Diophantus of Alexandria a study in the History of Greek Algebra

393 Pages·1910·37.82 MB·English

by  Sir Thomas L.Heath

DIOPHANTUS OF ALEXANDRIA
A STUDY IN THE HISTORY
OF
GREEK ALGEBRA
BY
SIR THOMAS L. HEATH, K.C.B.,
SC.D., SOMETIME FELLOW OF TRINITY COLLEGE, CAMBRIDGE
SECONT> EDITION
WITH A SUPPLEMENT CONTAINING AN ACCOUNT OF FERMAT'S
THEOREMS AND PROBLEMS CONNECTED WITH DIOPHANTINE
ANALYSIS AND SOME SOLUTIONS OF DIOPHANTINE
PROBLEMS BY EULER
Cambridge :
at the University Press
1910
(ffamtm&ge:
PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS
Library
PREFACE
r I "'HE first edition of this book, which was the first English
*
Diophantus, appeared in 1885, and has long been out of
print. Inquiries made for it at different times suggested to me
that it was a pity that a treatise so unique and in many respects
so attractive as the Arithmetica should once more have become
practically inaccessible to the English reader. At the same time
I could not but recognise that, after twenty-five years in which so
much has been done for the history of mathematics, the book
needed to be brought up to date, Some matters which in 1885
were still subject of controversy, such as the date of Diophantus,
may be regarded as settled, and some points which then had to
be laboured can now be dismissed more briefly. Practically the
whole of the Introduction, except the chapters on the editions of
Diophantus, his methods of solution, and the porisms and other
assumptions found in his work, has been entirely rewritten and
much shortened, while the chapters on the methods and on the
porisms etc., have been made fuller than before. The new text of
Tannery (Teubner 1893, 1895) has enabled a number of obscure
passages, particularly in Books V and VI, to be cleared up and,
as a basis for a reproduction ofthe whole work,is much superiorto
the text of Bachet. I have taken the opportunity to make my
version of the actual treatise somewhat fullerand somewhat closer
to the language of the original. In other respects also I thought
I could improve upon a youthful work which was my first essay in
the history of Greek mathematics. When writing it I was solely
concerned to make Diophantus himself known to mathematicians,
833088
vi PREFACE
and I did not pay sufficient attention to Fermat's notes on the
various problems. It is well known that it is in these notes that
many of the great propositions discovered by Fermat in the
theory of numbers are enshrined but, although the notes are
;
literally translated in Wertheim's edition, they do not seem to
have appeared in English; moreover they need to be supple-
mented by passages from the correspondence of Fermat and from
the Doctrinae analyticae Inventum Novum of Jacques de Billy.
The histories of mathematics furnish only a very inadequate
description of Fermat's work, and it seemed desirable to attempt
to give as full an account of his theorems and problems in
or connected with Diophantine analysis as it is possible to
compile from the scattered material available in Tannery and
Henry's edition ofthe OeuvresdeFermat(1891 1896). So much
of this material as could not be conveniently given in the notes
to particular problems of Diophantus I have put together in
the Supplement, which is thus intended to supply a missing
chapter in the history of mathematics. Lastly, in order to make
the book more complete, I thought it right to add some of the
more remarkable solutions of difficult Diophantine problems given
byEuler,forwhom such problems had agreat fascination; the last
section ofthe Supplement is therefore devoted to these solutions.
T. L. H.
October, 1910.
CONTENTS
INTRODUCTION
PAGES
CHAP. I. Diophantus and his Works !_i3
II. The MSS. ofand writers on Diophantus . . 14 31
III. Notation and definitions of Diophantus . . . 3253
IV. Diophantus' methods ofsolution . . - . . 54 98
V. The Porisms and other assumptions in Diopbantus m99110
VI. The place of Diophantus 127
THE ARITHMETICA
BOOK L
129-143
IL
143-156
IIL 156168
IV
- 168199
\V7* 200 225
VI. . - . 226246
On Polygonal numbers 247 2^9
Conspectus ofthe Arithmetica 260266
SUPPLEMENT
SECTIONS I V. NOTES, THEOREMS AND PROBLEMS BY FERMAT.
I. On numbers separable i.nto .integ.ral .squar.es .... 267277
II- Equation 3*-Ayi=\ 277292
III. Theorems and Problems on rational right-angled _
triangles 293 3,g
IV. Other problems by Fermat 318320
V. Fermat's Triple-equations 321328
SECTION VI. SOME SOLUTIONS BY EULER 329380
INDEX: I. GREEK 381382
II. ENGLISH 382 387

INTRODUCTION
CHAPTER
I
DIOPHANTUS AND HIS WORKS
THE divergences between writers on Diophantus used tobegin,
as Cossali said1, with the last syllable of his name. There is now,
however, no longerany doubt that the name was Diophantar, not
Diophanter2.
The question of his date is more difficult Abu'lfaraj, the
Arabian historian,inhisHistoryoftlteDynasties,placesDiophantus
under the Emperor Julian (A.D. 361-3), but without giving any
authority; and it may be that the statement is due simply to a
confusion of our Diophantus with a rhetorician of that name,
mentioned in another article of Suidas, who lived in the time of
Julian*. On the other hand, Rafael Bombelli in his Algebra,
1 Cossali,Origine, trasportoinItalia,primiprogressiinessaddr Algebra(Parma,
1797-9),I.p.61: "Suladesinenzadelnomecomincialadiversitatragliscrittori."
1 Greekauthorityisoverwhelminglyin favourofDiophant<w. Thefollowingisthe
evidence, which is collected in the second volume ofTannery'sedition ofDiophantus
(henceforward to be quoted as "Dioph.," "Dioph. n. p. 36" indicating page 36 of
Vol. ii., while "Dioph. II. 20" will mean proposition 20 of Book II.): Suidas s.v.
'TraTia (Dioph. n. p. 36), Theon ofAlexandria, on Ptolemy'sSyntaxis Book I. c.9
(Dioph.II.p.35),Anthology, EpigramonDiophantus(Ep.xiv.126; Dioph. II.p.60),
Anonymi prolegomena in Introductionem arithmeticam Nicomachi (Dioph. II. p. 73),
Georgii Pachymerae paraphrasis (Dioph. n. p. 122), Scholia of Maximus Planudes
(Dioph.n.pp.148,177,178etc.),Scholiumonlarublichus/ Nicomachiarithm.inirod.,
ed. PisteUi, p. 127 (Dioph. n. p. 72),aScholium on Dioph. n.8fromtheMS. "A"
(Dioph. n. p. 260), which is otherwise amusing (H ifwxn aw>Ai6#arre, eft;perk TOV
fStaArTami,-a"fYroeuxarTsroftulSitvo/cpoXeir'daistiToUnV,rDeioipXhXawnrtufsfo,vftofrewtphiej/duiifTfwircuKlCtyU oSif)yeocuurTpOrVobrltetpmosrrionsgeffncewrpajl-
andofthisoneinparticular");JohnofJerusalem(lothc.)alone(VitaloannisDamas-
ceni xi.: Dioph. n.p.36),ifthereadingofthe MS.Parisinus 1559 isright,wrote,in
theplural,wj UvOaydpcu.1}AIO^OJTOI,wherehoweverAio^oiraiisdearlyamistakefor
3 \i3dvios, ffoQurriis 'Arrtox/s, rwr tfl 'louXtoyoO TOV/ScurtAlcds
QeodoffiovTOV-rpfff^vripov Qajryariovxarp6j,/ua^TTjjAIOC><UTOI/.
H. D.
INTRODUCTION
2
published in 1572, says dogmatically that Diophantus lived under
Antoninus Pius (138-161 A.D.), but there is no confirmation ofthis
dateeither.
The positive evidence on the subject can be given very shortly.
An upper limit is indicated by the fact that Diophantus, in his
book on Polygonal Numbers, quotes from Hypsicles a definition
of such a number1. Hypsicles was also the writer of the sup-
plement to Euclid's Book XIII. on the Regular Solids known as
Book XIV. of the Elements hence Diophantus must have written
A ;
laterthan,say, 150 B.C. lower limit is furnished by the fact that
Diophantus is quoted byTheon ofAlexandria2 hence Diophantus
;
wrote before, say, 350 A.D. There is a wide interval between
150 B.C. and 350 A.D., but fortunately the limits can be brought
closer. We have a letter ofPsellus (nth c.) in which Diophantus
and Anatolius are mentioned as writers on the Egyptian method
of reckoning. "Diophantus," says Psellus3, "dealt with it more
accurately, but the very learned Anatolius collected the most
essential parts of the doctrine as stated by Diophantus in a
different way (reading erepwf) and in the most succinct form,
dedicating (irpoae^xav^a-e) his work to Diophantus." It would
appear, therefore, that Diophantus and Anatolius were contem-
poraries, and it is most likely that the former would be to the
latter in the relation of master to pupil. Now Anatolius wrote
about 278-9A.D., and was Bishop ofLaodicea about 280 A.D. We
may therefore safely say that Diophantus flourishedabout 250 A.D.
or not much later. This agrees well with the fact that he is not
quoted by Nicomachus (about 100 A.D.), Theon ofSmyrna (about
130A.D.) or lamblichus (endof3rd c.).
1 Dioph.I.p.470-2.
2 TheoAlexandrinusinprimumlibrumPtolemaeiMathematicaeCompositionis(onc.
IX.): seeDioph. II. p. 35, Ka.0' a KOI &i6<pavr6s0ij<rr TTJSyapparadosd/uera^Touo&o-ijs
/catfffTWffrjsirai/Tore,TOiro\\aTT\a(na'6/j.ei>ovelSosCTT'avrrjvavrbTOeZSoj&TTCUK.r.e.
3 Dioph. II.p. 38-9: ireplde rrjs aiyvimaKrjs fj,e668ov Tatirrjs Ai6<f>ai>Tos (j.ev di^\a^i>
aKpi^ffTfpov,654\oyiibraTos'AvctToXiosTCI ffweKTiKurara /J.^pij rrjsKOT' iKfivov (iriffTrnj.?)*
airo\el;a/J.evosertpw(PfWpwsoreraifnf)Aio^di'TyffwoTTTiKuraraTrpoffe<p&vr]ffe. TheMSS.
readrr^pw,whichisapparentlyamistakeforerfywsorpossiblyforeralpip. Tannerycon-
jecturesrif fralpif,but this is verydoubtful; ifthearticlehadbeenthere, Aio^avTyT
fralpq would have been better. On the basis of eratpy Tannery builds the further
hypothesis that the Dionysius to whom the Arithmeticaisdedicatedisnoneother than
Dionysius whowasat thehead ofthe Catechist school at Alexandria 232-247andwas
Bishopthere 248-265A.D. Tanneryconjectures then that Diophantuswasa Christian
and a pupil ofDionysius (Tannery, "Sur la religion des derniers mathematicians de
1'antiquite," Extrait des Annales de Philosophic Chretienne, 1896, p. 13sqq.). It is
howeverdifficulttoestablishthis(Hultsch,art. "DiophantosausAlexandreia"inPauly-
Wissowa'sReal-EncyclopddiederclassischenAltertumswissenchaften}.
DIOPHANTUS AND HIS WORKS
3
The only personal particulars about Diophantus which are
known are those contained in the epigram-problem relatingto him
in the Anthologyx. The solution gives 84 as the age at which he
died. His boyhood lasted 14 years, his beard grew at 21, he
married at 33; a son was born to him five years later and died, at
the age of42, when his father was 80years old. Diophantus' own
death followed fouryears later2. It is clear that the epigram was
written, not long after his death, by an intimate personal friend
with knowledge of and taste for the science which Diophantus
made his life-work3
.
The works on which the fame ofDiophantus rests are:
(1) The Arithmetica (originally in thirteen Books).
(2) A tract On PolygonalNumbers.
Six Books ofthe former and part ofthe lattersurvive.
Allusions in the Aritkmetica imply the existence of
(3) A collection of propositions under the title of Porisms;
in three propositions (3, 5 and 16) of Book V. Diophantus quotes
as known certain propositions in the Theory ofNumbers, prefixing
to the statement of them the words "We have it in thePorisms
that " (e^o/iey eV rot? TLopt,afj,acriv ort K.r.e.}.
A
scholium on a passage of lamblichus where he quotes a
dictum of certain Pythagoreans about the unit being the dividing
line (fj,e06piov) between number and aliquot parts, says "thus
Diophantus in the Moriastica* for he describes as 'parts' the
progression without limit in the direction of less than the unit."
Tannery thinks the Moptacrrucd may be ancient scholia (now
lost) on Diophantus I. Def. 3 sqq.5; but in that case why should
Diophantus be supposed to be speaking? And, as Hultsch
1 Anthology,Ep.xiv. 126; Dioph.n.pp.60-1.
2 The epigram actually says that his boyhoodlasted | ofhislife; his beardgrew
after T\more; afterfmorehe married,and his son was born fiveyears later; the son
livedtohalfhisfather'sage,andthefatherdiedfouryearsafterhis son. Cantor(Gesch.
d. Math. I3, p. 465) quotes a suggestion of Heinrich Weber that a bettersolution is
obtainedifweassumethatthesondiedatthetime whenhisfather'sagewasdoublehis,
notatanageequaltohalftheageatwhichhisfatherdied. Inthatcase
is would substitute lof for 14, i6 for 21, 25$ for 33, 30! for 42, 6i for 80,
and65^for 84above. I do not see anyadvantagein this solution. On the contrary,
4 think the fractional results are an objection to it, and it is tobe observed that the
^nholiasthasthesolution84,\xd+erTiv^eXd+ffrxom+t5h+e\eqxu+at4i=onx.
3 Hultsch,art.DiophantosinPauly-Wissowa'sReal-Encyclopadie.
jjs 45 lDaimobplhi.chnu.spI.n72Nincootme.achiarithm. introd.p. 127(ed.Pistelli); Dioph.II.p.72.
I 2
INTRODUCTION
4
remarks, such scholia would more naturally have been quoted
as a-yoXia and not by the separate title Mopiavriicd1. It may
have been a separate work by Diophantus giving rules for reckon-
ing with fractions; but I do not feel clear that the reference
may not simply be to the definitions at the beginning of the
Arithmetica.
With reference to the title of the Arithmetica, we may observe
that the meaningof the word dpid^TiKd here is slightly different
from that assigned to it by more ancient writers. The ancients
drew a marked distinction between dpidpijTiKij and \OJIO-TIKIJ,
though both were concerned with numbers. Thus Plato states
that dpid/j,rjTiKrj is concerned with the abstract properties of
numbers (as odd and even, etc.), whereas \oyia-TiKtj deals with the
same odd and even, but in relation to one another2. Geminus also
distinguishes the two terms3. According to him dpiO^riK^ deals
with numbers in themselves, distinguishing linear, plane and solid
numbers, in fact all the forms of number, starting from the unit,
and dealing with the generation of plane numbers, similar and
dissimilar, and then with numbers of three dimensions, etc.
\oyiariKij on the other hand deals, not with the abstract properties
of numbers in themselves, but with numbers of concrete things
(ala-drjTwv, sensibleobjects), whence it calls them by the names of
the things measured, e.g. it calls some by the names /^XtTi?? and
<iaXtT?794. But in Diophantus the calculations take an abstract
form (except in V. 30, where the question is to find the number
of measures of wine at two given prices respectively), so that the
distinction between XO^IO-TIKIJ and dpiQ/jLijTucr/ is lost.
We findjthe Arithmetica quoted under slightly different titles.
Thus the anonymous author of prolegomena to Nicomachus"
IntroductioArithmetica speaks of Diophantus' "thirteen Books of
Arithmetic5." A scholium on lamblichus refers to "the last
theorem of the first Book of Diophantus' Elements of Arithmetic
1 Hultsch,loc.cit.
2 Gorgias,451B, C: rd fjxv &\\aKadairep17dpiB'^T/TIKT;i]hoyicrTiKriHxfi' iffpiTO avrb
ydp dffTi,rb re apriov Kal TO irepiTTbv dia<p4pei 5 roffovrov,OTL Kal irpos avra Kal irpds
d\\ri\aTTWJ?xfiTXTjtfoustiriffKoirelTOirepiTrovKalTOapTiovi)\oyiaTiK7].
3 Proclus,Comment, onEuclidI.,p.39, 14-40, 7.
4 Cf.Plato,Laws819B,c,ontheadvantageofcombiningamusementwithinstruction
in arithmetical calculation, e.g. by distributing apples or garlands (^Xow T TLVUV
diavofMlKalffT<pdvuv)andtheuseofdifferentbowlsofsilver,gold,orbrassetc. (tfudXas
a/Mi xpvffov Kal X&XKOU *cai apytipov Kal rotoi/raw TIV&V aXXwc KepawuvTes,ol5 6'XasTTWS
5ia5t56rcj,oirepelirov,eijiraidiavtvapubrrovTesrasTUVdva
5 Dioph.II.p. 73,26.