Table Of ContentFROM DISCRETETOCONTINUOUS
AUSTRALASIAN STUDIES
IN HISTORY AND PHILOSOPHYOFSCIENCE
VOLUME16
GeneralEditor:
R.W.HOME,University0/Melboume
EditorialAdvisoryBoard:
W.R.ALBURY,University0/NewSouthWales
D.W.CHAMBERS,Deakin University
S.GAUKROGER,University0/Sydney
H.E.LEGRAND,University0/Melboume
A.MUSGRAVE,UniversityofOtago
G.C.NERLICH,University0/Adelaide
D.R.OLDROYD,University0/NewSouthWales
E.RICHARDS,Universityo/Wollongong
J.SCHUSTER,Universityo/Wollongong
R.YEO,Griffitb University
Thetitlespublishedinthisseriesarelistedattheendof thisvolume.
KATHERINENEAL
The University0/Sydney,
Sydney,Australia
FROM DISCRETE
TO CONTINUOUS
The Broadening 0/Number Concepts
inEarly Modem England
Springer-Science+Business Media, B.Y.
Library ofCongressCataloging-in-PublicationData
ISBN 978-90-481-5993-2 ISBN 978-94-017-0077-1 (eBook)
DOI 10.1007/978-94-017-0077-1
Printedonacid-freepaper
AllRightsReserved
©2002SpringerScience+BusinessMedia Dordrecht
Originallypublished byKluwerAcademic Publishersin2002.
Softcoverreprintofthehardcover 1stedition 2002
Nopartofthematerial protected bythiscopyrightnotice maybereproducedor
utilized inanyformorbyanymeans,electronicormechanical,
includingphotocopying,recordingorbyanyinformationstorageand
retrievalsystem,withoutwritten permissionfromthecopyrightowner.
For mygrandmotherKatherine Johnson and mydaughterElizabeth
TABLE OF CONTENTS
Acknowledgments: ix
Cbapter1: TransformationoftbeNumberConcept
Cbapter2: TheAncientSources 12
Aristotle 12
Euclid 17
Proclus 22
GreekAtomists 25
Cbapter3: TheContemporaryInßuences 28
TheAbacusTradition 28
SimonStevin 33
TheTechniquesofViete 36
Descartes 39
Conclusion 45
Cbapter4: EarlyModernEnglisbAlgebra 46
TheNatureoftheTexts 47
RobertRecorde 49
TheTreatmentofNumbersinArithmeticTexts 58
John Tapp 61
ThomasHarriot'sArtsAnalyticaePraxis 62
Oughtred'sClavisMathematicae 71
Conclusion 77
Cbapter5: TbeDevelopmentoftbeLogaritbms:NapierandBriggs 80
Napier'sBackground 84
Algebra 85
Napier'sLogarithms 87
Briggsand NumericalMethods 104
Briggs' MethodofConstruction 106
Conclusion 113
Cbapter6: IsaacBarrow 11S
BackgroundandEducation 116
Barrow'sMathematicalPractice 121
FoundationalIssues inBarrow'sCambridgeLectures 132
Conclusion 136
Cbapter7: JohnWallis 138
BackgroundandEducation 139
Wallis' MathernaticalPractice 143
TheCompositionofContinua 151
Conclusion 155
Cbapter8: Conclusion 158
References: 163
Indices: 171
ACKNOWLEDGMENTS
Ihave received financial help forthisbook fromthe UniversityofToronto inthe
formofaMay Fellowshipand from theUniversityofSydney.
Several peoplehave read thisbook initsmany forms. Iwould like tothank
Craig Fraser and AlexanderJones fortheirsupportinwriting the originalthesis,on
which this book isbased. The anonymous readers providedagreat deal ofvaluable
criticism, and Ithank them fortheir detailed comments. Iwould also like to thank
John Schusterforheroicallyreading theentire manuscriptand John Henryforhis
encouragementand support. Mythanks alsogotoRachel Ankenyformaking sure I
actually had time tofinish myrevisions, and Jason Tong forhis valiant efforts with
thepictures,diagrams,formatting and index. ChristopherNeal, myhusband,has
suppliedinvaluable emotional support. My daughter Elizabethmade my lifehappy
butbusy during the finalprocess.
CHAPTER 1
TRANSFORMATION OF THE NUMBER CONCEPT
In the early modem period a crucial transformation occurred in the c1assical
conception ofnumber and magnitude.' For the Greeks, the unit, or one, was not a
number; it was the beginning ofnumber and it was used to measure a multitude.
Numbers were merely collections of discrete units that measured some multiple.
Magnitude, on the other hand, was usually described as being continuous, or being
divisible into parts that are infinitely divisible. A distinction was made between
arithmetic and geometry; arithmetic dealt with discrete or unextended quantity while
geometry dealt with continuous or extended quantity.i Inthe early modernperiod a
transformation occurred in this classical conception ofnumber and magnitude. For
instance, Simon Stevin (1548-1620) insisted in his 1585 Arithmetique that the
traditional Greek notion of numbers was wrong: he believed that numbers were
continuousrather thandiscrete.' Stevinalsodeveloped asystem forindefinite decimal
expansions ofnumber that implicitly containedthe idea ofa numerical continuum."
Moreover, Franccis Viete (1540-1603) introduced an improved form ofsymbolism:
1Thetenn"number"iselusiveanddifficulttoworkwithbecauseitsmeaningchangedoftenthroughtime
(thisofcourseisoneoftheaspectsthatmakesthesubjecthistoricallyinteresting). Differentmeanings
mightbeusedbytheprotagonistsofthisbook,secondaryauthoritiescitedsuchasJacobKlein,and
differentlytrainedreaders.Forinstance,mathematicallytrainedreadersmayinterpret'number'asan
elementofthesetofrealnumbers,definedeitherconstructively(bytheusualconstructionsincludingan
equivalentofDedekindcuts)fromthenaturalnumbersassumedgiven,oraxiomaticallybyasetofaxioms
includingsomeequivalentofacompletenessproperty.Thesereaderswillthereforehaveawell-defined
continuuminmind,inwhichtheusualtypesofnumbersasnatural,integer,rationaletc.areembedded.
Readerswithoutthismathematicalbackgroundwillprobablyhaveadifferentnumber conceptinmind.
Mostlikelythiswouldbethecommon numberconceptofroutinepractitionersofletter-algebra:numbers
arewhatlettersstandforinalgebra,theycanbepicturedalongascaledstraightline(theso-callednumber
line).Oftentheprecisedefinitionofthesenumbers islefttotheexperts,asistheprecisedefinitionofthe
globalproperties(suchascontinuity)ofthenumberset.Lessmathematicallyorientedreadersusuallylocate
theabstractnessofnumbers inthefactthatitisnotnecessarytosaywhattheyare,andyetonecando
algebrawiththem.Inthisbook,termssuchasnumbersandcontinuitywillbemainlyusedasdefinedby
theactorsinquestion.
2OJafPedersen, "LogisticsandtheTheoryofFunctions: AnEssayintheHistoryofGreekMathematics,"
Archives Internationalesd'histoiredesSeiences24(1974):29-50,haspointedoutthatlogistics,or
practicalreckoning, althoughitwasdeniedthestatusofbeingatheoreticalscienceandthusnotwritten
aboutforitsownsake,wascarriedtoagreaterstateofperfectionthanhaspreviouslybeensuspected. For
example, Ptolemy'sAlmagestdependeduponnumericalcalculationswhosemethods andprocedures were
neverexplainedinthetext. However,noGreekpracticalmathematics textswereavaiJableintheearly
modernperiod.
3Continuous inanAristoteliansenseaswillbediscussed inchapter2.
4 MeaninginthiscasethatStevinnotedthatnumberscanbepicturedalonganumberline.
2 CHAPTER1
unknowns were distinguished from given magnitudes by being labelled with capital
vowels whilethegivenmagnitudes weredenotedbycapitalconsonants. Thispowerful
newsymbolismwasusedtodenotebothunknownmagnitudesandnumbers;itindicated
that numbers and magnitudes were, in asense, interchangeable.i This association of
numbers withmagnitudes encouragedthenotion thatnumbers could alsobe treatedas
though they were continuous in the Aristotelian sense. Thus the traditional idea of
discrete number versus continuous magnitude was challenged in the early modem
periodin several ways. Stevin belongedto atraditionofcomputational mathematics
thathad apractical, problem-solvingorientation. These practitioners wereindependent
ofthemore theoreticallyorientedmathematics ofthehumanists.? The demands made
bybookkeeping, accounting,measuring,navigation,andwarhadcreated dissatisfaction
with traditional methods. Stevin began his career as a bookkeeper; his first work,
published in 1582, was titled Tables ofInterest. This work, which includes many
examples of common business problems as well as tables of interest, shows that
mercantile computations werebecorningquitecomplex," Stevin's 1585edition ofThe
Disme, dedicated to the wish that "any man's business may be performed easily
thereby," wasmotivatedinpart by adesireto easepracticalcomputations.i Although
Stevin'sdevelopmentofan indefinite decimal expansionofnumber (with its implicit
idea ofa numerical continuum) did not explicitIy become apart ofthe theoretical
discussions aboutmathematical continuainEurope inthisperiod, itcertainly advanced
the notionofnumbers being continuousrather than discrete.
Viete, ontheotherhand,had adeepregard fortheancientGreek sourees, although
atthesametimehewascomrnittedtoalgebraasapowerfulproblem-solvingtechnique.
However Viete's methods differed substantially from the earlier work ofthe abacus
schools andofthecossic algebraists." Hisemphasis waspurelytheoretical; hedidnot
considerpracticalproblems. Indeed, hewent as faras giving up the name ofalgebra
altogether." Moreover, inthe earlier works symbols only denotednumbers and they
S Laterinthechaptertherewillbeadiscussionaboutthedistinction between geometricalandabstract
magnitude.
6 Forawonderful discussion oftheimpactofhumanismonmathematicsinthe15thand 16thcenturies,in
particularthecollectionandtranslationofGreekmathematicalauthorssuchasApollonius, Archimedes,
Diophantus,Euclid,Hero,PappusandProclus,seePaulLawerence Rose"HumanistCulture and
RenaissanceMathematics:TheltalianLibrariesoftheQuattrocento"StudiesintheRenaissance20(1973)
7SimonStevin, ThePrinciple WorksofSimonStevin,ed.D.J.Struik,vol.2(Arnsterdam: C.V.Swets and
Zeitlinger,1958). ThePrincipal Works0/SimonStevinincludeshisTablesofInterest. Intheearly
modemperiodbankinghouses pursued variedactivities involvingquestionsofinsurance,ofannuities and
otherpayments atsetintervals,ofdiscountingofsumsdueatalaterdateandrelatedtransactions. Many
oftheexamplesinStevin'sbookutilisefractionsandcomplicated(fortheperiod) interestcomputations.
SeealsoN.Z.Davis,"SixteenthCenturyFrenchArithmeticsandtheBusiness Life,"Journal0/History0/
Ideas21(1960):18-48,andthediscussioninDijksterhuis'SimonStevinfordetails.
8Thededicationreads,"ToAstronorners,Land-meters, MeasureresofTapestry,Gaugers, Stereometersin
general, Money-Masters,andtoallMerchants,SIMON STEVIN wisheshealth," TheMathematicalWorks
ofSimonStevin,p.389.
9Cossic algebraandVietewillbediscussed indetailinchapter three.
10Vietebelieved thatalgebra,asitwaspracticed inhisperiod, hadbeen"spoiledanddefiledbythe
barbarians."Introduction totheAnalyücArt,trans.ReverendJ.W.Smith, inJ.Klein'sGreek
