Fibonacci’s Liber Abaci ||

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


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


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


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


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


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;


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


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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36 II.LiberAbaci

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


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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38 II. LiberAbaci

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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40 II.LiberAbaci

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

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