Thābit ibn Qurra’s Restoration of Euclid’s Data: Text, Translation, Commentary

375 Pages·2018·4.942 MB·English

by  Nathan Sidoli, Yoichi Isahaya

Sources and Studies in the History of Mathematics
and Physical Sciences
Nathan Sidoli
Yoichi Isahaya
Thābit ibn Qurra’s
Restoration
of Euclid’s Data
Text, Translation, Commentary
Sources and Studies in the History of Mathematics
and Physical Sciences
ManagingEditor
JedZ.Buchwald
AssociateEditors
A. Jones
J.Lu¨tzen
J.Renn
AdvisoryBoard
C.Fraser
T.Sauer
A.Shapiro
Sources and Studies in the History of Mathematics and Physical Sciences was
inaugurated as two series in 1975 with the publication in Studies of Otto
Neugebauer’s seminal three-volume History of Ancient Mathematical Astronomy,
which remains the central history of the subject. This publication was followed the
next year in Sources by Gerald Toomer’s transcription, translation (from the
Arabic), and commentary of Diocles on Burning Mirrors. The two series were
eventually amalgamated under a single editorial board led originally by Martin
Klein (d. 2009) and Gerald Toomer, respectively two of the foremost historians of
modern and ancient physical science. The goal of the joint series, as of its two
predecessors, is to publish probing histories and thorough editions of technical
developments in mathematics and physics, broadly construed. Its scope covers all
relevant work from pre-classical antiquity through the last century, ranging from
Babylonian mathematics to the scientific correspondence of H. A. Lorentz. Books
in this series will interest scholars in the history of mathematics and physics,
mathematicians, physicists, engineers, and anyone who seeks to understand the
historical underpinnings of the modern physical sciences.
Moreinformationaboutthisseriesathttp://www.springer.com/series/4142
Nathan Sidoli • Yoichi Isahaya
Thābit ibn Qurra’s Restoration
of Euclid’s Data
Text, Translation, Commentary
Nathan Sidoli YoichiIsahaya
School for International Liberal Studies College of Arts
Waseda University Rikky oUniversit y
Tokyo, Japan Tokyo, Japan
ISSN 2196-8810 ISSN 2196-8829 (electronic)
Sources and Studies in the History of Mathematics and Physical Sciences
ISBN 978-3-319-94660-3 ISBN 978-3-319-94661-0 (eBook)
https://doi.org/10.1007/978-3-319-94661-0

Library of Congress Control Number: 2018951065

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“…forthetoweringdead
Withtheirnightingalesandtheirpsalms”
Preface
This book began in a reading group lead by myself and attended by Takatomo Inoue,
Yoichi Isahaya, and Masayo Watanabe. A(cid:3215)er reading a few other things, we began to
readThābit’sversionofEucid’sData(cid:3214)omimagesoftwomanuscripts. A(cid:3215)eramonthor
twohadpassed,InoueandWatanabemovedontobiggerandbetterthings,andIsahaya
andIdecidedtoseethetextthrough. Asweprogressed,Ibegantothink,naively,that
wecouldfairlyeasilyturnthisworkintoaneditionofthetext—maybewithatranslation.
OurworkingpracticewasthatIsahayawouldprepareadra(cid:3215)ofthetext,sendittome,
Iwouldprepareadra(cid:3215)ofthetranslation,thenwewouldmeetandcorrectbothagainst
themanuscripts. Wealsocomparedthismaterial—atfirstsomewhatsporadically—with
al-Ṭūsī’s version of the treatise. As the project developed, I slowly came to see that in
order to do it justice, we would need to produce a monograph study, including a new
interpretationoftheGreektext.
Anumberofscholarshavehelpedusinvariouswaysasweworkedonthisproject.
Fabio Acerbi, Mohammad Bagheri, Hamid Bohlul, Benno van Dalen, Richard Lorch,
andKenSaitoallkindlyaidedusinsecuringcopiesofmanuscripts. AndrewAranaand
Marco Panza discussed philosophical matters with Nathan Sidoli, both in person and
in correspondence, and thus helped us to clari(cid:3051) our thinking on a number of points.
Alexander Jones made a number of corrections and suggestions for clarification in the
editorial process. Fabio Acerbi made many critical comments on the introduction and
savedus(cid:3214)omanumberofblunders. SonjaBrentjesandTakanoriSuzukibothreadthe
completemanuscriptandmademanycommentsandcorrections. TimHicksproo(cid:3214)ead
theEnglishprose. Weareextremelygratefulfortheirgenerosity.
ThisbookwastypesetusingXETEX. Wearegratefultothemanydevelopersofthis
project,andespeciallytoLarsMadsenandPeterWilsonfortheirworkonthememoir
package,andtoVafaKhalighiforhisworkonthebidipackage. ThefontfortheRoman
scriptisPeterBaker’sJunicode;thatfortheArabicscriptisKhaledHosny’sAmiri;and
that for the Greek script is Gentium Plus. These fonts are available under the Open
FontLicense. Thededicationisaquotation(cid:3214)om“InMyCra(cid:3215)orSullenArt”byDylan
Thomas.
NathanSidoli
Tokyo,November(cid:2419)(cid:2417),(cid:2419)(cid:2417)(cid:2418)(cid:2424)
vii
viii Thābit’sRestorationoftheData
Mathematical notation
In order to discuss the mathematical content of the work, we introduce the following
notationalconventions:(cid:3124)
Magnitudes: Wedenotemagnitudeswithablackboard-boldtype,suchthatAdenotes
ageneralmagnitudeandadenotesthesamemagnitudewhenitisknown—that
is,knowninmagnitude.
Points: We denote points with italic type, such that A denotes a general point, and a
denotesthesamepointwhenitisknown—thatis,knowninposition. Hence,we
canalsodenoteagenerallineasAB,andthesamelineasAB ,AB orAB when
p m p,m
itisknown,sincealinecanbeknownineitherposition,magnitude,orboth. In
this way, aB is the name of any line passing through the given point a , while
p
aB isthenameofacertainlinegiveninpositionthatpassesthroughthegiven
p
pointa —thelabelB,however,isjustpartofthenameofthelineanddoesnot
p
indicateanyparticularpoint. Thenameab denotesalinewhoseendpointsare
p,m
bothgiveninposition.(cid:3125)
Lines: Wealsousecursivetypetodenotelineswithasingleletter,suchthatLdenotes
a general line, and l , l or l denotes the same line, as known in position, in
p m p,m
magnitude,orinboth.
Figures: We denote rectilinear figures with bold type, such that a general, rectilinear
figure,constructed(cid:3214)ompointsA,B,C,…isdenotedasF(ABC…),or(cid:3214)omlines
A, B, C as F(ABC…), a triangle as T(ABC), a square as S(ABCD), S(AB),
or S(A) a rectangle as R(AB,BC), or R(A,B), a parallelogram as P(ABCD),
P(AB,BC),orP(A,B),andagnomonasG(ABCDEF)orG(AB,BC,CD,AF).
Arectilinearfigurecanbeknowninmagnitude,F(ABC…) ,inform,F(ABC…) ,
m f
orboth,F(ABC…) .
f,m
Circles: A circle is denoted as C(A,R), where A is its center and R its radius; C(R),
whereRisitsradius,orasC(ABC),whereA,BandCarethreepointsthrough
whichitpasses. Acirclemaybeknowninmagnitude,C(A,r ) ,orinmagnitude
m m
andinposition,C(a,r ) .
m m,p
Angles: WeuseGreekletterstodenoteangles,suchthatΘdenotesageneralangle,and
θ , θ , denote an angle known in magnitude, or known in magnitude and in
m m,p
\ \
position. Wealsoo(cid:3215)endenoteananglewiththreepoints, ABCand ABC .
m
(cid:3124) OurnotationismodeledonthatintroducedbyDijksterhuis((cid:3124)(cid:3132)(cid:3131)(cid:3130),(cid:3128)(cid:3124)–(cid:3128)(cid:3125))inthe(cid:3124)(cid:3132)(cid:3126)(cid:3131)Dutchoriginal
ofhisworkonArchimedes,andusessomeaspectsofanotationemployedbyTaisbak((cid:3124)(cid:3132)(cid:3132)(cid:3124)).
(cid:3125) ThedistinctionbetweenABp,mandabp,mmayseempedantic,buttheDataalsodealswithlinesgiven
inpositionandmagnitudewhoseendpointsarenotthemselvesgiven,suchasthatcutoffbetweentwo
parallellinesgiveninpositiononalinethatfallsonthematagivenangle—considertheimplicationof
Data(cid:3125)(cid:3130).
Preface ix
Ratios: A ratio between two magnitudes A and B is denoted (A : B), and when it is
knownitisdenoted(A:B) .
r
These symbols are simply a shorthand for the concepts introduced verbally in the
text. Inparticular,theyaredesignedtoavoidtheambiguitiesthatcanarise(cid:3214)omusing
the algebraic symbols introduced by Descartes that distinguish clearly between known
andunknownquantitiesandlendthemselvestoanarithmeticreading.
Forexample,usingmodern algebraicnotation,onemightexpressProp.(cid:3125)—ifthere
is a known magnitude and its ratio to another magnitude is known, then the other
magnitudeisknown—assomethinglike
a,a:x=r⇒x=ra.(cid:3126)
In this case, however, it is difficult not to read this symbolic expression as arithmetic,
despitethefactthatthepropositionismeanttocoverabroaderrangeofinterpretations.
Inoursymbolism,Prop.(cid:3125)wouldbeexpressedas
a,(a:B) ⇒b,
r
which is nothing more than a shorthand for the verbal expression given above. In
particular, we are interested in expressing the fact that the two bs denote the same
magnitude, firstingeneralandthenasknown,andthatthesecondbisknowndespite
thefactthatitisnotexpressedintermsofknownmagnitudesgivenintheenunciation
oftheproposition,whichissimplywhatwereadintheancientproposition.
(cid:3126) Forexample,Thaer((cid:3124)(cid:3132)(cid:3129)(cid:3125),(cid:3129)(cid:3129))summarizesthispropositionas“Vona,κausκa”—whichtheabove
statedexpressionfleshesout.
x Thābit’sRestorationoftheData
Naming conventions
WerefertotheindividualdefinitionsandpropositionsofThābit’sRestorationoftheData
asDef.x andProp.x ,whilethoseoftheGreekData,aseditedbyMenge((cid:3124)(cid:3131)(cid:3132)(cid:3129)),are
a a
referredtoasDataDef.x ,Datax . Forexample,Prop.(cid:3128)(cid:3125)(Data(cid:3128)(cid:3127)),referstothe(cid:3128)(cid:3125)nd
g g
proposition of the Restoration, as edited and translated in this work, corresponding to
the(cid:3128)(cid:3127)thpropositionofMenge’sedition. Wewillo(cid:3215)enrefertotheArabicworkstudied
inthisbooksimplyastheRestoration.
Transcription of letter-names
FortranslationsofGreekandArabicmathematicalsources,weusethefollowingtran-
scriptionsforletter-names. Alltranslationsareourown.
Arabic Greek
A α A
ا
B β B
ب
G γ G
ج
D δ D
د
E ε E
ه
Z ζ Z
ز
H η H
ح
T θ Q
ط
K
ك
L λ L
ل
M
م
N
ن
S
س
C
ص