Sources and Studies in the History of Mathematics and
Physical Sciences
Editorial Board 1.Z. Buchwald 1. Llitzen G.J. Toomer
Advisory Board P.J. Davis T. Hawkins
A.E. Shapiro D. Whiteside
Springer Science+Business Media, LLC
SourcesandStudiesintheHistoryofMathematicsandPhysicalSciences
K.AndersenBrookTaylor'sWorkonLinearPerspective
H,j.M.BosRedefiningGeometricalExactness:Descartes'TransformationoftheEarlyModernConceptofConstruction
J.CannOlvS.DostrovskyTheEvolutionofDynamics:VibrationTheoryfrom1687to1742
B.ChandlerlW.MagnusTheHistoryofCombinatorialGroupTheory
A.I.DaleAHistoryofInverseProbability:FromThomasBayestoKarlPearson,SecondEdition
A.I.DaleMostHonourableRemembrance:TheLifeandWorkofThomasBayes
A.I.DalePierre-SimonLaplace,PhilosophicalEssayonProbabilities,TranslatedfromthefifthFrencheditionof1825,withNotesbytheTranslator
PJ. FedericoDescartesonPolyhedra:AStudyoftheDe Solidorum Elementis
B.R. GoldsteinTheAstronomyofLevibenGerson(1288-1344)
H.H.GoldstineAHistoryofNumericalAnalysisfromthe16thThroughthe19thCentury
H.H.GoldstineAHistoryoftheCalculusofVariationsfromthe17thThroughthe19thCentury
G.GraBhoffTheHistoryofPtolemy'sStarCatalogue
A.W.GrootendorstJandeWitt'sElementa Cu",arum Linearum, Liber Primus
T.HawkinsEmergenceoftheTheoryofLieGroups:AnEssayintheHistoryofMathematics1869-1926
A.HermanniK.vonMeyennIV.F.Weisskopf(Eds.)WolfgangPauli:ScientificCorrespondenceI:1919-1929
Continued after Index
Fibonacci'sLiber Abaci
ATranslationintoModernEnglishofLeonardoPisano'sBookofCalculation
Springer
Laurence Sigler (deceased) Mathematics
Bucknell University Lewisburg, PA 17837
Sources and Studies Editor: Gerald J. Toomer
2800 South Ocean Boulevard, 21 F Boca Raton, FL 33432
USA
Library of Congress Cataloging-in-Publication Data Fibonacci, Leonardo, ca. 1170-ca. 1240
[Liber abaci. English] Fibonacci's Liber abaci : a translation into modem English of Leonardo Pisano's Book
of calculation / [translated by] Laurence Sigler p. cm.-(Source and studies in the history ofmathematics and physical sciences)
Includes bibliographical references. 1. Mathematics, Medieval. 2. Mathematics-Early Works to 1800. I. Sigler, L.E.
II. Title III. Sources and studies in the history of mathematics and physical sciences. QA32 .F4713 2002 51O-de21
ISBN 978-0-387-40737-1 ISBN 978-1-4613-0079-3 (eBook) DOI 1O.1007/978-1-4613-0079-3 2001057722
Printed on aeid-free paper.
First softeover printing, 2003 2002 Springer Science+Busiess Media New York
Originally published by Springer-Verlag New York, Ine in 2002 AII rights reserved. This work may not be translated or copied in whole or in part without the written permis-sion of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. lJse in connection with any form ofinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
9 8 7 6 5 4 3 2 I SPIN 10953310
Typesetting: Pages created by the author using a Springer TEX macro paekage.
www.springer-ny.com
To Ettore, with admiration.
-L.E. Sigler
My heartfelt thanks to our friend Dr. Alex Khoury and to theDepartment ofMathematics at Bucknell University for their supportand encouragement in preparing this book for publication. I amespecially thankful to Professors Gregory Adams, George Exner,Paul McGuire, Howard Smith, Karl Voss and Ms. Abbe Sattesonfortheir contributions.
-lM. Sigler
PagefromoriginalmanuscriptLiber Abaci(courtesyofBibliotecaAmbrosiana,Milan).
Contents
IIntroduction 1
IILiberAbaci 13
1HereBeginstheFirstChapter 17
2OntheMultiplicationofWholeNumbers 23
3OntheAdditionofWholeNumbers 39
4OntheSubtractionofLesserNumbersfromGreaterNumbers45
5OntheDivisionsofIntegralNumbers 49
6OntheMultiplicationofIntegralNumberswithFractions77
7OntheAdditionandSubtractionandDivisionOfNumberswithFractionsandtheReductionofSeveralPartstoaSinglePart 99
8OnFindingTheValueofMerchandisebythePrincipalMethod127
9OntheBarterofMerchandiseandSimilarThings
10OnCompaniesandTheirMembers
11OntheAlloyingofMonies
179
213
227
viiiContents
12HereBeginsChapterTwelve 259
13OntheMethod ElchataymandHowwithIt NearlyAllProblemsofMathematicsAreSolved 447
14OnFindingSquareandCubicRoots,andontheMultiplication,Division,andSubtractionofThem,andOntheTreatmentofBinomialsandApotomesandtheirRoots 489
15OnPertinentGeometricRulesAndonProblemsofAlgebraandAlmuchabala 531
16NotesforLiberabaci 617
17Bibliography 635
PartI
Introduction
Liber abaci isoneofthemostimportantbooksonmathematicsoftheMiddleAges.ItseffectwasenormousindisseminatingtheHindunumbersys-temandthemethodsofalgebrathroughoutEurope.ThisisthefirsttranslationoftheLatinmanuscriptofLiber abaci intoamodernlanguage.Itishopedthatitsavailabilitytohistorians,mathematicians,andthepublicingeneralwillmakeacontributiontotheirknowledgeofthispartofourculturalheritage.Mathematicsandscienceare,afterall,asmuchapartofourcultureaslitera-ture,art,andmusic.Itisasimportantforapersontoknowabouttheclassicsofmathematicsandscienceasitistoknowabouttheclassicsofliteratureandart.
LeonardoPisano,knowntodaytomathematiciansandscientistsovertheworldbythenameFibonacci,wasacitizenofthemaritimecity-stateofPisafrom1170untilafter1240.ThiswasthetimeoftheCrusades,ofstrongpoliticalconflictsbetweentheEmperorFrederickIIoftheHolyRomanEmpireandthePapacy;itwasalsothetimeofthereligiousfervorofSt.FrancisofAssisi.TheItalianmaritimestatesofPisa,Genoa,VeniceandAmalfiwerelockedinintensetraderivalrythroughouttheMediterraneanworld,includingByzantiumandtheMuslimcountries.LeonardowasinstructedinmathematicsasayouthinBugia,atrading
enclaveestablishedbythecityofPisaandlocatedontheBarbaryCoastofAfricaintheWesternMuslimEmpire.Hecontinuedtodevelopasamathe-maticianbytravelingonbusinessandstudyinginsuchplacesasEgypt,Syria,Provence,andByzantium.HedevelopedcontactswithscientiststhroughouttheMediterraneanworld.HebecameproficientinEuclid'sElements, andtheGreekmathematicalmethodofdefinition,theorem,andproof.HelearnedfromtheArabicscientiststheHindunumbersandtheirplacesystem,andtheal-gorithmsforthearithmeticoperations.Healsolearnedthemethodofalgebraprincipallyfoundintheworkofal-KhwarizmI[K]. Throughhisstudyandtravelandlearneddisputationswithworldscientists,hebecameaverysuperiorcre-ativemathematician.HeparticipatedintheacademiccourtofFrederickIIwhosoughtoutandrecognizedgreatscholarsofthethirteenthcentury.Leonardowithhisscientificknowledgesawclearlytheadvantagesoftheusefulmathe-maticsknowntotheMuslimscientists,principallytheirHindunumeralsanddecimalplacesystem,theircalculatingalgorithms,andtheiralgebra.Knowl-edgeoftheHindunumeralsbegantoreachEuropeinthesecondhalfofthetenthcenturythroughtheArabsbywayofSpain,howevertheirusagewasstillnotageneralpracticeatLeonardo'stime.Leonardoresolvedtowritehisen-cyclopedicwork,Liber abaci, tobringtotheItalianpeopletheworld'sbestmathematicsinausableform.Calculationhasbeenanactivityofmankindsinceancienttimes.Itwas
facilitatedbyvariousmechanicaldevicesthatbyGreekandRomantimeshaddevelopedintotheabacus.Thebestknownformconsistsofawoodenframestrungwithwiresonwhicharemountedbeadsforcounters.Theefficiencyofthisabacusisattestedtobyitssurvivalanduseinsomepartsoftheworldeventoday.Therewerealsoearlyformsoftheabacusconsistingoftablesofwood
L. Sigler, Fibonacci's Liber Abaci Springer-Verlag New York, Inc. 2002
4 I. Introduction
ormarbleonwhichwereengravedlines.Onthelinesweremanipulatedsmallcountersofstone.Anotherformuseddustorpowderonthetableonwhichmarksweremadewiththefinger.DuringtheseventeenthcenturybothBlaisePascalandGottfriedLeibnizdesignedmechanicalcalculatingmachines.Todaywehaveelectroniccalculatorsandelaboratecomputerstoassistuswithourcalculations.Theinexpensiveelectronichandcalculatoristheabacusoftoday.TheHindusandArabsutilizedwrittennumberswithaplacesystemand
methodsforthebasicoperationsthatdidnotrequiretheabacus.Romannu-meralsandothersimilarsystemsofwritingnumbersdidnotfacilitatecalcula-tion.ThecalculationsweredonewiththeabacusandtheanswerswerewrittendowninRomannumerals.TheHindunumeralswiththeplacesystemareac-tuallyusedbothtomakethecalculationandtowritedowntheresult.Thesearetheproceduresthatchildren aretaughtinschoolwhentheylearntodoaddition,multiplication,subtraction,anddivisionwithpencilandpaper.IntheMiddleAgesinEuropethesenewwrittenprocedureswerecalledalgorithmstodifferentiatethemfromcalculatingwiththeabacus.LeonardoteachestheseproceduresinthisbookLiber abaci. Thesewrittenproceduresofcalculation,algebra,andpracticalmathematicsingeneralwereknowninItalyintheMiddleAgesasabaca.
Liber abaci, ortheBookofCalculation,appearedfirstin1202,andthenagaininasecondversionin1228.Leonardo'sstatedintentionwastointroducetheHindunumbersystemanditsoperationstotheItalianpeople.However,Liber abaci ismuchmorethanmerelyanintroductiontotheHindunumbersystemandthealgorithmsforworkingwithit.Liber abaci isanencyclope-dicworktreatingmuchoftheknownmathematicsofthethirteenthcenturyonarithmetic,algebra,andproblemsolving.It is,moreover,atheoreticalaswellaspracticalwork;themethodsemployedinLiber abaci Leonardofirmlyestab-lisheswithEuclideangeometricproofs.Onemustnotbemisledbythelackofmodernmathematicalsymbolismintothinkingthatthisworkisnotexcellentorrigorousmathematics.Onedoesnotjudgethequalityofmathematicsbythesymbolismwithwhichitiswritten.Liber abaci wasgoodmathematicswhenitwaswrittenanditisgoodmathematicstoday.Liber abaci isase-riousmathematicalworkwrittenonarithmeticandappliedmathematicsbyasuperiorcreativemathematician.Oneshouldhereagainmakethepoint,thatwhilederivedfromthewordaba-
custhewordabaci refersinthethirteenthcenturyparadoxicallytocalculationwithouttheabacus.ThusLiber abaci shouldnotbetranslatedasTheBookoftheAbacus.Amaestro d'abbaca wasapersonwhocalculateddirectlywithHindunumeralswithoutusingtheabacus,andabaca isthedisciplineofdoingthis.ItwasLeonardo'spurposetoreplaceRomannumeralswiththeHindunumeralsnotonlyamongscientists,butincommerceandamongthecommonpeople.Heachievedthisgoalperhapsmorethanheeverdreamed.Italianmer-chantscarriedthenewmathematicsanditsmethodswherevertheywentintheMediterraneanworld.ThenewmathematicsalsospreadintoGermanywhereitwaspropagatedbythecossists(acorruptionoftheItaliancosa, orthing,theunknownofalgebra).
1. Introduction5
ForthreecenturiesorsoacurriculumbaseduponLeonardo'sLiber abaciwastaughtinTuscanyinschoolsofabaconormallyattendedbyboysintendingtobemerchantsorbyothersdesiringtolearnmathematics.Otherinstructorsandsomeverygoodmathematiciansalsowrotebooksofabacoforuseintheschool.Thesebooksvaryfromprimitiverulemanualsuptomathematicsbooksofquality,butnonewassocomprehensive,theoretical,andexcellentastheLiber abaci ofLeonardoPisano.LeonardoPisanowroteotherbooksonmathematics:Liber quadratorum
(1225), Practica geometriae (1223),Flos andEpistola ad Magistrum Theodo-rum (1225).ItishisLiber quadratorum, orThe Book of Squares lSi]'thatoffersbesttestimonytohispowerasamathematician.ThisworkcanbesaidtostandbetweentheworkofDiophantusandtheworkofPierreFermatinthetheoryofnumbers.ItdemonstratesLeonardo'spowerasacreativemathemati-cian.
Liber abaci isanimpressiveworkonarithmetic,algebra,andappliedmath-ematicsbaseduponthetheoreticalfoundationofEuclid'smathematics.Gen-eralmethodsareestablishedbyusingthegeometricalgebrafoundprincipallyinBookIIoftheElements. LeonardoturnstoBookXforafoundationofatheoryofquadraticirrationalnumbers.ThroughoutLiber abaci proofsaregivenforoldmethods,methodsacquiredfromtheArabicworld,andformethodsthatareLeonardo'soriginalcontributions.Leonardoalsoincludesthosecommonplacenon-algebraicmethodsestablishedinthemediaevalworldforproblemsolving,atthesametimegivingthemmathematicallegitimacywithhisproofs.Amongotherstheyincludecheckingoperationsbycastingoutnines,variousrulesofproportion,andmethodscalledsingleanddoublefalseposition.Inadditiontoteachingallofthenecessarymethodsofarithmeticandalge-
bra,LeonardoincludesinLiber abaci awealthofapplicationsofmathematicstoallkindsofsituationsinbusinessandtrade,conversionofunitsofmoney,weight,andcontent,methodsofbarter,businesspartnershipsandallocationofprofit,alloyingofmoney,investmentofmoney,simpleandcompoundinterest.Theproblemsontradegivevaluableinsightintothemediaevalworld.Healsoincludesmanyproblemspurelytoshowthepowerandbeautyofhismathemat-ics;theseproblemsarenoteworthyforhischoiceofappealingvividimagesandhisingenuityinsolution.IntheprefacetoLiber abaci Leonardostateshowinhistravelsandstudies
hehasfoundtheHindunumbersystemanditsmethodsofcalculationtobesuperiortoallothermethods,andthathewishestobringthesetotheItalianpeopleinthiswork.HestressesthathegivesproofsforthevalidityofthemethodshehasusedbaseduponEuclideanprinciples.Heremindsthereaderofthenecessityforstudyandpracticetoachieveproficiencywithhismethods.Leonardogivesatableofcontentsforhisentirebook.Thistableofcontents
isamplifiedattheheadofeachchapterwithmoredetailedcontentlists.Inchapter1thetennumeralsoftheHindunumber systemarepresented,in-
cludingzero,thatiscalledzephirfromtheArabic.Theplacesystemisexplainedwherebynumbersofanysizecanberepresentedwithonlythe tennumerals.Thissystemisourfamiliardecimalplacesysteminwhichanyfigureinthefirst
6 1. Introduction
placecountsonlyforitself,butthefigureinthesecondplacetotheleftcountsassomanytens.Insequencethethirdplacefromtherightisvaluedinhundreds,thefourthinthousands,andsoforth.ThezeroorzephirasLeonardocallsitcountsfornothingandservesasaplaceholder.Largenumbersareorganizedbytriplestofacilitatereading.Accustomedaswearetotheuseofourdecimalsystemandouralgorithmsforadditionandtheotheroperations,itiseasytooverlookthatforEuropeinthethirteenthcenturythisbookbroughtanewandrevolutionarywaytodoarithmetic.Leonardosupplementsthewrittennumberswithasystemofremembering
numbersbyusingvariousfingerpositionsinthehands.WhenLeonardosaysthatanumberiskeptinthehand,hemeansitliterally.Thismediaevalmemorysystemofkeepingnumbersinhandwaswidelyused,buthasfallenoutofusetoday.Thisholdingofnumbersinthehandsallowedonetoperformthecomputationalalgorithmsmoreefficientlyandwithlesswriting.Todaywemakesmallnotationswithpencilorpenofnumberstobecarriedorborrowedorwesimplyrememberthemasweperformthecalculations.Additionandmultiplicationofsmallnumbersarepresentedwithtablestobememorizedbythelearner,justaschildrendotoday.Inchapter2analgorithmformultiplicationisgivenbeginningwithnum-
bersoftwoplacesbynumbersoftwoplaces,andnumbersofoneplacebythoseofmanyplaces.Thealgorithmsofmultiplication,addition,subtraction,anddivisiondiffersolittleinconceptfromthoseusedtodaythatitseemsentirelyunnecessarytodwelluponhowthenumbersareplaceddifferentlybelow,above,ortothesideofwherethereaderlearnedtoplacethemwhenhewasinele-mentaryschool.WeleaveittothereadertomakesuchcomparisonsandtodiscoverhowLeonardo'salgorithmswork.Manyofthealgoristsdidalotoferasingandreplacingofnumeralsandhand-heldnumbersastheyworkedalongonacalculation.Leonardointroducesandexplainsthemethodofcheckingbycastingout
nines.CastingoutninesisaveryoldmethodandprobablygoesbacktothePythagoreans.Leonardoshowsthattheresidueofanumbermodulonineisequaltotheresidueofthesumofitsdigits.InthisbookLeonardouseschecksnotonlywithresiduesmoduloninebutalsoseven,eleven,andotherprimes.Healsowarnsthatanydivisionbythemodulusnumberinviteserrorsincheck-ing.LeonardodemonstratesaconsiderableelementaryknowledgeofmodulararithmeticthatGausssocapablydevelopedinDisquisitiones arithmeticae.BuildingslowlytomorecomplicatedcomputationalsituationsLeonardonext
presentsthreeplacesbythreeplacesandtwoplacesbythreeplacesmultipli-cations.Thisisfollowedbymultiplicationoffourplacesbyfourplaces,andtwoplacesbyfourplaces,andthreeplacesbyfourplaces.Multiplicationofnumberswith000attheirheadisdiscussed.Henextpresentsfiveplacesbyfiveplacesmultiplication.Proportionsandpowersoftenareusedtoexplaintheplacesystemandhowtheoperationswork.thenfiveplacesbymanyplacesmultiplicationistreated.Henextshowshowtousehandmemorytofacilitatetwoplacesbytwoplacesmultiplication.Alsooneplacebymanyplacesmulti-plicationwithuseofthehandsistreated,andthenthreeplacesbythreeplaces
I. Introduction7
multiplication.Finallythemultiplicationofanynumbersofanysizeistaught.Inchapter3analgorithmforadditionofwholenumbersfornumbersof
arbitrarysizeisgiven.Amediaevalsystemcalledchessboardmultiplicationisexplained.Aproofforcastingoutninesisgiven.Heteachesaddingbycolumns.Aprocedureforkeepingexpensesinatablewithcolumnsforpounds,soldi,anddenariispresented.Inchapter4subtractionofwholenumbersisexplained.Inchapter5divisionsofsmallnumbersandsimplefractionsarepresented.
Besidecommonfractionsasweknowthem,Leonardoalsodevelopsaformofcomposedfractions;usageofsuchfractionsaretreacabletothescientistswritinginArabic.Thesearesumsoffractionsinacompactnotationinwhichsuccessivefractionshavedenominatorswhicharemultiplesofthepreviousones.Forexample,thecomposedfraction~ ~ ~ means2xjx5+3;5+ t whichisequaltoO.Thedivisionalgorithmispresentedaswellasdivisionusingheadandhand.Nextfollowsdivisionbyprimeswithtwofigures.Divisionischeckedbymodulararithmeticaswellasmultiplication.Factorizationofregularorcompositenumbersistreated.Acompositionruleforregularnumbersinwhichnumbersareexpressedastheproductoftheirfactorsisgiven.Heusesthecompositionrulefordivisionofregularnumbers;adivisionofanumberbyaregularnumbercanbeaccomplishedbydividingsuccessivelybythefactors.Thepresentationofthecomposedfractionsiscloselyconnectedwiththefundamentaltheoremofarithmetic,thatistheuniquefactorizationofanywholenumberintoprimefactors.SuchfactorizationsLeonardocallscompositionrulesorsimplytheruleforthenumber.Healsoproducescompositionrulesincludingfactorssuchas10,20,12,andotherstofitapplicationswhicharebaseduponcommonmeasurements.Suchfactorizationsare,ofcourse,notprimefactorizations,butderivedfromthem.ThecomposedfractionsasusedbyLeonardoinclude decimalfractions.For
example,thedecimalfraction28.2429536481occursinchapter12intheproblementitledA Man Who Travelled through Twelve Cities, andisexpressedbyLeonardowiththecomposedfraction
1846359242101010 10 1010101010 1028.
Inchapter6Leonardotreatsmixednumbersorwholenumberswithsim-plefractions.Theseproceduresinvolvechangingmixednumberstoimproperfractions,performingtheoperations,andthenreducingtheansweragaintoamixednumber.Theresultsareusuallyexpressedincomposedfractions.Re-sultsarecheckedbycastingoutninesorsomeothermodulus.Operationsforwholenumberswithtwoorthreefractionalpartsinonecomposedfractionarepresented.Hethenpresentsoperationsinorderforwholenumberswithtwofractionalpartsintwocomposedfractions,wholenumberswithtwofractionswithmanyparts,threefractions,andpurefractions.Leonardothenpresentsan-otherspecialnotation:numbersandfractionsterminatinginacircle.Fractionsofmixednumbersaretreated.
8 1.Introduction
Inchapter7practiceontheoperationsiscontinuedwiththeaddition,sub-traction,divisionofnumberswithfractionsandthereductionofseveralpartstoasinglepart.Addition,subtraction,anddivisionofonefractiontoanotheraregiven.Theninorderaretreatedadditionandsubtractionoftwofractionsfromtwofractions,divisionofintegersbymixednumbersandviceversa,ad-dition,subtraction,divisionofintegerswithfractions.Addition,subtraction,divisionoffractionsofmixednumbersistreated.Leonardodiscussesatlengththeseparationoffractionsintosumsofunitfractions,thatis,howanyfractioncanbewrittenasthesumoffractionswithunitnumerators.ThistopicgoesbacktotheEgyptianpreferenceforunitfractionsandforthisreasonthetopicisoftencalledEgyptianfractions.Inchapter8thevalueofmerchandiseisfoundbytheprincipalmethod
ofproportion.HereLeonardotreatsmanysimplebusinessnegotiationsusingproportions.Thereareproblemssuchas,if2poundsofbarleycost5soldi,thenhowmuchdo7poundscost?Leonardosystematizestheseproblemstosimplediagramsofproportionwhichhecallsthemethodofnegotiation.Intheseproblemsonelearnstheweightandmonetarysystemsprevalentinthethirteenthcenturyaswellastheproductsboughtandsold.Therearebusinessproblemsonthesaleofthingsbyhundredweight,onmonetaryexchange,onthesaleofcanes,bales,andtorcelli,ontheconversionofunitstoPisanrolls.TheexamplesaredrawnfromtheentireMediterraneanworld.Themonetary,volume,andweightunitsarereferencedandfurtherexplainedattheiroccurrenceinthechapters.AllnamesinthetranslationgiveninItalianareinLeonardo'stextinLatin.Inchapter9thebusinessnegotiationsofthepreviouschapterareextended
tomoreitemsthroughbarterbaseduponsomecommonmonetaryvalue.Thesystematicproportionbaseddiagrammethodusedbeforeisextended.Alsotreatedarethebarterofcommonthings,thesaleofmoneyalreadybartered,thepurchaseofmoneyaccordingtorule.Thereareproblemsonhorsesthateatbarleyinanumberofdays,menwhoplanttrees,andmenwhoeatcorn.Inchapter10isfoundananalysisofinvestmentsandprofitsofcompanies
andtheirmembers.Theproblemsaresolvedusingproportion.FUndamentalconceptsaredevelopedonhowprofitsaresharedamongshareholdersinbusinessventuresaccordingtothesizeoftheirindividualinvestments.Theseproblemscastlightupontheoperationofbusinessinthethirteenthcentury.Inchapter11Leonardodiscussesatlengththealloyingofmoniescontaining
silverandcoppertoobtainacoinresultwithsomefixedproportionofsilverandcopper.Theproblemsaresolvedusingproportions.Thereareoftenmul-tiplesolutionstotheproblemsfortheyinvolveindeterminatelinearequations.Afterproblemswithone,two,andmoremoniesaresolved,thenanalogousprob-lemsareintroducedwhicharesolvedusingthesameprinciples.Theseincludemixturesoffruit,gold,meat,grain,andbirds.Chapter12isprincipallyonthemethodoffalseposition.Theproblemsgiven
leadtooneormorelinearequationsinoneormanyunknowns.Themethodoffalsepositionworksbytheposingofargumentswhichareapproximationswhicharethencorrectedtogivetruesolutions.Themethodofsinglefalsepositionsolvesproblemswhichareequivalenttolinearequationsofthesimple
I. Introduction9
typeAx=B, anddoublefalsepositionwhichisusedonproblemsleadingtoequationsofthetypeAx + B =C.Doublefalsepositionproblemsarefoundinchapter13.BesidesthemethodoffalsepositionLeonardosolvesproblemsusingwhathecallsthedirectmethod.Thismethodinvolvescallingthesoughtquantitythething andcreatinganequationcontainingthething.Theequationisstatedinsentencesandnotwrittensymbolicallyaswedotoday.Theequationisthensolvedstepbystepforthething.Thisis,ofcourse,algebraasweknowit,andispreciselythemethoddescribedbyal-KhwarizmIinhisbookonalgebra.Chapter13openswithsomeresultsonsummingarithmeticserieswithappli-
cationstosomenotveryrealisticproblemsontravellers.Somemorediscussionofproportionoccursbeforethetreeproblemisintroduced.ThetreeproblemisthetypicalproblemrequiringthesolutionoftheequationAx =B. It issolvedbythemethodofsinglefalseposition.Manyvariationsaregiven.Manyingeniousandvividproblemsaregivenonsnakes,four-leggedanimals,eggs,businessventures,ships,vatsfullofliquidswhichemptythroughholes,andpurenumberproblems.Theimaginativenatureoftheproblemsmakesstimu-latingreading.Thereisextensivepresentationoftheproblemsofmenhavingdenari.Intheseproblemsonemanwillgiveanotherorotherssomenumberofhisdenariwhichwillproduceacertainratiooramount.Fromthegivingandtakingandsetconditionsthenumberofdenarithateachmanhascangenerallybefound.Theseproblemsleadtolinearequationswhichmustbesolvedinin-tegers.Thereareoftenmanysolutionsandsometimesnone.Anotherproblemslightlymoreelaborateisthefoundpurseproblem.Heremenhavingdenarifindapurseorpursescontainingdenari.Conditionsaregivenandonemustfindtheamountofeachman'sdenariandtheamountofdenariineachpurse.Againtheseproblemsrequiresolutionsinintegers.Asimilarproblempresentedisoneinwhichmenhavedenariandproposetobuyahorseundercertaincon-ditions.Fromthedescribedconditionsonemustfindhowmanydenarieachmanhasandwhatisthecostofthehorse.Generallyonefindspositiveintegralsolutions,butthereareseveralproblemsinwhichanegativeintegralsolution,calledadebitbyLeonardo,isnecessary.Asthereareusuallymanysolutionstotheproblemsoneoftenseekstheminimalsolutions.Sometimesanadditionalconditionisimposed.SuchequationsarecalledDiophantine,althoughintruthDiophantususuallysoughtfractionalsolutionswhereasLeonardoseeksintegralsolutionstotheseproblems.LeonardomakesfrequentuseofnegativenumbersinLiber abaci. Wewish
toemphasizethatLeonardowascompletelycapableofconceivingofnegativenumbersforsolutionstoequationsasreasonableonesinappropriatecircum-stances.Furthermore,therearegivencompleteruleswith proofs forthead-ditionandmultiplicationofpositiveandnegativenumbersandtheserulesareusedextensively,especiallyinchapter13.Therearenoteworthybusinessproblemsabouttravellerswithperiodicex-
pensesandprofits.Therearealsobankingproblemsaboutinvestments,simpleandcompoundratesofinterest,and futurevalueofinvestments.Thereareproject-relatedproblemswithlaborcostsandprofits.Andthereareproblemswhicharecontrivedwithbirds,fruits,andanimalswhichillustratetheclev-
10 1.Introduction
ernessoftheauthor,andbelongtotheloreofmathematics;includedisthefamousrabbitproblemwhichgeneratestheFibonaccisequence.Thereareanumberofdivinationproblems,asLeonardotermsthem.These
involverecoveringsomeunknownnumberafterseveraloperationsareperformeduponitandtheresultisgiven.Therecoveryofthegivennumbergenerallyinvolvessomeuseofmodulararithmetic.Therearefinallysomeproblemsin-volvingsummingseries.Chapter13useselchataym[4], themethodofdoublefalsepositionwhich
solvesnotonlythetreeproblem,Ax=B, butsolvesalsothemorecomplicatedequation,Ax +B =C.Bothsingleanddoublefalsepositionarebaseduponproportionorlinearextrapolation.InthischapterLeonardosolvesrathercom-plicatedsetsoflinearequationsbyiteratingtheelchataymprocessseveraltimesforthefindingofvaluesofseveralunknowns.Sometimeshesearchesforrationalsolutionswhenappropriateandsometimesforintegralsolutions.Heisperfectlyawarethatmanyproblemshavemultiplesolutionsandgenerallygivesustheminimalones.Thereareproblemswithnonumericalsolutions.Theproblemsrangeagainoversuchsubjectsasmoney,workers,travellerswithexpensesandprofits,menwithmoneyfindingapurse,horsebuying,andsoforth.Inchapter14Leonardocollectsinformationandtechniquesforhandling
roots.HeusestheclassificationsfoundinBookXofEuclid'sElements forthesumsanddifferencesofunlikeroots,namelybinomialsandapotomes.Hepresentstheresultsofoperationsonbinomialsandapotomesandsimplificationofsuchexpressions.Althoughhedealswithhigherrootsthantwo,hepresentsherenothingsignificantnotfoundintheElements.Inchapter15wefindareviewofproportionandacollectionofelementary
geometricproblems.ThePythagoreantheoremisusedandalsosimpleareasandvolumesarefound.Thetechniquesofalgebraareagainpresentedbutthistimeforquadraticinsteadoflinearequations.Thepresentationdifferslittlefromthatfoundinal-KhwarizmI'sbookonthesubject.Thisisnotplagia-rism,butratherfollowsthetraditionofshowingrespectforearlierworks.BookVIIoftheElements is,forexample,apresentationofPythagoreanmathemat-ics.LeonardoclearlylabelsthequadraticequationpresentationinLiber abaciasthatofal-KhwarizmIbywritingMaumeht[8] onthemargin[p406].Thesixstandardformswhichresultfrompostulatingthatallcoefficientsarenon-negativearetreatedandsolvedandanumberofappliedproblemsarepresented.Thetechniqueemployedforsolvingthequadraticequationiscompletingthesquare.Generallyonlypositivesolutionsarenoted,butLeonardoisperfectlyawarethattwosolutionsarepossible.ThisEnglishtranslationispreparedfromBaldassarreBoncompagni'sLatin
editionof1857[Bl.ThepagenotationsthroughouttheEnglishtext,[pI93]forexample,refertotheapproximate'beginningofeachnewpageoftheLatinedition.TheLatineditioncontainsmanymisprints,mostlynumericalones,anditselfnotesseveralmistakes(sic)withouttheobviouscorrectiontothem,butthereisnotonecasewherethemisprintormistakecausesanirresolubleambiguity.Contextisalwayssufficienttorestorecorrectvalues.Thetitlesinparenthesisareaddedforclarity.ThisEnglisheditionisthefirstpublished
1. Introduction11
translationoftheLatinworkintoamodernlanguage.Ihavetriedin thistranslationtoadhereascloselyaspossibletotheLatintextandtopresentaveryliteraltranslation.ThereexistanumberofmanuscriptsofLiber abaci inEuropewhichwereexaminedbyBoncompagniinpreparinghisdefinitivetext.TheBoncompagnitextiscompleteandunambiguous.Oneshouldnotwriteaboutthiseraofmathematicalhistorywithoutmaking
specialmentionoftheworkofMr.EttorePicutti.Someofhisworkislistedinthebibliography.HisworkstandsasamodelofclarityandprecisionforanypersonwhowishestowriteonearlyItalianmathematics.
PartII
LiberAbaci
Dedication and PrologueLiber abbaci, LeonardoPisano
CodiceMagliabechiano,C.I,2626,Badia Fiorentina, n.73BaldassarreBoncompagni
Roma,TipografiadelleScienzeMatematicheeFisicheViaLataNum.211,MDCCCLVII.
Here begins the Book of CalculationComposed by Leonardo Pisano, Family Bonaei,
In the Year 1202.
You,myMasterMichaelScott[1],mostgreatphilosopher,wrotetomyLord[2]aboutthebookonnumberswhichsometimeagoIcomposedandtranscribedtoyou;whencecomplyingwithyourcriticism,yourmoresubtleexaminingcir-cumspection,tothehonorofyouandmanyothersIwithadvantagecorrectedthiswork.In thisrectificationIaddedcertainnecessities,andIdeletedcertainsuperfluities.In itIpresentedafullinstructiononnumbersclosetothemethodoftheIndians,[3]whoseoutstandingmethodIchoseforthisscience.Andbe-causearithmeticscienceandgeometricscienceareconnected,andsupportoneanother,thefullknowledgeofnumberscannotbepresentedwithoutencounter-ingsomegeometry,orwithoutseeingthatoperatinginthiswayonnumbersisclosetogeometry;themethodisfullofmanyproofsanddemonstrationswhicharemadewithgeometricfigures[4].AndtrulyinanotherbookthatIcomposedonthepracticeofgeometry[5]Iexplainedthisandmanyotherthingspertinenttogeometry,eachsubjecttoappropriateproof.Tobesure,thisbooklooksmoretotheorythantopractice.Hence,whoeverwouldwishtoknowwellthepracticeofthisscienceoughteagerlytobusyhimselfwithcontinuoususeandenduringexerciseinpractice,forsciencebypracticeturnsintohabit;memoryandevenperceptioncorrelatewiththehandsandfigures,whichasanimpulseandbreathinoneandthesameinstant,almostthesame,gonaturallytogetherforall;andthuswillbemadeastudentofhabit;followingbydegreeshewillbeableeasilytoattainthistoperfection.AndtorevealmoreeasilythetheoryIseparatedthisbookintoxvchapters,aswhoeverwillwishtoreadthisbookcaneasilydiscover.Further,ifinthisworkisfoundinsufficiencyordefect,Isubmitittoyourcorrection.AsmyfatherwasapublicofficialawayfromourhomelandintheBugiacus-
tomshouseestablishedforthePisanmerchantswhofrequentlygatheredthere,hehadmeinmyyouth[6]broughttohim,lookingtofindformeausefulandcomfortablefuture;therehewantedmetobeinthestudyofmathematicsandtobetaughtforsomedays.TherefromamarvelousinstructionintheartofthenineIndianfigures,theintroductionandknowledgeoftheartpleasedmesomuchaboveallelse,andIlearntfromthem,whoeverwaslearnedinit,fromnearbyEgypt,Syria,Greece,SicilyandProvence,andtheirvariousmeth-ods,towhichlocationsofbusinessItravelledconsiderablyafterwardsformuch
16 II.LiberAbaci
study,andIlearntfromtheassembleddisputations.Butthis,onthewhole,thealgorithmandeventhePythagoreanarcs[7],Istillreckonedalmostaner-rorcomparedtotheIndianmethod.ThereforestrictlyembracingtheIndianmethod,andattentivetothestudyofit,frommineownsenseaddingsome,andsomemorestillfromthesubtleEuclideangeometricart,applyingthesumthatIwasabletoperceivetothisbook,Iworkedtoputittogetherinxvdis-tinctchapters,showingcertainproofforalmosteverythingthatIputin,sothatfurther,thismethodperfectedabovetherest,thisscienceisinstructedtotheeager,andtotheItalianpeopleaboveallothers,whouptonowarefoundwithoutaminimum.If, bychance,somethinglessormoreproperornecessaryIomitted,yourindulgenceformeisentreated,asthereisnoonewhoiswithoutfault,andinallthingsisaltogethercircumspect.[p2]
Here Ends the Prologue and Begin the Chapters.
OntherecognitionofthenineIndianfiguresandhowallnumbersarewrit-tenwiththem;andhowthenumbersmustbeheldinthehands,andontheintroductiontocalculations.Onthemultiplicationofwholenumbers.Ontheadditionofthem,onetotheother.Onthesubtractionoflessernumbersfromlargerones.Onthemultiplicationofwholenumberswithfractionsandalsofractions
alone.Ontheaddition,subtraction,anddivisionofwholenumberswithfractions
andalsothereductionoffractionalpartsintosingleparts.Onthebuyingandsellingofcommercialandsimilarthings.Onthebarterofcommercialthingsandthebuyingofcoin,andcertainrules
onthesame.Oncompaniesmadeamongparties.Onthealloyingofmoneyandalsotherulesthatarepertinenttoalloying.Onthesolutionstomanyposedproblemsthatwecallfalseposition.Ontheruleelchataymbywhichalmostallproblemsoffalsepositionare
solved.Onthefindingofsquareandcuberoots,andthemultiplication,division,or
subtractionofthem,andonthehandlingofbinomialsandapotomesandtheirroots.Onthepertinentrulesofgeometricproportions;onproblemsofalgebraand
almuchabala.[8]
Chapter 1
Here Begins the FirstChapter.
ThenineIndianfiguresare:
987654321.
Withtheseninefigures,andwiththesign0whichtheArabscallzephir[1]anynumberwhatsoeveriswritten,asisdemonstratedbelow.Anumberisasumofunits,oracollectionofunits,andthroughtheadditionofthemthenumbersincreasebystepswithoutend[2].First,onecomposesfromunitsthosenumberswhicharefromonetoten.Second,fromthetensaremadethosenumberswhicharefromtenuptoonehundred.Third,fromthehundredsaremadethosenumberswhicharefromonehundreduptoonethousand.Fourth,fromthethousandsaremadethosenumbersfromonethousanduptotenthousand,andthusbyanunendingsequenceofsteps,anynumberwhatsoeverisconstructedbythejoiningoftheprecedingnumbers.Thefirstplaceinthewritingofthenumbersbeginsattheright.Thesecondtrulyfollowsthefirsttotheleft.Thethirdfollowsthesecond.Thefourth,the third,andthefifth,thefourth,andthusevertotheleft,placefollowsplace.Andthereforethefigurethatisfoundinthefirstplacerepresentsitself;thatis,ifinthefirstplacewillbethefigureoftheunit,itrepresentsone;ifthefiguretwo,itrepresentstwo;ifthefigurethree,three,andthusinorderthosethatfollowuptothefigurenine;andindeedtheninefiguresthatwillbeinthesecondplacewillrepresentasmanytensasinthefirstplaceunits;thatis,iftheunitfigureoccupiesthesecondplace,itdenotesten;ifthefiguretwo,twenty;ifthefigurethree,thirty;ifthefigurenine,ninety.Andthefigurethatisinthethirdplacedenotesthenumberofhundreds,as
thatinthesecondplacetens,orinthefirstunits;andifthefigureisone,onehundred;ifthefiguretwo,twohundred;ifthefigurethree,[p3]threehundred,andifthefigurenine,ninehundred.Thereforethefigurewhichisinthefourthplacedenotesasmanythousandsasinthethird,hundreds,andasinthesecond,tens,orinthefirst,units;andthuseverchangingplace,thenumberincreases
L. Sigler, Fibonacci's Liber Abaci Springer-Verlag New York, Inc. 2002
18 II.LiberAbaci
byjoining.Andasthisprincipleisclearlyrevealed,itisshownwithfigures.Ifthefiguresevenisinthefirstplaceandthefigurethreeinthesecond,thenbothtogetherdenote37;orpermuted,thefigurethreeinthefirstandthefigureseveninthesecondwilldenote73.Againifthefigurefourisinthefirstplace,andtheunitinthesecond,thus14,undoubtedlyxiiiiwillbedenoted;orifthefigureoftheunitisinthefirstplace,andthefigurefourinthesecond,thus41,xliwillbedenoted.Againinthefirst2,andinthesecond7,make72;theoppositeindeedmake27.Howeverifonewillwishtowriteasmuchasseventy,thenoneputsinthefirstplace0,andafteritoneputsthefigureseven,thus70;ifeighty,thefigurezephirisfollowedbyeight,thus80;andthereforethisdemonstrationshowshowyoucanwriteanynumberfromtenuptoonehundredwithtwofigures.Withthreefromonehundreduptoonethousandcantrulybewritten;andifthefigureeightisinthefirstplace,andthefigurefiveinthesecond,andtheunitinthethird,then158,onehundredfifty-eightwillbedenoted;andpermuted,iftheunitisinthefirstplace,andthefigurefiveinthesecond,andthefigureeightinthethird,851,eighthundredfifty-onewillbedenoted;orpermuted,ifthefigureeightisinthefirstplace,theunitinthesecond,andthefigurefiveinthe third,then518willbedenoted.Againifpermuted,thefigurefiveisinthefirstplace,thefigureeightinthesecond,andtheunitinthe third,then185willbedenoted.Alsoiftheunitisinthefirstplace,thefigureeightinthesecond,andthefigurefiveinthe third,then581willbeundoubtedlydenoted;threeunits,infact,makeonehundredeleven.Trulyifsomuchasfivehundredyouwillwishtowrite,inthefirstandinthesecondplaceyouwillputthezephir,andinthethirdthefigurefive,inthisway,500;andthusyouwillbeabletowriteanynumberofhundredswithtwozephir.Andifyouwillwishtowritehundredswithtensorunits,thenyouputinthefirstplacethezephir,inthesecondtens,andinthethirdthehundredsthatyouwillwish.Forexample,ifinthefirstplaceisthezephir,andinthesecondthefigurenine,andinthethirdthefiguretwo,then290willbedenoted.Ifindeedyouwillwishtowritehundredswithunitsandwithouttens,youputinthesecondplace,namelyintheplaceofthetens,thezephir,andinthefirstthenumberofunitsthatyouwillwish,andinthe third,thefiguretwo,209;andthusaccordingtotheabovedemonstratedprincipleyouwritewiththreefigureswhatevernumberyouwillwishfromonehundreduptoonethousand.Andwithfour,fromonethousanduptotenthousand,andthenumbersnotedaboveareshownwithfiguresinthefollowing.
Mi MMxxiii MMMxxii MMMxx MMMMMdc10012023 3022 30205600
MMM Mcxi30001111
Mccxxxiiii1234
MMMMcccxxi4321
Andthusitiscontinuedwiththeremainingnumbers.Andwithfivefiguresarewrittenallnumbersbeginningwithtenthousandanduptoonehundred
1.HereBeginstheFirstChapter19
thousand.Withsix,trulyfromonehundredthousanduptoonethousandthousand,andthusinstepsadjoiningfiguretofigure,thenumberincreasesbyadjoininginsteps.Whenceifitwillhappenthatonecannotreadnorperceivesomenumberofmanyfiguresbyreasonofthegreatnumberoffigures,thenIshalltakecaretoshowhowitoughttobereadandunderstood.[p4]Therefore,forthefirstfigure,thatisthefigureofthefirstplace,onesays
one.
Ofthesecondthatisinthesecondplace,onesaysten.Ofthethirdthatwillbeinthethirdplace,onesaysonehundred,andadjoins
ittothehigherpart.Ofthefourthfigureofthenumber,onesaysthousand,andoneadjoinsitto
thelowerpart.Ofthefifthtrulyonesaystenthousand.Thereforeofthesixthonehundredthousand,andoneadjoinsittothehigher
part.Oftheseventhonesaysonethousandthousand,andoneadjoinsitagainto
thelowerpart.Oftheeighthonesaystenthousandthousand.Oftheninth,onehundredthousandthousand,andoneadjoinsittothe
higherpart.Ofthetenthonesaysonethousandthousandthousand,andoneadjoins
ittothelowerpart;andthuseverforthesethreenumbers,namelyforthethousands,andtenthousands,andhundredthousandstothehigherpart,onestrivestobuilduptothelastplaceofthenumber.Andthenceonebeginstoreadnumbersfromthelastplacebytheaforesaidadjoiningssayingeverofthelowestadjoiningthousandsofthousands,asmanyoneadjoinsisbeforeitinthelowerparttowardsthefirstplace,andofthehigheradjoined,sayingasmanyhundredthousandsthatareadjoinedbeforeitinthelowerpartsimilarlytowardsthefirstplaceofthenumber;andofthefiguresthatareonlyadjoinedafterthefourthplaceofthenumberonesaysasmanytenthousandsasareadjoinedbeforeitinthelowerpart;andthusonewillbeabletorecognizeandreadwhatevernumberofasmanyfiguresonewillwish.Andinorderthatthisisbetterunderstoodweproposeanumberofeightfigures,87654321.Andforthefigureonewhichisthefirstplace,onesaysone;forthefiguretwo,2,whichisinthesecond,Ishallsaytens;forthethree,3,whichisinthethirdplaceinthehigherpart,onesayshundreds.Forthefigurefour,4,whichisinthefourthplaceadjoinedinthelowerpartonesaysthousands,asisshownintheaforewrittennumber.Forthefigurefive,5,whichisinthefifthplace,onesaystenthousand;forthefiguresixwhichisinthesixthplace,adjoinedinthehigherpart,onesayshundredthousands;forthefigureseven,7,whichisintheseventhplaceadjoinedinthelowerpartonesaysthousandthousands;thereforeeighty-seventhousandthousandishadintheabovesaidnumberbyreasonofthetwoadjoinedlower,oneofwhichisthelower7andtheotheristhelower4,andfurthersixhundredfifty-fourthousand,andfurtherCCCxxi.Againweproposeanothernumberofninefigures,257604813,andbytheadjoiningorderitisrecognizedthatitcontainsinittwohundredfifty-seventhousandthousand,sixhundred
20 II.LiberAbaci
fourthousand,andeighthundredthirteen.Againanothernumberofthirteenfiguresisproposed,1007543289081;itisrecognizedthatthereisadjoinedtheonethousandthousandthousandthousand,seventhousandthousandthousand,fivehundredforty-threethousandthousand,twohundredeighty-ninethousand,andfurthereighty-one.Wecanindeedteachanothereasyrulesothatyoumostquicklywillbeabletoreadanumberofmanyfigures.Forexample,proposeanumberof15figures,678 35 784 165 296;youdetachthethreefirstfigures,namely296,aboveeverythreeyoudrawavirgulainthewayofanarcasinthetakenexample[3];andforanyvirgulayousay:andthethreefiguresthatareatfirstaredetached,youreadastheystand,andthusyousaysixhundredseventy-eightthousand thousandthousandthousand,astherearefourvirgulas,andninehundredthirty-fivethousand thousandthousand,asabovearethreevirgulasinnumberand[p5]sevenhundredeighty-fourthousandthousand,asabovearetwocurvesand105thousand,asthereisonevirgula,and296forthethreethataredetachedatthebeginning;andifforthelastremainsafigureortwo,youputthemunderalastvirgula,andyoureadthemallfourorallfivetogether,andthusyouwillbeabletoreadanumber,nomatterhowmanyfigures.Accordingtothematerialwrittenabove,withfrequentusetheaforesaid
figuresoftheplacesarewellrecognized;thosewhowishtoknowtheartofcal-culating,itssubtletiesandingenuities,mustknowcomputingwithhandfigures,amostwiseinventionofantiquity,accordingtoitsusebythemastersofmath-ematics.Thesignsarethese.Thecurvingofthelittlefingerofthelefthandoverthemiddleofthepalmofthehandsignifiesanddenotesone,1.IndeedbythecurvingofthesamefingerandtheringfingerandthemiddlefingeroverthemiddleofthepalmIshallmean4.Bythecurvingofthemiddlefinger,5.Theringfinger,6.Furtherbythepositioningofthelittlefingerupwardsabovethepalm,Ishallcertainlysignify7,andabovethatplaceifarepointedthelittleandringfingers,8isdenoted;indeedthepositioningofthemwiththemiddlefingerabovethesameplace,9.Withtheendsoftheforefingerandthethumbaremadeacircleinthejointofthethumb;thisdenotes10.Withthethumbandforefingerextendedandtouching,20.Withtheendsofthemmakingacircle,30.Withthethumbplacedovertheforefingerontheexteriorpartoftheindexfinger,40.Thecurvingofthethumboverthebeginningoftheforefinger,50.Thecurvingoftheforefingeroverthecurveofthethumb,60.Thecurvingoftheforefingerovertheendoftheextendedthumb,70.Thereforethecurvingoftheforefingeroverthecurveoftheextendedthumb,80.Againthecurvingoftheentireforefingeroveritself,90.Also,onehundredandonethousandaremadeintherighthandinthesameorder,namelythesignoftheunitmakes100intherighthand;twoindeed200;tenmoreoveronethousand,andthesignofninetymakes9000,asinthefollowingpagesareshownthepicturesofthehands[4].Allremainingnumbersfromtenuptotenthousandarethereforebuiltinthehandswiththesesignsinthisway;fromthesignoftwentyandfromthesignofthreearebuilt23;andfromthesignofthreethousandandfromthesignoffivehundredarebuiltintherighthandthreethousandfivehundred,andthusyouunderstandtherest.[p6]
1.HereBeginstheFirstChapter21
An Introduction to the Addition and Multiplication of Numbers.
2 and 2 make 4 Key for Three Key for Four2 3 5 3and 3 make 62 4 6 3 4 7 4 and 4 make 82 5 73 5 8 4 5 92 6 8 3 6 94 6 102 7 93 7 10 4 7 112 8 10 3 8 11 4 8 122 9 11 3 9 12 4 9 132 10 12 3 10 13 4 10 14
Key for Five Key for Six5 and 5 make 10 6 and 6 make 12 Key for Seven5 6 11 6 7 13 7 and 7 make 145 7 12 6 8 14 7 8 155 8 13 6 9 15 7 9 165 9 14 6 10 167 10 175 10 15
Key for Eight Key for Nine Key for Ten8 and 8 make 16 10 and 10 make 208 9 17 9 and 9 make 188 10 18 9 10 1920 and 20 make 40
20and 30make 50 30and 30make 60 40 and 40make 8020 40 60 30 40 7020 50 70 30 50 80 40 50 9020 60 80 30 60 90 40 60 10020 70 90 30 70 10040 70 11020 80 10030 80 no 40 80 12020 90 no 30 90 12040 90 130
50 and 50make 100 60and 60make 12050 60 110 60 70 13070 and 70make 14050 70 12060 80 14070 80 15050 80 130 60 90 15070 90 16050 90 140
80 and80
80 make90
160170 90 and 90make 180
22 II.LiberAbaci
Here End the Additions.
Here Begin the Multiplications.
On Two2 times 2 make 4 On Three On Four2 3 6 3 times 3 make 9 4 times 4 make 162 4 8 3 4 12 4 5 203 5 152 5 10 3 6 18 4 6 242 6 12 3 7 21 4 7 282 7 14 3 8 24 4 8 322 8 16 3 9 27 4 9 362 9 18 3 10 30 4 10 402 10 20
On Five On Six5 times 5 make 25 6 times 6 make 36 On Seven5 6 30 6 7 42 7 times 7 make 495 7 35 6 8 48 7 8 565 8 40 6 9 54 7 9 635 9 45 6 10 60 7 10 705 10 50
On8 times88
Eight8910
make 647280
On9 times9
Nine9 make 8110 90
On10 times
Ten I10 make 100
Here End the Multiplications.
[p7]Youthereforewritetheadditionsandmultiplicationsintables,alwaysmakinguseofthehandstoholdthenumbers,andonealikespiritedlymakesfreeuseofthehandstocarryouttheadditionsandmultiplicationsofanynumbers.
Chapter 2
Here Begins Chapter Twoon the Multiplication ofWhole Numbers.
Wedividechaptertwoonthemultiplicationofwholenumbersintoeightpartsinordertounderstandbettertheirpropertiesanddifferences.Thefirstpartwillbeonthemultiplicationoftwofiguresbytwo,andalsoonefigurebymany.Thesecond,onthemultiplicationofthreefiguresbythree,andalsotwofiguresbythree.Thethird,onthemultiplicationoffourfiguresbyfour,andmoreovertwoandthreefiguresbyfourfigures.Thefourth,onthemultiplicationoffivefiguresbyfive.Thefifth,onthemultiplicationofmanyfiguresbyfive,oranynumberbyitself.Thesixth,onthemultiplicationofnumbersoftwoplacesbynumbersofthesamenumberofplaces,thatistwofiguresbytwofigures,andalsoonefigurebymany,multiplyingwhateverisheldinthehands.Theseventh,onthemultiplicationofthreefiguresbythree,similarlywhateverisheldinthehandsismultiplied.Theeighth,onthemultiplicationofanynumbersinanotherway.
Here Begins the First Part on the Multiplicationof Two Figures by Two.
Anumberissaidtobemultipliedbyitselfwhenlikeismultipliedbylike,as12by12,or26by26.Anumberissaidtobemultipliedbyanothernumberwhenthenumbersmultipliedareunequaltoeachotheras12by37,and46by59;finally,aswepromised,weinstructhowtomultiplybyitselfourfirstnumberoftwoplaces,namelyfrom10upto100.Whenmoreoveryouwishtomultiplyanynumberoftwoplacesbyanynumberofthesamenumberofplaces,whetherthenumbersareequalorunequal,youwritethenumberbeneaththenumbersothatlikeplacesarebelowlikeplaces;andifthenumbersareunequal,letthe
L. Sigler, Fibonacci's Liber Abaci Springer-Verlag New York, Inc. 2002
IIllustration IFirst 4
1212
Second 441212
Last 1441212
Fi'rst 93737
Second 693737
The 1369residue 37
is 1. 37
24 II.LiberAbaci
greaterbebelowthelesser,andonebeginsthemultiplicationinthefirstplaceofthenumbers,asinthetableswrittenbefore.Thenonemultipliesthefigureinthefirstplaceoftheuppernumberintheaforewrittentablebythefigureinthefirstplaceinthelower,andtheunitsarewrittenoverthefirstplaceoftheaforewrittennumbers,andforeachtenaoneisheldinthelefthand;nextonemultipliesthefigureinthefirstplaceoftheuppernumberbythefigureinthesecondplace,namelybythelastfigureinthelowernumber,andviceversa;thefigureinthefirstplaceinthelowerismultipliedbythelastfigureintheupper,andallareaddedinhandwiththekepttens;andagaintheunitsarewrittenabovethesecondplace,andthetensareheldinthehand.Alsothelastfigureintheuppernumberismultipliedbythelastinthelower,andwhateverwillresultfromthemultiplicationisaddedtothetensheldinhandandtheunitsinthethirdplace,andthetensmadeabovewillbeputinthefourth,andthemultiplicationofanynumberswhatsoeverfromtenuptoonehundredwillbehad.Forexample,ifonewillwishtofindthemultiplicationof12by12,then12iswritteninthechalktableinwhichthelettersareeasilydeleted,asisshownwritteninthismargin;thefirstplaceinthelowernumberisbelowthefirstplaceintheupper,thatisthefiguretwobelowthefiguretwo,andthesecondplaceinthelowerbelowthesecondintheupper,namelythefigureonebelowthefigureone,andthetwoismultipliedbythetwo;therewillbe4thatisputabovebothofthetwos,asisplacedinthefirstillustration.Againtheupper2ismultipliedbytheonewhichisinthesecondplaceofthelowernumber;therewillbe2whichiskeptinhand,andagainthe2inthelowernumber[p8]ismultipliedbythe1intheupper;therewillbe2whichoneaddswiththeaboveheldtwo;therewillbe4thatisputovereachunitwhichmakesthe4inthesecondplaceafterthepriorputfigure4makingthefirstplace,asiswritteninthesecondillustration;andalsothe1intheuppernumberismultipliedbytheoneinthelowermaking1;thisiswritteninthethirdplace,namelyafterthewritten44,asisshowninthethirdandlastillustration.Andinthistotalresultsthemultiplicationof12byitself,namely144.Againthemultiplicationof37by37isclearlyillustrated.Indeedthe37is
writtenbeneaththe37,aswesaidaboveofthe12,andthe7ismultipliedbythe7;therewillbe49;thereforethe9isputaboveboth7,asisshowninthefirstillustration,andforthefigurefourinthetensplace,thatisinthe49,the4iskeptinhand,andthe7intheuppernumberismultipliedbythe3inthelower,andthe7inthelowerbythe3intheupper,andtheyareaddedtogether;therewillbe42whichyouaddwiththe4keptabove;therewillbe46;theunitsof46,whichare6,arewrittenaboveboth3asisshowninthesecondillustration.Andthe4,forthefourtensthatareinthe46,isheldinhand,andafterthisthe3intheuppernumberismultipliedbythe3inthelower;therewillbe9whichoneaddstothe4,bythewayofitsbeinginhand;therewillbe13;the3inthe13isputinthethirdplaceandthe1inthefourth,asiscontainedinthethirdandlastillustration.Andthusitwillbeknownwhetherthemultiplicationiscorrect:indeedthe
figuresthatareintheupper37,namelythe3andthe7,areadded;therewillbe10fromwhichissubtracted9;therewillremain1whichiskept.Againin
The 9604residue 98is1. 98
2.HereBeginsChapterTwo 25
thesamewaythefiguresofthelower37areadded,andthence9issubtracted;therewillremainlikewise1;thereforethe1whichremainsfromtheupper37andthe1whichremainsfromtheloweraremultiplied;1ismadewhichiscalledtheresidue,anditiskeptinthetableabovethemultiplication,asisdisplayedinthethirdillustration;afterwardsthefiguresthatareintheproductofthemultiplicationareadded,andfromtheproductsasmanymultiplesof9aresubtractedaswillbepossible,andif1willremainforthekeptresiduecertainlythemultiplicationwillbecorrect.Forexample,ifweaddthefiguresthatareintheproductofthemultiplication,namelythe1,3,6,and9,theywillmake19,fromwhichyousubtracttwicenine;therewillremain1fortheresidueaswesaidbeforemustremain;orfromthesaid19onetakes9thatisinthefirstplaceofit;therewillremainlikewise1.Andyounotewhenthefiguresof37,namelythe3andthe7areadded,thenifyoudividethe37by9,fromwhichdivisionremains1,thesameresultsasthereremainsfromthe10thatisproducedfromtheadditionofthe3andthe7,andfromthiswillbetaken9;fortheremainderthatisleftfromwhateverthenumberdividedby9,thatisthesumthatisputfromtheadditionofallthefiguresmakingthatsamenumber.Andnotingagain,asanynumberisdividedintoparts,andanysuchpartsaremultipliedbyanothernumber,themultiplicationintotalisequaltothesumofalltheproductsofthenumberseparatedintoparts.Thereforetheproductof36and37,addedtotheproductof1by37,isequaltotheproductof37by37.Butfromthemultiplicationof36by37resultsthenumberthatiscreatedfromamultipleofnine,as36isbuiltfromnines.Thereforethenumberarisingfromthe36timesthe37, ifitisdividedby9,nothingfromitwillremainindivisible.Alsothemultiplicationof1by37isequaltothesumofthemultiplicationof1by36andof1by1.Butthemultiplicationof1by36yieldsanumberthatisintegrallydivisibleby9;themultiplicationthereforeof1by1,namely1,remainsindivisibleby9.Thereforeoftheproductof37by37dividedby9thereremains1whichishadfromtheadditionofallthefiguresthatareintheproductof37by37,as[p9]wefoundabove;oriffromthesaidproductisdeleted9,thentherewillremain136,fromwhichisdeleted3and6,whichmakeasumof9;therewilllikewiseremain1;1369isindivisibleby9.Alsoifyouwishtomultiply98by98,thenthe98iswrittenbelowthe98as
I saidbefore;the8ismultipliedbythe8;therewillbe64:the4isputoverboth8,andthe6isheldinhandforthetens;andthe8ismultipliedbythe9;therewillbe72;andagainsymmetricallythe8inthelowerismultipliedbythe9in'---------'theupper;therewilllikewisebe72whichisaddedwiththeother72andaddedtothe6keptinhand;therewillbe150inhand;andastherearenounitsinthe150,azephirisputaboveboth9,andthe15whichisheldinhandforthetens;andthe9ismultipliedbythe9;therewillbe81whichisaddedtothe15keptinhand;therewillbe96fromwhich96the6iswritteninthethirdplaceandthe9inthefourth,asisdisplayedintheillustration.Weshallseeinthiswaywhetherthismultiplicationiscorrect;thefiguresoftheupper98,namelythe9andthe8,areadded,and9issubtracted;therewillremain8.Againthissamethingisdonewiththelower98;therewillremainlikewise8;andthe8ismultipliedbythe8;therewillbe64fromwhicharesubtractedalltheninesthat
[J]9007070The 1813
'residue 37is4. 49
26 II.LiberAbaci
areinthe64;therewillremain1fortheresidue;orinanotherway,thefiguresthatareintheaforesaid64areadded,namelythe6andthe4;therewillbe10fromwhich9issubtracted;therewilllikewiseremain1;afterwardsareaddedthefiguresthatareintheproductofthemultiplication,namelythe9,6,0,and4;howeveritisnotnecessarytoaddthefigurenineinallsuchsimilarchecks;withninesthesubtractionwillalwaysbedoneinadvance,whencethe6,0,and4areadded;therewillbe10fromwhich9issubtracted;therewillremain1forresidue,ashadtoremain.Andmoreoverifyouwillwishtomultiplyanynumberoftwoplacesnothavingunitsintheminthefirstplace,asin10or40or90,inwhichplacethezephirisalwaysnecessary,thenitwillbedonethus:youwritethenumberasIsaidabove,andthesecondplaceismultipliedbythesecond,andtwozephirareputbeforetheproduct,andthusweshallhavetheproductofanysuchgivenmultiplication.Ifyouseekthemultiplication70by70,thenboth70arethereforewritteninthemannergivenabove,andthefigureseventhatisinthesecondplaceintheuppernumberismultipliedbythe7inthelower;therewillbe49,beforewhichnumberthetwozephirareput,namelyforthosethatarebeforeeach7;4900ismadewhichistheproductofthesoughtmultiplication.Ifthemultiplicationof37and49issought,thenthe49iswrittenbeneaththe37,namelythelargernumberbeneaththesmaller,andthesameplacesbeneaththesameplaces,asisdisplayedinthemargin;andthe7ismultipliedbythe9;therewillbe63;the3isputabovethe7,andthe6iskeptinhandforthetens;andthe7iscrosswisemultipliedbythe4;therewillbe28whichisaddedtothe6heldinthehand;therewillbe34.Alsothe9ismultipliedbythe3;therewillbe27whichisaddedtothe34;therewillbe61;the1isputabovethe3,andthe6iskeptinhandforthetens;the3ismultipliedbythe4;therewillbe12whichoneaddstothe6;therewillbe18whichisputafterthe13intheupperposition;thisyields1813fortheproductofthegivenmultiplication,asisshownhere.Andthusitwillbeknownifthemultiplicationiscorrect:the37isdivided
by9;thatis,thefiguresin37areadded,namelythe3andthe7;therewillbe10fromwhich9issubtracted;therewillremain1whichiskept;likewisethefiguresin49areadded,namelythe4andthe9;therewillbe13fromwhich9issubtracted;therewillremain4whichismultipliedwiththekept1;therewillbe4whichiskeptfortheresidue,andthefiguresthatareintheproductofthemultiplicationareadded,namelythe1,8,1,and3;therewillbe13fromwhich9issubtracted;therewillremain4,asoughttoremainfortheresidue.OneproceedsinthemannerIspokeofabove,multiplyingbyseparating
numbersinparts,andsowiththemultiplicationofallsuchlargenumbers.Andthemultiplicationof37by49,[plO]isequaltothesumofthemultiplicationsofthe7bythe49andthe30bythe49.Butthemultiplicationof7by49isequaltothesumofthemultiplicationsofthe7bythe9andofthe7bythe40,andagainthemultiplicationof30by49isequaltothemultiplicationsofthe30bythe9,andofthe30bythe40.Thereforethemultiplicationof37by49isequaltosumoffourmultiplicationsthatare7by9,and7by40,and30by9,and30by40.AndtheIIII multiplicationsabovearetakeninorder:weshallmultiplyfirstthe7bythe9,andweput theunitsabovethefirstplacebecause
2.HereBeginsChapterTwo 27
whenthefirstplacemultipliesanyplaceitmakesthesameplaceoritendsinit.Second,wemultiplythe7bythe4;third,the9bythe3,andwetakethesumoftheseproducts;weputtheunitsinthesecondplacebecausewhenthefirstplacemultipliestheseconditmakesthesecondplace.Andnowaremultipliedthe7bythe40,andthe9bythe30;afterwardsattheendwemultiplythe3bythe4,namelythesecondplacebythesecond.Andtothisproductisaddedthekepttens;weputtheunitsinthethirdplace,andthetenswemadeaboveinthefourth;andthisishow30ismultipliedby40becausewithanysecondplaceonemakesthesecondplaceafteritthatismultiplied.Similarlythethirdplaceofanumberofanynumberofplacesthatonemultiplies,onemakesthethirdplaceafteritthatonemultiplies.Andthefourthmakesthefourthafteritthatonemultiplies,andthefifth,thefifth,andsoitis.Whichisthereforetosay,anyfirstplacewhatsoeveronemultiplies,thesameonemakes,asmakesanumberendinginitself.Andfigurebyfigureismultiplied,andfromthemultiplicationyieldsnotthelast.Alsothemultiplicationmakesthesameplace;andfromthemultiplicationofitresultsanumberoftwoplaces,as20or30,orcomposedfromthesecondandfirstas15and28;thenonemakesthenumberendinthesameplacethatthefirstplacemultiplies;andonthataccountwhenwemultiplythefirstplacebyanyplaceweputtheunitsofthatmultiplicationabovethesameplace,andthetenswekeepforthefollowingplace,hereyoulearnthesameofthemultiplicationoftheremainingplaces.
On the Multiplication of One Figure with Many.
Alsoifthemultiplicationofonefigurewithtwo,orwithmany,issought,thentheonefigureiswrittenabovethefirstplaceinthenumberthatonewillwishtomultiply,andtheonefigurealoneismultipliedbythefirstplaceofthenumber,andtheunitsareputoverit,andthetensarekeptinhand;andtheonefigureismultipliedbythesecondofthelowernumber,anditisaddedtothekepttens,andalwaystheunitsareput,andthetensarekept;andthesamefigureismultipliedinorderbythethirdandthefourth,andtheothersbysteps.Forexample,ifthemultiplicationof8with49issought,the8isputabovethe9,andthe8ismultipliedbythe9:therewillbe72;the2isputabovethe8andthe7iskeptinhand;andthe8ismultipliedbythe4;therewillbe32,andoneaddsthe7keptabove;therewillbe39,andthe9andthe3areput;392isyieldedforthesaidmultiplication,asinthemarginisshown.Alsoifthemultiplicationof7with308issought,thenthe7iswrittenabovethe8,andthe7ismultipliedbythe8;therewillbe56;the6isput,andthe5iskept;andthe7ismultipliedbythe0whichmakes0,whichoneaddstothekept5making5;anditisputaftertheput6;andonemultipliesthe7bythe3whichmakes21,andoneputsitaftertheput56;and2156resultswhichistheproductofthesaidmultiplication,andthusonefigurebyseveralismultiplied.
f39Il~1215~ I308
UJ6707081
First 345345
Second 5345345
Third 25345345
Fourth 025345345
Fifth 9025345345
Last 119025345345
28 II.LiberAbaci
On the Same.
Alsoifonewillwishtomultiply70by81,the0isdeletedfromthe7;thereisleftthe7,andthe7ismultipliedbythe81;[pH]therewillbe567whichisputbeforethenumber0whichweremovedfromthe70;therewillbe5670.
Here Begins the Second Part of the Second Chapter.
Howeverwhenonewillwishtomultiplythreefiguresbythreefigures,thenweeasilyteachauniversalruleforit.Namely,theplaceofonenumberiswrittenagainundertheplaceofanother,thatistheunitsbelowtheunits,thetensbelowthetens,andthehundredsbelowthehundreds;thefirstintheuppernumberismultipliedbythefirstinthelower,andtheunitsareputabovethefirstplacesofthenumbers,andthetensarekeptinhand;andonemultipliesthefirstintheupperbythesecondinthelower,andthefirstinthelowerbythesecondintheupper,andtheproductsandthekeptunitsareadded,andtheunitsareputandthetenskept;andonemultipliesthefirstintheupperbythethirdinthelower,andthefirstinthelowerbythethirdintheupper,andthesecondbythesecond,andthethreesaidproductsandthekeptnumberareadded;andtheunitsareputabovethethirdplace,andanytensarekeptinhand;andthesecondintheuppernumberismultipliedbythethirdinthelower,andthesecondinthelowerbythethirdintheupper;andfromtheaddedproductstheunitsareputandthetenskept;andthethirdismultipliedbythethird,anditisaddedtothekepttens;andtheunitsareput,andthetensareputafterwardsiftheyoverflowtheunits;andthuswillbehadthemultiplicationofanynumbersofthreefigures,whethertheyareequalorunequal.Evidentlyinthiscategoryaretheequalnumbers345and345,whichareto
bemultipliedtogether,andarelocatednexttoeachotherasdisplayedonthispage;andonemultipliesthe5bythe5;therewillbe25;the5isputaboveboth5asisdisplayedinthesecondillustration,andthe2iskeptinhandforthetens;andthe5intheuppernumberismultipliedbythe4inthelower,andthe5belowbythe4above;theproductsareaddedtothekept2;therewillbe42;the2isputaboveboth4,asiscontainedinthethirdillustration,andthe4iskeptforthefourtens;andthe5intheupperismultipliedbythe3inthelower,andthe5inthelowerbythe3intheupper,andthe4bythe44,andtheproductsofthethreemultiplicationsareaddedwiththe4keptinhand;therewillbe50;the0isputaboveboth3,asisshowninthefourthillustration,andthe5iskeptinhand;andonemultipliesthe4intheupperbythe3inthelower,andthe4inthelowerbythe3intheupper,andtheyareaddedwiththe4;therewillbe29;the9isputafterthe0,asisevidentinthefifthillustration,andthe2iskeptinhand;andthe3ismultipliedbythe3;therewillbe9whichoneaddswiththe2;therewillbeHthatisput,asisshowninthesixthandlastillustration.Andbytheabovesaidmethodsitwillbeverifiedwhetherthemultiplicationiscorrect;namelythefiguresofthe345aboveare
2.HereBeginsChapterTwo 29
added,andthence9issubtracted;therewillremain3;onedoessimilarlywiththe345belowandtherewillremainsimilarly3;andthe3ismultipliedbythe3fromwhich9issubtracted;thereremains0whichonehasfortheresidue;thenthefigureswhichareintheproductofthemultiplicationareadded,namelythe1,1,2,and5;therewillbe9,fromwhich9issubtracted;0remainsasoughttoremain.WhereforeIshalldeclare,infact,thatthemultiplicationofthesecondfigurebythesecondisaddedtothemultiplicationofthefirstfiguresbythethirdsbecause,aswassaid,thefirstplacemultipliesanyplacetomakethesameplace,andthesecondplacemultipliesanyplacetomaketheplaceaftertheplacewhichismultiplied.Andthusisthis,whenthefirstplaceismultipliedbythethird,thenthethirdplaceismade.Andwhenthesecondismultipliedbythesecond,thesameasbeforeismade,namelythethird,aftertheonethatismultiplied.Thereforetothemultiplicationofthesecondplacebythesecondplacemustbeadded[p12]theproductsofthefirstsbythethirds.It isfollowedbytheproductofthesecondfiguresbythethirds,fromwhichresultsthefourthplace,namelytheonethatfollowsthemthataremultiplied.Forthelastthethirdplaceismultipliedbythethird,fromwhichmultiplicationresultsthefifthplace,namelythethirdtothatwhichthethirdplacemultiplies.Andforthisreason,fromthisthatiscreatedfromthemultiplicationoffirstsbythirdsandsecondbysecond,weputtheunitsinthethirdplace,andwekeepthetensforthefourthplace.Andfromthisthatiscreatedfromthemultiplicationofthesecondsbythethirds,andfromthekepttensweputtheunitsinthefourthplace,andwekeepthetensforthefifthplace,andthetensareaddedtotheproductofthethirdplacebythethird,anditisputinthefifthplace,andthetensinthesixth,andthusishadtheabovemultiplication.
On the Same.
Alsoifitwillbewishedtomultiply607by607,thenthenumbersareadjacentlylocated;the7ismultipliedbythe7;therewillbe49;the9isput,andthe4iskept;andthe7ismultipliedbythe0andincrossing,the0bythe7;andthekept4isadded;therewillbe4whichoneputs;andthe7ismultipliedbythe6,andthe7bythe6,andthe0bythe0;therewillbe84;the4isput,andthe8iskept;andthe0ismultipliedbythe6,andthe0bythe6,andthezephirisaddedwith8;therewillbe8,andthe8isput,andthe6ismultipliedbythe6;therewillbe36;the6isput,andthe3,andthusyouwillhave368449fortheproductofthesaidmultiplication.
On the Same.
Alsoifitwillbewishedtomultiply780by780,thenthezephiraredeletedfromboth780;therewillremain78and78;andthe78ismultipliedbythe78;therewillbe6084beforewhichareputthetwozephir,and608400willbehadfortheproductofthesaidmultiplication.Alsoifitwillbewishedtomultiply900by900,thenthezephiraredeletedfromeachnumber,andthe9ismultipliedbythe9;therewillbe81beforewhichfourzephirareput,namelyforthefourdeletedzephirofboth900,and810000willbehadfortheproductofthesaidmultiplication.
368449607607
6084007878
The 56088residue 123
is O. 456
16687037451
1824003257
30 II.LiberAbaci
On the Same with Unequal Numbers.
Howeverifitwillbewishedtomultiplyunequalnumbers,thentheywillbemultipliedinthesamewayandorder;andifonewillhave123and456tomultiply,thenonenumberaftertheotheriswritten,aswassaidabove;andthe3ismultipliedbythe6;therewillbe18;the8isput,andthe1iskept;andthe3ismultipliedbythe5;therewillbe15whichisaddedwiththekept1;therewillbe16;andthe6timesthe2isaddedwith16;therewillbe28;the8isput,andthe2iskept;andthe3ismultipliedbythe4,andthe6bythe1,andthe2bythe5,andthesumisaddedwiththekept2;therewillbe30;the0isputandthe3iskept;andthe2ismultipliedbythe4,andthe5bythe1,andthesumisaddedwiththekept3;therewillbe16;the6isput,andthe1iskept,withwhichisaddedtheproductof1by4;therewillbe5whichisput,and56088willbehadfortheproductofthesaidmultiplication.Howeverifitwillbewishedtocheckthis,thenthefiguresof123areadded;therewillbe6,andthefiguresof456areadded;therewillbe15fromwhichnumber9issubtracted;therewillremain6,whichismultipliedby6;therewillbe36whichisdividedby9;thereremains0whichishadfortheresidue.Thenthefigureswhichareintheproductofthesaidmultiplicationareadded;therewillbe27whichisdividedbynine;thereremains0,asoneexpectstoremainfortheresidue.Alsoifitisproposedtomultiply370by451,thentheycanbemultipliedbytheabovesaidinstruction;howeversincethezephirisinthefirstplaceofoneofthenumbers,namelyofthe370,themultiplicationistaughtinanotherway,namelythesame0isdeletedfromthe370;therewillremain37whichismultipliedbythe451;therewillthereforebethemultiplicationoftwofiguresbythree,whichmultiplicationisyettobe[pI3]taught.The37iswrittenabovethe51ofthe451,andthe7ismultipliedbythe1;therewillbe7thatisput.Andthe7ismultipliedbythe5,andthe1ismultipliedbythe3;therewillbe38;the8isputandthe3iskept;andthe7ismultipliedbythe4,andthe3bythe4,andthesumisaddedtothekept3;therewillbe46;the6isputandthe4iskept;andthe3ismultipliedbythe4,andtheproductisaddedtothekept4;therewillbe16,andthe6andthe1areput,andweshallhave16687fortheproductofthesaidmultiplicationoftwofiguresbythree;whichisputbeforethe0,the0deletedfrom370;therewillbe166870;thereforeinthiswayanytwofiguresbyanythreefiguresaremultiplied.Alsoifthemultiplicationof320by570issought,then0isdeletedfromeachnumber;therewillremain32and57;thesenumbersaremultipliedtogether;therewillbe1824whichisputbeforetwozephir,and182400willbehadfortheproductofthesaidmultiplication.
The Third Part on the Multiplication of Four Figures.
Howeverwhenitwillbewishedtomultiplyfourfiguresbyfour,thenthenumbersarewritten,andsimilarplacesarelocatedbelowsimilarplaces;the
2.HereBeginsChapterTwo 31
firstismultipliedbythefirstanditisput,rememberingthenalwaystokeepthetens,andtheunitsareput,andthefirstismultipliedbythesecond,andthefirstbythesecond,andtheyareput;andthefirstbythethird,andthefirstbythe third,andthesecondbythesecond,andtheyareput;andthefirstbythefourth,andthefirstbythefourth,andthesecondbythe third,andthesecondbythethird,andtheyareput;andthesecondbythefourth,andthesecondbythefourth,andthethirdbythe third,andtheyareput;andthethirdbythefourth,andthethirdbythefourth,andtheyareput;andthefourthbythefourthanditisput;andthuswillbehadthemultiplicationofanynumbersoffourfigures,whethertheywillbeequalorunequal.Evidentlyinthiscategorythemultiplicationof1234byitselfisproposed,
andIwritedownthenumber;againthefirstismultipliedbythefirstaswesaidbefore,namelythe4bythe4;therewillbe16;andthe6isputoverboth4,andthe1iskept;andthe4ismultipliedbythe3,andthe4bythe3,andtheyareaddedtothekept1;therewillbe25;the5isputaboveboth3,andthe2iskept.Alsothe4oftheuppernumberismultipliedbythe2ofthelower,andthe4bythe2,andthe3bythe3,andtheproductsareaddedtothekept2;therewillbe27;the7isputaboveboth2,andthe2iskept;the4ismultipliedbythe1,andthe4bythe1,andthe3bythe2,andthe3bythe2,andthesefourproductsareaddedtothekept2;therewillbe22;the2isputaboveboth1,and2iskeptinhand;andthe3ismultipliedbythe1,andthe3bythe1,andthe2bythe2,andtheproductsareaddedtothekept2;therewillbe12;the2isput,andthe1iskeptinhand;andthe2ismultipliedbythe1,andthe2bythe1,andtheproductsareaddedtothekept1;therewillbe5whichisput;andthe1ismultipliedbythe1;therewillbe1whichisput;andthus1522756willbehadfortheproductofthemultiplication.
On the Same.
152275612341234
Againastheinformationisunderstood,themultiplicationof2345by6789isproposed;thereforethenumbersarewrittendown;the5ismultipliedbythe9;therewillbe45;the5isput,andthe4iskept;andthe5ismultipliedbythe8,andthe9bythe4,andtheproductsareaddedtothekept4;therewillbe80;the0isputandthe8iskept;andthe5ismultipliedbythe7,andthe9bythe3,andthe4bythe8,andtheproductsareaddedtothekept8;therewill""--r-h-e-1-59-20-2-05---'be102;the2isput,andthe10iskeptinhand;andthe5ismultipliedbytheresidue 23456,andthe9bythe2,andthe4bythe7,andthe8bythe3,andtheproductsis6. 6789areaddedwiththekept10;therewillbe110;the0isput,andthe11iskept;L- --'andthe4ismultipliedbythe6,andthe8bythe2,andthe3bythe7,andtheproductsareaddedwiththekept11;therewillbe72;the2isput,the7iskept;andthe3ismultipliedbythe6,andthe7bythe2,andtheproductsareaddedtothekept7;therewillbe39;the9isput,andthe3iskept,whichisaddedtotheproduct[pI4]ofthe2bythe6;therewillbe15,andthe5andthe1areput,andthuswillbehadthemultiplicationofthesaidnumbers,ashereisshown.
9252500253701
32 II. LiberAbaci
The Check.
Andthusitischeckedwhetherthemultiplicationiscorrect:theresidueof2345,whichis5,ismultipliedbytheresidueof6789,whichis3;therewillbe15fromwhichissubtracted9;thereremains6,anditistheresidueoftheproductofthemultiplication.Althoughitissaidallnumbersoffourfiguresaremultipliedthus,thereare
howeveramongthemthosewhichcanbemultipliedinanotherandeasierway,namelythosewhichhaveattheirheadzephir;andifthemultiplicationof5000and7000issought,thenthe5ismultipliedbythe7;therewillbe35,beforewhichisputasmanyzephirasareinthenumbers,whicharesix,andthus35000000willbehadfortheproductofthesaidmultiplication.Alsoifthemultiplicationof5100by7430issought,thenthe51ismultiplied
bythe743;therewillbe37893,beforewhichareputthethreezephirwhichareattheheadofbothnumbers,andthus37893000willbehadfortheproductofthesaidmultiplication.Alsoifthemultiplicationof2500and3701issought,onedeletesthetwo
zephirthatareattheheadof2500;therewillremain25whichonemultiplieswith3701,namelythetwofigureswiththefour,whichinturnisthis;onewritesthe25abovethe3701,asisdisplayedbelow,andonewillmultiplythe5bythe1;therewillbe5whichoneputs,andthe5bythe0,andthe1bythe2;therewillbe2whichoneputs;andthe5bythe7,andthe2bythe0;therewillbe35;oneputsthe5,andkeepsthe3;andonemultiplies5by3and2by7andoneaddstheproductswiththekept3;therewillbe32;andthe2isput,the3iskept,andthe2bythe3;therewillbe6whichoneaddswiththekept3;therewillbe9whichoneputs.Andthus92525ishadforthemultiplicationof25by3701,asisshownintheillustration,beforewhichisputtwozephir,andtheproductofthemultiplicationsoughtbeforewillbehad.
The Fourth Part of the Second Chapter.
Howeverwhenitwillbewishedtomultiplyanynumberoffivefiguresbyanynumberofthesamenumberofplaces,namelyfivefiguresbyfive,onemultipliesthelocatednumbersfirstplacebyfirst,andoneputs;andthefirstbythesecond,andthefirstbythesecond,andoneputs;andthefirstbythethirdandthefirstbythethird,andthesecondbythesecond,andoneputs;andthefirstbythefourth,andthefirstbythefourth,andthesecondbythethird,andthesecondbythethird,andoneputs;andthefirstbythefifth,andthefirstbythefifth,andthesecondbythefourth,andthesecondbythefourth,andthethirdbythethird,andoneputs;andthesecondbythefifth,andthesecondbythefifth,andthethirdbythefourth,andthethirdbythefourth,andoneputs;andthethirdbythefifth,andthethirdbythefifth,andthefourthbythefourth,andoneputs;andthefourthbythefifth,andthefourthbythefifth,andoneputs;
2.HereBeginsChapterTwo 33
andthefifthbythefifth,andoneputs.Andthusthemultiplicationofanynumbersoffiveplaces;andasthisisevidentlydemonstrated,amultiplicationisproposed,andforthose,equalorunequalmultiplicationsofthesameplacesareperceived:ifonewillwishtomultiply12345by12345,thenumbersarewrittendown,asistaughtabove;onemultipliesthe5bythe5;therewillbe25;oneputsthe5andkeepsthe2;andthe5bythe4,andthe5bythe4,andoneaddstheproductstothekept2;therewillbe42;oneputsthe2,andkeepsthe4;andthe5bythe3,andthe5bythe3,andthe4bythe4,andoneaddstheproductstothekept4;therewillbe50;oneputsthe0,andkeepsthe5;andthe5bythe2,andthe5bythe2,andthe4bythe3,andthe4bythe3,andoneaddstheproductstothekept5;therewillbe49;oneputsthe9,andkeepsthe4;andthe5bythe1,andthe5bythe1,andthe4bythe2,andthe4bythe2,andthe3bythe3,andoneaddstheproductstothekept4;therewillbe39;oneputsthe9,andkeepsthe3;andthe4bythe1,andthe[pI5]4bythe1,andthe3bythe2,andthe3bythe2,andoneaddsthemtothekept3;therewillbe23;oneputsthe3,andkeepsthe2;andthe3bythe1,andthe3bythe1,andthe2bythe2,andoneaddstheproductstothekept2;therewillbe12;oneputsthe2,andkeepsthe1;andthe2bythe1,andthe3bythe1,andoneaddstheproductstothekept1;therewillbe5,thatoneputs;andthe1bythe1willbe1,whichisput;andthustheproductofthesaidmultiplicationwillbehad.AgainIshallshowthiswayofmultiplyingtoproceedfromthatwhichoccursamongnumberswhichareproportional.Forifthreenumbersareproportional,asthefirstistothesecond,soisthesecondtothethird; thentheproductofthefirstbythethirdisequaltotheproductofthesecondbyitself.Andiffournumbersareproportional,asthefirstistothesecond,soisthethirdtothefourth.Thentheproductofthefirstbythefourthisequaltotheproductofthesecondbythethird,asisfoundinEuclid.Anumbertrulyascendsthroughconnectedplaceswithoutend;thereforeasthefirstplaceistothesecond,sothesecondistothethird,andthethirdtothefourth,andsoeachantecedenttoitsconsequence.Therefore,theproductofthesecondplacebyitselfmakesthesameplacemadebytheproductofthefirstbythethird.Andthemultiplicationofthesecondbythethirdmakestheplacemadebythemultiplicationofthefirstbythefourth.Indeed,themultiplicationisbegunbythefiguresofthefirstplace,fromwhichmultiplicationeitherresultsanumberofthefirstplace,orendsinitself.Andforthatreasonfromthemultiplicationofthefirstfigurebythefirsttheunitsareputabovethefirstplace,andthetensarekeptforthesecond,towhichareaddedthemultiplicationsofthefirstsbytheseconds,andanumberofthesecondplaceresults,orterminatinginthesameplace.Thereforetheunitsareputabovethesecondplace,andforeachtenthatishad,1iskeptforthethirdplace.Nextthefirstismultipliedbythethird,andtheproductisaddedtothemultiplicationofthesecondbythesecondbecausethemultiplicationofthesecondplacebythesecondmakesthesameplacethatismadebythemultiplicationofthefirstplacesbythethirds.Andforthatreasonfromthemultiplicationofthefirstfiguresbythethirds,andthesecondsbytheseconds,theunitsareputabovethethirdplace;afterthis,thefirstismultipliedbythefourth,andthesecondsbythethirds,asareinthe
1523990251234512345
34 II.LiberAbaci
fourproportionalplacesbecauseasthefirstistothesecond,soisthethirdtothefourth,andfromthesamemultiplicationsresultsanumberterminatinginthefourthplace.Andforthatreasontheunitsareputabovethefourthplace,andafterwardsthefirstsbythefifthsaremultiplied,andthesecondsbythefourths,andthethirdsbythethirdsbecauseasisthefirstplacetothesecond,soisthefourthtothefifth.Becausethemultiplicationofthesecondplacebythefourthmakestheplacemadefromthemultiplicationofthefirstbythefifth,namelythefifthplace;andagainasissecondplacetothethirdsoisthethirdtothefourth.Thereforethemultiplicationofthethirdplacebythethirdmakestheplacemadebythemultiplicationofthesecondsbythefourths,namelythefifthplace.Andforthatreasontheunitsareputoverthefifthplace,andthus,followingproportionality,theproductiseffectedforthemultiplicationofanynumbers.Andthiscanbemanifestlyunderstoodbythisthatfollows.Andnotingforthatreasonasthefirstplaceistothesecond,soisthepenultimatetothelast;andasthefirstistothethird,soisthethirdfromthelasttothelast;andasthefirstistothefourth,soisthefourthfromthelasttothelast,andsoforth.Inthisfollowingmultiplicationoffivefiguresbyfive,afterputtingthefivefiguresabovethefive,thesecondsbythefifthsaremultiplied,andthethirdsbythefourths;andthemultiplicationsgotomakethesixthplace;andthesecondplacemultipliesthefifth,whichhelpsmakethesixthplace,andonemakesthemultiplicationofthethirdsbythefourths,andasisthesecond[p16]placetothethird,sothefourthtothefifth.Nextthethirdsaremultipliedbythefifths,andthefourthbythefourth,andtheseventhplaceresultsbecausewiththethirdplaceonemultipliesthefifth,onemakeswiththethirdplaceandthefifth,namelytheseventh;nextthefourthsaremultipliedbythefifths,whichmaketheeighthplace.Tothelast,thefifthismultipliedbythefifth,whichmakestheninthplace;andthustheproductofthesaidmultiplicationishad.Indeedafterthiswhateverissaidaboutthemultiplication,whateveringenuityonecanhavefortheabovesaidmultiplicationinstruction,howeverunskilledisthecompletedinstruction,Imanagedtoshowthemultiplicationofeightplaces.
The Fifth Part of the Second Chapter.
Howeverwhenanyonewillwishtomultiplyanynumberofeightfiguresbyanynumberofthesamenumberofplaces,hemultipliesthefirstbythefirst,andheputstheresult;andthefirstbythesecond,andthefirstbythesecond,andheputsthesum;andthefirstbythethird,andthefirstbythethird,andthesecondbythesecond,andheputsthesum;andthefirstbythefourth,andthefirstbythefourth,andthesecondbythethird,andthesecondbythethird,andheputsthesum;andthefirstbythefifth,andthefirstbythefifth,andthesecondbythefourth,andthesecondbythefourth,andthethirdbythethird,andheputsthesum;andthefirstbythesixth,andthefirstbythesixth,andthesecondbythefifth,andthesecondbythefifth,andthethird
2.HereBeginsChapterTwo 35
bythefourth,andthethirdbythefourth,andheputsthesum;andthefirstbytheseventh,andthefirstbytheseventh,andthesecondbythesixth,andthesecondbythesixth,andthethirdbythefifth,andthethirdbythefifth,andthefourthbythefourth,andheputsthesum;andthefirstbytheeighth,andthefirstbytheeighth,thesecondbytheseventh,andthesecondbytheseventh,namelythosethatarewiththefirstandtheeighth,andthethirdbythesixth,andthethirdbythesixth,andthosethatarewiththesecondsandthesevenths,andthefourthbythefifth,andthefourthbythefifth;andsoonwiththosethatarewiththethirdandthesixth,andheputsthesum.Andthusalwaysinallmultiplicationsthefiguresthatemergefromtheinteriorpartsaremultipliedalternatelyfrombothparts;thusmultiplyingonebytheothertheyareaddedtogether;andthentheunitsareputandthetenskeptinhand.Andwiththemultiplicationofthefirstfigures,ascendinginorderintherestoftheplacestheyarecompleteduptothelast;thenthefirstfiguresofbothnumbersareleftcompletelybehind,andthesecondbythelastmultiplied,thatis,inthisproblemonemultipliesthesecondbytheeighth,andthesecondbytheeighth,andthethirdbytheseventh,andthethirdbytheseventh;whichareaddedwiththesecondandtheeighth;andthefourthbythesixth,andthefourthbythesixthwhichareaddedwiththethirdandtheseventh;andthefifthbythefifthwhicharebetweenthefourthandthesixth,andoneputsthesum;andthenthesecondsareleft;andonemultipliesthethirdbytheeighth,andthethirdbytheeighth,andthefourthbytheseventh,andthefourthbytheseventh,andthefifthbythesixth,andthefifthbythesixth,andoneputsthesum;andthethirdsareleft,andonemultipliesthefourthbytheeighth,andthefourthbytheeighth,andthefifthbytheseventh,andthefifthbytheseventh,andthesixthbythesixth,andoneputsthesum;andthefourthsareleft,andonemultipliesthefifthbytheeighth,andthefifthbytheeighth,andthesixthbytheseventh,andthesixthbytheseventh,andoneputsthesum;andthefifthsareleft,andonemultipliesthesixthbytheeighth,andthesixthbytheeighth,andtheseventhbytheseventh,andoneputsthesum;andtheseventhbytheeighth,andtheseventhbytheeighth,andoneputsthesum;andtheeighthbytheeighth,andoneputstheresult;andthusthemultiplicationofallnumbersofeightfigureswillbehad;anditwillbeclearlyunderstoodinnumbers;letthenumbersbe12345678and87654321,whicharemultiplied[pI7]onebytheotherasisdescribedfollowingthatwhichissaidabove;and'-10-8-21-52-02-23-7-46-38--'onemultipliesthe8bythe1;therewillbe8thatoneputs;andthe8bythe2, 12345678andthe1bythe7;therewillbe23;oneputsthe3andkeepsthe2;andthe8 87654321bythe3,andthe1bythe6,andthe7bythe2,andtheproductsareadded'-- ....--Jwiththekept2;therewillbe46;the6isput,andthe4iskept;andthe8bythe4,andthe1bythe5,andthe7bythe3,andthe2bythe6addedwiththekept4willbe74;the4isput,andthe7iskept;andthe8bythe5,andthe1bythe4,andthe7bythe4,andthe2bythe5,andthe6bythe3addedwiththekept7willbe107;the7isput,andthe10iskept,andthe8bythe6,andthe1bythe3,andthe7bythe5,andthe2bythe4,andthe6bythe4,andthe3bythe5,addedwiththekept10willbe143;the3isputandthe14iskept;andthe8bythe7,andthe1bythe2,andthe7bythe6,andthe
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2bythe3,andthe6bythe5,andthe3bythe4,andthe5bythe4addedwiththekept14willbe182;the2isput,andthe18iskept;andthe8bythe8,andthe1bythe1,andthe7bythe7,andthe2bythe2,andthe6bythe6,andthe3bythe3,andthe5bythe5,andthe4bythe4addedwiththekept18willbe222;the2isput,andthe22iskept;andthe7bythe8,andthe2bythe1,andthe6bythe7,andthe3bythe2,andthe5bythe6,andthe4bythe3,andthe4bythe5addedtothekept22willbe190;the0isput,andthe19iskept;andthe6bythe8,andthe3bythe1,andthe5bythe7,andthe4bythe2,andthe4bythe6,andthe5bythe3addedtothekept19willbe152;the2isput,andthe15iskept,andthe5bythe8,andthe4bythe1,andthe4bythe7,andthe5bythe2,andthe3bythe6addedtothekept15willbe115;the5isput,andthe11iskept;andthe4bythe8,andthe5bythe1,andthe3bythe7,andthe6bythe2addedtothekept11willbe81;the1isput,andthe8iskept;andthe3bythe8,andthe6bythe1,andthe2bythe7addedtothekept8willbe52;the2isput,andthe5iskept;andthe2bythe8,andthe7bythe1addedtothekept5willbe28;the8isput,andthe2iskept;andthe1bythe8addedtothekept2willbe10,thatisput;thustheproductofthesaidmultiplicationwillbehad.'frulyiftherearezephirattheheadsofanynumbers,andallofthezephir
thatexistattheheadsaredeletedfromthenumbers,andtheremainingfiguresaremultipliedtogether,andthedeletedzephirareputbeforetheproductwith-outthezephir,thentheproductofthemultiplicationwillbehad,aswedenotedinthemultiplicationsinthesecond,third,andfourthplaces;andifitisnotknownhowtomultiplyafewfigureswithmanybytheabovedemonstrationsofmultiplications,thenthenumbersarewrittendown,greaterbelowlower,thatisthenumberofmanyfiguresbelowthenumberwithfew,locatinginthefirstplaceofonebelowthefirstoftheother,andoneaftertheother,aswesaidabove,everyplaceproperlylocated;andthereareputafterthenumberoffewfiguresasmanyzephirasthefiguresofthegreaternumberexceedthelesser,andthusonewillhaveequallysizednumbersinmultiplication;andifonewillseektomultiplythreefigureswithsix,oneputsthenumberofsixfiguresbelowthenumberofthreefigures,andoneputsthreezephirafterthethreefigures,asareinthemultiplicationofsixfigureswithsixwhichonemultipliesaccordingtotheaforesaidinstruction.Forexample,whenitfollowstomultiply345by698541onewritestheminthisorder,namely,threezephirafter345.'frulywhatissaidofthepositionofthezephirafterthefigureswillbejudgedonlyacrudenecessity,becausesubtlythepositionofsuchzephirisnotneeded.
The Sixth Part of the Second Chapter.
'frulywiththeinstructionwrittenaboveformultiplying,onewillknowhowtooperatebyfrequentuseofthetable,andhewillwishtoknowthesameinstructionbyheart,andbyhand,withoutthewrittentablefornumbersoftwoandthreeplaces;hewillkeepinmemorythewritingofthenumbersthathewillwishtomultiply,andhewillbegintomultiplyaccordingtotheprescribed
2.HereBeginsChapterTwo 37
order,andhewillputinthefirstpositioninthelefthandtheplaceoftheunits,andinthesecondposition,namelyinthesamehandtheplaceofthetens.Thethirdhetrulyputsintherighthandtheplaceofthehundreds.Hetrulystrivestolearntoputinthefourththeplaceofthethousands.Hetrulykeepsthefifthandafterwardsinmemory;onecannotkeepitinhand;andthusthemultiplicationofanynumberswhatsoeverwillbehad.Forexample,ifitwillbewishedtomultiply12by12,thenthewritingofthemiskeptinmemory,andthe2ismultipliedbythe2making[pI8]4,andthe4oneputsinthelefthandintheplaceoftheunits,andonemultipliesthe2fromtheupper12bythe1fromthelower,andthe2ofthelowerbythe1fromtheupper,andoneaddsthemtogether;therewillbe4whichoneputsinthesamelefthandintheplaceofthetens,thatisinthesignforforty;andonemultipliesthe1bythe1,namelythesecondfigurebythesecondmaking1whichoneputsintherighthandintheplaceofthehundreds.Andthus144willbehadforthesoughtmultiplication,asisdisplayedonthispage.Againifonewillwishtomultiply48by48withoutwriting,onemultiplies
the8bythe8;therewillbe64;thereforeoneputsthe4inthelefthandintheplaceoftheunits,andkeepsthe6intherighthandintheplaceofthehundreds.Andonemultipliesthe8bythe4,andthe8bythe4,andoneaddstheproductstogether;therewillbe64whichoneaddswiththe6keptintherighthand;therewillbe70;oneputsthe0,thatisnothing,inthelefthandintheplaceofthetens,andthe7onekeepsintherighthand,towhichoneaddsthemultiplicationofthe4bythe4,namely16;therewillbe23;oneputsthe3intherighthandintheplaceofthehundreds.Andoneputsthe2inthesamehandintheplaceofthethousands,thatisthesignoftwothousand.Andthus2304willbehadforthesoughtmultiplication.Alsoifonewillwishtomultiply23by57,thenonekeepsthewritinginmemory,andonemultipliesthe3bythe7;therewillbe21;oneputsthe1intheplaceoftheunitsinthelefthand,andonekeepsthe2intherighthand;andthe3bythe5,andthe7bythe2;andoneaddstheproductstothekept2;therewillbe31;oneputsthe1intheplaceofthetens,andkeepsthe3intherighthand;andthe2bythe5,andoneaddstheproducttothekept3;therewillbe13;oneputsthe3intheplaceofthehundredsintherighthand,andthe1intheplaceofthethousandsandthus1311willbehadforthismultiplication.
The VIIth Part of the Second Chapter.
Alsoifonewillwishtomultiply347by347withoutwriting,thenonemul-tipliesthe7bythe7;onekeepsthewritingofthenumbersinmemory;therewillbe49;oneputsthe9inthelefthandintheplaceoftheunits,andintherightkeepsthe4;andtwicethe7bythe4,andoneaddstheproductstothekept4;therewillbe60;oneputsthe0intheplaceofthetensinthetensinthelefthand,thatisnothing,andkeepsthe6intheright;andtwicethe7bythe3;andthe4bythe4;andaddedtogethertherewillbe64;oneplusthe4intherightintheplaceofthehundreds,andthe6onekeepsintheplaceofthe
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thousands,orinmemory;andtwicethe4bythe3,andoneaddstheproductstothe6;thereiskept0,nothingintheplace;onemultipliesthe6andoneputsthesameforthe0;andonekeepsinmemorythe3;andthe3bythe3,andoneaddstheproducttothe3keptinmemory;therewillbe12whichagainonekeeps,asonecannotputitinthehand;andthus120409willbehadforthismultiplication.Andthusifoneknowshowtokeepthenumbersin memory,inthiswayoneiseducatedtoproduceresultsmoreeasilythanwiththetable.Onewillbeabletofindthemultiplicationsofanynumbersoftwoplacesandthreeplacesusing memoryandhands.
Chapter 3
Here Begins the ThirdChapter on the Addition ofWhole Numbers.
Moreoverwithanynumbers,nomatterhowmanyonewillwishtoadd,onewritestheminatableaccordingtothatwhichwesaidbeforewiththemulti-plicationofnumbers,thatisthefirstplacesofallthenumbersthatonewillwishtoaddbelowthefirstplaceofthenumberswhichoneplacedtogetherfortheaddition.Andthesecondbelowthesecond,andoneaftertheotherwhichfollow.Andthenonebeginstoaddinthehandsthefiguresofthefirstplacesofallthenumbersthatwereplacedtogetherfortheaddition,fromthelowernumberuptothehigher,ascending;onethereforeputstheunitsabovethefirstplaceofthenumbers,andkeepsthetensinhand;tothesetensoneaddsabovethenumberswhichexistinthesecondplaces,andoneputstheunitsabovethesecondplace,andagainonekeepsthetens.Withthemoneaddsabovethesumofthethirdplacesofthenumbers,andthusputtingtheunits,andkeepingthetens,[pI9]stepbystepaddingthenumbers,onecanhavethesumofallthenumberswithoutend.Andinordertoperceivebettertheadditionsoftwonumbers,andevenathird,andevenmore,areshown.Thereisindeedanotherwayofmultiplicationgreatlypraised,bestformul-
tiplyinglargenumbers,whichIshallshowinthemultiplicationof567by4321.Arectangleisconstructedintheformofachessboardhaving5pointsinlength,namelyonemorethanthenumberoffiguresofthegreaternumber,andhaving3pointsinwidth,astherearethreefiguresinthesmallernumber,andthegreaternumberisputovertheabovesaidrectangle,andthesmallerisputbe-foreit,andthisisdisplayed.Andthefirstfigureofthesmallernumber,namelythe7,ismultipliedbythe1,namelybythefirstofthegreaternumber;thismakes7whichisputinthefirstpointoftheupperline,namelyunderthe1,andthe7ismultipliedbythesecondfigureofthegreaternumber,namelybythe2;therewillbe14;the4isputbeneaththe2aftertheput7,namelyin
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thesecondpointoftheupperline,andthe1iskept;andaddedtoitisthemultiplicationofthe7bythe3;therewillbe22;the2isputinthethirdpointaftertheput4,andthe2iskept;toitisaddedthemultiplicationofthe7bythe4,namelytothelastfigureofthelongernumber;therewillbe30;the0isputinthefourthpoint,andthe3inthefifth.Alsoinasimilarwaythe6willbemultipliedsinglybythe1,andbythe2,andbythe3,andbythe4;therewillbe6inthefirstpointofthesecondline,and2inthesecond,and9inthethird,and5inthefourth,and2inthefifth;andagainonedoesthemultiplicationwiththefivethatisinthelastplaceofthesmallernumber,and5willbehadinthefirstpointofthethirdline,and0inthesecond,and6inthethird,and1inthefourth,and2inthefifth.Nextforthe7thatisputinthefirstpoint,7isputabovethe1,andthe6andthe4thatarediagonallyadjacenttooneanotherafterthe7areadded;therewillbe10;the0isputabovethe2,andthe1iskept;andtoitareaddedthe5,andthe2,andthe2,whichagainarelocateddiagonallyadjacentaftertheaforesaid6and4;therewillbe10;againthe0isputoverthethirdplace,namelyoverthe3;andagainthe1iskept,andaddedwiththe0,andthe9,andthe0,whichagainarelocateddiagonallyadjacentafterthesaid5and2and2;therewillbe10;againthe0isputoverthe4,namelyoverthelastplaceofthelargernumber,andagainthe1iskept;itisaddedtothe6,andthe5,andthe3,whicharediagonallyinsequence;therewillbe15;the5isputinthefifthplace,andthekeptoneisaddedwiththe1and2whichareindiagonalseque