John Wallis's Account of Mathematical Learning in Medieval England

50 Pages·2001·0.22 MB·English

by  Stedall, J. A.

HistoriaMathematica28(2001),73–122
doi:10.1006/hmat.2000.2307,availableonlineathttp://www.idealibrary.comon
Of Our Own Nation: John Wallis’s Account of Mathematical
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JacquelineA.Stedall
TheQueen’sCollege,OxfordOX14AW
InAtreatiseofalgebrabothhistoricalandpractical(London1685),JohnWalliswrotethefirstsurvey
ofthestateofmathematicallearninginmedievalEngland,anddiscussedwithparticularcarethearrival
andsignificanceoftheHindu–Arabicnumeralsystem.Thispaperoffersadetailedcommentaryon
Wallis’saccountinrelationtothesourcesheusedandthe17th-centuryOxfordcontextinwhichhe
wrote.ThepaperalsosupplementsWallis’streatmentwherepossiblewithsomeofthefindingsof
modernscholarship.Itthereforeprovidesontheonehandanoverviewofthespreadofmathematical
learning into medieval England, and on the other an insight into late 17th-century historiography.
Wallispioneeredseveralnewhistoriographicalmethodsandcanperhapsbeclaimedasthefirstmodern
historianofmathematics. (cid:176)C2001AcademicPress
DansAtreatiseofalgebrabothhistoricalandpractical(Londres1685)JohnWallisae´te´lepremiera`
e´criresurl’etatdese´tudesdemathe´matiquesdansl’AngleterreduMoyenAge,etiladiscute´avecunsoin
particulierdel’arrive´eetdusensdusyste`menume´riquehindou–arabe.Cetexpose´offreuncommentaire
de´taille´surl’essaideWallisencequiconcernelessourcesutilise´esetlecontexted’Oxfordaudix-
septie`mesie`cle,e´poquea`laquelleile´crivit.Cetexpose´ajouteenplusautraitementdeWalliscertaines
de´couvertesd’eruditionmoderne.Ilfournitdoncd’unepartunaperc¸uge´ne´raldel’e´tenduedese´tudes
demathe´matiquesdansl’Angleterremoyenaˆgeuseetd’autrepartunaperc¸udansl’historiographiea`
lafindudix-septie`mesie`cle.Wallisa`e´te´lepionierdeplusiersme´thodesd’historiographieetpourrait
eˆtreacclame´commelepremierhistorienmodernedemathe´matiques. (cid:176)C2001AcademicPress
MSCsubjectclassifications:01A35,01A45,01A85.
KeyWords:mathematics;history;medieval;English:Hindu-Arabicnumerals;JohnWallis.
WhenJohnWallis(1616–1703)wrotehisgreathistoricalstudy,Atreatiseofalgebraboth
historicalandpractical(1685),hedevotedthree-quartersofits100chapterstoexploring
the work of five 17th-century English mathematicians: Oughtred, Harriot, Pell, Newton,
andhimself[Stedall2000].Hisprideintheachievementsofhisowncountrymeninhisown
centurywasfromthebeginningaprimereasonforwritinghisbooknotjustasmathematics
butashistory,ahistorydesignedtomaketheEnglishcontributionplain.Thefirst14chapters
ofAtreatiseofalgebraareapreludetoWallis’slargerplan:inthemheexploredtheorigins
andearlydevelopmentofalgebraandtheevolutionofthemodernnumbersystemwithout
which,heargued,algebracouldneverhaveprogressed.Butaboveall,Wallisbegantoset
his story in an English context, by pointing to the existence of mathematical learning in
England since early postclassical times, and especially during the later medieval period
1100–1450. Commentators on A treatise of algebra have tended to pass over these early
73
0315-0860/01$35.00
Copyright(cid:176)C2001byAcademicPress
Allrightsofreproductioninanyformreserved.
74 JACQUELINEA.STEDALL HMAT28
chapters,1 yet they reveal Wallis at his finest as a historian, for here he displayed greater
objectivityandatruersenseofthecomplexitiesofhistoricaldevelopmentthaninalmost
anythingelsehewrote,andinhisinvestigationsintotheoriginsofthenumbersystem,he
wasthefirsttoapplymodernhistoriographicalstandardsandmethodstobothprimaryand
secondarymathematicalsources.
Auniquecombinationofcircumstancesin17th-centuryOxfordmadeWallis’sresearch
possible.Historiesofmathematicsareperhapswrittenonlywhenmathematiciansperceive
markedchangesinthenatureandscopeoftheirsubject,2andbythesecondhalfofthe17th
centuryitwasplainthatmathematicswassteadilyliberatingitselffromtheconstraintsofits
classicalpastandtakingonalifeandmomentumofitsown.Wallishadseenthisrevolution
atfirsthandduringhislongtenureoftheSavilianprofessorship,andindeedhaddonemuch
tobringitabout.Hewasalsowellplacedinasecondandmorematerialway,throughhis
accesstotheunprecedentedaccumulationofbooksandmanuscriptsinOxford’sBodleian
Library.Fromtheopeningofthelibraryin1602therehadbeenenergeticandwide-ranging
effortstocollectandpreservetextsfromEnglandandabroad[AppendixI;Philip1983]and
theconcentrationofthiswealthofmaterialinasingleplacebothreflectedandencouraged
newattitudestohistoricalstudy.TherangeofWallis’sreadingwillbecomeevidentinthe
courseofthispaper:hislongstandinginterestsingrammar,etymology,cryptanalysis,music,
astronomy,calendarreform,andgeneralhistoryallinformedhisaccountofthemedieval
period. He knew his classical sources thoroughly but also recognized, thanks to the new
proliferation of oriental studies in Oxford [Russell 1994; Toomer 1996; Feingold 1997],
the debt of European mathematics to Indian and Islamic sources, and the main theme of
Chaps. 2–4 of A treatise of algebra was the transmission of learning from Islamic Spain
tonorthernEurope.Hehadalreadypublishedsomeofthematerial30yearsbeforeinhis
Mathesis universalis [Wallis 1657, Chaps. 6–9] but there was also much in A treatise of
algebrathatwasnew.
ThispaperfollowsWallis’sChaps.2–4insomedetailandwithadoublepurpose:first,to
discoverwhatwasknownandunderstoodduringthe17thcenturyaboutmathematicsinthe
medievalperiod;second,tolookatWallis’smethodsofresearch,andhisestablishmentand
use of new historiographical techniques. Each of Wallis’s paragraphs, numbered for ease
ofreference,willbequotedinfullfollowedbyanaccompanyingcommentary.3 Thestory
istakenuppartwaythroughChapter2,atparagraph10,wherehavingbrieflyconsidered
what(slender)evidenceofalgebracouldbeprisedfromthewritingsofEuclid,Archimedes,
Pappus,andDiophantus,4WallisturnedtoArabicmathematics.
1Wallis’schiefbiographer,J.F.Scott,disposedoftheearlychaptersinonesentence:‘[Wallis’s]accountof
thehistoryofmathematicsinantiquityisverycomprehensiveandgivesevidenceofaclosestudyoftheclassical
literatureofthesciences.’ScottmadenomentionofWallis’sresearchesonthemedievalperiod[Scott1936,335;
Scott1938a,133].ForcommentaryonselectedparagraphsfromWallis’schapters1,2and6see[Molland1994,
215–218].
2ThefirsthistoryofmathematicswaswrittenbyEudemus(latefourthcenturyB.C.)who,likeWallisinthe
17thcentury,wasawareofthemanynewdiscoveriesmadebyhispredecessors[FauvelandGray1987,46–47].
3WallismadeanumberofhandwrittencorrectionsandannotationstohistextinhisowncopyofAtreatiseof
algebra(SavileA.3).Mostarecorrectionsoftypographicalerrorsandareincorporatedinthetranscriptsgiven
herewithoutcomment.Lengthierannotationsareshownincurlybracketsfg.
4TheexistenceofaGreek“geometricalalgebra”hasbeendiscussedatsomelengthduringthetwentiethcentury:
see,forinstance[Unguru1975,1979;VanderWaerden1976;Freudenthal1977;Weil1978;Mueller1981,43–44,
50–52;Berggren1984,394–410].
HMAT28 WALLISASAHISTORIANOFMATHEMATICS 75
x2.10AfterDiophantus(ifnotbefore,also)thislearningwaspursuedbyArabicauthors(butlittle
knowninEuropeforalongtime).FromthemithadthenameofAlgebra;not(assomewouldhave
it)fromGeber,whomtheyconjecture(withoutanygoodgroundthatIknowof)tohavebeenitsfirst
inventer;but(aswassaidbefore)fromitsArabicname,Al-gja¯brW’al-moka¯bala.
The term “Arabic authors” here and throughout should be taken, as Wallis intended, to
meanwritersfromanywhereintheIslamicworldwhousedArabic,thecommonlanguage
ofIslamicculture.
ImmediatelyapparentinthisparagraphisWallis’sinterestinwordderivation.Itwasa
longstandingpreoccupation:hisreputationasamathematicianhasovershadowedthefact
that in 1653 he published a highly regarded treatise on English grammar [Wallis 1653],
a substantial chapter of which was concerned with etymology. He was correct in tracing
thewordalgebratotheAl-jabrwa’lmuqa¯bala,theseminaltextofMuhammadibnMu¯sa¯
al-Khwa¯rizm¯ı(c.770–850)[Karpinski1915;Grant1974,106–111].Ontheopeningpage
ofAtreatiseofalgebraWallishadalreadytracedthemeaningsoftheArabwordsgja¯bara
(torestoreorsetabrokenbone)andka¯bala(tosetonethingagainstanother)[Wallis1685,
2].AliteraltranslationoftheArabictitlewouldbe“Restorationandbalancing,”perhaps
originallyreferringtothewellknownmethodofsolvingaquadraticequationbycompleting
(restoring)ageometricsquareandcomparing(balancing)theresultwithaknownquantity,
thoughlaterthetermswereusedofoperationsonequations[Saliba1973].Themistaken
associationofalgebrawithGeber(the12th-centuryastronomerJa¯biribnAflah)wasmade
byGirolamoCardano(1501–1576)who,inhislistof12greatscientists,notedMahometius
Mosisfilius(al-Khwa¯rizm¯ı)astheinventorofalgebrabutsupposedthatasaresultofhis
inventionhetookthenameGeber[Cardano1553,1011f].5 Wallishadearlierconsidered
thepossibleidentityofal-Khwa¯rizm¯ıandGeber,butseemstohaveconfusedJa¯biribnAflah
withtheearlyninth-centuryalchemistJa¯biribnHayya¯n,andhadbeenunabletodrawany
firmconclusion[Wallis1685,4];6inthisparagraphherejectedsuchahypothesis.
x 2.11 Divers writers (’tis said) there are of Algebra in that Language, and from them (I suppose)
theDenominationsofDiophantus(iffromhimtheylearnedit)cametobechanged;and(besidethe
DenominationsofRoot,square,andcube,)thatofSursolids(first,second,third,&c.)introduced.But
IratherthinktheArabs,eitherofthemselves,orfromsomeothers,haditlongbeforeDiophantus,and
thinkthisreckoningofPowers(bySursolids,&c.)differentfromDiophantusftobeagoodArgument
foritg.
Wallis had already described Diophantus’ method of naming powers as Movfi´& (unit),
A‰ı(cid:181)„o´& (number),1(cid:192)´”fi„ı& (power,orsquare), K(cid:192)´flo& (cube),denotedby„;&;–;•;
––;–•;•• andsoon[Wallis1685,4;Heath1931,476–478].Thiswasanadditivesystem
5ThewriterslistedbyCardanowereArchimedes,Aristotle,Euclid,IoannesScotus,IoannesSuisset,Apollonius
ofPerga,ArchytasTarantinus,MahometusMosisfilius,Alchindus,Heber[Geber]Hispanus,Galen,Vitruvius.
WallisreadandannotatedCardano’slistintheSavileLibrarycopy,nowSavileS.11.TheidentityofGeberisnot
obviousfromCardano’slistbutinhisEncomiumgeometriae[Cardano1663,IV,443],CardanoreferredtoGeber
asawriterontrianglesandcircles,aclearreferencetoJa¯biribnAflah.Seealso[Cifoletti1996,128;Høyrup1996,
113].ForotherascriptionsofalgebratoGebersee[Cifoletti1996,128–135].
6WallisknewJa¯biribnAflah’scommentaryonPtolemy’sAlmagest,publishedatNurembergin1534,inthe
copynowknownasSavileX.3,butdespitethisthoughtthatJa¯biribnAflahlivedintheninthcentury.The
alchemicalwritingsofJa¯biribnHayya¯n(lateeighthtoearlyninthcentury)werepublishedinLondonin1686.
Wallisdiscoveredadditionalinformationonal-Khwa¯rizmiin[Abu¯’l-Faraj1663,161;Eutychius1656,II,447],
bothtranslatedbytheOxfordArabistEdwardPococke.
76 JACQUELINEA.STEDALL HMAT28
inwhichhigherpowersareexpressedassumsofprecedingones.Wallissupposedthatthe
Arab writers introduced the alternative multiplicative system [Wallis 1657, Chap. 11] in
whichpowerswerecalled“root”(R),“zensus”(fromcensus,literallywealthoRrexcess,for
asquare,denotedZ),“cubus”(C),“zensRo-zensus”(ZZ),“firstsursolid”(e.g., Z),“censo-
cubus”(ZC),“secondsursolid”(e.g.,b Z ),andsoon,7 whereeveryprimepowerhasto
begivenanewnameandsymbol.Thiswasthesystemgenerallyusedbythe16th-century
cossistwriters,andonlywiththerediscoveryofDiophantusdidtheadditivesystemcome
back into use alongside it (leading to potential confusion as to whether A meant A5 or
qc
A6).WallislaterinAtreatiseofalgebraclaimedthatthemultiplicativesystemwasused
by “all our European Algebrists before Vieta, having learned it from the Moors” [Wallis
1685,91],butinthishewasmistaken:bothsystemswereusedin15th-centuryItalyinthe
earliestattemptstodealwithpowershigherthan3[Reich1994].
x2.12WiththeArabsallsortsofMathematicalLearningflourished,andwasimproved,foralong
timetogether,whileinEuropeitwasverymuchneglected.AmongstwhomwereMaimon,Almeon,
Alchindus,Albumasar,Alfraganus,Alfarabius,Geber,MahometesBagdadinus,MahometesbenMusa¯s,
Thebit,Haly,Alchabitius,Alhazen,anddiversothers.TowhomImayaddalsosome Persiansand
Tartars,asAl-suphi,Nasir-eddin,Shah-colgius,Uleg-beig,&c.whoseAstronomicalTablesareyetin
being.
AtseveralpointsinhisaccountWallispresented,ashere,alistofnameswithlittleadditional
information:inthiscasethereisnotevenadate.Ineverycasesuchlistsweredrawnfrom
Wallis’smainsourceforthisperiod,theDescientiismathematicisofJohnGerardVossius
(1577–1649),thethirdbookinhistrilogyoncontemporaryartsandsciences[Vossius1650].
VossiuswasarenownedDutchscholarwhohadspentsomeyearsinEnglandfrom1629to
1633(andhadbeenacanonofCanterbury)buthadreturnedtotakeupthechairofhistory
inthenewuniversityofAmsterdam.ThereheknewtheEnglishmathematicianJohnPell,
whotaughtinAmsterdamfrom1643to1646,anditmayhavebeenPellwhointroducedthe
workofVossiustoWallis.InDeScientiisdescriptionsandhistoriesofdifferentbranches
ofmathematics(arithmetic,geometry,logistics,music,optics,andsoon)arefollowedby
anextensivechronologicallistofmathematicians,foreachofwhomVossiusgave,asfar
aspossible,adateanddetailsofextantworks.Vossiuswasnot,however,amathematician
and did not discuss mathematical content. As he said himself [Vossius 1650, 37]: Neque
enimipsamtradimusscientiam;seddeeascribimus(“NordoIteachthescienceitself,but
only write about it”). His information was drawn from other authorities (whom he cited
frequently);hisowncontributionwastocollateandorderthefactsathisdisposal.
Wallis’scopyofDescientiisispreservedintheBodleianLibraryandhisfrequentand
detailedannotationsshowhowthoroughlyhereadit.TheArabicwriterslistedinx2.12are
alltobefoundinDescientiisthoughnotintheordergivenbyWallis.Thefirsttoappear
isMahometesbenMusas(al-Khwa¯rizm¯ı,c.770-850),whomVossiusmentionedbrieflyat
theendofhisaccountofGreekandLatinwritersonarithmetic[Vossius1650,41].Vossius
seemstohaveusedCardanoashissourcehere:bothreferredtoal-Khwa¯rizm¯ıasMahomet
sonofMoses(MahometMosisfilius),andVossiusstatedthatCardanolistedMahometninth
(actuallyeighth)inhislistof12greatscientists.8 AfewpageslaterMahometBagdadinus
7Theexactsymbolismvariedfromwritertowriterbutthisscheme,fromRecorde1557,istypical.
8Seenote5.TogetherwithMahometesbenMusas,VossiusalsomentionedoneAbrahamCai,aJew.Inthe
indextoDeScientiisAbrahaminsteadofMahometiswronglydescribedastheinventorofalgebraandninthin
Cardano’slist.Wallismadeahandwrittencorrectioninhiscopy.
HMAT28 WALLISASAHISTORIANOFMATHEMATICS 77
(al-Baghda¯d¯ı,fl.c.1230)andAlchindus(al-Kind¯ı,c.801–c.866)weredescribedbyVossius
aswritersongeometry[Vossius1650,61],9andthenAlchabitius1410(al-Quab¯ıs¯ı,fl.c.950)
andAlhazen1100(al-Hasanoribnal–Haytham,965–c.1040)aswritersonoptics[Vossius
1650,109].
The remaining writers, Maimon 827 and Almeon 838 (both references to Caliph al-
Ma’mu¯n,809–883,founderoftheHouseofWisdominBaghdad),10Albumasar884(Abu¯-
Ma’shar, c. 810–886), Alfraganus 879 (al-Fargha¯n¯ı, d. 861), Alfarabius 940 (al-Fa¯ra¯b¯ı,
c.870–950),Geber(Ja¯biribnAflah,fl.1145),Thebit1300(Tha¯bitibnQurra,836–901),
andHaly1202(Abu¯-’l-Hasan,fl.1020–1040)arealllistedasastronomers[Vossius1650,
173–181]. Also in Vossius but curiously missing from Wallis’s list are Albategnius 888
(al-Batta¯n¯ı, 850–929), Arzachel 1080 (al-Zarqua¯l¯ı or Azarquiel, d. 1100), and Abenezra
1145(RabbiAbrahambenMeiribnEzra,1092–1167).AsWallismentionedallthreelater
inconnectionwiththeirastronomicaltables,theirexclusionfromthispreliminarylistwas
perhaps deliberate. There are no other omissions: every Arab writer recorded by Vossius
was also noted by Wallis. In Wallis’s copy of De scientiis the pages on Arab astronomy
are particularly heavily annotated11 and it is clear that he read them carefully. But he
also extracted Arab writers from other sections and ordered the entire list more or less
chronologicallyforhisowntext.
Wallis’sbrieflistof“PersiansandTartars”camefromadifferentsource.In1648John
Greaves,ascholarofbothPersianandArabic,andSavilianProfessorofAstronomy(1643–
1649), had published the geographical tables of the Persian Nas¯ır al-D¯ın al-Tu¯s¯ı (1201–
1274)andofthe“Tartar”UlughBeg(1394–1449),kingofSamarkandandfounderofits
observatory[Greaves1648].12TwoyearslaterGreavestranslatedandpublishedastronom-
ical and chronological tables of Ulugh Beg, and the astronomy of the Persian al-Ka¯sh¯ı
(Shah-colgius)(d.1429)whoassistedUlughBeginSamarkandandmadeimprovements
totheastronomicaltablesofNa¯siral-D¯ınal-Tu¯s¯ı[Greaves1650a;1650b].Thecatalogue
of fixed stars compiled by Ulugh Beg also drew on the earlier observations of Al-Su¯f¯ı
(903–986).
TothemodernreaderWallis’slistofnames,devoidofhistoricalcontext,raisesfarmore
questionsthanitanswers,butWalliswasfollowingalongestablishedparadigmofhistorical
writing,whichconcentratedonauthorsratherthanideas,andonstabilityratherthanchange.
Theunderlyingassumption,thoughonethatWalliscouldnolongercompletelyshare,was
thatmathematicalknowledgederivedfromdivinerevelationorancientauthority,sothatthe
historyofmathematicswasessentiallythehandingon(traditio)ofsuchknowledgefrom
onegenerationorculturetothenext.
ThereareseveralhistoriesofthiskindbymedievalandRenaissancewriters.Assertions
thatmathematicswashandedfromtheBabyloniansortheHebrewstotheEgyptiansand
thencetotheGreekscanbeseenintheearliestpost-Classicalhistories,thoseofIsidorof
9TheDesuperficierumdivisionibusofal-Baghda¯d¯ıwaspublishedbyCommandinoin1570fromamanuscript
suppliedbyJohnDee,whoconjecturedthatitwasalostbookofEuclid,thoughthereisinfactonlyanindirect
connectionwithEuclid’swork.
10Vossius mentioned a second, later, Maimon who can be identified as the philosopher Rabbi Moshe ben
Maimonides(1135–1204).Wallisomittedhim,perhapsincorrectlyassumingduplication.
11Page177,inparticular,fromal-Fargha¯n¯ıtoibnEzra,isheavilyannotatedateveryparagraph.
12TheTartarscameoriginallyfromtheeastAsiansteppes;thedescriptionisprobablyusedindiscriminately
hereforthevariousMongoltribesthatoverrancentralAsiaintheearlythirteenthcentury.
78 JACQUELINEA.STEDALL HMAT28
Seville(570–636)[Isidor,III;MigneLXXXII,cols.153–184(155–169)]andBede(672–
735)[MigneXC,cols.647–653(650)].Latermedievalaccountsbecamemoresophisticated
butpresentedmuchthesamestory.Bythe13thcenturyBaconsawthehistoryofscienceas
aprocessofdeclineinwhichancientknowledgewasoccasionallyrecoveredonlytobelost
again[Molland1995,214,221–223].Twocenturieslater,in1464,Regiomontanuswrote
ahistoryofmathematicsinwhichthemainthemewasnotchange,butthecontinuityand
stability of mathematics as handed from one mathematician to another [Regiomontanus
1464; Rose 1975, 95–98]. Cardano’s 1553 list of 12 great writers has already been men-
tioned:theresixGreeks,twoBritons,aRoman,andthreeArabsareorderedbyeminence
ratherthanchronology,andwithoutanysuggestionofhistoricaldevelopmentorcontext.
BernardinoBaldi(1553–1617),inhisVitedematematiciwrittenalmostattheendofthe
16thcentury[Baldi1998;Rose1975,253–269;Moyer1999],stillconveyedmathematics
asacontinuoustraditionrunningfromtheBabylonians(Chaldeans)andEgyptiansthrough
the Greeks, Romans, and Arabs to his own time, in which the greatest achievement was
the restoration of Archimedes, and his Cronica de matematici listed an unbroken line of
mathematiciansfrom600B.C.to1596[Baldi1707].Asimilarlist,whichwasavailableto
Wallisbutwhichheappearsnottohaveused,wascompiledbyHenrySavilein1570and
ispreservedintheSavileLibrary.Savile’slistbeganwiththesonsofSethandcontinued
through the Druids and Zoroastrians to Abraham, Joseph, Homer, and Pythagoras before
reachingthefirmerhistoricalgroundofClassicalGreece[MSSavile29;Goulding1999,
123–125].13 Savile, like his predecessors, was chiefly concerned with demonstrating the
deeprootsofmathematicsinitsClassicalpast.
Onlyverygraduallydidtherebegintoemergeideasofmathematicalprogress,asensethat
modernmathematicianscouldaddtoorevenimproveupontheexistingbodyofknowledge
[Zilsel1945;Lilley1958,3–37;Crombie1975;Molland1978;Molland1983,141–148].
Writinginthe1640s,Vossiusstillpresentedthevariousbranchesofmathematicsasexisting
largelyindependentofageorculture,sothatdetailedtracingofideaswaslessimportant
thanidentifyingthecarriersofthetradition,whodidnotthemselvesneedtobeinnovators.
Wallis’s lists of names, here and elsewhere, were quite compatible with this established
styleofhistoricalwriting;itmakeshisnewmethodologylaterallthemoreremarkable.
x2.13 From those Arabians we have the names of Almagest, Azimuth, Almicanter, Zenith, Nadir,
Almanack,Algorism,Algebra,&c.anddiversotherArabicwords(nowdisused)wefindretainedin
Regiomontanus,Purbachiusandothersbeforethem,whoeithertranslatedArabicAuthors,oratleast
derivedtheirLearningfromthem.AsIfindindiversofthoseManuscriptAuthors,whichIhaveseen,
concerningtheAstrolabe(whosePartstheydescribebyArabicnames),andotherMathematicalLearning.
HereagainWallis’sinterestinetymologyisevidentandhewascorrectintracingallthese
wordstoIslamicscientific,andespeciallyastronomical,writing.GeorgPeuerbach(1423–
1461),humanistandastronomeroftheUniversityofVienna,wastheteacherofJohannes
Muller,Regiomontanus(1436–1476).Theywereregardedbytheircontemporariesasbeing
responsiblefortherenaissanceofastronomyinEurope,andRegiomontanuscompleteda
translationandcritiqueofPtolemy’sAlmagestbegunbyPeuerbach[Regiomontanus1550;
Rose1975,90–94].TheSavileLibraryheldacopyofthe1550editionwhichwouldhave
beenknowntoWallis.
13SaviledidpresentsoundhistoricalargumentsabouttheidentityofEuclid;see[Goulding1999,96–103].
HMAT28 WALLISASAHISTORIANOFMATHEMATICS 79
x2.14TheytranslatedEuclid,Ptolemy,Aristotle,anddiversothersoftheGreekAuthorsintoArabic;and
outoftheArabicwehadourfirstTranslationsofEuclid,Ptolemy,andotherGreekAuthors,intoLatin,
beforethoseoutoftheGreek.Athingofitselfnotorious,andsoalsoattestedbyVossius,(afterSirHenry
Savil:)EuclidemLatiniTranslatumhabueruntpriusexArabicoquamexGraecofonte.Quemadmodum
&anteCC,&infra,annos,nonaliaAristotelis,Galeni,Ptolemaei,aliorumquemultorum,interpretatio
inmanibuserat,quamexArabicaversioneLatine,velSemibarbaro,potius,expressa.AndbySirHenry
Savil,inhissecondLectureonEuclid,almostinthesamewords.Andfromthemwereceivednotonly
ourAlgebra,butotherpartsofMathematicalLearning;broughtbytheMoorsintoSpain,andfrom
thencepropagatedtootherpartsofEurope;abouttheyearofourLord1100,orsomewhatsooner.
The passage from Vossius translates as: “They had a Latin translation of Euclid from
Arabic before any from a Greek source. Just as, for up to two hundred years before that,
they had no translation of Aristotle, Galen, Ptolemy or many others, other than Latin,
or rather semibarbarous, versions from Arabic.” Vossius did not acknowledge Savile as
the source of his information but Wallis clearly knew Savile’s 1619 lectures on Euclid
well,andrecognisedtherelevantpassage[Savile1621,35]:14 Etquidemnosoccidentales
EuropaeiArabibusprimusomniumdebemusAristotelem,Euclidem,Galenum,Ptolemaeum,
caeterosGraecorumPrincipes,cumantecentumannosaliaeversionesnullae,praeterquam
exArabico,fuerintinmanibusnostrorumhominum,quodGraecaelinguaecognitonondum
in Italiam et Occidentem immigrasset. Wallis assumed that this information was already
wellknown(notorious).
Walliswaslessthancarefulhereandelsewhereindistinguishingbetweendifferentperi-
odsandgeographicallocationsofIslamicculture,andprobablyused“Moors”inthegeneral
sense of “Muslims.” Mathematical learning was brought not only by the true Moors, the
northAfricaninvadersandsettlersofSpain,butalsobylatertravellersandscholarsfrom
elsewhereintheIslamicworld,fromtheoldHellenisticregionsoftheeasternMediterranean
andthenewcentreoflearningatBaghdad[Fletcher1992].
ThefirstLatintranslationsofEuclidfromArabicweremadebyAdelardofBath(c.1130),
HermannofCarinthia(c.1143),andGerardofCremona,whoalsodidthefirsttranslation
fromArabicofPtolemy’sAlmagest(c.1175).InfactatranslationoftheAlmagestdirectly
fromGreekwasdoneinSicilyin1165byananonymousstudentfromSalernowhohasalso
been credited with a translation of Euclid’s Elements from Greek, but neither translation
was well known until the 20th century and they would not have been known to Savile,
Vossius,orWallis[Rose1975,76–79].
x2.15Uponthisaccount,IfindthatdiversofourownNation,aboutthetwelfthandthirteenthCentury,
(notsatisfiedwiththePhilosophyoftheSchoolmen,)wereinquisitiveintotheArabicLanguage,and
theMathematicalLearningthereincontained.
Walliswashere,perhapsdeliberately,pointingtothemedievalforebearsofthe17th-century
OxfordinterestinArabiclanguageandscience.Hewasalsointroducingtwonewthemes
hewasabouttoexploreindetail.Amoreprecisedeterminationofthedateoftransmission
of Arabic learning through Spain to the rest of Europe was to be the subject of his next
twochapters.TheroleofEnglishmeninacquiringandpropagatingthenewideas,andthe
subsequentrevitalisationofEnglishmathematicsfillstheremainderofWallis’sChapter2.
x2.16AsAdelardus,(amonkofBath)whomVossiusplacethabouttheyear1130whoforthatpurpose
travelledintoSpain,Egypt,andArabia;and(asVossiustellsus)translatedEuclid(andsomeotherArabic
14SavilepresentedtheBodleianLibrarywithaninscribedcopyofhisPraelectiones,now40S.39Art;Savile’s
originalmanuscriptisalsopreserved,asMSSavile37.
80 JACQUELINEA.STEDALL HMAT28
authors)outofArabicintoLatin,AnnohocMCXXX.AthelardussiveAdelardus,Anglus,Monachus
Bathoniensis,EuclidisGeometriamexArabicovertitLatine.Nec,Arabicescivisse,mirandum:Quando
nonmodoGalliam,Germaniam,Italiam,adiit;sedetiamHispaniam,Ægyptum,Arabiamipsam.
The quotation from Vossius translates as: “The year 1130 Athelard, or Adelard, an En-
glishman,amonkofBath,translatedthegeometryofEuclidfromArabictoLatin.Noris
itanywonderheknewArabicwhenhehadbeennotonlytoFrance,Germany,andItaly
butalsotoSpain,Egypt,andArabiaitself.”Modernscholarshiphasmodifiedthisaccount:
Adelard(c.1080–1150)travelledwidelyinFrance,Sicily(before1116),Cilicia(inwhatis
noweasternTurkey),Syria,andPalestinebutthereisnofirmevidencethatheevervisited
Spain [Burnett 1987]. He is best known for the first translations of Euclid from Arabic
to Latin, and three versions are ascribed to him.15 Wallis knew one version in the Savile
Library and another in Trinity College.16 Adelard also translated the astronomical tables
of al-Khwa¯rizm¯ı, Ezich elkaurizmi; the Bodleian Library owns a copy that is richly and
beautifullyillustratedinred,green,andgold[MSAuctF.19].
x2.17AndRobertusRetinensis(RobertofReading)whotravellingintoSpainontheaccountofthe
Mathematics,didtheretranslatetheAlcoranoutofArabicintoLatin,intheyear1143.(Asappearsby
hisEpiloguetothatTranslation,andthePrefaceofPetrusCluniacensisthereunto.)
ThereisnomentionofRobertinthepagesofDescientiisandhisinclusionhereisaresult
of Wallis’s own researches among Bodleian manuscripts: a copy of Robert’s translation
of the Koran, made for Peter, Abbot of Cluny, was acquired by the library as part of the
Seldencollection[MSSeldenSupra31,ff.32–204].17Inthepreface(whichappearsinthe
Bodleianmanuscriptasacolophonorendpiece)Robertwrotethathenowintendedtoreturn
tohischiefinterest,mathematics[MSSeldenSupra31,ff.32–33;MigneCLXXXIX,col.
657f],andforWallisthiswasreasonenoughtocounthimamongtheEnglishtranslatorsof
Arabmathematicallearning.
Robert’snameactuallyappearsattheendofthetranslationasKetenensisbutthelooped
‘K’ was read by Wallis as ‘R.’ Robert was in fact Robert of Chester whose name has
mutated through the forms Cestrensis, Kestrensis, Ketenensis, and Retinensis leading to
confusionwhichpersiststothepresentday:theBodleianLibrarycatalogueentryforthe
manuscriptdescribestheauthoras“probablyofKettoninRutland”(whereasWallistrans-
lated“Retinensis”as“ofReading”).TheDictionaryofnationalbiographystillcarriestwo
articles on Robert, headed “Chester, Robert (fl. 1182)” and “Robert the Englishman, (de
Ketenes,deRetines)(fl.1143).”Thedate1182intheformerarisesfromthedatingsystem
theninuseinSpain;itwasinfacttheyearwewouldnowdenoteas1144,whichatleast
bringsthetwoRobertsintothesametimeframe.
LittleisknownofRobert’slife.HewasinSpainfromabout1140andlivedneartheriver
Ebrointhenortheast.Heworkedcloselywithanothertranslator,HermannofCarinthia,
whoappearstohavecomefromtheregionthatbordersmodernAustriaandSlovenia(heis
alsosometimesknownasHermanofDalmatia),whichsuggestsatrans-Europeandimension
15AdelardIisaclosetranslationoftheentirework;AdelardIIwasthemostpopularversionbutomitsmany
oftheproofs;AdelardIIIisacommentaryratherthanatranslation.Recentscholarshiphasquestionedthetrue
authorshipofAdelardIIandhassuggestedthatitshouldbeascribedtoRobertofChester;see[BusardandFolkerts
1992].
16MSSavile19(AdelardII);MSTrinityCollege47(AdelardI).
17PetercommissionedthetranslationsothathecouldrefuteIslam;see[MigneCLXXXIX,col.649f;col.659f].
HMAT28 WALLISASAHISTORIANOFMATHEMATICS 81
tothetranslationprogrammethatWalliseitherfailedtoseeorchosetoignore[Thorndike
1923,II,14–98,155–187;Haskins1924,3–66,113,129;Rose1975,76–89;Lindberg1978,
52–90]. As a translator Robert was far more important than Wallis knew. By misreading
hissurnameWallisfailedtorecognisehimasthetranslatoroftheCanonsofArzachel(to
bediscussedinx4.6).Robertisnowalsothoughttobethepossiblewriterof“AdelardII”
[BusardandFolkerts1992].Heisbestknown,however,forthefirstLatintranslationofal-
Khwa¯rizm¯ı’sAl-jabrwa’lmuqa¯bala,thekeytextintheevolutionofArabicandEuropean
algebra.NocopyofRobert’stranslationreachedEngland[Karpinski1915,49–63;Hughes
1982], and unfortunately Wallis never knew of this important English contribution to the
earlydevelopmentofalgebra.
x2.18Aboutthesametime(orsomewhatsooner)GuilielmusdeConchis(WilliamShelley)issaidto
havetravelledintoSpaintofurnishhimselfwithArabicandMathematicalLearning;andbroughtfrom
thencediversArabicBooks.
Guillaume de Conches (d. ?1154) was a natural philosopher, born in Normandy.18 He
studied at Chartres, taught at Chartres or Paris, and retired to Anjou where he wrote the
philosophicalworkforwhichheisbestknown,hisDragmaticon.Thereisnoevidencethat
heeverwenttoSpain,northathewasfamiliarwithArabiclanguageorphilosophy,orwith
astronomicaltablesofanykind.NorwashementionedbyVossius.HisinclusionbyWallis
isthereforepuzzlinguntilwelookatthenextname,DanielMorley.
x2.19And,soonafter,DanielMerlacus(Morley),abouttheyear1180madeseveralJourneysintoSpain
onthelikeaccount,where(atToledo)ArabicandMathematicalLearningwereingreatrequest(brought
thitherbytheMoors)whichinotherpartsofEuropewerescarceknown.Andthesebroughtwiththem
thatkindofLearningintoEnglandveryearly,withstoreofArabicBooks.
ThisinformationaboutDanielMorley(fl.1170–1190)isnottobefoundinDescientiis,but
inthepreface[Halliwell1839,84–85]toMorley’sonlywork,hisLiberdenaturisinferiorum
et superiorum. There Morley helpfully gave a brief account of his life and travels which
tallieswiththesummarygivenbyWallis.First,hesaid,hewenttoParisbutfoundthatthe
teachers there carried only “leaden pens” with which they marked asterisks and obelisks
reverently in their texts, so he went on south to Toledo in search of something better, in
particularthecontentsofthequadrivium,thefourclassicalbranchesofmathematics.On
returningtoEnglandwithagoodcollectionofbooks,hewasdepressedbytheneglectof
PlatoandAristotlethere,anddecidedtoreturntoSpain,butwaswaylaidbyJohn,Bishop
ofNorwich(1175–1200),forwhomhewrotehistreatise.Theonlymanuscriptcopyofthe
prefacenowsurvivingisintheBritishLibrary[Sudhoff1918;Birkenmajer1970,45–51],
butWallisknewofanothercopythathadbeeninOxfordafewyearsearlier:
x2.20AparticularaccountoftheseTravelsofShelleyandMorleywasawhilesincetobeseenintwo
Prefaces,totwoManuscriptBooksoftheirsintheLibraryofCorpus-ChristiCollegeinOxford,but
hathlately(bysomeunknownhand)beencutout,andcarriedaway;whichPrefaces(oneorbothof
them)didalsomakementionoftheTravelsofAthelardusBathoniensis,andare,tothatpurpose,cited
byVossiusoutoftheManuscriptCopy.Whoeverhaththem,woulddoakindness(bysomewayor
other)torestorethem,oratleastaCopyofthem.
18ItwastheEnglishhistorianJohnBale(1506–1552)whoclaimedthatConcheswasborninCornwalland
whointroducedtheanglicizedformofhisname,Shelley.
82 JACQUELINEA.STEDALL HMAT28
TheCorpusChristimanuscripttowhichWallisreferredisthatnowknownasMSCCC
95. It includes a copy of the Liber de naturis from which the preface has, as Wallis de-
scribed, been neatly cut out, but the contents page lists the opening work as Philosophia
magistriDanieldeMerlac.Morley’sbookisfollowedwithoutabreakbytheDragmaticon
ofdeConches,whichends“ExplicitWilldeConchys,”anattributionwhichledtoamis-
takenidentificationinHenryCoxe’s1852catalog[Coxe1852].19 Wallis(orhisunknown
informant)musthavebeensimilarlymisled,andWallisassumedthatthemissingpreface
describedajourneytakenbybothmen,eventhoughhedatedthem40yearsapart.
TheprefacerelatingtoAdelardcomesfromadifferentCorpusChristimanuscript,MS
CCC86,containingAdelard’sDecausis[MSCCC86,f.163].ItwasusedbyVossiusto
dateAdelard’sactivitiesto1130,andWalliswouldhaveknownofitfromthereferencein
Descientiis[Vossius1650,176].
x2.21AboutthesametimewereJohannesSarisburiensis,RogerusInfans,anddiversothersofthe
English.
JohnofSalisbury(d.1180)wasoneofthebestknownscholarsoftheday.Hetravelledas
farassouthernItaly,butknewlittleGreekandnoArabic,andemployedanItalianGreekto
maketranslationsofAristotle.Hewasprimarilyatheologianandnoloverofmathematics,
whichtohimmeantastrology:inhisPolycraticushedefinedmathematiciansasthose“who
fromthepositionofthestarsandthemotionoftheplanetsforetellthefuture,”andclassed
mathematicswithchiromancy,sortilege,andauguryasoneofthemagicarts,andasource
ofevil[MigneCIC,cols.407–409].
Roger Infans was the scholar Roger of Hereford (fl. 1178), but Wallis never used the
second, more usual, form of his name. Roger was a natural philosopher, computist, and
astrologer, with special knowledge of mines and minerals, and was familiar with some
Arabic texts, but it is not known whether he made his own translations [Russell 1932;
French1996].ThereisnomentionofhiminDescientiisandWallismusthavecomeacross
the unique occurrence of “Infans” in MS Digby 40, one of the few instances where we
can be sure that Wallis consulted the Digby collection. Roger’s Tractatus de computo in
MSDigby40isheaded“TractatusRogeriInfantis,”apparentlybecauseRogersaidthathe
wroteitwhilestillayoungman.Asaresult,he,likeRobertofChester,hasacquiredtwo
entries in the Dictionary of national biography: “Roger Infans (fl. 1124)” and “Roger of
Hereford(fl.1178).”“Themismatchingdatesstemfromthefigure1124whichappearsin
themarginoftheTractatusdecomputo,butwhichwasmeantaspartofthecalculation,not
asadateofwriting.Roger’snamewasanglicizedbythehistorianJohnLeland(1506–1522)
to“Yonge;”Wallisinthe1693LatintranslationofAtreatiseofalgebrawentfurtherand
gavehisnameas“RogerChild”[Wallis1693,6].
19ThemanuscriptisdescribedbyCoxe1852as“threebooksoftheNormanphilosopherWilliamdeConches,
aliasShelley.”TheerrorwaspointedoutbyH.NashinalettertoCorpusChristilibrarian,CharlesPlummer,
written25March1889.Theletter(preservedwithMSCCC95)begins:“IhavebeentoseetheBMMS(Arundel
377)ofDanieldeMorley.Itisthesamebookastheoneinyourlibraryanditisthenalsofollowedbyadialogue
betweentheDukeofNormandy(D)andthePhilosopher(P)ofGul.DeConchis.Coxeconfoundedthetwo.A
passagewhichIcopiedfromyourMSfol15boccursonthelastfoliooftheArundel377,wherethedivision
betweenthetwoisquitedistinct.Youwillseethe‘incipit’ofGuldeConchisintheArundelcatalogue.Imention
thisasyoumayliketomakeanoteofthefactinyourcopyofCoxe’scatalogue.Thebeginning(missinginthe
CCCMS)containsadelightfullittlepieceofautobiography.”