paolo mancosv aristotelian logic and euclidean mathematics

25 Pages·2002·1.72 MB·English

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.