti setangoc redner aifia dna sti ;setangoc eht lanoitidart noitalsnart si .”esuac“ s’eltot&rA smynonys rof aitia era or dioti dna to diu fi ,yllaretil( eht“ ”eroferehw dna eht“ esuaceb fo ”tahw - I etalsnart eht“ nosaer ;)”yhw suht ot evig eht aitia fo gnihtemos si ot yas yhw ti si eht ,esac dna X si aiton fo Y tsuj ni esac Y si esuaceb fo X .fc( ].H[ ,ztinoB Index ,sucilefotsirA ,nilreB ,0781 .)2-05’771 ecneH ,”esuac“ sa ti si desu ni laiuqolloc ,hsilgnE si a ylriaf doog noitalsnart fo aifia .fc( eht noitcnujnoc .)”esuaceb“ lacihposolihP ,egasu ,revewoh smees yllareneg ot esab flesti no a naemuH sisylana fo ;noitasuac dna na aitia si ton a naemuH .esuac roF siht nosaer ti si ylbaborp elbasivda ot tpoda a tnereffid ;noitalsnart ”noitanalpxe“ smees retteb naht ’.”nosaer“ .dutS .tsiH .lihP ,.icS loV ,32 .oN ,2 .pp ,562-142 .2991 29/1863-9300 00.5S OO.O+ detnirP ni taerG .niatirB @ .2991 nomagreP sserP .dtL 241 242 Studies ni History and yhposolihP of Science the Quaestio de certitudine mathematicarum.2 ehT latnemadnuf seussi desiar yb siht etabed were yllaitnesse the gniwollof :owt (a) tahW si the pihsnoitaler neewteb nailetotsirA cigol dna naedilcuE -ehtam ?scitam nI other words, nac scitamehtam eb ,deredisnoc as was netfo ,thguoht the mgidarap noitacifilpmexe of the 1aedi of nailetotsirA‘ ,’ecneics debircsed ni the roiretsoP Analytics, or does ti llaf short of ?ti sihT del ot a luferac ,sisylana at tsael yb ecnassianeR ,sdradnats of the erutan of lacitamehtam .snoitartsnomed )b( fI scitamehtam does ton evired sti ytniatrec yb the mrof of sti -artsnomed ,snoit how are we ot yfitsuj sti ytniatrec dna ?ecnedive ylralohcS work, yllaicepse yb ,ebbocaiG has nwohs that the Quaestio de certitudine mathematicarum crossed the nailatI seiradnuob ot reach as far as lagutroP dna .ecnarF nI noisulcnoc ot sih elcitra no ,aryereP ebbocaiG derutcejnoc that the Quaestio thgim evah had a noisuffid dna ecnatropmi that tnew dnoyeb the part of the etabed he had .derevocnu ehT melborp of the enutrof of the Quaestio has osla neeb desiar yb salohciN enidraJ ni noitcennoc htiw the melborp of ytiunitnoc neewteb“ the ’wen‘ secneics dna seigolometsipe of the htneetneves yrutnec dna reilrae .”stnempoleved nI ralucitrap enidraJ sevig the gniwollof lasiarppa of the noitautis for :scitamehtam ehT secruos dna senutrof fo eht yrutnec-htneetxis nailatI snoissucsid fo eht sutats fo lacitamehtam snoitartsnomed dna eht sdnuorg fo ytniatrec ni scitamehtam evah neeb elttil .deiduts esehT ,setabed ,era ,revewoh detcelfer ni eht stnemtaert fo eht sutats fo scitamehtam yb surohpotsirhC ,suivalC eppesuiG inacnaiB dna s’oelilaG roF* a tsrif yrammus noitcudortni ot eht etabed eht redaer si derrefer ot .N ,enidraJ ehT‘ ygolometsipE fo eht ,’secneicS ni The Cambridge History of Renaissance Philosophy, sde .C .B ,ttimhcS .Q .R .D ,rennikS E. relsseK ,egdirbmaC( ,)8891 .pp 685-711, yllaicepse .pp 693-697. roF lacihpargoilbib noitamrofni dna a deliated sisylana fo eht srohtua tsuj denoitnem ees eht gniwollof skrow yb .G .C :ebbocaiG ‘II muiratnemmoc ed enidutitrec muracitamehtam ,-iipicsid muran id ordnasselA ,’inimolocciP Physis 14 (1972), 162-193; ocsecnarF‘ izzoraB e al oitseauQ ed enidutitrec ’muracitamehtam Physis 14, (1972), 357-374; aL‘ enoisselfir acitametamatem id orteiP ,’anetaC Physis 15 .)3791( ;691-871 enuclA‘ enitneceuqnic itnadraugir li 0ssecorp id enoizatulavir acigolometsipe alled acitametam otibma’llen alled enoizulovir acifitneics ,’elatnemicsanir ni La Eerio, 31 ,)3791( ;44-7 aL‘ oitseauQ ed enidutitrec muracitamehtam onretni’lla alled aloucs ,’anavodap ni Atti del convegno di storia della logica ,audaP( ,)4791 .pp l-59 ;21 inogipE‘ len otnecies alled oitseuQ“ ed enidutitrec :”muracitamehtam eppesuiG ,’inacnaiB Physis 18 (1976), 540; nU‘ atiuseg a&ssergorp allen oitseauQ“ ed enidutitrec ”muracitamehtam :elatnemicsanir otineB ,’aryereP Physis 19 (1977), 5 l-86; Alle radici della rivoluzione scientifica rinascimentale: le opere di Pietro Catena sui rapporti rra matematica e Iogica ,asiP( .)1891 See osla .M ,izzardeP luS‘ ovitatnet id ordnasselA inimolocciP id errudir a omsigollis al I enoizartsomid ilged itnemelE id ,’edilcuE ni Culmra e Scuola 13, 1974, .tsaf ,25 ;032-122 no itteloM dna onatimoT ees eht selcitra ni .L ineivilO ,).de( Aristotelismo Veneto e scienza moderna, 2 vols, ,audaP( ,)3891 ylevitcepser yb ,oguraC .pp ,715-905 dna ivaD ,eleinaD .pp .126-706 ehT gniwollof era emos fo eht yramirp :secruos .A ,inimolocciP Commenrarium de cerritudine mathematicarum, ;7451 .F ,izzoraB Opusculum, in quo una Oralio. & duae Questiones: altera de certitudine, & altera de medielare Marhematicarum conlinentur, ;0651 .P ,anetaC Universa loca in logicam Aristorelis in marhematicas disciplinas, 1556; Super loca malhemalica contenta in Topicis et Elenchis Aristotelis, 1561; Oratio pro idea merhodi, 1563; .B ,aryereP De communibus omnium rerum naluralium principiis et affeclionibus libri quindecim, ;6751 muigelloC ,esnecirbminoC Commenlarii In oclo libros Physicorum Arisroteli, 1594; G. ,inacnaiB De mathemalicarum nafura dissertatio, .5161 Aristotelian Logic and Euclidean Mathematics 243 dneirf dna rotnem Jacopo ,inozzaM stnemtaert hcihw ylgnitseretni enibmoc -sisni ecnet no the ytniatrec dna ycnellecxe of scitamehtam dna lacitamehtam -artsnomed noit htiw sisahpme no the laitnatsbus elor of scitamehtam ni the yduts fo ’.erutan ehT mia of siht paper si ot show that the Quaestio de certitudine mathemati- carum had a noisuffid hcihw reached, ,yllacihpargoeg as far as dnalgnE dna dnaloP dna yllacigolonorhc as far as the 1670s. I lliw esylana the snoitubirtnoc ot the Quaestio yb ,suicelgimS ,sillaW ,sebboH Barrow dna 4.idnessaG tI si ym mialc that the lairetam I tneserp sevig na yllacoviuqenu evitisop rewsna ot the melborp of ytiunitnoc desiar ni s’enidraJ :elcitra great snoitrop of the -etsipe ,lacigolom dna erom yllareneg ,lacihposolihp noitcelfer no scitamehtam ni the htneetneves yrutnec tonnac eb dootsrednu tuohtiw gnirrefer kcab ot the ecnassianeR setabed no the Quaestio. ehT erutcurts of the paper si as .swollof ehT tsrif noitces lliw edivorp emos dnuorgkcab ot the ecnassianeR snoitubirtnoc ot the Quaestio. sihT lliw set the stage for the snoitubirtnoc of the srohtua rednu .noitaredisnoc noitceS 2 lliw esylana the oitseauQ 14, oitces 14 of the Logica yb ’.suicelgimS noitceS 3 lliw ebircsed ’sillaW snoitcaer ot the theses dleh yb suicelgimS dna emos of the snoitcejbo desiar yb sebboH tsniaga ’sillaW srewsna ot .suicelgimS noitceS 4 lliw esylana the snoitubirtnoc of caasI Barrow tsniaga the theses dleh ynam sraey erofeb yb ;aryereP I lliw osla eugra that the laer target of Barrow’s attacks was ton aryereP tub .idnessaG 1. ehT Quaestio de certitudine muracitamehtam ehT Quaestio de certitudine mathematicarum detanigiro htiw the noitacilbup ni 1547 of a esitaert yb ordnasselA inimolocciP (15081578) deltitne Commen- tarium de certitudine mathematicarum disciplinarum hcihw nac eb ylthgir -noc deredis eno of the tsom tnatropmi ecnassianeR snoitubirtnoc ot the yduts of the erutan of .scitamehtam s’inimolocciP tcejorp nac eb deziretcarahc as na tpmetta ot etufer a daerpsediw tnemugra hcihw demia at gniwohs the edutitrec eeS‘ ,enidraJ .po ,.tic eton ,2 .p .907 ehT suoiverp noitatouq si no ,p .807 lacihpargoilbiB‘ noitamrofni no ,sillaW ,sebboH worraB dna idnessaG si ylisae .elbaliava roF suicelgimS ees Universal Lexicon. da ;mecov dna ,legovremmoS Biblioth?que de la Compagnie de J&us, da .mecov suicelgimS deid ni 1618 dna eh si tnatropmi ni eht yrotsih fo msicilohtaC rof sih sseltneler ngiapmac tsniaga eht .snainicoS ehTS Loaica’s tsrif noitide saw dehsilbup ta tdatslognI ni .8161 ehT rehto eerht snoitide erew dehsilbup ni drofxO ni ,4361 .,8361 8561 .yGvitcepser I evah desu eht 8561 noitide esohw eltit egap :sdaer IICELGIMS/INITRAM/ACIGOL SITATEICI-OS .S.USE1 /.sirotcoD/ELGOLOEHT Selectis Disputationibus & qurestionibus illusfrata./Et ni soud somoT /:atubirtsib nI diuqci&/auq ni ociletotsirA 0nagro lev utingoc lev/,muirassecen etatirucsbo ,muxelprep mat eralc & ,eucipsrep mauq edil/-os ca esovren muC/.rutatcartrep Indice Rerum mertsullireP/DA/osoipoc ca -ingaM mucif .mD/munimoD MAMOHT ,MUICSYOMAZ tabeducxE/,IINOXO/.c& .A LichJeld, .dacA .rgopyT sisnepmI .H .J/,SPPIRC NIWDOG & .R ,EVARGALB An. Dom. 1658./Gum Privilegie. 244 Studies in History and Philosophy of Science of scitamehtam (asserted yb eltotsirA dna detaretier yb seorrevA dna a gnol tsil of nailetotsirA )’srotatnemmoc gniugra morf the noitpmussa that -ehtam scitam sekam esu of the tsehgih epyt of citsigollys ,snoitartsnomed hcihw ni ecnassianeR ygolonimret were dellac demonstrationes potissimae (see .)woleb s’inimolocciP noitubirtnoc was .dlofowt tsriF he deugra that the -artsnomed snoit of scitamehtam are ton potissimae dna that therefore the tnemugra deniltuo evoba si sugob ecnis the ssimerp si .eslaf He deveihca siht yb a luferac noitinifed of what stnuoc as a demonstratio potissima dna yb gniwohs yhw lacitamehtam snoitartsnomed do ,ton dna tonnac ,ylbissop tif otni hcus a .yrogetac s’inimolocciP noitisop was :evitavonni yb gnitarapes the citcatnys serutaef of naedilcuE scitamehtam morf those of nailetotsirA ,cigol he nac eb nees as na tnatropmi tnemom ni the seires of stneve that del ot the ecnegreme fo“ scitamehtam as the citsiugnil loot of the wen ’.”ecneics ,revewoH -olocciP inim deveileb ni the ytniatrec of .scitamehtam suhT ni the tsal part of the esitaert he deugra for the ytniatrec of scitamehtam yb gnizisahpme the lautpecnoc erutan of the stcejbo of scitamehtam ,hcihw gnieb created yb the namuh ,dnim evah the tsehgih level of ytiralc dna .ytniatrec nI order ot proceed ti si laitnesse ot ecudortni the lacigolonimret snoitcnitsid gninrecnoc tnereffid sepyt of snoitartsnomed decudortni yb eltotsirA ni the Posterior Analytics dna rehtruf detarobale yb seorrevA ni sih Proemium ot sih yratnemmoc no s’eltotsirA Physics. eltotsirA (Posterior Analytics, Book 1, noitceS 13) had dehsiugnitsid two sepyt of :snoitartsnomed roO i& dna ro8 ~5~6 htrofecneh( hoti dna dioti), or ni the nitaL ygolonimret quia dna propter quid, .e.i of the ‘fact’ dna of the denosaer‘ fact’. ehT tsrif epyt of noitartsnomed proceeds morf effects ot sesuac whereas the dnoces epyt proceeds morf sesuac ot effects.8 nI ’seorrevA Proemium the noitcnitsid semoceb ;dlofeerht -nomed snoitarts are deifissalc rednu the areneg of quia, propter quid dna potissima. sihT si the noitcnitsid proposed yb inimolocciP who deziretcarahc the demon- strati0 potissima as a noitartsnomed hcihw sevig htob the esuac dna the effect of na tneve (simul et quia et propter quid). He smees ot yfitnedi ti htiw a msigollys of the tsrif erugif htiw lasrevinu .ssimerp More ,yllacificeps -olocciP inim deriuqer the elddim ot evah the mrof of a noitinifed dna ot enimreted the etamixorp esuac of the effect ni a euqinu .yaw Chapter 11 of sih esitaert was inimolocciPh ,setouq gnoma ,srehto ,treblA ,samohT ,suilisraM ,arramiZ ,ofiN .oloiaiccA ,ebbocaiG aL‘ enoisselfir ,’acitametamatem .po ,.tic eton ,2 .p .853 esehTX erew semitemos ,deifitnedi rof elpmaxe ni ,allerabaZ htiw eht evitisopnroc dna evituloser sdohtem desu yb eht .snaicitamehtam roF na sisylana fo siht melborp ees E. ,itreB aznereffiD‘ art li odotem ovitulosir ilged iciletotsira e al ”oituloser“ ied ,’icitametam ni omsilefo~sirA ,oteneV op. ,.tic eton ,2 .pp .754-534 ehT nailetotsirA noitacifissalc sah neeb laitneulfni neve ni tnecer krow ni yhposolihp fo ecneics no .noitanalpxe See rof elpmaxe .B .A ,ydorB sdrawoT‘ na nailetotsirA yroehT fo cifitneicS ,’noitanalpxE Philosophy of Science 39 (1972). 20-31; dna .B nav ,nessaarF ehT‘ scitamgarP fo ,’noitanalpxE naciremA lacihposolihP ylretrauQ 41 .)7791( .151-341 Aristotelian Logic and Euclidean Mathematics 245 detoved ot gniwohs that snoitartsnomed ni scitamehtam do ton mrofnoc ot yna of the evoba 9.snoitcirtser erehT were lareves snoitcaer ot s’inimolocciP work. lareveS sralohcs agreed htiw mih that snoitartsnomed ni scitamehtam did ton mrofnoc ot the tsetcirts nailetotsirA sdradnats for demonstrationes potissimae; siht puorg ,dedulcni for ,elpmaxe anetaC dna .aryereP ,revewoH rieht noitavitom for gniyned that lacitamehtam snoitartsnomed were potissimae were etiuq .tnereffid saerehW aryereP smialc siht ot eb the case ni order ot etargined ,meht inimolocciP dna anetaC do so ni order ot ezisahpme rieht ymonotua dna ytniatrec deedni( anetaC smialc that lacitamehtam snoitartsnomed evres as the ledom for sdohtem ni lla .)senilpicsid yB ,tsartnoc elpoep ekil ,izzoraB inacnaiB dna onatimoT deugra that at tsael emos snoitartsnomed ni scitamehtam did mrofnoc ot the stnemeriuqer for demonstratio potissima, hguohtla ton lla of meht (for elpmaxe izzoraB dna inacnaiB ylticilpxe dedulcxe proofs yb -artnoc noitcid morf the mlaer of demonstrationes potissimae).” eW lliw see that the niam seussi desiar yb the Quaestio were llits evila ni the dnoces part of the htneetneves .yrutnec ehT snoitisop dleh yb ,inimolocciP anetaC dna aryereP dellac otni noitseuq dehsirehc sfeileb ni scitamehtam as the mgidarap of ;ecneics or worse, dedulcxe scitamehtam morf the mlaer of ecneics ”.rehtegotla snaicitamehtaM of the erbilac of sillaW dna Barrow tlef siht mialc dluoc ton go .degnellahcnu eroM‘ sliated nac eb dnuof ni ,enidraJ op. cif., eton .2 enidraJ sezirammus eht noitcnitsid edam yb ,eltotsirA neewteb dioti dna hofi snoitartsnomed :suht enO‘ fo sih selpmaxe fo eht remrof ylthgils( )dednapxe :si ylnevaeh seidob hcihw era raen eht htrae od ton ;elkniwt eht stenalp era raen eht ;htrae ecneh eht stenalp od ton .elkniwt sihT msigollys setartsnomed eht ecneserp fo na devresbo ,tceffe ton ,gnilkniwt ni a ,tcejbus the planets; dna ti seod os yb snaem fo a elddim ,mret being near the earth, hcihw setutitsnoc eht etamixorp esuac fo taht .tceffe yB tnemegnarraer fo smret a noitartsnomed‘ fo eht ’tcaf [hot11 si deniatbo ni hcihw eht elddim mret seificeps eht tceffe rehtar naht eht .esuac suhT ew :evah ylnevaeh seidob hcihw od ton elkniwt era raen eht ;htrae eht stenalp od ton ;elkniwt ecneh eht stenalp era raen eht .htrae traP fo s’eltotsirA noitnetni ni siht egassap roiretsoP[ Analytics, kooB I, noitceS 131 ,si ti ,smees ot hsiugnitsid evitartsnomed smsigollys morf detaler smsigollys esohw ,sessimerp tslihw ,eurt liaf ot nialpxe eht ’.noisulcnoc Ibid., .p .686 nOO‘ eht eussi fo sfoorp yb noitcidartnoc ni eht htneetneves yrutnec ees .P ,usocnaM nO‘ eht sutatS fo sfoorP yb noitcidartnoC ni eht htneetneveS ,’yrutneC esehrnyS 88 )1991( .1451 sihT repap osla slaed htiw rehto snoitacifimar fo eht Quaestio sa rof elpmaxe ni sutlaviR dna .nidluG ehT“ tsom lufecrof tnemetats fo hcus snoitisop si dedivorp yb .aryereP aeM“ oinipo .tse sacitamehtaM sanilpicsid non esse eirporp :saitneics ni mauq menoinipo rocudda nrut sjila nrut coh onu 6mixam .otnemugra ericS tse mer rep massuac erecsongoc retporp mauq ser ;tse & aitneics tse sinoitartsnomed ;sutceffe oitartsnomed metua rouqol( ed omissitcefrep sinoitartsnomed )ereneg eratsnoc tebed xe sih eauq tnus rep es & airporp suie douq ;rutartsnomed eauq brev tnus rep ,snedicca & ,ainummoc rutnudulcxe a sitcefrep ,subinoitartsnomed des ,sucitamehtaM euqen taredisnoc maitnesse ,sitatitnauq euqen senoitceffa suie tatcart tuorp tnanam xe ilat ,aitnesse euqen taralced sae rep sairporp ,sassuac retporp mauq tnusni ,itatitnauq euqen ticifnoc senoitartsnomed saus xe sitacidearp sjirporp & rep ,es des xe ,subinummoc t8 rep ,snedicca ogre anirtcod scitamehtaM non tse eirporp :aitneics roiaM suiuh imsigollys non tege ,enoitaborp minete ttrepa ruticile xe sih eauq atpircs tnus ba .tsirA I. .tsoP oitamrifnoC sironiM ruticud xe ,sih eauq tibircs otalP ni .7 .bil ed .lbupeR snecid socitamehtaM erainmos acric ,metatitnauq & ni sidnatcart sius subinoitartsnomed non 6cifitneics des xe madsubiuq subinoitisoppus .eredecorp merbomauQ non tluv manirtcod muroe eralleppa maitnegilletni tua ,maitneics des mutnat :menoitatigoc ni mauq SPIW D-Z%Z 246 Studies in History and Philosophy of Science 2. suicelgimS tI dluohs eb tnedive morf what has neeb dias so far that tnemeergasid tuoba the Quaestio de certitudine mathematicarum dertnec dnuora the eussi whether lacitamehtam snoitartsnomed were demonstrationes potissimae. deednI -occiP inimol ylticilpxe degdelwonkca that siht melborp had dedivorp the niam nosaer for gnitirw the Commentarium. oT siht yrev eussi the hsiloP naicigol nitraM suicelgimS detoved the oitseauQ 14, oitces 14, of sih Logica (1618): rehtehW“ lacitamehtam snoitartsnomed are tsom perfect dna evah the serutaef of potissimae 2‘.”snoitartsnomed ’suicelgimS citsalohcs elyts of noitisopxe detneserp ni na ylredro dna citamehcs noihsaf the lareves snoitisop hcihw had neeb dleh htiw respect ot the erutan of lacitamehtam .snoitartsnomed tI dluohs eb dekramer that suicelgimS did ton etaicossa the tnereffid snoitisop htiw cificeps .seman suicelgimS nageb yb gnidnuopxe the tsrif‘ ’noitisoporp gnitressa that potis- simae demonstrationes, fi yeht tsixe at ,lla nac ylno appear ni .scitamehtam ehT tnemugra degnih nopu a lacitpecs noitisop gninrecnoc ruo ytiliba ot wonk the secnesse of larutan .sgniht He neht proceeded ot noitnem a ralimis tnemugra desu ot edulcxe the ytilibissop that yna other ecneics naht scitamehtam dluoc ylbissop eb tuoba yrassecen sgniht (de re necessaria) dna proceed morf yrassecen .selpicnirp 3I maitnetnes atlum tib&s s&orP ni I. .bil murous muroiratnemmoC ni .medilcuE ,mureV istemat euqen menotalP euqen mulcorP euqen saila sohposolihP ,sevarg sumerebah serotcua suiuh ,eaitnetnes nemat di rep es mutsefinam tif siviuc iuq lev retivel odom tiregitta mutidure mulli murocitamehtaM .merevlup maN is siuq muces tetuper euqta retnegilid teredisnoc -artsnomed senoit ,sacirtemoeg eauq rutnenitnoc sirbil murotnemelE .dilcuE tnalp tegilletni sae cis esse satceffa tu etna :sumixid ca tu ed sitlum munu tua muretla mareforp ,mulpmexe retemoeG tartsnomed mulugnairt erebah sert solugna selauqea suboud ,sitcer aeretporp douq sulugna ,sunretxe iuq ruticiffe xe eretal suilli ilugnairt ,otcudorp tis silauqea suboud silugna medsuie ilugnairt ibis :sitisoppo siuQ non tediv coh muidem non esse massuac suilli sinoissap eauq ?rutartsnomed muc suirp arutan tis mulugnairt ,esse & erebah sert solugna selauqea suboud ,sitcer mPuq lev icudorp sutal ,suilli lev ba oe eretal ireif mulugna melauqea suboud ,sitcer mauq lev icudorp sutal suilli lev ba oe eretal ireif mulugna melauqea suboud ?sinretni ,aeretearP elat muidem tebah es oninmo rep snedicca da malli ;menoissap man evis sutal ,rutacudorp & taif sulugna ,sunretxe evis ,non ommi istemat sumagnif menoitcudorp suilli ;siretal ,qmenoitceffe ilugna inretxe esse ,melibissopmi sunimolihin nemat alli oissap tesseni ;olugnairt ,ta diuq duila rutinifed esse snedicca mauq douq tsetop esseda & esseba ier retearp suie ?menoitpurroc dA ,ceah salli ,senoitisoporp mutoT tse suiam aus ,etrap selauqea esse saenil eauq rutnucud a ortnec da ,maitnerefmucric dulli sutal esse ,suiam douq rutinoppo iroiam ,olugna & di suneg ,aila mauq orberc taprusu ni ?odnartsnomed ni mauq sitlum subinoitartsnomed sae orp oidem t.ebihda & taclucni ?sucitamehtaM tu essecen tis xe sih subinoitartsnomed eauq tnatsnoc sitacidearp -ummoc ,subin non engig matcefrep .”maitneics De subinummoc , .po ,.tic eton ,2 .pp .52-42 ehT evoba egassap sniatnoc ynam fo eht semeht taht deziretcarahc eht Quaestio: )a( eht yticifitneics fo ;scitamehtam (b) eht lasuac erutan fo eht ;msigollys )c( eht esu fo noitisoporP 23.I morf s’dilcuE .stnemelE ,suicelgimS‘ ,acigoL .po ,.tic eton ,5 .pp .385-085 “Ibid., .p .085 It si ot eb dekramer taht hcus lacitpecs snoitisop gninrecnoc ruo ytiliba ot wonk eht ecnesse fo larutan sgniht erew ydaerla daerpsediw ni eht .ecnassianeR nailetotsirA Logic and Euclidean Mathematics 247 .giF I gnivaH detneserp the tnemugra hcihw dedulcxe the ytilibissop that other ,secneics ni ralucitrap ,scisyhp nac evah demonstrationes potissimae, suicelgimS proceeded ot ssucsid the stnemugra ni ruovaf of the siseht that lacitamehtam snoitartsnomed do deedni possess the serutaef of demonstrationes potissimae. ehT tsrif eno seiler no the ytirohtua of eltotsirA who, gnidrocca ot siht ,noitaterpretni ni the Posterior Analytics had denifed the potissima -artsnomed noit as a msigollys of the tsrif erugif esuaceb the lacitamehtam secneics esu that erugif ni rieht proofs. dnA siht tnemugra dluow ton dloh tuohtiw na ticilpmi ,noitpmussa no s’eltotsirA part, that lacitamehtam snoitartsnomed are potis- simae. ehT dnoces tnemugra slaeppa ot the fact that demonstrationes potis- simae dluohs eb Ilasuac dna tuoba yrassecen ;stcejbo dna htob snoitidnoc nac eb dnuof ni lacitamehtam .snoitartsnomed ,revewoH whereas there does ton mees ot eb yna melborp tuoba the ytissecen of lacitamehtam ,snoitartsnomed the melborp tuoba ytilasuac si erom ,etaciled ecnis emos mialc that -ehtam lacitam snoitartsnomed are ton desab no laer sesuac tub ylno sesuac evitaler ot ruo egdelwonk (“causas . . . cognoscendi”). I5 nI yna case, yeht are lasuac as si detartsulli yb gnisu a melborp morf dilcuE hcihw had neeb a sucol classicus ni the suoiverp snoissucsid no the ytilasuac of lacitamehtam .snoitartsnomed tI si melborp 1.I morf s’dilcuE Elements where ti si nwohs how ot tcurtsnoc na laretaliuqe elgnairt revo a nevig .tnemges ehT noitcurtsnoc sesu two yrailixua selcric hcihw evah rieht sertnec at the stniopdne of the nevig tnemges dna iidar lauqe ot the tnemges (see .giF 1). Now the elgnairt CBA si laretaliuqe ecnis sti sedis are lauqe ot the suidar of the emas elcric dna suht are lauqe ot each other. ehT lasuac erutan of the proof si deugra as .swollof rehtiE siht ytreporp of the r%‘ sti yramirp gninaem eht laeppa ot ytilasuac seiler no eht aedi taht eht ecnesse fo a lacirtemoeg erugif netfo( deifitnedi htiw sti ,)noitinifed yas a ,elgnairt yllasuac senimreted sti rehto non( )laitnesse .seitreporp ,revewoH eht lareves sgninaem fo ytilasuac ni noitaler ot lacitamehtam snoitartsnomed lliw egreme ni eht esruoc fo eht .repap eD“>‘ sisuac orev iste madiuq tnetibud sae non erebah sarev sasuac ,idnesse des ,idnecsongoc nemat ,arever selat tnebah ,sasuac subiuq sitisop rutiuqes silat .”sateirporp Op. ,.tic eton ,5 .p .185 sihT noitisop saw dleh yb eht naedemihcrA rotatnemmoc divaD sutlaviR ni sih Archimedis opera quae extant novis demonstrationibus commentariisque illustrata per Davidem nrutlaviR a Flwantia (1615). 248 Studies in History and Philosophy of Science elgnairt nac eb nwohs ot dneped yllasuac no the erutan of the elgnairt or .ton fI so neht eno nac evorp the thguos ytreporp morf the laitnesse erutan of the elgnairt neve fi on hcus noitartsnomed si tey .elbaliava fI there si on hcus esuac neht we evah a noitcidartnoc ecnis no eurt effect ni the dlrow nac eb tuohtiw emos eurt 61.esuac enO dluohs eton that siht noitisop sdnet ot edecnoc hcum ot those who detsetnoc that lacitamehtam snoitartsnomed were potissimae. ehT tnemugra smia at gnihsilbatse the ytilibissop for emos lacitamehtam -nomed snoitarts ot eb potissimae neve fi de f&to there yam eb enon htiw hcus .ytreporp suicelgimS neht proceeded ot tneserp the dnoces‘ :’noitisoporp lacitamehtam snoitartsnomed are ton potissimae, esuaceb yeht do ton eugra morf eurt dna yrassecen .sesuac sihT noitisop nac eb deugra morf two tnereffid sepyt of .snoitpmussa ehT tsrif eno proceeds morf a rebmun of smialc tuoba the lacigolotno sutats of lacitamehtam :stcejbo dnA deedni emos eugra siht tniop morf the fact that lacitamehtam seititne ekil seititnauq dna ,serugif as yeht are deredisnoc ni ,scitamehtam era ton ot eb dnuof ni erutan [in rerum natura] . ,revoeroM ti si deriuqer fo a eurt dna tcefrep noitartsnomed ot eb tuoba a laer ytitne dna ton tuoba na yranigami .eno ,esiwrehtO ti lliw ton evah ylurt dna yllaer yna seitreporp tub ylno hguorht ”.noitanigami ,revewoH suicelgimS sdnif siht rettal noitisop .elbanetnu ,deednI asserts -gimS ,suicel lla that si deriuqer si the ytilibissop of the ecnetsixe of the tcejbus of a lacitamehtam ,noitartsnomed dna siht si deetnaraug yb s’doG potentiam. tI si ton deriuqer fo a noitartsnomed taht sti tcejbus tsixe ni actual ;ytilaer ,esiwrehto ni retniw ereht dluoc eb on ecneics fo eht ;esor dna ereht dluoc eb on ecneics fo a erutuf .espilce tI seciffus ni tcaf taht ti dluoc tsixe ni .ytilaer deednI ti si ton ni tbuod taht esoht tcaxe ,serugif sa yeht era denifed yb ,snaicitamehtam nac eb nevig yb s’doG power.‘* roF ,elpmaxe gnihton stneverp doG gnitaerc a enil yltnednepedni of a ,enalp .e.i a erem htgnel tuohtiw .htdaerb ehT eurt noitatnemugra ni ruovaf of the dnoces‘ ’noitisoporp si that lacitamehtam snoitartsnomed do ton possess the laer sesuac of gnieb (non continent ni se veras causas essendi) dna yltneuqesnoc kcal the ytissecen that nac ylno etanigiro morf eurt .sesuac nI other words, lacitamehtam -artsnomed snoit are tneicifed ni htob serutaef of a potissima ,noitartsnomed .e.i ytilasuac 161bid. tE““ mediuq illunnon tnaborp di xe ,oe douq aitne ,scitamehtam tu setatitnauq & ,earugif morp a scitamehtaM ,rutnaredisnoc non rutned ni murer arutan orroP da marev & matcefrep menoitartsnomed ;rutiriuqer tu tis ed etne ,ilaer non ed etne ;oiranigami niuqoila non tibebah erev & remaer sallu ,setateirporp des mutnat rep ”.menoitanigami Ibid. euqeN““‘ orev da menoitartsnomed rutiriuqer tu mutcejbus utca ilaer ,tatsixe iuqoila( ed asor ni ,emeyh & ed ispylcce arutuf non tessop esse )aitneics des sitas ,tse tu retilaer eretsixe .tissop noN tse metua muibud satcaxe salli ,sarugif selauq tnuinifed icitamehtaM rep Dei maitnetop irad ”.essop Ibid., .p .285 Aristotelian Logic and Euclidean Mathematics 249 dna .ytissecen erehT are two sepyt of stnemugra for the proof of siht .noitressa enO proceeds a posteriori (inductio) dna the other eno a priori. Let su redisnoc the tnemugra a posteriori. fI eno sesylana s’dilcuE melborp 1.I denoitnem ,evoba ti si raelc that the nosaer yhw the elgnairt si laretaliuqe si ton esuaceb of the two selcric desu ni the :noitcurtsnoc roF ni the tsrif noitartsnomed ni ,dilcuE the elgnairt si nwohs ot eb laretaliuqe morf the fact that ti si detcurtsnoc neewteb two selcric dna all sti sedis era nward morf eht ertnec ot eht .ecnerefmucric ydoboN nac liaf ot ees taht siht seod ton enimreted eht eurt esuac fo gnieb [veram causam essendi]. nI tcaf eht elgnairt si ton laretaliuqe no tnuocca fo sti gnieb detcurtsnoc neewteb owt .selcric roF ti dluow llits eb laretaliuqe neve fi ti erew ton detcurtsnoc neewteb two .selcric morF ecnehw ti swollof that siht esuac si latnedicca ot that 91.ytreporp rehtonA elpmaxe hcihw si deredisnoc ni siht ,noitcennoc dna hcihw si rehtona sucol classicus of the Quaestio, si s’dilcuE noitisoporp 23.I hcihw shows that the mus of the lanretni selgna of a elgnairt slauqe two thgir selgna yb gnitiolpxe the lanretxe :elgna ylralimiS ni noitisoporp 32 fo eht tsrif koob fo dilcuE ti si nwohs taht a elgnairt sah eht eerht selgna lauqe ot owt thgir .selgna ,roF gnicudorp eno ,edis eht lanretxe elgna si lauqe ot eht owt lanretni .selgna But siht si ton eht eurt esuac fo .gnieb roF eht elgnairt dluow llits evah eht eerht selgna lauqe ot owt thgir seno neve fi eht lanretxe elgna saw ton ereht non[ esset].‘O ehT dnoces tnemugra proceeds a priori. enO seugra that ni the potissima noitartsnomed the esuac of the ytreporp si the ecnesse of the tcejbus morf hcihw the ytreporp ;setanigiro ,revewoh ni scitamehtam eno does ton eugra morf the ecnesse of the tcejbus tub morf the s’tcejbus pihsnoitaler [habitudinem] ot other .serugif suhT lacitamehtam ,snoitartsnomed sedulcnoc the ,tnemugra do ton eugra yb veram causam essendi. ehT tnemugra for the dnoces ssimerp slaeppa ot the mialc that serugif dna seititnauq are lacisyhp seitreporp [acci- dentia] dna the ecneics that slaed htiw meht si .scisyhp suhT whereas scisyhp sevorp yb eurt ,sesuac scitamehtam seugra morf the pihsnoitaler that eno erugif has ot other .serugif ,revewoH siht yaw of gniugra proceeds morf cisnirtxe sesuac ecnis the erugif tuoba hcihw we are gnivorp gnihtemos does ton dneped for sti gnieb no the yrailixua serugif desu ot show sti .seitreporp ehT tnemugra tsniaga ytissecen si ylpmis a noitairav of the suoiverp .eno “+‘Nam ni amirp enoitartsnomed ,sidilcuE rutartsnomed mulugnairt esse ,muretaliuqea xe ,oe douq tis mutcurtsnoc artni soud ,solucric ;qtebah ainmo aus aretal a ortnec da :maitnerefmucric ibU omen non ,tediv non irangissa marev masuac ,idnesse non mine mulugnairt ocricdi tse ,muretaliuqea aiuq tse mutcurtsnoc artni soud :solucric maN ismaite non tesse artni soud solucric ,mutcurtsnoc cuhda tesse ,muretaliuqea ednu silat ,asuac tse silatnedicca illi ”.itateirporp Ibid. retilimiSY‘O‘ ni .23 enoitisoporp imirp irbil ,sidilcuE ,rutartsnomed mulugnairt erebah sert solugna selauqea siboud :sitcer aiuq 0tcudorp onu ,eretal sulugna sucesnirtxe tse silauqea suboud silugna :sinretni ta ceah non tse asuac arev :idnesse aiuq ismaite non tesse sullu sulugna ,sucesnirtxe terebah sunimolihin sulugnairt sert selauqea suboud ”.sitcer Ibid. 250 Studies in History and Fhilosophy of Science tA siht ,tniop after gnivah deyevrus the tnereffid ,snoitisop suicelgimS si ydaer ot evig sih noinipo no the elohw .eussi tsniagA the tnemugra yb ,ytirohtua elihw gnidecnoc that eltotsirA evag lacitamehtam selpmaxe for the seitreporp of the potissima ,noitartsnomed he seugra that eltotsirA reven evag na elpmaxe where lla these seitreporp dleh at .ecno ,revoeroM eltotsirA reven asserted that lacitamehtam proofs dluohs etanigiro ex veras causas essendi. gninrecnoC the ytilasuac of snoitartsnomed suicelgimS seined that ni -ehtam scitam snoitartsnomed proceed yb gnisu the eurt sesuac of the :tcejbus gninrecnoC eht dnoces ,tnemugra ti tsum eb deined taht eurt sesuac fo gnieb era ni .scitamehtam roF neve fi yrassecen seitreporp evah eurt sesuac fo ,gnieb yleman eht ecnesse fo eht ,tcejbus tey hcus sesuac era ton deredisnoc yb eht ,naicitamehtam ecnis eh swonk taht yeht gnoleb ot .scisyhP eH sredisnoc sih ssenisub ylno ot etartsnomed seitreporp[ ]fo eht erugif hguorht eht ,erugif ro hguorht gnihtemos cisnirtxe ot eht ’*.erugif ,revoeroM there si na tnatropmi ecnereffid neewteb scitamehtam dna the other .secneics nI the other secneics there thgim ton eb de facto potissimae demonstra- tiones tub yeht are .elbissop nI scitamehtam hcus ytilibissop si ton .nevig roF ni rehto ,secneics neve fi de facto ereht era ton potissimae ,snoitartsnomed ereht dluoc ,eb sa raf sa eht erutan fo eht stcejbo dna fo eht ecneics .swolla roF eht stcejbo evah eht eurt sesuac fo gnieb fo rieht seitreporp dna ot etartsnomed hguorht hcus sesuac seod ton og dnoyeb eht lamrof erutan fo eht .tcejbo But ni ,scitamehtam rehtien era eht eurt sesuac fo gnieb fo lareves seitreporp ,nevig ron si ti eht ssenisub fo eht naicitamehtam ot etartsnomed hguorht meht tub taht fo eht .tsicisyhp roF ,elpmaxe gnieb laretaliuqe si a erem latnedicca ytreporp fo eht ,elgnairt hcihw nac gnoleb ro ton gnoleb ot .ti ,yltneuqesnoC ti sah on esuac fo gnieb rehto naht na latnedicca ,eno taht ,si eht tcaxe noitcurtsnoc fo eht 22.elgnairt suicelgimS dedulcnoc the oitseauQ yb gnisop the melborp whether s’eltotsirA work ni gniyal nwod snoitidnoc rof eht demonstratio potissima had neeb ni .niav Not at ;lla s’eltotsirA tnatropmi ,noitubirtnoc sedulcnoc ruo ,naicigol stsisnoc ni gnivah nevig a perfect laedi toward hcihw snoitartsnomed dluohs .evirts dA“‘> ,mudnuces mudnageN tse ni sicitamehtaM esse sare“ sasuac :idnesse maN iste -eirporp setat eairassecen tnaebah sare“ saus sasuac ,idnesse epmen maitnesse ,itcejbus nemat selat sasuac non taruc ,sucitamehtaM muc taics sae da macisyhP :erenitrep ius orev iiciffo tesnec esse ,mulos marugif rep marugif ,erartsnomed ues rep diuqila earugif ”.mucesnirtxe ,.dibI .p .385 maN“2‘ ni siila ,siitneics iste ed otcaf non tnis senoitartsnomed ,eamissitop tnussop nemat ,esse mutnauq tse xe arutan murotcejbo & :eaitneics maN & atcejbo tnebah sare“ sasuac idnesse muraus ,mutateirporp & rep selat sasuac ,erartsnomed non tidecxe menoitar melamrof ;itcejbo ta ni ,sicitamehtaM euqen rutnad esrev easuac idnesse muramirulp ,mutateirporp euqen rep sae erar&nomed da mucitamehtaM ,tatceps des da .mucisyhP maN ,mulugnairt ibrev ,aitarg esse ,muretaliuqea tse sateirporp me& .silatnedicca eauq tsetop ,esseda & ,esseba & xe itneuqesnoc non tebah maila masuac ,idnesse isin ,melatnedicca coh ,tse maspi menoitcurtsnoc matcaxe ilugnairt . ” Ibid.