DIOPHANTOS OF ALEXANDRIA.
Honlion : C. J. CLAY and SOX,
CAMBEIDGE UNIVERSITY TRESS WAREHOUSE,
AVE :^rARIA LANE.
CAMBRIDGE: I>i:i(;ilToN. lil'.l.l.. .\M> co.
LEIPZIG: r. A. liKocKllArs.
DIOPHANTOS OF ALEXANDRIA
A STUDY IN THE lIISToUV
OF
GREEK ALGEBRA
BY -ClJUilM
T. L. HEATH, B.A.
SCHOLAR OF TRINITY COI.LKCiK, CAMHRIlxiK.
EDITED FOR THE SYNDJC&^^^^XH^ UNIVEPiSITY PliESS.
-^-V3 ^-'-- 3^"\S>.
((UHIVEP.'ITY
QTambriligc :
AT THE UNIVERSITY PKK.SS.
1885
[All Ei'jhts A'.^.Tc;./.]
CTambritigr :
rniNTF.ri dy c. j. clay, m.a. and sox,
AT THK rXIVERSITY PRKSS.
PREFACE.
The scope of the prosont book is sufficiently indicatod Ity
the title and the Table of Contents. In the chapter on
" Dioijhantos' notation and definitions" several suggestions
are made, which I believe to be new, with regard to the
origin and significance of the symbols employed by Diophantos.
A few words may be necessary to explain the purp(».se of the
Appendix. This is the result of the compression of a large
book into a very small space, and claims to have no inde-
pendent value apart from the rest of my work. It is in-
tended, first, as a convenient place of reference for mathe-
maticians who may, after reading the account of Diophantos'
methods, feel a desire to see them in actual operation, and,
secondly, to exhibit the several instances of that variety of
peculiar devices which is one of the most prominent of the
characteristics of the Greek algebraist, but which cannot l)o
brought under general rules and tabulated in the same way
as the processes described in Chapter V. The Appendix, then,
is a necessary part of the whole, in that there is much in
Diophantos which could not be introduced elsewhere ; it must
not, however, be considered as in any sense an alternative to
the rest of the book: indeed, owing to its extremely con-
densed form, I could hardly hope that, by itself, it would
even be comprehensible to the mathematician. I will merely
add that I have twice carefully worked out the .«;<.lution of
H. D. ^
VI PREFACE.
every problem from tlic proof-sheets, so that I hope and be-
lieve that no mistakes will be found to have escaped me.
It would be mere tautology to enter into further details
here. One remark, however, as to what the work does not,
and does not profess to, include may not be out of place.
No treatment of Diophantos could be complete without a
thorough revision of the text. I have, however, only cursorily
inspected one MS. of my author, that in the Bodleian Library,
which unfortunately contains no more than a small part of
the first of the six Books. The best Mss, are in Paris and
Rome, and I regret that I have had as yet no opportunity of
consulting them. Though this would be a serious drawback
were I editing the text, no collation of MSS. could afifect my
exposition of Diophantos' methods, or the solutions of his
problems, to any appreciable extent; and, further, it is more
than doubtful, in view of the unsatisfactory results of the
collation of three of the MSS, by three different scholars in
the case of one, and that the most important, of the few ob-
scure passages which need to be cleared up, whether the text
in these places could ever be certainly settled.
I should be ungrateful indeed if I did not gladly embrace
this opportunity of acknowledging the encouragement which
I have received from Mr J. W, L. Glaisher, Fellow and Tutor
of Trinity College, to whose prospective interest in the work
before it was begun, and unvarying kindness while it was
proceeding, I can now thankfully look back as having been
in a great degree the " moving cause " of the whole. And,
finally, I wish to thank the Syndics of the University Press
for their liberality in undertaking to publish the volume.
T. L. HEATH.
11 May, 1885.
LIST OF BOOKS OH PAl'KKS KKAD ()I{ KKKKlMtKI) 'K >.
SO FAR AS THEY CON'CERN OK AUK ISKFIL
TO THE SUBJECT.
1. Bookg directlif upon Dinphautois.
Xylander, Diopliaiiti Alexambini Reruni Arithmetit-arum Libri sex
Item Liber de Numcri.s Polygonis. Opus incoiupiirabile Latino
redditum et Commeutariis explanatum Biusileae, 1575.
Bachet, Diophanti Alexandrini Arithmeticoioim Libri sex, et de niuueri.s
multaugulis liber uiiu.s. Lutetiae Parisiorimi, 1G21.
Diophanti AJexandi-ini Ai-ithmeticorum libri sex, et de uumeris multaugu-
lis liber unus. Cum commeutariis C. G. Bacheti V.C. et oWrua-
tionibus D. P. de Fermat Senatoris Tolcsani. Tolosae, 1G70.
ScHULZ, Diophantus von Alexandria arithmetische Aufgaben nebst desseu
Schrift liber die Polygon-zahlen. Aus dem Griecbi-scheu iibersetzt
und mit Anmerkungeu begleitet. Berlin, 18-22.
PoSELGER, Diophantus von Alexandrien iiber die Polygon-Zahlen.
Uebersetzt, mit Zusiitzen. Leipzig, 1810.
Crivelli, Elementi di Fisica ed i Problemi aritlmietici di Diofanto
Alessandrino analiticamente dimostrati. In Venczia, 1744.
P. Glimstedt, Forsta Boken af Diophanti Arithmetica algebraisk Ocfvcr-
sattning. Lund, 1855.
Stevin and Girard, " Translation " in Les Oeuvres mathematiques de
Simon Stevin. Leyde, 1684.
2. M'orha indirectly fluridati)i<j Diftj'lnmtitg.
BoMBELLi, L' Algebra diuisa in tre Libri Bologna, 1579.
F'ermat, Opera Varia mathematica. Tolixsai', H;7l>.
Brassinne, Precis des Oeuvres mathematicpies de P. Fcrnuit et de I'Aritlj-
metique de Diophante. P'""is l''*-'>3-
CossALi, Origine, traspoi-to in Italia, prinii progre.s.si in e-ssa dell' Algebni
Storia critica Parnm, 17U7.
Nesselmanx, Die Algebra der Griechcn. Berlin, IM2.
John Kersey, Elements of Algebra. London, 1674.
Walms, Algebra (in Opera Mathematica. Ox.»iiittC, 161)5 9 .
Saundek.son, N., Elements of Algebra. >"»''
Vlll LIST OF AUTIlulUTIKS.
3. Buuks ic/tich iiifiitivii or (/ice infurmation about Dio^laiiUof,
including historiiis of mathematics.
CuLEUHOOKE, AlgeVira with Arithmetic and ^Mensuration from the Sanscrit
of Brahmaguptii and Bhiiscara. London, 1817.
SriDAs, Lexicon (ed. G. Bernhardy). Ilalis et Brunsvigae, 1853.
Fabricii.s, Bibliotheca Graeca (ed. Harless).
AuCLEARAJ, History of the Dynasties (tr. Pococke). Oxon. 16C3.
Ch. Th. v. Murr, Memorabilia Bibliothecarum publicarum Norimbergen-
sium et Universitatis Altdorfinae. Norimbergae, 1786.
DoPPELMAYR, Historische Nachricht von den Xiirnbergischen Mathema-
ticis und Kiinstlern. (Nliruberg, 1730.)
Vos.siis, De universae mathesius natiira et coustitutione
Amstelaedami, 16G0.
Hkilbronneh, Historia matheseos universae. Lipsiae, 1742.
MuNTLCLA, Histoire des Math(5matiques. Paris, An 7.
IviAEUEL, Matheniatisches "\Vorterl)uch. Leipzig, 1830.
Kaestner, Geschiclite der Matheniatik. Giittingen, 1796.
BussuT, Histoire G(5uerale des Mathematiques. Paris, 1810.
Hankel, Zur Geschichte der Mathematik in Altertlium und Mittelalter.
Leipzig, 1874.
Cantor, Vorlesungen Uber Geschichte der Mathematik, Band L
Leipzig, 1880.
Dr Heinrich Slter, Gesch. d. :^Lathematischen Wisseuschaften,
Zurich, 1873.
Jame.s Gow, a short History of Greek Mathematics.
Camb. Univ. Press, 1884.
4. Papers or Pamphlets read in connection with Diophantos.
Poselger, Beitriige zur Unbestimmten Analysis.
(Berlin xihhandhmgen, 1832.'i
I.. RoDET, L'Algebre d'Al-Kharizmi et les methodes indienne et grecque.
{Journal AHiatitjite, Janvier, 1878.)
WoEPCKE, Extrait du Faklni, traitc^ d'Algebrc par Abou Bekr ^[ohammed
ben Alhayan Alkarkhi, precede d'un memoiresurralgebre indeterminet;
chez los Arabes. Paris, 1853 .
WoEi'CKE, Mathematiques chez les Orientaux.
1. Journal Asiatique, Fdvrier-Mars, 1855.
2. Journal Asiatique, Avril, 1855.
I'. Tanxehv, "A <iuelque epocpie vivait Dioi)hante /" {Bulletin des iSciences
Mat/ufm. ct Astronom. 1879.)
I'. Tax.nery, L'Arithm(5ti(iue dans Pajtpus {Bordeaux Memoirs, 1880.)
lIusEN, Tiie Algel>ra of .Mohammed ben Musa. London, 1831.
1Ii:ii>er<;, Quacstiones Archimedeae. llauniae, 1879.
CONTENTS.
CHAPTER I.
HISTORICAL INTRODUCTION'.
PAGES
§ 1. Diophantos' name and particulars of his life .... i
§ 2. His date. Different views 3
(«) Internal evidence considered 4_S
{b) External evidence 8 — IG
§ 3. Results of the preceding investigation 16—17
CHAPTER II.
THE WORKS OF DIOPHANTOS ; THEIR TITLES AND GENERAL
CONTENTS; THE PORTIONS WHICH SURVIVE.
§ 1. Titles : no real evidence that 13 books of Aritliiiietics ever existed
corresponding to the title IS — 23
No trace of lost books to be restored from Arabia. Corruption
must have taken place before 11th cent, and probably before
950 A.D 23— "iC,
Poiisms lost before 10th cent. a.d. 2<)
§ 2. What portion of the Arithmetics is lost? The contents of the
lost books. The Polygonal Numbers and Porism.i may have
formed part of the complete ArithmcticK. Objections to this
theory 2(>— 3.">
Other views of the contents of thf lost Books .... 3J — 37
Conclusion 37
CHAPTER III.
THE WlllTEKS UPON J»lolMIA.\ T« ),s.
§ 1. (heck 38-39
§ 2. Arabian 39 — 12
§ 3. European gencially 42— 5('>
CONTENTS.
CHAPTER IV.
\OT.\TI(»N AND DEFINITIONS (»F DlOPH.\NTOS.
VAC.KS
§ 1. Introduction ,57
§ 2. Sign for the unknown quantity discubsed 57 — 67
§ 3. Notation for powers of the unknown G7— 09
§ i. Objection that Diophantos loses generality by the want of
more algebraic symbols answered 69
Other questions of notation : operations, fractions, dc. . . 69—76
§ 5. General remarks on the historical development of algebraic
notation : three stages exhibited 76—80
§ 6. Ou the influence of Diophantos' notation on his work . . 80—82
CHAPTER V.
§1.
§3.
SI.
diophantos' METHODS OF SOLUTION.
General remarks. Criticism of the positions of Hankcl and
Ncsselmann
Diophantos' treatment of equations .....
(A) Determinate equations of different degrees.
(1) Pure equations of different degrees, i.e. equations con
taining only one power of tlie unknown
(2) Mixed quadratics
(3) Cubic equation .......
Indeterminate equations.
'.. Indeterminate equations of the first and second degrees.
(li)
(1)
(2)
Single equation (second degree) •
1. Those which can always be rationally solved
2. Those which can be rationally solved only
under certain conditions
II,
Double equations.
1. First general method (first degree) .
Second method (first degree) .
2. Double equation of the second degrei
Indeterminate equations of liigher degrees.
(1) Single ecjuations (first class)
,, (second class)
(2) Double equations .
Summary of the prerediiiji incestiijntioii
Transition .....
Mitiiod of limits ....
Method of appro.\imation to limits .
88—114
88-
-90
90-
-93
93-
-94
95-
-98
95
95—98
99—105
105—107
107
108—111
111—112
112—113
113—114
114—115
115—117
117—120
CONTKNTS.
CHAPTER VI.
PAOEH
1. The PonsHis of Diophantos 121 I2.'i
2. Other theorems assumed or implied 12.>— 132
('/) Numbers as the sum of two squares 127— 1:<0
(h) Numbers as the sum of three squares l:{0 — l:{l
(c) Numbers as the sum of four squares 131— 1H2
§1.
§2.
§3.
§4.
§5.
§6.
§7.
CHAPTER VII.
HOW FAR WAS DIOPHANTOS ORIGINAL?
Preliminary 133—134
Diophantos' algebra not derived from Arabia .... 134—135
Reference to Hypsikles 13.") — 130
The evidence of his language 13G— 138
Wallis' theory of Greek Algebra 138
Comparison of Diophantos with his Greek predecessors . . 139—142
Discussion in this connection of the Cattle-prohlem . . . 142 — 147
CHAPTER VIII.
DIOPHANTOS AND THE EARLY ARABIAN ALGEBRAISTS.
§ 1. Preliminary
§ 2. Comparison of Diophantos with Mohammed ibn Musfi . . 149-
§ 3. Diophantos and Abu'1-Wafa
§ 4. An anonymous Arabic ms. 155-
§ 5. Abu Ja'far Mohammed ibn Alhusain
§G. Alkarkhi 156-
14S
-155
155
-156
156
APPENDIX.
ABSTRACT OF DIOPHANTOS.
A rithmetics. Book I
. 16.S— 171
Bookn
. 172—181
Bookm
. 181-189
Book IV
. 189-208
BookV
. 209-224
Book VI
. 225-237
Polygonal Numbers
. 238—244
EREATUM.
On p. 78, last line but one of note, for " Targalia" read "Tartaglia"
DIOPHANTOS OF ALEXANDRIA.
CHAPTER I.
Historical Introduction.
§ 1. The doubts about l^iophantos begin, as has been
remarked by Cossali^ with his very name. It cannot be posi-
tively decided whether his name was Diophanfos or Diophan^es.
The preponderance, however, of authority is in favour of the
view that he was called Diophantos.
(1) The title of the work which has come down to us under
his name gives us no clue. It is Aiocpdvrov 'A\€^avBp€Q}<i 'AptOfj,-
rjTLKwv ^i/SXia ly. Now Atocjjdvrov may be the Genitive of
either Ai6(f)avTo<i or A,io^dvrr]<;. We learn liowever from this
title that he lived at Alexandria,
(2) In Suidas under the article "TiraTia the name occurs
in the Accusative and in old editions is given as Aio<f)dvTr}v ;
but Bachet'^ in the Preface to his edition of Diophantos assures
us that two excellent Paris MSS. have Aio^avrov. Besides this,
Suidas has a separate article At6(})avTo<i, ovofia Kvpiov. More-
over Fabricius mentions several persons of the same name
Ai6<f)avTo<;, but the name /lio(f)dvTr]<i nowhere occurs. It is
on this ground probable that the correct form is AiotpavTo^.
We may compare it with "EK<f)avTo<;, but we cannot go so far
as to say, with Bachet, that Aio<f)dvT'r]^ is not Greek ; for we
have the analogous forms 'lepot^dvrrjt;, o-vK0(f)dvTr]<;.
1 "Su la desinenza del nome coniincia la divcisitii tni gli scrittori " (p. 61).
- '-Ubi moueudus es imprimis, in editis Suidae libris male habori, <«j Ato-
(pavT-qv, ut ex duobus probatissimis codicibus manu exaratis (lui extant in
Bibliotheca Regia, depraehendi, qui veram exhibcut Icctionem d% Aw^ayrof."
H. D. ^
2 DIOPIIAXTOS OF ALEXANDRIA.
(3) In the only quotation from Diophantos which we
know Tlioon of Alexandria (fl. 3G5 — 390 A.D.) speaks of him
as At6<f)avTo<i.
(4) On the other hand Abu'lfaraj, the Arabian historian,
in his History of the Dynasties, is thought to be an authority for
the form Diophanfcs, and certainly in his Latin translation of the
two passages in which D. is mentioned by Abu'lfaraj, Pococke
writes Diophantes. But, while in the first of the two passages
in the original the vowel is doubtful, in the second the name is
certainly Diophantos. Hence Abu'lfaraj is really an authority
for the form Diophantos.
(5) Of more modern writers, Rafael Bombelli in his
Algebra, published 1572, writes in Italian "Diofanie" corre-
sponding to Aio(f)dvT7]<;. But Joannes Regiomontanus, Joachim
Camerarius, James Peletarius, Xylander and Bachet all write
Diophan^ws.
We may safely conclude, then, that Diophanios was the
name of our author. Far more perplexing than the doubt as
to his name is the question of the time at which he lived. As
no certainty can even now be said to have been reached on this
point, it will be necessary to enumerate the indications which
bear on the question. Before proceeding to consider in order
the internal and external evidence, it will be well to give the
only facts which are known of his personal history, and which
can be gathered from an arithmetical epigram upon Diophantos.
This epigram, the probable date of which it will be necessary to
consider later along with the question of its authorship, is as
follows :
Ovt6<; tol ^t6(f>avToi> e'^et rd(f)o<;, d /jiiya Oavfia,
Kal T«</)09 €K Te^i/779 fierpa ^10 to \i<y€L.
"Ektt]v Kovpi^eiv ^toTou ^eo? cutraae fioipijv,
AoyBeKUTrj S" iTriOeU firjXa Tropev '^(Xodetv.
Tfj 8' ap' e'</)' e^BofuiTT} to yafirjXiov i'jyfraTO <f>iyyo<;,
E/c 8e ydficov TrifMirrcp TralS' iirev^vaev eret.
At ai r7]\vy€Tov BetXov Te«09, VjfXLav irarpo^,
Tov 8e Kal 7; Kpvepo^ fierpov eXoov ^iotov.
Iler'^o? 3' av iriavpecro'L Trapijyopecov eviavTOL<i
Tj]8e TToaov ao(piT) repfx eTreprja-e /3iOV.
HISTORICAL IXTUODUCTIOX. 3
The solution of this epigram-problem gives 84 as the age
at Avhich Diophautos died. His boyhood lasted 14 years, his
beard grew at 21, he married at 33/; a son was born to him
5 years later and died at the age of 42, when his father was
80 years old. Diophanto.s' own death followed 4 years later
at the age of 84. Diophantos having lived to so great an age,
an approximate date is all that we can expect to find for
the production of his works, as we have no means of judg-
ing at what time of life he would be likely to write his
Aiithmetics.
§ 2. The most important statements upon the date of
Diophantos which we possess are the following :
(1) Abu'lfaraj, whom Cossali calls "the courageous compiler
of a universal history from Adam to the 13th century," in his
History of the Dynasties before mentioned, places Diophantos,
without giving any reason, under the Emperor Julian (3G1 —
368 A.D.). This is the view which has been ordinarily held.
It is that of Montucla.
(2) We find in the preface to Rafael Bombclli's Algebra,
published 1572, a dogmatic statement that Diophantos lived
under Antoninus Pius (138 — 161 A.D.). This view too has
met with considerable favour, being adopted by Jacobus de
Billy, Blancanus, Vossius, Heilbronner, and others.
Besides these views we may mention Bachet's conjecture,
which identifies the Diophantos of the Arithmetics with an
astrologer of the same name, who is ridiculed in an epigram
attributed to Lucilius ; whence Bachet concludes that he
lived about the time of Nero (54—68) (not under Tiberius,
as Nesselmann supposes Bachet to say). The three views
here mentioned will be discussed later in detail, as they are
all worthy of consideration. The same cannot be said of a
number of other theories on the subject, of which I will quote
only one as an example. Simon Stevin* places Diophantos
later than the Arabian algebraist Mohammed ibn Miisa
1 Les Oeuvrcs Mathcin. de Sim. Stevin, augm. par Alh. Girard, Loyden, 1634,
"Quant h, Diophant, il semblc iiu'cn son temps los inventions de Mahomet
ayent seulement tsto cognues, commc bo poult colligcr de sea six premiers
livres."
1—2
DIOPHANTOS OF ALEXANDRIA.
Al-Kliarizmi who lived in the first half of the 9th century, the
absurdity of which view will appear.
We must now consider in detail the
(a) Internal evidence of the date of Diophantos.
(1) It would be natural to hope to find, under this head,
references to the works of earlier or contemporary mathema-
ticians. Unfortunately there is only one such reference trace-
able in Diophantos' extant writings. It occurs in the fragment
upon Polygonal Numbers, and is a reference to a definition
given by a certain Hypsikies\ Thus, if we knew the date of
Hypsikles, it would enable us to fix with certainty an upper
limit, before which Diophantos could not have lived. It is
particularly unfortunate that we cannot determine accurately
at what time Hypsikles himself lived. Now to Hypsikles is
attributed the work on Regular Solids which forms Books
XIV. and xv, of the Greek text of Euclid's Elements. In the
introduction to this work the author relates'^ that his father
knew a treatise of Apollonius only in an incorrect form, whereas
he himself afterwards found it correctly worked out in another
book of ApoUonios, which was easily accessible anywhere in
his time. From this we may with justice conclude that Hypsikles'
father was an elder contemporary of ApoUonios, and must have
died before the corrected version of ApoUonios' treatise was
given to the world. Hypsikles' work itself is dedicated to a
friend of his father's, Protarchos by name. Now ApoUonios
died about 200 B.C.; hence it follows that Hypsikles' treatise
' Polyg. Numbers, prop. 8.
"Kal iirtdelxOri t6 waph. 'typiKkeX iv 8p(p Xeyd/J-evov.^'
'^ <Tvvairob(ixOivTo% oZv koI tov 'T^iacX^ouj 8pov, k.t.X."
' "Kal TTOTf SteXoOfTfj (sc. Basileides of Tyre and Hypsikles' father) rb virb
' AwoWuviov ypa<p^u wepi ttjs (TvyKplaems roO SwStKaiSpov Kal tov tlKoffa^dpov tlov
(U TT)v avTj]v <j<paipav iyypa<{>onivuv, rlva \oyov ix^i vpbs dWijXa, fSo^af raOra
fMT] dpOQi ytypaipivai rbv 'ArroWwi/iof. aCrrol di ravra SiaKaddpavres fypa\j/av wj
Tjj/ iKoveiv TOV naTpos. ^yCo Si vartpov irtpUtreaov iTlpi^ /3t/JA/v i"r6 ' AiroWuvlov
iKbfhotxivtfi, Kal TTtpUxovTi anobuiiu ijyiwi (?) irepl tov i/iroKfifj^vov. Kal fieydXtji
i\J/vxaywy^Or)v iirl Ty irpofiXi^fxaTos ^T-qati. Tb fj.iv viro ' AwoWwi'lov iKbodh (oiKf
KOiff, OKOirdv. Kal yap irtpKpiptrai, k. t. X."
HISTORICAL INTRODUCTION. 5
on Regular Solids was probably written about 180 B.C. It
was clearly a youthful productiou. Besides this we have another
work of Hypsikles, of astronomical content, entitled in Greek
dva^opLK6<i. Now in this treatise we find for the first time
the division of the circumference of a circle into 360 degrees,
which Autolykos, an astronomer a short time anterior to Euclid,
was not acquainted with, nor, apparently, Eratosthenes who
died about 194 B.C. On the other hand Hypsikles used no
trigonometrical methods : these latter are to some extent em-
ployed by the astronomer Hipparchos, who made observations
at Rhodes between the years 101 and 126. Thus the discovery
of trigonometrical methods about 150 agrees well with the
conclusion arrived at on other grounds, that Hypsikles flourished
about 180 B.C.
We must not, however, omit to notice that Nesselmann,
an authority always to be mentioned with respect, takes an
entirely different view. He concludes that we may with a fair
approach to certainty place Hypsikles about the year 200 of our
era, but upon insufficient grounds. Of the two arguments used
by Nesselmann in support of his view one is grounded upon
the identification of an Isidores whom Hypsikles mentions'
as his instructor with the Isidores of an article in Suidas:
'lo-tSwpo? ^tXocro0O9 09 e^Ckocro^ae fiev vtto TOt«? dS€\<f)oi<;,
eiirep rt? dWo<i, iv fiaOrjfiaaLv: and, further, upon a conjecture
of Fabricius about it. Assuming that the two persons called
Isidoros in the two places are identical we have still to deter-
mine his date. The question to be answered is, what is the
reference in viro roh dB€X(j)oi<; ? Now Fabricius makes a con-
jecture, which seems hazardous, that the dBe\(f)ot are the
brothers M. Aurelius Antoninus and L. Aurelius Verus, who
were joint-Emperors from 160 to 169 A.D. This date being
assigned to Isidoros, it would follow that Hypsikles should
be placed about A. d. 200.
In the second place Nesselmann observes that according to
Diophantos Hypsikles is the discoverer of a proposition respect-
ing polygonal numbers which we find in a rather less perfect
' Eucl. XV. 5. "77 5^ evpejii, u$'l<xl5wpos 6 Ti/x^Tepos vip-nyqaaTo fi^yat 5i6dcK-
fiXos, ^x" ■'■0" '■poVoi/ TovTov,"
6 DIOPHANTOS OF ALEXANDRIA.
form in Nikomachos and Theon of Smyrna ; from this he
argues that Hypsikles must have been later than both these
mathematicians, adducing as further evidence that Theon (who
is much given to quoting) does not quote him. Doubtless, as
Theon lived under Hadrian, about 130 A.D., this would give a
date for Hypsikles which would agree with that drawn from
Fabricius' conjecture ; but it is not possible to regard either
piece of evidence as in any way trustworthy, even if it w^ere
not contradicted by the evidence before adduced on the other
side.
We may say then with certainty that Hypsikles, and there-
fore a fortiori Diophantos, cannot have written before 180 B.C!.
(2) The only other name mentioned in Diophantos' writings
is that of a contemporary to whom they are dedicated. This
name, however, is Dionysios, which is of so common occurrence
that we cannot derive any help from it whatever.
(3) Diophantos' work is so UTiique among the Greek trea-
tises which we possess, tliat he cannot be said to recal the style
or subject-matter of any other author, except, indeed, in the
fragment on Polygonal Numbers ; and even there the reference
to Hypsikles is the only indication we can lay hold of.
Tiie epigram-problem, which forms the last question of the
5th book of Diophantos, has been used in a way which is rather
curious, as a means of determining the date of the Arithmetics,
by M. Paul Tannery \ The enunciation of this problem, which
is different from all the rest in that (a) it is in the form of an
epigram, (6) it introduces numbers in the concrete, as applied
to things, instead of abstract numbers (with which alone all
the other problems of Diophantos are concerned), is doubtless
borrowed by him from some other source. It is a question
about wine of two different qualities at the price respectively of
8 and 5 drachmae the %o{;9. It appears also that it was wine of
inferior quality as it was mixed by some one as drink for his
servants. Now M. Tannery argues (a) tliat the numbers 8 and
5 were not hit upon to suit the metre, for, as these are the only
numbers which occur in the epigram, and both are found in
* lUilh'tin (ten Sciences mathnnntiqiiis et astronomif/ucs, 1879, p. 201.
HISTORICAL INTRODrCTION. 7
the same line in the compounds 6KTa8pd^^f^ov<: and irein-eSpdx-
fiov<i, some other numerals would serve the purposes of metre
equally well, (b) Neither were they taken in view of the solu-
tion of the problem, for each number of ;^6e? which it was
required to find are found to contain fractions. Hence (c) the
basis on which the author composed his problem must have
been the price of wines at the time. Now, says M. Tannery*,
it is evident that the prices mentioned for wines of poor quality
are famine prices. But wine was not dear until after tlie time
of the Antonines. Therefore the composer of the epi^-am, and
hence Diophantos also, is later than the period of the Antonines.
This argument, even if it is correct, does no more than give
us a later date than we before arrived at as the upper limit.
Nor can M. Tannery consistently assert that this determination
necessarily brings us at all near to the date of Diophantos ; for
in another place he maintains that Diophantos was no original
genius, but a learned mathematician who made a collection of
problems previously known ; thus, if so much had already been
done in the domain which is represented for us exclusively by
Diophantos, the composer of the epigram in question may well
have lived a considerable time before Diojihantos. It may be
mentioned here, also, that one of the examples which M. Tan-
nery quotes as an evidence that problems similar to, and even
more difficult than, those of Diophantos were in vogue before
his time, is the famous Problem of the Cattle, which has been
commonly called by the name of Archimedes ; and this very
problem is fatal to the theory that arithmetical epigrams must
necessarily be founded on ftict. These considerations, however,
though proving M. Tannery to be inconsistent, do not neces-
sarily preclude the possibility that the inference he draws from
the epigram-problem solved by Diophantos is correct, for (a) the
date of the Cattle-problem itself is not known, and may be
later even than Diophantos, (6) it does not follow that, if
M. Tannery's conclusion cannot be proved to be necessarily
right, it must therefore be wrong.
1 "II est d'ailleurs facile de se rendre comptc que ccs prix n'ont pas 6t6
choisis en vue de la solution: on doit done supposcr qu'ils sent rt-els. Or ce
sent evideininent, pour los vius de basse quality, do prix de famine."
8 DIOPHAXTOS OF ALEXANDRIA.
On the vexed question as to how far Diophantos was original
we shall have to speak later.
Wo pass now to a consideration of the
(b) External evidence as to the date of Diophantos.
(1 ) We have first to consider the testimony of a passage of
Suidas, which has been made much of by writers on the ques-
tion of Diophantos, to an extent entirely disproportionate to its
intrinsic importance. As however it does not bear solely upon
the question of date, but upon another question also, it cannot
be here passed over. The passage in question is Suidas' article
'TTTartaV The words which concern us apparently stood in
the earliest texts thus, eypay^rev vTro/xvrj/jLa et? Aiocfxivrrjv
Tov darpovofiiKov. Kavova et<? to. KOiVLKa' ^AiroWcovlov
viro/jiVTjfia. With respect to the reading A.io(f)dvr'}]v, we have
already remarked that Bachet asserts that two good Paris MSS.
have A.i6(f>avTov.
The words as found in the text cannot be right. Aiocfjdvrrjv
TOV da-TpovofiiKov should (if the punctuation were right) be
Aio(f>dvTr)v TOV d(TTpov6^ov, the former not being Greek.
Ku.ster's conjecture '^ is that we should read vTrofivrjfia ek
Aio(f)dvTov da-TpovofjLLKov Kavova' et? to, KwviKa ^ AttoWojvlov
vTToiJ.vnp.a. If this is right the Diophantos here mentioned must
have been an astronomer. In that case the person in question
is not our Diophantos at all, for we have no ground whatever to
imagine that he occupied himself with Astronomy. It is cer-
tain that he was famous only as an arithmetician. Thus John
of Jerusalem in his life of John of Damascus^ in speaking of
some one's skill in Arithmetic compares him to Pythagoras and
' tiraTla rj O^wvos tov TtufUTpov Ovyarrip tov ' A\f^avSp^wi <pi\oa6^ov xal avTr]
<f)iK6(To<poi, KoX woWoh yvwpifioi' yvvr] 'IcridiJopov tov <t>i.\oao<pov iJKnaafv iirl t^s
PaaiXdai 'ApKailoV (ypayptv vir6fu/r)na th \io<p(xvTr)v tov dtXTpovopuKov. Kaxoi'a eh
Ta KuviKa' ' AiroWwvlov v-ir6fj.injfia.
' Suidae Lexicon, Cantabiigiac, 1705.
3 Chapter xi. of the Life as Kiven in Sancti patris iiostri Joannis Damasceni,
Monaclii, et rreshyteri Ilierusulymitani, Opera omnia quae exstant ft ejus nomine
circumferuntur. Tonms primus. Parisiis, 1712. 'AvaXoyla^ di'ApidfxrjTiKii ovtu^
i^rfaKr/Kacif ti'^uwr, wi UvOayopai t} Ai6(pavToi,
HISTORICAL INTRODUCTION. f)
Diophantos, as representing that science. However, Baclict
has proposed to identify our Diophantos with an astrologer of
the same name, who is ridiculed in an epigram' supposed to he
written by Lucilius. Now the ridicule of the epigram would
be clearly out of place as applied to the subject of the epigram
mentioned above, even supposing that Lucilius' ridiculous
hero is not a fictitious personage, as it is not unreasonable to
suppose.
Bachet's reading of the passage is vTro/ivrjfia eh Aio^afToz/,
t6i> darpovofiLKOv Kavova, etf ra koovlku AttoWcoviov vTro^vr,-
fia'^. He then proceeds to remark that it shows that Hypatia
wrote a Canon Astronomicus, so that she evidently was versed
in Astronomy as well as Geometry (as shown by the Commen-
tary on Apollonios), two of the three important branches of
Mathematics. It is likely then, argues Bachet, that she was
acquainted with the third. Arithmetic, and wrote a commentary
on the AritJtmetics of Diophantos. But in the first place we
know of no astronomical work after that of Claudius Ptolemy,
and from the way in which 6 da-rpovo^iLKO'; Kavwv is mentioned
it would be necessary to suppose that it had been universally
known, and was still in common use at the time of Suidas, and
yet was never mentioned by any one else whom we knjULUUi
inexplicable hypothesis.
' 'ISipixoyivt) Tov larpov 6 affrpoXoyoi Ai6<f>ai'T(X
Eln-e /xovovi ^wfjs ivvia pLrjvas ^X^'"-
KcLKeivos ycXdaai, Ti /jl(v 6 KpSvos ivvia. /xrjvwy,
^■qal, \^y€L, (TV voef Ta/xa 5i ci'inofxa. aoc
Elwe Kal ^KTslvas fwvov Tjxj/aro' Kal AiO(pain-os
'AWov dve\iri^u)v, avrbs awf (TKapKrev.
"Ludit non innenustus poeta turn in Diopbantum AstroloRum, turn in niccli-
cum Hermogenera, quem et alibi saepe false admodum perstringit, qniVl solo
attactu non aegros modo, sed et ben(^ valentes, velut pestifero sidere afflntoa
repente necaret. Itaque nisi Diopbantum nostrum Astrologiae iieritum fuissc
negemus, nil prohibet, quo minus eum aetate Lucillij extitisse dicanius."
Bacbet, Ad Urtorrm.
- From tbis reading it is clear that Bachet did not rest his view of the
identity of our Diophantos with the astrologer upon the i)as8age of Suidas.
M. Tannery is therefore mistaken in supposing this to be the case, "Bachet,
ayant lu dans Suidas qu'Hy^mtia avait commentu le Canon astronomique d©
notre auteur..."; that is precisely what Bacbet did rmt read there.
10 DIOPHANTOS OF ALEXANDRIA.
Next, the expression ek Aio^avrov has been objected to by
Nesselmann as not being Greek. He maintains that the Greeks
never speak of a book by the name of its author, and therefore
we ought to have Atocfxivrov dpidfiijTiKa, if the reference were
to Diophantos of the Arithmetics. M. Tannery, however, de-
fends the use of the expression, on the ground that similar
ones are common enough in Byzantine Greek. M. Tannery,
accordingly, to avoid the difficulties which we have mentioned,
supposes some words to have dropped out after ^lo^avrov, and
thinks that we should read et? At6<f)avTov . . .rov aa-rpovofiiKov
Kavova. et9 ra KwviKa WiroWcoviov virofivrjixa, suggesting that
before tov acrrpovoiiiKov Kavova we might supply et? and under-
stand TlroXe^iaiov.
It will be seen that it is impossible to lay any stress upon
this passage of Suidas. We cannot even make sure from this
that Hypatia wrote a commentary upon Diophantos, though it
has been very generally asserted by historians of mathematics
as an undoubted fact, even by Cossali, who in speaking of the
corrupt state into which the text of Diophantos has fallen
remarks that Hypatia was the most fortunate of the commen-
tators who have ever addressed themselves to his writings.
(2) I have already mentioned the epigram which in the
form of a problem gives us the only facts we know of Dio-
phantos' life. If we only knew the exact date of the author of
this epigram, our difficulties would be much lessened. It is
commonly assigned to Metrodoros, but even then we are not
sure whether Metrodoros of Skepsis or Metrodoros of Byzan-
tium is meant. It is now generally supposed that the latter
was the author ; and of him we know that he was a gram-
marian and arithmetician who lived in the reign of Constantine
the Great.
(8) It is satisfactory in the midst of so much uncertainty to
find a most certain reference to Diophantos in a work by Theon
of Alexandria, the fatherof Hypatia, which gives us a loiuer limit
for the date, more approximate than we could possibly have
derived ironx the article of Suidas. The ftict that Theon quoted
Diopiiantos was first noted by Peter Ramus* ; " Diophantus,
' Schold Mathnntitirii, Book i, p. Su.
HISTORICAL INTRODUCTION. \\
cujus sex libros, cum tamcn author ipso tredccim poUiceatur,
graecos habemus de arithmcticis admirandac subtilitatis artcm
coniplexis, quae vulgo Algebra arabico nomine appellatur : cum
tamen ex authore hoc antique (citatur enim a Theone) anti-
quitas artis appareat. Scripserat et Diophantus harmonica."
This quotation was known to Montucla, who however draws an
absurd conchision from it* which is repeated by Klucrel in his
Worterbuchl The words of Theon which refer to Diopliantos
are koI Ai6(})avT6<i (f)r](riv on, rf]<; fxovdSo'i (iixeraderov ova-T]<:
Kal earcoay]^ Trcivrore, to 7roWa7rXaaia^u/j,evov elSo'i eV avTTjf
avTo TO ei8o9 earat. We have only to remark that these words
are identically those of Diophantos' sixth definition, as given in
Bachet's text, with the sole difference that iravTore stands in
the place of the equivalent dei, in order to see that the refer-
ence is certain beyond the possibility of a doubt. The name of
Diophantos is again mentioned by Theon a few lines further on.
Here then we undoubtedly have a lower limit to the time of
Diophantos, supplied by the date of Theon uf Alexandria, and
one which must obviously be more approximate than we could
have arrived at from any information about his daughter
Hypatia, however trustworthy. Theou's date, fortunately, we can
determine with accuracy. Suidas^ tells us that he was con-
temporary with Pappos and lived in the reign of Theodosius I.
The statement that he was contemporary with Pappos is almost
1 "Theon cite une autre ouvrage de cet analyste, oil il ctoit question dc la
pratique de I'arithm^tique. Je soupv'onnerois que c'etoit la qu'il expliquoit plus
au long les regies de sa nouvelle arithnietique, sur quoi il ue s'ctoit pas assez
^tendu au commencement de ses questions." Montucla,
Apparently translated word for word in Eosenthal's EncyclopUdie d. reinen
Mathem. iii. 195.
- I. 177, under Arithmetik: "Diophantus hab ausser seincm grossen arith-
metischen Werke aucb ein Werk iiber die praktiscbe Arithmetik gescbriebcn,
das aber verloren ist."
To begin with, Montucla quotes the passage as occurring in the 5th Book of
Theon's Commentary, instead of the first. The work of Diophantos which
Theon quotes is not another work, but is identically the Arithmeticn vihich wo
possess.
^ Qiuv 6 iK Tov ^lovjelov, Alyvirrioi, (pi\6(TO<f>oi, Ji'yxP°^°^ ^^ UdTririft ri^ <pi\oc6^
Kal avrc^ 'AXe^avdper irOyxavov di an<por{poi iirl Qfodoaiov fiaffi\^ws tov Tfxff^vri-
poV iypafe 'MaOTj/xariKd, ' ApiO/xrjTtKo, k. t. \.
12 DIOPHANTOS OF ALEXANDRIA.
certainly incorrect .and due to a confusion on the part of Suidas,
for Pappos probably flourished under Diocletian (a. D, 284 —
305) ; but the date of a certain Commentary of Theon has been
definitely determined' as the year 372 A. D. and he undoubtedly
flourished, as Suidas says, in the reign of Theodosius I. (379 —
395 A. D.).
(4) The next authority who must be mentioned is the
Arabian historian Abu'lfaraj, who places Diophantos without
remark under the emperor Julian. This statement is important
in that it gives the date which has been the most generally ac-
cepted. The passage in Abu'lfaraj comes after an enumeration
of distinguished men who lived in the reign of Julian, and is
thus translated by Pococke : "Ex iis Diophantes, cuius liber
A. B. quem Algebram vocat Celebris est."
It is a difiicult question to decide how much weight is to be
allowed to Abu'lfaraj's dogmatic statement. Some great autho-
rities have unequivocally pronounced it to be valueless. Cossali
attributes it to a confusion by Abu'lfaraj of our author with
another Diophantos, a rhetorician, who is mentioned in another
article'^ of Suidas as having been contemporary with the em-
peror Julian (361 — 363); and assumes that Abu'lfaraj made the
statement solely on the authority of Suidas, and confused two
persons of the same name. Cossali remarks at the same time
upon a statement of Abu'lfaraj's translator, Pococke, to the effect
that the Arabian historian did not know Greek and Latin.
Colebrooke too' {Algebra of the Hindus) takes the same view.
Ncnv it certainly seems curious that Cossali should remark upon
Abu'lfaraj's ignorance of Greek and yet suppose that he made a
statement merely upon the authority of Suidas ; and the ques-
tion suggests itself: had Abu'lfaraj no other authority? We
1 "On the date of Pappus," Ac., by Hermann Usener, Neues Rheinisches
Museum, 1873, Bd. xxviii. 403.
* Ai^dvios, (TO(piaTr)i AvTioxf'y. twv ivl toO 'lovXiavov toO Uapa^aTov xp<5»'w»'.
Kal fJi^XP^ Qeo5offlov tov vpea^vripov, '^aayaviov Trarpoj, fiadrjrr]^ Aio(pdin-ov.
* Note M. p. LXiii. "The Armenian Abu'lfaraj places the Algebraist Dio-
phantus under the emperor Julian. Ihit it may be (luestioned whether he ha8
any authority for that date, besides the mention by Greek authors of a learned
person of ihc name, the instructor of Libanius, who was contemporary with tht^t
pmperor,"
HISTORICAL INTR()incTR)X. l.S
must certainly, as was remarked by Schulz, admit that he must
have had ; for he gives yet another statement about Diophaiitos,
which certainly comes from another source, that his work was
translated into Arabic, or commented upon, by Mohanuiicd
Abu'1-Wafa. There would seem however to be but one possibility
which would make Abu'lfaraj's statement trustworthy. Is it
possible that the two persons, whom he is supposed to have
confused, are identical ? Is it a sufficient objection that Liba-
nius distinguished himself chiefly as a rhetor and not as a
mathematician ? In fact, in the absence of any evidence to the
contrary, why should the arithmetician Diophantos not have
been a rhetorician also ? This question has given occasion to
some jests on the compatibility of the two accomplishments.
M. Tannery, for example, quotes Fermat, who was " Conseiller
de Toulouse " ; and Nesselmann mentions Aristotle, arriving
finally at the conclusion that the two may be identical, and so,
while Abu'lfaraj's statement has nothing against it, it has a
great deal in its favour. But M. Tannery thinks he has made
the identification impossible by finding Suidas' authority, namely
Eunapios in the Lives of the Sophists, who mentions this other
Diophantos as an Arabian, not an Alexandrian, and professing
at Athens \ Certainly if this supposition is correct, we cannot
identify the two persons, and therefore cannot trust the state-
ment of Abu'lfaraj. There is a further consideration — that the
reign of Julian (361 — 363) could certainly only have been the
end of Diophantos' life, as we see by comparing Theon's date,
above mentioned, to whom Diophantos is certainly anterior;
he may indeed have been much earlier, because (1) Theon
quotes him as a classic, and (2) the absence of quotations before
Theon does not necessarily show that the two were nearly
contemporary, for of previous writers to Theon who would have
been likely to quote Diophantos ?
(5) In the preface to his Algebra, published A.D. 1572,
Rafael Bombclli gives the bare statement that Diophantos lived
^ "II uous donne ce Diophante, qu'il a connu et dont il ne fait d'ailleure
pas grand cas, comme nO, non pas a, Alexandrie, ainsi que le matht-maticien,
mais en Arable (AiocpavTos 6 'Apd^ios), et, d'autre part, conime prolessant A
Athenes."
14 DIOPIIANTOH OF ALEXANDRIA.
in the reign of Antoninus Pins', giving no proof or evidence of
it. From the demonstrated incorrectness of certain other state-
ments of Bombelli concerning Diophantos we may infer that we
ought not hastily to give credence to this ; on the other hand it
is scarcely conceivable that he would have made the assertion
without any ground whatever. The question accordingly arises,
whether we can find any statement by an earlier writer, which
might have been the origin of Bombelli 's assertion. M. Tannery
thinks he has found the authority while engaged in another
research into the evidence on which Peter Ramus ascribes to
Diophantos a treatise on Harmonics '^ an assertion repeated by
Gessner and Fabricius^. As I cannot follow M. Tannery in his
conjectures — for they are nothing better, but are rather con-
jectures of the wildest kind, — I will give the substance of his
remarks without much comment, to be taken for what they are
worth. According to M. Tannery Ramus' source of information
was a Greek manuscript on music ; this there is no reason to
doubt; and in the edition of Antiquae musicae auctores by
Meibomius we read, in the treatise by Bacchios 6 <yepo)v, that
there were five definitions of rhjjthm, attributed to Phaidros,
Aristoxenos, Nikomachos, Ae6(f)avT0<; and Didymos. Now the
name Aed<^ai/T09 is not Greek; the form Aea)(f)avTo<; however is,
but M. Tannery argues that a confusion between Aeo and Atw is
much less likely than a confusion between Aco and Aio. (I may be
allowed to remark here that I cannot agree with this view.
Of course A and A are extremely likely to be confounded, but
that I should have been at the same tmie changed into € seems
to me anything but probable. Besides, this involves two changes,
whereas the change of Ae« into Aeo involves only one variation.
This latter change then is the smaller one, and why should it
1 "Qucsti aniii passati, csscndosi ritrouato una opera greca di qucsta dis-
ciplina nclla libraria di Nostra Siguore in Vaticano, coniposta da un certo
Diofantc Alessandrino Autor Greco, il quale fh a tempo di Antonin Pio..."
[I quote from the edition published in 1579, which is in the British Museum.
I have not seen the original edition of 1572.]
2 "Scripserat et Diophantus harmonica."
=* "Harmonica Diophanti, quae (icsiwrus et alii memorant, iuteUige de har-
monicis numcris, uou dc scripto quoJam musici argumeuti," though what is
meant by "harmonic numbers," as Nessclmann remarks, is not quite clear.
HISTORICAL INTRODUCTION'. lo
be less likely than the other ? I confess that it seems to mc hy
far the more likely of the two ; for the long ami short vowx-ls o,
M must have been closely associated, as is proved by the fact
that in ancient inscriptions^ we find O written for both O and il
indiscriminately, and in others H used for both sounds.) Tlicii,
according to M. Tannery, Ramus probably took the name for
Ai6(f)avTo<;, and was followed by other writers. Admitting that
the identification with the arithmetician Diophantos is hypo-
thetical enough, M. Tannery goes on to say that it is confirmed
by finding the name of Nikomachos next to Ae6(f)avTo<i, and by
observing that Euclid and Ptolemy also were writers on music,
which formed part of the fiadrjixara. Now in enumerations of
this sort the chronological order is generally followed, and the
dates of many authors have been decided on grounds no more
certain than this. (It is an obvious remark to make to M.
Tannery that " two wrongs do not make a right " : it does not
follow that, because other dates have been decided on insufficient
grounds, we should determine Diophantos' date in the same
manner ; wKfiught rather to take warning by such unsatisfactory
determinations. But to proceed with M. Tannery's remarks) —
In the present case we know that Aristoxenos was a disciple of
Aristotle, and that Nikomachos was posterior to Thrasyllos who
lived in the reign of Tiberius. Thus we can prove the chrono-
logical order for two of the five names. Again, Nikomachos
must be anterior to his commentator Apuleius who was con-
temporary with Ptolemy, and Ptolemy speaks in his Harmonics
of a tetrachord due to a neo-Pythagorean Didymos. Of Phaidros
we know nothing. Hence if we admit that the names are given
in chronological order, and remember that Diophantos lived to
be 84 years of age, we might say that, coming between Niko-
machos and Didymos, he lived in the reign of Antoninus Pius,
as Bombelli states, i.e. 138 — IGl A.D.
M. Tannery, however, is conscious of certain objections to
this theory of Diophantos' date. This determination would, he
says, have great weight if Bacchios 6 '^epoiv had been an author
' I mean, of course, inscrr. later than the introduction of Q, before which
time one sign was necessarily used for both letters. Further, I lay no strcBH
upon this fact except as an illustration.
16 DIOPHANTOS OV ALEXANDRIA.
sufficiently near in point of" time to Diophantos and the rest in
order to know their respective ages. Unfortuoately, however,
that is far from certain, Bacchios' own date being very doubtful.
He is generally supposed to have lived in the time of Constantine
the Great ; this is however questioned by M. Tannery who
thinks that the epigram given by Meibomius, in which Bacchios
is associated with a certain Dionysios, refers to Constantine
Porphyrogenetes, who belongs to the sixth century. Next,
grave doubts may be raised concerning the determination by
means of the supposed chronological order; for the definitions
of rhythm given by Nikomachos and Diophantos (?) are very
nearly alike, that of Diophantos being apparently a development
of that of Nikomachos : kutu 8e NiKOfia^ov, '^povcou evTUKTO'i
avvd€<TC<i' Kara 8e Ai6(f)avTov (?), -^povcov avvdeai<; kut dvaXo'^iav
re Kol (TVfifierplav irpo'i eavTov<;. The similarity of the two
definitions might itself account for their juxta-position, which
might then after all be an inversion of chronological order.
Again the age of Didymos must be fixed differently. By
" Didymos " is meant the son of Herakleides Ponticus, gramma-
rian and musician, whom Suidas places in the reign of Nero.
Thus, if we assume Bacchios' order to be chronological, we must
place Diophantos in the reign of Claudius, and Nikomachos in
that of Caligula.
§ 3. Results of the preceding investigation.
I have now reviewed all the evidence we have respecting the
time at which Diophantos lived and wrote, and the conclusions
arrived at, on the basis of this evidence, by the greatest autho-
rities upon the subject. It must be admitted the result cannot
be called in any sense satisfactory ; indeed the data arc not
sufficient to determine indisputably the question at issue. The
latest determination of Diophantos' date is that of M. Tannery,
and there has been no theory propounded which seems on the
whole preferable to his, though oven it cannot be said to have
been positively established ; it has, however, the merit that, if it
cannot be proved, it cannot be impugned ; as therefore it seems
HISTORICAL INTROnrcTION. 17
open to no objection, it would seem best to accept it provisionally,
as the least uusatistactory theory. We shall therefore be not
improbably right in placing Diophantos in the second half of the
third century of our era, making him thus a contemporary of
Pappos, and anterior by a century to Theon of Alexandria and
his daughter Hypatia,
One thing is quite certain: that Diophantos lived in a
period when the Greek mathematicians of great original power
had been succeeded by a number of learned commentators, who
confined their investigations within the limits already reached,
without attempting to further the development of the science.
To this general rule there are two most striking exceptions, in
different branches of mathematics, Diophantos and Pappos.
These two mathematicians, who would have been an ornament
to any age, were destined by fate to live and labour at a time
when their work could not check the decay of mathematical
learning. There is scarcely a passage in any Greek writer
where either of the two is so much as mentioned. The neglect
of their works by their countrymen and contemporaries can be
explained only by the fact that they were not appreciated or
understood. The reason why Diophantos was the earliest of the
Greek mathematicians to be forgotten is also probably the
reason why he was the last to be re-discovered after the Revival
of Learning. The oblivion, in fact, into which his writings and
methods fell is due to the circumstance that they were not
understood. That being so, we are able to understand why
there is so much obscurity concerning his personality and the
time at which he lived. Indeed, Avhen we consider how little
he was understood, and in consequence how little esteemed, we
can only congratulate ourselves that so much of his work has
survived to the present day.
CHAPTER II.
THE WORKS OF DIOPHANTOS ; THEIR TITLES AND GENERAL
CONTENTS; THE PORTIONS OF THEM WHICH SURVIVE.
§ 1. We know of three works of Diophantos, which bear
the following titles.
(1) Wpi6fxr]TtKci}v /Si/SXia ly.
(2) Trep] TToXvyoovcov apidfioov.
(3) TropiafMara.
With respect to tlie first title we may observe that the
meaning of "dpid/jbrjTiKa' is slightly different from that assigned
to it by more ancient writers. The ancients drew a marked
distinction between dpidfiijTiKT] and \ 0740- rt/c?;, both of which
were concerned with numbers. Thus Plato in Gorgias 451 B*
states that dpidfirjrtKy'] is concerned with the abstract properties
of numbers, odd even, and so on, whereas XoytaTCKij deals with
the same odd and even, but in relation to one anotlier. Geminos
also gives us definitions of the two terms. According to him
dpidfxrjTLKij deals with abstract properties of numbers, while
XoyiariKi] gives solutions of problems about concrete numbers.
From Geminos we see that enunciations were in ancient times
concrete in such problems. But in Diophantos the calculations
' £1 tIs fjie fpoiTo..!'(l SwAcpares, tL^ eariv rj dptOfirjTiKr] t^x*''?> cI'toim' S**
avTip, tSairep <ri> dpri, 6ti twv 5id \6you tis t6 Kvpoi ixovauv. Kal et /xe iwavip-
ono Twf TTipl tL ; etiroifi' Av, 6ti twv vtpl rb Apribv tc koI irtpiTTov Sj dp
(Kdrepa Ti^yx'**'^' ^"'■a- «' 5' av fpoiTO, Trjv 5^ XoyiariKriv rlva KoKds rix^riv ;
ilvoiix &v 6ti Kal ai>T7) iarl tCiv \6yifi t6 trdv Kvpovp-ivuv. Kal el IwavipoiTo 'H
iTfpl tI ; etiToifJ.^ hv wainp o\ iv rc^ StJ^v <iv-f^pa.<^6iXivoi, Sri ra fikv &\\a KaOdwep
rj dpiOixrjTiKTi T} XoyuTTiKT] ^X"' ""fpi TO avTO yap icTL, to re dpTiov Kal to irepiTToV
diatpitid Sk ToaovTov, oti Kal tt/jos aina Kal Trpos aX\i)\a ttuis ?x** irXridovi iiriffKOirei
TO TTCpiTTOV Kal TO df>Tl0V 7] XoyKTTlKrj. (.tOnjlUg, 451 B,C.
HIS WORKS. 10
take an abstract form, so that the distinction between XoyiaTiKij
and apidfjiriTLKr] is lost. We thus have W.pid/xr}TiKd given as
the title of his work, whereas in earlier times the term could
only properly have been applied to his treatise on Polygonal
Numbers. This broader use by Diophantos of the term arith-
metic is not without its importance.
Having made this preliminary remark it is next necessary to
observe that of these works which we have mentioned some
have been lost, while probably the form of parts of others has
suffered considerably by the ravages of time. The Arithmetics
should, according to the title and a distinct statement in the
introduction to it, contain thirteen Books. But all the six
known MSS.^ contain only six books, with the sole variation
that in the Vatican MS. 200 the same text, which in the rest
forms six books, is divided into seveii. Not only do the MSS.
practically agree in the external division of the work ; they
agree also in an equally remarkable manner — at least all of
them which have up to the present been collated — in the lacunae
and the mistakes which occur in the text. So much is this
the case that Bachet, the sole editor of the Greek text of
Diophantos, asserts his belief that they are all copied from one
original ^ This can, however, scarcely be said to be established,
^ The six mss. are :
1—3. Vatican mss. No. 191, xiii. c, cbarta bombycina.
No. 200, XIV. c, charta pergamena.
No. 304, XV. c, charta.
4. MS. in Nat. Library at Paris, that used by Bachet for his text.
5. MS. in Palatine Library, collated for Bachet by Claudius Salmasius.
6. Xylander's ms. which belonged to Andreas Dudicius.
Colebrooke considers that 5 and 6 are probably identical.
- "Etenim neque codex Eegius, cuius ope banc editionem adornavimus;
neque is quern prae manibus habuit Xilander; neque Palatinus, vt doctissimo
viro ClauLlio Salmasio refcrente accepimus ; neque Vaticanus, quern vir suniniua
lacobus Sirmondus mihi ex parte transcribendum curauit, quicquam amplius
continent, quam sex hosce Arithmcticorum libros, et tractatum de iiumeris
multangulis imperfectum. Sed et tarn infeUcitcr hi omnes codices inter ae
consentiunt, vt ab vno fonte manasse et ab eodem exemplari dcscriptos fuisso
non dubitem. Itaque parum auxilij ab his subministratum nobis esse, veris-
simu allirmare possum," Epintola ad Lectorem.
It will be seen that the learned Bachet spells here, as everywhere, Xylander's
name wrongly, giving it as Xilander.
O 9
20 DIOPHANTOS OF ALEXANDRIA.
for Bachet had no knowledge of two of the three Vatican MSS.
and had only a few readings of the third, furnished to him by
Jacobus Sirmondus. It is possible therefore that the collation
of the two remaining mss. in the Vatican might even now lead
to important results respecting the settling of the text. The
evidence of the existence in earlier times of all the thirteen
books is very doubtful, some of it absolutely incorrect. Bachet
says * that Joannes Regiomontanus asserts that he saw the
thirteen books somewhere, and that Cardinal Perron, who had
recently died, had often told him that he possessed a MS.
containing the thirteen books complete, but, having lent it
to a fellow-citizen, who died before returning it, had never re-
covered it. Respecting this latter MS. mentioned by Bachet
we have not sufficient data to lead us to a definite conclusion
as to whether it really corresponded to the title, or, like the
MSS. which we knoAv, only announced thirteen books. If it
really corresponded to the title, it is remarkable how (in the
words of Nesselmann) every possible unfortunate circumstance
and even the " pestis " mentioned by Bachet seem to have
conspired to rob posterity of at least a part of Diophantos'
works.
Respecting the statement that Regiomontanus asserts that
he saw a MS. containing the thirteen books, it is clear that
it is founded on a misunderstanding. Xylander states in two
passages of his preface " that he found that Regiomontanus
1 "loannes tamen Regiomontanus tredecini Diophanti libros se alicubi
vidisse asseverat, et illustrissimus Cardinalis Perronius, quern nupei- ex-
tinctum niagno Christianae et literariae Rcipublicae detrimeuto, conquerimur,
mihi saepe testatus est, se codicem manuscriptum habuisse, qui tredeeim Dio-
phanti libros integros contineret, quern cilm Gulielmo Gosselino conciui suo,
qui in Diophantum Commentaiia meditabatur, perhumauiter more suo exhi-
buisset, pauUo post accidit, ut Gossclinus peste correptus iuteriret, et Diophanti
codex codem fato nobis criperetur. Cum enim prccibus meis motus Cardi-
nalis amplissimus, nullisque sumptibus pai-cens, apud heredes Gosselini codicem
ilium diligenter exquiri mandassct, et quouis pretio redimi, nusquam repertus
est." Ad lectorcm.
■•* "Inueni deinde tanquam exstantis in bibliothecis Italicis, sibique uisi
mentionem a Regiomontano (cuius etiam nominis memoriam ueneror) factam."
Xylander, Epistola nuncupatoria.
"Sane tredeeim libri Arithmeticae Diophanti ab aliis perhibentur exstare in
bibliotheca Vaticana; quos Regiomontanus illo uiderit." Ibid.
HIS WORKS. 21
mentioned a MS. of Diophantos which he liad seen in an Italian
library; and that others said that the thirteen books were
extant in the Vatican Library, " which Regiomontanus saw."
Now as regards the latter statement, Xylander was obviously
wrongly informed ; for not one of the Vatican Mss. contains
the thirteen books. It is necessary therefore to inquire to what
passage or passages in Regiomontanus' writings Xylander refers.
Nesselmann finds only one place which can be meant, an Oratio
habita Patavii in praelectione Alfragani^ in which Regiomon-
tanus remarks that " no one has yet translated from the Greek
into Latin the thirteen books of Diophantosl" Upon this
Nesselmann observes that, even if Regiomontanus saw a MS.,
it does not follow that it had the thirteen books, except on
the title-page ; and the remarks which Regiomontanus makes
upon the contents show that he had not studied them thoroughly ;
but it is not usually easy to see, by a superficial examination,
into how many sections a Ms. is divided. However,- this passage
is interesting as being the first mention of Diophantos by a
European writer; the date of the Speech was probably about
1462. The only other passage, which Nesselmann was acquaint-
ed with and might have formed some foundation for Xylatider's
conclusion, is one in which Regiomontamis (in the same Oratio)
describes a journey which he made to Italy for the purpose
of learning Greek, with the particular (though not exclusive)
1 Printed in the work Eudimenta astronomica Alfrarfani. "Item Alba-
tegnius astronomus peritissimus de motu stellarum, ex observationibus turn
propriis turn Ptolemaei, omnia cum demonstrationibus Geometricis et Addi-
tionibus Joannis de Eegiomonte. Item Oratio introductoria in omnen scientias
Mathematicas Joannis de Reijiomonte, Patavii habita, cum Alfraganum pnblice
praelegeret. Ejusdem utilissima introductio in elementa Euclidis. Item Epis-
tola Philippi Melanthonis nuncupatoria, ad Senatum Noribergensem. Omnia
jam recens prelis publicata. Norimbergae anno 1537. 4to."
- The passage is: "Diofanti autem tredecim libros subtilissimos nemo osqne-
hac ex Graecis Latinos fecit, in quibus flos ipse totius Arithmeticae latet, are
videlicet rei et census, quam hodie vocant Algebram Arabico nomine."
It does not follow from this, as Vossius maintains, that Kegiomontanus sup-
posed Dioph. to be the inventor of algebra.
The "ars rei et census," which is the solution of determinate quadratic
equations, is not found in our Dioph. ; and even supposing that it was given in
the MS. which liegiomontanus saw, this is not a point which would des4.•r^•o
special mention.
22 DIOPHANTOS OF ALEXANDRIA.
object of turning into Latin certain Greek mathematical works\
But Diopliantos is not mentioned by name, and Nesselmann
accordingly thinks that it is a mere conjecture on the part
of Cossali and Xylander, that among tlie Greek writers mentioned
in this passage Diophantos was included ; and that we have
no ground for thinking, on the authority of these passages,
that Regiomontanus saw the thirteen books in a complete form.
But Nesselmann does not seem to have known of a passage
in another place, which is later than the Oration at Padua,
and shows to my mind most clearly that Regiomontanus never
saw the complete work. It is in a letter to Joannes de Blan-
chinis^ in which Regiomontanus states that he found at Venice
" Diofantus," a Greek arithmetician who had not yet been
translated into Latin ; that in the proemium he defined the
several powers up to the sixth, but whether he followed out
all the combinations of these Regiomontanus does not know ;
'^ for not more than six books are found, though in the proemium
he promises thirteen. If this book, a wonderful and difficult
luork, could be found entire, I should like to translate it into Latin,
for the knowledge of Greek I have lately acquired would
suffice for thisV' &c. The date of this occurrence is stated
1 After the death of his teacher, Georg von Peurbach, he tells us he went
to Eome &c. with Cardinal Bessaiion. "Quid igitur rehquum crat nisi ut
orbitam viri clarissimi sectarer? coeptum felix tuum pro viribus exequerer?
Duce itaquo patrono communi Romam profectus more meo Uteris exerceor, ubi
scripta plurima Graecorum clarissimorum ad literas suas disceudas me invitant,
quo Latinitas in studiis praesertim Mathematicis locupletior redderetur."
Peurbach died 8 April, llGl, so that tlie journey must have taken place
between 1-lGl and 1171, when he permanently took up his residence at Niim-
berg. During this time he visited in order Eome, Ferrara, Padua (where he
delivered the Oration), Venice, Rome (a second time) and Vienna.
2 Given on p. 135 of Ch. Th. v. Murr's Memorabilia, Norimbergae, 1786, and
partly in Doppelmayr, Ilistorischc Nachricht von der Kiirnbergischen Mathe-
vuiticis uml Kiimtlcrn, p. 5. Note y (Niiruberg, 1730).
3 The whole passage is :
" Hoc dico dominationi uestrae me reperisse nunc uenetiis Diofantum aritli-
meticum graecum nondum in latinum traductura. Hie in prohemio diiliniendo
terminos huius artis ascendit ad cubum cubi, primura cnim uocat uumcrum,
quern numeri uocant rem, secundum uocat potentiam, ubi uumeri dieunt
censum, deinde cubum, deinde potentiam poteutiae, uocant numerum censum
de ceusu, item cubum de ccusu ct taudom cubi. Ncscio tamen si oumes com-
HIS WORKS. 2li
in a note to be 1463. Here then we have a distinct contradicti-.u
to the statement that Regiomontanus speaks of having si-eu tliir-
teen books ; so that Xylander's conchisions must be abandoned.
No conclusion can be arrived at from the passage in F'ermat's
letter to Digby (15 August 1G57) in which he says: The nanu'
of this author (Diophantos) " me donne I'occasion de vous faire
souvenir de la promesse, qu'il vous a pleu me faire de recouvrer
quelque manuscrit de c^t Autheur, qui contienne tous les treize
livres, et de m'en faire part, s'il vous pent tomber en main."
This is clearly no evidence that a complete Diophantos existed
at the time.
Bombelli (1572) states the number of books to be seven\
showing that the MS. he used was Vatican No. 200.
To go farther back still in time, Maximus Planudcs, who
lived in the time of the Byzantine Emperors Andronicus I. and
II. in the first half of the 14th century, and wrote Scholia to
the two first books of the Arithmetics, given in Latin in
Xylander's translation of Diophantos, knew the work in the
same form in which we have it, so far as the first two books
are concerned. From these facts Nesselmann concludes that
the corruptions and lacunae in the text, as we have it, are due
to a period anterior to the 14th or even the 13th century.
There are yet other means by which lost portions of Diophan-
tos might have been preserved, though not found in the original
text as it has come down to us. We owe the recovery of some
Greek mathematical works to the finding of Arabic translations
of them, as for inststnce parts of Apollonios. Now we know
binationes horum proseeutus fuerit. non enim reperiuntur nisi 6 eius libri qui
nunc apud me sunt, in prohemio autem pollicetur se scripturum tredecim. Si
liber hie qui reuera pulcerrimus est et diflicilimus, integer inueniretur [Doppel-
mayr, inueHi'atur] curarem eum latiuum facere, ad hoc enim sufficereut mihi
literae graecae quas in domo domini mei reuerendissimi didici. Curate et uos
obsecro si apud uestros usquam inueniri possit liber ille integer, sunt enim in
urbe uestra non nulli graecarum litterarum periti, quibus solent inter caetoros
tuae facuitatis libros huiusmodi occurrere. Interim tamen, si suadebitis. Hex
dictos libros traducere in latinum occipiam, quatenus latinitas hoc nouo et
pretiosissimo munere non careat. "
1 "Egli e io, per arrichire il mondo di cosi fatta opera, ci dessimo i\ tradurlo
e cinque libri {delU settc che sotio) tradutti ue abbiamo." Bombelli, pref. to
Algebra.
24 DIOPHANTOS OF ALEXANDRIA.
that Diophantos was translated into Arabic, or at least studied
and commented upon in Arabia. Why then should we not
be as fortunate in respect of Diophantos as with others ? In
the second part of a work by Alkarkhi called the Fakhrl^
(an algebraic treatise) is a collection of problems in deter-
minate and indeterminate analysis which not only indicate
that their author had deeply studied Diophantos, but are,
many of them, directly taken from the Arithmetics with the
change, occasionally, of some of the constants. The obliga-
tions of Alkarkhi to Diophantos are discussed by Wopcke in
his Notice sur le Fakhrl. In a marginal note to his MS. is a
remark attributing the problems of section iv. and of section
III. in part to Diophantos^. Now section IV. begins with pro-
blems corresponding to the last 14 of Diophantos' Second Book,
and ends with an exact reproduction of Book ill. Intervening
between these two parts are twenty-five problems which are not
found in our Diophantos. We might suppose then that we have
here a lost Book of our author, and Wopcke says that he was
so struck by the gloss in the MS, that he hoped he had dis-
covered such a Book, but afterwards abandoned the idea for the
reasons : (1) That the first twelve of the problems depend upon
equations of the first or second degree which lead, with two
exceptions, to irrational results, whereas such were not allowed
by Diophantos. (2) The thirteen other problems which are
indeterminate problems of the second degree are, some of them,
quite unlike Diophantos ; others have remarks upon methods
employed, and references to the author's commentaries, which
we should not expect to find if the problems were taken from
Diophantos.
It does not seem possible, then, to identify any part of
1 The book which I have made use of on this subject is: "Extrait dn Fakhrl,
traits d' Algl'bre par Abou liekr Mohammed ben Alhavan Alkarkhi (mauuscrit
1)52, supplement arabe de la bibliothequc Imperiale) pr^ced6 d'un m<?moire sur
I'Algebre ind<5termiiiee chez les Arabes, par F. Woepckc, Paris, 1858."
2 Wopcke's translation of this gloss is: "J'ai vu en cet endroit une glose de
I'dcriture d'Ibn Alsir&dj en ces termes : Je dis, les probli'mes de cette section et
une partie de ceux de la section pr^c(5dente, scut pris dans les livres de Dio-
phante, suivunt I'ordre. Ceci fut 6crit par Ahmed IJen Abi 13eqr Ben Ali Ben
Alsiiiulj Alkclaueci."
HIS WORKS. 2.'>
the Fakhrl as having formed a part of Diophantos' work now
lost. Thus it seems probable to suppose that the form in which
Alkarkhi found and studied Diophantos was not different from
the present. This view is very strongly supported by the follow-
ing evidence. Bachet has already noticed tliat the solution
of Dioph. II. 19 is really only another solution of ii. 18, and
does not agree with its own enunciation. Now in the Faklu^l
we have a problem (iv. 40) with the same enunciation as
Dioph. II. 19, but a solution which is not in Diophantos' manner.
It is remarkable to find this followed by a problem (iv, 41)
which is the same as Dioph. ii. 20 (choice of constants always
excepted). It is then sufficiently probable that il. 19 and
20 followed each other in the redaction of Diophantos known
to Alkarkhi ; and the fact that he gives a non-Diophantine
solution of II. 19 would show that he had observed that the
enunciation and solution did not correspond, and therefore set
himself to work out a solution of his own. In view of this
evidence we may probably assume that Diophantos' work had
already taken its present mutilated form when it came into
the hands of the author of the Fakhrl. This work was written
by Abu Bekr Mohammed ibu Alhasan Alkarkhi near the
beginning of the 11th century of our era ; so that the cor-
ruption of the text of Diophantos must have taken place before
the 11th century.
There is yet another Arabic work even earlier than this
last, apparently lost, the discovery of which would be of the
greatest historical interest and importance. It is a work upon
Diophantos, consisting of a translation or a commentary by Mo-
hammed Abu'1-Wafa, already mentioned incidentally. But it
is doubtful whether the discovery of his work entire would
enable us to restore any of the lost parts of Diophantos. There
is no evidence to lead us to suppose so, but there is a piece
of evidence noted by Wopcke* which may possibly lead to
an opposite conclusion. Abu'1-Wafa does not satisfactorily deal
with the possible division of any number whatever into four
squares. Now the theorem of the possibility of such divi.siou
1 Journal Asiatique. Ciuqui^me s^rie, Tome v. p. 231.
2b DIOPHANTOS OF ALEXANDRIA.
is assumed by Diophantos in several places, notably in iv. 31.
We have then two alternatives. Either (1) the theorem was
not distinctly enunciated by Diophantos at all, or (2) It was
enunciated in a proposition of a lost Book. In either case
Abu'1-Wafa cannot have seen the statement of the theorem ia
Diophantos, and, if the latter alternative is right, we have an
argument in favour of the view that the work had already been,
mutilated before it reached the hands of Abu'1-Wafa. Now
Abu'l-Wafa's date is 328—388 of the Hegira, or 940—988 of
our Era.
It would seem, therefore, clear that the parts of Diophantos'
Arithmetics which are lost were lost at an early date, and
that the present lacunae and imperfections in the text had
their origin in all probability before the 10th century.
It may be said also with the same amount of probability
that the Porisms were lost before the 10th century a.d. We
have perhaps an indication of this in the title of another work
of Abu'1-Wafa, of which Wopcke's translation is " Demonstra-
tions des thdoremes employes par Diophante dans son ouvrage,
et de ceux employes par (Aboul-Wafa) lui-meme dans son com-
mentaire." It is not possible to conclude with certainty from
the title of this work what its contents may have been. Are
the " theorems " those which Diophantos assumes, referring for
proofs of them to his Porisms ? This seems a not unlikely sup-
position ; and, if it is correct, it would follow that the proofs
of these propositions, which Diophantos must have himself
given, in fact, the Porisms, were no longer in existence in
the time of Abu'I-Wafa, or at least were lor him as good as lost.
It must be admitted then that we have no historical evidence
of the existence at any time subsequent to Diophantos himself
of the Porisms.
Of the treatise on Polygonal Numhers we possess only a
fragment. It breaks off' in the middle of the 8th proposition.
It is not however probable that much is wanting; practically
the treatise seems to be nearly complete.
§ 2. The next (juestion which naturally suggests itself is :
As we have apparently six books only of the Arithmetics out of
thirteen, where may we suppose the lost matter to have been
HIS WORKS. 27
placed in the treatise? Was it at tlie beginning, micUHe, or
end? This question can only be decided when we have come
to a conclusion about the probable contents of the lost p<jrtion.
It has, however, been dogmatically asserted by many who have
written upon Diophantos — often without reading him at all, or
reading him enough to enable them to form a judgment on the
subject — that the Books, which we have, are the Jirst 9ix and
that the loss has been at the end; and such have accordingly
wondered what could have been the subject to which Diophantos
afterwards proceeded. To this view, which has no ground save
in the bare assertions of incompetent or negligent writers,
Nesselmann opposes himself very strongly. He maintains on
the contrary, with much reason, that in the sixth Book
Diophantos' resources are at an end. If one reads carefully
the last four Books, from the third to the sixth, the conclusion
forces itself upon one that Diophantos moves in a rigidly defined
and limited circle of methods and artifices, that any attempts
which he makes to free himself are futile. But this fact can
onl}^ be adequately appreciated after a perusal of his entire
work. It may, however, be further added that the sixth Book
forms a natural conclusion to the whole, in that it is made up
of exemplifications of methods explained and used in the pre-
ceding Books. The subject is the finding of right-angled
triangles in rational numbers, such that the sides satisfy given
conditions, Arithmetic being applied to Geometry in the geo-
metrical notion of the right-angled triangle. As was said
above, we have now to consider Avhat the contents of the lost
Books of the Arithmetics may have been. Clearly we must
first inquire what is actually wanting which we should have
expected to find there, either as promised by the author
himself in his own work, or as necessary for the elucidation or
completion of the whole. We must therefore briefly indicate
the general contents of the work as we have it.
The first book contains problems leading to determinate
equations of the first degree'; the remainder of the work being
1 As a specimen of the rash way in which even good writers speak of Dio-
phantos, I may instance here a remark of Viucenzo Riccati, who says: "De
problematibus determiuatis quae rcsulutis aequatiouibus dignoscuutur, nilill
28 DIOPH.\^TOS OF ALEXANDRIA.
a collection of problems which, with scarcely an exception, lead
to indeterminate equations of the second degree, beginning with
simpler cases and advancing step by step to more complicated
questions. These indeterminate or semideterminate problems
form the main feature of the collection. Now it is a great step
from determinate equations of the first degree to semideter-
minate and indeterminate problems of the second; and we must
recognise that there is here an enormous gap in the exposition.
We ought surely to find here (1) determinate equations of the
second degree and (2) indeterminate equations of the first.
With regard to (2), it is quite true that we have no definite
statement in the work itself that they formed part of the
writer's plan; but that they were discussed here is an extremely
probable supposition. With regard to (1) or determinate
quadratic equations, on the other hand, we have certain
evidence from the writer's own words, that the solution of the
adfected or complete quadratic was given in the treatise as it
originally stood ; for, in the first place, Diophantos promises a
discussion of them in the introductory definitions (def. 11)
where he gives rules for the reduction of equations of the
second degree to their simplest forms; secondly, he uses his
method for their solution in the later Books, in some cases
simply giving the result of the solution without working it out,
in others giving the irrational part of the root in order to find
an approximate value in integers, without writing down the
actual root\ We find examples of pure quadratic equations
oninino Diophantus (!); agit duntaxat de eo problematum semidetenninatorum
genere, quae respiciimt quadrata, aut cubos numerorum, quae problemata ut
resolvantur, (juantitates radicales de industria sunt vitandae." Pref. to ana-
litiche istituzioni.
^ These being tbe indications in the work itself, what are we to think of a
recent writer of a History of Mathematics, who says: "Hieraus und aus dem
Umstand, dass Diophant nirgends die von ihm versprochene Theorie dcr
Auflosung der quadratiscben Gleichungen gibt, schloss man, er habe dieselbe
nicht gekannt, und bat desshalb den Arabern stets den Ruhm dieser ErtinJuug
zugctlieilt," and goes on to say that "nevertheless Nesselmann after a thorough
study of the work is convinced that D. knew the solution of the quadratic"?
It is almost impossible to imagine that these remarks are serious. The writer
is Dr Heinricli Suter, (Jcschichte d. Mathetmitischen WissemchaJ'ten. Zweite
Autliigf. Ziiricb, 1873.
HIS Wol^KS. 29
even in the first Book : a fact which shows that Diophantos
regarded them as in reality simple equations, taking, as he does,
the positive value of the root only. Indeed it would seem that
Diophantos adopted as his ground for the classification of these
equations, not the index of the highest power of the unknown
quantity contained in it, but the number of terms left in it
when it is reduced to its simplest form. His words are': "If
the same powers of the unknown occur on both sides but with
different coefficients we must take like from like until we have
one single expression equal to another. If there are on both
sides, or on either side, terms with negative coefficients, the
defects must be added on both sides, until there are the same
powers on both sides with positive coefficients, when we must
take like from like as before. We must contrive always, if
possible, to reduce our equations so that they may contain one
single term equated to one other. But afterwards we will
explain to you also hoiu, luhen two terms are left equal to a
- Diophantos' actual words (which I have trauslated freely) are: MtrA 5^
Tavra eav d-rrb irpo^Xr^^iaTos tlvos -yh-qrai virap^ii eldeffi rots avroh jurj ofioTrXTjBfj
5^ dirb eKar^puv twv fiepuiv, deriaa a.<paipe'it> to. ofxoia dir6 twu 6/xoiwv, ?a)S &v ^»'ds(!)
elSoj €pI eidei tjov -yiv-qTaf eav de ttws if OTror^pu} ivvirapxTJ^^), ^ ^v dfKporipois
iveWei^f/r] (?) rivk etSr), de-qaei irpoaBelvaL to, Xeivovra etOT) if dfjLcpor^poii roh
fxipeaii', ews Slp eKarepij) tQv fxepQiv rd ei5r] ivvirdpxovra. y^vrjTai. Kal TraXi;' a'^e-
Xelv Ta 6p.oi.a diro rwc onoiuf, ?ws &v eKarepij) tCiv fxepQ)V if eTooi KaTa\(i<p6rj.
ne(pCKoTex''''i)<^6w S^ tovto eu rats vvoffrdaecn twv irpordaiwv, lav eVS^^'n'ot. ?wi
hv if eldos €vl etdei tffov KaraKucpdrj. vcrrepov 5^ aoi del^o/xev Kal nUk dvo ddwv tcuv
ivi KaraXeKpO^vTUV to toiovtov XvcTai.
I give Bachet's text exactly, marking those places where it seems obviously
WTong. KCLTaXeL^drj should of course be KaTaXeicpO^.
It is worth observing that L. Kodet, in Journal Asiatique, Janvier, 1878, on
"L'Algebre d'Al-Ivliarizmi et les muthodes indienne et grecque," quotes this
passage, not from Bachet's text, but from the MS. which Bachet used. His
readings show the following variations :
Bachet. L. Rodet.
■yiv7)Tai yevq(T€TaL [?? How about the construc-
tion with idv ?]
virap^is Tiva icra
if elSos
iv X(L\l/ei.
ivbs el8os
ivtXXeixpri
[I doubt the latter word very much,
compounded as the verb is with the
prep, iu twice repeated.]
so DIOPHANTOS OF ALEXANDRIA.
third, such a question is solved." That is to say, "reduce when
possible the quadratic to one of the forms x = a, or x^ = b. I
will give later a method of solution of the complete equation
x^±ax=± b." Now this promised solution of the complete
quadratic equation is nowhere to be found in the Arithmetics
as we have them, though in the second and following Books
there are obvious cases of its employment. We have to decide,
then, where it might naturally have come; and the answer is
that the suitable place is between the first and second Books.
But besides the entire loss of an essential portion of Dio-
phautos' work there is much confusion in the text even of that
portion which remains. Thus clearly problems 6, 7, 18, 19 of
the second Book, which contain determinate problems of the
first degree, belong in reality to Book I, Again, as already re-
marked above, the problem enunciated in ii. 19 is not solved at
all, but the solution attached to it is a mere " dXKco^" of ii. 18.
Moreover, problems 1 — 5 of Book il. recall problems already
solved in i. Thus il. l = l. 34: ii. 2 = 1. 37: ii. 3 is similar
to I. 33 : II. 4 = I. 35 : li. 5 = I. 36. The problem i. 29 seems
also out of place in its present position. In the second Book a
new type of problem is taken up at il. 20, and examples of it
are continued through the third Book. There is no sign of a
marked division between Books ii. and ill. In fact, expressed
in modern notation, the last two problems of li. and the first
of III. are the solutions of the following sets of equations :
II. 35. x''+[x + y + z) = a^
y^+{x + y + z)=h''
:^ + [x -ir y -\- z) = c"
II. 36. x^—{x + y-\-z)= a- \
y--{x + y + z) = lA
z" -{x+y + z)=c- ]
III. 1. (x + y + z) -.!■' = a"
{x + y + z)~y' = U'
{x + y + z)-2' = c'
These follow perfectly naturally upon each other; and
therefore it is quite likely that our division between the two
HIS WORKS. 31
Books was not the original one. In fact tlie frequent occur-
rence of more definite divisions in tlie middle of the Books,
coupled with the variation in the Vatican Ms. which divides our
six Books into seven, seems to show that the work may have
been divided into even a larger number of Books originally.
Besides the displacements of problems which have probably taken
place there are many single problems which have been much
corrupted, notably the fifth Book, which has, as Nesselmann
expresses it\ been "treated by Mother Time in a very step-
motherly fashion". It is probable, for instance, that between
V. 21 and 22 three problems have been lost. In several other
cases the solutions are confused or incomplete. How the im-
perfections of the text were introduced into it we can only con-
jecture. Nesselmann thinks they cannot be due merely to the
carelessness of a copyist, but are rather due, at least in part, to
the ignorance and inexpertness of one who wished to improve
upon the original. The view, which was put forward by
Bachet, that our six Books are a redaction or selection made
from the complete thirteen by a later hand, seems certainly
untenable.
The treatise on Polygonal Numbers is in its subject related '
to the Arithmetics, but the mode of treatment is completely
different. It is not an analytical work, but a synthetic one ;
the author enunciates propositions and then gives their proofs ;
in fact the treatise is quite in the manner of Books vil. — X. of
Euclid's elements, the method of representing numbers by
geometrical lines being used, which Cossali has called linear
Arithmetic. This method of representation is only once used in
the Arithmetics proper, namely in the proposition v. 13, where
it is used to prove that if a; + 7/=l, and a; and y have to be so
determined that aj + 2, ?/ + 6 are both squares, we have to divide
the number 9 into two squares of which one must be > 2 and
< 3. From the use of this linear method in this one case in the
Anthmetics, and commonly in the treatise on Polygonal Numbers,
we see that even in the time of Diophantos the geometrical
representation of numbers was thought to have the advantage
1 "Namentlich ist in dicser Hinsicht daa fuufte Buch stiefmutterlich von dcr
Mutter Zeit behandelt woiden." p. 2GB.
3.2 DIOPHANTOS OF ALEXANDRIA.
of greater clearness. It need scarcely be remarked how opposed
this Greek method is to our modern ones, our tendency being
the reverse, viz., to the representation of lines by numbers. The
treatise on Polygonal Numbers is often, and probably rightly,
held to be one of the thirteen original Books of the Arithmetics.
There is absolutely no reason to doubt its genuineness ; which
remark would have been unnecessary but for a statement by
Bossut to the effect: "II avoit dcrit treize livres d' arithmetiques,
les six premiers (?) sont arrives jusqu'a nous : tons les autres
sont perdus, si, ndanmoins, un septieme, qu'on trouve dans
quelques(!) editions de Diophante, n'estpas de lui"; upon which
Reimer has made a note : " This Book on Polygonal Numbers is
an independent work and cannot possibly belong to the Collection
of Diophantos' Arithmetics^" This statement is totally un-
founded. With respect to Bossut's own remark, we have seen
that it is almost certain that the Books we possess are not the
first six Books ; again, the treatise on Polygonal Numbei's does
not only occur in some, but in all of the editions of Diophantos
from Xylander to Schulz ; and, lastly, Bossut is the only person
who has ever questioned its genuineness.
We mentioned above the Porisms of Diophantos. Our
knowledge of them is derived from his own words ; in three
places in the Arithmetics he refers to them in the words exo/j-ev
iv Tot<? iropiaixacnv : the places are V. 3, 5, 19. The references
made to them are for proofs of propositions in the Theory of
Numbers, which he assumes in these problems as known. It is
probable therefore that the Porisms were a collection of propo-
sitions concerning the properties of certain numbers, their
divisibility into a certain number of squares, and so on ; and it
is reasonable to suppose that from them he takes also the many
other propositions which he assumes, either explicitly enunciating
I them, or implicitly taking them for granted. May we not then
reasonably suppose the Porisms to have formed an introduction
to the indeterminate and semi-determinate analysis of the
second degree which forms the main subject of the A 7'ithmetics?
And may we not assume this introduction to have formed an
' "Dieses Buch de numeris multanguUn ist cine fiir sich bestehendc Schrift
und gehort keinesweges in die Sammluug der Arithmeticorum Diophant'e."
Ills WORKS. 3«?
integral part, now lost, of the original thirteen books ? If this
supposition is correct the Po7'isms also must have intervened be-
tween Books I. and ll., where we have already said that probably
Diophantos treated of indeterminate problems of the first
degree and of the solution of the complete quadratic. The
method of the Ponsms was probably synthetic, like the Poly-
gonal iVwrnfters, not (like the six Books of the Anthmetics)
analytical ; this however forms no sufficient reason for refusing
to include all three treatises under the single title of thirteen
Books of Arithmetics. These suppositions would account easily
for the contents of the lost Books ; they would also, with the
additional evidence of the division of our text of the Arithmetics
into seven books by the Vatican MS., show that the lost portion
probably does not bear such a large proportion to the whole as
might be imagined. This view is adopted by Colebrooke \ and
after him by Nesselmann, who, in support of his hypothesis
that the Arithmetics, the Porisms and the treatise on Polygonal
Numbers formed only one complete work under the general
title of dptd/jLTjTLKa, points out the very significant fact that we
never find mention of more than one work of Diophantos, and
that the very use of the Plural Neuter term, dpid/xrjTiKa, would
seem to imply that it was a collection of different treatises on
arithmetical subjects and of different content. Nesselmann, how-
ever, does not seem to have noticed an objection previously urged
^ Algebra of the Hindus, Note M. p. lxi.
"In truth the division of manuscript books is very uncertain: and it is by
no means improbable that the remains of Diophantus, as we possess tlicni, may
be less incomplete and constitute a larger portion of the thirteen books an-
nounced by him (Def. 11) than is commonly reckoned. His treatise on polygon
numbers, which is surmised to be one (and that the last of the thirteen), follows,
as it seems, the six (or seven) books in the exemi)lar8 of the work, as if the
preceding portion were complete. It is itself imperfect: but the manner is
essentially different from that of the foregoing books: and the solution of
problems by equations is no longer the object, but rather the demonstration
of propositions. There appears no gi-ouud, beyond bare surmise, to presume,
that the author, in the rest of the tracts relative to numbers which fulfilled
his promise of thirteen books, resumed the Algebraic manner: or in short,
that the Algebraic part of his performance is at all mutilated in the copies
extant, which are considered to be all transcripts of a single imperfect
exemplar."
H. D. 3
34 DIOPllANTOS OF ALEXANDRIA.
against the theory that the three treatises formed only one work,
by Schulz, to the effect that Diophantos expressly says that his
work treats of arithmetical problems^. This statement itself
does not seem to me to be quite accurate, and I cannot think
that it is at all a valid objection to Nesselmann's view. The
passage to which Schulz refers must evidently be the opening
words of the dedication by the author to Dionysios. Diophantos
begins thus: "Knowing that you are anxious to become ac-
quainted with the solution [or ' discovery,' eupecri?] of problems
in numbers, I set myself to systematise the method, beginning
from the foundations on which the science is built, the pre-
liminary determination of the nature and properties in numbers^."
Now these "foundations" may surely well mean more than is
given in the eleven definitions with which the treatise begins,
and why should not the "properties of numbers" refer to the
Porisms and the treatise on Polygonal Numbers .? But there is
another passage which might seem to countenance Schulz's
objection, where (Def. 11) Diophantos says "let us now proceed
to the propositions'... which we will deal with in thirteen Books\"
The word used here is not problem {Trpo/SXTj/xa) but proposition
(TrporaaL^;), although Bachet translates both words by the same
Latin word " quaestio," inaccurately. Now the word irporaaL^;
does not only apply to the analytical solution of a problem : it
applies equally to the synthetic method. Thus the use of the
word here might very well imply that the work was to contain
1 Schulz remarks on the Porisms (pref. xxi.): "Es ist daher nicht uuwahr-
Bcheinlich dass diese Porismeu eine eigene Schrift uuseres Diophautus wareu,
welche vorziiglich die Zusammensetzung dcr Zahlen aus gew-issen Bestaud-
theilen zu ihrem Gegeustando hattc. Kunnte man diesc Schrift gar als eine
Bestandtheil des grossen in dreizehn Biichern abgefassten arithmetischen
Werkes anseheu, so wiire es sehr erkliirbar, dass gerade dieser Theil, der den
blossen Liebhaber weniger anzog, verloren ging. Da indess Diophantus aus-
driickiich sagt, sein Werk behandele arithmetische Probleme, so hat weuigstens
die letztere Annahme nur einen geringen Grad von Wahrscheinhchkeit."
* Diophantos' own words are: Tiju tvptcnv twv iv roh apid/ioTs Trpo^XijfidTuiv,
TifU(l)TaT^ fiOL AiovOffie, -yivilKTKtiiv ae cnrovdalus ^xovra naOuv, opyavwcrai r^c /j^dodov
iweipdOrji', dp^ofKifOS d(f>' uv avviarrjKe rd Trpaynara 0e(jif\lii)v, vTroaTTJffai Trjv iv tois
dptOfioh tpvffiv T« Kal Swaniv.
•* vvv 5^ iirl rds irpordans x^RV'^'^t^^"' '^- '''■ ^•
* T^s irpay/xareiai avrQv kv TpiffKaldfKa fii^Xlois yiyivripiivr)s.
HIS WORKS. ;i5
not only problems, but propositions on numbers, i.e. miglit
include the Po7'isms and Polygonal Numbers as a part of the
complete Arithmetics. These objections which I have made
to Schulz's argument are, I think, enough to show that his
objection to the view adopted by Nesselmaun has no weight.
Schulz's own view as to the contents of the missing Books of
Diophantos is that they contained new methods of solution in
addition to those used in Books I. to vi., and that accordingly
the lost portion came at the end of the existing six Books. In
particular he thinks that Diophantos extended in the lost Books
the method of solution by means of what he calls a double-
equation {Bi7r\r] laorr]^ or in one word hi,Tr\ola6rr}<;). By means
of this double-equation Diophantos shows how to find a value
of the unknown, which will make two expressions containing it
(linear or quadratic) simultaneously squares. Schulz accordingly
thinks that he went on in the lost Books to show how to make
three such expressions simultaneously squares, i.e. advanced to a
triple-equation. This view, however, seems to have nothing to
recommend it, inasmuch as, in the first place, we nowhere find
the slightest hint in the extant Books of anything different or
more advanced which is to come ; and, secondly, Diophantos'
system and ideas seem so self-contained, and his methods to
move always in the same well-defined circle that it seems
certain that we come in our six Books to the limits of his art.
There is yet another view of the probable contents of the
lost Books, which must be mentioned, though we cannot believe
that it is the riglit one. It is that of Bombelli, given by Cossali,
to the effect that in the lost Books Diophantos went on to solve
determinate equations of the third and fourth degree; Bombelli's
reason for supposing this is that Diophantos gives so many
problems the object of which is to make the sum of a square
and any other number to be again a square number by finding
a suitable value of the first square ; these methods, argues
Bombelli, of Diophantos must have been given for the reason
that the author intended to use them for the solution of the
equation x*-\-px = q^. Now Bombelli had occupied himself
1 Cossali's words are (p. 75, 76):..."non tralascier<i di notarc 1' opinionc, di
cui fu teutato Bombelli, chc nclli soi libri cioe diil tempo, di tutto distrufe'gitore,
3—2
36 DIOPHAXTOS OF ALEXANDRIA.
much, almost during his whole life, with the then new methods
of solution of equations of the third and fourth degree ; and, for
the solution of the latter, the usual method of his time led to
the making an expression of the form Ax^+ Bx + C a. square,
where the coefficients involved a second unknown quantity.
Nesselmann accordingly thinks it is no matter for surprise that
in Diophantos' entirely independent investigations Bombelli
should have seen, or fancied he saw, his own favourite idea.
This solution of the equation of the fourth degree presupposes
that of the cubic with the second term wanting ; hence Bombelli
would naturally, in accordance with his view, imagine Diophantos
to have given the solution of this cubic. It is possible also that
he may have been influenced by the actual occurrence in the
extant Books [vi. 19] of a cubic equation, namely the equation
x^ + x = 4x^ + 4, of which Diophantos at once writes down the
solution a; = 4, without explanation. It is obvious, however,
that no conclusion can be drawn from this, which is a very
easy particular case, and which Diophantos probably solved^ by
simply dividing out by the factor x^+l. There are strong
objections to Bombelli's view. (1) Diophantos himself states
(Def. XI.) that the solution of the problems is the object in itself
of the work. (2) If he used the method to lead up to the
solution of equations of higher degrees, he certainly has not gone
to work the shortest way. In support of the view it has been
asked "What, on any other assumption, is the object of defining
in Def ll. all powers of the unknown quantity up to the sixth ?
rapitici, si avanzasse egli a sciogliere 1' equazionc x*+px-q, parendogli, die
nei libri riinastici, con proporsi di trovar via via numeri quadrati, cammini una
strada a qucU' intento. Egli e di fatto procedendo sn queste tracce di Diofanto,
che Vieta deprime 1' esposta equazione di giado quarto ad una di secondo.
Siccome pen"!* cio non si effettua che mediante una cubica mancante di secondo
termine; cosi il pcnsiero sorto in auimo a Bombelli iniporterebbc, che Diofanto
nei libri perduti costituito avesse la regola di sciogliere questa sorta di equa-
zione cubicbe prima d' innoltrarsi alio scioglimento di quella equazione di quarto
grado."
' This is certainly a simpler explanation than Bachet's, who derives the
solution from the proportion ar* : .x-=x : 1.
Therefore x' + x : x- + l = .r : 1.
Therefore x^ + x : 4x^ + i = x : i.
But the equation being .r''-)-.r = 4j-'-' + 4, it follows that x-i.
HIS WORKS. 37
Surely Diophantos must have meant to use them." The answer
to which is that he has occasion to use them in the work, but
reduces all the equations which contain these higher powers by
his regular and uniform method of analysis.
In conclusion, I may repeat that the most probable view is
that adopted by Nesselmann, that the works which we know
under the three titles formed part of one arithmetical work,
which was, according to the author's own words, to consist of
thirteen Books. The proportion of the lost parts to tlie whole
is probably less than it might be supposed to be. The Ponsnis
form the part, the loss of which is most to be regretted, for
from the references to them it is clear that they contained
propositions in the Theory of Numbers most wonderful for the
time.
CHAPTER III.
THE WRITERS UPON DIOPHANTOS.
§ 1. In this chapter I purpose to give a sketch of what has
been done directly, and (where it is of sufficient importance)
indirectly, for Dioph antes, enumerating and describing briefly
(so far as possible) the works which have been written on the
subject. We turn first, naturally, to Diophantos' own country-
men ; and we find that, if we except the doubtful " commentary
of Hypatia," spoken of above, there is only one Greek, who has
written anything at all on Diophantos, namely the monk Maxi-
mus Planudes, to whom are attributed the scholia attached to
Books I. and ii. in some MSS., which are printed in Latin in
Xylander's translation of Diophantos. The date of these scholia
is the first half of the 14th century, and they represent all that
we know to have been done for Diophantos by his own country-
men. How different his fate would have been, had he lived a
little earlier, when the scientific spirit of the Greeks was still
active, what an enormous impression his work would then have
created, we may judge by comparing the effect which it had
with that of a far less important work, that of Nikomachos.
Considering then that up to the time of Maximus Planudes
nothing was written about Diophantos (beyond a single quota-
tion by Theon of Alexandria, before mentioned, and an occa-
sional mention of the name) by any Greek, one is simply
astounded at finding in Bossut's history a remark like the
following : " L'auteur a eu parmi les anciens une foule d'inter-
prfetes (!), dont les ouvrages sont la plupart (!) perdus. Nous
regrettons, dans ce nombre, le commentaire de la cdlebre
Hipathia (sic)." Comment is unnecessary. With respect to
THK WRITKHS UPON DK H'HANTc )S. :][)
the work of Maximus Planudes itself, he has only commented
upon the first two Books, the least important and most olomen-
tary, nor can his scholia be said to have any importance.
Bachet speaks contemptuously of them\ and even the modest
Xylander has but a low opinion of their value^
§ 2. I have, in first mentioning Maximus Planudes, de-
parted a little from chronological order, for the greater con-
venience of giving first the Greek writers upon Diophantos.
But long before the time of Maximus Planudes, the work of
Diophantos had found its way to Arabia, and there met with
the respect it deserved. Unfortunately the actual works writ-
ten in Arabia directly upon Diophantos are all lost, or at lea.st
have not been discovered up to the present time. So far there-
fore as these are concerned we have to be contented with the
notices on the subject by Arabian historians or bibliographers.
It is therefore necessary to collect from the earliest and best
sources possible the scattered remarks about Diophantos and
his works. The earliest and therefore presumably the best and
most trustworthy authority on the subject of Diophantos in
Arabia is the Kifab Alfihrist of al-Nadim', the date of which
is as early as circa 990 a,d. The passages in this work which
refer to Diophantos are :
(a) p. 269, "Diophantos [the last vowel, however, being
I = t; in one codex, in the rest undetermined] the Greek of
1 Bachet says: "Porro Graeci Sclioliastae in duos priores libros adnota-
tiones edi non curauimus, vt quae nullius sint momenti, easque proinde
Guilielmus Xilander(!) censura sua meritd perstrinxerit, si cut tamen oleum
operamque perderc a(le7) leue est, vt miras GraecuU huius ineptias peruidcre
cupiat, adeat Xilandrum."
- Xylander says the Scholia are attributed to Maximus Planudes, and com-
bats the view that they might be Hjiiatia's thus: "Sed profecto si ea tanta
fuit, quantam Suidas et alij perhibent, istae annotationes cam autorcm non
apnoscunt, de quibus quid senserim, raeo more liberfe dixi suis locis." Kpistola
Nuncupatoria.
3 This work has been edited by Fliigel, 1871. The author himself dates
it 987, and Wcipcke (Journal Asiatique, F^vrier-Mars, 18.55, p. 2.56) states that
it was finished at that date. This is, however, not correct, for in his preface
Fliigel shows that the work contains references to events which are certainly
later than <)87, so that it seems best to say simply that the date is cir<-a
990 A.I).
40 DIOPHANTOS OF ALEXANDRIA.
Alexandria. He wrote Kitab Sina'at al-jabr," i.e. "the book of
the art of algebra."
(h) p. 283, Among the works of Abii'1-Wafa is mentioned
"An interpretation' (tafsir) of the book of Diophantos about
algebra."
(c) On the same page the title of another work of Abu'l-
Wafa is given as " Demonstrations of the theorems employed
by Diophantos in his work, and of those employed by (Abu'l-
Wafa) himself in his commentary " (the word is as before
tafsir).
{d) p. 295, On Kosta ibn Luka of Ba'lbek it is mentioned
that one of his books is tafsir on three-and-a-half divisions
(Makalat) of the book of Diophantos on " questions of numbers."
We have thus in the Fihrist mentions of three separate
works upon Diophantos, which must accordingly have been
written previously to the year 990 of our era. Concerning
Abu'1-Wafa the evidence of his having studied and commented
upon Diophantos is conclusive, not only because his other works
which have survived show unmistakeable signs of the influence
of Diophantos, but because the proximity of date of the Fihrist
to that of Abu'1-Wafa makes all mistake impossible. As I have
said the Fihrist was written circa 990 A.D. and the date of
Abu'1-Wafa is 328—388 a.h. or 940—998 A.D. He was a
native of Buzjan, a small town between Herat and Nishapur in
Khorasan, and was evidently, from what is known of his works
one of the most celebrated astronomers and geometers of his
time^. Of later notices on this subject we may mention those
' There is a little doubt as to the exact meaning of tafsir — whether it means
a translation or a commentary. The word is usually applied to the literal exe-
gesis of the Koran ; how much it means in the present case may perhaps be
ascertainable from the fact that Abu'1-Wafa also wrote a tafsir of the Algebra of
Mohammed ibn Musa al-Khruizml. It certainly, according to the usual sense,
means a commentary not a mere translation — e.g. at p. 249 al-Nadim clearly
distinguishes translators of Aristotle from the mufassirln or makers of tafsir, i.e.
commentators.
For this information I am indebted to the kindness of Professor Robertson
Smith.
- Wcipcke, Journal Asiatiqne, Ft'vrier-Mars, 1855, p. 244 foil.
Abu'l-Wafa's full name is Mohammed ibn Mohammed ibn Yahya ibn Ismail
ibn Al'abbfis Abu'1-Wafa Al-Iiuzjani.
THE WRITERS UPON DIOPHANTOS. 41
in the Tarlkh Ilokoma (Hajji-Khalifa, No. 2204), by tho Imam
Mohammed ibn. 'Abd al-Karim al-Shahrastani wlio died A.ii. 548
or A.D. 1153\ Of course this work is not so trustworthy an
authority as the Fihrist, which is about 160 years earlier, and
the author of the Tarlkh HoJcoma stands to the Fihrist in the
relation of a compiler to the original source. In the Tarikh
Hokoma we are told {a) that Abu'1-Wafa " wrote a commen-
tary on the work of Diophantos concerning Algebra," (6) that
'• Diophantos, the Greek of Alexandria, conspicuous, perfect,
famous in his time, wrote a famous work on the art of
Algebra, which has gone over into Arabic," i. e. been trans-
lated. We must obviously connect these two notices. Lastly
the same work mentions (c) another work of Abu'1-Wafa,
namely '• Proofs for the propositions given in his book by
Diophantos."
A later writer still, the author of the History of the Dynas-
ties, Abu'lfaraj, mentions, among celebrated men who lived in
the time of Julian, Diophantos, with the addition that " His
book^..ou Algebra is celebrated," and again in another place
he says upon Abu'1-Wafa, " He commented upon the work of
Diophantos on Algebra."
The notices from al-Shahrastani and Abu'lfaraj are, as I have
^ The work Bibliotheca arabico-hispana Escurialensis op. et studio Mich.
Casiri, Matriti, 1760, gives many important notices about mathematicians
from the Ta'rikh Hokoma, which Casiri denotes by the title Bibliotheca philo-
sophorum.
Cossali mentions the Ta'rikh Hokoma as having been written about a.d. 119^!
by an anonymous person: "II hbro piti antico, che ci fornisca tratti relativi
all' origine dell' analisi tra gli arabi e la Bihlioteca arabica de' jilosoji, scritta
circa 1' anno 1198 da anonimo egiziano" (Cossali, i. p. 174). There is however
now apparently no doubt that the author was al-Shahrastani, as I have said in
the text.
2 After the word "book" in the text comes a word Ab-kismet which is un-
intelligible. PocoQke, the Latin translator, simply puts A. B. for it: "cuius liber
A. 13. quern Algebram vocat, Celebris est." The word or words are apparently
a corruption of something ; Nesselmann conjectures that the original word was
an Arabic translation of the Greek title, Arithmetics— a supposition which, if
true, would give admirable sense. The passage would then mark the Arabian
perception of the discrepancy (according to the accepted meaning of termn)
between the title and the subject, which is obviously rather algebra than arith-
metic in the strict sense.
42 DIOPHANTOS OF ALEXANDRIA.
said, for obvious reasons not so trustworthy as those in the
Fihrist. They are, however, interesting as showing that Dio-
phantos continued to be kno^vn and recognised for a consider-
able period after his work found its way to Arabia, and was
commented upon, though they add nothing to our information
as to what was done for Diophantos in Arabia. It is clear that
the work of Abu'1-Wafa was the most considerable that was
written in Arabia upon Diophantos directly ; about the obliga-
tions to Diophantos of other Arabian writers, as indirectly
shown by similarity of matter or method, without direct refer-
ence, I shall have to speak later.
§ 3. I now pass to the writers on Diophantos in Europe.
From the time of Maximus Planudes to a period as late as
about 1570 Diophantos remained practically a sealed book, and
had to be rediscovered even after attention had been invited to
it by Regiomontanus, who, as was said above, was the first
European to mention it as extant. We have seen (pp. 21, 22)
that Regiomontanus referred to Diophantos in the Oration at
Padua, about 1462, and how in a very interesting letter to
Joannes de Blanchinis he speaks of finding a MS. of Diophantos at
Venice, of the pleasure he would have in translating it if he could
only find a copy containing the whole of the thirteen books, and
his readiness to translate even the incomplete work in six books,
in case it were desired. But it does not appear that he ever
began the work ; it seems, however, very extraordinary that the
interest which Regiomontanus took in Diophantos and tried to
arouse in others should not have incited some of his German
countrymen to follow his leading, at least as early as 1537,
when we know that his Oration at Padua was published. Hard
to account for as the fact may appear, it was left for an Italian,
Bombelli, to rediscover Diophantos about 1570; though the
mentions by Regiomontanus may be said at last to have borne
their fruit, in that about the same time Xylander was en-
couraged by them to persevere in his intention of investigating
Diophantos. Nevertheless between the time of Regiomontanus
and that of Rafael Bombelli Diophantos was once more for-
gotten, or rather unknown, for in the interval we find two
mentions of tlie name, (a) b} Joachim Camerarius in a letter
THE WRITERS UPON DIOPHANTOS. 43
published 1556\ in which he mentions that there is a MS. of
Diophantos in the Vatican, which he is anxious to see, (6) by
James Peletarius^ who merely mentions the name. Of the
important mathematicians who preceded Bombelli, Fra Luca
Pacioli towards the end of the loth century. Cardan and Tar-
taglia in the 16th, not one so much as mentions Diophantos^
The first Italian to whom Diophantos seems to have been
known, and who was the first to discover a MS. in the Vatican
Library, and to conceive the idea of publishing the work, was
Rafael Bombelli. Bachet falls into an anachronism when he
says that Bombelli began his work upon Diophantos after the
appearance of Xylander's translation*, which was published in
1575. The Algebra of Bombelli appeared in 1572, and in the
^ De Graecis Latinisque tiumerorum notis et praeterea Saracenis sett Indicts,
etc. etc., studio Joachimi Camerarii, Papeberg, 1556.
In a letter to Zasius : " Venit mihi in nientem eorum quae et de bac et aliis
liberalibus artibus dicta fuere, in eo convivio cujus in tuis aedibus me et Peuce-
rum nostrum participes esse, suavissima tua invitatio voluit. Cum autem de
autoribus Logistices verba fierent, et a me Diophantus Graecus nominaretur, qui
extaret in Bibliotbeca Vaticana, ostendebatur turn spes quaedam, posse nobis
copiam libri illius. Ibi ego cupiditate videndi incensus, fortasse audacius non
tamen iiifeliciter, te quasi procuratorem constitui negotii gerendi, mandate
voluntario, cum quidem et tu libenter susciperes quod imponebatur, et fides
solenni festivitate firmaretur, de illo tuo et poculo elegante ct vino optimo.
Neque tu igitur oblivisceris ejus rei, cujus explicationem tua benignitas tibi
commisit, neque ego non meminisse potero, non modo excelleutis \-irtuti8 ct
sapientiae, sed singularis comitatis et incredibilis suavitatis tuae."
- Arithmeticae practicae methodus facilis, per Gemmani Frisium, etc. Hue
accedunt Jacobi Peletarii annotationes, Coloniae, 1571. (But pref. of Peletarius
bears date 1558.) P. 72, Nota Peletarii: "Algebra autem dicta videtur a Gebro
Arabe ut vox ipsa sonat ; hujus artis si non inventore, saltern excultore. Alii
tribuunt Diophanto cuidam Graeco."
'■' Cossali I. p. 59, "Cosa pero, che reca la somma maraviglia si 6, che largo
in Italia non si spandesse la cognizionc del codice di Diofanto : che in fiore
essendovi lo studio della greca lingua, non veuisse da qualche dotto a coman
vantaggio tradotta; che per 1' opposto niuna menzione ne faccia Fra Luca verso
il fine del secolo xv, e niuna Cardano, e Tartaglia intorno la metA del secolo
XVI ; che nelle biblioteche rinianesse sepolto, ed andassc dimenticato per modo,
che poco prima degli anni 70 del secolo xvi si riguardasse per una scoperta
1' averlo rinvonuto nella Vaticana liiblioteca."
■» "Non longo post Xilandrum interuallo llaphael Bombellius Bononiensis,
Graecum e Vaticana Bibliotheca Diophanti codicem nactus, omnes priorum
quattuor librorum quaestiones, et 6 libro quinto nonuuUas, probk-matibus uiub
iuseruit, in Algebra sua quam Italico sermono conwcripsit."
44 DIOPHANTOS OF ALEXANDRIA.
preface to this work * the author tells us that he had recently
discovered a Greek book on Algebra in the Vatican Library,
written by a certain Diofantes, an Alexandrine Greek author
who lived in the time of Antoninus Pius ; that, thinking highly
of the contents of this work, he and Antonio Maria Pazzi de-
termined to translate it ; that they actually translated five
books out of the seven into wliich the MS. was divided ; but
that, before the whole was finished, they were called away from
it by other labours. The date of these occurrences must be a
few years before 1572. Though Bombelli did not carry out his
plan of publishing Diophantos in a translation, he has neverthe-
less taken all the problems of Diophantos' first four Books and
some of those of the fifth, and embodied them in his Algebra,
interspersing them with his own problems. Though he has
taken no pains to distinguish Diophantos' problems from his
own, he has in the case of Diophantos' work adhered pretty
closely to the original, so that Bachet admits his obligations to
Bombelli, whose reproduction of the problems of Diophantos he
maintains that he found in many points better than Xylander's
translation'^ It may be interesting to mention a few points of
1 This book Nesselmann tells ns that he has never seen, but takes his infor-
mation about it from Cossali. I was fortunate enough to find a copy of it
published in 1579 (not the original edition) in the British Museum, the title
being U Algebra, opera d'l Rafael BomheUi da Bolorjiia diiiisa in tre Libri In
Bologna, Per Giovanni Rossi. MDLXXIX. I have thus been able to verify the
quotations from the preface. The whole passage is :
"Questi anni passati, essendosi ritrouato una opera greca di questa disciplina
nella libraria di Nostro Signore in Vaticano, composta da un certo Diofante
Alessandrino Autor Greco, il quale fCl a tempo di Antonin Pio, e havendo mela
fatta vedere Messer Antonio Maria Pazzi Reggiano publico lettore delle Matema-
tiche in Roma, e giu dicatolo con lui Autore assai intelligente de numeri (an-
corche non tratti de numeri irrationali, ma solo in lui si vcde vn perfetto ordine
di opcrare) egli, ed io, per arrichirc il mondo di cosi fatta opera, ci dessimo a
tradurlo, e cinque libri (delli sette che souo) tradutti ne habbiamo ; lo restaute
non haueudo potuto finire per gli trauagli aueuuti all' uno, e all' altro, e in detta
opera habbiamo ritrouato, ch' egli assai volte cita gli Autori Indiani, col che mi
ha fatto conoscere, che questa disciplina appo gl' indiani prima ih, che a gli Arabi."
The parts of this quotation which refer to the personality of Diophantos, the
form Diofante, &c., have already been commented upon ; the last clauses we
shall have occasion to mention again.
'^ Continuation of quotation in note 4, p. 43 :
"Sed suas Diophanteis quaestionibus ita immiecuit, ut has ab illis distiu-
THE WKITKKS UPON Dl< )1'|IANT()S. 45
notation in this work of Bombclli. At bef,dnning of Book ii he
explains that he uses the word "tanto" to denote the unknown
quantity, not "cosa" like his predecessors; and his symbol for
it is i, the square of the unknown (x-) is c., the cube i; and so on.
For plus and minus (pm and meno) he uses the initial letters p.
and m. Thus corresponding to x + Q we should find in Bom-
belli 11 p. 6, and for .r + ox-4<, 11 p. 5i m. 4. This notation
shows, as will be seen later, some advance upon that of Dio-
phantos in one important respect.
The next writer upon Diophantos was ^Vilhelm Holznianu
who published, under the Graecised form of his name Xylander
by which he is generally known, a work bearing the title :
Diophanti Alexandrini Rerum Arithmetical' um LibH sex, quo-
rum primi duo adiecta hahent Scholia Maximi {ut coniectura
est) Plamtdis. Item Liber de Numeris Polygonis seu Multan-
gulis. Opus incomparahile, uerae Arithmeticae Logisticae per-
fectionem continens, paucis adhuc uisum. A Guil. Xylandro
Augustano incredihili lahore Latine redditum, et Commentariui
explanatum, inque lucem editum ad Illustriss. Principem Ludo-
vicum Vuirtemhergensem Basileae per Eusebium Episcopium, et
Nicolai Fr. haeredes. mdlxxv. Xylander was according to his
own statement a " public teacher of Aristotelian philosophy in
the school at Heidelberg\" He was a man of almost universal
culture ^ and was so thoroughly imbued with the classical litera-
ture, that the extraordinary aptness of his quotations and his
wealth of expression give exceptional charm to his writing
whenever he is free from the shackles of mathematical formulae
and technicalities. The Epistola Nuncupatoria is addressed
to the Prince Ludwig, and Xylander neatly introduces it by
the line "Offerimus numeros, numeri sunt principe digni." This
preface is very quaint and interesting. He tells us how he
first saw the name of Diophantos mentioned in Suida.s, and
guere non sit in promptu, neque vefD se fidum satis interpretem pracbuit, cum
passim verba Diophanti immutet, bisque pleraque addat, plcraque pro arbitrio
detrahat. In multis nihilominus interprctationem Bombellii, Xilandriana prac-
stare, et ad banc emendandam me adjuvisse iu(,'enue fateor." Ad Icctorem.
' " Publicus pbilosophiae Aristoteleae in schola Hcidolbergcnsi doctor."
- Even Bachet, who, as wo shall see, was no favourable critic, calls him " Vir
omnibus disciplinis excultus."
46 DIOPHANTOS OF ALEXANDRIA.
then found that mention had been made of his work by Regio-
montanus as being extant in an Italian Library and having
been seen by him. But, as the book had not been edited, he
tried to reconcile himself to the want of it by making himself
acquainted with the works on Arithmetic which were actually
known and in use, and he apologises for what he considers to
have been a disgrace to him\ With the help of books only he
studied the subject of Algebra, so far as was possible from what
men like Cardan had written and by his own reflection, with
such success that not only did he fall into what Herakleitos
called olrjo-Lv, lepav voaov, or the conceit of " being somebody " in
the field of Arithmetic and " Logistic," but others too who were
themselves learned men thought him (as he modestly tells us)
an arithmetician of exceptional merit. But when he first
became acquainted with the problems of Diophantos (he con-
tinues) his pride had a fall so sudden and so humiliating that he
might reasonably doubt whether he ought previously to have
1 I cannot refrain from quoting the whole of this passage :
"Sed cum ederet nemo : cepi desideriimi hoc paulatim in animo consopire, et
eorum quos consequi poteram Arithmeticorum librorum cognitions, et medita-
tionibus nostris sepelire. Veritatis porro apud me est autoritas, ut ei con-
iunctum etiam cum dedecore meo testimonium lubentissime perhibeam. Quod
Cossica seu Algebrica (cum his enim reliqua comparata, id sunt quod umbrae
Homeric^ in Necya ad aniniam Tiresiac) ca ergo quod nou assequebar modo,
quanquam mutis duntaxat usus preceptoribus caetera ai)To5/5a/cTos, sed et augers,
uariare, adeoque corrigere in loco didicissem, quae summi et fidelissimi in
docendo uiri Christifer Eodolphus Silesius, Micaolus, Stifelius, Cardanus, No-
nius, aliique litteris mandauerant : incidi in otrjaiv, lepav vbaov, ut scitfe appel-
lauit Heraclitus sapientior multis aliis philosophis, hoc est, in Arithmetica, et
uera Logistica, putaui me esse aliquid: itaque de me passim etiam a multis,
iisque doctis uiris iuJicatum fuit, me non de grege Arithmeticum esse. Verum
ubi primum in Diophantca incidi : ita me recta ratio circumegit, ut flenddsne
mihi ipsi antca, an uero ridendus fuissem, haud iniuria dubitaucrim. Operas
preciuni est hoc loco et meam inscitiam inuulgarc, et Diophantei operis,
quod mihi ncbulosam istam caligincm ab oculis detcrsit, immo cos in coenum
barbaricum defossos eleuauit et repurgauit, gustum aliquem exhibere. Surdorum
ego uumerorum tractationem ita tenebam, ut etiam addere alioriun inuentis
aliquid non poenitendum auderem, atque id quidem in rebus arithmeticis mag-
num habetur, et difficultas istarum rerum multos a mathematibus deterret.
Quanto autem hoc est praeclarius, in iis problematis, quae surdis etiam numsris
uix posse uideutur explicari, rem eo deduccre, ut quasi solum arithmeticum
ucrtere iussi obsurdescant illi plane, et ne mentio quidem eorum in tractatione
inguniosissimarum quaustiouum admittatur."
THE WRITERS I'PON DlOl'llANTOS. 47
bewailed, or laughed at himself. He considers it therefore
worth while to confess publicly in how disgraceful a condition
of ignorance he had previously been content to live, anil to do
something to make known the work of Diophantos, which had
so opened his eyes. Before this critical time he was so familiar
with methods of dealing with surds that he actually had ventured
to add something to the discoveries of others relating to them ;
these were considered to be of great importance in questions
of Arithmetic, and their difficulty was of itself sufficient to
deter many from the study of Mathematics. " But how much
more splendid " (says Xylander) " the methods which reduce
the problems which seem to be hardly capable of solution even
with the help of surds in such a way that, while the surds, when
bidden (so to speak) to plough the arithmetic soil, become true
to their name and deaf to entreaty, they are not so much as
mentioned in these most ingenious solutions ! " He then de-
scribes the enormous difficulties which beset his work owing
to the coiTuptions in his text. In dealing, however, with the
mistakes and carelessness of copyists he was, as he says, no
novice; for proof of which he appeals to his editions of Plutarch,
Stephanus and Strabo. This passage, which is delightful read-
ing, but too long to reproduce here, I give in full in the note '.
> " Id uero mihi accidit durum et uix superabile incommodum, quM mirificft
deprauata omnia inueni, ctim neque problematum expositio interdum integra
esset, ac passim numeri (iu quibus sita omnia esse in hoc arRumento, quia
ignorat?) tarn problematum quam solutionum siue cxplicatiouum corruptissimi.
Non pudebit me ingenue fateri, qualem me heic gesserim. Audacter, et summo
cum feruore potius qusim alacritate auimi opus ipsum initio sum aggressus,
laborque mihi omnis uoluptati fuit, tantus est meus rernm arithmoticarum amor,
quin et gratiam magnam me apud omnes liberalium scientiarum amatores ac
patronos initiirum, et praeclare de rep. litteraria nierituriim intclligebam,
eamque rem mihi laudi (quam ii bonis profectam nemo prudens aspernatur)
gloriaeque fortasse etiam emolumento fore sperabam. Progressus aliquantulum,
in salebras incidi : quae tantum abest ut alacritatem meam retuderint, ut etiam
animos milii addiderint, neque enim mihi novum aut insolons est aduersus
librariorum incuriam certamen, et hac in re militaui, (ut Horatii nostri uerbis
utar) non sine gloria, quod me non arroganter dicere, Dio, Plutarchu.^, Strabo,
Stephanusque nostri testantur. Sed cum mox in ipsum pelagus nionstris scatons
me cmsus abripuit: non dcspondi equidem animum, neque manus dedi, scd
tamen saepius ad cram undo soluissem respeju, qujmi portum in quem csaet
euadeudum cogitandu prospicerem, depracheudiquc non minus uerii quum ele-
48 DIOPHANTOS OF ALEXANDRIA.
Next Xylander tells us how he came to get possession of a manu-
script of Diophantos. In October of the year 1571 he made
a journey to Wittenberg ; while there he had conversations on
mathematical subjects with two professors, Sebastian Theodoric
and Wolfgang Schuler by name, who showed him a few pages
of a Greek manuscript of Diophantos and informed him that
it belonged to Andreas Dudicius whom Xylander describes as
" Andreas Dudicius Sbardellatus, hoc tempore Imperatoris Ro-
manorum apud Polonos orator." On his departure from Witten-
berg Xylander wrote out and took with him the solution of
a single problem of Diophantos, to amuse himself with on his
journey. This he showed at Leipzig to Simon Simonius Lucen-
sis, a professor at that place, who wrote to Dudicius on his
behalf. A few months afterwards Dudicius sent the MS. to
Xylander and encouraged him to persevere in his undertaking
to translate the Arithmetics into Latin. Accordingly Xylander
insists that the glory of the whole achievement belongs in
no less but rather in a greater degree to Dudicius than to
himself. Finally he commends the work to the favour of the
Prince Ludwig, extolling the pursuit of arithmetical and alge-
braical science and dwelling in enthusiastic anticipation on the
influence which the Prince's patronage would have in help-
ing and advancing the study of Arithmetic \ This Epistola
ganter ea cecinisse Alcaeum, quae (si possum) Latino in hac quasi uotiua mea
tabula scribam.
Qui uela uentis uult dare, dum licet,
Cautus futuri praeuideat modum
Cursus. mare ingressus, marine
Nauigct arbitrio necesse est.
Sand, quod de Echeueide pisce fertur, eum nauim cui se adplicet remorari, poenh
credibile fecit mihi mea cymba tot mendorum remoris retardata. Expediui
tamen me ita, ut facilS omnes mediocri de his rebus iudicio praediti, iutellecturi
sint incredibilem me laborem et aerumnas difticilimas superasso : pudore etiam
stimulatum oneris quod ultro milii imposuissem, non perferendi. Paucula quae-
dam non plane explicata, studio et certis de causis in alium locum reiecimus.
Opus quidem ipsum ita absoluimus ut ueque eius nos pudere debeat, et Arith-
meticae Logisticesque studiosi nobis se plurimum debere sint baud dubie
professuri."
' "Hoc non modo tibi Princeps Illustrissimc, honorificum erit, atque glori-
osum ; sed tc labores nostros approbantc, arithmcticae studium cum alibi, tum
in tua Academia et Gymuasiis, excilabitur, confirmabitur, prouebetur, et ad
THE WRITERS UPON DIOPIIANTOS. 49
Nuncnpatoria boars the date 14th August, 1574'. Xylandcr
(lied on the 10th of February in the year following that of the
publication, 1576. Some have stated that Xy lander published
the Greek text of Diophantos as well as the Latin translation.
There appears to be no foundation for the statement, which
probably rests on a misunderstanding of certain passages in
which Xylander refers to the Greek text. It is possible that
he intended to publish the Greek original but was prevented
by his death which so soon followed the appearance of his trans-
lation. It is a sufficient proof, how^ever, that if such was his
purpose it was never carried out, that Bachet asserts that he
himself had never seen or found any one who had ever seen
such an edition of the Greek text ^
Concerning the merits of Xylander and his translation of
Diophantos much has been written, and chiefly by authors who
were not weW acquainted with the subject, but whose very
ignorance seems to have been their chief incitement to startling
statements. Indeed very few persons at all seem to have
studied the book itself: a fact which may be partly accounted
for by its rarity. Nesselmaun, whose book appeared in 1842,
tells us honestly that he has never been able to find a copy,
but has been obliged to take all information on the subject
at second hand from Cossali and Bachet '. Even Cossali, so far
as he gives any opinion at all upon the merits of the book,
seems to do no more than reproduce what Bachet had said
before him. Nor does Schulz seem to have studied Xylander' s
work : at least all his statements about it are vague and may
very well have been gathered at second hand. Both he and
perfectam eiu.s scientiam multi tuis auspiciis, nostro labore pcrducti, niognam
hac re tuis in remp. beneficiis accessionem factam esse gratissima commemora-
tione praedicabunt."
1 "Heidelberga. postrid. Eidus Sextiles cio lo lxxiv."
- "An vero et Graece a Xilandro editus sit Diophantus, nondum certti com-
perire potui. Videtur sanb in multis suorum Commcntariorum locis, de Graeco
Diopbanto tanquam a se cdito, vcl mox edendo, verba facere. Sed banc cditi-
onem, neque mihi vidisse, neque aliquem qui viderit hactcnus audivisse contigit."
Bachet, Epist. ad Led.
3 There is not, I believe, a copy even in the British Museum, but I had
the rare good fortune to find the book in the Library of Trinity College,
Cambridge.
H. D. *
50 DIOPHANTOS OF ALEXANDRIA.
Nesselmann confine themselves to saying that it was not so
worthless as many writers had stated it to be (Nesselmann on
his part confessing his inability to form an opinion for the
reason that he had never seen the book), and that it was
well received among savants of the period, while its effect on
the growth of the study of Algebra was remarkable \ On
the other hand, the great majority of writers on the subject
may be said to shout in chorus a very different cry. One
instance will suffice to show the quality of the statements that
have been generally made : to enumerate more would be waste
of space. Dr Heinrich Suter in a History of Mathematical
Sciences (Zurich 1873) says^ " This translation is very poor,
as Xylander was very little versed in Mathematics." If Dr Hein-
rich Suter had taken the trouble to read a few words of
Xylander's preface, he could hardly have made so astounding
a statement as that contained in the second clause of this
sentence. This is only a specimen of the kind of statements
which have been made about Xylander's book ; indeed I have
been able to find no one who seems to have adequately studied
Xylander except Bachet ; and Bachet's statements about the
work of his predecessor and his own obligations to the same
have been unhesitatingly accepted by the great majority of
later writers. The result has been that Bachet has been uni-
versally considered the only writer who has done anything
considerable for Diophantos, while the labours of his prede-
cessor have been ignored or despised. This view of the relative
merits of the two authors is, in my vieAv, completely erroneous.
From a careful study and comparison of the two editions I
have come to the conclusion that honour has not been paid
where honour was due. It would be tedious to give here in
1 Schulz. "Wie uuvollkomiucn Xylanders Ai-beit auch ausfiel, wie oft cr
auch den rcchten Sinn verfehlte, und wic oft auch seine Aumerkuugeii den
Laser, der sich Eathes eiholen will, im Stichc lasseu, so gut war dock die
Aufnalimc, welche sein Uuch bei den Gelehitcn damaligcr Zcit fand; dcnn in
der That giog den Matlicmatikcru durch die Erscheinung dieses Werkes ein
neues Licbt auf, und es ist mir schr wabrscbcinlicb, dass er viel dazu beigetrageu
hat, die allgemeiue Arithmetik zu ihrer nachnialigen Hohe zu erheben."
- "Dicse Uebersetzung aber ist sebr schwach, da Xylander in Mathematik
sebr wenig bewaudcrt war."
THE WIUTEUS UPON DluPlIANToS. ol
detail the particular facts which led me to this conchision. I
will only say in this place that my suspicions were first aroused
by reading Bachet's work alone, before I had seen tlie earlier
one. From perusing Bachet I received the impression that his
repeated emphatic and almost violent repudiation of obligation
to Xylander, and his disparagement of that author suggested
the very thing which he disclaimed, that he was under too
great obligation to his predecessor to acknowledge it duly.
I must now pass to Bachet's work itself It was the first
edition published which contained the Greek text, and appeared
in 1621 bearing the title: Diophanti Alexandrini Anthmetico-
7-um libri sex, et de numeris nndtancjulis liber unus. JVtoic
2)riinuin Graece et Latine editi, atque absolidissimis Commentariis
illitstrati. Auctore Claudio Gaspare Bacheto Mezinaco Sebusiano,
V.G. Lidetiae Pansiorum, Surnptibus Hieronymi Drovart^, via
Jacobaea, sub Scuto Solari. MDGXXI. (I should perhaps
mention that we have a statement^ that in Carl von Montchall's
Library there was a translation of Diophantos which the mathe-
matician "Joseph Auria of Neapolis" made, but did not ap-
parently publish, and which was entitled "Diophanti libri sex,
cum scholiis graecis Maximi Planudae, atque liber de numeris
polygonis, collati cum Vaticanis codicibus, et latine versi a
Josepho Auria." Of this work we know nothing; neither
Bachet nor Cossali mentions it. The date would presumably
be about the same as that of Xylauder's translation, or a little
later.) Bachet's Greek text is based, as he tells us, upon a MS.
which he calls "codex Regius", now in the Bibliotheque Na-
tionale at Paris; this MS. is his sole authority, except that
Jacobus Sirmondus had part of a Vatican MS. transcribed for
him. He professes to have produced a good Greek text, having
spent incalculable labour upon its emendation, to have inserted
1 For "surnptibus Hierouymi Drovart" Nesselinann has "surnptibus Sebas-
tiani Cramoisy, 1021 " which is found in some copies. The former (as given
above) is taken from the title-page of the copy which I have used (from the
Library of Trinity College, Cambridge).
- Schulz, Vorr. xliii.: "Noch erwuhnen die Litteratorcn, dass eich in der
Bibliothek eines Carl von Moutchall einc Bearbeitung des Diophantus von dem
beruhmten Joseph Auria von Neapel (vermuthlich doch uur handschriftUoli)
befuudeu habe, welche den Titel I'uhrte u. s. w." (see Text).
52 DIOPHANTOS OF ALEXANDRIA.
in brackets all additions which he made to it and to have
given notice of all corrections, except those of an obvious or
trifling nature; a few passages he has left asterisked, in cases
where correction could not be safely ventured upon. In spite
however of Bachet's assurance I cannot help doubting the
quality of his text in many places, though I have not seen
the MS. which he used. He is careful to tell us what pre-
vious works relating to the subject he had been able to con-
sult. First he mentions Xylander (whom he invariably quotes
as Xilander), who had translated the whole of Diophantos, and
commented upon him throughout, "except that he scarcely
touched a considerable pai't of the fifth book, the whole of the
sixth and the treatise on multangular numbers, and even the
rest of his work was not very successful, as he himself admits
that he did not thoroughly understand a number of points."
Then he speaks of Bombelli (already mentioned) and the
Zetetica of Vieta (in which the author treats in his own way a
large number of Diophantos' problems : Bachet thinks that he
so treated them because he despaired of restoring the book
completely). Neither Bombelli nor Vieta (says Bachet) made
any attempt to demonstrate the difficult porisms and abstruse
theorems in numbers which Diophantos assumes as known in
many places, or sufficiently explained the causes of his opera-
tions and artifices. All these omissions on the part of his
predecessors he thinks he has supplied in his notes to the
various problems and in the three Books of "Porisms" which he
prefixed to the work\ As regards bis Latin translation, he
says that he gives us Diophantos in Latin from the version of
Xylander most carefully corrected, in which he would have us
know that he has done two things in particular, first, corrected
^ On the nature of some of Bacliet's proofs Nicholas Saunderson (formerly
Lucasian Professor) remarks in Elements of Algebra, 1740, apropos of Dioph.
III. 17. "M. Bachet indeed in the IGth and 17th props, of his second book of
Porisms has given us demonstrations, such as they are, of the theorems in the
problem: but in the first place he demonstrates but one single case of those
theorems, and in the next place the demonstrations he gives are only synthetical,
and so abominably perplexed withal, that in each demonstration he makes uso
of all the letters in the alphabet except I and 0, singly to represent the quantities
he has there occasion for."
THE WRITERS UPON DIuPllANTOS. 53
what was wrong and supplied the numerous lacunae, secondly,
explained more clearly what Xylander had given in obscure or
ambiguous language: "I confess however", he says "that this
made so much change necessary, that it is almost more fair
to attribute the translation to me than to Xilander. But if
anyone prefers to consider it as his, because I have held fast,
tooth and nail, to his words when they do not misrepresent
Diophantus, I do not care'". Such sentences as these, which
are no rarity in Bachet's book, are certainly not calculated to
increase our respect for the author. According to Montucla",
"the historian of the French Academy tells us" that Bachet
worked at this edition during the course of a quartan fever, and
that he himself said that, disheartened as he was by the diffi-
culty of the work, he would never have completed it, had it
not been for the stubbornness which his malady generated iu
him.
As the first and only edition of the Greek text of Dio-
phantos, this work, in spite of any imperfections we may find in
it, does its author all honour.
The same edition was reprinted and published with the
addition of Fermat's notes in 1G70. Diophanti Alexandrini
Arithmeticorwni lihri sex, et de numeris multangidis liher itmis.
Cum commentariis G. G. Bacheti V. G. et ohseruationibus D. P.
de Fermat Senatoris Tolosani. Accessit Doctrinae Amdyticae
inuentum nouum, collectum ex variis eiusdem D. de Fermat
Epistolis. Tolusae, Excudehat Bernardus Bosc, ^ Regione CuUegii
Societatis Jesu, MDGLXX. This edition was not pubhshed
by Fermat himself, as certain writers imply ^ but by his son
'■"Deinde Latinum damus tibi Diophantum ex Xilandri versione accura-
tissime castigata, in qua duo potissimum nos praestitisse scias velim, nam
et deprauata correximus, hiantesque passim lacunas repleuimus : et quae sub-
obscure, vel ambigue fuerat interpretatus Xilander, dilucidius exposuimus; fateor
tamen, inde tantam inductam esse mutationem, vt propemodum aequius sit ver-
sioneni istam nobis quam Xilandro tribuere. Si quis autem potius ad eum \^t-
tinere contendat, qu5d eius verba, quatenus Diophanto fraudi non erant, niordicus
retinuimus, per me licet."
2 I. 323.
' So Dr Hcinrich Suter: "Diese Am(fahe witrde 1G70 ditrch Fernuit ernnt^rt,
der sie mit seinen eigenen algebraischep Untersuchungen und Erfindungen
^asstattete,"
54 DIOPHANTOS OF AT.KXAXDRIA.
after his death. S. Fermat tells us in the preface that this
publication of Fermat's notes to Diophantos was part of an
attempt to collect together from his letters and elsewhere his
contributions to mathematics. The "Doctrinae Analj'ticae In-
uentum nouum" is a collection made by Jacobus de Billy from
various letters which Fermat sent to him at different times.
The notes upon Diophantos' problems, which his son hopes will
prove of value very much more than commensurate with their
bulk, were (he says) collected from the margin of his copy of
Diophantos, From their brevity they were obviously intended
for the benefit of experts \ or even perhaps solely for Fermat's
own, he being a man who preferred the pleasure which he had
in the work itself to all considerations of the fame which might
follow therefrom. Fermat never cared to publish his investiga-
tions, but was always perfectly ready, as we see from his letters,
to acquaint his friends and contemporaries with his results. Of
the notes themselves this is not the place to speak in detail.
This edition of Diophantos is rendered valuable only by the
additions in it due to Fermat; for the rest it is a mere reprint
of that of 1621. So far as the Greek text is concerned it is
very much inferior to the first edition. There is a far greater
number of misprints, omissions of words, confusions of numerals;
and, most serious of all, the brackets which Bachet inserted in
the edition of 1621 to mark the insertion of words in the text
are in this later edition altogether omitted. These imperfec-
tions have been already noticed by Nesselmannl Thus the
reprinted edition of 1670 is untrustworthy as regards the text.
^ Lectori Beneuolo, p. iii. : "Doctis quibus tantum pauca sufficiunt, harum
obseruationum auctor scribebat, vel potius ipse sibi scribens, his studiis exerceii
malebat quam gloriari ; adco autem ille ab omni ostentationo alienus erat, vt nee
lucubratioues suas ty]iie mandari curauerit, ct suonim qiiandoquc resjionsorum
autographa nullo scruato exemplari pctentibus vitro miserit ; iiorunt scilicet ple-
rique celeberrimorum huius saeculo Geomctrarum, quam libenter ille et quaut&
bumanitate, sua iis inuenta patefecerit."
2 "Was dieser Abdruck an iiusserer Eleganz gewounen hat (denn die Ba-
chet'sche Ausgcbe ist niit ausserst unangcnehmen, nanientlich Griechischeu
Lettern gedruckt), das hat sie an inncrm Werthe in Bczug auf den Text ver-
loren. Sie ist nicht bloss voller Diuckfchler in cinzelnen Worten und Zeichen
(z. B. durchgehends ir statt "?>), 900) sondern audi ganze Zeilen sind ausgelassen
Oder doppelt gedruckt, (z. B. iii. 12 cine Zeile doppelt, iv, 25 eine doppelt und
THE WRITERS UPON DIOPHANTOS. 55
I omit here all mention of works which are not directly
upon Diophantos (e.g. the so called "Translation" by Stevin and
Alb. Girard). We have accordingly to pass from 1670 to 1810
before we find another extant work directly upon Diophantos.
In 1810 was published an excellent translation (with additions)
of the fragment upon Polygonal Numbers by Poselger : Dio-
phantus von Alexandrien iiher die Polygonal-Zahlen. Uebersetzt
mit Zusdtzen von F. Th. Poselger. Leipzig, 1810.
Lastly, in 1822 Otto Schulz, professor in Berlin, published a
very meritorious German translation with notes: Diupliantus
von Alexandria arithmetische Aufgahen nebst dessen Schrift iiber
die Pohjgon-Zahlen. Aus dem Griechischen ilbersetzt iind mit
Anmerkungen begleitet von Otto Schulz, Professor am Berlinisch-
Colnischen Gymnasium zum grauen Kloster. Berlin, 1822. In
der Schlesingerschen Buck- und Musikhandlung. The former
work of Poselger is with the consent of its author incorporated
in Schulz's edition along with his own translation and notes
upon the larger treatise, the Arithmetics. According to Nessel-
mann Schulz was not a mathematician by profession: he pro-
duced, however, a most excellent and painstaking edition, with
notes chiefly upon the matter of Diophantos and not on the
text (with the exception of a very few emendations) : notes
which, almost invariably correct, help much to understand the
author. Schulz's translation is based upon the edition of
Bachet's text published in 1670; so that nothing has been done
for the Greek text since the original edition of Bachet (1621).
I have now mentioned all the extant books which have been
written directly upon Diophantos. Of books here omitted
which are concerned with Diophantos indirectly, i.e. those
which reproduce the substance of his solutions or solve his
gleich hinterher eine ausgelassen, rv. 52 eine doppclt, v. 11 eine aup^'clnpsen,
desgleichen v. 14, 2.5, 33, vi. 8, 13 und so weiter), die Zalileu Verstiimmcit, was
aber das Aergste ist, die Bacbet'schen kritischen Zeicheu sind fast iiberall, die
Klammer durcbgtingig weggefallen, so dass diese Ausgabe als Text des Diophant
vcillig unbrauchbar geworden ist," p. 283.
Accordingly Cantor errs when he says "Die beste Textamijabe ist die von
Bachet de Meziriac mit Anmerkungen von Format. Toulouse, 1G70." (Getch.
p. 31)0.)
56 DIOPHANTOS OF ALEXANDRIA.
problems or the like of them by different methods a list has
been given at the outset. As I have already mentioned a
statement that Joseph Auria of Naples wrote circa 1580 a
translation of Diophantos which was found (presumably in MS.
form) in the library of one Carl von Montchall, it is necessary
here to give the indications we have of lost works upon Dio-
phantos. First, we find it asserted by Vossius (as some have
understood him) that the Englishman John Pell wrote an un-
published Commentary upon Diophantos. John Pell was at
one time a professor of mathematics at Amsterdam and gave
lectures there on Diophantos, but what Vossius says about his
commentary may well be only a recommendation to undertake
a commentary, rather than a historical assertion of its comple-
tion. Secondly, Schulz states in his preface that he had lately
found a note in Schmeisser's Orthodidaktih der Mathematik that
Hofrath Kausler by command of the Russian Academy pre-
pared an edition of Diophantos \ Of this nothing whatever is
known; if ever written, this edition must have been only for
private use at St Petersburg.
I find a statement in the New American Cyclopaedia (New
York, D. Appleton and Company), vol. VI. that "a complete
translation of his (Diophantos') works into English was made
by the late Miss Abigail Lousada, but has not been published."
^ The whole passage of Schmcisser is: "Die mechanische, geistlose Behand-
lung der Algebra ist ins besondere von Herru Hofrath Kausler stark geriigt
worden. In der Vorrede zu seiner Ausgabe des Vjlakerschen ExempcUmclis
beginnt er so : ' Seit mehreren Jahren arbeitete ich fiir die Kussisch-Kaiserliche
Akademie der Wissenschaften Diophants unsterbliches Werk iiber die Arithnietik
aus, und fand darin einen solchen Schatz von den feinsten, scharfsinnigsten
algebraischcn Auflosungen, dass mir die mechanische, geistlose Methode der
neuen Algebra mit jedem Tage mehr ekelte u. s. w.' " (p. 33.)
CHAPTER IV.
NOTATION AND DEFINITIONS OF DIOPHANTOS.
§ 1. As it is my inteution, for the sake of brevity and
perspicuity, to make use of the modern algebraical notation
in giving my account of Diophantos' problems and general
methods, it will be necessary to describe once for all the
machinery which our author uses for working out the solutions
of his problems, or the notation by which he expresses the
relations which would be represented in our time by algebraical
equations, the extent to which he is able to manipulate unknown
quantities, and so on. Apart, however, from the necessity of
such a description for the proper and adequate comprehension
of Diophantos, the general question of the historical develop-
ment of algebraical notation possesses great intrinsic interest.
Into the general history of this subject I cannot enter in this
essay, my object being the elucidation of Diophantos ; I shall
accordingly in general confine myself to an account of his
notation solely, except in so far as it is interesting to compare it
with the corresponding notation of his editors and (in certain
cases) that of other writers, as for example certain of the early
Arabian algebraists.
§ 2. First, as to the representation of an unknown quantity.
The unknown quantity, Avhich Diophantos calls ttXj/^o? fiovdBoiu
aXoyov i.e. "a number of units of which no account is given,
or undefined " is denoted throughout (def. 2) by what is uni-
versally printed in the editions as the Greek letter ? with an
accent, thus ?', or in the form s°'. This symbol in verbal
description he calls u aptOfxo'^, "the number" i.e. by inipli-
58 DIOPHANTOS OF ALEXANDRIA.
cation, the number par excellence of the problem in question.
(In the cases where the symbol is used to denote inflected
forms, e.g. accusative singular or dative plural, the terminations
which would have been added to the stem of the full word
dpi6fi6<i are printed above the symbol 9 in the manner of an
exponent, thus 9'' (for dpidfxov, as r' for t6v), <?°", the symbol
being in addition doubled in the plural cases, thus 99°'', 99°"'^ 99""
99°'« for dpidfjLOL K.T.X. When the symbol is used in practice, the
coefficient is expressed by putting the required Greek numeral
immediately after it, thus 99°' Td corresponds to 11a-, 9'a to sc
and so on.
Respecting the symbol 9 as printed in the editions it is
clear that, if 9' represents dpiOixo^, this sign must be different
in kind from all the others described in the same definition, for
they are clearly mere contractions of the corresponding names\
The opinion which seems to have been universally held as to
the nature of the symbol of the text by the best writers
on Diophantos is that of Nesselmann and Cantor ^ Both
authors tell us that the final sigma is used to denote the
unknown quantity representing upi6p6<i, the complete word for
it ; and they imply in the passages referred to that this final
sigma corresponds exactly to the x of modern equations, and
that we have here the beginning of algebraical notation in the
strict sense of the term, notation, that is, which is purely
conventional and shows in itself no necessary connection be-
tween the symbol and the thing denoted by it. I must observe,
however, that Nesselmann has in another place ' corrected the
impression which the reader might have got from the first
passage referred to, that he regarded the use of the sign for
dpi6fji6<; as a step towards genuine algebraical notation. He
makes the acute observation that, as the symbol occurs in
many places where it represents dpidfio^ used in the ordinary
untechnical sense, and is therefore not exclusively used to
designate the unknown quantity, the technical dpi6fi6<i, it
must after all be more of the nature of an abbreviation than
' Vide infra S", k", 55", &c. contractions for Suva/jLi^, kv^os, dwafioSufafxis, &c.
•■! Nesselmann, pp. 290, 291. Cantor, p. 400.
3 pp. 300, 301,
NOTATION AND DEFINITIONS OF DIOPHANTOS. 59
an algebraical symbol. This view is, I think, undoubtedly
correct ; but the question now arises : bow can the final signia
of the Greek alphabet be an abbreviation for dpiOfio^ ? The
difficulty of answering this question suggests a doubt which,
so far as I am aware, has been expressed by no writer upon
Diophantos up to the present time. Is the sign, which Bachet'.s
text gives as a final sigma, really the final sigma at all ?
Nesselmann and Cantor seem never to have doubted it, for
they both assign a reason why the final 9 was appropriated for the
designation of the unknown quantity, namely that it was the
only letter of the Greek alphabet which was not already in
use as a numeral. The question was suggested to me princi-
pally by the doubt whether the final sigma, 9, was developed
as distinct from the form cr as early as the date of the MS. of
Diophantos which Bachet used, or rather as early as the first
copy of Diophantos, for the explanation of the sign is given
by the author himself in the text of the second definition.\
This being extremely doubtful, if not absolutely impossible,
in what way is its representation as a final sigma in Bachet's
text to be accounted for ? The MS. from which Bachet edited
his Greek text is in the Bibliotheque Natiouale, Paris, and I
have not yet been able to consult it : but, fortunately, in a paper
by M. Kodet in the Journal Asiatique (Janvier 1878), I found
certain passages quoted by the author from Diophantos for
the purpose of comparison with the algebra of Mohammed
ibn Musti Al-Kharizmi. These passages M, Rodet tells us that
he copied accurately from the identical MS. which Bachet used.
On examination of these passages I found that in all but two
cases of the occurrence of the sign for (ipi6/j.6<; it was given
as the final sigma. In one of the other cases he writes for
6 dpiOfiof (in this instance untechnical) the abbreviation o d\
and in the other case we find ijTj"' for dpt6fj.oi In this last place
Bachet reads 99°'. But the same symbol cji|" which M. Rodet
gives is actually found in three places in Bachet's own edition.
(1) In his note to iv. 3 he gives a reading from his MS. which
he has corrected in his own text and in which thr signs i\d and
qi|^ occur. They must here necessarily signify npidp.6<i d and
dpidfiol 7) respectively because, although tlic sense rcciuire.s
GO DIOPHANTOS OF ALEXANDRIA.
1 8
the notation corresponding to - , - , not x, 8x, we know, not
only from Bachet's direct statement but also from the trans-
lation of certain passages by Xylauder, that the sign for dpi6fjb6<;
is in the MSS. very often carelessly written for dpid/MoaTov and
its sign. (2) In the text of iv. 14 there is a sentence (marked
by Bachet as interpolated) which has the expression l| ij? where
again the context shows that i|L| is for dpcO/xoL (3) At the
beginning of v. 12 there is a difficulty in the text; and Bachet
notes that his MS. has o SfjrXaaifoi' avroO l|... where a Vatican
MS. reads 6 hijfkaa-iwv avrov dpLdjjiov... Xylander also notes
that his MS. had firjTe o hiir\.aaL(ov avrov ap.... It is thus clear
that the MS. which Bachet used sometimes has the sign for
dpiOfMo^; in a form which is at least sufficiently like q to
be taken for it. This last very remarkable variation as com-
pared with 9?°' seemed at first sight inexplicable ; but oq refe-
rence to Gardthausen, Griechische Falaeur/raphie, I found under
the head " hieroglyphisch-conventionell " an abbreviation 9, 9^
for dpidfio'^, dpiO^JbOL, which the author gives as occurring in
the Bodleian MS. of Euclid '. The same statement is made
by Lehmann'^ {Die tacky graphischen Ahkurzungen der grie-
chischen Handscltriften, 1880) who names as a sign for dpcd^io^,
found in the Oxford MS. of Euclid, a curved line similar to
that used as an abbreviation for Kai He adds that the ending
is placed above it, and the simple sign is doubled for the
plural. Lehmann's facsimile of the sign is like the form given
by Gardthausen, except that the angle in the latter is a little
more rounded by Lehmann. The form ijq°' above mentioned
as given by M. Rodet and Bachet is also given by Lehmann
with a remark that it seems to be only a modification of the
other. If we take the form as given by Gardthausen, the change
necessary is the very slightest possible. Thus by assuming
this conventional abbreviation for dpi,6/j.6<i it is easy to see
1 D'Orvillo MSH. X. 1 inf. 2, 30.
" p. 107: "Von Sigeln, welchcn ich audi nnderwarts begegnet bin, sind zu
uennen apiOixb^, das in der Oxfordcr Euclidhandsclirift niit eiucr der Note
Kal ahnlichen Schlangenlinie bezeicbuet wird. Die Enduug wird dariiber
gesetzt, zur Bezejchuung des Plurals wird das ejnfache Zeichen verdoppelt,"
NOTATION AND DEFINITIONS OF DIOrHANTOS. 01
how it was thought by Bachet to be a final sigma and Iwiw
also it might be taken for the isolated form given by M. Rodet.
As I have already implied, I cannot think that the symbol
used by Diophantos is really a final sigma, 9. That the con-
ventional abbreviation in the Euclid MS. and the sign in
Diophantos are identical is, I think, certain; and that neither
of the two is a final sigma must be clear if it can be proved
that one of them is not. Having consulted the Ms. of the first
ten problems of Diophantos in the Bodleian Library, I conclude
that the symbol in this work cannot be a final sigma for the
following reasons. (1) The sign in the Bodleian MS. is written
thus, '<^° for dpcdfMO'i; and though the final sigma is used uni-
versally in this MS. at the end of words there is, besides
a slight difference in shape between the two, a very distinct
difference in size, the sign for dpc6/ji6<: being always very much
larger. There are some cases in which the two come close
together, e.g. in the expression eh '<^° Tee, and the difference is
very strongly marked. (2) As I have shown, the breathing is
prefixed before the sign. This, I think, shows clearly that the
symbol was regarded as an abbreviation of certain letters be-
ginning with a the first letter of dpiO/xo^. It is interesting also
to observe that in the Bodleian MS. there are certain cases in
which dpcdfio^ in its untechnical, and dpt0fj,6<; in its technical
sense follow each other as in era^a to tov Seurepov 'S^ dptdfiov
et'o's, where (contrary to what might be expected) the sign is
used for the untechnical dpidfjio'i and the other is written in
full. This is a very remarkable piece of evidence to show that
the sign is an abbreviation and in no sense an algebraical
symbol. More remarkable still as evidence of this view is the
fact that in the same MS. the luord dpi6fi6<; in the definition
6 Be firjBeu tovtcop toov ISicofjidTayv Kr'r}adfievo<;...e-)(Oiv he...
dpidfjb6<; KaXeiTUL is itself denoted by the symbol, so that in
the MS. there is absolutely no difference between the full name
and the symbol.
My conclusion therefore being (1) that the sign given as ?
in Bachet's text of Diophantos is not really tlic final .sigma,
(2) that it is an abbreviation of some kind for dpiOfMot, the
question arises. How was this abbreviation arrived at ? If it is
62 DIOPHANTOS OF ALEXANDRIA.
uot a hieroglyph (and I have cot yet found any evidence of its
hieroglyphic origin), I would suggest that it might very well
be a corruption, after combination, of the two first letters of the
word, Alpha and Rho.
Before I go on to state when and how I conceive this
contraction may liave come about, I may observe that, given
its possibility, my supposition has, it seems to me, every-
thing in its favour. (1) It would explain, and is countenanced
by, the solitary occurrence in M. Kodet's transcription of the
contraction a*. (2) It would also explain the remarkable
variation in the few words quoted from Xylander's note on
v. 12, fMT]T€ 6 hiTrKaaioiv avTov ap fio a... These words are
important because in no other sentence which he quotes in
the Greek does any abbreviation of dpt9iJL6<i occur. As his
work is a Latin translation he rarely quotes the original Greek
at all: hence we might have doubted whether the sign for
dpi6fx6(; occurred in his MS. in the same form as in Bachet's.
That it did occur in the same form is, however, clear from the
note to III. 12\ That is to say, both ap and <h are used in one
and the same MS. to signify apt^/xo?. This circumstance is easily
explained on my hypothesis ; and I do not see how it can be
explained on any other. But (3) the most important advantage
that my theory would hav« is that it would establish uniformity
between the different abbreviations used by Diophantos. It
would show him to have proceeded on one invariable principle
in fixing those abbreviations which we should naturally have
expected to be parallel. Diophantos, in fact, appears to have
proceeded thus. He took in all cases the first letter of the
corresponding words i.e. a, B, k, fi. Then, as these could not be
used alone for the reason that they all represented numbers, he
added another letter to each. Now, as it happened, the second
letter in each of the four words named occurred later in the
1 In tills problem it evidently occurred wrongly instead of the sign for the
fraction apiOfioarbv (as was commonly the case in the mss.), for after stating
that the context showed the reading apidfids to be wrong Xy lander says: "Est
sane in Graeco nota senarii S". Sed locum habere non potest." Now s and r
are so much alike that what was taken for one might easily be taken for the
other.
NOTATION AND DEFINITIONS OF Dl< )|'HANToS. Q:)
alphabet than the respective first letters. Thus a with p addctl,
B with V added, k with v added, and /i with o added gave abbre-
viations luhich could not he confounded with particidar numbers.
No doubt, if the two letters in each case were not written in the
same line by Diophantos, but the second raised above the other,
the signs might, unless they or the separate letters were dis-
tinguished by some special marks, have been confused with
numerical fractions. There would however be little danger of
this ; such confusion would be very unlikely to arise, for (a) the
context would nearly always render it impossible, as also would
(6) the constant recurrence of the same sign for a constantly
recurring term, coupled with the fact that, if on any particular
occasion it denoted a numerical fraction, it could and would
naturally be expressed in lower terms. Thus, if 8", /c", /i° were
numerical fractions, they would be as unlikely to be written thus
as we should be unlikely to write -^, ■^, ^. Indeed the
only sign of the four which, written with the second letter
placed as an exponent to the second, could reasonably be supposed
to represent a numerical fraction is a", which miglit mean yi^.
But, by a curious coincidence, confusion is avoided in this case ;
and the contraction, which I suppose to have taken place, might
very well be an expedient adopted for the purpose : thus we
may have here an explanation why only one of the four signs
ap, 3u, Kv, fio is contracted, j (4) Again, if we assume <^ to be a
contraction of ap, we can explain the addition of terminations to
mark cases and number in the place where the second letter of
the other abbreviations is written. The sign '<^ having no
letter superposed originally, this addition of terminations was
rendered practicable without resulting in any confusion. On
its convenience it is unnecessary to enlarge, because it is clear
that the symbol could then be used instead of the full word far
more frequently than the others. Thus oblique cases of Bvvafii'i
are written in full where oblique cases of apidfi6<; would be
abbreviated. For 8", /c", fi° did not admit of the addition of
terminations without possible confusion and certain clumsiness.
A few words will suffice to explain my views concerning the
evolution of the sign for dpid^xo^. There are two alternatives
possible. (1) Diophantos may not himself have made the con-
64 DTOPHANTOS OF ALEXANDRIA.
traction at all ; he may have written the two letters in full. In
that case I suppcse the sign to he a cursive contraction used by
scribes. I conceive it would then have come about through a
tolerably obvious intermediate form, 'p. The change from this
to either of the two forms of the symbol used in MSS, for dpid/j-o^;
is very slight, in one case being the loss of a stroke, in the
other the loss of the loop of the p. (2) Diophantos may have
used a sign approximately, if not exactl}^ like the form which
we now find in the MSS. Now Gardthausen divides cursive
writing into two kinds, which he calls " Majuskelcursive " and
" Minuskelcursive." One or other of these terms would be
applied to a type of writing according as the uncial or cursive
element predominates. That in which the uncial element pre-
dominates is the " Majuskelcursive," which is intermediate be-
tween the uncial and the cursive as commonly understood.
Gardthausen gives examples of MSS. which show the gradations
through which writing passed from one to the other. Among
the si^ecimens of the " Majuskelcursive " writing he mentions
a Greek papyrus, the date of which is 154 A.D., i.e. earlier than
the time of Diophantos. From this MS. he quotes a contraction
for the two letters a and p, namely up. This may very well be
the way in which Diophantos wrote the symbol ; and, after
being copied by a number of scribes successively, it might very
easily come into the MSS. which we know in the slightly simplified
form in which we find it\
1 Much of what I have written above concerning the symbol for dpi6/MS
appeared in an article "On a point of notation in the Arithmetics of Dio-
phantos," which I contributed to the Journal of Philolofjy (Vol. xin. No. 25,
pp. 107 — 113). Since that was written I have considered the subject more
thoroughly, and I have been able to profit by a short criticism of my theory,
as propounded in the article alluded to, by Mr James Gow in his recent History
of Greek Mathematics (Camb. Univ. Press, 1884). In the Addenda thereto
Mr Gow states that he does not think my suggestion that the supposed final
sigma is a contraction of the first two letters of apiOfi-os is true, for three reasons.
It is right that I should answer these objections in this place. I will take them
in order.
1. Mr Gow argues:— "The contraction must be supposed to be as old as
the time of Diophantus, for he describes the symbol as tA s instead of to, or
Tw ap. Yet Diophantus can hardly (as Mr Heath admits) have used cursive
characters." Upon this objection I will remark that I do not think the descrip-
NOTATION AND DEFINITIONS OV DMniANToS. C".
In the following paq^es, as it is impossible to say for cort;iin
what this sign really is, I shall not hesitate, where it is neces-
tion of the symbol as to s proves that the supposed contraction must be ns oM
as the time of Piophantos himself. I see no reason, even, why Diophnnfoi
liimself shoiikl not have written Kal lanv avrov cnjfxe'ov to dp. For (a) it seems
to me most natural tliat the article should be in the same number as ffrniuoy.
Mr Gow might, I think, argae with equal force that the Greek should run, kcI
((TTiv wuTov ar]/M€7a ra dp. And yet o-nixeTov is not disputed. Saj)posing, then.
that we have assumed on other grounds that Diophantos used the first two
1( tters, contracted or uncontracted, of apiOp.6i as his symbol for it, I do not seo
that the use of the article in the singular constitutes any objection to our
assumption. (Ji) Besides the censideration that to dp is perfectly possible
grammatically, we have yet other evidence for its possibility in expressions
which we actually find in the text. The symbol for the fifth power of the
unknown, or for 5vvaiJ.6Kv^os, is described thus: Kal {<xtiv avrov arjftuov t6 ii
twiarjixov /^cra v, ok". In this case much more than in the supposed case of
dp should we have expected the plural article with 5k instead of the singular ; but
5k iTria-rjfiov ?xo»'to i* is in apposition with ok" and is looked upon as a single
expression, and therefore preceded by the singular article to. If we give full
weight to these considerations, it must, I think, be admitted that Mr Gow's
conclusion that the contraction must be as old as the time of Diophantoa,
whether true or not, is certainly not established by his argument from the
description of the symbol as tA r. Hence, as one link in the reasoning em-
bodied in Mr Gow's first objection fail=;, the olijection itself breaks down.
Mr Gow appeai-s to have misunderstood me w^hen he attributes to me tlic
inconsistency of supposing Diophantos to have used cursive characters, while
in another place I had disclaimed such a supposition. It will be sufficiently
clear from the explanation which I have given of the origin of the contraction
that I am very far from assuming that Diophantos used cursive characters such
as we now use in writing Greek. At the same time it is possible that Mr Gow's
apparent mistake as to my meaning may be due to my own .inadvertence in
saying (in the article above-mentioned) "If it [the symbol] is not a hiero-
glyph (and I have not found any evidence of its hieroglyphic origin), I would
suggest that it might very well be a corruption of the two letters dp" (printed
thifs), where, however, I did not mean the cursive letters any more than
uncials.
2. I now pass to Mr Gow's second objection to my theory.
"The abbreviation s° for dpi.0p.6s in its ordinary sense is very rare indeed.
It is not found in the mss. of Nicoraachus or Pappus, where it might most
readily be expected. It may therefore be due only to a scribe who had some
reminiscence of Diophantus." The meaning of this last sentence does not seem
quite clear. I presume Mr Gow to mean "/h the rare cases where it does occur,
it may be due, &c." I do not know that I am concerned to prove that to" for
dpiOixos is of very frequent occurrence in mss. other than those of Diophanlon.
Still the form (,, which I have no hesitation in stating to be the same as t-,.
occurs often enough in the Oxford lis. of Euclid to make Gardthausen and
H. D. J
CG DIOPHAXTOS OF ALEXANDRIA.
sary to designate it, to call it the final sigma for convenience'
Lebmann notice it. And, even if its use in that ms. is due to a scribe wbo bad
some reminiscence of Diophantos, I do not see that this consideration affects
my tbeory in the least. In fact, it is not essential for my theory that tliis si^n
should occur in a single instance elsewhere than in Diophantos. It is really
quite sufficient for my purpose that o," occurs in Dioi>hantos for apiOfwi in its
ordiiKn-y sense, which I hold that I have proved.
3. Mr Gow's tliird objection is stated thus: "If s is for dp. then, by
analogy, the full symbol should be s' (like 5", k'") and not j°." (a) I must
first remark that I consider that arguments from analogy are inapplicable in
this case. The fact is that there are some points in which all the five signs of
which I have been speaking are undoubtedly analogous, and others in which
some are not; therefore to argue from analogy here is futile, because it would
be equally easy to establish by that means either of two opposite conclusions.
I might, with the same justice as Mr Gow, argue backwards that, since there
is undoubtedly one point in which s° and 5" are not analogous, namely the
superposition in one case of terminations, in the other case of the second letter
of the word, therefore the signs must be differently explained : a result which,
so far as it goes, would favour my view, (b) Besides, even if we admit the
force of Mr Gow's argument by analogy, is it true that s' (on the supposition
that s is for dp) is analogous to 5" at all? I think not; for s does not corres-
pond to 5, but (on my supposition) to 8v, and I only partially corresponds to v,
inasmuch as t is the tliinl letter of the complete word in one case, in the other
i; is the second letter, (c) As a matter of fact, however, I maintain that my
suggestion does satisfy analogy in one, and (I think) the most important respect,
namely that (as I have above explained) Diophantos proceeded on one and the
same system in making his abbreviations, taking in each case the two first letters
of the word, the only difference being that in one case only are the two letters
contracted into one sign.
Let us now enquire whether my theory will remove the difficulties stated by
Mr Gow on p. 108 of his work. As reasons for doubting whether the symbol for
dpid/xoi is really a final sigma, he states the following. "It must be remembered :
(1) that it is only cursive Greek which has a final siijma, and that the cursive
form did not come into use till the 8th or 9th century : (2) that inflexions are
appended to Diophantus' symbol s' (e.g. s°", ss°S etc.), and that his other symbols
(except f) are initial letters or syllables. The objection (1) might be disposed
of by the fact that the Greeks had two uncial sigmas C and 1, one of which
might have been used by Dioi)hantus, but I do not see my way to dismissing
objection (2)." First, with regard to objection (1) Mr Gow rightly says that,
supposing the sign were really s, it would be possible to dismiss this objection.
On my tlieory, however, it is not necessai-y even to dismiss it : it does not exist.
Secondly, my theory will dismiss objection (2). "Diophantus' other symbols
(except /;>) are inititil letters or syllables." I answer "So is to." "Inflexions
are appended to Diophantus' symbol s'." I answer "True; but the nature of
the sign itself made this convenient," as I have above explained.
NOTATION AND DKriNITIoNS uV 1 )li .pll VNToS. CyJ
sake, subject to the remarks which 1 have here ma^le on the
subject.
§ 3. Next, as regards the notatiou Avhich Diupliaiitus used
to express the different powers of the unknown quantity, i.e.
corresponding to x^, x^ and so on. The square of tlie unknown
is called by Diophantos SviafiK: and denoted by the abbrevia-
tion* B". Now tiie word Bvvafiiq ("power") is commonly used in
Greek to express a square number. The first occurrence of the
word in its technical sense is probably as early as the second
half of the fifth century B.C. Eudemos uses it in quoting from
Hippokrates (no doubt word for word) who lived about that
time. The dilBference iu use between the words Bvvafj.i<; and
T€Tpnycovo<i corresponds, in Cantor's view^, to the difference
l)etween our terms "second power" and "square" respectively,
the first having an arithmetical signification as referring to a
number, the second a geometrical reference to a plane surface-
area. The difference which Diophantos makes in their use is,
however, not of this kind, and Sui/a/i/.? in a geometrical sense,
is not at all uncommon; hence the correctness of Cantor's
suggestion is not at all certain. Both terms are used by
Diophantos, but in very different senses. hvvafii<i, as we have
said, or the contraction 8" stands for the second power of the
unknown quantitt/. It is the square of the unknown, apidfio'i
or '<h°, only and is never used to express the square of any other,
i.e. any known number. For the square of any known number
Diophantos uses TeTpdj(ovo<;. The higher powers of the un-
known quantity which Diophantos makes use of he calls Kvfio<:,
BvvafioSvva/jic<;, Bvva/j,6Kv^o<;, KVfS6Kv/3o<i, corresponding respec-
tively to x^, X*, x\ a.". Beyond the sixth power he does not go,
having no occasion for higher powers in the .solutions of his
1 I should observe with respect to the mark over the v that it is given in the
Greek text of Bachet as a circumtiex accent printed in the form ~. By writers
on Diophantos later than Bachet the sign has been variously printed as 3", a" or
5". I have generally denoted it by 5", except in a few special cases, when
quoting or referring to writers who use either of the other forms. The same
remark applies to /tt°, the abbreviation for ixovdSn, as well as to the circumflex
written above the denominators of Cheek numerical fractions piven in this
chapter as examples from the text of Diophantos.
' Gesrhichte der Mtitheiiuitik, p. 178.
GS DIOPH\NTOS OF ALEXANDRIA.
problems. For these powers he uses the abbreviations k^, ht'\
S/c", kkP respectively. There is a difference between Diophantos'
use of the complete words for the third and higher powers and
that of Bvvafit<;, namely that they are not always restricted like
Bvvafxt<; to powers of the unknown, but may denote powers of
ordinary known numbers as well. This is probably owing to
the fact that, while there are two words hvvafii<i and TeTpdywvo^
which both signify "square", there is only one word for a third
power, namely kv^o'?. It is important, however, to observe that
the abbreviations «", 8S", 8/c", /c/c" are, like Bvva/j,i<i and S", onlij
used to denote powers of the unknown. It is therefore ob-
viously inaccurate to say that Diophantos "denotes the square
of a number {hvvajxi^) by S", the cube by «", and so on", the
only number of which this could be said being the 9' {dpi,6fM6<;)
of the particular problem. The coefficients which the different
powers of the unknown have are expressed by the addition of
the Greek letters denoting numerals (as in the case of dpidfj.o'i
itself), thus Sk^ kW corresponds to 26ir^ Thus in Diophantos'
system of notation the signs 8" and the rest represent not merely
the exponent of a power like the 2 in x^, but the whole ex-
pression, x\ There is no obvious connection between the symbol
S" and the symbol 9' of which it is the square, as there is be-
tween x^ and X, and in this lies the great inconvenience of the
notation. But upon this notation no advance was made by
Xylander, or even by Bachet and Fermat. They wrote N
(abbreviation of Numerus) for ?' of Diophantos, Q {Quadratas)
for 8", C for «" (cubus) so that we find, for example, \Q■\■oN='l^f,
corresponding to x^ -\- 5x= 24. Thus these writers do in fact no
more than copy Diophantos. We do, however, find other symbols
used even belore the publication of Xylander's Dioiiliantos, e g.
in 1572, the date of Bombelli's Algebra. Bombelli denotes the
unknown and its powers by the symbols i, t, ^, and so on.
But it is certain that up to this time the common symbols had
been Ji {Radix or Res), Z {Zensus i.e. square), C {Cuhvs).
Apparently the first important step towards x'^, x^ &c. was
taken by Vieta, who wrote Aq, Ac, Aqq, &c. (abbreviated fur
A quadratus and so on) for the powers of A. This system,
besides showing in itself the connection between the difforfut.
NOTATION AND DEFINITIONS OF DIOI'HANTOS. GO
powers, has the infinite advantage that by means of it we can
use in one and the same solution any number of unknown
quantities. This is absolutely impossible with the notati(.n
used by Diophantos and the earlier algebraists. Diophantos
does in fact never use more than one unknown quantity in the
solution of a problem, namely the apLdfi6<i ur <;' .
§ 4. Diophantos has no symbol for the operation of multi-
plication: it is rendered unnecessary by the fact tiiat his
coefficients are all definite numerals, and the results are simply
put down without any preliminary step wdiich would make a
symbol essential. On the ground that Diophantos uses only
numerical expressions for coefficients instead of general symbols,
it would occur to a superficial observer that there must be a
great want of generality in his methods, and consequently that
these, being (as might appear) only applicable to the particular
numbers which the author uses, are necessarily interesting only
as clever puzzles, but not general enough to be valuable to the
serious student. To this objection I reply that, in the first
place, it was absolutely impossible that Diophantos should have
used any other than numerical coefficients for the reason that
the available symbols of notation were already employed, the
letters of the Greek alphabet always doing duty as numerals,
with the exception of the final <?, which Diophantos was supposed
to have used to represent the unknown quantity. In the second
place T do not admit that the use of numerical coefficients only
makes his 'solutions any the less general. This will be clearly
seen when I come to give an account of his problems ami
methods. Next as to Diophantos' symbols for the operations
of Addition and Subtraction. For the former no symbol at all is
used: it is expressed by mere juxta-position, thus K^dh^ly^i
corresponds to x^ + V^x" + ox. In this expression, however, there
is no absolute term, and the addition of a simple numeral, as
for instance /3, directly after e, the coefficient of vv, would cause
confusion. This ."act makes it necessary to have some term to
indicate an absolute term in contradistinction to the variable
terms. For this purpose Diophantos uses the word ^ovdha, or
units, and denotes them after his usual manner by the abbre-
viation lA?. The number of monads is expressed as a c«<offiiipnf.
DIOPHANTOS OF ALEXANDRIA.
Thus correspouding to the above expression x^ i-ISou^ + 5x+ 2
we should find in Diophantus «'' a 8" r-y 99 e fi° $. As Bachet
uses the sign + for addition, he has no occasion for a distinct
symbol to mark an absolute term. He would accordingly write
IC +1'3Q + 5X+2. It is worth observing, however, that the
Italians do use a symbol in this case, namely K (Xumero), the
first power of the unknown being with them li (Rudice).
Cossali* makes an interesting comparison between the terms
used by Diophantos for the successive powers of the unknown
and those employed by the Italians after their instructors, the
Arabians. He observes that Fra Luca, Tartaglia, and Cardan
begin their scale of powers from the power 0, not from the
power 1, as does Diophantos, and compares the scales thus :
Scala Diojantea. Sc((la Anibii.
1 . Numero. . .il Noto.
X 1. Numero... riguoto. 2. Cosa, Radice, Lato.
X- 2. Podesta. :?. Censo.
.!■' '6. Ciibo. 4. Cubo.
.T^ 4. Podesta-Podesta. 5. Censo di Censo.
^5 5. Podesta Cubo. C). Eelato V.
x* 6. Cubo-Cubo. 7. Ccuso di Cubo, o Cubo di Censo.
x7 7 S. Relato2«.
.1-8 8 1). Censo di Censo di Ceuso.
.1-9 [) 10. Cubo di Cujo.
and so on. So far, however, as this is meant to be a comparison
between Diophantos and the early Arabian algebraists them-
selves (as the title Scala Araba would seem to in»ply), there
appears to be no reason why Cossali should not have placed
some term to express Diophantos fiovdBe<i iu the same line
with Numero in the other scale, and moved tlie numbers
1, 2, 3, S:c. one place upwards in the first scale, or downwards
' Upon Wallis' comparison of the Diophantiue with the Arabian scale
Cossali remarks: "ma egli non ha riflettuto a due altre diflerenze tra le scale
medesime. La prima si e, che laddove Diofanto denomina con singolarita
Numero 11 numero ignoto, denominanJo Monade il numero dato di compara-
zione : gli antichi italiani degli arabi seguaci denominano questo il Numero ;
e Radice, o IjuIo, o Cosa il numero sconosciuto. La scconda e, che Diofanto
comincia la scala dal numero ignoto; e Fra Luca, Tartaglia, Cardano la in-
coiiiinciano dal numero noto. Ecco le due scale di rincontro, onde meglio
risaltino all" occhio le diffcrcnzc loro." i. p. I'.lo.
i
NOTATION AND DEFINITIONS <»K l)It>l'IfANT(»s. 71
in tlie secoiul. As Diopliantos does nut go "beyond the sixtli
power, the hist three phices in the tirst scale are left blank.
An examination of these two scales will show also that the
generation of the successive powers dirt'ers in the two systems.
The Diophantine terms for them are based on the addition of
exponents, the Arabic on their multipliaition \ Thus the "cube-
cube" means in ])iophantos of, in the Italian and Arabic system
x". The first method of generation may (says Cossali) be
described as the method by which each power is represented
by the product of the two lesser powers which are nearest
to it, the method of viidtiplication ; the second the method of
elevation, i.e. the method which forms by raising to the second
or third power all powers which can be so formed, or the ^ih,
Cth, 8th, 9th, &c. The intermediate powers which cannot be
so formed are called in Italian Relati. Thus the fifth power is
Relato 1", x' is Relato 2", a-'" is Censo di Relato 1", .r" is Relato 3",
and so on. Wall is calls these powers supevsolida, reproduced by
Montucla as sursolides.
For Subtraction Diophantos uses a symbol. His full terra
for Negation is Xei-\\r.<;, corresponding to inrap^i<i, which denotes
the opposite. Thus Xeiylrei (i.e. with the want of) stands for
minus, and the symbol used to denote it in the MSS. is an
inverted i/r or ^ (Def. 9 /cat t^? Xet>|re&)9 (njfietov yfr €X\nre<i
Kara) vevov >>jt) with the top shortened. As Diophantos uses
no distinct sign for +, it is clearly necessary, to avoid confusion,
that all the negative terms in an expression should be placed
together after all the positive terms. And so in fact he
does place them^ Thus corresponding to x^ — ox' + Sx — 1,
1 This statement of Cossali's needs qualification however. There is at least
one Arabian algebraist, Alkarkhi, the author of the Fakhri referred to above
(pp. 24, 25), who uses the Diophantine system of powers of the unknown de-
pending on the additioji of exponents. Alkarkhi, namely, expresses all powers
of the unknown above the third by means of nUil, his term for tlie square, and
ka% his term for the cube of the unknown, as follows. The fourth power is
with hun mdl mdl, the fifth null Jca'b, the sixth ka'b ka'b, the seventh null mal
ka'b, the eighth nial ka'b ka'h, the ninth ka'b ka'b ka'b, and bo on.
» Dr Heinrich Suter however has the erroneous statement that Diopliantos
would express j?-5x^ + 8.r - 1 by k" d /A «"« »• ij ^ M° «'. which is exactly what h«
would not do.
72 DIOPHANTOS OF ALEXANDKIA.
Diophantos would write «" a s°"' ?; >//v S" e ^° d. With respect
to this curious sign, given in the MSS. as ^ and described as an
inverted truncated ■\\r, I must here observe tliat I do not believe
it to be what it is represented as being. I do not believe that
Diophantos used so fantastic a sign for minus as an inverted
truncated -v/r. In the first place, an inverted -^^ seems too
curious a sign, and too far-fetched. To one wlio was looking
for a symbol to express mimis many others more natural
and less fantastic than j/v must have suggested themselves.
{Secondly, given that Diophantos used an inverted -x/r, why
should he truncate it ? Surely that must have been unneces-
sary ; Ave could hardly have expected it unless, without it,
confusion was likely to arise; but ^ could hardly have been
confused with anything. It seems to me that this very trunca-
tion throws doubt on the symbol as we find it in the MS.
Hence I believe that the conception of this symbol as an
inverted truncated -^ is a mistake, and that the description of
it as such is not Diophantos' description ; it appears to me to be
an explanation by a scribe of a symbol which he did not under-
stand*. It seems to me probable that the true explanation is
the following : Diophantos proceeded in this case as in the others
which we have discussed (the signs for apt^/xd?, ^vvaiJii<;, etc.).
As in those cases he took for his abbreviation the first letter of
the word with such an addition as would make confusion with
numbers impossible (namely the second letter of the word,
which in all happens to come later in the alphabet than the
corresponding first letter), so, in seeking an abbreviation for
\et>|rt9 and cognate inflected forms developed from Xtir, he
first took the initial letter of the word. The uncial" form is
A. Clearly A by itself would not serve his purpose, since it
denotes a number. Therefore an addition is necessary. The
second letter is E, but AE is equally a number. The second
1 I am not even sure that the description can be made to mean all that it is
intended to mean. AXiWs scarcely seems to be sufliciently precise. Might it
not be applied to ^ with any part cut off, and not only shortened at the top?
* I adhere to the uncial form above for clearness' sake. If Diophantos used
the "Majuskelcursive" form, the explanation will equally apply, the difference
of form being for our purpose lU'Rlifiihle.
NOTATION AND DEFINITIONS OF DIOl'llANTt )S. 7:{
letter of the stem Xnr is I, but Al is open to objectiuii \slnii
so written. Hence Diophantos placed the I inside the A, thus,
A. Of the possibility of this I entertain no doubt, because
there are indubitable cases of combination, even in uncial
writing, of two letters into one sign. I would refer in par-
ticular to X, which is an uncial abbreviation for TAAANTON.
Now this sign. A, is an inverted and truncated i/r (written in
the uncial form, ^j; and we can, on this assumpti(tn, easily
account for the explanation of the sign for minus which is given
in the text.
For Division it often happens that no symbol is necessary,
i.e. in the cases where one number is to be divided by another
which will divide it without a remainder. In other cases the
division has to be expressed by a fraction, whether the divisor
be an absolute number or contain the variable. Thus the case
of Division comes under that of Fractions. To express nume-
rical fractions Diophantos adopts a uniform system, which is
also seen in other writers. The numerator he writes in the
ordinary line like a number; then he places the denominator
above the line to the right of the numerator, in the same place
as we should write an exponent, usually placing a ciicumflex
a-cent over the eud of it. Thus i| is represented by tf'^ , yJ^j^
is aP, -Win- is (v. 12) ^eTZT?""^", ^[\%' is (iv. 17) y.S'xicd^'^^'.
Diophantos, however, often expresses fractions by simply putting
iv fjLopiro or [lopiov between the numerator and the denominator,
i.e. one number divided by another. Cf. IV, 29 pF-.^f ^ttS fiopiov
HS^.fipixh, i.e. Vif/nV' ^^^'J "^'- -'^ /5-,^X ^^ p-opiw pK^.aKi, i.e.
TyiC^o* There is a peculiarity in the way in which Diophan-
tos expresses such complex fractions as ,.,,'• It will be
best understood by giving a typical case. This jiarticular
fraction Diopliantos writes thus, aoyXB^^^.u^, that is, it is as if
he had written with our notation y^Y i. Instances of this
- " - . fiH[)ii
notation occur passim, cf. V. 2 T7r6i"'^.d^ is e(iuivalent to -p.f •
Bachet reproduces Diophantos' notation by writing in the.se
cases '/jY h ^^^^ iti h respectively.
74 DIOPHANTOS OF ALEXANDHIA.
But there is another kind of fraction, besides the purely
numerical one, which is continually occurring in the Arith-
metics, such fractions namely as involve the unknown quantity
in some form or other in their denominators. The simplest
case is that in which the denominator is simply a power of the
unknown, 9'. Concerning fractions of this kind Diophantos
says (Def. 3) "As fractions named after numbers have similar
names to those of the numbers themselves (thus a third is
named from three, a fourth from four), so the fractions ho-
monymous with what are called dpiO/iot, or unknowns, are
called after them, thus from dpi6fx6<; we name the fraction to
dpiOfjioaTov [i.e. - from ;r], to Suvafioarov from BvvafiL<;, to
Kv^oarov from Kv^o<i, to SwafjuoBwafioaTov from SuvafioBvva-
/i.t<?, TO Bvva/xoKv/SoaTOv from Bwap-oKv^o^, and to kv^okv-
^oarov from Kv^oKv/So'i. And every such fraction shall have
its symbol after the homonymous number with a line to indi-
cate the species" (i.e. the order or power)*. Thus we find, for
8 .
example, IV. 3, 77"*" corresponding to , or with the genitive
termination of dpiOfioaTov, e.g., IV, 16, Xe''°^-^ or — . Cf. av^"-^
250
or-V- ^iJe by side with the employment of the symbols to
express fractions corresponding to -, —2, &c., we find the terms
dpiOfioarov, Bwafioarov k.t.X. used in full : this is regularly
the case when the numerator of the fraction itself contains a
numerical fraction. Thus in v. 31 dptdfjioaTov d d^ corre-
sponds to -^ and hwapLoarov T d^ to - ^ •
Diophantos extends his use of fractions still further to more
complicated ca.ses in which the numerator and denominator
1 The meaning of the last sentence is not quite clear. I am inclined to
think there is something wrong with the text, which stands in Bachet as follows :
?$ft 5^ iKaarov avTwv iirl Toy< ofiuvi'ifiov dpiO/xov arjiii'iov ypdij.fj.r]v Ixo" SiaffT^Wovaav
rb el5os. This he translates, "Habebit autem quaelibet pars a sibi cognomitie
numero notarn et literam superscriptam quae speciem a specie distinguat."'
Here apparently literam corresponds to ypdfifxr]v.
XOTATIOX AND DKFIMTK t.NS OF UK )l'll ANTi )S. 7o
may be compound expressions themselves, involving the un-
known ([uantity. Thus, iv. 37, we liave /u." ^ ^ 9°'' «"''", i.e.
' — — . AVhen, however, tlie denominator is a compound ex-
pression Diophantos uses the expedient which he adopts in the
case of Large numbers occurring as numerator or denominator,
namely, the insertion between the expressions denoting tlio
numerator and denominator of the term iv fjLopi(p or fxoplov.
Thus in VI. 13 we find, h" ^./u," j34>k iv fiopup B"6" a /x" "^ \ei\Jrei
., . . 60^-' + 2520 - .' ^, -'5 c>- c. - .
' sjc - - 5 s - r ^ 2j-' + ox- + ix + \
fiopiM o" a >i^ p /j.° a corresponding to ^ — ^^ .
To connect the two sides of an equation Diophantos uses
Avords (cro9 or i'ao9 iari, or the oblique cases of i<to<; when they
are made necessary by grammatical construction. It would
appear, at least from Bachet's edition of Diophantos, that the
equations were put down in the ordinary course of writing,
and that they were not placed in separate lines for each step
in the process of simplification, being in fact written in the
same way as the propositions of Euclid. We have, however,
signs of a system by which the steps were tabulated in a
manner very similar to that of modern algebraical work, s<^
that by means of a sort of skeleton of the procedure we get a
kind of bird's-eye view of its course, in the manuscript of Dio-
])hantos which Bachet himself used. We have it on the
authority of M. Rodet, who in an article in the Journal Asia-
tique^ has occasion to quote certain passages from the text of
J)iophantos, that to certain problems is attached a tabular
view of the whole process, which Bachet has not in his editinn
reproduced at all. M. Rodet gives from the MS. several in-
iitances. In these we have equations set down in a form very
like the modern, the two members being connected by the
letter I (abbreviated for Xaoi) as the sign of e<iualit\'. Besides
1 Janvier, 1878.
- Here again the abbreviation is explicable on the .same jirinciiik- as 11k»»c
which I have previously discnssed. ( by itself means 10, but a distin^ishinp
mark is ready to Land in the breathing' phmcd over it.
70
DIOPHANTOS OF ALEXANDRIA.
the equations written in this form there are on the left
side words signifying the nature of tlie operation in passing
from one particular step to the next. To illustrate this I will
give the table after Rodet for the very simple problem i. 32.
" To find two numbers whose sum, and the ditference of whose
squares are given." (The sum is supposed to be 20, the dif-
ference of squares is 80.) Diophantos assumes the difference
of the two numbers themselves to be two dpidfioL I will put
the Greek table on the left side, and on the right the modern
equivalent. The operations will be easily understood.
iKdeais s" /uoT : yuoif. set]
virepox-f)
si IX
I ps>.^
Ixfpt(j/ji6i
S a
I p.o.-^
inrop^is
^Pp.oI?
(^ po.q
Put for the numbers x + 10, 10 - x.
Squaring we have .r2+20x+ 100,
x2 + 100-20x.
Thediff., 40x = 80.
Dividing, x = 2.
Result, greater is 12, less is 8.
The comparison of these two forms under which the same
operations appear is most interesting. It is indeed obvious
that if we take the skeletons of work given in the MS, the
similarity is most striking. It is true that the Greek notation
for the equations is very much inferior to the modern, but on
the other hand the words indicating the operations make the
whole very little less concise than the modern work. The
omission of these tabular skeletons supplied in the MS. is a
very grave defect in Bachet's edition, and thanks are due to
M. Rodet for his interesting quotations from the original
source. The same writer quotes two other such tables, which,
however, for brevity's sake, we omit here. Though in the Ms.
the sign I is used to denote equality, Bachet makes no use of
any symbol for the purpose in his Latin translation. He uses
throughout the full Latin word. It is interesting however to
observe that in his earlier translation (1575) Xy lander does
use a symbol to denote equalit3^ namely ||, two short vertical
parallel lines, in his notes to Diophantos. Thus we find, for
example (p. 7G) \Q -¥ 12 || \Q + 0^7 + 9, which we should ex-
press by x' +12= x^ + Qx + ^d.
§ 5. Now that we have described in detail Diopliantos'
method of expressing algebraical quantities and relations, we
NOTATION AND DEFINITIONS OF DIOPIIANToS. 77
may remark on the general system which he uses that it is
essentially different in its character from the modern notation.
While in modern times signs and symbols have been developed
which have no intrinsic relationship to the things which they
syml)olise, but depend for their use upon convention, the case is
quite different with Diophantos, where algebraic notation takes
the form of mere abbreviation of words which are considered a.s
pronounced or implied. This is partly proved by the symbols
themselves, which in general consist of the first letter or letters
of words (so written as to avoid confusion), the only possible
exception being the supposed final sigma, 9, for dpiBfxof or the
unknow^n quantit3^ Partly also it is proved by the fact that
Diophantos uses the symbol and the complete word very often
quite indifferently. Thus we find often in the same sentence
9 or 99 and dpt6fi6<;, dpiO/xol, S" and hvvaixi<;, ^ and \eiyjrei, and
so on. The strongest proof, however, that Diophantos' algebraic
notation >vas mere abbreviation is found in the fact that the
abbreviations, which are his algebraical symbols, are used for
the corresponding words even when those words have a quite
different signification. So in particular the symbol 9 is used as
an abbreviation for dpL6fi6<;, when the word is used, not in its
technical Diophantine sense for the unknown, but in its ordinary
meaning of a number, especially in enunciations ^\here dpi$fi6<;
in its ordinary sense naturally occurs oftenest. Similarly ^ is
not used only for Xelyjret but also for other inflexional forms of
the stem of this word, e.g. for Xiiruv or \€Lyjra<; in ill. 3: Evpeh'
Tpel<; dpiOfiov^ o7rco<; 6 diro tov avy/ceifievov iic rcov rpLiov JJ^
eKaarov nroifj rerpdycovoi'. Other indications are (1) the sepa-
ration of the symbc.ls and coefficients by particles [cf. I. 4:J
99°"' dpa I] ; (2) the addition of terminations to the symbol to
represent the different cases. Nesselmann gives a good instance
in which many of these peculiarities are combined, 99°' dpa
I fx° \ laot elalv 99°'* Td fiovdcrc Tt. I. ad Jin.
In order to determine in what place, in respect of .systems
of algebraic notation, Diophantos stands, Nesselmann observes
that we can, as regards the form of exposition cf algebraic
operations and equations, distinguish three historical stages of
development, well marked and easily discernible. 1. Tho first
<S DIorilANTOS OF ALKXANDlilA.
Stage Nesselinann repiesonts by the name Rltetoric Algebra,
or "reckoning by Complete words." The characteristic of this
stage is the absolute want of all symbols, the ■svhole of the
calculation being carried on by means of complete words, and
forming in fact continuous prose. As representatives of this
first stage Nesselmann mentions lamblichos (of whose algebrai-
cal work he quotes a specimen in his fifth chapter) "and all
Arabian and Persian algebraists who are at present known."
In their works we find no vestige of algebraic symbols ; the
same may be said of the oldest Italian algebraists and their
followers, and among them Regiomontanus. 2. The second
stage Nesselmann proposes to call the Syncopated Algebra.
This stage is essentially rhetorical and therein like the first in
its treatment of questions, but we now find for often-recurring
operations and quantities certain abbreviational symbols. To
this stage belongs Diophantos and after him all the later
Europeans until about the middle of the seventeenth century
(with the exception of the isolated case of Vieta, who, as we
have seen, initiated certain changes which anticipated later
notation to some extent; we must make an exception too,
though Nesselmann does not mention these cases, in favour of
certain symbols used by Xylandcr and Bachet, j| being used by
the former to express equality, + and — by both, as also the
ordinary way of representing a fraction by placing the numera-
tor above the denominator separated by a Hue drawn horizon-
tally'). 3. To the third stage Nesselmann gives the name
Symbolic Algebra, which uses a complete system of notation
by signs having no visible connection with the words or things
which they represent, a complete language of symbols, which
supplants entirely the rhetorical system, it being possible to
work out a solution without using a single word of the ordinary
written language, with the exception (for clearnes.s' sake) of
1 These are only a few scattered instances. Nesselmann, though he does
not mention Xylauder's and Bachet's symbols, gives other instances of isolated
or common uses of signs, as showing that the division between the dilTerent
stages is not shfirphj marked. He instances the use of one operational algebraic
symbol by Diophantos, namely ^, for which Lucas de Bnrgo uses m (and p for
plKn), Targalia (p. Vieta has + and -, also = for ~. Oughtred uses x , and
Harriot expresses multiplication by juxtaposition.
NOTATION AND DKl'lMTK )NS (.F Llopll ANTuS. 70
a conjunction here ami there, and so on. Neither is it thr
Europeans posterior to the middle of the seventeenth century
who were the first to use Si/mbolic forms of Algebra, In this
they were anticipated many centuries by the Indians.
As examples of these three stages Nesselmann gives three
instances quoting word for word the solution of a quadratic
equation by Mohammed ibn Musa as an example of the first
stage, and the solution of a problem from Diophantos to illus-
trate the second. Thus :
First Stage. Example from Molmmmed ibn Musa (ed.
Rosen, p, 5). "A square and ten of its roots are equal to nine
and thirty dirhems, that is, if you add ten roots to one square,
the sum is equal to nine and thirty. The solution is as follows :
halve the number of roots, that is in this case five ; then
multiply this by itself, and the result is five and twenty. Add
this to the nine and thirty, which gives sixty-four; take the
square root, or eight, and subtract from it half the number
of roots, namely five, and there remain three : this is the root
of the square whicli was required and the square itself is
nine^."
Here we observe that not even are symbols used for num-
bers, so that this example is even more "rhetorical" than the
w^ork of larablichos who does use the Greek symbols for his
numbers.
Second stage. As an example of Diopliantos I give a trans-
lation word for word ^ of II. 8, So as to make the symbols
correspond exactly I use S {Square) for h" {8vvaf/.c<i), X (Xiun-
her) for 9, U for Units {fiovdhe<i).
" To divide the proposed square into two squares. Let it be
proposed then to divide 16 into two squares. And let the first
' Thus Mohammed ilm Mfisfi states in words the solution
x2+10j + 25 = C4,
therefore x + 5 = 8,
x = 3.
' I have used the full words whenever Diophantos does so, and to avoid con-
fusion have written Siium-c and Xiimbfi- in the technical sense with a capital
letter, and italicised tlicm.
80 DIOPHANTOS OF Al.EXANDIUA.
be supposed to be Oue Square. Thus 16 miuus One Square
must be equal to a square. I form the square from any number
of iV's minus as many U's as there are in the side of 16 U's.
Suppose this to be 2 K's miuus 4 U's. Thus the square itself
will be 4 Squares, 16 U. minus 16 ^V.'s. These are equal to
16 Units minus Oue Square. Add to each the negative term
(Xeti/ri?, deficiency) and take equals from equals. Thus
5 Squares are equal to 16 Numbers; and the Number is
16 fifths. One [square] will be 256 twenty-fifths, and the other
144 twenty-fifths, and the sura of the two makes up 400
twenty-fiftlis, or 16 Units, and each [of the two found] is a
square.
Of the third stage any exemplification is unnecessary.
^>^ § 6. To the form of Diophantos' notation is due the fact
that he is unable to introduce into his questions more than one
unknown quantity. This limitation has made his procedure
often very different from our modern work. In the first place
he performs eliminations, which we should leave to be done in
the course of the work, before he prepares to work out the
problem, by expressing everything which occurs in such a way
as to contain only one unknown. This is the case in the great
majority of questions of the first Book, which are cases of the
solution of determinate simultaneous equations of the first order
with two, three, or four variables; all these Diophantos ex-
presses in terms of one unknown, and then proceeds to find it
from a simple equation. In cases where the relations between
these variables are complicated, Diophantos shows extraordinary
acuteness in the selection of an unknown quantity. Secondly,
however, this limitation affects much of Diophantos' work in-
juriously, for while he handles problems which are by nature
indeterminate and woukl lead with our notation to an inde-
terminate equation containing two or three unknowns, he is
compelled by limitation of notation to assign to one or other of
these arbitrarily-chosen numbers which have the effect of
making the problem a determinate one. However it is but
fair to say that Diophantos in assigning an arbitrary value to
a quantity is careful to tell us so, saying " for such and such
a quantity we put any number wliatever, say such and such
NOTATION AND DKFINITION'S (tF 1)F< )riIANT()S. Hi
a one." Thus it can lianlly be said tliat there is (in LfiMieral)
any loss of universality. We may say, then, that in general
Diophantos is obliged to express all his unknowns in terms,
or as functions, of one variable. There is something exce.ssively
interesting in the clever devices by which he contrives .so to
express them in terms of his single unknown, <?, as that by that
very expression of them all conditions of the problem are
satisfied except one, which serves to complete the solution by
determining the value of 9. Another consequence of Diophan-
tos' want of other symbols besides 9 to express more variables
than one is that, when (as often happens) it is necessary in the
course of a problem to work out a subsidiary problem in order
to obtain the coefficients &c. of the functions of 9 which express
the quantities to be found, in this case the required unknown
which is used for the solution of the new subsidiary problem is
denoted by the same symbol 9 ; hence we have often in the
same problem the same variable 9 used with two diti'erent
meanings. This is an obvious inconvenience and might lead to
confusion in the mind of a careless reader. Again we find two
cases, II. 29 and 30, where for the proper working-out of the
problem two unknowns are imperatively necessary. We should
of course use x and y ; but Diophantos calls the first 9 as u.sual ;
the second, for want of a term, he agrees to call " one unit,"
i.e. 1. Then, later, having completed the part of the solution
necessary to find 9 he substitutes its value, and uses 9 over
again to denote what he had originally called " 1 " — the second
variable — and so finds it. This is the most curious case I have
met with, and the way in which Diophantos after having
worked with this " 1 " along with other numerals is yet able to
pounce upon the particular place where it has passed to, so as to
substitute 9 for it, is very remarkable. This could only be pos-
sible in particular cases such as those which I have mentioned :
but, even here, it seems scarcely possible now to work out the
problem using x and 1 for the variables as originally taken by
Diophantos without falling into confusion. Perhaps, however,- it
may not be impo.ssible that Diophantos in working out the
problems before writing them down as we have them may have
given the " 1 " which stood for a variable some mark by which
H. D. ^
82 DIOPHANTo.S OF ALKXANDl'JA.
he could recognise it and distinguish it from otlier numbers.
For the problems themselves see Appendix.
It may be in some measure due to the defects of notation in
his time that Diophantos will have in his solutions no numbers
/ whatever except rational numbers, in which, in addition to
surds and imaginary quantities, he includes negative quantities.
Of a negative quantity i^er se, i.e. without some positive quan-
tity to subtract it from, Diophantos had apparently no con-
ception. Such equations then as lead to surd, imaginary, or
negative roots he regards as useless for his purpose : the solu-
tion is in these cases dSvvaro'i, impossible. So we find him
describing the equation 4 = 4.r + 20 as utotto^ because it would
give ^ = — 4. Diophantos makes it throughout his object to
obtain solutions in rational numbers, and we find him fre-
quently giving, as a preliminary, conditions which must bo
satisfied, which are the conditions of a result rational in Dio-
phantos' sense. In the great majority of cases when Diophan-
tos arrives in the course of a solution at an equation M'hich
would give an irrational result he retraces his steps and finds
out how his equation has arisen, and how he may by altering
the previous work substitute for it another which shall give
a rational result. This gives rise, in general, to a subsidiary
problem the solution of which ensures a rational result for the
problem itself. Though, however, Diophantos has no notation
for a surd, and does not admit surd results, it is scarcely true to
say that he makes no use of quadratic equations which lead to
such results. Thu.s, for example, in v. 33 he solves such an
equation so far as to be able to see to what integers the
solution would approximate most nearly.
CHAPTEIi V.
DIOPHANTOS' METHODS OF SOLUTION.
§ 1. Before I give an accouut in detail of the differojit
methods which Diophantos employs for the sohition of his pro-
blems, so far as they can be classified, I must take exception to
some remarks which Hankel has made in his account of Dio-
phantos {Zur Geschichte der Mathematik in Alterthum vnd
Mittelalter, Leipzig, 1874, pp. 164 — 5). This account does
not only possess literary merit : it is the work of a man who
has read Diophantos. His remarks therefore possess excep-
tional value as those of a man particularly well qualified to
speak on matters relating to the history of mathematics, and
also from the contrast to the mass of writers who have thought
themselves capable of pronouncing upon Diophantos and his
merits, while they show unmistakeably that they have not
studied his work. Hankel, who has read Diophantos with aj)-
preciation, says in the place referred to, "The reader will now
be desirous to become acquainted with the classes of inde-
terminate problems which Diophantos treats of, and his methods
of solution. As regards the first point, we must observe that
in the 130 (or so) indeterminate questions, of which Diophantos
treats in his great work, there are over 50 different classes of
questions, which are arranged one after the other without any
recognisable classification, except that the solution of earlier
questions facilitates that of the later. The first Book only con-
tains determinate algebraic equations; Books ir. to v. contain
for the most part indeterminate questions, in which expressions
which involve in the first or second degree two or more variables
are to be made squares or cubes. Lastly, Book vi. is concerned
G— 2
84 DIOPIIANTOS OF ALEXANDRIA.
Avith riglit-augled triangles regarded purely arithmetically, in
which some one linear or quadratic function of the sides is to
be made a square or a cube. That is all that we can pronounce
about this elegant series of questions vnthout exhibiting singhj
each of the fifty classes. Almost more different in kind than
the questions are their solutions, and we are completely unable
to give an even tolerably exhaustive review of the different
varieties in his procedure. Of more general comprehensive
methods there is in our author no trace discoverable : every ques-
tion requires an entirely different method, winch often, even in
the problems most nearly related to the former, refuses its aid.
It is on that account difficult for a more modern mathematician
even after studying 100 Diophantine solutions to solve the lOlsi
question ; and if we have made the attempt and after some vain
endeavours read Diophantos' own solution, we shall be astonished
to see how suddenly Diophantos leaves the broad high-road,
dashes into a side-path and with a quiet turn reaches the
goal : often enough a goal with reaching which we should not
be content ; we expected to have to climb a difficult path, but
to be rewarded at the end by an extensive view ; instead of
which our guide leads by narrow, strange, but smooth ways to
a small eminence ; he has finished ! He lacks the calm and
concentrated energy for a deep plunge into a single important
problem : and in this way the reader also hurries with inward
unrest from problem -to problem, as in a succession of riddles,
without being able to enjoy the individual one. Diophantos
dazzles more than he delights. He is in a wonderful measure
wise, clever, quick-sighted, indefatigable, but does not penetrate
thoroughly or deeply into the root of the matter. As his ques-
tions seem framed in obedience to no obvious scientific necessity,
often only for the sake of the solution, the solution itself also lacks
perfection and deeper signification. He is a brilliant performer
in the art of indeterminate analysis invented by him, but the
science has nevertheless been indebted, at least directly, to this
brilliant genius for few methods, because he was deficient in
speculative thought which sees in the True more than the
Correct. That is the general impression, which I have gained
from a thorough and repeated study of Diophantos' arithmetic."
DIOPIIANTOS METHODS OF SOLUTION. S.")
Now it will be at once obvious that, if Ilaiikcl's representa-
tion is correct, any hope of giving a general account of Dio-
phantos' methods such as I have shown in the heading of this
chapter would be perfectly illusory. Hankel clearly asserts
tliivt there are no general methods distinguishable in the Arith-
metics. On the other hand we find Nesselmann saying (pp.
308 — 9) that the use of determinate numerals in Diophajitos'
problems constitutes no loss of generality, for throughout he
is continually showing how other numerals than those which
he takes will satisfy the conditions of the problem, showinn-
" that his whole attention is directed to the explanation of the
method, for Avhich purjjose numerical examples only serve as
means " ; this is proved by his frequently stopping short, when
the method has been made sufficiently clear, and the remainder
of the work is mere straightforward calculation. Cf. v. 14, IS,
19, 20 &c. It is true that this remark may only apply to the
isolated " method " employed in one particular problem and in
no other ; but Nesselmann goes on to observe that, though the
Greeks and Arabians used only numerical examples, yet they
had general rules and methods for the solution of equations, as
we have, only expressed in words. "So also Diophantos, whose
methods have, it is true, in the great majority of cases no such
universal character, gives us a perfectly general rule for solving
what he calls a double-equation." These remarks Nesselmann
makes in the 7th chapter of his book ; the 8th chapter he
entitles "Diophantos' treatment of equations'," in which he
gives an account of Diophantos' solutions of (1) Deterniinato,
(2) Indeterminate equations, classified according to their kind.
Chapter 9 of his book Nesselmann calls "Diophantos' methods
of solution^" These "methods" he gives as fcjlJows^ : (1) "The
adroit assumption of unknowns." (2) "Metho<l of reckoning
^ "Diophant's BehandlunR der Gleichunt,'tn."
2 "Diophant's AuflosuriKsmethodeu."
3 (1) "Die gescbickte Annahme dcr Unbekannten." (2) '•Mflhode ilcr
Zuriickrechming und Nebenaufgabe." (3) "Gebrauch dcs Symbols fiir dio
Unbekannte in verschiedenen Bedeutungen." (4) "Metliodc der Gren/.fu."
(5) "Auflosung durch blo.sse Reflexion." (G) "Autt.isung in ollKemeinPn
Ausdriicken." (7) " Willkiihrliche Bestiniraungen und Annuhnien." (8) "tie-
branch dcs rcchtwinkhu'cn Dipiecks."
86 DIOPHANTOS OF ALEXANDRIA.
backw artls and auxiliary questions." (3) " Use of the symbol
for the unknown in different significations." (4) "Method
of Limits." (5) " Solution by mere reflection." (G) " Solution
in general expressions." (7) "Arbitrary determinations and
assumptions." (8) " Use of the right-angled triangle."
At the end of chapter 8 Nesselmann observes that it is not
the solution of equations that we have to wonder at, but the per-
fect art which enabled Diophantos to avoid such equations as he
could not technically solve. We look (says Nesselmann) with
astonishment at his operations, when he reduces the most
difficult questions by some surprising turn to a simple equation.
Then, when in the 9th chapter Nesselmann passes to the
" methods," he prefaces it by saying : " To represent perfectly
Diophantos' methods in all their completeness would mean
nothing else than copying his book outright. The individual
characteristics of almost every question give him occasion to
try upon it a peculiar procedure or found upon it an artifice
which cannot be applied to any other question Mean-
while, though it may be impossible to exhibit all his methods in
any short space, yet I will try to give some operations which
occur more often or are by their elegance particularly notice-
able, and (where possible) to make clear their scientific prin-
ciple by a general exposition from common stand-points." Now
the question whether Diophantos' methods can be exhibited
briefly, and whether there can be said to be any methods in his
work, must depend entirely upon the meaning we attach to the
word "method." Nesselmann's arrangement seems to me to be
faulty inasmuch as (1) he has treated Diophantos' solution of
equations — which certainly proceeded on fixed rules, and there-
fore by " method " — separately from what he calls " methods of
solution," thereby making it appear as though he did not look
upon the " treatment of equations " as " methods." Now cer-
tainly the " treatment of equations " should, if anything, have
come under the head of " methods of solution " ; and obviously
the very fact that Diophantos solved equations of various kinds
by fixed rules itself disproves the assertion that no metJwds
;nc discernible. (2) The classification under the head of
" Mctliods of solution" seems unsatisfactory. In tlic first
DIOPHANTOS METHODS UF SoLlTIoX. S7
place, some of the classes can hardly be said to be nu'thmls of
solution at all; thus the third, "Use of the symbol for the
unknown in different significations", might be more justly
described as a "hindrance to the solution"; it is vlxx inconve-
nience to which Diophantos was reduced owing to the want of
notation. Secondly, on the assumption of the eight " methods"
as Nesselmann describes them, it is really not surprising that
" no complete account of them could be given without copying
the whole book." To take the first, "the adroit assumption of
unknowns." Supposing that a number of distinct, ditferent
problems are proposed, the existence of such differences makes
a different assumption of an unknown in each case absolutely
necessary. That being so, how could it be possible to give a
rule for all cases ? The best that can be done is an enumera-
tion of typical instances. The assumption that the methods
of Diophantos cannot be tabulated, on the evidence of this
fact, i.e., because no rule can be given for the " adroit assump-
tion of unknowns" which Nesselmann classes as a "method," is
entirely unwarranted. Precisely the same may be said of
"methods" (2), (5), (C), (7). For these, by the very nature of
things, no rule can be given : they bear in their names so much
of rule as can be assigned to them. The case of (4), "the
Method of Limits", is different; here we have the only class
which exemplifies a "method" in the true sense of the term,
i.e. as an instrument for solution. And accordingly in this case
the method can be exhibited, as I hope to show later on :
(8) also deserves to some extent the name of a " method."
I think, therefore, that neither Nesselmann nor Hanktl has
treated satisfactorily the question of Diophantos' methods, the
former through a faulty system of classification, the latter by
denying that general methods are anywhere discernible in
Diophantos. It is true that we cannot find in Diophanto.s' work
statements of method put generally as book-work to be applied
to examples. But it was not Diophantos' object to write a text-
book of Algebra. For this reason we do not find the separate
rules and limitations for the solution of different kinds of equa-
tions systematically arranged, but we have to seek them out
laboriously from the whole of his work, gathering .«5cattcred
88 DIOPHANTOS OF ALEXANDRIA.
indications here and there, and so formulate tlicni in the best
way we can. Such being the case, I shall attempt in the follow-
ing pages of this chapter to give a detailed account of what may
be called general methods running through Diophantos. For
the reasons which I have stated, my arrangement will be different
from that of Nesselmann, who is the only author who has
attempted to give a complete account of the methods. I shall
not endeavour to describe as methods such classes of solutions as
are some which are, by Nesselmann, called "methods of solution":
and, in accordance with his remark that these " methods" can
only be adequately described by a transcription of the entire
work, I shall leave them to be gathered from a perusal of
my reproduction of Diophantos' book which is given in my
Appendix.
§ 2. I sluili begin my account with
Diophantos' tueatmknt of equations.
This subject falls naturally into two division.s : (A) Deter-
minate equations of different degrees. (B) Indeterminate
equations.
(A.) Determinate equations.
Diophantos was able without difficulty to solve determinate
c( [nations of the first and second degree ; of a cubic equation we
Hnd in his Arithmetics only one example, and that is a very
special case. The solution of simple equations we may pass
over; hence we must separately consider Diophantos' method
of solution of (1) Pure equations, (2) Adfected, or mixed
quadratics.
(1) Pare determinate equations.
By pure equations I mean those equations which contain
oidy one power of the unknown, whatever the degree. The
solution is effected in the .same way whatever the exponent of
the term in the unknown ; and Diophantos regards pure eijuations
of any degree as though thfy were simple C(iuations of the first
DIOPHANTOS METHODS OF SOLLTION. M)
degree'. He gives a general rule for this case without regard to
the degree: "If we arrive at an equation containing on each
side the same term but with different coefficients, we must take
equals from equals, until we get one term equal to another
term. But, if there are on one or on both sides negative terms,
the deficiencies must be added on both sides until all tiie terms
on both sides are positive. Then we must take equals from
equals until one term is left on each side." After these opera-
tions have been performed, the equation is reduced to the form
Ax"* = B and is considered solved. The c^vses which occur in
Diophantos are cases in which the value of x is fouud to be
a rational number, integral or fractional. Diophantos only
recognise one value of x which satisfies this equation ; thus if
m is even, he gives only the positive value, a negative value jjer
se being a thing of which he had no conception. In the same
way, when an equation can be reduced in degree by dividing
throughout by any power of x, the possible values, x=0, tlius
arising are not taken into account. Thus an equation of the
form x^ = ax, which is of common occurrence in the earlier part
of the book, is taken to be merely equivaleut t'> the simj»Ie
ecpiation x = a.
It may be observed that the greater proportion of the pro-
blems in Book l. are such that more than one unknown quantify
is sought. Now% when there are two unknowns and two condi-
tions, both unknowns can be easily expressed in terms of one
symbol. But when there are three or four quantities to be
found this reduction is much more difficult, and Diophanto-s
manifests peculiar adroitness in effecting it : the result being
that it is only necessary to solve a simple equation with one
unknown quantity. With regard to pure eciuations, .some have
asserted that pure quadratics were tlie only form «>f ([uailratic
' Dof. 11: MerA 5^ Tavra iav airb irpofi\r)naT6s rii'ot yivijTai Crapiit ilitai
To?s aiiToh fir} bixoirX-qdr) Si dirb iKaripwv tCiv fitpCiv, Strati i^xupfiy rd Ofioia aw6
Twv ofioiwu, iws av ifbs (?) eZSos ivl etdfi toov y^vrrrai. 4cu> bl Twt (V oTOTifHfi im-
irdpxv^^) V ^v diJ.<poT^poii ivtWei^f/ri TifiL etSr), Sc^ati ■Kp<(jOtlvai ri. Xiiworra itSif
(i> dn(l>oT^pois Tois ixipeaiv, ?wj dv inaTtpip tu.'v nfpwv to. ttorj iwwdpxovTa finfrai.
Kal trdXiv d^eXfiv rd cp.oia dirb rwf ofioluv tm ay cKor^/x^ twi' fitpif In tlSot
Kara\ui>0^. Bachet's text (lO-il). p. 10.
90 DIOPHANTOS OF ALEXANDRIA.
solved in Diuplianto;;' : a statement entirely without foundation.
We proceed to consider
(2) Mixed quadratic equations.
After the remarks in Def. 11 upon the reduction of pure
equations until we have one term equal to another term,
Diophantos adds*: "But we will show you afterwards how, in
the case also when two terms are left equal to a single term,
such an equation can be solved." That is to say, he promises to
explain the solution of a mixed quadratic equation. In the
Arithmetics, as we possess the book, this promise is not fulfilled.
The first indications we have on the subject are a number of
cases in which the equation is given, and the solution written
down, or stated to be rational without any work being shown.
Thus, IV. 23, "x = 4a; — 4, therefore cc = 2": vi. 7, "8-^x^ — lx = 7,
hence a; = ^ " : vi. 9, " 630.^'^ — 73.r = 6, therefore x = f^": and, vi.
8, " G30j;*^ + 7Sx = G, and x is rational." These examples, though
proving that somehow Diophantos had arrived at the result,
are not a sufficient proof to satisfy us that he necessarily was
acquainted with a regular method for the solution of quadratics ;
these solutions might (though their variety makes it somewhat
unlikely) have been obtained by mere ti'ial. That, however,
Diophantos' solutions of mixed quadratics were not merely
empirical, is shown by instances in v. 3.1 In this problem he
shows pretty plainly that his method was scientific, in that he
indicates that he could approximate to the root in cases where
it is not rational. As this is an important point, I give the
substance of the passage in question : "x has to be so determined
•^■'-GO , x'-iJO . 2 ,.n -
that it must be > — - — and < -^ , i.e. .x - 00 > o.r, and
o O
x' - GO < S.r.
Therefore x" = Hx + some number > GO, therefore x must
be not less than II, and x^ <8x + 60, therefore a; must be not
greater than 12."
1 Cf. Iloimer, translation of Bossut's Ge.^rh. d. Math. i. 55. Kliigel's
Dictionary. Also Dr Hcinrich Siitcr's doubts in Oesch. d. Math. Zurich, 1873.
* vaTfpov hi (TOL Sci^ofxev Kal ttois 8i'o iidJiv lawv iv\ KaraXfifdivTwv t6 towvtov
\vtTai.
DIOrilAXTOS' MKTIloDS UF SOLUTION. 1»1
Now by examining the roots of these two equations we rtnJ
x> ^, and .'•<4 + V<(),
or .v> lO-G'-ldi and x< Il'TITN.
It is clear therefore that x inai/ be < 1 1 or > 12, and there-
fore Diophantos' limits are not strictly accurate. As however
it was doubtless his object to find integral limits, the limits 11
and 12 are those wdiich are obviously adapted for his purpose,
and are a fortiori right. Later in the same problem he makes
an auxiliary determination of x, which must be such that
x^+m>1±c, a,-' + GO < Sir,
which give x>\l+J(ji, a;<12 + J^\.
Here Diophantos says x must be > 10, <21, wliioli again
are clearly the nearest integral limits.
The occurrence of these two examples which we have given
of equations whose roots are irrational, and therefore could not
be hit upon by trial, show's that in such cases Diophantos must
have liad a method by which he approximated to these roots.
Thus it may be taken for granted that Diophantos had a definite
rule for the solution of mixed quadratic equations.
We are further able to make out the formula or rule by
which Diophantos solved such equations. Take, for example,
the equation ax^ ■\- hx -\- c = 0. In our modern method of solution
we divide by a and write the result originally in tlic form
7 /~T^
x= — y~ + A / —J . It docs not appear that Diophantos
divided throughout by a. Rather he first multiplied by a so as
to bring the equation into the form aV + ahx + ac = ; tlicn
solving he found ax= — \h ±J\i}^ — ac, and regarded the
result in the' f..rm ,^_^ ^ ± ^'l ^'-«^, Whether the inter-
a
mediate procedure was as we have described it is n(H certain ;
but it is certain that he used the result in the form given. One
remark however must be made upon the form of the root.
' Nessi'luiann, p. :ilO. Also IJo.li-t, Jountdl A^iniiijiir. .Iiiiivi.r, I'-T-.
92 DIOPHANTOS OF ALEXANDRIA.
Diophantos takes no account of tlie existence of two roots,
according to the sign taken before the radical. Diophantos
ignores always the negative sign, and takes the positive one as
giving the value of the root. Though this perhaps might not
surprise us in cases where one of the roots obtained is nega-
tive, yet neither does Diophantos use both roots when both arc
positive in sign. In contrast to this Nessehnann points out
that the Arabians (as typified by Mohammed ibn Milsa) and
the older Italians do in this latter case recognise both roots.
M. llodet, however, remarks upon this comparison between
Diophantos and the Arabians, so unfavourable to the former, as
follows (a) Diophantos did not write a text-book on Algebra,
and in the cases where the equation arrived at gives two
positive solutions one of them is excluded a priori, as for ex-
ample in the case quoted by him, v. 13. Here the inequality
72.r > llx'^ + 17 would give a; < f f or else x<-^. But the other
inequality to be satisfied is *l^x<l^x^ + 19, which gives x>\%
ov x> f^. As however -^j < /\j, the limits x<^\> fjj are
impossible. Hence the roots of the equations corresponding
to the negative sign of the radical must necessarily be rejected.
(6) Mohammed ibn Mu^a, althougli recognising in theory two
roots of the equation x^ + c = hx, in practice only uses one of
the two, and, curiously enough, always takes the value cor-
responding to the negative sign before the radical, whereas
Diophantos uses the positive sign. But see Chapter viii.
From the rule given in Def. 11 for compensating by addition
any negative terms on either side of an equation and taking
equals from equals (operations called by the Arabs aljahr and
almulcahala) it is clear that as a preliminary to solution
Diophantos so arranged his equation that all the terms were
positive. Thus of the mixed quadratic equation we have three
cases of which we may give instances : thus,
-^,p + Jip' + m<i
Case 1. YoYm mx^ -»rpx = q; the root is
1)1
according to Diophantos. An instance is afforded by VI. G.
Diophantos arrives namely at the equation C./-I- 3u: = 7, which,
if it is to be of any service to his solution, should give a rational
value of X ; whereupon Diophantos says " the S(]uarc of half tlic
DIOPHANTOS' MrnioDS OF SOLUTION. [):}
coefficient' of a; togetlier with tlie pn.duot ul" the absohite term
and the coefficient of x^ must be a square number; but it is not,"
i.e. ]/>"+ mq, or in tliis case (•^)'' + 42, must be a scjuare in onh-r
that the root may be rational, which in tliis case it is not.
Case 2. Form mx^ = px + q. Diopliantos takes
i u + V i i)^ + tun
X = - f^ iii 1 . An example is IV. 45, where 2x-* > Gx + 1 S.
Diophantos says : " To solve this take the square of half the co-
efficient of X, i.e. 9, and the product of the ftbsolute term and
the coefficient of x', i.e. od. Adding, we have 45, the square
root^ of which is not^ < 7. Add half the coefficient of x and
divide by the coefficient of x"^ ; whence x < 5." Here the form
of the root is given completely; and the whole operation by
which Diopbantos found it is revealed.
Case 3. Form mx^ + 3' = P'' '• Dioplianto.s' root is
— M— - — /. Cf. in V. 13 the equation already mentioned,
17.6-^+ 17 <72x. Diophantos says: "Multiply half the coeffi-
cient of X into itself and we have 1296 : subtract the i)roduct
of the coefficient of x^ and the absolute term, or 2<S9. The
remainder is 1007, the square root of which is not^ > 31. Achl
half the coefficient of x, and the result is not > 07. Divide
by the coefficient of x^, and x is not > f f ." Here again we have
the complete solution given.
(3) Cubic equation.
There is no ground for supposing that Diophantos was
acquainted with the solution of a cubic ct[uation. It is true
there is one cubic e(piation which occurs in the Arithmetics,
but it is only a very particular case. In vi. 19 the equation
arises, a.'^ + 2^ + 3 = a;^ + 3a; - 3a;'^ - 1, and Diophantos says
simply, "whence x is found to be 4." All that can be said of
1 For "coeflicient" Diophantos uses simply irX^t'os, luiinbcr: thus "number
of apiOfiol " = coeff . of x.
- Diophantos calls the "square root" irXei'pd or side.
2 7, though not accurate, is clearly the nearest integral limit which will servo
the purpose.
* As before, the nearest intajruJ limit.
O-i DIOPHANTOS OF ALEXANDRIA.
this is that if we write the equation in true Diophantine fashion,
so that all terms are positive,
x^ + oc = 4'X^+ 4.
This equation being clearly equivalent to x{x^ + l) = 4(.r'^ + l),
Diophantos probably detected the presence on both sides of the
equation of a common factor. The result of dividing by it is
a; = 4, which is Diophantos' solution. Of the two other roots
X = ±J — 1 no account is taken, for reasons stated above.
From this single example we have no means of judging how
far Diophantos was acquainted with the solutions of equations
of a degree higher than the second.
I pass now to the second general division of equations.
(B.) Indeterminate equations.
As has been already stated, Diophantos does not in his
Arithmetics, as we possess them, treat of indeterminate equa-
tions of the first degree. Those examples in the First Book
which would lead to such equations are, by the arbitrary
assumption of one of the required numbers as if known, con-
verted into determinate equations. It is possible that the
treatment of indeterminate equations belonged to the missing
portion which (we have reason to believe) has been lost between
Books I. and II. But we cannot with certainty dispute the
view that Diophantos never gave them at all. For (as Nessel-
mann observes) as with indeterminate quadratic equations our
object is to obtain a rational result, so in indeterminate simple
equations we seek to find a result in ivhole numbers. But the
exclusion of fractions as inadmissible results is entirely foreign
to our author; indeed we do not find the slightest trace that he
ever insisted on such a condition. We take therefore as our
first division indeterminate equations of the second degree.
I. Indeterminate equations of the second degree.
The form in which these equations occur in Diophantos is
universally this : one or two (and never more) functions of the
uid<nown quantity of the form A.x^ ■{■ Bx + G are to be made
rational s(|iiaro mnnbers by finding a suitable value for x.
DIOPIIANTCJS MKTIloDS OF SOUTlnX. <).-,
Tims we have to deal with one or two equations ol the fuini
Ax^+Bx+ C = y\
(1) Single equation.
The single equation of the form Ax" -\- Bx -\- C = if takes
special forms when one or more of the coefficients vanish, «»r
are subject to particular conditions. It will be well to give in
order the different forms as they can be identified in Dio-
phantos, and to premise that for "=/" Diophantos sim})l\'
uses the formula laov rerpaycovcp.
1. Equations which can always be solved rationallv. This
is the case when ^ or C or both vanish.
Form Bx = f. Diophantos puts y^ = any arbitrary square
number = nr, say therefore x= - . C'f iir. 5 : 2x = f, i/ = 1 (I,
x=S.
Form Bx + C = f. Diophantos puts for f any value m'',
and x= — T-, — • He admits fractional values of x, onlv takin<f
care that they are "rational," i.e. rational and positive. E.x.
III. 7.
Form Ax' + Bx = f. For i/ Diophantos puts any multiple
of.r, -a:; whence Ax+B= .,x, the factor x disappearinf?
n n tin
and the root x=0 being neglected as usual. Therefore
x= ,^"\ ,. Exx. II. 22, 34.
Hi — A ir
2. Equations whose rational solution is only possible under
certain condition.s. The cases occurring in Diophantos are
Form Ax"^ + C = >/. This can be lationally solved accord-
ing to Diophantos
(a) When A is positive and a square, say a'.
Thus oV -f C = _?/*, In this case ?/* is put = (ax ± in)* ;
therefore a'x' + C = (ax ± m)^
a - m'
96 DIOPHANTOS OF ALEXANDRIA.
(m and the doubtful sign being always assumed so as to give
£c a positive value).
{/3) When C is positive and a square number, say c*.
Tlius Aa;' + r = y^. Here Diophantos puts ?/ = (mx ± c) ;
therefore Ax^ + c^ = (mx ± cf,
A — lit'-
(7) When one solution is known, any number of other
solutions can be found. This is enunciated in vi. 16 tlms,
though only for the case in which C is negative: "when
two numbers are given such that when one is multiplied
by some square, and the other is s\ibtracted from the product,
the result is a square number; another square also can be
found, greater than the first taken square, which will have the
same effect," It is curious that Diophantos does not give a
general enunciation of this proposition, inasmuch as not only
is it applicable to the cases ± Ax^ ±C^ = if, but to the general
form Ax^ -\- Bx -\- G = y"'.
In the Lemma at vi. 12 Diophantos does prove that the
equation Ax^ + C = y can be solved when ^ + C is a square,
i.e. in the particular case when the value x = l satisfies the
equation. But he does not always bear this in mind, for in
III. 12 the equation o2x^ + 12 = y^ is pronounced to be impos-
sible of solution, although 52 -f 12 = G4, a square, and a rational
solution is therefore possible. So, ill. 13, 2(j(jx^ — 10 = 3/* is said
to be impossible, though a; = 1 satisfies it.
/ It is clear that, if a; = satisfies the ciiuation, (7 is a
square, and therefore this case (7) includes the previous case (/?).
It is interesting to observe that in VI. 15 Diophantos
states that a rational solution of the equation
Ax' -€' = 7/
is impossible unless A is the sum of two squares^.
' Nesselmann compares Lpj,'cmlro, Tlirorie des Xomhrrs, p. GO.
DIOPHANTOS MI-:TH01)S Ol' SOLI TloN. !)7
Lastly, we must consider the
Form Ax^ + Bx+C = y\
This equation can be reduced by means of a change of
variable to the previous form, wanting the second term. Thus
if we put x = z — ^ . , the transformation gives
^- + ^A - y ■
Diophantos, however, treats this form of the equation quite
separately from the other and less fully. According to him the
rational solution is only possible in the following cases.
(a) When A is positive and a square, or the equation is
aV+ Bx + C=y"; and Diophantos puts if= {ax + mf, whence
Exx. II. 20, 21 &c.
2am -B'
(yQ) When C is positive and a square, or the ccjuation is
Ax^ + Bx -\- c' = y-; and Diophantos writes y' = {mx + c)^ whence
X = —J Y • Exx. IV. 0, 10 &c.
A - m'
(7) When \B'^ — AG is positive and a square number.
Diophantos never expressly enunciates the possibility of this
case: but it occurs, as it were unawares, in iv. 33. In this
problem 3« + 18 — a;^ is to be made a square, and the ec|uation
^x-\-\^ — a? = y'^ comes under the present form.
To solve this Diophantos assumes 3a;+ 18 — j;* = 4x* which
leads to the quadratic 3a; + 18 — ox^ = 0, and " the equation is
not rational". Hence the assumption 4a/' will not do : "and we
must seek a square [to replace 4] such that 18 times (this
square + 1) + (f)"'' may be a square". Diophantos then solves
this auxiliary eciuation 18 {x" + 1) + ;,' = if, finding x = IS. TIr-u
he assumes
3x+l8-a,'=(18)V,
which gives 325a;^ - 3a; - 18 = 0, whence Jc = -^^.
H. D. 7
98 DIOPHANTOS OF ALEXANDRIA.
It is interesting to observe that from this example of Dio-
phantos we can obtain the reduction of this general case to the
form At? + G^ = y^, wanting the middle term.
Thus, assume with Diophantos that Ax^+Bx-^ C = m^x^,
therefore by solution we have
B
^ +
and X is rational provided ~t- — AC+ Cnf is a square. This
B'^
condition can be fulfilled if — - AC he a square by a previous
case. Even if that is not the case, we have to solve (putting,
for brevity, D for ~ — AG) the equation
D + Cm' = f.
Hence the reduction is effected, by the aid of Diophantos alone.
(2) Double-equation.
By the name "double-equation" Diophantos designates the
problem of finding one value of the unknown quantity x which
will make two functions of it simultaneously rational square
numbers. The Greek term for the "double-equation" occurs
variously as Bnr\ola-6T7](; or BtTrXi} caorij'i. We have then to
solve the equations
mx^ + ax + a= u'\
nx'-{- ^x + b = w')
in rational numbers. The necessary preliminary condition is
that each of the two expressions can severally be made squares.
This is always possible when the first term (in x') is wanting.
This is the simplest case, and wc shall accordingly take it first.
y
DIOPHANTOS MKTIKJDS OF SOLUTION. })0
1. Double equation of the first degree.
Diophautos has one distinct method of solving the ociuations
ax + a= II'
^x + b=w'
taking slightly different forms according to the nature of the
coefficients.
(a) First method of solution of
^a; + b = wy
This method depends upon the equation
\ 2 )
'-^J-P<I.
If the difference between the two functions can be separated
into two factors p, q, the functions themselves are equated to
( — ~ ] . Diophantos himself states his rule thus, in ii. 12:
" Observing the difference between the two expressions, seek
two numbers whose product is equal to this ditfercncc ; then
either equate the square of half the difterence of the factors to
the smaller of the expressions, or the square of half the sum to
the greater." We will take the general case, and investigate
what particular cases the method is applicable to, from Dio-
phantos' point of view, remembering that his cases are such
that the final quadratic equation for w arising reduces always to
a simple one.
Take the equations
ax + a= ii\ '
/S./; + i = n)\
and subtracting we have (a — /8) x + (a — 6) = i^ — w*.
Let a— 13 = 8, a - 6 = e for brevity, thou 8x + e = «' - ui\
7 •>
Thus
100 DIOPHANTOS OF ALEXANDRIA.
We have then to separate B:c + e into two factors ; let these
factors be «, — + - , and \vc accordins^ly write
p p
8x €
u + v = 1- ,
p p
/hx e Y
, ^ SV 2hxfe \ fe V ,, s
therefore —5- -\ {- + p] + { +p] =4 (ax + a).
p p \p / \p /
Now in order that this equation may reduce to a simple one,
either
(1) the coefficient of x^ must vanish or 8 = 0, therefore
a = ^, or
(2) the absolute term must vanish.
Therefore ( ~ + i^ ) = 4a,
or p' + 2e/ + e' = 4ap%
i. e. / +2{a-h-2a) pi' + (a - 6)" = 0.
Therefore {p^ — a + hf = 4a6,
whence ah must be a square number.
Therefore either both a and h are squares, in which case we
may substitute for them c' and d\ p being then equal to c ±d,
or the ratio a : 6 is the ratio of a square to a square.
With respect to (1) we observe that on one condition it is
not necessary that S should vanish, i.e. provided we can, before
solving the equations, make the coefficients of x' in both equal
by multiplying either equation or both by a square number, an
operation which does not affect the problem, for a square multi-
plied by a square is still a square.
DIOPHANTOS MI-yniODS OF SOLUTION. 101
Thus if =^ or aii^ = ^m^, the coudition 8 = will bo
jo il
satisfied by multiplying the equations respectively by n' and
i/r ; and thus we can also solve the equations
like the equations
•'■ + « = '^^l
,v + b = w')
in an infinite number of ways.
Again the equations under (2),
ax+ 6^= ^i^
^x + d' = w\
can be solved in two different ways, according as we write them
in this form or in the form
^c'x + c'd' = 2U"
obtained by multiplying them respectively by (T, c* in order
that the absolute terms may be equal.
We now give those of the possible cases which are found
solved in Diophantos' own work. These are equations
(1) of the form
ant'x + a= ii^]
a case which includes the more common one, when tiie co-
efficients of X in both are equal.
(2) of the form
^x + cP = 2uy
solved in two different ways according as they are thus written,
or in the alternative form,
ad\c + c'd^ = u'
^c'x + c"d' = w
102 DIOPHANTOS OF ALEXANDRIA.
General solution of Form (1), or,
am
an
Multiplying respectively by if, nf, we have to solve the
equations,
am^n^x + an^ = il^\
amVic + hm^ = w'^) '
The difiFercnce = aif — hm^. Suppose this separated into
two factors p, q.
Put It' ± %d — p,
n T w' = q,
whence ..■^ ^ (P ±3)\ ^r- = (P-^)' ,
therefore am'^ifx + an^ = ( — 9- j ,
or a'm^n^x + hm'^= \—y^
Either equation will give the same value of x, and
p^ + q^ arf + hmf
i 2
X— —
•xmSi^
since yq = arf — hnf.
Any factors p, q may be chosen provided the value of x
obtained is positive.
Ex. from Diophantos.
65- G«=w')
G5 - 24^ = to'} '
,, . 2G() - 24.7- = w'-)
therefore ... ^, .
Oo — 24a; = w
The difference = 195 = 15 . 13 say,
therefore / 15 - 13 y ^ ^^^ _ ^^^^ 24a; = 64, or a; = §.
C-^^)
DIOPHANTOS MKTHODS OF SOLUTION. 1()3
General solution (first method) of Form (2), or,
oc + c" = ii'^l
^x + d'' = 2o'y
In order to solve by this method, we multiply by rf*, c*
respectively and write
(xd^x + c'dr = n")
^c'x + c\V = w") '
u being the greater.
The diti: = {id' - /3r) x. Let the factors of this be px, 7,
therefore ii^=(&i+lJ\\
Hence x is found from the equation
This equation gives
j9V + 2x (pq - 2a^) + q" - ^c'd' = 0,
or, since pq = ad^ — y8c^
p'x' - 2x {ad' + /Sr ) + q' - ^c'd' = 0.
In order that this may reduce to a simple equation, as
Diophantos requires, the absolute term must vanish.
Therefore q' - ^c\l' = 0,
whence q = 2cd.
Thus our method in this case furnishes us with only one
solution of the double-equation, q being restricted to the value
2cd, and this solution is
_ 2 {ad:" + I3c') _ Sc'ff (gff + )9 c')
Ex. from Diophantos. This method is only used in one
particular case, IV. 45, w^here c' = d' as the equations originally
stand, namely
8x+-i = u'\
(jx+4> = iv'] '
104- DIOPHANTOS OF ALEXANDRIA.
the difference is 2x and q is necessarily taken = 2>/i = 4, and
the factors are ^ , 4,
therefore 8a; + 4 = r^ + 2 j and rr = 11 2.
General solution (second method) of Form (2), or
Here the difference ={ol - ^) x + (c^ - (T) = hx -\- € say, for
brevity.
Let the factors oi dx + e\iQ p, h . Then, as before proved
(p. 100), p must be equal to (c + d).
Therefore the factors are
«-/3 - J 7
, oj + c + a, c + ft,
c + fZ
and we have finally
\c±d J
kC ± d.
two. (^^)^'^+4.f<;;/>-fo,
which equation gives two possible values for x. Thus in this
case we can find by our method two values of x, since one of the
factors, p, may be c + d.
Ex. from Diophantos, III. 17 : to solve the equations
10a; + 9 = w'^)
5a; + 4 = 10^ } '
The difference is here 5x + 5, and Diophantos chooses as
the factors 5, a; + l. This case therefore corresponds to the
value c + d of jj. The solution is given by
(i-y=
lOx + !), whence x - 28.
DIOPHANTOS METHODS OF SOLUTION. 105
The other value c — (Z of jj is in this case excluded, because
it would lead to a negative value of x.
The possibility of deriving any number of solutions of a
double-equation when one solution is known does not seem
to have been noticed by Diophantos, though he uses the prin-
ciple in certain special cases of the single equation. Fermat
was the first, apparently, to discover that this might always
be done, if one value a of x were known, by substituting in
the equations x+ a for .v. By this means it is possible to find
a positive solution even if a is negative, by successive appli-
cations of the principle.
But nevertheless Diophantos had certain peculiar artifices
by which he could arrive at a second value. One of these
artifices (which is made necessary in one case by the unsuit-
ableness of the value found for x by the ordinary method),
employed in iv. 45, gives a different way of solving a double-
equation from that which has been explained, used only in
a special case.
(/S) Second method of solution of a double-equation of the
first degree.
Consider only the special case
hx + if = u^,
{h+f)x + n^ = iu\
Take these expressions, and ?^^ and write them in order uf
magnitude, denoting them for convenience by A, B, C.
A = {h+f)x + n\ B = hx + n\ G = n\
,, „ A-B f , A-B=fx
therefore F^ = ^ ^^^ 5- (7 = 1...
Suppose now hx + if = {}j + nf,
therefore hx = y~ + 2ny,
therefore A-B = ^(if + ^nij),
f
or ^=(2/+«)' + ^(2/'+2ny),
thus it is only necessary to make this expression a s(|uarc.
106 DIOPHANTOS OF ALEXANDRIA.
Write therefore
(l + {) / + 2'^ ({ + 1)2/ + '^' = (Vy - '')"'
whence any number of values for y, and therefore for x, can be
found, by varying p.
Ex. The only example in the Arithmetics is in iv. 45.
There is the additional condition in this case of a limit to the
value of X. The double-equation
8a; + 4 = u^^
Qx + 4! = vf
has to be solved in such a manner that x<1.
A— B
Here ^ — ^ ~ 3 > ^^^ ^ ^^ taken ' to be {y + 2)^
therefore ^-5=^^^,
therefore ^ = 2^!+^^^ + ^^ + 4 1/ + 4 = ^|-V -|^ + 4
which must be made a square, or, multiplying by f ,
3^/^ -I- 12t/ + 9 = a square,
where y must be < 2.
Diophantos assumes
32/^+123/ + 9 = (m2/-3r,
6w + 12
whence ?/ = -^ ., ,
and the value of m is then determined so that ?/ < 2.
As we find only a special case in Diophantos solved by
this method, it woidd be out of place to investigate the con-
1 Of course Diophantos uses the same variable .r where I have for clearness
used y.
Then, to express what I have called in later, he says:
"I form a square from 3 minus some number of x's and .r becomes some
number multiplied by 6 together with 12 and divided by the dillerence by which
the square of the number exceeds three,"
DIOPHANTOS METHODS OF SOLUTION. 1()7
ditions under which more general cases might be solved in this
manner \
2. Double equation of the second degree,
or the general form
Ace' + Bx + C = 2i\
A'x'' + B'x + C' = w\
These equations are much less thoroughly treated in Diophan-
tos than those of the first degree. Only such special instances
occur as can be easily solved by the methods which we have
described for those of the first degree.
One separate case must be mentioned, which cannot be
solved, from Diophantos' standpoint, by the preceding method,
but which sometimes occurs and is solved by a peculiar method.
The form of double-equation being
ax' + ax = u'] (1),
^x'+bx = w'\ (2),
Diophantos assumes
whence from (1)
a
X = -2
m — a
and by substitution in (2)
a \~ ha , ,
+ a must be a square,
m —aj 7n —a
a'^ + ha{m'-a)
IS a square.
Therefore we have to solve the equation
abm^ + a (/9a — ah) = if,
and this form can or cannot be solved by processes already
given according to the nature of the coefficients^
, . OJ+bi
• Bachet and after him Cossali proved the pos.Kibility of solving ^^^ j\ "V
this method under two conditions.
- Diophantos did not apparently observe that this form of e<iualion could be
108 DIOPHANTOS OF ALEXANDRIA.
II. Indeterminate equations of a degree liigher than the second.
(1) Single Equations.
These are properly divided by Nesselmann into two classes ;
the first of which comprises those questions in which it is re-
quired to make a function of a; of a higher degree than the
second a square ; the second comprises those in which a rational
value of X has to be found which will make any function of cc,
not a square, but a higher power of some number. The first
class of problems is the solution in rational numbers of
Aar + Bx""' + + Kx + L = f,
the second the solution of
Ax'' + Bx"-' + 4-Kx + L = y\
for Diophantos does not go beyond making a function of j; a
cube. Also in no instance of the first class does the index n
exceed 6, nor in the second class (except in a special case or two)
exceed 3.
First class. Equation
Ax'' + 5a;""' + + Kx + L = y\
We give now the forms found in Diophantos.
1. Equation Ax"" + Bx^ + Cx + d"- = y\
Here we might (the absolute term being a square) put for
y the expression mx + d, and determine ni so that the coefficient
of X in the resulting equation vanishes, in which case
Q
%nd = G, m = ^ ,
and we obtain in Diophantos' manner a simple equation for x,
giving
^ ^ _ G'-U'B
reduced to one of tbe first degree by dividing by x- and putting ?/ for , iu
wbich case it becomes
a + ay = u''- )
p + by^w'^S'
Tbis reduction was given by Lagrange.
DIOPHANTOS METHODS OF SOLUTION. l(j!)
Or we might put for y an expression mV + H.r + (/, and (k-ter-
inine m, n so that the coefficients of x, x'' in the resultiin'
equation both vanish, whence we should again have a simple
equation for x. Diophantos, in the only example of this furm of
equation which occurs, makes the first supposition. Thus in
VI. 20 the equation occurs,
x'-Sx' + Sx + l = i/,
and Diophantos assumes ?/ = i] x- + 1, whence x=^K
2. Equation Ax* + Bx^ + Cx" + Dx + E=tf.
In order that this equation may be solved by Diophantos'
method, either A ox E must be a square. If A is a square and
equal to a^ we may assume y = ax^ + — x-\- n, determining n so
that the term in x' vanishes. If ^ is a square (e^ we may write
y — Tm?+ ^x + e, determining m so that the term in a-' may
vanish in the resulting equation. We shall then in either case
obtain a simple equation for x, in Diophantos' manner.
The examples of this form in Diophantos are of the kind,
a V + Bx^ + 6V + Dx + e' = y;
where we can assume y = ± ax^ + kx ± e, determining k so that
in the resulting equation (in addition to the coefficient of x*,
and the absolute term) the coefficient of a;', or that of a;, may
vanish, after which we again have a simple equation.
Ex. IV. 29 : 9x' - 4a;' + 6a;' - 12a; + 1 = y'. Here Diophantos
assumes y = ox" — Ga; + 1, and the equation reduces to
32a-' - 3Ga;' = anda; = U.
Diophantos is guided in his choice of signs in the ex-
pression ± aaf+kx±ehy the necessity for obtaining a "rational"
result.
But far more difficult to solve are those e<[uations in which
(the left expression being bi-quadratic) the odd powers of x arc
wanting, i.e. the eciuations Ax* + Cx' + E= y', and Ax* + A* = y*,
110 DIOPHANTOS OF ALEXANDRIA.
even when -<4 or ^ is a square, or both are so. These cases
Diophantos treats more imperfectly.
3. Equation Ace* + Ccc'' + E = if.
Of this form we find only very special cases. The type is
which is written
aV-cV + e'=2/',
X
Here ii is assumed to be ax or - , and in either case we
^ X
have a rational value of x.
25
Exx. V. 30 : 25*^ - 9 + — = 2/^ This is assumed to be
equal to 25a;^
V. 31 : -^ c^ - 25 + 2 = y"- f assumed to be = -7-^ .
4. Equation Ax''^E = if.
The case occurring in Diophantos is a;'* + 97 = if. Diophantos
tries one assumption, _y = a;^ — 10, and finds that this gives
^ = -i^, which leads to no rational result. Instead however
of investigating in what cases this equation can be solved, he
simply shirks the equation and seeks by altering his original
assumptions to obtain an equation in the place of the one first
found, which can be solved in rational numbers. The result is
that by altering his assumptions and working out the question
by their aid he replaces the refractory equation, a;* + 97 = y^, by
the equatioQ x* + 337 = f, and is able to find a suitable sub-
stitution for y, namely o^ — 25. This gives as the required
solution cc = '^. For this case of Diophantos' characteristic
artifice of retracing his steps' — "back-reckoning," as Ncsscl-
mann calls it, see Appendix v. 32.
5. Equation of sixth degree in the special form
a;« _ Ax" + Bx + c' = f.
1 "Methode der Zuriickrccbnung und Ncbcnaufgabe. "
DIOPHANTOS METHODS OF SOLUTION. Ill
It is only necessary to put y = .7/ + c, whence — A.r' + B= 2cx*
and ^^'^ = 7", 9 • This gives Diophantos a rational solution
if . IS a square.
G. If however this last condition does not hold, as in the
case occurring IV. 19, x'^ - 16^' + a; + 64 = ?/^ Diophantos
employs his usual artifice of " back-reckoning," by which he is
enabled to replace this equation by .«*' — 128a;' + a; + 409G = y*,
which satisfies the condition, and (assuming y = x^-\- 04) x is
found to be -j^r.
Second Class. Equation of the form
Ax'' + Bx''~^ + +Kx + L = y^.
Except for such simple cases as Ax^='if, Ax* = if, where it is
only necessary to assume y = mx, the only cases which occur in
Diophantos are Ax^ -^ Bx-\- C= y^, Ax^ + Ba? + Cx+D = if.
1. Equation Ax' + Bx + C = y\
There are of this form only two examples. First, in vi. 1
a;* — 4a: + 4 is to be made a cube, being at the same time already
a square. Diophantos therefore naturally assumes ;c — 2 = a
cube number, say 8, whence x= 10.
Secondly, in vi. 19 a peculiar case occurs. A cube is to be
found which exceeds a square by 2. Diophantos a.ssumes
(a;-l)^ for the cube, and (a^ + l)^ for the square, obtaining
a;' - 3a;" + 3x - 1 = a;^ + 2a; + 3, or the equation
a;' + a; = 4a;* + 4,
previously mentioned (pp. 36, 93), which is satisfied by x = 4.
The question here arises: Was it accidentally or not that this
cubic took so special and easy a form? Were a;— l,a; + l
assumed with the knowledge and intention of finding such an
equation ? Since 27 and 25 are so near each otlier and are, as
Fermat observes \ the only integral numbers which satisfy the
1 Note to VI. I'J. Fermatii Opera Math. p. I'.i'i.
112 DIOPHANTOS OF ALEXANDRIA.
conditions, it seems most likely that it was in view of these
numbers that Diophantos hit upon-the assumptions x + 1, x—1,
and employed them to lead back to a known result with all the
air of a general proof. Had this not been so, we should probably
have found, as elsewhere in the work, Diophantos first leading
us on a false tack and then showing us how we can in all cases
correct our assumptions. The very fact that he takes the right
assumptions to begin witli makes us suspect that the solution is
not based upon a general principle, but is empirical merely.
2. The equation
Ax^+Bx-' + Gx + D^f.
If -4 or D is a cube number this equation is easy of solution.
For, first, if A = a^ we have only to write y = ax+ -^ , and we
arrive in Diophantos' manner at a simple equation.
C
Secondly, if D = d?, we put y = ^^ x + d.
If the equation is a^x^ + Bx^ + Cx + d^ = y^, we can use either
assumption, or put y=ax + d, obtaining as before a simple
equation.
Apparently Diophantos only used the last assumption ; for
he rejects as impossible the equation y^=8x' — x^+8x—l
because y = 2x — l gives a negative value a; = — ^y, whereas
either of the other assumptions give rational values \
(2) Double-equation.
There are a few examples in which of two functions of x one
is to be made a square, the other a cube, by one and the same
rational value of x. The cases arc for the most part very
simple, e.g. in vi. 21 we have to solve
4x + 2=y''
2x + l=z''
therefore ?/' = ^z^, and z is assumed to be 2.
1 There is a special case in which C aud 1) vanish, Ax'^ + Bx- = y'K Here y
is put —mx and x- ., , . Cl'. iv. 0, 30.
»«•* - A
DIOPHANTOS' METHODS OF SOLUTION. \l:\
A rather more complicated case is vi. 23, where we have
the double equation
of + 2x- + X = 2^\'
2
Diophantos assumes )j = moc, wheuce x= —^ , and we have
to solve the single equation
/ 2 Y .^/^^_y 2
W-2) ^■^\m^-2) "^ nT^l
^ = ^.
(m«-2f
To make 2m* a cube, we need only make 2in a cube, or put
m =■ 4. This gives for x the value f .
The general case
Ax' + Bx^+ Cx = z\
hx^ + ex =y^>
would, of course, be much more difficult ; for, putting i/ = nix,
we find X = —, — r .
and we have to solve
or Ccm* + c(Bc- 2b C) m" + hc{hC-Bc) + A c' = u\
of which equation the above corresponding one is a very parti-
cular case.
§ 3. Summari/ of the preceding investigation.
We may sum up briefly the results of our investigation of
Diophantos' methods of dealing with equations tiius.
1. Diophantos solves completely equations of the first
degree, but takes pains beforehand to secure that the solution
shall be positive. He shows remarkable address in reducing a
number of simultaneous equations of the first degree to a single
equation in one variable.
H. D. 8
114 DIOPHANTOS OF ALEXANDRIA.
2. For determinate equations of the second degree Dio-
phantos has a general method or rule of solution. He takes
however in the Arithmetics no account of more than 07ie root,
even when both roots are positive rational numbers. But his
object is always to secure a solution in rational numbers, and
therefore we need not be surprised at his ignoring one root of a
quadratic, even though he knew of its existence.
3. No equations of a higher degree than the second are
found in the book except a particular case of a cubic.
4. Indeterminate equations of the first degree are not
treated in the work as we have it, and indeterminate equations
of the second degree, e.g. Aaf + Bx -\-G = y', are only fully treated
in the case where ^ or C vanishes, in the more general cases
more imperfectly.
5. For " double-equations " of the second degree he has a
definite method when the coefficient of x^ in both expressions
vanishes ; this however is not of quite general application, and is
supplemented in one or two cases by another artifice of particular
application. Of more complicated cases we have only a few
examples under conditions favourable for solution by his
method.
6. Diophantos' treatment of indeterminate equations of
higher degrees than the second depends upon the particular
conditions of the problems, and his methods lack generality.
7. After all, more wonderful than his actual treatment of
equations are the extraordinary artifices by which he contrives
to avoid such equations as he cannot theoretically solve, e.g. by
his device of " Back-reckoning," instances of which, however,
would have been out of place in this chapter, and can only
be studied in the problems themselves.
§ 4. I shall, as I said before, not attempt to class as methods
what Nesselmann has tried so to describe, e.g. "Solution by mere
reflection," "solution in general expressions," of which there
are few instances definitely described as such by Diophantos,
and " arbitrary determinations and assumptions." It is clear that
the most that can be done to formulate these " methods " is the
DIOPHANTOS METHODS OF SOLUTION. 1 1 .')
enumeration of a few instances. This is what Ncssclmann
has done, and he himself regrets at the end of his chapter on
"Methods of solution" that it must of necessity be so incomplete.
To understand and appreciate these artifices of Diophantos it is
necessary to read the problems themselves singly, and for these
I refer to the abstract of them in the Appendix. As for the
" Use of the right-angled triangle," all that can be said of a
general character is that rational right-angled triangles (whose
sides are all rational numbers) are alone used in Diophantos,
and that accordingly the introduction of such a right-angled
triangle is merely a convenient device to express the problem of
finding two square numbers whose sum is also a square number.
The general forms for the sides of a right-angled triangle are
c^ -f h^, a^ — 1>\ 2ab, which clearly satisfy the condition
{a' + by={a'-b'f + {2abY.
The expression of the sides in this form Diophantos calls "form-
ing a right-angled triangle from the numbers a and b." It is
by this time unnecessary to observe that Diophantos does not
use general numbers such as a, b but particular ones. " Forming
a right-angled triangle from 7, 2 " means taking a right-angled
triangle whose sides are 7'^ + 2', 7" — 2'^ 2 . 7 . 2, or o3, 4o, 28.
§ 5. Method of Limits.
As Diophantos often has to find a series of numbers in
ascending or descending order of magnitude : as also he does
not admit negative solutions, it is often necessary for him to
reject a solution which he has found by a straightforward method,
in order to satisfy such conditions ; he is then very frequently
obliged to find solutions of problems which lie within certain
limits in order to replace the ones rejected.
1. A very simple case is the following: Required to find a
value of X such that some power of it, x", shall lie between two
assigned limits, given numbers. Let the given numbers bo a, b.
Then Diophantos' method is : Multiply a and 6 both succes.sively
by 2", 3", and so on until some (nf' power is seen which lies be-
tween the two products. Thu.s suppose c" lith between up' and /*// ;
6—2
Il6 DIOPHANTOS OF ALEXANDRIA.
then we can put « = - , in which case the condition is satisfied,
P
for(-| lies between a and 6.
&
Exx. In IV. 34 Diopbantos finds a square between f and 2
thus : he multiplies by a square, 64 ; thus we have the limits 80
and 128; 100 is clearly a square lying between these limits ;
hence (lo)'^ or f| satisfies the condition of lying between | and 2.
Here of course Diophantos might have multiplied by any
other square, as 16, and the limits would then have become 20
and 82, between which there lies the square 25, and so we
should have f§ again as the square required.
In VI. 23 a sixth power (a " cube-cube ") is required which
lies between 8 and 16. Now the sixth powers of the first
natural numbers are 1, 64, 729, 409 6... Multiply 8 and 16 (as
in rule) by 2° or 64 and we have as limits 512 and 1024, and
729 lies between them ; therefore "^^-^^ is a sixth power such as
was required. To multiply by 729 in this case would not give
us a result.
2. Other problems of finding values of x agreeably to
certain limits cannot be reduced to a general rule. By giving,
however, a few instances, we may give an idea of Diophantos'
methods in general.
Q
Ex. 1, In IV, 26 it is necessary to find x so that ,
x' + x
lies between x and x+l. The first condition gives 8 > a;'' + a;^
Diophantos accordingly assumes
8 = {x+lY = x' + x' + '^^ + Jj,
which is >x'' + .c\ Thus x = ^ satisfies one condition. It also
Q
is seen to satisfy the second, or -5-— < a; -f- 1 : but Diophantos
X + X '■
practically neglects this condition, though it turns out to be
satisfied. The method is, therefore, hero imperfect.
Ex. 2. Find a value of x such that
x>y^x''-C)0)<l-(x'-(y()),
or x^ — 60 > ox, x^ - 60 < 'Sx.
DIOPIIANTOS' MKTIIODS ()F SOLrTIuX. II7
Hence, says Diophantos, rr is <(: 11 nor > 12. Wi- liave
already spoken (pp. 00, 91) of the reasoning by which h.-
arrives at this result (by taking only one root of the quadratic,
and taking the nearest integral limits). It is also required
that a? — 60 shall be a square. Assuming then
a;^ - 60 = {x - mf, x = — ^ — ,
2m
which must be > 11 < 12, Avhence
m' + 60 > 22m, m" + 60 < 2hn,
and (says Diophantos) in must therefore lie between 10 and 21.
Accordingly he writes x'' - 60 = (x - 20)^ and x = llh, which is
a value of x satisfying the conditions.
§ 6. Method of Approximation to Limits.
We come now to a very distinctive method called by Dio-
phantos 7rapia6T7]<i or irapiaorrjro'; dywyjj. The object of this
is to find two or three square numbers whose sum is a given
number, and each of which approximates as closely as possible
to one and the same number and therefore to each other.
This method can be best explained by giving Diophantos'
two instances, in the first of which two such squares, and in the
second three are required. In cases like this the principles
cannot be so well described with general symbols as with con-
crete numbers, whose properties are immediately obvious, and
render separate expression of conditions unnecessary.
Ex. 1. Divide 18 into two squares each of which > 6.
Take \^ or 6i and find what small fraction -^ added to it
makes it a square : thus 6^ + -^ must be a square, or 26 4- -,
X y
is a square. Diophantos puts
26 + \_ = [^+l)\ or 26/ + 1 =(5^+ 1)',
whence ?/ = 10 and „ = t.W. •»!" - = in,' •""' *'' *" i^>'> ~ ^
J y' '"" J-
118 DIOPHANTOS OF ALEXANDRIA.
square = (f l)^ [The assumption of {oy + 1)^ is not arbitrary,
for assume 26?/^ + 1 = {py + l)^ therefore y = ^ ^ ^ , and, since
- should be a small proper fraction, therefore 5 is the most
y
suitable and the smallest possible value for ^^, 26 — p- being < Ip
or p^ + 2jj + 1 > 27.] It is now necessary (says Diophantos)
to divide 13 into two squares whose sides are each as near
as possible to |^.
Now the sides of the two squares of which 18 is by nature
compounded are 3 and 2, and
3 is > fi by ^1
2 is < f^ by l^i ■
Now if 3 — -g^, 2 + ^ were taken as the sides of two squares
their sum would be
2^^601
^^^ 400 '
which is > 13.
Accordingly Diophantos puts
3 - 9a;, 2 + ll.r,
for the sides of the required squares, where x is therefore not
exactly ^ but near it.
Thus, assuming
(3 -9a:)' + (2 + 11a;)' = 13,
Diophantos obtains x = y^.
Thus the sides of the required squares are \^\, f^.
Ex. 2. Divide 10 into three squares such that each square
is >3.
Take ^o or 3^ and find what fraction of the form ^ added
9
to it will make it a square, i.e. make 30 + -^ a square or Wy'^ + 1,
I 3 1
where - = - .
X y
DIOPHANTdS' METHODS OF SOLUTION. Ill)
Diophautos writes 30/ + 1 = (r)y + 1)-, whence y = 2 and
^*-
And 3^ + ^V = a square = '^' .
[As before, if we assume 30//^ + 1 = {py + 1)*, 7 = , Z"' ^ ,
and since - must be a small proper fraction, 30 —;:>'' should < 2y;
or p^+ 2jj + 1 > 81, and 5 is the smallest possible value of p.
For this reason Diophantos chooses it.]
We have now (says Diophantos) to make the sides of the
required squares as near as may be to y.
Now 10=9+l = 3^+(f)^+(|)^
and 3, f, 4 are the sides of three squares whose sum = 10.
Bringing (3, f , 4) and y to a common denominator, wo have
(f^, ^, M) and M.
Now 3is>ffbyf§,
f is<M by M.
f is<Mby M-
If then we took 3 - f^, f + f§, f + fi for the sides, the sum
of their squares would be 3 (y )* or ^{f, which is > 10. Diophantos
accordingly assumes as the sides of the three required squares
3 - Zox, f + 37a-, I + 31a-,
where x must therefore be not exactly ^'^, but near it.
Solving (3 - rox)' + (f + 37a,f + (4 + 3U-)^ = 10,
or 10-llG.t-+3555a;'=10,
we have x = ^-^ ;
the required sides are therefore found to be
sw. w. sw>
and the squares 'UM^^> VW^'. "Mi^-
The two instances here given, though only instiinces, serve
perfectly to illustrate the method of Diophantos. To have put
them generally with the use of algebniical symbols, nistead <»f
120 DIOPHANTOS OF ALEXANDRIA.
concrete numbers, would have rendered necessary the intro-
duction of a large number of such symbols, and the number of
conditions (e.g. that such and such an expression shall be a
square) which it would have been necessary to express would
have nullified all the advantages of this general treatment.
As it only lies within my scope to explain what we actually
find in Diophantos' work, I shall not here introduce certain
investigations embodied by Poselger in his article " Beitrage zur
TJnbestimmten Analysis," published in the Ahhandlungen
der Koniglichen Akademie der Wissenschaften zu Berlin Aus dem
Jahre 1832, Berlin, 1834. One section of this paper Poselger
entitles "Annaherungs-methoden nach Diophantus," and obtains
in it, upon Diophantos' principles \ a method of approximation
to the value of a surd which will furnish the same results as the
method by means of continued fractions, except that the approxi-
mation by what he calls the " Diophantine method " is quicker
than the method of continued fractions, so that it may serve to
expedite the latter ^
^ "Wenn wir den Weg des Diophantos verfolgen."
2 "Die Diopliantisehe Mctbode kann also dazii diencn, die Convergcnz der
Partialbriiche des Kettenbruchs zu beschlcunigen."
K^r.r^.
CHAPTER VI.
§ 1. THE PORISMS OF DIOPIIANTOS.
We have already spoken (in the Historical Introduction)
of the Porisms of Diophantos as having probably foiined a
distinct part of the work of our author. We also riiscussed the
question as to whether the Porisms now lost formed an
integral portion of the Arithmetics or whether it was a com-
pletely separate treatise. What remains for us to do under the
head of Diophantos' Porisms is to collect such references to
them and such enunciations of definite porisms as are directly
given by Diophantos. If we confine our list of Porisms to those
given under that name by Diophantos, it docs not therefore
follow that many other theorems enunciated, assumed or implied
in the extant work, but not distinctly called Porisms, may not
with equal propriety be supposed to have been actually pro-
pounded in the Ponsm^. For distinctness, however, and in
order to make our assumptions perfectly safe, it will be better
to separate what are actually called porisms from other theorems
implied and assumed in Diophantos' problems.
First then with regard to the actual Porisms. I shall not
attempt to discuss here the nature of the proposition which was
called a porism, for such a discussion would be irrelevant to
my purpose. The Porisms themselves too have been well
enumerated and explained by Nesselmann in his tenth chapter;
he has also given, mth few omissions, the chief of the other
theorems assumed by Diophantos as known. Of necessity,
therefore, in this section and the next I shall have to cover
very much the same ground, anil shall acconliugly bo a.s brief us
may be.
122 DIOPHANTOS OF ALEXANDRIA.
Porism 1. The first porism enunciated by Diopbantos
occurs in v. 3. He says " We bave from tbe Porisms tbat if
eacb of two numbers and tbeir product wben severally added to
tbe same number produce squares, the numbers are the squares
of two consecutive numbers \" This theorem is not correctly
enunciated, for two consecutive squares are not the only two
numbers which will satisfy the condition. For suppose
x + a = m\ y + a = n^, xy+a= p^.
Now by help of the first two equations we find
xy + a= m^n^ — a {m^ + n^ — \)+ a^
and this is equal to ]f. In order that
m\^ — a {m^ + n^ — l)-\-d^
may be a square certain conditions must be satisfied. One
sufficient condition is
m^ + ?i" — 1 = 2inn,
or m — n = + 1,
and this is Diopbantos* condition.
But we may also regard
2«,8
rrrn
a {m? + 71" - 1) + a'' = p^
as an indeterminate equation in m of which we know one
solution, namely m = n ± 1.
Other solutions are then found by substituting z + {ii ±1)
for m, whence we have the equation
{re -a)z' + 2 {n' (n ±l)-a{n±l)}z + {ii' - a) {n ± If
- a(?r— 1) + a'^ = p^,
or {7i' - a) / + 2 {n' - a) (w ± 1) ^ + {n (n ± 1) - af = p\
which is easy to solve in Diopbantos' manner, the absolute term
being a square.
But in the problem V. 3 tlwee numbers are required such
that each of them, and the product of each pair, severally added
^ Kal iirel ^xoM*" ^'' '''O'S irophixaffiv, on (of hvo dpiO/Jiol iKarepds re Kai 6 vir^
avTuiv /nerd rod avTov SoOivTos ttoltj Tfrpdycovov, ■yt-ybvaaiv dwb Svo reTpayuvuv rwv
KOLTk Tb ^f ijs.
THE PORISMS OF DIOPHANTOS. 123
to a given number produce squares. Thus, if the third number
be z, three more conditions must be added, namely, z + «, zx-\-a,
yz + a should be squares. The two last conditions are satisfied,
if m + 1 = n, by putting
z = 2{a; + y) — 1 = 4/m" + 4??i + 1 — 4a,
when xz + a= {m {2m + 1) - Sa}",
yz + a= {m {2m + 3) - (2a - l)f ,
and this means of satisfying the conditions may have affected the
formulating of the Porism.
V. 4 gives another case of the Porism with — a for + a.
Porism 2. In V. 5 Diophantos says* , " We have in the
Porisms that in addition to any two consecutive squares we can
find another number which, being double of the sum of both
and increased by 2, makes up three numbers, the product of
any pair of which ^lus the sum of that pair or the third
number produces a square," i.e.
m\ m^ + 2??i + 1 , 4 ( 7?i' + m + l),
are three numbers which satisfy the conditions.
The same porism is assumed and made use of in the follow-
ing problem, v. 6.
Porism 3 occurs in v. 19. Unfortunately the text of the
enunciation is corrupt, but there can be no doubt that the
correct statement of the porism is " The difference of two cubes
can be transformed into the sum of two cubes." Diophantos
contents himself with the mere enunciation and does not pro-
ceed to effect the actual transformation. Thus we do not know
his method, or how far he was able to prove the porism as
a perfectly general theorem. The theorems upon the trans-
formation of sums and differences of cubes were investigated by
Vieta, Bachet and Format.
1 Kal txoiJ^o irdXi;/ (v toZs wopifffMacriv on iraai 56o Ttrpaywvoif toTj (tori t6 iint
irpo<TevplaK€Tai irepos dptOubs 8i a)c divXaaiuv <jvv<ifj.<t>oripov Kal SvUt. fxtiiuiv, t/xji
apidfiodi TTOtei tSv 6 vwd 6iroiu}vo!}i> idvTe irpoffXdiir] avvan<p!)Ttpov, iatnt Xoiw-di- «■<>•«
TeTpd-ywvov.
124 DIOPHANTOS OF ALEXANDRIA.
Vieta gives three problems on the subject ^ (Zetetica iv.).
1. Given two cubes, to find in rational numbers two others
whose sum equals the difference of the two given ones. As
a solution of a^-I/ = x^ H- y^, he finds
_ a{a'-^h') _ hj^a'j- If)
^- a' + h' ' y~ a' + b' '
2. Given two cubes, to find in rational numbers two others,
a +0 -X y, X- ^3_^, , y ^,_^, .
3. Given two cubes, to find in rational numbers two
others, whose difference equals the difference of the given ones;
a -u -X y, X- ^3_^^3 , y- ^^,_^^, .
In 1 clearly x is negative if 2b^>a^', therefore, to secure a
"rational" result, (v-j > 2. But for a "rational" result in 8 we
must have exactly the opposite condition, t5 < 2. Fermat, who
apparently was the first to notice this, remarked that in con-
sequence the processes 1 and 3 exactly supplement each other,
1 Poselger {Berlin Abhandhuigcn, 1832) has obtained tlicse results. He gets,
e.g. the first as follows:
Assume two cubes {a-xf, (mx-j3f, which are to be taken so that their
sum = a3 - /33.
Now (a-x)^ = a^-3a-x + 3ax--x\
(fftx - /3)3 = - /33 + 3?»/3-x - 3»i''/3x2 + m^a?.
If then
G)'
and
3(TO2/3-a) Sap'
^~ m3-l a3 + /33'
(a-x)3 + (mx-/3)8 = a3-/3»,
a(a3-2^)l
THB PORlSMS OP DIOPHANTOS. 125
SO that by employing them successively we can effect the trans-
formation of 1, even when
Process 2 is always possible, therefore by the suitable com-
bination of processes the transformation of a sum of two cubes
into a difference, or a difference of two cubes into a sum of two
others, is always practicable.
Besides the Po)nsins, there are many other propositions
assumed or implied by Diophautos which are not definitely
called porisms, though some of them are very similar to the
porisms just described.
§ 2. Theorems assumed or implied by Diophantos.
Of these Nesselmann rightly distinguishes two classes, the
first being of the nature of identical formulae, the second
theorems relating to the sums of two or more square num-
bers, &c.
1. The first class do not require enumeration in detail. We
may mention one or two examples, e.g. that the expressions
C^-) - ah and a" (a -h 1)' + a' + (a + 1)" are squares, and that
a {a^ — a) -\- a + (a^ — a) is always a cube.
Again, Nesselmann thinks that Diophantos made use of the
separation of a' — 1/ into factors in the solution of v. 8, in
which he gives the result without clearly showing his mode
of procedure in obtaining it ; though its separability into
factors is nowhere expressly mentioned, and is not made use of
in certain places where we should most naturally expect to find
it, e.g. in iv. 12.
2. But ftxr more important than these identical formulae
are the numerous propositions in the Theory of Numbers which
we find stated or assumed as known in the Arithmetics. It is,
in general, in explanation or extension of these that Fermat
wrote his famous notes. So far as Diophantos is concerned it
is extremely difficult, (jr rather impossible, to .say how far these
126 DIOPHANTOS OF ALEXANDRIA.
propositions rested for him upon rigorous mathematical demon-
stration, and how far, on the other hand, his knowledge of them
was merely empirical and derived only from trial in particular
cases, whereas he enunciates them or assumes them to hold
in all possible cases. But the objection to assuming that
Diophantos had a completely scientific system of investigating
these propositions, as opposed to a merely empirical knowledge
of them, on the ground that he does not prove them in the
present treatise, would seem to apply equally to Fermat's o-svn
theorems set forth in these notes, except in so far as we might
be inclined to argue that Diophantos could not, in the period
to which he belongs, have possessed such machinery of demon-
stration as Format. Even supposing this to be true, we should
be very careful in making assertions as to what the ancients
could or could not prove, when we consider how much they
did actually accomplish. And, secondly, as regards machinery
of proof, we have seen that up to Fermat's time there had
been very little advance upon Diophantos in the matter of
notation.
It will be best to enumerate here in order the principal
propositions of this kind which we find in Diophantos, observing
in each case any indication, which is perceptible, of the extent
which we may suppose Diophantos' knowledge of the Theory of
Numbers to have reached. It will be necessary and useful
to refer to Fermat's notes occasionally.
The question of the merits of Fermat's notes themselves
this is not the place to inquire into. It is well known
that he almost universally enunciates the theorems contained
in these notes without proof, and gives as his reason for not
inserting the proofs that his margin was too small, and so on.
It is considered, however, that as his theorems are always true,
he must necessarily have proved them rigorously. Concerning
this statement I will only remark that in the note to v. 25
Format addresses himself to the solution of a problem which
was " most difficult and had troubled him a long time," and
says that he has at last found a general solution. Of this
he gives a demonstration wliich is hopelessly wrong, and which
vitiates the solution completely.
THE PORISMS OF DIOPHANTOS. ] 27
(a) Theorems in Diophantos respecting the comjyosition of
numbers as the sum of two squares.
1. Any square number can be resolved into two squares in
any number of ivays, li. 8, 9.
2. Any number luhich is the sum of two squares can be
resolved into two other squares in any number of ways, u. 10.
N.B. It is implied throughout that the squares may be
fractional, as well as integral.
3. If there are two whole numbers each of tuhich is the
sum of tiuo squares, their product can be resolved into the sum of
two squares in two ways, iii. 22.
The object of ill. 22 is to find four rational right-angled
triangles having the same hypotenuse. The method is this.
Form two right-angled triangles from (a, b), (c, d) respectively,
viz.
a== + b\ a' - b\ 2ab,
c' + d', c'-cr, 2cd.
Multiplying all the sides of each by the hypotenuse of the
other, we have two triangles having the same hypotenuse,
{a' + h'){c'+d^), {a'-b'){c' + d?l 2ab{c' + d'),
(a' + b') (c'+ cf), (a' + b') {& - d'), 2cd (a^ + b').
Two other triangles having the same hypotenuse are got by
using the theorem enunciated, viz.
(a' + ¥) (c' + d') = (ac ± bdf + (ad + be)',
and the triangles are formed from ac ± bd, acl + be, being
(a' + b') (c* + d'), 4abcd + (rr - b') (c' - d"), 2 (ac + bd) {ad - be),
{a' + 6") {c' + d'), ^abcd - {a' - b') (c' - d'), 2 {ac - bd) {ad + be).
In Diophantos' case
a' + b' = V + 2"- = 5,
d' + d'=2' + :i'=U;
and the triangles are
(65, 52, 39), (65, 60, 2o), (65, 63. 16), (65, c>i>, 33).
128 DIOPHANTOS OF ALEXANDRIA.
[If certain relations hold between a, h, c, d this method
fails. Diophantos has provided against them by taking two right-
angled triangles viro iXax^o-Toov dptOfiwv (3, 4, 5), (5, 12, 13)].
Upon this problem Fermat remarks that (1) a prime number
of the form ^n + 1 can only be the hypotenuse of a right-angled
triangle in one way, the square of it in two ways, «&;c.
(2) If a prime number made up of two squares be multiplied
by another prime also made up of two squares, the product can
be divided into two squares in two ways ; if the first is mul-
tiplied by the square of the second, in three ways, &c.
Now we observe that Diophantos has taken for the hypotenuse
of the first two right-angled triangles the first tiuo prime numbers
of the form 4n + 1, viz. 5 and 13, both of which numbers are the
sum of two squares, and, in accordance with Format's remark,
they can each be the hypotenuse of one single right-angled
triangle only. It does not, of course, follow from this selection
of 5 and 13 that Diophantos was acquainted with the theorem
that every prime number of the form 4?i + 1 is the sum of two
squares. But, when we remark that he multiplies 5 and 13
together and observes that the product can form the hypotenuse
of a right-angled triangle in four ways, it is very hard to resist
the conclusion that he was acquainted with the mathematical
facts stated in Format's second remark on this problem. For
clearly 65 is the smallest number which can be the hypotenuse
of four rational right-angled triangles ; also Diophantos did not
find out this fact simply by trijing all numbers up to G5 ; on
the contrary he obtained it by multiplying together the first
two prime numbers of the form 4?j-f- 1, in a perfectly scientific
manner.
This remarkable problem, then, serves to show pretty con-
clusively that Diophantos bad considerable knowledge of the
properties of numbers which arc the sum of two squares.
4. Still more remarkable is a condition of possibility of
solution prefixed to the problem v. 12. The object of this
problem is "to divide 1 into two parts such that, if a given
number is added to either part, the result will be a square."
Unfortunately the text of the added condition is very much
THE PDKISMS (^F Dlnl'HANTOS. 1l>!»
corrupted. There is no doubt, however, about the first few
words, " The given number must not be odd."
i.e. No number of the form 4/i-|- 3 [or 4« - 1] can be the sum
of two squares.
The text, however, of the latter half of the condition is, in
Bachet's edition, in a hopeless state, and the point cannot be
settled without a fresh consultation of the Mss.^ The true con-
dition is given by Fermat thus. " The given number mu.st not
be odd, and the double of it increased by one, when, divided by the
greatest square which measures it, mu^st not be divisible by a
pnme number of the form 4h— 1." (Note upon v. 12; also in a
letter to Roberval). There is, of course, room for any number
of conjectures as to what may have been Diophantos' words'.
There would seem to be no doubt that in Diophantos' condition
there was something about "double the number" (i.e. a number
of the form 4n), also about "greater by unity" and "a prime
number." From our data, then, it would appear that, if Dio-
phantos did not succeed in giving the complete sufficient and
necessary condition stated by Fermat, he must at all events have
made a close approximation to it.
1 Bachet's test has bel 8r] tov diS6iJLevoi> n-qn irepiaaov iXvai, firire 6 BiirXafflui'
avTov q fi^a. fjceL^ova ^r) fxipos 5 . fj fxerpdrai viro rot a°". s°".
He also says that a Vatican ms. reads /nTjre 6 diirXajlujv avrov api9iJ.ov fioudSa
d. fiell^ova ^XV M^pos Tiraprov, fj neTpelrai viro tov irpwrov apiO/j-ov.
Neither does Xylander help us much. He frankly tells us that he cannot
understand the jmssage. ' ' Imitari statueram bonos grammaticos hoc loco, quorum
(ut aiunt) est multa nescirc. Ego vcru noscio hoic non multa, scd pacnc omnia.
Quid enim (ut reliqua taccam) est /xrjTe 6 onrXaalijsv avrov ap no a, &c. quae
causae liuius irpocoi.opi.dfiov, quae processus ? immo qui processus, quae operatic,
quae solutio?"
* Nesselmann discusses an attempt made by Schulz to correct the text, and
himself suggests nrfre rbv 8nr\affiova avrov apiOnov /xovdSi fitl^ova fx^iv, 6s fie-
rpeirai vir6 rivoi irpurov api.6y.ov. But this ignores /i^pos riraprov and is not
satisfactory.
Haukel, however (Gesch.d. Math. p. 169), says: "Ich zweifele nicht, dass die
von den Msscr. arg entstellte Determination so zu lesen ist: Sei Si) rbv 5iW/i*ror
Urire ntpicabv ehai, /J-rire rov dnrXacrlova ai/Tou apiOnbv fjLovdSi a fitl^ova fitrptiadcu
iino TOV irpil)Tov apid/iov, 8s aj' /louadi d ixd^uv IxV t'-^po^ TirapTov." Now this cor-
rection, which exactly gives Fermat's condition, seems a decidedly probable one.
Here the words p.ipos rirapTov find a place; and, secondly, the rept'tition of
liovaSi d nd^uv might well confuse a copyist, tov for tov is of course natural
enough ; Nesselmann reads nvos for tov.
H. I). i»
130 DIOPHANTOS OF ALEXANDRIA.
We thus see (a) that Diophantos certainly knew that no
number of the form 4??. + 3 could be the sum of two squares,
and (b) that he had, at least, advanced a considerable way to-
wards the discovery of the true condition of this problem, as
quoted above from Fermat.
(6) On numbers luhich are the sum of three squares.
In the problem v. 14 a condition is stated by Diophantos
respecting the form of a number which added to three parts of
unity makes each of them a square. If a be this number,
clearly 3a + 1 must be divisible into three squares.
Respecting the number a Diophantos says "It must not be
2 or any multiple of 8 increased by 2."
i. e. a number of the form 24>i + 7 cannot be the sum of three
squares. Now the factor 3 of 24 is irrelevant here, for with
respect to three this number is of the form 3w + 1, and this so
far as 3 is concerned might be a square or the sum of two or
three squares. Hence we may neglect the factor 3 in 24/i.
We must therefore credit Diophantos with the knowledge of
the fact that no number of the form 8n + 7 can be the sum of
three squares.
This condition is true, but does not include all the numbers
which cannot be the sum of three squares, for it is not true
that all numbers which are not of the form 871 + 7 are made up
of three squares. Even Bachet remarked that the number a
might not be of the form 32?i -)- 9, or a number of the form
9G>i+ 28 cannot be the sum of three squares.
Fermat gives the conditions to which a must be subject
thus:
Write down two geometrical series (common ratio of each 4),
the hrst and second series beginning respectively with 1, 8,
14 16 C4 256 1024 4096
8 32 128 512 2048 8192 32768
then a must not bo (1) any number obtained by taking twice
liny term of tlu' ii[)per scries ami adding all the preceding terms,
THE PORISMS OF DIOPHAN'TOS. l.'H
or (2) the number found by adding to the numbers so obtained
any multiple of the correspondino: term of the second series.
Thus (a) must not be,
8?i + 2.1 = 8" + 2,
32/1 + 2.4 + 1 = 32« + 9,
128n + 2.1G + 4<+ 1 =128n + 87,
ol2n + 2.64 + 16 + 4 + 1 = 512n + 149,
&c.
Again there are other problems, e.g. v. 22, in which, though
conditions are necessary for the possibility of solution, none are
mentioned; but suitable assumptions are tacitly made, without
rules by which they must be guided. It does not follow from
the omission to state such rules that Diophantos was ignorant
of even the minutest points connected with them ; as however
we have no definite statements, it is best to desist from specula-
tion in cases of doubt.
(c) Goinposition of naniberti an the sum of four squares.
Every number is either a square or the sum of two, three or
four squares. This Avell-known theorem, enunciated by Format
in his note to Diophantos iv. 31, shows at once that any number
can be divided into four squares either integral or fractional,
since any square number can be divided into two other squares,
integral or fractional. We have now to look for indications in
the Arithmetics as to how far Diophantos was acquainted with
the properties of numbers as the sum of four squares. Un-
fortunately it is impossible to decide this question with any-
thing like certainty. There are three problems [iv. 31, 32 and
V. 17] in which it is required to divide a number into four
squares, and from the absence of mention of any condition to
which the number must conform, considering that in both cases
where a number is to be divided into three or two Sipiares [v. 14
and 12] he does state a condition, we should probably be right
in inferring that Diophantos was aware, at least empirically, if
not scientifically, that any number could be divided into four
squares. That he was able to prove the theorem scientifically
it would be rash to assert, though it i.s not impossible. But wc
9—2
132 DIOPHANTOS OF ALEXANDRIA.
may at least be certain that Diophantos came as near to the
proof of it as did Bachet, who takes all the natural numbers up
to 120 and finds by trial that all of them can actually be ex-
pressed as squares, or as the sum of two, three or four squares
in whole numbers. So much we may be sure that Diophantos
could do, and hence he might have empirically satisfied himself
that in any case occurring in practice it is possible to divide
any number into four squares, integral or fractional, even if he
could not give a rigorous mathematical demonstration of the
general theorem. Here again we must be content, at least in
our present state of knowledge of Greek mathematics, to remain
in doubt.
CHAPTER VII.
HOW FAR WAS DIOPHANTOS ORIGINAL ?
§ 1. Of the many vexed questions relating to Diophantos
none is more difficult to pronounce upon than that which we
propose to discuss in the present chapter. Here, as in so many
other cases, diametrically opposite views have been taken by au-
thorities equally capable of judging as to the merits of the ca.se.
Thus Bachet calls Diophantos "optimum praeclarissiniumque Lo-
gisticae parentem," though possibly he means no more by this
than what he afterwards says, "that he was the first algebraist of
whom we know." Cossali quotes "T abate Andres" as the most
thoroughgoing upholder of the originality of Diophantos. M.
Tannery, however, whom we have before had occasion to men-
tion, takes a completely opposite view, being entirely unwilling
to credit Diophantos with being anything more than a learned
compiler. Views intermediate between these extremes are
those of Nicholas Saunderson, Cossali, Colebrooke and Nessel-
mann; and we shall find that, so far as we are able to judge
from the data before us, Saunderson's estimate is singularly
good. He says in his Elements of Algebra (1740), "Diophantos
is the first writer on Algebra we meet with among the ancients ;
not that the invention of the art is particularly to be ascribed
to him, for he has nowhere taught the fundamental rules and
principles of Algebra; he treats it everywhere as an art already
known, and seems to intend, not so much to teach, as to culti-
vate and improve it, by applying it to certain indeterminate
problems concerning square and cube numbers, right-angled
triangles, &c., which till that time seemed to have been either
not at all considered, or at least not regularly treateil c»f. These
134 DIOPHANTOS OF ALEXANDRIA.
problems are very curious and ratertaining; but j^et in the
resolution of them there frequently occur difficulties, which
nothing less than the nicest and moc^ refined Algebra, applied
with the utmost skill and judgment, could ever surmount: and
most certain it is that, in this way, no man ever extended the
limits of analytic art further than Diophantos has done, or dis-
covered greater penetration and judgment; whether we consider
his wonderful sagacity and peculiar artifice in forming such
proper positions as the nature of the questions under considera-
tion required, or the more than ordinary subtilty of his reason-
ing upon them. Every particular problem puts us upon a new
way of thinking, and furnishes a fresh vein of analytical treasure,
which, considering the vast variety there is of them, cannot but
be very instructive to the mind in conducting itself through
almost all difficulties of this kind, wherever they occur."
§ 2. We will now, without anticipating our results further,
proceed to consider the arguments for and against Diophantos'
originality. But first we may dispose of the supposition that
Greek algebra may have been derived from Arabia. This is
rendered inconceivable by what we know of the state of learning
in Ai'abia at different periods. Algebra cannot have been
developed in Arabia at the time when Diophantos wrote ;
the claim of Mohammed ibn Musa to be considered the first
important Arabian algebraist, if not actually the first, is ap-
parently not disputed. On the other hand Rodet has shown
that M(jhammed ibn Musii was largely indebted to Greece.
There is moreover great dissimilarity between Greek and Indian
algebra ; this would seem to indicate that the two were evolved
independently. We may also here dispose of Bombelli's strange
statement that he found that Diophantos very often quoted
Indian authors \ We do not find in Diophantos, as we have
him, a single reference to any Indian author whatever. There
is therefore some difficulty in understanding Bombelli's positive
statement. It is at first sight a tempting hypothesis to suppose
that the "frequent quotations" occurred in parts of Diophantos'
^ "Ed in detta opera abbiamo ritrovato, ch' egli assai volte cita gli autori
indiani, col cho rui lia fatto conoscere, che questa disciplina appo gl' indiani
prima lu che agli arabi."
HOW FAR WAS DIOPHANTOS ORIGINAL? 135
work contained only in the MS. which Bombclli used. But wo
know that not a single Indian author is mentioned in that MS.
We can only explain the remark by supposing that Bombelli
confused the text and the scholia of Maximus Planudes ; for in
the latter mention is made of an " Indian method of multiplica-
tion." Such must be considered the meagre foundation for
Bombelli's statement.
There is not, then, mucli doubt that, if we are to find any
writers on algebra earlier than Diophantos to whom he was
indebted, we must seek for them among his own countrymen.
§ 3. Let us now consider the indications bearing upon the
present question which are to be found in Diophantos' own work.
Distinct allusions to previous writers there are none with the
sole exception of the two references to Hypsikles which occur
in the fragment on Polygonal Numbers. These references, how-
ever, are of little or no importance as affecting the question of
Diophantos' originality; for, so far as they show anything, they
show that Diophantos was far in advance of Hypsikles in his
treatment of polygonal numbers. And, so far as we can judge
of the progress which had been made in their theoretical treat-
ment by writers anterior to Diophantos from what we know of
such arithmeticians as Nikomachos and Theon of Smyrna, we
must conclude that (even if we assume that the missing part of
Diophantos' tract on this subject was insignificant as compared
with the portion which has survived) Diophantos made a great
step in advance of his predecessors. His method of dealing
with polygonal numbers is new ; and we look in vain among
his precursors for equally general propositions with regard to
such numbers or for equally scientific proofs of known pro-
perties. Not that previous arithmeticians were uuaccjuaiuted
with Diophantos' propositions as applied to particular jwlygonal
numbers, and even as applicable generally ; but of their general
application they convinced themselves only empirically, and by
the successive evolution of higher and higher orders of sucli
numbers.
We may here remark, with respect to the term "arithmetic"
which Diophantos applies to his whole work, that he is making
a new use of the term. According to the previously ucc.-j»t.-<l
136 DIOPHANTOS OF ALEXANDRIA.
distinction of apLdfiijriKi] and XoytariKT] the former treats of the
abstract properties of numbers, considered apart from their
mutual relations, XoyiaTCKi] of problems involving the relations
of concrete numbers. XoyiartKi], then, includes algebra. Ac-
cording to the distinction previously in vogue the term dpid-
fiTjTiKi] would properly apply only to Diophantos' tract on
Polygonal Numbers ; but, as in the six books of Diophantos the
numbers are treated as abstract, he drops the distinction.
§ 4. Next to direct references to the names of predecessors,
we must look to the language of Diophantos, in order to see
whether there is any implication that anything which he teaches
is new. And in this regard we might naturally expect that
the preface or dedication to Dionysios would be important. It
is as follows : Trjv evpeaiv rwv iv Toh dpi6p.ol<i irpo^X'qiicnoiv,
TifiicoTari fioL Aiovvcrte, jivooaKcov ere a7rovSai(o<; e')(0VTa ixadeiv,
opyavajcrat rrjv fieOoSov €7reipd6r)v, dp^(i/ji€po<i d(f wv crvi'iaTrjKe
rd Trpdy/jiaTa Oefxekicov, v7roar7]aai Tt)v iv Tol<i dpi6fioL<; <pvaiv
re Kol hvvafJiLV. Laa)<i fiev ovv hoKel to Trpdyfjba Sva-^^^epearepov,
eVeiS?) iMrjTTW yvwptyiov iari, hvaekinaTOL yap et? KaropOuxxLV
elaiv al rwv dp'^^ofjiiicov -v^up^ai, o/iw? 8' evKardXTjTrrov aoc jevt]-
aerat Btd rrjv arjv irpodvp.Cav koI Tr)v e^rjv dirohet^Lv' ra^eta
fydp et9 fx,d6r](rLV eTriOufiia irpoaXa^ovaa ScSa^yjv. The first
expression which would seem to carry with it an indication of
the nature of the work as conceived by Diophantos himself is
opyavwxrat rrjv fiedoSov. The word opyavwaat has of itself been
enough to convince some that the whole matter and method of
the Arithmetics were originaP. Cossali and Colebrooke are of
opinion that the language of the preface implies that some part
of what Diophantos is about to teach is new". But Montucla
1 Cf. the view of " 1' abate Andres " as stated by Cossali: "Diofanto stesso
parla in guisa, che sembra mostrare assai cLiaramente d' essere stata sua iuven-
zione la dottrina da lui proposta, e spiegata nulla sua opera."
" "A me par troppo il dire, che da quelle cspressioni non ne esca alcuu
lume; mi pare troppo il restingcre la novita, che annunziano, al metodo, che nell'
opera di Diofanto regnas si mira ; ma parmi anche troppo il dedurne essere state
Diofanto in assoluto sonso inventor dell' analisi." Cossali.
"He certainly intimates that some part of what he proposes to teach is new:
Tffwj fxh oiV doKel rd irpayfia 5v<TXf p^<rTepoi' iireiSri firiiru) yvdpi/xoi' iari: while in
other places (Def. 10) he expects the student to be previously exercised in the
HOW FAR WAS DIOPHANTOS ORIGINAL ? 137
does not go too far when he says that the preface does not give
us any chie. The word op-^avwaai is translated by Bachet as
" fabricari," but this can hardly be right. It means " to set forth
in order", to "systematise"; and such an expression may per-
fectly well apply if there were absolutely nothing new in the
work, and Di(iphantos were merely writing a text-book simply
giving in a compact and systematic form the sum and substance
of previous labours. The words eVetS?} /u,7;7r&) ^vwpiixov iariv
have also been made use of by advocates of Diophantos' claim
to originality; but, looked at closely, they clearly imply no
more than that the methods were unknown to Dionysios. The
phrase is subjective, as is shown by the following words, SuaiX-
TTicTTOC yap eh Karopdcocriv eicriv al twv dp^^^ofievav ylrv^ai.
The language of the definitions also has been variously
understood. "L' abate Andres" concluded from their very
presence at the beginning of the book that Diophantos is
minutely explaining preliminary matter as if he were speaking
of a new science as yet unknown to others. But the fact is
that he does not minutely explain preliminary matter ; he gives
an extremely curt summary of the necessary preliminaries.
Moreover he makes no stipulations as to what he will choose to
call by a certain name. Thus a square KaXelrai 8vvafii<i :
the unknown quantity is called dpidfio'i, and its sign is ?•*.
Again, he says Xeiyfri^ iirt Xel^jrtv TroXXaTrXaaiaadelcra Troiei
virap^iv, Minus multiplied hij minus (jives j)lus^. In the 10th
algorithm of Algebra. The seeming contradiction is reconciled by conceiving tlic
principles to have been known, but the application of thorn to a certain class of
problems concerning numbers to have been new." Ci'lebrooke.
• I adhere to this translation of the Greek because, tiiough not quite literal,
it serves to convey the meaning intended better than any other version. It is
not easy to translate it literally. Mr James Gow (History of Creek Mnthemntict,
p. 108), says that it should properly be translated "A difference multiplied by
a difference makes an addition." This translation seems unfortunate, because
(1) it is difficult, if not impossible, to attach any meaning to it, (2) Xc^jj and
vvap^is are correlatives, whereas "difference" and "addition" are not. If
either of these words are used at all, we shoidd surely say either "A dillerenco
multiplied by a diffirence makes a sum", or "A subtraction multiplietl by a
subtraction makes an addition." The true meaning of Xer^ii must be " a falling-
short" or "a wanting", and that of i-Vapfis "a presence" or "a forthcoming".
If, therefore, a literal translation is desired, I would suggest " A wanting multi-
138 DIOPHANTOS OF ALEXANDRIA.
Definition he says how important it is that the beginner should
be familiar with the operations of Addition, Subtraction, &c.;
and in the 11th Definition the rules for reducing a quadratic to
its simplest form are given in a dogmatic authoritative manner
which would only be appropriate if the operation were generally
known : in fine, the definitions, in so far as they have any bear-
ing on the present question, seem to show that Diophantos does
not wish it to be understood that they contain anything new.
He gives them as a short but necessary resume of known prin-
ciples, more for the puriDose of a reminder than as laying any
new foundation.
To assert, then, that Diophantos invented algebra is, to say
the least, an exaggeration, as we can even now see from the
indications above mentioned. His notation, so far as it is a nota-
tion, is apparently new ; but, as it is merely in the nature of
abbreviations for complete words, it cannot be said to constitute
any great advance in algebra.
§ 5. I may here mention a curious theory propounded by
Wallis, that algebra was not a late invention at all, but that it
was in common use by the Greeks from the time of their earliest
discoveries in the field of geometry, that in fact they disco-
vered their geometrical theorems by algebra, but were extremely
careful to conceal the fact. But to believe that the gi'eat Greek
geometers were capable of this sj^stematic imposition is scarcely
possible\
plied by a wanting makes a forthcoming". But, thongb this would be correct,
it loses by obscurity more than it gains by accuracy.
1 "De Algebra, prout apud Euclideu Pappum Diophantum et scriptores
habetur. Mihi quidem extra omne dubium est, veteribus cognitam fuisse et
usu comprobatam istiusmodi artcm aliquam Investigaudi, qualis est ea quam
nos Algebram dicinius : Indequc derivatas esse quae apud cos conspiciuntur
prolixiores et intricatae Demonstrationcs. Aliosque ex recentioribus mecum
bae in re sen tire comperio. ...Hanc autem artem Investigaudi Veteres occu-
luerunt sedulo: contenti, per demonstrationes Ajjagogicas (ad absurdum seu
impossibile ducentcs si quod assorunt negetur) asscnsum cogere; potins quam
directum methodum indicare, qua fuerint inventae propositiones illae quas ipsi
aliter et per ambages demonstrant." (Wallis, Opera, Vol. ii.)
Bossut is certainly right in bis criticism of this theory. "Si cette opinion
6tait vraic, elle inculperait cch grands hommes d'unc charlatancrie systc^matique
et traditionolle, ce qui est invraiscmblable en soi-nieme et ne pourrait Stre admis
sans les preuves les plus rvidentes. Or, sur quoi une telle opinion cst-elle
HOW FAR WAS DIOPHANTOS ORIGINAL? 139
§ 6. What remains to be said may, perhaps, be best arranged
under the principal of Diophantos' methods as headings ; and it
will be advisable to take them in order, and consider in each
case whether anything is anticipated by Greek authors whose
works we know. For it would seem useless to speculate on
what they might have written. If we once leave the safe
ground of positive proveable fact, such an investigation as the
present could lead to no useful result. It is this fact which
makes so much of what M. Tannery has written on this stibject
seem unsatisfactory. He states that Diophantos was no more
than a learned compiler, like Pappos : though it may be ob-
served that this is a comparison by no means discreditable to
the former ; he does not think it necessary to explain the com-
plete want of any other works on the same subject previous to
Diophantos. The scarcity of information respecting similar
previous labours, says M. Tannery, is easily explicable on other
grounds which do not concern us here\ The nature of the
work joined to what we know of Diophantos would seem to
prove his statement, thinks M. Tannery ; thus the work is very
unequal, some operations being even clumsy*. But we are not
likely to admit that inequality in a work is any evidence against
originality ; for what great genius always equalled himself?
Certainly, if we cannot find any certain traces of anticipation of
Diuphantos by his predecessors, he is entitled to the benefit of
any doubt. Besides, given that Diophantos was not the in-
ventor of any considerable portion of his science, the merit of
having made it known and arranged it scientifically is little less
than that of the discoverer of the whole, and very much greater
than that of the discoverer of a small fraction of it.
First with regard to the use of the unknown quantity by
fondce? Sur quelques anciennes propositions, tirc^es principaleraent du trei-
ziiSme livre d'Euclide, oti Ton a cru reconnaitre I'alK^bre, mais qui ne siipposont
r^ellement que I'analyse gtom^trique, dans laquelle les anciens C-taient fort exer-
ccs, commc je I'ai d6ja marqu6. II parait certain que Ics (Jroca n'ont commcuc<5
a connaitre I'alt^ubre qu'au temps de Diophante." (Ilistoire Gin(rale tUn Math/-
viatiques par Charles Bossut, Paris, 1810.) The truth of the last sentence is not
so clear.
' lluUrtin (ipx Srifiirfx miithi'mnl'ujnex ct dgtronomiqnt's, 1870, p. 261.
2 Ibid.
140 DIOPHANTOS OF ALEXANDRIA.
Diophaiitos. There is apparently no indication that dpidfi6<i, in
the restricted sense appropriated to it by Diophantos, was em-
ployed by any other extant writer without an epithet to mark
the use, and certainly Bvvafii<; as restricted to the square of the
unknown is Diophautine. But the employment of an unknown
quantity and calculations in terms of it are found before Dio-
phantos' time. To find a thing in general expressions is, with
Diophantos, to find it iv dopiaro). Cf. the problem IV. 20.
But the same word is used in the same sense by Thymaridas in
his Epanthema. We know of him only through lamblichos,
but he probably belongs to the same period as Theon of Smyrna.
Not only does Thymaridas distinguish between numbers which
are wpiafMevoL (known) and doptaroc (unknown), but the Epan-
thema gives a rule for solving a particular set of simultaneous
equations of the first degree with any number of variables.
The artifice employed is the same as in i. 16, 17, of Diophantos.
This account which lamblichos gives of the Epanthema of Thy-
maridas is important for the history of algebra. For the essence
of algebra is present here as much as in Diophantos, the " nota-
tion" employed by him showing only a very slight advance.
Thus we have here another proof, if one were needed, that
Diophantos did not invent algebra.
Diophantos was acquainted with the solution of a mixed or
complete quadratic. This solution he promises in the 11th
Definition to explain later on. But, as we have before remarked,
the promised exposition never comes, at least in the part of his
work which we pos.se.ss. He shows, however, sufficiently plainly
in a number of problems the exact rule which he followed in
the solution of such equations. The question therefore arises :
Did Diophantos himself discover and formulate his purely
arithmetical rule for solving complete determinate quadratics,
or was the method in use before his time ? Cossali points out
that the propositions 58, 59, 84 and 85 of Euclid's BeSofieva
give in a geometrical form the solution of the equations
ax — x^= b, ax -i x' = b, and of " A .
xy = b)
It was only necessary to transform the geometry into algebra, in
HOW FAR WAS DIOPHANTOS ORIGINAL? 141
order to obtain Diophantos' rule ; and this might have been
done by some mathematician intermediate between Euclid and
Diophantos, or by Diophantos himself. It is quite possible that
it may have been in this manner that the rule arose; and, if
that is the case, it is probable that the transformatii^n referred
to was accomplished by some matliematician not much later than
Euclid himself; for Heron of Alexandria {circa 100 R.C.) already
used a similar rule\ We hear moreover of a work on quadratic
equations by Hipparchos (probably ciixa IGl — 126 B.C.)*. Thus
we may conclude that Diophantos' rule for solving complete
quadratics was not his discovery.
§ 7. But it is not upon Diophantos' solution of determinate
equations that the supporters of his claim to originality rely ;
it is rather that part of his work which forms its main subject, ^
namely, Indeterminate or Semi-determinate Analysis. Accord- /
ingly it is to that that the term Diophantine analysis is applied.
We should therefore look more especially for anticipations of
Diophantine analysis, if we would be in a position to judge as to
Diophantos' originality.
The foundation of semi-determinate analysis was laid by
Pythagoras. Not only did he propound the geometrical theorem
that in a right-angled triangle the square on the hypotenuse is
equal to the sum of the squares on the other two sides, but he
applied it to numbers and gave a rule — of somewhat narrow
application, it is true — for finding an infinite number of right-
angled triangles whose sides are all rational numbers. His
rule, expressed in algebraical form, asserts that if there are
three numbers of the form 2m'^ - 2wi -t- 1, 2m^ — 2/n, and 1m — 1,
they form a right-angled triangle. This rule applies clearly to
that particular case only in which two of the numbers differ by
unity, i.e. that particular case of Diophantos' general form for
a right-angled triangle (m* -^-n^ w' - n"", 2mn) in which in-ii = l.
But Pythagoras' rule is an attempt to deal with the general
problem of Diophantos, II. 8, 9. Plato gives another form f.ir a
» Cf. Cantor, pp. 341, 342. The solution of a quailratio \va.>< for Heron no
more than a matter of arithmetical calcnlation. He solved such e<iuationH by
making both sides complete squares.
2 Cf. Cantor, p. 313.
142 DIOPHANTOS OF ALEXANDRIA.
rational right-angled triangle, namely (w^ + 1, m^—l, 2;/i), which
is that particular case of the form used by Diophantos in which
71=1. Euclid, Book x. prop. 29 is the same problem as Dioph.
II. 8, 9, Diophantos improving upon Euclid's solution. In compar-
ing, however, Euclid's arithmetic with that of Diophantos we
should remember that with Euclid arithmetic is still geometry:
a fact which accounts for his marvellously-developed doctrine of
irrational and incommensurable numbers. In Diophantos the
connection with arithmetic and geometry is severed, and irrational
numbers are studiously avoided throughout his work.
There is another certain case of the solution of an indeter-
minate equation of the second degree in rational numbers before
Diophantos. Theon of Smyrna, in his work Twv Kara fiaOrjfia-
TLKTjv '^(^p'rjcrl/xcov et9 r})v rov TI\dTcovo<i dvayvcoaLv [sc. expositio,
say the editors], gives a theorem Trepl irXsvpiKoov Kal Sia/xerpiKcov
dpLd/xoov. From this theorem we derive immediately any number
of solutions of the equations
provided that we can find, by trial or otherwise, one solution of
either. Theon does not make this application of his theorem :
he solved a somewhat important problem of the second degree
in indeterminate analysis without knowing it. There is an
allusion to the doctrine of Side- and Diagonal-numbers in
Proclus, Comment, on Euclid iv. p. 111.
§ 8. Such are the data upon which Nesselmann founded his
view as to the originality of our author. But M. Tannery has
tried to show, by reference to a famous problem, that still more
difficult questions in indeterminate analysis had been propounded
before the time of Diophantos. This problem is known by the
name of the " Cattle-problem " ; it is an epigram, and is com-
monly attributed to Archimedes. It was discovered by Lessing,
and his discussion of it may be found in Zur Geschichte und
Litteratur (Braunschweig, 1773), p. 421 seqq. I have quoted it
below according to the text given by Lessing'. The title does
1 I have unfortunately not been able to consult llic critical Wdik on this
epigram by Dr J. Struvc anil Dr K. L. Stiuvc, father and son (Altona, 1H21).
s
HOW FAR WAS DIOPHANTOS ORIGINAL? 143
not actually imply that Archimetles was the author. Of the
two divisions into which it falls the second leads to an induter-
My information about it is derived at second-hand from Nesselmann. LessinR's
text can hardly be perfect, but it seems better to give it as it is without emen-
dation.
nPOBAHMA
oirep 'APXIMHAH2 iv ^Trtypd/jL/j.affiv fvpwv
TOiS iv 'AXe^avdpeigi wepl ravra Trpayp-aTov/idfoi^ t'tfTeiv aTricTciXtv
if TTJ TTpbs "RpaToaOivrjv rhv Kvprivaiov
eTn(7To\fj.
nXijdiii' ijeXloio (SoCoi>, u ^eive, pArp-qffov,
<f>povTlb' eiricrTrjcras, el /xer^x^'J ffO(pir]s,
iroffffij dp' iv ireStois HiKeXrjs ttot' (^6ffKeTo vrjffov
Qpivad-qs, rerpaxv (rri^ea Saffffafi^v-q
Xpo^V" oXXaaaovra' t6 iih XevKoio yaXaKTos,
Kvav4(i> 8' irepov xpw/toTt Xa/iirdfievov,
dXXoye /xiv ^avObf, t6 di iroiKlXoy. 'Ev 5e eKaartf)
aricpet i<rav ravpoi Tr\i}de<yL ^pidbiievoL,
av/x/jLeTpi-qs TOLTJade TerenxoTer dpydrpixas fxiv
Kvaviuv Taupojv r]/j.icr€L rjdi rplrif),
Kal ^av^ots (T6/jLTra(Tiv taovs, w ^eive, vb-qaov.
Avrdp Kvaviovs t^ rerpdrip fiipei
/MiKTOXpbuv Kal ir^/XTTTif), ?Tt ^avdo'ial re iraji.
Toys 5' viroXeiiro/xifOis iroiKiXbxjx^o.^ adpei
dpyfvvCiv ravpwv ^KTip p-epei, f^So/udrtf. re
Kal ^avOoh avrous irdcriv laa^op.ivovs.
OrfXeiaiai Si j3ov<Ti rdo' ^TrXero" XevKorpixfS p-fv
r)(Tav ffvpLirdffrjs Kvavirfs dyeXtjs
rtfi TpiTdT(f re fiipei Kal T€TpdT(f} drptKis Iffat.
A&rip Kvdveai t<J) Tirpdri^j re waXiv
/UKTOXpowu Kal irifiirrifi 6/j.ov pJpei Iffd^ovro
<Ti>v TaipoLS irda'r)% eh vofiov ipxop^vris.
Savdorpix^^y dyeXijs W/urrT(f> fi^pei rjoi Kai <\T<f)
iroiKiXai ia-dpiOp-of ttXtjOos Ix^"- '^^'''P^-XTI
^avdal o' 7]pid/xiOvTo p-ipovi rpiTov rjpxaei. laon
dpyevprjs dyiXrjs e^oonaTi^ re fxipei.
SeTve, tri) 5' ijeXioio /So'ej iroaai drpeKh elirwv
X^^pls M^f raijpuv ^aTpe(piui> dpiOp-bv,
Xwpis 5' oB drjXeiai Saai Kari xpoidv iKacTai,
oiiK d'idpb Ke Xiyoi, oi)5' dpi0p.Qv dSa^i,
oil pirjv iribye <TO<po7s iv dpiOfioiV ctW tdi (ppd^ev
Kal TaSe irdvra ^bwv ijeXloio iradij,
'Apyorpixes ravpoi fiiv iirel m^alaro wXijOC'v
Kvav^ois taravT ffiireSov laofxerpoi
eh ^dOoi eh evpo^ re' rd 5' av irepifxriKea xdyryj
wl/jLirXavTO tXivOov QpivaKt-qt TreSi'o.
^afdoi o' av r' eh ff Kai Troi/nXoi dOpoiaOimtf
144 DIOPHANTOS OF ALEXANDRIA.
minate equation of the second degree. In view of this fact it is
important for us to discuss briefly the matter and probable date
of this epigram. Struve does not admit that it can pretend to
that antiquity which is claimed for it in the title. This we
may allow without going so far as Kliigel, who makes it as late
as the introduction of the present decimal system of numeration.
Nesselmann's view is that the heterogeneous conditions, which
are thrown together to render the problem difficult, show that
the author (if the whole is due to one author) could have had
no idea how to solve it. Nesselmann is of opinion that the
editors of the anthology were justified in refusing a place to this
epigram, that the most one could do would be to admit the
first part and condemn the latter part as corrupt, and that we
might fairly regard the whole as unauthentic because even the
first part could not belong to the age of Archimedes. The first
part, which falls into two divisions, gives seven equations of the
first degree for determining eight unknown quantities, namely
the number of bulls and cows of each of four colours. The
solution of the first part gives, if {X YZW) are the numbers of the
bulls, {xyziu) the corresponding numbers of cows,
X = 10366482 n, x = 7206360 n,
Y= 7460514 w, 2/ = 4893246 ?i,
Z= 7358060 w, 2=3515820??,
W= 4149387 w, «<; = 54.39213 w,
where n is an integer. If we take the smallest possible value
the number of cattle is sufficiently enormous. The Scholiast's
solution corresponds to the value 80 of n, the result being
" truly," as Lessing observes, " a tolerably large herd for Sicily."
The same might be said of the solution arising from putting
w = 1 above. This is surely a curious commentary on M.
Tannery's theory above alluded to (pp. 6, 7), that the price of
the wine in vi. 33 of Diophantos is a sufficient evidence of the
i(xravT' afx^o\aZr)v i^ ivbi apxa/J-efOi
ffxvM-o- reXtiovvTes t6 rpiKpacnredov oUre irpoaovruu
aSXoxpouv ravpwv, oUt^ iTriXenro/JL^vuv.
TaOra cvvf^fvpilji' Kal ivi TrpairiSfcrcriv ddpolaas
Kal ir\t)Oi.o}v diro5ous, u) ^^i>(, iravra fiirpa.
^PX^o Kv5i6o}v inKri(p6po%' taOi Tf iravTUii
KCKpifi^fot rairrxi uinrvLO% cp ffocply.
now FAR WAS DIOl'HANTOS OHKMNAL? U.',
date of the epigram. If the " Cattle-problem " of which we arc
now speaking were really due to Archimedes, we should, sup-
posing M. 'i'aunery's theory to hold good, scarcely have found
the result in such glaring contradiction to what cannot but
have been the facts of the case, Nesselmann further argues in
ftivour of his view by pointing out (1) that the problem is
clearly at an end, when it is said that he who solves the
problem must be not unskilled in numbers, i.e. where I have
shown the division ; and the addition of two new conditions
with the preface "And yet he could not pretend to proficiency
in wise calculations" unless he could solve the rest, shows
the marks of the interpolator on the face of it, and, moreover,
of a clumsy interpolator who could neither solve the complete
problem itself, nor even conceal his patchwork. (2) The lan-
guage and versification are against the authenticity. (3) The
Scholiast's solution does not, as it claims, satisfy the whole
problem, but only the first part. (4) The impossibility of
solution with the Greek numeral notation and the absurdly
large numbers show that the author, or authors, could not
have seen what the effect of the many heterogeneous conditions
would be. Nesselmann draws the conclusion above stated ; and
we may safely assume, as he says, that this ej)igram is from the
historical point of view worthless, and could not, even if it
were shown to be earlier than the date of Diophantos, be held
to prove anything against his originality.
M. Tannery takes the opposite view and uses the epigram
for the express purpose of proving his assumption that Dio-
phantos was not an original writer. M. Tannery takes a passage
attributed to Geminos in which he is describing the distinction
between XoytariKi] and apLOfirjTiKt'j. XoyiariKi] according to
Geminos dewpet to /zef KXijOki' vtt \\pxiM^ov<i ^oIkov trpu-
l3Xi]/j.a, toOto' he p-ifKira^i fcuL (jjiaXLTu^i ctpiO/xovs'- Ul the two
1 I do not read rovs as M. Tannery does. He alters tovto, the original
reading, into rods, simply remarking that tovto is an "inadmissible reading."
TovTo Si is certainly a reading which needs no defence, being exactly what we
should expect to have. The passage appears to be taken from the Scholia to
Plato's Cluinnideii, where, however, Stallbaum and Heiberg read Otupti o^ roOro
Hkv rb K\r}6iv i'lr" 'A/ixtM')'Joi'S (ioeiuoi' wij6ft\)ina, touto bi li'iMrai «o» 0ioXirat
H. D. 10
146 DIOPHANTOS OF ALEXANDRIA.
kinds of problems which are here distinguished as falling
within the province of Xoyca-TiK/j M. Tannery understands
the first to be indeterminate problems, the type taken {kXvO^v
vir 'Apxi-M^ovi ^o'Ckou Trpo^Xiifia) being nothing more or less
than the very problem we have been speaking of. He states
that Nesselmanu has not appreciated the problem properly,
and finally that we have here an indubitable reference to an
indeterminate problem of the second degree (viz. the equation
8Ax^ + l=if, where yl is a very large number) more difficult
than those of Diophantos. But this statement would seem
simply to beg the question. For, if the expression of Geminos
refers to the problem which we are speaking of, it may even
then only refer to the first part, that is, an indeterminate
problem of the first degree : M. Tannery has still to show that
the whole problem is one, and a genuine product of antiquity.
But I have not found that M. Tannery makes any attempt
to answer Nesselmann's arguments ; and, unless they are an-
swered, the conclusion which the latter draws from them cannot
be said to be invalidated.
But Nesselmann's view is also opposed by Heiberg {Quae-
stiones Archimedeae, 1879). I do not think, however, that his
arguments in favour of the authenticity are conclusive ; and,
though answering some, he does not answer all of Nesselmann's
objections. With regard to the language Heiberg observes that
the dialect need not surprise us, for the use by Archimedes of
the Ionic instead of the Doric dialect for this epigram would
easily be explained by the common use of the Ionic dialect for
epic and elegiac poetry \ And he further suggests that, even if
dpidfiovs, which Beems better than the reading quoted by M. Tanuery from
Hultsch, Ileronis Reliquiae, and given above.
^ Heiberg admits that the language of the title is not satisfactory. He
points out that tV iiriypd/xfxaaii' shoukl go, not with eipuv, but with airiffreiXe,
though so far separated from it, and that the use of the plm-al i -my pd/ifiaaLv is
unsatisfactory. Upon the reading noiKlXai iaapUlixov ttXtjOos ^x"" '''^''^po-XV (1- 21)
he observes that by symmetry z should not be equal to four times (l + l){1V+w),
but to (j + i) (W+ic) itself, and, even if that were the case, we should require
TerpdKis. Hence he suggests for this lino iroiKlXai ladpiOfiov nX^Ooi ?xo*'<^' ^<pdvrj.
(Apparently, to judge from his punctuation, Lessing understood Terpaxij in
the sense of "fourthly.") Heiberg explains irXivOov (1. 36) as " quadraugulum
sohdum," by which is mtaut dimply " a square," as is clearly indicated by 1. 34.
now FAR WAS DlOrilANTdS OUICINAL ^ 147
the difficulties as to the hmguagc arc considered too great, we
may suppose the problem itself to have been the work of
Archimedes, the language of it that of some later author. But,
if Heiberg will go so far as to admit that the language may be
the work of a later author than Archimedes, it would be no
more unnatural to suppose that the matter itself of the latter
part of the problem was also of later date. The suggestion that
Archimedes could not have solved the whole problem (as com-
pleted by the two last conditions) Heiberg meets with argu-
ments which appear to be extremely unsafe. He says that
Archimedes' approximations to the value of J^, although we
cannot see by what process he arrived at them, show plainly
that his arithmetic was little behind our modern arithmetic,
and that, e.g., he possessed means of approximation little in-
ferior to the modern method by continued fractions. Heiberg
further observes that Archimedes possessed machinery for deal-
ing with very large numbers. But we are not justified in
assuming on these two grounds that Archimedes could solve
the indeterminate equation 8Ax^+l=y^, where (Nesselmann,
p. 488) A = 51285802909803, for the solution of which we
should use continued fractions. I do not think, therefore, that
Heiberg has made out his case. Hence I should hesitate to
assume that the problem before us is an indubitable case,
}n-evious to Diophantos, of an indeterminate equation of the
second degree more difficult than those treated by him.
The discussion of the " Cattle-problem " as possibly throwing
some light on the present question would seem to have adiled
nothing to the arguments previously stated ; and the (juestiou
of Diophantos' originality may be considered to be uuaftectod by
anything that has been said about the epigram.
We may therefore adopt, with little or no variation, Ncssel-
mann's final result, that he is far from believing that Diophantos
merely worked up the materials of others. On the contrary he
is convinced that the greater part of his propositions and his
ingenious methods are his own. There is moreover an " ludivi-
duum" running through the whole work which strongly confirms
this conclusion.
lU-2
CHAPTER VIII.
DIOPHANTOS AND THE EARLY ARABIAN ALGEBRAISTS.
§ 1. I propose in this chapter to examine briefly the indica-
tions which are to be found in certain Arabian algebraists of in-
debtedness to, or points of contact with, Diophantos. And in
doing so I shall leave out of consideration the Arabic translations
of his work or commentaries tiiereupou. These are, so far as
we know, all lost, and such notices ot them as we have I have
given in Chapter ill. of this Essay (pp. 39 — 42). Our histori-
cal knowledge of the time and manner in which Diophantos
became known to the Arabs is so very scanty as to amount
almost to nothing: hence the importance of careful comparison
of the matter, methods, and mode of expression of Diophantos
with those of the important representatives of early Arabian
algebra. Now it has been argued that, since the first transla-
tion of Diophantos into Arabic that we know of was made by
Abu'i-Wafa, who lived A.D. 940 — 998, while Mohammed ibn
Musa's algebraical work belongs to the beginning of the 9th
century, Arabian algebra must have been developed inde-
pendently of that of the Greeks. This conclusion, however, is
not warranted by this evidence. It does not follow from the
want of historical proof of connection between Greek and
Arabian algebra that there was no such connection; and it is to
internal evidence that we must look for the correction of this
misconception. I shall accordingly enumerate a number of
points of similarity between the Arabian algebraists and Dio-
phantos which would seem to indicate that the Arabs were
ac(|uainted with Diophantos and Greek algebra before the time
of Mohammed ibn Musu, and that in Arabian algebra generally,
DIOPHANTOS AND THE EARLY AllAHIAN AMJKHUAISTS. 14;)
at least iu its beginnings, the Greek element greatly j.rr.lomi-
nated, though other elements were not wanting.
§2. The first Arabian who concerns us liere is Mohammed
ibn Musa Al-Kharizmi. He wrote a work which he callfd
Aljahr wahnukUhala, and which is, so far as we know, not only the
first book which bore such a title, but (if we can trust Arabian
notices) was the first book which dealt with the subject indi-
cated thereby. Mohammed ibn Musa uses the words aljahr ami
almukiihala without explanation, and, curiously enough, there
is no application of the processes indicated by the words in the
theoretical part of the treatise: facts which must be held to
show that these processes were known, to some extent at least,
even before his time, and were known by those names. A mere
translation of the two terras jabr and mukdbala does not of itself
give us any light as to their significance. Jabr has been trans-
lated in Latin by the words restauratio and restitutio, and in
German by "Wiederherstellung"; mukabala by oppusitio, or
"comparison," and in German by "Gegeniiberstellung." Fortu-
nately, however, we have explanations of the two terms given
by later Arabians, who all agree as to the meaning conveyed by
them\ When we have an algebraical equation in which terms
affected with a negative sign occur on either side or on both
sides, the process by which we make all the terms positive, i.e.
adding to both sides of the equation such positive terms as will
make up the deficiencies, or absorb the negative ones, is jabr or
restauratio. When, again, we have by jabr transformed our
equation into one in which all the terms are positive, the
process by which we strike out such terms as occur on both
sides, with the result that there is, finally, only one term con-
1 Rosen gives, in his edition of The Algebra of Mohammed ben Miua, a
number of passages from various authors explaining aljabr and almukabala.
I shall give only one, as an examijle. Itosen says " In the Kholaset iil Hisub, a
compendium of arithmetic and geometry by Baha-Eddiu Mohammed ben Al
Hosain, who died A.n. 1031, i.e. 1575 a.i>., the Arabic text of which, together
with a Persian commentary by Roshan Ali, was printed at Calcutta (IHTi, Svo),
the following explanation is given : ' The side (of the e<iuation) on which bouic-
thing is to be subtracted, is made complete, and as much is to be added to the
other side; this is jebr; again those cognate (luautitiea which are equal on both
sides are removed, and this is iW)kdbalah\"
150 DIOPIIANTOS OF ALEXANDRIA.
taining each power of tlie unknown, i.e. subtracting equals from
equals, is mukdhala, oppositio or "comparison." Such was the
meaning of the terms ja6?* and mukdhala) and the use of these
words together as the title of Mohammed ibn Musa's treatise is
due to the continual occurrence in the science there expounded
of the processes so named. It is tiTie that in the theoretical
part of it he assumes that the operations have been already
completed, and accordingly divides quadratic equations at once
into six classes, viz.
aa? = hx, ax^ = c, hx = c,
x^ + hx = c, x^ + c = hx, x^ = bx-{-c,
but the operations are nevertheless an essential preliminary.
Now what does Diophantos say of the necessary preliminaries
in dealing with an equation? "If the same powers of the un-
known with positive but different coefficients occur on both
sides, we must take like from like until we have one single ex-
pression equal to another. If there are on both sides, or on
either side, negative terms, the defects must be added on both
sides, until the different powers occur on both sides with posi-
tive coefficients, when we must take like from like as before.
We must contrive always, if possible, to reduce our equations so
that they may contain one single term equated to one other.
But afterwards we will explain to you how, when two terms are
left equal to a third, such an equation is solved." (Def. 11.)
Here we have an exact description of the operations called by
the Arabian algebraists aljahr and almukahala. And, as we
said, these operations must have been familiar in Arabia before
the date of Mohammed ibn Musa's treatise. This comparison
would, therefore, seem to suggest that Diophantos was well
known in Arabia at an early date.
Next, with regard to the names used by Mohammed ibn
Musa for the unknown quantity and its powers, we observe that
the known quantity is called the "Number"; hence it is no
matter for surprise that he has not used the word corresponding
to npi6fx6<i for his unknown quantity. He uses shai ("thing")
for this purpose or jidr ("root"). This last word may be a
translation of the Indian mfda, or it may be a recollection of the
DIOPIIANTOS AND THE EARLY ARAIUAN ALGEBRAISTS. 151
pl^t] of Nikomachos. But avg can say nothing with certainty
as to the connection of the three words. For the square of the
unknown he uses mal (translated by Cantor as "Vcrmogen,"
" Besitz," equivalent to "power"), which may very well be a
translation of the Svvafii'i of Diophantos.
M. Bodet comments in his article Ualgehre (VAl-Khh'izmi
(Journal Asiatique, 1878) upon the expression used by Moham-
med ibn Musa for minus, with the view of proving that it is as
likely to be a reminiscence of Diophantos as a term derived
from India \
The most important point, however, for us to examine here
is the solution of the complete (juadratic equation as given by
Diophantos and as given by Mohammed ibn Miisa, The latter
gives rules for the solution of each of the forms of the quadratic
according to his distinction ; and each of these forms we find in
Diophantos. After the rules for the three forms of the complete
quadratic Mohammed ibn Miisa gives geometrical proofs of
them. Now in Greece it was the practice to work out theorems
1 He says (pp. 31, 32) "Le mot dont il se sert pour dtisigner lestermcs d'une
equation affectes du signe -est naqis, qui signifie, comme on le sait, ' manquant
de, prive de ' : un ampute, par exemple, est iwqis de son bras ou de sa jambc ;
c'est done tres-improprement qu'Al-Kharizmi cmploie cette expression ^wur
designer ' la partie eulevec'...Aussi le mot en question n'a-t-il plus c'te cmployo
par ses successeurs, et Behri ed-Din qui, au moment d'cxposer la regie dcs siijncs
dans la multiplication algebriquc, avait dit : ' s'il y a soustraction, on appcllo
CO dont on soustrait zaid (additif), et ce que Ton soustrait naqis (manquant
de), ne nomme plus dans la suite les termes n^gatifs que ' les s^par^s, mis H
part, retranches.'
D'oti vient ce mot 7iaqis? II repond, si Ton vcut, au Sanscrit ii;i<M oa au
pr6fixe vi- au moyen desquels on iudique la soustraction : njekas ou ckoiias veut
dire 'dont on a retranchc,' niais I'adjectif viias se rapportc ici au ' ce dont on
a retranche ' de Beba ed-Din, et non a la quautito rctrancbil-e. Or, lo groc
possede et emploie en langage algebriquc une expression tout A fait ana-
logue, c'est I'adjectif AXi7rr)s, dont Diopbant se sert, par excmplo, pour
definir le signe de la soustraction 71 : xj/ tWtir^i Kdru vtvoy, ' un \f/ iiicomplct
incline vers le bas.' L'arabe, j'en prcnds a temoin tous les arabisauts, tra-
duirait iWinrji par en-m'iqis. Dans I'indication dcs operations algubriquca
Diopbant lit, li la place de son signe -71, if \d^pu : fjiowdSft /3 if \d4^u dpiOfxou
iv6i, dit-il; mot-amot : "2 unites manquant d'une inconune,' pour cxprinier
2-x. Done, s'il est possible qu'Al-Kbarizmi ait emprunt6, sauf I'emploi qu'il
en fait, son HtJ<//.s- au sauscrit uiias, il pourrait tout aussi bicu so fairo iju'il
I'eAt pris au grec iv Xti^et."
152 DIOPHANTOS OF ALEXANDRIA.
concerned witli numbers by the aid of geometry; even in Dio-
phantos we find the geometrical method employed for the
treatise on Polygonal Numbers and a trace of it even in the Arith-
metics, although the separation between geometry and algebra
is there complete. On the other hand, the Indian method was
to employ algebra for working out geometrical propositions, and
algebra reached a far higher degree of development in India
than in Greece, though it is probable that even India was in-
debted to Greece for the first principles. Hence we should
naturally consider the geometrical basis of early Arabian algebra
as a sign of obligation to Greece. This supposition is supported
by a very remarkable piece of evidence adduced by Cantor. It
is based on the letters used by Mohammed ibn Miisa to mark
the points in the geometrical figures used to prove his rules.
The very use of letters in a geometrical figure is Greek, not
Indian ; and the letters which are used are chosen in what
appears to be, at first sight, a strange manner. The Arabic
letters here used do not follow the order of the later Arabian
alphabet, an order depending on the form of the letters and the
mode of writing them, nor is their order quite explained by the
original arrangement of the Arabian alphabet which corresponds
to the order in the other Semitic languages. If however we
take the Arabic letters used in the figures and change them
respectively into those Greek letters which have the same nu-
merical value, the series follows the Greek order exactly, and
not only so, but agrees with it in excluding 5" and t. But what
reason could an Arab have had for refusing to use the particular
letters which denoted 6 and 10 for geometrical figures? None,
so far as we can see. The Greek, however, had a reason for
omitting the two letters 5" and i, the former because it was
really no longer regarded as a letter, the latter because it was a
mere stroke, I, which might have led to confusion. We can
hardly refuse to admit Cantor's conclusion from this evidence
that Mohammed ibn Musa's geometrical proofs of his rules for
solving the different forms of the complete quadratic are Greek.
And it is, moreover, a reasonable iufereuce that the Greeks
themselves discovered the rules for the solution of a complete
quadratic by means of geometry. We thus have a confirmation
Vi
DIOPIIANTOS AND THE EARLY ARABIAN ALfJEBRAlSTS. l.'>3
of the supposition as to the origin of the rules used by Diophan-
tos, which was mentioned above (pp. 140, 141), and we may pro-
perly conclude that algebra, as we find it in Diophantos, was the
result of a continuous development which extended from the
time of Euclid to that of Heron and of Diophantos, and was
independent of external influences.
I now pass to the consideration of the actual rules which
Mohammed ibn Mfisii gives for the solution of the complete
quadratic, as compared with those of Diophantos. We remarked
above (p. 91) that Diophantos would appear, when solving the
equation ax' -\- bx = c, to have first multiplied by a throughout,
so as to make the first term a square, and that he would, with
b
our notation, have given the root in the form — — .
Mohammed ibn Musa, however, first divides by a throughout :
" The solution is the same when two squares or three, or more
or less, be specified ; you reduce them to one single square and
in the same proportion you reduce also the roots and simple
numbers which are connected therewith \" This discrepancy
between the Greek and the Arabian algebraist is not a very
striking or important one; but it is worth while to observe that
Mohammed ibn Musa's rule is not the early Indian one ; for
Brahmagupta (born 598) sometimes multiplies throughout by a
like Diophantos, sometimes by 4a, which was also the regular
practice of (^'rldhara, who thus obtained the root in the form
. This rule of Crldhara's is quoted and followed
2a '
by Bhaskara. Another apparent discrepancy between Moham-
med ibn Musa and Diophantos lies in the fact that Diophantos
never shows any sign, in his book as we have it, of recognising
two roots of a quadratic, even where both roots are positive and
real, and not only when one of them is negative: a negative or
irrational value he would, of course, not recognise ; unless an
equation has a real positive root it is for Diophantos "impossible."
Negative and irrational roots appear to be tacitly ])ut aside by
' Rosen, The Al<jtl>r<i of Muluimtiu-U bfii Miisu, p. '.».
164 DIOPHAIITOS OF ALEXANDRIA.
Mohammed ibn Musa and the earliest Indian algebraists, though
both Mohammed ibn Musii and the Indians recognise the exist-
ence of two roots. The former undoubtedly recognises two roots,
at least in the case where both are real and positive. His most
definite statement on this subject is given in his rule for the
solution of the equation x^ + c = hx, or the case of the quadratic
in which we have "Squares and Numbers equal to Roots; for
instance, ' a square and twenty-one in numbers are equal to ten
roots of the same square.' That is to say, what must be the
amount of a square, which when twenty-one dirhems are added
to it, becomes equal to the equivalent of ten roots of that square?
Solution : Halve the number of the roots ; the moiety is five.
Multiply this by itself; the product is twenty-five. Subtract
from this the twenty-one which are connected with the square ;
the remainder is four. Extract its root; it is two. Subtract this
from the moiety of the roots, which is five ; the remainder is
three. This is the root of the square which you required, and
the square is nine. Or you may add the root to the moiety of
the roots; the sum is seven; this is the root of the square which
you sought for, and the square itself is forty-nine. When you
meet with an instance lohich refers you to this case, try its solu-
tion by addition, and if that do not serve, then subtraction cer-
tainly will. For in this case both addition and subtraction may
be employed, which will not answer in any other of the three
cases in which the number of the roots must be halved. And
know that, when in a question belonging to this case you have
halved the number of roots and multiplied the moiety by itself,
if the product be less than the number of dirhems connected
with the s(piare, then the instance is impossible; but if the pro-
duct be equal to the dirhems by themselves, then the root of
the square is equal to the moiety of the roots alone, without
either addition or subtraction. In every instance where you
have two squares, or more or less, reduce them to one entire
square, as I have explained under the first case\" This defi-
nite recognition of the existence of two roots, if Diophantos
could be proved not to have known of it, would seem to show
' Quoted from The Algebra of Mohavimed ben Musa (ed. Rosen), pp. 11, 12.
DTOPIIANTOS AND THE EARLY ARABIAN ALGEBRAISTS. 155
that Mohammed ibu Musii could here have been indebted to
India only. Rodet, however, remarks that we are not justified
in concluding from the evidence that Diophantos did not know
of the existence of two roots: in the cases where one is negative
we should not expect him to mention it, for a negative root is
for him "impossible," and in certain cases mentioned above (p.
92) one of the positive roots is irrelevant. Rudet further ob-
serves that Mohammed ibn Musa, while recognising in theory
two roots of the equation ,v^+c = bx, uses in practice only one,
and that (curiously enough) in all instances the root correspond-
ing to the sign minus of the radical. This statement however
is not quite accurate, for in some examples of the rule which
we quoted above he gives two possible values \
Mohammed ibn Milsa, being the first writer of a treatise
on algebra, so far as we know, is for obvious reasons the most
important for the purposes of this chapter. If the influence of
Diophantos and Greek algebra upon the earliest Arabian algebra
is once established, it is clearly unnecessary to search so carefully
in the works of later Arabians for points of connection with our
author. For, his influence having once for all exerted itself,
the later developments would naturally be the result of other
and later influences, and direct reminiscences of Diophantos
would disappear or be obscured. I shall, therefore, mention
only a few other Arabian authors, and those with greater
brevity.
§ 3. Abu'1-Wafa Al-Buzjani wc have already had occasion
to mention (pp. 40, 41) as a translator of Diophantos and a
commentator on his work. As then he studied our author so
thoroughly it would be only natural to expect that his works
would abound in reminiscences of Diophantos. On Abu'1-Wafa
perhaps the most important authority is Wopcke. It must suftice
to refer for details to his articles '^
§ 4. An Arabic MS. bearing the date 972 is concerned with
the theory of numbers throughout and particularly with the
formation of rational right-angled triangles. Unfortunately the
• Cf. Rosen's edition, p. 42.
2 Cf. in particular the articles on MalMmntiqucs chez let Arabt-» {Journal
Asiatique for LS.'}")).
156 DIOPHANTOS OF ALEXANDRIA.
beginning of it is lost, and with the beginning the name of the
author. In the fragment we find the problem To find a square
which, when increased or diminished by a given number, is again
a square proposed and solved. Tlie author of the fragment
was undoubtedly an Arabian, and it would probably not be rash
to say that much of it was based on Diopliautos.
§ 5. Again, Abu Ja'far Mohammed ibn Alhusain wrote
a treatise on rational right-angled triangles at a date probably
not much later than 992. He gives as the object of the whole
the investigation of the problem just mentioned. It is note-
worthy (says Cantor) that a geometrical explanation of the
solution of this problem makes use of similar principles to those
which we could trace in Mohammed ibn Musa's geometrical
proofs of the solution of the complete quadratic, and he further
definitely alludes to Euclid II. 7. If we consider the use of
right-angled triangles as a means of finding solutions of this
problem, and c^, c^ be the two sides of a right-angled triangle
which contain the right angle, then c^' + c^ is the square of the
hypotenuse, and c'^ + c^ + '^c^c^ is a square. Hence, says Ibn Al-
husain, c^ -f c'^ is a square which, when increased or diminished
by the same number 'Ic^c^, is still a square. Diophantos says
similarly that " in every right-angled triangle the square of the
hypotenuse remains a square when double the product of the
other two sides is added to, or subtracted from, it." (ill. 22.)
§ 6. Lastly, we must consider in this connection the work of
Alkarkhi, already mentioned (pp. 24, 25). We possess two
treatises of his, of which the second is a continuation of the
first. The first is called Al-Kafi fll hisUb and is arithmetical,
the second is the Fakhrl, an algebraic treatise. Cantor points
out that, when we compare Alkarkhi's arithmetic with that of
certain Arabian contemporaries and predecessors of his, we see
a marked contrast, in that, while others used Indian numeral
signs and methods of calculation, Alkarkhi writes out all his
numbers as words, and draws generally from Greek sources
rather than Indian. The advantages of the Indian notation as
compared with Greek in securing clearness and compactness of
work were so great that we might naturally be surprised to see
Alkarkiii ignoring them, and might wonder that he could have
DIOPHANTOS AND THE EARLY ARABIAN ALGEBRAISTS. 157
been unaware of them or have undervalued them so much.
Cantor, however, thinks that the true explanation is, not that he
was ignorant of the Indian arithmetical methods and notation or
underestimated their advantages, but that Alkarkh! was a repre-
sentative of one of two mathematical schools in Arabia, the
Greek and the Indian. Alkarklii was not the solitary repre-
sentative of the Greek arithmetic; he was not merely an excep-
tion to an otherwise universal acceptation of the Indian method.
He was rather, as we said, a representative of one of two schools
standing in contrast to each other. Another representative of
the Greek school was Abu'1-Wafa, who also makes no use of
ciphers in his arithmetic. Even in Alkarkhi's arithmetical
treatise, as in the works of Abu'1-Wafa, there are not wanting
certain Indian elements. These could hardly by any means
have been avoided, at any rate as regards the matter of their
treatises; but the Greek element was so predominant that, prac-
tically, the other may be neglected.
But the real importance of Alkarkhi in this connection centres
in his second treatise, the Fakhri. Here again he appears as
an admiring pupil of the Greeks, and especially of Diophantos,
whom he often mentions by name in his book. The Fakhri
consists of two parts, the first of which may be said to contain
the theory of algebra, the second the practice of it, or the
application to particular problems. In both parts we find
Diophantos largely made use of Alkarkhi solves in this treatise
not only determinate but indeterminate equations, so that he
may be taken as the representative of the Arabian indeter-
minate analysi.s. In his solutions of indeterminate ccjuations
of the first and second degrees we find no trace of Indian methods.
Diophantos is the basis upon which he builds, but he has also
extended the Greek algebraist. If we refer to the account
which the Italian algebraists give of the evolution of the
successive powers of the unknown quantity in the Arabian
system, we shall see (as already remarked, p. 71, n. 1) that
Alkarkhi is an exception to the adoption of the Indian .system
of generation of powers by the vviltiplicution of indices. He
uses the additive .system, like Diophantos. The square of the un-
ku.iwu bein^ mal, and the cube ka'b, the succeeding powers are
158 DIOPHANTOS OF ALEXANDRIA.
mal mrd, null ha'h, ka'b ka'h, iiuil mal ha'h, rncil ka'h Jca'b, ka.h ka'b
ka'b, &c. Alkarklii speaks of the six forms of the quadratic which
Mohammed ibn MusJi distinguished and explains at the same
time what he understands hy jab?' and mukabala. He appears
to include both processes under jabr, understanding rather by
mukdbala the resulting equation written in one of the six
forms. Among the examples given by Alkarkhi are .r^+10a;=39,
and 0;'^+ 21 = 10a;, both of which occur in Mohammed ibn Miisa.
Alkarkhi has two solutions of both, the first geometrical, the
second (as he expresses it) "after Diophantos' manner." The
second of the two equations which we have mentioned he
reduces to x^ — 10a; + 25 = 4, and then, remarking that the first
member may be either (x — 5f or (5 — xf, he gives the two
solutions x = 7, and x = 2. The remarkable point about his
treatment of this equation is his use of the expression " after
Diophantos' manner" applied to it. We spoke above (p. 92)
of the doubt as to whether Diophantos knew or did not know
of the existence of two roots of a quadratic. But Alkarkhi's
expression " after Diophantos' manner " would seem to settle
this question beyond the possibility of a doubt; and perhaps
it would not be going too far to take his words quite literally
and to suppose that the two examples of the quadratic of which
we are speaking were taken directly from Diophantos. If so,
we should have still more du'ect proof of the Greek origin of
Mohammed ibn Miisa's algebra. On the other hand, however,
it must be mentioned that of two geometrical explanations of
the equation a;'^ + 10^=39 which Alkarkhi gives one cannot
be Greek. In the first of the two he derives the solution directly
from Euclid, ii. 6 ; and this method is therefore solely Greek.
But in the second geometrical solution he employs one line to
represent x'^, another to represent 10j7, and a third to represent
100. This confusion of dimensions is alien from the Greek
manner ; we must therefore suppose that this geometrical
solution is an Arabian product, and probably a discovery of
Alkarkhi himself
As an instance of an indeterminate ecpiation treated by
Alkarkhi we may give the equation mx^ + nx + p = y\ He
gives as a cuudition for the solution that either m or p must ])o
DIOPHANTOS AND THE EARLY ARABIAN ALGEBRAISTS. 150
a square. He then puts for y a binomial expression, of which
one term is either Jlna? or JJ). This is, as we have seen,
exactly Diophantos' procedure.
With regard to the collection of problems, whicli forms the
second part of the Fakhrl, we observe that Alkarklii only
admits rational and positive solutions, excluding even the value
0. In other cases the solution is for Alkarkhi as for Diophantos
" impossible." Many of the problems in indeterminate analysis
are taken directly from Diophantos, and are placed in the order
in which they are there found. Of a marginal note by Alsiraj
at the end of the fourth section of the second part of the treatise
we have already spoken (p. 25).
ADDENDUM.
In the note beginning on p. 6i I discussed three objections urged by
Mr James Gow in his History of Greek Mathematics against my suggestion
as to the origin of the symbol <>) for dpi0/x6s. The second of these objections
asserted that it is of very rare occurrence, and is not found in the >iss. of
Nikomachos and Pappos, where it might most naturally be expected. In reply
to this, I pointed out that it was not in the least necessary for my theory that
it should occur anywhere except in Diophantos ; and accordingly I did not
raise the question whether the symbol was found in mss. so rarely as Mr Gow
appears to suppose. Since then I have thought that it would be interesting to
inquire into this point a little further, without, however, going too far afield.
■\;\liile reading Heiberg's Quaestiones Archimeikae in connection with the Cattle-
problem discussed in chapter viii. it occurred to me that the symbol for apiOnos
would be likely to be found, if anywhere, in the mss. of the De areiiae mimero
Ubclhts of Archimedes, which Heiberg gives at the end of the book, and that, if
it did so occur, Heiberg's textual criticisms would i^lace the matter beyond
doubt, without the necessity of actually collating the mss. My expectation
proved to be fully justified ; for it is quite clear that the symbol occun-cd in the
MSS. of this work of Archimedes rather frequently, and that its form had given
rise to exactly the same confusion and doubt as in the case of Diophantos. I
will here give references to the places where it undoubtedly occurred. See the
following pages in Heiberg's book,
p. 172.
p. 174. Heiberg reads dpL6fwv, with the remark "/cat omnes." But the
similarity of the signs for api6fi6s and Kai is well known, and it
could hardly be anything else than this similarity which could
cause such a difference of readings.
p. 187. Heiberg's remark " apiOfxiov om. codd. Bas. E ; excidit ante s (Kai)"
speaks for itself. Also on the same page "dpiOnwu] si FBC."
p. 188. js three times for dpiBixuv.
p. 191. Here there is a confusion between 5" (six) and dpiOfxis, where
Heiberg remarks, "Error ortus est ex compendio illo uevhi dpiOfios,
de quo dixi ad I, 3."
p. 192. iXaTTuv and dpid/xos given as alternative readings, with the obser-
vation, " Confusa sunt compendia."
Thus it is clear that the symbol in question occurs tolerably often in the mss.
of another arithmetical treatise, and that the only one which I have investigated
in this connection : a fact whicli certainly does not support Mr Gow's statement
that it is veiy rarely found.
APPENDIX.
ABSTRACT OF THE ARITHMETICS AND THE
TRACT ON POLYGONAL NUMBERS.
H. D.
u
J
DIOPHANTOS. ARITHMETICS.
BOOK I.
Introduction addressed to Dionjjsios.
Definitions.
1. "Square" and "side," "cube," "square-square," etc.
2. " Power." Notation 8", k", 88", Sk", kk", /a", c^.
3. Corresponding fractions, the reciprocals of the foi-mer ; names
used corresponding to the " nimibers."
4. "]S'umber"x"Number"= square. Square x square = "squarc-
squai-e," &c.
5. " Number " x corresponding fraction = unit (/xoi-ds).
6. "Species" not changed by multiplication with monads.
* I Reciprocal x reciprocal - reciprocal scpiare, etc.
9. Minus multiplied by minus gives j^lus. Notation for minus, /p.
10. Division. Remark on familiarity with processes.
11. Simplification of equations.
Frohlenis.
1. Divide a given number into two having a given difference.
Given nund)er 100, given difference 10.
Lesser number required x. Therefore
2.«+ 40-100,
X - 30.
The required numbers an- 70, 30.
2. To divide a given number into two having a given nitio.
Given number GO, given ratio ."3 : 1.
Two numbers x, 2>x. Therefore ./j- ir».
Till" numbers arc 45, 15.
11—2
164 DIOPHANTOS OF ALEXANDRIA,
3. To divide a given number into two having a given ratio and
diffei-ence '.
Given number 80 ; ratio 3:1; diflference 4.
Smaller number x. Therefore the larger is 3x + 4, x = 1 9.
The numbers are 61, 19.
4. Find two numbers in a given ratio, their difference also being
given.
Given ratio 5:1. Diffei'ence 20.
Numbers 5x, x. Therefore x-6, and the numbers are 25, 5.
5. To divide a given number into two such tliat the sum of
given fractions (not the same) of each is a given number.
Necessary condition. The latter given number must lie between
the numbers arising when the given fractions are taken of the first
given number.
First given number 100, given fractions ;^ and -, given
sura 30.
Second part 5a-. Therefore first part = 3 (30 - x).
Therefore 90 + 2a; = 100, x = 5.
The required parts are 75, 25.
6. To divide a given number into two parts, such that a given
fraction of one exceeds a given fraction of the other by a given
difierence.
Necessary condition. The latter number must be less than that
which arises when that fraction of the first number is taken which
exceeds the other fraction.
Given number 100 ; fractions - and -respectively : excess 20.
4 G
Second part Gx Therefore 10.v+80=100, x = 2, and the
parts are 88, 12.
7. From the same (rcqiiired) number to take away two given
numbers, so that the remainders are in a given ratio.
Given numbers 100, 20; ratio 3:1.
X required number. Therefore
a;- 100 : a- 20 = 1 : 3, .«;^ 140.
8. To two given numbers to add tlie same (required) number, so
that the sums ai-e in a given ratio.
1 By thiH Diojihantos means " such that one is so many times the other ylm
a given number."
ARITHMETICS. BOOK I. 165
Cotulilion. This ratio must bo less tlmu that of the greater given
number to tlie smaller.
Given numbers 100, 20, given ratio 3:1.
X required number. Therefore
3a; + 60 = a: +100, and x=20,
9. From two given numbers to subtract the same (required) one
so that the two remainders are in a given ratio.
Condition. This ratio must be greater than that uf the greater
given number to the smallei'.
Given numbers 20, 100, ratio 6 : 1.
X required number. Therefore
1 20 - 6x- = 100 - X, and a; - 4.
10. Given two numbers, to add the same (required) number to
the smaller, and subtract it from the lai-ger, so that the sum in
the first case may have to the difference in the second a given
ratio.
Given numbers 20, 100, given ratio 4:1.
X required number. Therefore
20 + x- : 100-cc = 4 : 1, and a; = 76.
11. Of two given numbers to add the first to, and subtract the
second from, the same (required) number, so that the numbers which
arise may have a given ratio.
Given numbers 20, 100 respectively, ratio 3:1.
X required number. Therefore
3a; - 300 = a; + 20, and x = 160.
12. To divide a given number twice into two parts, such that
the fii-st of the first pair may have to the first of the second a given
ratio, and also the second of the first pnir to the second of the second
another given ratio.
Given number 100, ratio of greater of fii-st parts to less of
second 2:1, ratio of greater of second parts to less of
first 3:1.
X smaller of second parts. The parts then are
, J"" . 1 'int^ ^^^ ~ ^■" [ • Therefore 300 - 5x- = 1 00, x - 40,
100- 2a; j X J
and the parts are (80, 20), (60, 40).
13. To divide a given number thrice into two parts, such that one
of the first parts and one of the second pai-ts, the other of the second
166 DIOPUANTOS OF ALEXANDRIA.
parts and one of the third parts, the other of the tliird parts and the
remaining one of the first parts, are respectively in given ratios.
Given number 100, ratio of gi-eater of first parts to less of
second 3 : 1, of greater of second to less of third 2:1,
and of greater of third to less of first 4:1.
X smaller of third parts. Therefore greater of second = 2x,
less of second = 100 - 2x, greater of first = 300 - Gx.
Therefore less of first = Gx- 200.
Hence greater of third = 2ix - 800.
Therefore 25^-800 = 100, a; =36,
and the respective divisions are (84, 16), (72, 28), (64, 36).
li. To find two numbers such that their product has to their
sum a given ratio. [One is arbitrarily assumed subject to the]
Condition. The assumed value of one of the two must be greater
than the numerator of the ratio [the denominator being 1].
Ratio 3:1. x one number, the other 12 ( > 3). Therefore
12a; = 3a; +30, a; = 4,
and the numbers are 4, 12.
15. To find two numbers such that each after receiving from the
other a given number may bear to the remainder a given ratio.
Let the first receive 30 from the second, ratio being then 2:1,
and the second 50 from the first, ratio being then 3:1.
cc 4- 30 the second. Therefore the first = 2x - 30,
and a; + 80 : 2a;- 80=3 : 1.
Therefore x = 64, and the numbers arc 98, 94.
16. To find three numbers such that the sums of each pair are
given numbers.
Condition. Half the sum of all must be greater than any one singly.
Let (1) + (2) = 20, (2) + (3) = 30, (3) + (l) = 40.
x the sum of the three. Therefore the numbers are
X - 30, X - 40, X - 20.
Hence the sum x = 3x' - 90. Therefore a; = 45,
and the numbers ai-e 15, 5, 25.
17. To find four numbers such that the sums of all sets of three
are given.
Condition. One third of the sum of all must be greater than
any one singly.
Sums of threes 22, 24, 27, 20.
J
ARITHMETICS. HOOK I. 107
X the sum of all four. Therefore the mimhera uro
.T-22, a; -24, a; -27, a; -20.
Therefore 4x- - 93 = x, .r = 31 ,
and the numbers are 9, 7, 4, 11.
18. To find three numbers such that the sum of any puir
exceeds the third by a given number.
Given excesses 20, 30, 40.
2x sum of all, x = 45. The numbers arc 25, 35, 30.
19. [A different solution of the foregoing problem.]
20. To find four numbers such that the sum of any three exceeds
the fourth by a given number.
Condition. Half the sum of the four given dilterences must be
greater than any one of them.
Given differences 20, 30, 40, 50.
2a; the sum of the four required numbers. Tlierefoi-c the
numbers are
a;- 10, a;- 15, x-20, x-2b.
Therefore 4x- 70 = 2.f, and .^•= 35.
Therefore the numbers are 25, 20, 15, 10.
21. [Another solution of the foregoing.]
22. To divide a given number into three, such that the sum
of each extreme and the mean has to the remaining extreme a given
ratio.
Given number 100 i (1) + (2) = 3 . (3), (2) + (3) 4.(1).
X the third. Hence the sum of first and second = 3a-. There-
fore 4a; =100.
X = 25, and the sum of the first two = 75.
y the first '. Therefore (2) + (3) = 4v/. Therefore by = 100,
y = 20. The required parts are 20, 55, 25.
23. To find three numbers such that the greatest exceeds the
middle number by a given fraction of the least, the middle exceeds
the least by the same given fraction of the greatest, but the least
exceeds the same given fraction of the middle number by a given
number.
1 As already remarked on pp. 80, 81, Diophantos docs not uso a second
syllable for the uuknowu, but uses dpiO/xos for the second oiHiratiou as well ob
for the lirtit.
168 TJlOPHANTOS OF ALEXANDRIA.
Condition. The middle number must exceed the least by such a
fraction of the greatest, that if its denomiuator be multiplied into
the excess of the middle number over the least, the result is gi-eater
than the middle number.
Greatest exceeds middle by ^ of least, middle exceeds least by
o
of greatest, least exceeds ^ of middle by 10.
aj + 10 the least. Therefore middle = 3aj, greatest = 6a; -30.
Therefore x- 12^,
and the numbers are 45, 37^, 22|^.
24. [Another solution of the foregoing.]
25. To find three numbers such that, if each give to the next
following a given fraction of itself, in order, the results after each has
given and taken may be equal.
Let first give ^ of itself to second, second . of itself to third,
third - of itself to first.
5
Assume the second to be a number divisible by 4, say 4.
3a; the first, and x = '2. The numbers are 6, 4, 5.
26. Find four numbers such that, if each give to the next
following a given fraction of itself, the results may all be equal.
Let first give ^ of itself to second, - , -, ^ being the other
fractions.
Assume the second to be a multiple of 4, say 4.
3a; the first. The second after giving and taking becomes a; + 3.
Therefore first after giving x to second and receiving of
fourth = a; + 3.
Therefore fourth =18- 6a;. And fourth after giving 3 - a;
to first and receiving r^ of third = a; + 3. Therefore third
= 30a; - 60.
Lastly the third after giving 6.x- - 1 2 to fourth and receiving
1 from second = a; + 3. Therefore
50
24a; - 47 - a; + 3, a; - ^o •
ARITHMETICS. ]J0OK I. 100
m, I- *i 1 1^0 , 120 111
Therefore the numbers are ^ , 4, -^ * ~ok '
or, multiplying by the common denominator, 150, 92, 120, 114.
27. To find three numbers such that, if each receives a given
fraction of the sum of the other two, the results are all equal.
The fractions being , -r, _ , the sum of the second and third
is assiimed to be 3, and x put for the first.
The numbers are, after multiplying by a common denominator,
13, 17, 19.
28. To find four numbers such that, if each receives a given
fraction of the sum of the remaining three, the four results are
eqiial.
The e;iven fractions being 77 , 7 , ^ , ;r , we a^suvie the sum of
° ° 3 4 6
the last three numbers to be 3.
Putting X for the first, Diophantos finds in like manner that
numbers are 47, 77, 92, 101.
29. Given two numbers, to find a third which, when multiplied
by each successively, makes one product a square and the other the
side of that square.
Given numbers 200, 5.
X required number, 200a; = (oxY, a; = 8.
30. To find two numbers ivhose sum and whose product are
given.
Condition. The square of ludf the sum 7nust exceed tlie product
by a square number, Iuti Se tovto irXaaixaTiKov '.
Given sum 20, product 9G.
2x the difierence of the required numbers.
Therefore numbers are 10 + x, 10 -x.
Hence 100-aj'=96.
Therefore x ^ 2, and the difierence = 4. The required numbei-s
are 12, 8.
1 There has been much controversy as to the moaninR of this diflicult
phrase. Xylancler, the author of the Schoha, Bachct, Cossali, Schulz, NcbhoI-
mann, all discuss it. As I do not profess here to bo commcntinR on the
text I shall uot criticise their respective views, hut ouly remark tliat I think
it is best to take TrXaff/xariKov in a passive sense. "And this condition can
(easily) be formed," i.e. can be investigated (and shown to bo tnu), or </i.«.
covered.
170 DIOPIIANTOS OF ALEXANDRIA.
31. 2^0 find two numbers, luiving given their sum and the sian
of their squares.
Condition. Double the sum of the squares must exceed the square
of their sum by a sqxiare, Icrri Se KaX tovto irXaa-fxaiLKov.
Sum 20, sum of squares 208.
2x the difference.
Therefore the numbers arc 10 + .r, 10 - cc.
Thus 200 + 2x- = 208. Hence x = 2,
and the numbers are 12, 8.
32. To find two numbers, having given their sum and the
difference of their squares.
Sum 20, difference of squares 80.
2x difference of the numbers,
and we find the numbers 12, 8.
33. I'ofind two munbcrs whose difference and product are given.
Condition. Four times the 2)roduct together with square of differ-
ence must produce a complete square, Icttl Se koI tovto TrXao-ynariKo'v.
Difference 4, product 96.
2x the sum. Therefore the numbers are found to bo 12, 8.
31. Find two numbers in a given ratio sucli that the sum of
their squares is to their sum also in a given ratio.
Ratios 3 : 1 and 5 : 1 respectively.
X lesser number, x—1] the numbers are 2, 6.
35. Find two numbers in a given ratio such that the sum of
their squares is to then- difference in a given ratio,
Ratios being 3 : 1, 10 : 1, the numbers are 2, 6.
36. Find two numbers in a given ratio such that the difference
of their squares is to their sum in a given i*atio.
Ratios being 3 : 1 and 6:1, the numbers arc 3, 9.
37. Find two numbers in a given i\atio such that the difference
of their squares is to their difference in a given ratio.
Ratios being 3 : 1 and 12 : 1, the numbers are 3, 9.
Similarly by this method can be found two numbers in a given
ratio (1) such tliat their product is to their sum in a given ratio, or
(2) such that their product is to their diflerence in a given ratio.
38. To find two numbers in a given ratio such that the square
of the smaller is to the larger in a given ratio.
Ratios 3 : 1 and 6:1. Numbers 54, 18.
ARITHMETICS. BOOK I. 171
39. To find two numbers in a given ratio such that tliu square
of the smaller is to the smaller itself in a given ratio.
Ratios 3 : 1 and G : 1. Numbers 18, 6.
40. To find two numbers in a given ratio such that the sf^uaro
of the less has a given ratio to the sum of both.
Ratios 3:1,2:1. Numbers 24, 8.
41. To find two numbers in a given ratio such that the square
of the smaller has a given ratio to their diUcrcncc.
Ratios 3 : 1 and G : 1. Numbers 36, 12.
42. Similarly can be found two numbei-s in a given ratio,
(1) such that square of larger has a given ratio to the smaller.
(2) such that square of larger has to larger itself a given i-atio,
(3) such that squai'e of larger has a given ratio to the sum or
difference of the two.
43. Given two numbers, to find a tliird such that the sums of
the several paii's multiplied by the corresponding third give three
numbers in A. p.
Given numbers 3, 5.
X the required number. Therefore the three expressions are
3a; +15, 5a; +15, 8a;.
Now 3a; + 15 must be either the middle or the least of the
three, 5a; + 15 cither the greatest or the middle.
15
(1) 5a;+ 15 greatest, 3x+ lo least. Therefore x=- -^ •
15
(2) 5x+ 15 greatest, 3a;+ 15 middle. Therefore x^ ^ .
(3) 8a; greatest, 3a; + 15 least. Therefore x = 15.
172 DIOPHANTOS OF ALEXANDRIA.
BOOK 11.
[The first five questions of this Book arc identical with questions in
Book I. In each case the ratio of one required number to the
other is assumed to be 2 : 1. The enunciations only are here
given.]
1. To find two numbers whose sum is to the sum of their
squares in a given ratio.
2. Find two numbers whose difference is to the difference of
their squares in a given ratio.
3. Find two numbers whose product is to their sum or difference
in a given ratio.
4. Find two numbers such that the sum of their squares is to
the diffei'ence of the numbers in a given ratio.
5. Find two numbers such that the difference of their squares is
to the sum of the numbers in a given ratio.
6. Find two numbers having a given difference, and such that
the difference of their squares exceeds the difference of the numbers
themselves by a given number.
Condition. The square of their difference must be less than the
sum of the two given differences.
Difference of numbers 2, the other given number 20.
X the smaller number. Therefore x + '2 is the larger and
4a; + 4 = 22.
a; = 4|, and the numbers ai*e 4|, 6 J.
7. Find two numbers such that the diflTerence of their squares
may be greater than their diflTerence by a given number and in a
given ratio (to it)'. [Difference asswned.]
Coiulition. The ratio being 3:1, the square of tlie difference
of the numbers must be < sum of three times that difference and the
given number. ui.
' By this Diophautos mcaiiK "may exceed a given proportion or fractio f
it by a given number. "
ARITHMETICS. BOOK II. 17:^
Given number 10, difference of nunibei-s required 2.
X the smaller number. Therefore the hirger x + 2,
and 4a; + 4 = 3 . 2+10.
Therefore a; = 3,
and the niimbei-s are 3, 5.
8. To divide a square mnnber into two squares.
Let the square number be IG.
x^ one of the required squares. Therefore IG-.r* must be
equal to a square.
Take a square of the form' {nix - 4)-, 4 being taken as the
absolute term because the square of 4 = 1 G.
i.e. take (say) (2a; -4)* and equate it to 16 -a;'.
Therefore 4.'c- - 1 Gx = - x'.
Therefore a^ = "5" ,
5 '
^.v 'A 256 144
and the squares required are -^ , -^ .
9. [Another solution of the foregoing, practically equivalent.]
10. I'o divide a number which is the sum of two squares into two
other squares.
Given number 13 = 3'+2^
As the roots of these squares are 2, 3, take [x + 2)* a.<? the first
square and {mx - 3)" as tlie second required, say (2x- - 3)*.
Therefore (x + 2 )^' + (2a; - 3)^' = 1 3 .
324 1
Therefore the required squares are -^^ > 05 "
11. To find two square numbers diflWing by a t/iven ntimber.
Given difference GO.
Side of one number x, side of the other x phis any number
whose square < GO, say 3.
Therefore (x- + 3)^' - a;' = 60,
and the required .squares are 72], 132 j.
» Diophantos' words are: "I form the s(iuare from any number of apiBnol
minus as many units as are contained in the side of U)." The prt-eftution im-
plied throuRhout in the choice of m is that we must assume it so tli-ii tli. r. suli
may be rational in Diophantos' sense, i.e. rational and positive.
174 DIOPHANTOS OF ALEXANDRIA.
12. To add such a number to each of two given numbers that
the results shall both be squares.
(1) Given numbers 2, 3, required number x.
x + 2)
Therefore ^M^^^^st each be squares.
x+ o) ^
This is called a double-equation.
To solve it, take the difference between them, and resolve it into
tioo factors^ : in this case say 4 and -. .
Then take either
(a) the square of half the difference bettveen these factors
and equate it to the smaller expression,
or (b) the sqriare of half the sum and equate it to tlie larger.
225
In this case (a) the square of half the diflference = -^ .
m r o 225 - 97
Iheretorc x+ ^= yrr > ^^^ ^ — wr,
o4 d4
... ^. 225 289
Avhilc the squares are -rr- , -ttt •
(2) In order to avoid a double-equation,
First find a number which added to 2 gives a square, say x^ — 2.
Therefore, since the same number added to 3 gives a square,
x' + 1= square = {x- 4y say,
the absolute term (in this case 4) being so chosen that the
solution may give x^>2.
15
Therefore x=-^ ,
97
and the required number is ^ , as before.
13. Fro77i two given numbers to take the same (required) number
so tluit both the remainders are squares.
Given numbers 9, 21.
Assuming 9 - x^ as the required number we satisfy one condi-
tion, and it remains that 12 + a;" = a square.
Assume as the side of this square x minus some number whose
square > 12, say 4.
1 Wc must, as usual, choose suitable factors, i.e. such as will give a "ra-
tioual" result. This must always be premised.
ARITHMETICS. HOOK II. 175
Therefore (x - 4)^ = 1 2 + x\
1
^ = 2'
and the required number is 8|.
14. Fro7n the same {required) number to subtract sui-cessiv:/;/ two
given munbers so that the remainders may both be squares.
6, 7 the given numbers. Tlien
(1) let cc be required number.
Tlierefore cb - G~| , ,
^ \ are both squares.
The difference = 1, which is the product of 2 and .^ ;
and, l)y the rule for solving a double-equation,
121
(2) To avoid a double-equation seek a number which exceeds
a square by 6,
i. e. let a;" + G be the required number.
Therefore also x- - 1 : square = {x- 2)' say.
Hence ^ ~ I '
and the number required = tr •
15. To divide a given number into tioo ])arts, and tojind a square
number tohich when added to either of the two parts gives a square
number.
Given number 20. Take two numbci-s the sum of whose
squares < 20, say 2, 3. Add x to each and square.
"We then have x^ + 4a; + 41
X' + Gx + 'd) '
and if 4.x- + 4]^
6a; + 9j
are respectively subtracted the remaindei-s are the same s<|uiire.
Let then x- be the square required,
and therefore 4a; + 4"|
Gx + Oj
the required parts of 20.
Then 10X-+13-20,
and X ^ iV, .
176 DIOPHANTOS OF ALEXANDRIA.
(68 132\
iTi ' ~i7r ) '
49
and the required square rrrrr .
16. To divide a given number into two parts and find a square
which exceeds either part by a square.
Given number 20.
Take (x + rtif for the required square, where m' < 20,
i.e. let (x + 2Y be the required sqi;are (say).
This leaves a square if either 4a; + 4) . , ^ ^ ,
^ ^ „ ?■ 1*5 subtracted,
or 2.r + 3J
Let these be the parts of 20,
and x=-7, .
b
/76 44\
Therefore the parts required are (-w, /. ) >
.625
and the required square is -^^ .
ob
17. Find two numbers in a given ratio such that cither together
with an assigned square produces a square.
Assigned square 9, ratio 3:1.
If we take a square whose side is mx + 3 and subtract 9 from
it, the remainder will be one of the numbers required.
Take e. g. (x + 3)^ - 9 = cc^ + 6a; for the smaller number.
Therefore 3a;* + 18a; = the larger number,
and 3x* + 18a; + 9 must be made a square - (2.x-- 3)" say.
Therefore x = 30,
and the required numbers are 1080, 3240.
18. To find three numbers such that, if each give to the next
following a given fraction of itself and a given number besides, the
results after each has given and taken may be equal.
First gives to second - of itself + 6, second to third ^ of itself
i> b
+ 7, third to first = of itself + 8.
Assume that the first two are 5x, Gx [equivalent to one con-
,.,. -, , ,. , ,, , ^ - 90 108 105
ditionj, aiul we find the numbei"s to be - , - , -=- .
ARITHMETICS. BOOK H. 177
19. Divide a number into throe parts .sati-sfyinfj tlio conditions
of tlie preceding problem.
Given number 80. First gives to second p of itself + G ic,
and results are equal.
[Diophantos a.ssumes 5x, 12 for tlie first two nunilM-rs, and his
,^ . 170 228 217 ,
result IS -y^- , — , — ; but the solution does not cor-
respond to the question.] (See p. 2.5.)
20. To find three squares such that the difference of the great-
est and the second is to the diffei-ence of the .second and the lea.st in
a given ratio.
Given ratio 3:1.
Assume the least square ^- .r^, the middle = .r* -I- 2x -f 1.
Therefore the greatest = x° + 8x + 4 = square = {x + SYsay.
Therefore x ^ '■ ,
and the squares are 30|, 12^, 6;^.
21. To find two numbers such that the square of either added
to the other number is a square.
X, 2a; -1- 1 are assumed, which by their form satisfy one con-
dition. The other condition gives
4x- + 5x + \ ^ square = (2.c - 2)" say.
3
Therefore ,-« = --,
3 19
and the numbers are :^ , ^^ .
Id lo
22. To find two numbers such that the square of eitlier utitntu
the other number is a square.
a; -f- 1 , 2.r -f- 1 are assumed, satisfying one condition.
Therefore 4.7;^ -i- 3x - square - 9,r* say.
3
Therefore -^ = ? >
and the numbers are ^ . c •
5
23. To find two numbers such that the sum of the square of
either and the sum of both is a square.
Assume x, x+l for the numbers. These satisfy one condi-
tion.
H. D. 12
178 DIOPnANTOS OF ALEXANDRIA.
Also a;* + ix + 2 must be a square = {x - 2)* say.
Therefore x = -, .
4
1 5
Hence the numbers are t , ^ .
4 4
24. To find two numbers such that the clifTerence of the square
of either and the sum of both is a square.
Assume x + 1, x for the numbers, and we must have
a;^ - 2a; - 1 a square = {x- 3)- say.
Therefore x=^,
and the nvimbei's are 3|, 2^.
25. To find two numbers such that the sum of either and the
square of their sura is a square.
Since 03" + 3x^, x^ + 8x' are squares,
let the numbers be 3x-, Sx- and their sum x.
Therefore 1 Icc^ = a; and a; = yy .
3 8
Therefore the numbers are y^ , ynT •
26. To find two nunibers such that the difference of the square
of the sum of both and either number is a square.
If we subtract 7, 12 from 16 we get squares.
Assume then 12a;', 7x^ for the numbers, 1 6a;" = square of sum.
4
Therefore Idoif = ix, x = ^1) f
192 112
and the numbers are ^„^ , ^r^^ .
obi obi
27. To find two numbers such that the sum of either and their
product is a square, and the sum of the sides of the two squan s
so arising equal to a given number, 6 suppose.
Since x (4a; - 1) + a; = square, let a;, 4a; - 1 be the numbers.
Therefore 4a;* + 3a; - 1 is a square, whose side is 6 - 2a;.
37
Therefore x = --,
in 1 37 121
and the numbers are - , .
^1 til
28. To find two numbers sndi that tlu; difference of their pro-
duct and either is a square, and the sum of the sides of the two
squares so arising equal to a number, 5.
ARITIIMKTICS. ROOK II. 17!»
Assume 4a; + 1, x for tlie nuinheis, wliicli tlicit.'foie satisfy on*-
condition.
Also 4.V* - 3x- - 1 - {') - 2,<f . Thoivforc .r : "''
17'
26 121
and the numbers are - ,
29. To find two square numhrrs such that the sum of the product
and either is a square.
Let the numbers ' be x", y".
Therefore
o „ ..\ are both snuares.
a;V + .r-j '
To make the first a square we make x" + 1 a square, putting
x^ + 1 = (x - 1)-. Therefore x = '-.
4
We have now to make y— (_j/^ + 1) a square [and y must be
different from x\.
Put 9/ + 9 = (3^ - 4)' sav.
Therefore y=-^-:.
9 49
Therefore the numbers are .— , ^^ .
lb 0/b
30. To find two square numbers such that the difference of their
pi*oduct and either is a square.
Let a;', y' be the numbers.
Therefore aJ*2/^ — '/) , ,,
2 2 zf ^^^ ^^<^Vi\ squares.
X y — X )
, 2;")
A solution of x? -\ ^ square is re ^ .^ ,
and a solution of y- - 1 = square is _?/= — .
m, . , 1 25 289
Therefore the numbers are , „ , -^ . •
lb 04
31. To find two numbers such that their product ± their sum
gives a square.
1 DIophantos docs not use two unknowns, but assumes the numl)crs to bo
x"- and 1 until ho has found x. Then he uses the same unknown to find wliat be
had first called unity, as explained above, p. 81. The same remark applies to tho
next problem.
12-2
180 DIOPHANTOS OF ALEXANDRIA.
a' + ^= ± 2ab is a square. Put 2, 3 for a, b, and 2- + 3' ± 2 . 2 . 3
is a square. Assume then product = (2' + 3") .x*^ = 13a;",
the numbers being x, 1 3x, and the sum 2 . 2 . 3aj° or 1 2x-°.
7
Thei-efore 14a; = 1 2x^, and x= ^.
7 91
Therefore the numbers are ^ , -^ .
b
32. To find two numbers whose sum is a square and ha\dng the
same property as the numbers in the preceding problem.
2 . 2m . m = square, and 2m\^ + «i|^ ± 2 . 2m. in = square.
Ifw-2, 4'+2'±2.4.2 = 36or 4.
Let then the product of numbers be (4' + 2') x^ or 20a;' and
their sum 2 . 4 . 2x- or 16a;*, and let the numbers be 2a;, lOx.
3
Therefore 1 2a; = 1 Qx', x = -,
6 30
and the numbers are ^ , -t-.
4 4
33. To find three numbers such that the sum of the square of
any one and the succeeding number is a square.
Let the first be x, the second 2a; + 1, the third 2 (2a'+ 1) + 1
or 4a; + 3, so that two conditions are satisfied.
Lastly (4a; + 3)' + x = square = (4a; - 4)" say.
7
Therefore a; - ^^ ,
57
7 71 199
and the numbers are _^ , ^ , -^ .
34. To find three numbers such that the difieronce of the square
of any one and the succeeding number is a square.
Assume first a; + l, second 2a: +1, third 4a; + 1. Therefore
two conditions are satisfied, and the third gives
1 Ga;' +7x = square = 25a;* say.
7
Therefore "^ ^ n >
16 23 37
and the numbers are n , -q- , -q •
35. To find three numbers such that, if the square of any one be
added to the sum of all, the result is a square.
/7>i-n\ ^ ^^^^ .^ ^ square. Take a number soparublo into
ARITHMETICS. BOOK 111. IM
two factors (m, u) in three ways, say 12, whicli is tlio pro
duct of (1, 12), (2, 6), (3, 4).
The values then of — ^ — are 5.\, 2, - .
Let now o\x, 2x, ^x be the numbers. Their sum is 12.r'.
Therefore So; = 1 2.t*, a: = !^ ,
and the numbei-s are - , ^ , q •
O ij <J
30. To find three numbers such that, if the sum of all be sub-
tracted from the square of auy one, the result is a square.
— ^r— J —mn is a square. Take 12 as before, and let G.^x,
Ax, Zlx be the numbers, their sum being \'2x*.
Therefore x = - ,
G
, ,, , 91 28 49
and the numbers are vt, , - , ,^^ .
12' 6 ' 12
BOOK III.
1. To find three numbers such that, if the square of any one be
subtracted from the sum of all, the remainder is a square.
Take two squares af, 4a;* whose sum = 5x".
Let the sum of all three numbers be 5.*;^, and two of the numbers
X, 2x. These assumptions satisfy two couditions.
4 121
Next divide 5 into the sum of two squares [ii. 10] ^-^, -^,
2
and assume that the third nuuiber is t x.
2 17
Therefore x+2x+ ^x = 5x^ Therefore a; = ,, . ,
17 34 34
and the numbers are — , , ^-^ •
2. To find three numbers such that, if the scjuare of the sum be
added to any one of them, the suui is a s(|uan'.
L^^ =
ix,x =
l7'
28
289
48
' 289
60
' 289
182 DIOPHANTOS OF ALEXANDRIA.
Let the square of the sum be x', aud the numbers 3x', Sx", 15a;'.
Hence 2Gx^ - x, sc = ^ ,
and the numbers are ^^ , g^g , ^.
3. To find three numbers such that, if any one be subtracted
from the square of their sum, the result is a square.
Sum of all 4a;, its square 16a;", the numbers 7x^, 12.v*, \5x'.
Therefore
and the numbers are
4. To find three numbers such that, if the square of their sum be
subtracted from any one, the result is a square.
Sum X, the three numbers 2a;'', 5x^, lO.f'^.
Therefore x = ^r^ ,
It
and the numbers are ir^r^ , r-r^ , ^r—z^ .
289' 289' 289
5. To find three numbers such that the sum of any pair exceeds
the third by a square, and the sum of all is a square.
Let the sum of the three be (x+l)"; let first + second=third+ 1,
x^
so that third .- '^ + a; ; let second + third - first + x",
1 a;* 1
so that first = a; + ^ . Therefore second ='^ + ^ .
But first + third = second + square, therefore 2.*; = square = 16,
suppose. Therefore x = S, and (8^, 32^, 40) is a solution.
6. [The same otherwise.]
First find three squares whose sum is a square. Find e.g.
what square number + 4 + 9 gives a square, i.e. 36.
Therefore (4, 36, 9) are such squares.
Next find three numbers such that sum of a pair = third + given
number, say, first + second- third = 4, second + third - first
= 9, third + first - second = 36, by the previous problem.
7. To find three numbers whose sum is a square, and such that
the sum of any pair is a square.
ARITHMETICS. BOOK III. 183
Let the sum be a;* + 2.«+l, sum of fh-st and sccoml x*, and
therefore the third 2a; + 1 ; let second + third = (x- 1)*.
Therefore the first is 4a:, and therefore the secontl x* - 4x.
But first + third = square, or G.t- + 1 = square -Vl\ say.
Therefore s; = 20,
and t]ie numbers are (80, 320, 41).
8. [The same otherwise.]
9. To find three numbers in A. p. such that the sum of any pair
is a square.
First find three square numbers in A. p. any two of which are
together > the third. Let x^, {x + 1 )* be two of these ;
therefore the tliird is a;* + 4a; + 2 = (a; - 8)* say.
31
Therefoi-e x - ...,
or we may take as the squares 961, 1G81, 2401.
We have now to find three numbers, the sum.s of pairs being
these numbers.
Suin of the three = ^-^ = 252 U-,
and we have all the three numbers.
10. Given one number, to find three others such that the sura of
any pair of them and the given number is a square, and also the sura
of the three and the given number is a square.
Given number 3. Suppose first + second = x* + 4a; + 1, second
+ third - x" + G.f + 6, sum of all three = x* + 8x + 13.
Therefore third = 4.« + 1 2, second = x-° + 2x - 6, first = 2x + 7.
Also third + first + 3 = square, or 6x+22 = squai-e=K)0sui)i)Ose.
Therefore o: = 1 3,
and the numbers are 33, 189, 64
11. Given one number, to find three othci-s such that, if the
given number be subtracted from the sum of any pair of them or
from the sum, the results are all squares.
Given number 3. Sum of first two x* + 3, of next pair
x' + 2x + 4, and sum of the three x* + 4x + 7. Therefore
third = 4x + 4, second = x* - 2x, first = 2x + 3. Therefore,
lastly, 6x + 4 = square -- 64 say. Therefore x=10, and
(23, 80, 44) is a solution.
184 DIOPHANTOS OF ALEXANDRIA.
12. To find three numbers such that the sum of the product of
any two and a given nmnher is a square.
Let the given number be 12. Take a square (say 25) and sub-
tract 12. Take the difference (13) for the product of the
first and second numbers, and let these numbers be ISrc, - .
X
Again, subtract 1 2 from another square, say 1 6, and let the
diffei-ence 4 be the product of the second and third
numbers. Therefore the third number = ix.
Hence the third condition gives 52.7;^+ 12 = square, but
52 = 4. 13, and 13 is not a square, therefore this equa-
tion cannot be solved by our method.
Thus we must find two numbers to replace 13 and 4 whose
product is a square, and such that either +12 = square.
Now the product is a square if both are squares. Hence
we must find two squares such that either + 12 = square.
The squares 4 and ^ satisfy this condition.
Retracing our steps we put 4.r, - , - for the numbers, and we
have to solve the equation
X' +12 = square = (as + 3)^ say.
Therefore « = n ,
and (2, 2, ^) is a solution.
o
13. To find three numbers sui;h that, if a given number is sub-
tracted from the product of any pair, the result is a square.
Given number 10.
Put product of first and second =a square + 10=4 + 10 say,
and let first = 14.r, second = - . Also let product of second
andthird=19. Therefore third = 19a;. Whence 2G6x-'- 10
must be a square; but 266 is not a square.
Hence, as in the preceding problem, we must find two squares
each of which exceeds a square by 10.
Now ( — - — 1 -10-[--^-], therefore 30| is one sucli
square. If vi' be another, ta'-XO must be a square
= (m - 2)" say, therefore in = 3^.
AUITHMETICS. BOOK III. 185
Thus, putting 30 jr, -, 12;iu; for the uumbei-s, we have, from
the third coudition, [)929x-^- 1 GO ^square =^(7 7a;- 2)* say.
Therefore x = ^r= ,
1 1
1 ,1 , 1240] 77 5021
and the numbers arc „„ , ., , ■
77 '41' 77
U. To find three numbers such that the product of any two
added to tlie third gives a square.
Take a square and subtract part of it for tlie tliinl number.
Let x^ +6x+d be one of the sums, and let the third number
be 9. Therefore product of fii-st and second = x* + 6x.
Let the fii'st = x, therefore the second = a; + G.
From the two remaining conditions
10a: + 54) . .
10a; + 6/ are both squares.
Therefore we have to find two squares differing l)y 48, wliich
are found to be IG, G4.
and (1, 7, 9) is a solution.
15. To tind three numbers such that the product of any two
exceeds the third by a square.
First X, second x + 4, therefoi-e their product is x- + Ix, and
we suppose the third to be 4a;.
Therefore by the other conditions
4x- + 15a; ) , ^,
-2 a\ ^^'^ **^ squares.
The difference = IGa; + 4 = 4 (4a; + 1),
and f -^-TT — ) = 4 a;' + 15a;.
/ 4.r + 5 Y
25
Therefore ^'' = ^,^, 'ind the numbers are found.
IG. To find throe numbers such that tlie product of any two
added to the square of the third gives a square.
Let first be x, second 4x + 4, third 1. Two conditions arc
thus satisfied, and the remaining one gives
a; + (4a; + 4)^ ^ a square - (4a; - 5)' say.
9
Tlierefore a^ = ;r:t >
and the numbers are 9, 328, 73.
i^^)'-
186 DIOPIIANTOS OF ALEXANDRIA.
17. To find three numbers such that the product of any two
added to the sum of those two gives a square.
Leimna. The squares of two consecutive numbers liave this
property.
Let 4, 9 be two of the numbers, x the third.
Therefore 10a; + 9) , , ,
, I must both be squares,
5a; + 4 j ^
and tlie difference = 5a; + 5 = 5 (a; + 1).
Therefore by Book ii.,
10a: + 9 and a; =28,
and (4, 9, 28) is a solution.
18. \_Another solution of the foo'egoinff problem.]
Assume the first to be x, the second 3.
Therefore 4a; + 3 = square = 2-5 say, whence a; = 5^, and 5^, 3
satisfy one condition. Let the third be a;, 5^ and 3
being the first two.
Therefore 4a; + 3 ) , , ., ,
-, _, > must both be squares,
C^a; + 5iJ ^
bid, since the copfficients in one expression are both greater
than those in the other, but neither of the ratios of corre-
sponding ones is that of a square to a square, our method
will not solve them.
Hence (to replace 51, 3) we must find two numbers such that
their product + their sum = square, and the ratio of the
numbers each increased by 1 is the ratio of a square to
a square.
Let them be y and 4?/ + 3, which satisfy the latter con-
dition ; and so that product + sum = square we must have
4y^ + 81/ + 3 - square = (2^ - 3)'', say.
3
Therefore y = --
3
Assume now ^tt, 45, a; for the numbers.
Therefore oja; + 4M
13x 3 \- ave both squares.
To" "^ 10)
or 130a; +1051 , .
130X+ 30J -- 1-^1-^"--^'
ARITHMETICS. BOOK III. 187
the ditlerence = 75, -wliicli has two factors 3 ami 2"),
7
and X - :^A gives a solution,
3 7
the numbers being , 4J, .
19. To fniil three numbers siu-li tliat the iiroduct of any two
exceeds the sum of those two by a square.
Put first = X, second any number, and we fall into the same
difficulty as in the preceding. We have to find two
numbers such that their product minus their sum =
square, and when each is diminished by one they have
the ratio of squares. 4?/+l, y+l satisfy the latter
condition, and it/- - 1 - square -{2i/ — 2)' say.
5
Therefore !/ = :>•
c
13 28
Assume then as the numbers -^- , ~, x.
b b
Therefore 2}^x-3U
5 , _ ^ /• are both squares,
8 ''*'' ~ 8 )
or lOx-U] . ^.
10a;-2Gj "'"^ 'l"'^'"^^'
the difterence = 12 = 2 .G, and x =^ 3 is a solution.
13
The numbers are -^ , 3h, 3.
o
20. To find two numbers such that their product added to both
or to cither gives a square.
Assume x, ix - 1 ,
since x{ix- 1) + x = ix^ = square.
Therefore also 4a,-' + 3.x- - 1 ) , ^,
, > arc both squares,
4x^ + ix-l)
the diflference = x = 4a; , 7 ,
4
and X = ,j^ gives a solution.
21. To find two numbers such that the product exceeds tho
sum of both, and also cither severally, V)y a stjuare.
Assume x + 1, ix,
since 4a; (x- + 1 ) - 4a; = scpiare.
188 DIOPIIANTOS OF ALEXANDRIA.
Therefore also 4a;* + 3.« - 1 ~) , ,
2 'j- are both squares,
the difference = ix = 4x. 1.
Therefore a;=l],
and (2|, 5) is a solution.
22. To find four numbers such that, if we take the square of the
sum ± any one singly, all the resulting numbers are squares.
Since in a rational right-angled triangle square on hypotenuse
= squares on sides, square on hypotenuse =*= twice product
of sides = square.
Therefore we must find a square which will admit of division
into two squares in four ways.
Take the right-angled triangles (3, 4, 5), (5, 12, 13). Multiply
the sides of the first by the hypotenuse of the second and
vice versa.
Therefore we have the triangles (39, 52, 65), (25, 60, 65).
Thus 65^ is split up into two squares in two ways.
Also G5 = 7' + 4' = 8^-hP.
Therefore 65^ = {T - ^J + 4 . 7^ 4" = (8^ - 1')^ + 4 . 8M* .
= 33'' + 56^= 63* +16',
which gives two more ways.
Thus 65* is split into two squares in four ways.
Assume now as the sum of the numbers 65a;,
first number = 2 .39 . 52a;* = 4056a;*^
second „ =2 25 . 60x* = 3000a;*
third „ =2.33. 56a;* = 3696a;* i
fourth „ =2.16.63.«*=2016a;*j
65
Therefore 12768x-* = 65a; and x=i21Q9>'
and the numbers arc found, viz.
17136600 12675000 15615600 8517600
163021824' 163021824' 163021824' 163021824"
23. To divide a given number into two j)arts, and to find a
sijuare which exceeds either of the parts by a scpiare.
Let the given number be 10, and the square x" + 1x + 1.
Put one of the parts 2.'b+1, the other 4a;. Therefore the
conditions are satisfied if fix- -I- 1 = 10.
and the sum = 12768x*
Tlierefore
ARITHMETICS. HOOK IV. 189
3
X-
and the parts are 6, 4, the square 6].
24. To divide a given number into two parts, and to find ii
square which added to either of the parts produces a square.
Given number 20. Let the square be x^ + 2x+\.
Tliis is a square if we add 2a; + 3 or 4a; + 8.
Therefore, if these ai'e the parts, the conditions are .satisfied
when 6a; + 1 1 = 20, or a; ~ H.
Therefore the numbers into which 20 is divided an- (6, 14)
and the required square is 6^.
BOOK IV.
1. To divide a given number into two cubes, such that the sum
of their sides is a given number.
Given number 370, sum of sides 10.
Sides of cubes 5 + x,5-x. Therefore 30x' 4 250 = 370, x = 2,
and the cubes are 7^, 3^.
2. To find two numbers whose difference is given, and also the
difference of their cubes.
Difference 6. Difference of cubes 504. Let the numbcra be-
a; + 3, a; - 3. Therefore 1 8a;* + 54 = 504.
Tlierefore a;' = 25, a; = 5,
and the sides of the cubes are 8, 2.
3. A number multiplied into a square and its side makes the
latter product a cube of which the foi-mer product is the side; to find
the square.
Let the square be a;". Therefore its side is x, and let tho
Q
number be - .
Hence the products are 8a;, 8, and {Sxy =
rr,, . . 1 1
Therefore a; = ^ , ^ = 7-
190 DIOPnANTOS OF ALEXANDRIA.
4. To add the same number to a square and its side and make
them the same, [i.e. make the first product a square of which the
second product is side]'.
Square cc^, whose side is x. Let the number added to x' be
such as to make a square, say 3a;".
Therefore Zx" + x = side of 4x° = 2x and x = ^.
The square is 3 and the number is - .
5. To add the same number to a square and its side and make
them the opposite.
Square x^, the number ix° - x.
Hence 5a;^ -x = side of 4a;" = 2.r, and x--=.
6. To add the same square number to a square and a cube and
make them the same.
Let the cube be o;^ and the square any square number of x^'s,,
say Ox--. Add to the square 16a;^ (The 16 is arrived
at by taking two factors of 9, say 1 and 9, subtracting
them, halving the remainder and squaring.)
1 o
Therefore x^ + 16.«" = cube = S.c'' suppose and a; = — .
Whence the numbers are known.
7. Add to a cube and a square the same square and make them
the opposite.
[Call the cube (1), first square (2), and the added square (3)].
Now suppose (2) + (3) = (1) [since (2) + (3) = a cube\.
Now a' + y^2ah is a square. Suppose then {\) =a- + 1/,
(3) = 2ah. But (3) must be a square.
Therefoi-e 2ab must be a square ; hence we put « = 1, & = 2.
Tlius suppose (1) = 5x^, (3) = 4,r-, (2) = x^. Now (1) is a cube.
Therefore a? - 5,
and (1)- 125, (2) = 25, (3) = 100.
' 111 this aud the following enunciations I have kept closely to the Greek,
partly for the purpose of showing Diophautos' mode of expression, and partly
for the brevity gained thereby.
"To make them the same" means in the case of -1 what I have put in
brackets; "to make them the opposite" means to make the first product a side
of which the second product is the square.
ARITHMETICS. BOOK IV. 191
8. [Another solution of the foregoing.]
Since (2) + (3) = (l), a cube, and (1) + (3) = .square, I havn
to find two squares whose .sum + one of thoni - a S(iuan',
and whose sum = (l). Let the fii-st square be ar*, the
second 4.
Therefore 2.r + 4 = a square = (2^; - 2)' say. Therefore x = i,
and the squares are IG, 4.
Assume now (2) = ix\ (3) = IGx'.
Therefore 20a;' = a cube, and x = 20,
thus (8000, 1600, 6400) is a solution.
9. To add the same number to a cidje and its side, and make
them the same.
Added number x, cube 8x-^, say. Therefore second sum = 3a;,
and this must be the side of cube Sx^ + x, or 8a;^ + a; = 27x'.
Therefore 19x' = a;.
But 19 is not a square. Hence we must find a square to
replace it. Kow the side 3a; comes fx-om the assumed 2j;.
Hence we must find two consecutive numbers whose cubes
differ by a square. Let them he y, y+\.
Therefore 3v/" + 3v/ + 1 =square = (l — 2?/)* say, and y = l.
Thus instead of 2 and 3 we must take 7 and 8.
Assuming now added number = x, side of cul)e = Ix, side of
new cube — 8a;, we find 343.<;' + x = 512x''\
Therefore a;^ = ^J^, a; = l.
(343 7 1 \
2197'T3'13;^'^'"^"^^""-
10. To add the same number to a cube and its sid'! and i/uikc
them the opjwsite.
Suppose tlie cube 8x-^, its side 2x, the number 27x-' - 2a;.
Therefore 35x' - 2a; = side of cube 27j;', therefore 3r)x*-5 = 0.
This gives no rational value. Now 35 = 27 + 8, 5 = 3 + 2.
'^ Therefore we must find two numbers the sum of wlioso cubes
bears to the sum of the numbers tlie ratio of a square
to a square.
Let sum of sides = anything, 2 say, and side of first cul>c = :.
Therefore 8 - 12^3 + 6s' = twice a square.
Therefore i-(jz + 3z^ = a .square = (2 - 45)* sjiy, and : = on-
192 DIOPHANTOS OF ALEXANDRIA.
of the sides = ^ ^ , and the other side = j^ . Take for
them 5 and 8.
Assuming now as the cube \'25x^, and as the number
5 1 2^' - 5x, we get 637a;^ -5x = 8x, and a; = - ,
, /125 5 267\ .
11. To find two cubes whose sum equals the sum of their sides.
Let the sides be 2x, Sx. This gives 35a;' = 5x. This equation
gives no rational result. Finding as in the preceding
problem an equation to replace it, 637a;' = 1 Sx, a; = = ,
1.1 1 125 512
and the cubes are ^^ , -^^^ .
12. To find two cubes whose difference equals the difference of
their sides.
Assume as sides 2x, 3x. This gives Idx^ = x. Irrational; and
their difference
we
have to find two cubes such that
difference of sides
= ratio of squares. Let them be (z + 1)*, z\
Therefore Sz' + 3;^ + 1 = square = (1 - 2z)- say.
Therefore z = 7.
Now assume as sides 7a;, Sx. Therefore 169a;* = a;, and .'c=t^.
1 o
Therefore the two cubes
(i^> a
13. To find two numbers such that the cube of the greater + the
less = the cube of the less + the greater.
Assume 2a;, 3a;. Therefore 27a;' + 2a; = Sa;' + 3a;.
Therefore 19.x' = a;, which gives an irrational result. Hence,
as in 12th problem, we must assume 7a;, 8a;,
7 8
and the numbers are as there v:^ , ^g •
1 4. To find two numbers such that either, or their sum, or their
difference increased by 1 gives a square.
Take unity from any square for the first number ; let it be,
say, 9a;" + 6.r.
ARITHMETICS. BOOK IV. 19:^
But the second + 1 =a square. Therefore wc must Cnd a squar.-
such that the square found + 9x* + 6x = a square. Taku
factors of 9j;* + 6.r, nz, (9x+6, x). Square of half dif
ference -16xf + 24a; + 9.
Therefore, if we put the second number IG.c" + 24.C + 8, threo
conditions are satisfied, and the remaining condition gives
difference + 1 = square.
Hence 7a;* + 1 Sx + 9 = square = (3 - 3x-)- say.
Therefore x = 18,
and (3024, 5624) is a solution.
15. To find three square numbers such that their sum equals
the sum of their differences.
8umo{diSerences=A-B+B-C+A-C = 2{A-C) = A+B+C,
by the question.
Let least (C) = 1, greatest = a;' + 2.x- + 1. Therefore sum of the
three squares = 2x^ + ix = x^ + 2a; + 2 + the middle one.
Therefore the middle one (JB) = a;' + 2a; - 2. This is a
9
square, = (x - 4)" say. Therefore a; = - ,
and the squares are (s^, ~nr ? 1 ) '^'' (196, 121, 25).
16. To find three numbers such that the sum of any two multi-
plied by the third is a given number.
Let (fii-st + second) . third = 35, (second + third) . first = 27, and
(third + first) . second = 32, and let the third = x.
3.3
Therefore first
+ second =
Assume first =
1*^
— , second
X
250
Therefore ^^^
If
+ 10 = 27~
+ 25 = 32
These equatians are inconsistent, but if 25 - 10 u-ere t^ual to
32 - 27 or 5 they would be right. Therefore we have t<»
divide 35 into two parts differing by 5, i.e. 15 and 20.
Thus first number = — , second = — . Therefore . +15-27,
r*^ X *
II. 1).
X--5, and (3, 4. 5) is a solution.
13
194 DIOPHANTOS OF ALEXANDRIA.
17. To fiiid three numbers whose sum is a square, and such th<(t
the sum 0/ the square of each and the succeeding number is a square.
Let the middle number be 4a;. Therefore I must find what
square + ^x gives a square. Take two numbers whose
product is ix, say 2ic and 2. Therefore {x-\y is the
square. Thus the first number = £c- 1.
Again 16x-* + third = square.
Therefore third = a square - 1 6a;* = (4a; + Vf—YQtx? say, = 8a;+ 1 .
Now the three together = square, therefore 13a;=square=1697/-
say. Therefore x=\ 3^. Hence the numbers are
ISy^'-l, 52/, 104/ +1.
Lastly, (third)* + first = square.
Therefore 10816/ + 221/ = a square or 10816/ + 221= a
220 55
square = (104y/ +1)° say. Therefore y = -"- = — ,
, /36621 157300 317304\ . , .
18. To find three numbers whose sum is a square, and such that
the difference of the square of any one and the succeeding number is
a square.
The solution is exactly similar to the last, the numbers being
in this case 13/ +1, 52/, 104/ -1. The resulting
equation is 10816/ - 221 = square = (104^ - 1)-,
whence 2/ =-^^-^,
, /170989 640692 1270568\ .
^^^ (T08I6-' T0816 ' 10816 ; ^^ " ^^^"*^°'^-
19. To find two numbers such that the cube of the first + thr
second - a cube, the square of the second + the first = a square.
Let the first be x, the second 8-a;'', therefore a;''-16.r''+64 + a' =
a square = (x* + 8)* say, whence 32a;^ = x. This gives an
irrational result since 32 is not a square. Now 32 = 4. 8.
Therefore we must put in our assumptions 4 . 64 insteail.
Then the second number is 64 -a;^, and we get, as an
equation for x,
256a;» = 1 . Therefore x = — ,
lb
1 262143
and tlu. numbers are jg, ^^gg .
ARITHMETICS. UOOK IV. 10.",
20. To jlud three numbers imfffinite/i/^ s?/<7( (/uU the ]>ro<liict «/
ani/ two increased by 1 is a square.
Let the product of first and second be x* + 2x, whence on»i
condition is satisfied, if second = x, first = x + 2. Now th«
product of second and third + 1 =a square ; let this pro-
duct be Ox* + 6a;, so that third number =^x + G. Also lh(?
product of third and first + 1 = square, i.e. 9x* + 24j; + 13 ^
a square. Nov.\ if 13 ivere a square, and tfie coefficient of
X v}ere 6 times the side of this square, the problem icoubl
be solved indefinitelij as required.
Now 13 comes from 6.2 + 1, the 6 from 2 . 3, and the 2 from
2.1. Therefore we want a number to replace 3 . 1 such
that four times it + 1 = a square ; therefore we need only
take two numbers whose difierence is 1, say 1 and 2
[and 4. 2.1+ (2 — 1)- = square]. Then, beginning again,
we put product of first and second = x-* + 2a-, second x,
first x+2, pi'oduct of second and third = 4x* + 4.r, and
third = ix + 4. [Then first x third + 1 = 4«- + 1 2x + 0. ]
And (x + 2, X, 4x + 4) is a solution.
21. To find four numbers sicch that the product of ani/ two,
ina'eased by 1, becomes a square.
Assume that the product of first and second = a;* + '2x, fii-st = x,
second = x + 2, and similarly third = 4x + 4, fourth =9a:+ G,
but (4a; + 4)(9a; + 6) + 1 = square = 36a;- + 60x + 2;3.
Also for second and fourth,
9x-' + 24x + 1 3 = square = (O.v- - 24..- + 1 6), say.
Therefore x = -r-^.
lb
All the conditions are now satisfied*,
68 105\
^ /J^ 33
^° V16' 16'
the solution bein^ i ^t, > t^ > -i « > if.
22. Find three numbers which are proportional and such that
the difference of any two is a square.
Assume a; to be the least, x + 4 the midiUe, x + 1 3 the greatoist.
therefore if 13 were a square we shoidd have an indefi-
nite solution satisfying three of the conditions. We muHt
1 I.e. in general expressions.
2 Product of second and third + 1 = (a: + 2) (ix + 4) + 1 - i-c' ♦ l'^-^ + 9. ^hidi i«
a square.
1.1—2
196 DIOPHANTOS OF ALEXANDRIA.
therefore replace 13 l)y a squai-e wliich is tlie sura of two
squares.
Thus if \vc assume x, x + 9, x+ 25, three conditions are
satisfied, and the fourth gives x (x + 25) = {x + 9)", there-
fore x = -;^ ,
I
. ,81 144 256\ . . ^.
I i IT , „- . r,- IS a solution.
/«i 144 ZDt)\
[j ' T ' 7')
23. To find three numhers such that the sum of their solid content^
and any one of them is a square.
Let the product be x' + 2x, and the first number 1, the second
£C*+ 2x
ix + 9 ; therefore the third = :, — -^ . This cannot be
divided out generally unless x^: 4:X=2x : 9 or a;" : 2.^'=4a; : 9,
and it could be done if 4 were half of 9.
Now ix comes from (jx - 2x, and 9 from 3^, therefore we have
to find a number m to replace 3 such that 2»i-2 = -^,
therefore m^ = 4w - 4 or m = 2. ^
We put therefore for the second number 2x + 4, and the third
then becomes Ix. Therefore also [third condition]
5
x^ + 2x + \x - square = ix^ say, whence x= ,
. (. 34 1\\ .
solution.
24. To find three numbers such that tJie difference of their solid
content and any one of them is a square.
Fii'st x, solid content x^.+ x; therefore the product of second
and third = 03 + 1 ; let the second = 1 .
Therefore the two remaining conditions give
a i both squares [Double equation.]
Difference = cc = ^ . 2a;, aay ; therefore (.t + :^)' = vX-*+k-1, x-
.™.(^;,i,
8
-— ) is a .solution.
I.e. the continued product of all three.
Observe the solution of a mixed quadratic.
X-
27'
26
27
136
ARITHMETICS. UooK IV. l!l7
25. Divide a givou number into two jiarts wljose prcKluct i.s a
cube mltius its side.
Given number 6. First part x ; therefore second = 6 - x, and
6a; - a;* = a cube minus its side = {^x-lf- (2x - 1 ) say,
so that 8x^ - 1 2.t' + ix = Gx - x'. This wouUl reduce to a
simple equation if the coefficient of x were the same on
both sides. To make tliis so, since G is lixcd, we must
put m for 2 in our assumption, where
3m - )n = G, or m = 3.
Therefore, altering the assumption,
(3x•-l)^-(3x•-l)=6x•-.t•^
, 26
whence
and the parts are
2G. To divide a rjiven number into three parts sii.<7i that their con-
firmed 2)roduct equals a cube lohose skle is the sum oj' their differencen.
Given number 4. Let the product be 8a;' : now the sum of
differences = twice difference between third and firat;
therefore difference between third and first parts = x.
Let the first be a multiple of a;, say 2a;. Therefore the
third = 3a;.
Hence the second = r. ^, ii"d, if the second had lain between
o
the first and third, the problem would have been solved.
Now the second came from dividing 8 by 2 . 3,- so that we have
S
to find two consecutive numbers such that ^, . , ^
their product
lies between them. Assume m, m + 1 ; therefore -,
7/t + m
lies between m and m + 1.
g
Therefore —5 + 1 > ?« + L
m + m
Therefore m* + m + 8 > 7?i' + 2«t' + m, or 8 > m" i in\
1\'
Take
(IN '
"i + ., ) ) which is > m"" + m, and p.|u:itc it t<
Therefore m + ;^ - 2, and m = \^
27
cube.
198 DIOPHANTOS OF ALEXANDRIA.
Hence we assume for the numbers
5 9 8
3^' 5'^'' 3^'
or (25x, 27a;, -iOx), multiplying throughout by 15.
Therefore the sum = d2x= 4, and •'k = oo j
and f -^ , , „ j are the three parts required.
o
[N.B. The condition —^ <m+l is ignored in the work,
and is incidentally satisfied.]
To find two numbers whose product added to either gives a
Suppose the first number equals a cube number of a;'s, say 8a;.
Second a;^ - 1, (so that 8a;^ - 8a; + 8a; = cube);
also 8a;' - 8a; + a;^ - 1 must be a cube = (2a; — 1)^ say.
14
Therefore 1 2>x* = 1 4,x-, x = yoi
1 o
^"^ (13-' res) i« ^ soi^^ti^"-
28. To find two numbers such that the difference between the
product and either is a cube.
Let the first be 8a;, the second a;^ + 1 (since 8a;'+8a; - 8a;= cube) ;
also 8if^ + 8a; - a;* - 1 must be a cube, which is "im-
possible " [for to get rid of the third power and the abso-
lute term we can only put this equal to (2a;- 1)^ which
gives an " irrational " result]. Assume then the first
= 8a; + 1, the second = a;* (since 8a;^ + a;^ - x^ = cube).
Therefore 8a;' + a;" - 8.r - 1 = a cube = {2x - If say.
14
Therefore x = y^ ,
wi ^ 125 196
and the nunilx'rs are „- , .
Id 169
29. To find two numbers such that their product =t their sum - a
cube.
Let the first cube be G4, the second 8. Therefore twice the
sum of the numbers = 64 - 8 = 56, and the sum of the
numbers = 28, but thi-ii- product + their sum = 64.
Therefore their product - 36.
ARITHMETICS. BOOK IV. 199
Therefore we have to tind two numbers whose sum -• 28, and
whose product = 36. Assuming 14 +a;, 14 -x for these
numbers, 196 -x':::^ 36 and x* = 160, and if 160 were a
a square we could solve it rationally.
Now 160 arises from 1 4'' -36, and 14= J. 28 --.56
2 4
= T (diflference of cubes) ; 36 = ^ sum of cubes.
Therefore we have to find two cubes sucli that
( - of their difiereuce ) - ^^ their sum = a square.
Let the sides of these cubes be s + 1 , s - 1 .
1 3 1
Therefore - of their difference = r> ~' + ;j> '^^^^ the square of this
9,3,1
=r ^2^'^4-
Hence ( - . differer cc ) ^ .^ • sum = t ~ + ;> =' + a~~) v"^ "*" "')•
Therefore
dz' + Gz' + 1 - 4^' - 1 2s = a square = (3;:' + 1 - 6c)» say,
9
whence 32s^ ■= 36s*, and ~ ^ o •
Therefore sides of cubes are
17 1 , , ^ 4913 1
-, g, and the cubes .^,,--
4913
Now put product of numbers + their sum = -,-
product - sum = ^y^
therefore their sum
2456
~ 512 '
their product
2457
~ 512 ■
Then let the first number =
1228
X + half sum = x + — - - ,
second
1228
= 512 -"•
-ru r 1507984
Therefore -j^-^^-^-^
-x* '•*"
""- 512-
Therefore 2621 4 4x
•- 250000.
200 DIOPHANTOS OF ALEXANDRIA.
„ 500
Hence x = ^—- ,
512
, /1728 728\ .
""'"^ V^12 ' 512 j '' ^ '"^"*'°''-
30. To find two numbers such that their product ± their
sum = a cube [same problem as the foregoing].
Every square divided into two parts, one of tohich is its side,
makes the 2)roduct of these ])arts + their sum a cube.
[i.e. x(x^ - x) + x' — x + x= a cube.]
Let the square be x^ ; the parts are x, x^ — x,
and fi-om the second part of the condition
x^ -x^ -X- = x^ - 2x^ = a cube = [7,] say.
Therefore 5 *^ = 2a;'', x = -^ ,
o <
, /16 144\ .
and ( -^ , -jq 1 IS a solution.
31. 7'o find four square numbers such that their sum + the su?n
of their sides - a yiven number.
Given number 1 2. Now x* + a; + j = a square.
Therefore the sum of four squares + the sum of their sides
+ 1 = 13.
Thus we have to divide 13 into 4 squares, and if from eacli of
their sides we subtract ^ we shall have the sides of the
required squares.
10 . n G4 36 144 81
Now l3 = 44-9 = --f25+-25+05'
and the sides of the required squares are
11 ^ 19 13
10' lO' 10' 10"
32. To find four squares such that their sum ininus the sum of
their sides equals a given number.
Given number 4, Then similarly f side of first - - j + ... = 5.
-,.,..,-. 9 IG G4 36
and 5 is divided into ^-^ , ^rz , ^^ . ?,- >
2y 25 25 2;)
and the sides of the squares arc ( . -
13 21 17^
10 ' 10 ' lOy
ARITHMETICS. BOOK IV. 201
33. To divide unity into two parts snch that, if given numbers
be added to each, the j)roduct of the resulting expressions may be a
square.
Let 3, 5 be the numbers to be added, aud let the parts be ,
1 -xj
Therefore (a; + 3) (6 - a;) = 18 + 3x — a;' - a square = ■l.c* say.
Hence 18 + 3a; = 5a;*; but 5 comes from a S([uare+1, and
the roots cannot be rational unless
(this square +l)18 + [7jj =a square.
Put (m- +l)18 + f^j =a square,
or 72/?r + 81 = a square = (8ni + 9)* say.
Therefore «i = 1 8.
Hence we must put
(x + 3) (6 - a;) = 18 + 3a; - a;' - 3•24a;^
Therefore 325x'' - 3a; - 18 = 0.
78 6
Therefore
sohition.
31. [Another solution of the foregoing.]
Suppose the first a; - 3, the second i -x; therefore
a; (9 - a;) = square = 4a;* say,
9
and 5a;- = 9a;, whence a;=p, but I cannot take 2 from
9
- , and X must be > 3 < 4.
9
Now the value of a; comes from - — r . Therefore, since
a square + 1
a; > 3, this S(piare + 1 < 3, therefore the square < 2. It is
5
also > -. .
4
Therefore I mustfind a square between -and 2, or ^^ and — .
And -TT-r- or ^ will satisfy the conditions.
64 16
Put now x(9-x)- j^, .c.
202 DIOPIIANTOS OF ALEXANDRIA.
Therefore
U4
IT
21 20^
/21 20\ . , ^.
and ( .,- , . 1 IS a solution,
35. To divide a given number into three j)arts such that the pro-
duct of the first and second, with the third added or subtracted, may be
a square.
Given number G, tlie third part x, the second any niimher
less than G, say 2, Therefore the first — ^ - x.
Hence 8 - 2x' ± a; = a square. \^Doid)le-equation.^ And it cannot
be solved by our method since the ratio of tlie coefficients
of X is not a ratio of squares. Therefore we must find a
number y to replace 2, such that
?/ + 1 ,
r = a squai-e = 4 say.
2/-1
5
Therefore 2/ + 1 = -iy — 4, and y = q •
5 13
Put now the second part = - , therefore the first - ~ —x.
o o
G5 5
Therefore -^ -^x^x^^s. square.
Thus „„ o . }• are both squares,
or „„^ ~ i^ .^l- are both squares : difference =195 = 15. 13.
2G0-24:x)
Hence (^^^ ~ Y = 65 - 24a;, and 24a; = 64, a; = | .
/5 5 8\
Therefore the parts arc ( ^ , ^ , ^ j .
36. To find two numbers such that the first with a ce^-tain fraction
of the second is to the remainder of the second, and the second with the
same fraction of the first is to the remainder of the first, each in given
ratios.
Let the first with the fraction of the second = 3 times the
remainder of the second, and the second with the same
fraction of tlie first = 5 times the remainder of the first.
Let the second = x + 1, and let the part of it received by the first
be 1. Therefore the first - 3.c - 1 [for 3.f - 1 + 1 - 3.c].
ARITHMETICS. 1U)0K IV. 203
Also fii-st + second - ix, uiul first + second = sum of tlie numbers
after interchange, therefore J^I$±^^S^l_ ^ q
remainder of tiret
2
Therefore the remainder of the first = ^x, and hence the second
receives from the first 3.0 - 1 - ^ x- = r a; - 1 .
o o
Hence l^^^^, = r, tlierefore r,x^ + tx~\ = 3.c - 1 ,
6x-\ x+\ 3 3
and x = ^ .
8 12
Therefore the first number = -;; , and the second = -''- : and 1 is
< t
rr- of the second.
Multiply by 7 and the numbers are 8, 12; and the fraction is
^ ; but 8 is not divisible by 12, so multijily by 3,
and (24, 36) is a solution.
37. To find two numbers indefinitely such their product + their
sum = a given number.
Given number 8. Assume the first to be x, the second 3,
Therefore 3a; + a; + 3 = given number = 8.
Therefore x-'j, and the numbers are ( . , 3 j .
5 8 — 3
Now - arises from -. — !r . Therefore we may put mx + n for
4 3+1 "^ ^
1 1 1 1 1 z' 8 ~ ("*•*-' + '" )
the second number, and tiie nrst = r- .
mx + n+l
38. To find three numbers such that (the product + the sum) of
any two equals a given number.
Condition. Each number must be 1 less than some squai-o.
Let product + sum of first and second = 8, of second and tbinl
- 15, of third and first = 24.
Thi-n , — , = the first: let the second = x- 1.
second + 1
Therefore ^-"•^- first --1. Similarly third — -1.
X X •'-■
204 DIOPHANTOS OF ALEXANDRIA.
Therefore (^^ - 1 V^ - l) + ^ - 2 = 24, and ^.^ - 1 = 24,
12
therefore aj = -. ,
o
, /33 7 68\ . , ^.
and I _ , ^ , 1 IS a solution,
39. To find two numbei-s indefinitely such that their product
exceeds their sum by a given number.
Let the first number be x, the second 3. Therefore product
- sum = 3.V - a; - 3 = 2a; - 3 = 8 (say). Therefore x = -- .
Thus the first = — , tlie second = 3. But — = — :r .
x + d
Hence, putting the second = a; + 1 , the first = — — .
40. To find three numbers such that the product of any two
exceeds their sum by a given number.
Condition. Each of the given numbers must be 1 less than
some square.
Let them be 8, 15, 24.
Therefore first number = :; — =- = , say. Therefore
second - 1 a;
9 IG
the first = - + 1 , the second = a; + 1 , and the third = — + 1 .
X X
Therefore ('^ + 1 V^ + l) - ^ - 2 = 24.
Orl4i-ll24, Ij',
X- 5
md ( q-s ,,.,,_) is a solution.
/57 17 9Z\ .
(l2' 5'12;^^^
41. To find two numbers indefinitely whose product has to their
sum a given ratio.
Let the ratio be 3 : li the first number x, the second 5.
15
Therefore 5x = 3 (5 + x), and x = — .
r,ut -^ = r~~:^ , and, putting x for 5,
the indefinite solution is: first = -^_ , second =x.
X — o
ARITUMETICS. BOOK IV. 205
42. To find three numbers such that for any two their product
bears to theii- sum a given ratio.
_ first and second multiplied . , , ,
Let 5 — 7 , — = 6, and let the other ratios be
nrst + st^cond
4 and 5, the second number .r. Therefore first = — , third
X — 3
4a;
~ x-i'
3x ix . 3x 4a; \ , ^ ,
Also -. j = 5 { —5+ r) or 12x-
x-Z a;-4 \a;-3 a;- 4/
35.0-' - 1 20.r.
\a; — o x — -±/
120
Therefore x = -^, ,
, /360 120 480\ . . , .
""n^' 23' 28;"''^'''"^^'^""-
43. To find three numbers such that the product of any two h«s
to the sum of the three a given ratio.
Let the ratios be 3, 4, 5. First seek three numbers such that
the product of any two has to an arbitrary number (say 5)
the given ratio. Of these, let the product of the first
and the second =15.
15
Therefore if x = the second, the first = ■ — .
X
But the second multiplied by the third = 20.
20 20 15
Therefore the third - — , and "^—^^ — = 25.
x or
Therefore 25.v'=20.15.
And, if 20 . 15 were a square, what is required would be done.
Now 15 = 3.5 and 20 = 4 . 5, and 15 is made up of the ratio
3 ; 1 and the arhilrary number 5.
12»r
Therefore we must find a number m such that — = ratio of
a square to a square.
Thus 1 2/?r . 5»i = GOm' - square = 900/»', say. Tliereforo m = 15.
Let then the sum of the three =15,
and the product of the first and sccon<l = 15, therefore tli«»
first = — .
X
GO , , 45. GO _. , (.
Simihirlv tlio third = — ; therefore , ^ < •> and x ^ G.
X X
206 DIOPUANTOS OF ALEXANDRIA.
45
Therefore the first number =- , the third = 10,
47
and the sum of the three = 23| =— . Now, if this loere 15,
the j)rohlevi loould he solved.
Put therefore 15a;° for the sum of the tlirce, and for the
numbers l^x, Qx, \0x.
47
Therefore 2^x = 15a;-, and x=---,
/705 282 470\
whence fgQ-,3Q,3QJ
is a sohition.
44. To find three numbers such that the product of their sum and
the first is a triangular number, that of their sn7n and the secoml a
square, and that of their su7h and the third a cube.
Let the sum be x', and the first —, , the second -, , the third — ^ ,
X X .r
which will satisfy the three conditions.
1 8
But the sum =—3 = .r* or 18 = x\
X
Therefore loe must rejilace IS by a fourth iiower.
But 18 = sum of a triangular number, a square and a cube;
let the fourth power be x^, which must be made up in the
same way, and let the square be x* -1o? + 1. Therefore
the triangular number + the cube = 2a;*— 1; let the cube
be 8, therefore the triangular number = 2a;^ - 9. But 8
limes a triangular number + 1 = a square.
Therefore IG.x'^ - 71 = a square = (4a;- 1)^ say; therefore x = 9,
and the triangular number = 153, the square =6400 and
the cube = 8.
Assume tlien as the first number -^-, as the second — „ ,
x' ' a'
as the third -3 .
Therefore — „- = a;* and x = 9.
a;
,,,, /153 6400 8\ . , ^.
ARITHMETICS. BOOK IV. 207
45. To find three numbers such that the dijj'crence of the greatest
and the middle has to tlie difference of the middle and the least a given
ratio, and also the sinn of any pair is a square.
Ratio 3. Since middle number + lea.st = a square, let them = l.
Therefore middle > 2 ; let it be x + 2, so that least = 2 - x.
Therefore the interval of the greatest and the middle = 6a-,
whence the greatest = Tx + 2.
Therefore ' > are both squares [Donble equation] : take
two numbers Avhose product = 2x, say - and 4, and pro-
ceed by the rule. Therefore x= 112, biit I cannot take
\\2 from 2; therefore x must be found to be < 2, so
that 6.f+4<lG.
Thus there are to be three squai-es 8a; + 4, Gx + 4, 4 ; ami
difference of greatest and middle = ^ of difference between
middle and least.
Therefore we must find three squares having this proi)erty,
such that the least = 4 and the middle one < 1 6.
Let side of middle one be s + 2, wlience the gi'eatest is equal to
2^ + 42 „ , , 4 2 16,
_3- + .^ + 4. + 4^-- + -3-c--.4.
Therefore this is a square, or 3^* + 1 2s; + 9 = a square; but the
middle of the required squares < 1 6, therefore z <'2.
Put now 3i' + 1 2s + 9 = {mz - 3)» = mV - (jmz + 9.
Therefore z ^ „ — " , which must be < 2.
m — o
Hence 6m + 12 < 2m- - 6, or 2m» > 6m + 18,
and 18 . 2 + 3- = 4.5 ; therefore we may put in-'- + ^.
Thus we have 3;:- + 12^ + 9 = (3 - bz)'.
Hence s = yi , -'^'i^l the side of the middle square ■--■ ... aii«l
the square itselr - .
Turning to the original problem, wo i>ut y^j'^ ^•'-" + ^•
Therefore x = , ' , which is < 2.
<26
208 DIOPHANTOS OF ALEXANDRIA.
Hence the greatest of the required numbers
= 7x + 2 - -
11007
726 '
2817
and the second of them = as + 2 = - ,
lab
87
and the thii'd = 2-x = ^^z .
46. To find three numbers such that the difference of the squares of
the f/reatcst and the middle numbers has to the difference of the middle
and the least a given ratio, and the sums of all 2}airs are severally
squares.
Ratio 3. Let greatest + middle = the square 1 (Sx^. Therefore
greatest is > 8a;^, say 8a;" + 2, Hence middle = Bx* - 2,
and greatest + middle > greatest + least, therefore great-
est + least < 1 6a;* > Sx^ = 9a;-, say; therefore the least
number = a;® - 2.
Now difference of squares of greatest and middle = 64a;*, and
difference of middle and least = 7a;", but 64 ^ 21.
Now 64 comes from 32 . 2, so that I must find a number m
21
such that 32m = 21. Therefore ««. = ^ .
Assume now that the gi'eatest of the numbers sought
21 ,21 „ 21
= 8a;* + -^ , the middle = 8a; - — , the least = a; - — .
[Therefore difference of squares of greatest and middle
= 21a;* = 3. 7a;*.]
The only condition left is
21 21
8a;* - p + a;* - -gT, ^ a square
9.1;
— = a square = {S.r - 6)" say.
r,., r 597
Therefore x = yr:^ .
5/6
/3069000 2633544 138681\ . , ^.
"^'"^^ (331776' 331776 ' 331776;^^''^ ^°^^^^^'^"-
AIUTILMETICS. BOOK V.
BOOK V.
1. To find three numbers in c. p. such that each exceeds a given
number by a square.
Given number 12. Find a square which exceeds 12 by a
square [by ii. 11], say 42^. Let the first number be 42^,
the third x^, so that the middle one = 6i.r.
a;2_ 12")
Therefore „. ,_> are both squai-es : their diflference
therefore as usual we find the value of x, viz. — - ,
A^, 2346 i 130321\ .
^^ (^21,^^, -10816 j-^
solution.
2. To find three numbers in g. p. such that each together icith the
same given number equals a square.
Given number 20. Take a square whicli exceeds 20 by a
square, say 36, so that IG + 20 - 3G = a square.
Put then one of the extremes 16, the other x*, so that the
middle term = \x.
^ + 20 "i
Therefore , ^^> are both squares : their difference
\x + 20j
= y? -ix^x{x- 4),
whence we have 4a; + 20 = 4, which gives an irrational
result,
but the 4^1(16), and we should have in i)lace of 4 some
number > 20. Therefore to replace 16 we must find
some square > 4 . 20, and such that with the addition of
20 it becomes a square.
Now 81>80; therefore, putting for the nijuired square
{m + 9)-, (»4 + 9)' + 20 = square -{m-U)' .siiy. Therefore
m = .i, and the square = (9A)'' OOj.
H.D. " " 1*
210 DIOPIIANTOS OF ALEXANDRIA.
Assuming now for the numbers 90^, O^x, a?, we have,
^^ > are both squares : and the difference =a;(a;-9i),
9^a;+20j
whence we derive x = -r— - ,
152 '
/nm 389.1 1681 \ .
^^ V ^^' T52 ' 23I04J ^^ ^ '°^^^^^"^-
3. Givoi one numhei; to find three others such that any one of
them or the product of any two, when added to the given number, pro-
duces a square.
Given number 5. Porism. If of two numbers each and their
pi'oduct together with the same number make squares,
the two numbers arise from two consecutive squares.
Assume then {x + 3)-, {x + 4)-, and put for the first number
a;^+6cc + 4, and for the second a;° + 8x+ll, and let the
third equal twice their sum minus 1, or ix^ + 28a; + 29.
Therefore 4a;' + 28a; + 34 = a square = (2x - 6)^ say.
Hence a; = ^r^; ,
26
/2861 7645 20336\ . , ,.
and I -;r=-;r , -sv^/T . Wr,,^ IS a solution.
V676 ' 676 ' 676 /
4. Given one number, to find three others such that each, and
the product of any two exceed the given number by some square.
Given number 6. Take two consecutive squares x", a;* + 2a; + 1,
add 6 to each, and let the first number = a;* + 6, the
second number = a;^ + 2a; + 7, the third being equal to twice
the sum of first and second mhiics 1, or 4a;^ + 4a; + 25.
Therefore third minus 6 =4a;' + 4a; + 19 = square =(2a; -6)* say.
17
Therefore a; = — : ,
/4993 6729 22660\ .
*^^ (-784' T84' -78r)-^-'^<^l"*^°^^-
[Observe in this problem the assumption of the Porism numbered
(1) above (pp. 122, 123).]
5. To find three sqiiares such thai the procbict of any tivo, added
to the sum of those two, or to the remaining one, gives a square.
Porism. If any two consecutive squares be taken, and a third
number which exceeds twice their sum by 2, tliese three
ARITHMETICS. BOOK V. 211
numbers have the property of the numbers recpiired by
the problem.
Assume as the first a;^ + '2a; + 1 , and as the second x" + 4x + 4.
Therefore the third = Ax^ + 1 2x- + 1 2.
Hence x- + 3.v + 3 = a square -{x- 'df say, and a; = ^ .
m c /25 64 196\ .
iherefore ( ^ > 'K ■> ~q~ ) ^^ '^ solution.
C. To find three numbers such that each exceeds 2 by a square,
and the product of any two minus both, or minus the remaining
one = a square.
Add 2 to numbers found as in 5th problem. Let the first be
x^ + 2, the second x^ +'2x+ 3, the third ■^if + -ix + 6, and
all the conditions are satisfied, except
4x^ + 4a; + 6 - 2 = a square = 4 (a; - 2)* say.
3
Therefore x = -^ ,
o
/59 114 246\ .
^^^ (,25' 25' 25 J '' ^' '°^"^^""-
7. To find two numbers such that the sum of their product and
the squares of both is a square. [^Lemma to the following j)roblem.^
First number x, second any number {m), say 1.
3
Hence a;' + a; + 1 = a square = {x — 2)- say, and -c = r .
Therefore
K , 1] is a sokitiou, or (3, 5).
8. To find three right-angled triangles ' whose areas are equal.
First find two numbers such that their product + sum of their
squares = a square, i.e. 3, 5, as in the preceding problem
[15 + 3' + 5^'= 7'].
Now form three right-angled triangles from
(7, 3), (7, 5), (7, 3 + 5),
respectively, i,e. the triangles
(7« + 3S 7*- 3', 2.7.3), Ac.
^ I.e. rational right-angled triangles. Ax <tll the triangles ichich Diophantos
treats of are of this kind, I shall sometime.'^ use mimphj the tcord "triannU" to
represent "rational right-angled triangle," for the purposes of brevity, where
the latter expression is of very frequent occurrence.
14-2
212 DIOPHANTOS OF ALEXANDRIA.
and we have the triangles
(40, 42, 58), (24, 70, 74), (15, 112, 113)
and the area of each = 840.
[7^ - 3=) 7 . 3 - (7' - 5^) 7 . 5 = (8= - 7') 8 . 7].
['For if ah + 0?-^ 1)^=0%
(c- - a') ca = {c' - 6=) cb = {{a + bf - c'] (a + b) c,
since each = abc (a + 6)].
9. To find three numbers such that the square of any one =t the
sum of the three = a square.
Since, in a right-angled triangle, (hypotenuse)^ ± twice product
of sides = a square, we make the three numbers hypotenuses,
and the sum of the three four times the area.
Therefore I must find three triangles having the same area, i.e.
as in the preceding problem,
(40, 42, 58), (24, 70, 74), (15, 112, 113).
Therefore, putting for the numbers 5 803, 74a;, 113£C, their sum
= 245a; = four times the area of any one of the triangles
= 33G0a;^anda; = Qg.
Therefore [-^ , W WJ ^' ^ '°^^^^^°"-
10. Given three squares, it is possible to find three numbers
such that the products of the three pairs are respectively equal to
those squares.
4 9
Squares 4, 9, IG. One number x, the second - , the third - ,
a; X
and -^=16, and a; = li.
x^ ' ^
1 Ncsselmann suggests that Diophantos discovered this as follows. Let
the triangles formed from (n m), (q in), (r m) have their areas equal,
therefore n (m^ - 11^) = q {vi^ - q'') = ?• (r" — //(-), therefore m-n - Jt-' — m-q - q^,
m^=- — i =,r- + nq + q^.
n-q
Again, given [q m ?i). to find r;
q [m"- - q'^) = r(r- - m"), and in- -q'' — n- + nq
from above, therefore q (n- + nq) = r (;•- - n^ - nq - q"),
or q {n'^ + nr) + q'^(n + r) = r{r^-}i-).
Dividing by r + n, qn + q- = r^ - rn,
therefore {q + r)n = r-- q-,
and r = q + n.
ARITHMETICS. BOOK V. 213
Therefore the niuiibers are (l.l, „, 6).
We observe that 'C = -, where G = product of 2, 3, and
4 = side of 1 G.
Hence rule. Take the product of the sides, 2, 3, divide by tlio
side of the third square, and divide 4, 9 again by the
result.
11. To find three numbers such that tlie product of any two
the sum of the three = a square.
As in 9th problem, find three riglit-anglcd triangles having
equal areas : the squares of the hypotenuses are 3364,
5476, 127G9. Now find as in 10th problem three num-
bers, the products of paii-s of which equal these squares,
which we take because each ± (4. area) or 3360 = a square;
the three numbers then are
4292 3277 4181
"113 '^' 37 ""' ^ ^'
It only remains that the sum = 3360a;".
m f 32824806 ._. ,
Therefore -^t^t-^-t?^ x- 3360x".
121249
^, , 32824806
Therefore
407396640'
whence the numbers are known.
12. To divide unity into two parts such tJvat if the same given
number he added to eitlier part the result will be a square.
Condition. The added number must not be odd [the text of
this condition is discussed on p. 129 and note.]
Given number 6. Therefore 13 must be divided into two
two squares so that each > 6. Thus if I divide 13 into
squares whose difference < 1, this condition is satistied.
13
Take -^ = 6|, and I wish to add to 6i a small fraction which
will make it a square,
or, multiplying by 4, I wish to make ;, + 2G a square, or
26x* + 1 = a square = {5x+\ )' Siiy,
whence x = 10.
214 DIOPHANTOS OF ALEXANDRIA.
Therefore to make 2G into a square I must add y^,
or to make 6^ into a square I must add
400^
1 13 /51\'
^^^ 4o-o-'-2=Uo;-
Tlierefore I must divide 1 3 into two squares such tluit their sides
51
Tnay be as nearly as possible equal to ^ . [TrapicronjTos
dywyrj, above described, pp. 117 — 120.]
Now 13 = 2^ + 3^ Therefore I seek two numbers such that
51 9
3 minus the first = ^ , or the first = — , and 2 plus
51 11
the second = ^ , so that the second = — .
I write accordingly (11a +2)°, (3 - 9a;)° for the required
squares substituting x for -— .
Therefore the sum = 202a;- - lO-i; + 13 = 13.
5 ,^, ., 257 258
Hence x = y^ , and the sides are y^. ' Tni »
and, subtracting 6 from the squares of each, we find as the
pai-ts of unity
/ 4843 ^358-
V10201 ' 10201,
13. To divide unity into two jyarts such that, if we add given
numbers to each, the results are both squares.
Let the numbers be 2, G, and let them be represented in the
figure. Suppose DA = 2, AB = l, BE=G, G a point in
A£ so chosen that DG, GE may both be squares. Now
Tlierefore I have to divide 9 into tioo squares such that one of
them lies between 2 and 3.
Let the hitter square be x^. Therefore the second square
= 9 - a;', wliere x-" > 2 < 3.
Take two squares, one > 2, tlie other < 3, [the former square
ARITHMETICS. BOOK V. 21')
being the smaller], say ^ ■ , -^^ . Therefore, if we can
make x" lie between these, what wtis required is done
We must have ^^^,<\l
Hence, in making 9 - x^ a square, we must find
17 19
67)1
^?Tl
17 19
>12"I2-
Thus 72w>17m'' + 17, and 36*- 17. 17 = 1007 which' is
'^ Sl°, hence m is :}► — . Similarly ?« is -j; tt; •
Let m = 3i Therefore 9 -x' = h - '-x\ ,
and x = =Ti •
53
TT » 7056 ,^, ^ „, /1438 1371\
Hence ar ^ ^^^^ , and the segments of 1 are (^^gog ' 2809J "
14. To divide unity into three pm-ts such that, if we add the same
number to the three parts severally/, the results are all squares.
Comlition. Given number must not be 2 [Condition remarked
upon above, pp. 130, 131.]
Given number 3. Thus 10 is to be divided into three squares
such that each > 3.
Take ^ of 10, or 3^, and find x so that ^^7,+ 3J may be a square,
or 30x° + 1 = a square = {6x +\f say.
Therefore a; = 2,
1 121
and 36 '*' ^^ " "36" " ^ s^^''^^"^-
Therefore we have to divide 10 into three squares each near to
— . , [7rapio-OT7;T09 aywyr;'].
00
1 I.e. the integral part of the root is ^31. The limits taken arc .1 fortiori
limits as explained on p. 93, n. 3 and 4. Strictly speaking, wc could only say,
taking integral limits, that x/Iu07<32, but this limit is not narrow enough to
secure a correct result in the work which follows.
21 G DIOPHANTOS OF ALEXANDRIA.
9 16
Now 10 = 3^ + 1' = the sum of the three squares 9, ^, ^ .
3 4 11
Comparing the sides 3, -, p with -^, or (multiplying by 30)
90, 18, 24 with 55, we must make each side approach 55.
Put therefore for the sides
3-35CC, 31a; + g, S7x+-^
[35 = 90-55, 31 = 55-24, 37 = 55-18],
we have the sum of the squares
= 35550;^- 116rK+ 10 = 10.
Therefore x = ^rv^- ,
3555
and this solves the problem.
15. To divide unity into three parts such that, if three given
numbers be added, each to one of the parts, tlie results are all
squares.
Given numbers 2, 3, 4. Then I have to divide 10 into three
squares such that the first > 2, the second > 3, the third
> 4. Let us add - unity to each, and find three squares
whose sum is 10, the first lying between 2, 2^, the
second between 3, 3i, and the third between 4, 4|.
Divide 10 into two squares, one of which lies between 2 and 2^.
Then this square minus 1 will give one of the parts of unity.
Next divide the other square into two, one lying between
3, 3J ; this gives the second part, and therefore the third.
16. To divide a given number into three parts such that the
sums of all pairs are squares.
Number 10. Then since the greatest + the middle jmrt
= a square, &c., the sum of any pair is a square < 10, but
twice the sum of the three = 20, There/ore 20 is to he
divided into three squares each of lohich < 10. Now
20=16 + 4. Therefore we must divide 16 into two
squares, one of which lies between 6 and 10; we then
have three squares each of which is < 10, and whose sum
= 20, and by subtracting each of these squares from 10
we get the parts required;
[16 must be divided into the two squares by v. 13.]
ARITHMETICS. ROOK V. 217
17. To divide a given number into four parts such that the sum
of any three is a square.
Number 10. Then three times the sum = the sum of four
squai'es.
Hence 30 must be divided into four squares, eacli of wliich
< 10. If we use the method of Tra^icro'-nj? and make each
near 7|, and then subtract each square found from 10, wc
have the required i)arts.
But, observing that 30-1G + 9 + 4+1, I take i, 9 and divide
17 into two squares each of which < 10 > 7. Then sub-
tract each of the four squares from 10 and we have the
required parts.
18. To find three numbers S2ich that, if we add any one of them
to the cube of their sum, the result is a cube.
Let the sum be x, the numbers 7x^, 26a;', G3x^. Hence, for the
last condition, 9Ga;^ = x. But 9G is not a square. There-
fore it must be replaced. Now it arises from 7 + 2G + G3.
Therefore I have to find three numbers, each 1 less than a cube,
whose sum is a square. Let the sides of the cubes be
wi+ 1, 2-m, 2, whence the numbers are
m^ + 3'w' + 3m, 7- 12ni+ Gnr-m'', 7,
aud the sum = ^m" - dm + 14 = a square = (3?« - 4)-.
2
Therefore 7n = -r-. ,
15
1538 18577 ^
and the numbers are qq^t- > qqtF ' '•
Therefore, putting the sum - x, and the numbers of the problem
3375^' 3375 '"'''*''
15
we find X = ^ : therefore, «fec.
54
1 9. To find tloree numbers such t/uif, if we subtract any one of them
from t/ie cube of the sum, the result is a cube.
Let the sum be x, the numbex"8 - x', " x', . x'.
rr. r 4877 a /
Therefore ^r^;-- x" ^ x ; |(
n .io \
4877
but irz^TT-, ^ 3 - the sum of three cubes.
1728
218 DIOPHANTOS OF ALEXANDRIA.
Therefore we must find three cubes, each < 1, such that (3 -their
sum) = a square =2,^ say. Tlierefore we liavetofind three
3 162
cuhes whose sum is -=^, or we have to divide 162
into three cubes. But 162 = 125 + 64 - 27.
Now (Porism) the difference of two cuhes can be transformed
into the sum, of two cuhes. Having then found the three
cubes we start again,
2
and x = 1\x^, so that x = -^, which, with the three cubes,
determines the result.
20. To find three numbers such that, if we subtract the cube of
tlieir sum from any one of them, the result is a cube.
Sum =x, and let the numbers be 2x', 9a;^, 28a;^. Therefore
39 x^ = 1, and we must replace 39, which = sum of three
cubes + 3.
Therefore we must find three cubes whose sum + 3 = a
square.
Let their sides be m, 3 -m, and any number, say 1.
Therefore 9m^ + 31 - 277/2- = square = (3m - 7)" say, so that
6 , , .. ^.1 , 6 9
m-
- , and the sides of the cubes are ■= , - , 1.
Starting again, let the sum be x, and the numbers
341 3 854 3 250 3
Wb^' 125*' 125*'
25 5
so that 1445a;' =^ 125, ^''=289' ^ " 17 ^
thus the numbers are known.
21. To find three numbers, whose sum is a square, such that the
cube of their sum added to any one of them gives a square.
Let the sum be x^, the numbers 303', Bx", IS.'c". Therefore
26a;* = 1 ; and, if 26 were a fourth power, this would
give the result.
To replace it by a fourth power, wc must find three squares
whose sum diminished by 3 = a fourth power, or thi-ee
numbers such that each increased by 1 - a square, and
the sum of the three - a fourth power. Let these he
ARITHMETICS. ROOK V. 219
m* - 2ni^, m^ + 2m, m^ - 2m [sura ^ m*] ; then if we put m
anything, say 3,
the numbers are 63, 15, 3.
Thus, putting for the sum x^, and for the numbers 3j;', 15x',
GSx", a; = 5 , and the problem is solved.
22. To find three numbei-s whose sum equals a square, and such
that tlie cube of the sum exceeds any one of them by a square.
[Incomplete in the text.]
23. To divide a given fraction into three parts, such that each
exceeds the cube of the sum by a square.
Given fraction - . Therefore each = — + a square. Therefore
3
the sum of the three = sum of three squares + — .
13
Therefore we have simply to divide — into three squares.
24. To find three squares such that tJieir contimced product
added to any one of them gives a square.
Let the "solid content" = x", and we want three squares such
that each increased by 1 gives a square. They can be got
from right-angled triangles by dividing the square of
one of the sides about the right angle by the square of
the other. Let the squares then be
9 , _25 , ^
16 '^' 144*' 225 '*^-
14400
Therefore the solid content = x". This = x'.
Olo'iUU
120
Therefore " oq ** ~ ^ '
120 .
but _g, IS not a square.
Thus we must find three right-angled triangle.s such that,
if 6's are their bases, ;/s arc tlieir p<'q)endicular8,
p H/9^6,6, 6^ == square, or assuming one tnangle arbitrarily
3»,6,
(3, 4, 5), we have to make l2pj)J>J'j ^ square, or
220 DIOPHANTOS OF ALEXANDRIA.
a square. " This is easy" (Diophantos ') and the triangles
(3, 4, 5), (8, 15, 17), (9, 40, 41),
satisfy the condition,
and 03 = -^ ;
25 256
.1 .1 /25 256 9\
the squares then are ( -7- , -5, , Vr / •
25. To find three squares such that their continued product exceeds
any one of them hy a square.
Let the " solid content " = a;*, and let the numbers be got fx'om
right-angled triangles, being namely
16 ^ 2 _64 ,
25^^' 169 '^' 289^'
m r 4.5.8 , ,
Therefore — r^ a; = 1,
and the first side ought to be a square.
As before, find three triangles, assuming one (3, 4, 5) such that
hjiji^2^^pj>^= Q. square^, [letters denoting hypotenuses
and bases], or such that 20 v^-^ — a square.
[For the rest the text is in a very unsatisfactory state.]
1 Diophantos does not give the work here, but merely the results. Moreover
there is a mistake in the text of (5, 12, 13) for (H, 15, 17), and the problem is not
finished.
Schulz works out this part of the problem thus :
Find two right-angled triangles whoso areas are in the ratio vi : 1. Let the
Bides of the first be formed from {2m + 1, m - 1), and of the second from (m -f- 2,
7K-1), BO that two sides of the first are Am" -2m -2, ^m- + (im and the area
=6m'» + 9Hi3-9,/i2-6m.
Two sides of the second are 2m'^ + 2m-i, G?;! + 3, and m times the area
= Cm'* + 9m' - 9m- - Om. Now jmt e.g. jft = 3, therefore the first k-ianglc is
formed from 7, 2, viz. (28, 4,5, ,'53) ; second from 5, '2, viz. (20, 21, 29).
2 Cossali remarks: "Construct the triangles (/, h,p) [i = ipotenusa],
[-b ' —b ' ir=^^j'
/ib^' + iip^ bAbp-p(b^-ip^) p Abp + b {b^ - ip-) ,,\
^""^ [^b—' b ' b ^^)'
ARITHMETICS. HOOK V. 221
26. To find three squares such that each exceeds tlicir continued
product by a square.
Let the "solid content " = a;', and the squares have to be found
by means of the same triangles as before. We put 25x*,
625a;-, 1 4784a;- for them, ic.
[Text again corrupt.]
27. To find three squares such that the product of any two
increased by 1 is a square.
Product of fii-st and second + 1 = a square, and the third is
a square.
Therefore solid content + each = a square ; and the pnjblem
reduces to the 24th above.
28. To find three squares such that the product of any two
diminished by 1 is a square.
[Same as 25th problem.]
or the solid content of the three hypotenuses has to that of the three perpen-
diculars the ratio of a square to a square.
It is in his note on this imperfect problem that Fermat makes the error
which I referred to above. He says on the problem of finding tico triiuujks such
that the products of hij2)otcnuse and one t<ide of each have a given ratio "This
question troubled me for a long time, and any one on trying it will find it very
difficult : but I have at last discovered a general method of sohing it.
"Let e.g. ratio be 2. Form triangles from (ab) and (a d). The rectangles
under the hypotenuses and the perpendiculars are respectively 2ba^ + 2lPa,
2da^ + d^a, therefore since the ratio is 2, ba^ + b^a = 2(da' + d^a), therefore by
2(P _ Ij3
transposition 2<P -P = ba^ - 2da^ ; therefore, if - — — y- be made a square, tho
problem will be solved. Therefore I have to find two cubes <P, IP such that
2(P-h^ divided or multiplied by b-2d-a. square. Let x + 1, 1 be the sides,
therefore
2d3-63 = 2x3 + Cx= + Cx + l, 2b-d^l-x^
therefore
(l-a;)(l + 6a; + 6x' + 2x3) = l + 5x-4x3-2r«-square=^|x + l-^V) ,
and everything is clear."
[Now Fermat makes the mistake of taking 2b - d instead o{ h- 2d, and thus
he fails to solve tho problem. Brassinnc (author of a Pr<Jci3 of DiophantoH and
Fermat) thinks to mend the matter by milking (1 +Gx + 6x» + 2r>)(l +2x) a
square, whereas, the quantity to be made a square is (1 + 6x + Ox' + 2x^1 { - 1 - 2x).
The solution is thus incurably wTong.]
Fermat seems afterwards to have discovered that his solution did not help to
solve this particular problem of Diophantos, but docs not seem to have seen that
the solution is inconsistent with his own problem itself.
222 DIOPHANTOS OF ALEXANDRIA.
29. To find three squares such that unity diminished by the
product of any two = a square,
[Same as (26).]
30. Given a number, find three squares such that the sum of any
two together with the given number jyroduces a square.
Given number 15. Let one of the required squares =9.
Therefore I must find two other squares, such that each
+ 24 = a square, and their sum + 15 = a square. Take
two pairs of numbers whose product = 24, and let them
be the sides of a right-angled triangle' which contain
4
the right angle, say - , Gx ; let the side of one square be
2
half the difierence, or - - 3a;.
X
3
Again, take other factoi'S - , 8x, and half the difierence
3
= 4:X = side of the other square, say.
2x
/ 3 \^ /2 \"
Therefore (^ — 4ccj +( — 3a; j +15 = a square,
or -f + 25a;^ - 9 = a square = 25a;^ say.
g
Therefore x = ^ , and the problem is solved.
31. Given a number, to find three squares such that the sum of
any pair exceeds the given number by a square.
Given number 13. Let one of the squares be 25. Therefore
we must seek two more such that each + 12 = a square, and
(sum of both) - 13 ^ a square. Divide 12 into products
(3a;, -) and (4a;, -J, and let the squares
/3 2\" / 3 \"
Therefore (^ a; - - j + ( 2a; - .^ j - 13 = a square,
or ^ + 61 a;" - 25 = a square = -| say.
a;" * '^ XT
Therefore x = 2, and the problem is solved.
1 I. c. corrcspoudiug factors in the two pairs, in this case G.r, Sj.
be
AllITHMETICS. BOOK V. 223
32. To find three squares such that the sum of thdr squares is a
mare.
Let one be x", the second 4, the third 9. Therefore
a;* + 97 = a square = (x' - 10)* say.
3
Therefore ^'' = 2o> ^"* *^--^ ^^ ^^ot a square and must be
replaced.
Hence I have to find p*, q* and m such that "^~ ^ ^^ = a
2m
square. Let ;r = s-, q' = \, and //t = ;:' + 4. Therefore
m''-;/-5' = («* + 4)*-s--16 = 8c'. Hence we must
, 8«' 4;:^
^ave, ggT^g = ^ square, or - — - = a square.
Put s' + 4 = (;s+l)«say.
Therefore s = ^ , and the squares are i^" = t , q^ = 4, and
25
m= -Y, or, taking 4 times each, ^r = 9, (7' = IG, 7?i =: 25.
Starting again, put the first square = :<r, the second ^ 9, the
third = 16, whence the sum of the squares =- x* + 337
= {x' - 25)^
12
Therefore
'144
/144 \
and f -^ ,9, 16 j is a sol
ution.
33. [Fpigra7n-problem].
'OKTaSpd^ov; Koi. TrevraSpa^ovs ^oeas tis ffii^f.
Toi? TrpoTToXolcn ttuIv XPV^"^' a^rora^a/xcvos.
Kat Tifxrjv airihoiKcv vwlp iravruiv jiTpaymvov,
Tas iTTLTaxOeLcra^ Be^d/Jievov /iovaSas,
Kat TToiovvTa ttuXiv erepoV (re (f)ipiiv TtTpdywvov
KT7]adfj.evov nkivpdv crvvdeixa twv ;(0£'a)v.
'flare StacrreiXov toOs OKTaSpd^fiov^ ttoVoi ^crav,
Kai TrdXc tovs Irepovs irai kiye irevTiSpaxfiovi.
Let the given number {iinTaxOela-ai. fiovaScs) be 60. The
meaning is : A man buys a certain number of ;^dc? of
wine, some at 8 draclnnas, the rest at 5 eacli. He pays
for them a square number of drachmas. And if we add
60 to this number the result is a stjuare whose side = the
224 DIOPHANTOS OF ALEXANDRIA.
whole number of xo'es. Required how many ho bought
at each price.
Let X = the whole number of xo'es. Therefore a" - 60 = the price
paid, wliich is a square, (x - m)" say. Now - of the price
of the five-drachma xoes + o of the price of the eight-
drachma xo'es = X. We cannot have a rational solution
unless ic > Q (x" - GO) < g {x^ - 60).
Therefore a;' > 5a; + 60 < 8x + 60.
Hence x" = 5a; + a number > 60, or x is' -^ 11.
Also a;- -f: 8a; + 60.
Therefore a; is :|» 12, so that x must lie between 11 and 12.
But a;'-60=(a;-«^)^
therefore a;= —^ — , which > 11, < 12,
whence vf + 60 > 22m < 24??t.
From these we find, m is not > 21, and not < 19.
Hence we put x' - 60 = {x - 20)',
and a; = 11^.
Thus a;^-132i, a;^- 60 = 721,
and 72^ has to be divided into two numbers such that - of
the first + Q of the second = 11 A. Let the first = z.
o
Therefore
|.^(72i-.) = lH,
or
and
5. 79
"' ~ 1 o
79
Therefore the number of xocs at five-drachmas = j^ -
59
eight „ -j2-
[At the end of Book v. Bachet adds 45 Greek arithmetical epi-
grams collected by Salmasius, which however have nothing to do
with Diophantos.J
1 See pp. 'JO, *J1 for uu uxijlaualiuu of thusu liiuils.
225
BOOK VI.
1. To find a rational right-anfjkd triamjle such that the hypote-
nuse exceeds each side by a cube.
Suppose a triangle formed from tlie two numbei-s x, 3.
Therefore hypotenuse =.x--+ 9, perpendicular = G.r, base=x'-9.
Therefore by the question x^ + 9 - (x* - 9) should l)e a culx?, or
18 should be a cube, which it is not. Now 18 = 2. 3*,
therefore we must replace the number 3 by m, where
2»r = a cube ; i. e. m = 2,
Thus, forming the triangle from .r, 2, viz. (x" + 4, \x, x^ - 4),
we must have a;" - 4a; + 4 a cube.
Therefore {x - 2)- = a cube, or x-2 - a cube - 8 say.
Hence x =10,
and the triangle is (40, 96, 104).
2. To find a right-angled triangle such that the sum of the
hypotenuse and either side is a cube.
Form a triangle as before from two numbers, and one of them
must be a number twice whose square - a cube, i.e. 2.
Therefore, forming a triangle from x, 2, or (x' + 4, 4x, 4 - x*)
we must have a;* + 4x + 4 a cube, and x^ < 4.
■ " 27
Hence a; + 2 = a cube, which must be < 4 > 2 -^ — say.
o
Therefore ^ "^ "s" '
. , . /135 352 377\
and the triangle is (^_ , -^ - , — j .
3. To find a right-angled triangle such that the sum of the area
and a given number is a square.
Let 5 be the given number, (3a;, 4a;, o.^) the triangle.
Therefore 6 1* + 5 = square = 9x' say.
5
Hence 3x-^ = 5, and ^ is not a square ratio.
Hence I must find a triangle and a iuiml)er such that the
difference of the square of the number and the area of
the triangle has to 5 a square ratio, L e. - ^ of a square.
.. .. 13
226 DIOPHANTOS OF ALEXANDRIA.
Form a triangle from x, - : then the area = a;* — 5, and let the
° ' «' a;'
1 2.5 ^. _ . 101 1 -
number =x-i , so that i . 5 -\ — j- = - of a square,
or, 4 . 2o + — 15- = a square = (10 + - j
24
Whence cc = -^ .
o
24 5
The triangle must therefore bo formed from -^ j ni >
and the number is -7:7; .
oU
Put now for the original triangle (Jix, 2>x, bx), where (hj^b) is
24 5 pbx' 170569 ,
and we have the solution.
4. To find a right-angled triangle such that its area exceeds a
given number by a square.
Number 6, triangle {3x, ix, 5x).
Therefore Gx^ - 6 = square = 4aj* say.
Hence, as before, we must find a triangle and a number such
that the area of the triangle - (number)^ = -^ of a square.
Form the triangle from ?», — .
° 7)1
1 fi 1
Therefore its area = ju' :,, and let the number he m--^. — .
1)1, z m
Hence G (G s
\ m.
or, 36;>i" - GO = a square = (6»i - 2)*.
Therefore vi = .. , and the triangle must be formed from (-^, -A,
the number being ^ .
5. To find a riglit-anglcd tri;nigle such that a given number
exceeds the area by a S(juare.
Number 10, triangle (3x', ix, 5x). Therefore lO-Gx-'-a
square,
ARITHMETICS. BOOK VI. 227
and a triangle and a number must be found sudi tli:it (nund.fr)*
+ area of triangle = -- of a square. Form a triangle frum
m, - , and let the number be - + 5m.
Dt m
Therefore 260«i' + 100 = a square,
or 65»i* + 25 ^- a square = (8ni + 5)^ say.
Therefore m - 80.
The rest is obvious.
6. To find a right-angled triangle such that th'' sum of the area
and one side* about the right angle is a given number.
Given number 7. Triangle {3x, 4x, 5x),
therefore 6x"+3£c=7.
■^j +6.7 not being a square, is not possible.
Hence we must siibstitute for (3, 4, 5) a right-angled triangle
, , /onesideX* ^ . ,
sucli that ( — ^ j + I times the area = a square. Let
one side be x, the other 1.
7 1
Therefore ^x + j = a. square,
or lix + 1 = a square)
Also, since the triangle is rational, x'+l = a squarei '
Now the difference — x^ — lix = x(x~ 14). Therefore, putting
24
7^ = 14a; + 1, we have x- -=-. Therefore the triangle is
/24 25\
(-^ , 1, -;^ ), or we may make it (24, 7, 25).
Going back, we take as the triangle (24.c, 7x, 25x).
Therefore 84a;" + lx-l, and x^ - .
4
. / 7 25
Hence the triangle is ( 6, t , .
7. To find a right-angled triangle sucii that its area exceeds ouo
of its sides by a given number.
1 N.B. For brevity and distinctness I slmll in future call llic flidcs about
the ri«ht angle simply "sides," and not apply the term to the hyiwUjnuMC. which
will always be called "hypotenuse."
1 .;— 2
228 DIOPHANTOS OF ALEXANDRIA.
Given number 7. Therefore, as before, we have to find a
right-angled triangle such that ( ~ ) + ^ times area
= a square, i.e. the triangle (7, 24, 25).
Let the triangle of the problem be (7a;, 24a;, 25a;).
Therefore 84a;^ - 7a; = 7, and x= ^.
8. To find a right-angled triangle such that the sum of its area
and both sides = a given number.
Number 6. Again I have to find a right-angled triangle such
/sum of sidesX'
that ( ~ J -f times area = a square. Let «i,
1 1 .1 -1 ..1 p fm+l\- „ m" 7m 1
1 be the sides; therefore ( -— j +3m = ^ +-^ + - = a
square, and m" -f- 1 = a square.
Therefore vi' + 1 im + 1) , , ,
, V are both squares,
m- + 1 J ^ '
and the difierence = 2«i . 7.
45
Therefore "'' ^ 28 '
(45 53\
Assume now for the triangle of the problem (45a;, 28a;, 53a;).
Therefore G30a;' + 73a; = 6,
and X is rational,
9. To find a right-angled triangle such that its area exceeds
the sum of both sides by a given number.
Number 6. As before we find subsidiary triangle (28, 45, 53).
Therefore, taking for the required triangle (28a;, 45a;, 53a;), we
find 6S0x- - 73a; = 6,
and x= ^ .
65
10. To find a right-angled triangle such that the sian of its area,
hypotenuse, and one side is a given number.
Given number 4. Assuming hx, px, bx, ,-, - + hx ^-bx=i,
and in order that this equation may have a rational solu-
tion I must find a triangle such that
/hypotenuse -f one sideV
-f- 4 times area = a square.
I
AlUl'lI MIOTICS. BOOK VI. 229
^[ake a right- angled triangle from ;«, m+ 1. Therefore
/hyiJOtenuse + one sicle\ * /2«t* + 2//i + 1 + '2m + 1\*
V 2 ; =v 2 )
= Hi* + im' + Gin' + 4//t + 1
and 4 times area = im {/a + 1) (2m + 1)
which = Sm^ + 12/«- + im.
Therefore
m* + I2iu^ + ISm- + Sm + 1 ^ a square = (7/4- + Gm - 1)^ say.
Hence m=,,
4
and the triangle must be formed from ( , - j, or (5, 9).
Thus we must assume for the triangle of the problem the
similar triangle {2Sx, 45.f, ~^3.c), and G30.C* + 81a-= 4.
4
Therefore x = — — .
lOo
11. To find a right-angled triangle such that its area exceeds the
sum of the hypotenuse and one side by a given number.
Number 4. As before, Vjy means of the triangle (28, 45, 53)
we get G30a;" — 81.'; = 4.
Therefore x = ^ .
6
12. To find a right-angled triangle such that the difference of
its sides is a square, and also the greater alone is a square, and,
thirdly, its area -1- the less side - a square.
Let the triangle be formed from two numbei-s, the gi-eater side
being twice their product. Hence I must find two
numbers such Uiat twice their product is a square and also
exceeds the difference of their squares by a square. This
is true for any two numbers of which the gi-eater - twice
the less.
Form then the triangle from x, 2x, and two conditions are
fulfilled. The third condition gives Gx* + 3x* - a wjuarc,
or 6x^ -I- 3 = a square. Therefore we must seek a number
such that six times its square with 3 produces a square,
i.e. 1, and an infinite number of others.
Hence the triangle required is formed from 1, 2.
Lemma. Given two numbers whose sum is a s(iuai-o, an infinite
number of squares can be found which by multiplication with one of
230 DIOPHANTOS OF ALEXANDRIA.
the given ones and the addition of the other to this product give
squares.
Given numbers 3, G. Let x" + 2a; + 1 be the square required,
which will satisfy
3 (x^ + 2a; + 1) + 6 = a square, or 2>x- + G.r + 9 = a square.
This indeterminate equation has an infinite number of
solutions.
1 3. To find a right-angled triangle such that the sum of its area
and either of its sides = a square.
Let the triangle be (5a;, 12a;, 13x).
Therefore 30.«' + 1 2a; = a square = 36a;^ say.
Therefore 6a; = 12, and a; = 2.
But SOx^ + 5x is not a square when x = 2. Therefoi'e I must
find a square m^a;^ to replace 36a;" such that the value
12
—5 — :ryr of X IS veol and satisfies 30a;- + 5a; = a square,
m - 30
rru- • 1 1 .-. .• 60m^+2520
This gives by substitution -, — z^^ ,, . . = a square.
*' ^ m* - 60m + 1)00 ^
Therefore 60??r + 2520=a square. If then [by Lemma'] we
had 60 m' + 2520 equal to a square, the equation could he
solved.
Now 60 arises from 5, 12, i.e. from the product of the sides
of (5, 12, 13); 2520 is the continued product of the
area, the greater side and the difference of the sides
[30. 12.1235].
Hence we must find a subsidiary triangle such that the pro-
duct of the sides + the continued product of greater side,
difierence of sides and area - a square.
Or, if we make the greater side a square, we must have [dividing
by it], less side + product of difference of sides and area
= a square. Therefore we must, given two numbers
(area and less side), find some square such that if we
multiply it by the area and add the less side, the result is
a square. 2'his is done hy the Lemmas^ and the auxiliary
triangle is (3, 4, 5).
1 Diophantos has expressed this rather curtly. If (h p b) bo the triangle
(b>p), we have to make hp + ^bp . (b -p) b
a Kquarc, or if b is a square, 2' + i ^'P U'-p) must be a square.
ARITHMETICS. I50<)K VI. liSl
Thus, if the original triangle is {Zx, ix', 5x),
we have Gx-* + 4a;) , .
4
Let !»= — a —„ be the solution of the first equation.
9G 12
Therefore the second gives — ; — q-^; — 7. — Trr. + . " ^ = a S(iuarc.
" 7Ji'-12//i' + 30 m'-G '
Hence 12 m^ + 24 = a square,
and we must find a square such that twelve times it + 24 = a
square [as in Lemma],
Therefore m^ = 25,
and a; = Tj-jT .
^. . , . , . , . /12 16 20\
Therefore the triangle required isLq, jq, -.q)-
14. To find a right-angled triangle such that its area exceeds
either side by a square.
The triangle found as before to l)c similar to (3, 1, 0), i.e.
(3a;, ix, 5x).
Therefore 6.r - 4a; = square ^ m" {< G).
4
Hence x =
Q-m"
96 12
'''''^ (6 - my 6 - wr
a square, or 24 + 12»t-
a stjuaiv
Let m = 1 say.
Therefore
4
X ~ ^ ,
/12
16 A
and the triangle i« v g . 5 ' - / •
Or, putting m = z+l, we tiiid 3r + 6c+9 a square, ami
13 22 .1
z^^, 3+1^0 , SO that X- is rational.
This relation can be satisfied in an infinite number of ways it b- pin a «,uaro,
and also ;; + i />i). ., ,._
Therefore wo liave to find a triant^le such that Krcatcr side ^^luore. difference
of sides = s(iuare, less side + area = square.
Form the triangle from (u, h), therefore greater side =2.1,. which ib a Hqimro.
if a -26, difference of sides =16^-36^= a square, less side + area -3&> + 0fc»=
a square.
232 DIOPHANTOS OF ALEXANDRIA.
15. To find a right-angled triangle such that its area exceeds
either the liypotenuse or one side by a square.
Let the triangle be (3a;, ix, 5x).
Therefore 6a;* -5a;) , ^,
V are both squai-es.
\)X — ox\
3
Making the latter a square, we find x = ;, Cm? < 6).
Therefore from the first ,^ ^, - t, ; = a square, or
(G - m)- - m
15wt* - 36 = a square.
This equation we cannot solve, since 15 is not the sum of two
squares.
Now 15??i^=the product of a squai-e less than the ai-ea, the
hy})0teuuse, and one side ; 36 = the continued product of
the area, one side, and the difference between the hypote-
nuse and that side.
Hence we must find a right-angled triangle and a square such
that tlie square is < 6, a^id the continued pi'oduct of the
square, the hyj)otenuse of the triangle, and one side of it
exceeds the continued product of the area, the said side
and the difference hetioeen the hypotenuse and that side by
a square.
[Lacuna and coiTuption in text'].
Foi*m the triangle from two "similar plane numbers" [numbers
of the form ah, oir], say 4, 1. This will satisfy the con-
ditions, and let the square be 36. (< area.)
The triangle is then (8*, \bx, 17a;).
Therefore GO.r- - 8a; = 360;^^ say.
1
Thus x = y^,
and the triangle ^^ (3' ^' y )
1 Schulz works out the subsidiary part of this problem thus, or rather only
proves the result given by Diophantos that the triangle must be formed from
two "similar plane numbers'' a, aU- [i.e. a. 1 and ah. h.] ; and hyp. h = a-h^^-a-.
greater side ij = a-b* - «-, less side k = 2a-b'\ area /= ^ kg. Now
h-k = a^b* - 2a-b'^ + a" = {ab"^ - ay,
ft square ; and hkz'^ - Jcfih - ft) is a square i{ z-=k (h - k) k, for, if we then divide
by the square h - k and twice by the square kk, we get 2 (k-(i)^ia\ which is a
square.
AUlTilMETlCS. I'.uolv VI. 233
16. Given two numbers, if some square be multiplictl by one of
them, and the other be subtracted, the result being u square, then
another square can be found greater than the tii-st square wliich han
the same property. [Leitwia to the following problem.]
Numbei-s 3, 11, side of square 5, so that 3. 25 -11= 64 = a
square. Let the required square be (.« + 5)*.
Therefore
3 (a; + 5)- - 1 1 = 3.v" + 30x + 64 = a square = (8 - 2.r)» say.
Hence x = 62.
The side of the square = 67, and the square it.self = 4489.
17. 2h find a riyht-amjled triamjle such that the sum vf the area
ami either the hypotenuse or one side = a square.
We must first seek a triangle {h, k, (j) and a square s' such
that hkz- - ka {h -k) = a, squai-e, and z' > «, the area.
Let the triangle be formed from 4, 1, and the square be 36, but,
the triangle being (8, 15, 17), the square is not > area.
Therefore we must find another square to replace 36 by
the Lemma in the preceding. But
hk = 136, ka {h - i) = 480 . 9 = 4320.
Thus 36 . 136 - 4320 = a square, and we want to find a larger
square {m') than 36 such that 136?«' - 4320 - a square.
Putting m = z+ 6, (s- + 122 + 36) 136 - 4320 = square,
or, 136^' + 16322 + 576 = a square = (»z - 24)* say.
This equation has any number of solutions, of which one
gives 676 for the value of {z + 6)' [putting n = 16].
Hence, putting for the triangle (8x, 15.i-, 17x), we get
60x* + 8a; = 676x-*,
Therefore ^ ^ 77 "
18. To find a right-anfied triangle such that the Hue hiscctiw/ nti
acute angle is rational.
Let the bisector (A D) = 5a; and one .section of tlu- has,- ( Itli) .ij,
so that the perpendicular \x.
234 DIOPIIANTOS OF ALEXANDRIA.
Let the whole base be some multiple of 3, say 3. Then
CD = 3-3x.
But, since AD bisects the i BAC, the hypotenuse = - (3 - 3a;),
therefoi-e the hypotenuse = 4 - 4a;.
Hence IGa;' - 32a; + 16 = 16a;= + 9, and a; = ^ .
Multiplying throughout by 32, the perpendicular = 28, the base
= 96, the hypotenuse = 100, the bisector =: 35.
19. To find a right-angled triangle such that the sum of its area
and hypotenuse = a square, and its perimeter = a cube.
Let the area = x, the hypotenuse = some square minus x, say
16 -a;; the product of the sides = 2x. Therefore, if one
of the sides be 2, the other is x, and the perimeter = 18,
which is not a cube.
Therefore we must find a square which by the addition of 2
becomes a cube.
Let the side of the square be {x+ 1), and the side of the cube
(a:-l).
Thei-efore a;^ - 3a;^ + 3a; - 1 = a;^ + 2a; + 3, from which a; = 4.
Hence the side of the square is 5, and of the cube 3.
Again, assuming area = x, hypotenuse = 25 - a?, we find that
the perimeter = a cube (sides of triangle being x, 2).
But (hypotenuse)'' = sum of squares of sides.
Therefore of - 50a; + 625 = a;- + 4,
621
and x=-^.
20. To find a right-angled triangle such that the sum of its area
and hg2>otenuse = a cube, and the perimeter = a square.
Area x, hypotenuse some cube mimis x, sides x, 2.
Therefore we have to find a cube which by the addition of 2
becomes a square. Let the side of the cube = m-1.
(3 \*
^m + lj say.
<¥)'• '
Put then the area a;, the sides x and 2, the hypotenuse ^y^ -x.
(4913 \*
—rrr xj = a;^ + 4 gives a;.
21
Thus wi = -J-, and the cube ■■
4913
AlUTIIMETICS. ROOK VI. 235
21. To find a right-angled triancjle mch that thf sum of its area
and one side is a square and its perimeter is a cube.
Make a riglit-angk'il triangle from .r, x + 1.
Therefore the i)erpendicular -2x+\, the base = 2x* + 2x, the
hypotenuse = 2x* + 2x* + 1 .
First, Ax- + Ga; + 2 = a cube, or (4a; ■(- 2) {x + 1 ) = a cube. If wo
divide all tlie sides by x + 1 we have to make 4x + 2
a cube.
Secondly, area + perpendicular = a square.
^, - 2x^ + 3xVx 2x+l
Ihereiore — ; r-r^ — + -^ = a square.
(x + 1)* X + 1 ^
2x' + 5x* + 4x+l „ ,
Hence ^39" i " ^ 2x+ 1 =a square.
But 4x+ 2 = a cube. Therefore we must find a cuU- which is
double of a square.
3
Tlierefore 2x + 1 = 4, x = - ,
and
.1 . ■ 1 • /8 15 17\
the triangle IS (^g, ^ , -j
22. To find a right-angled triangle such that tlie sum of its
area and one side is a cube, while its perimeter is a square.
Proceeding as before, we have to make
4x + 2 a squarej
2x + 1 a cube / '
Therefore the cube = 8, the square = IG, »-• = .,,
and the triani
, . /16 63 65\
23. To fiml a right-angled triangle such that its perimeter is a
square, and the sum of its perimet^ and area is a cube.
Form a right-angled triangle from x, 1.
Therefore the sides are 2.7;, x-*- 1, and the hypotenuse x* + 1.
Hence 2x* + 2x should be a square, and x' -I- 2x* -f x a cul>c.
It is easy to make 2x'' + 2x a square : let it ^ 7«V.
2
Therefore x- ^"' ^ , and from the second condition
m* -2
8 8 _2^
{m' - 2)' "^ {m' - 2)' "^ m' - 2
must be a cube, i.e. 7-^ — rr-3 = a culto.
(w -J)
236 DIOPHANTOS OF ALEXANDRIA.
Therefox'e 2m* = a cube, or 2m = a cube = 8 say.
2 1 I
Thus ??i = 4, .X' = r7 = - . ^^^ *" ^ 7a •
14 7 49
But foi* one side of the triangle we have to subtract 1 from
this, which is impossible.
Therefore I must find another value of a; > 1 : so that
m" > 2 < 4.
And I must find a cube such that \ of the square of it
> 2 < 4.
Let it be n^, so that ?i" > 8 < 1 G, This is satisfied by
, 729 3 27
-"=G4''^==T-
97 729 512
Therefore m=^^, nr = . _ _ , x = ^^^ , and the square of this
16 25G 21/
> 1. Thus the triangle is known.
24. 7'o Jiiid a right-angled triangle such that its 2)erimeter is a
cube and the sum of its perimeter and area = a square.
(1) We must first see how, given two numbers, a triangle
may be formed whose perimeter = one of the numbers,
and whose area - the other.
Let 12, 7 be the numbers, 12 being the perimeter, 7 the area.
Therefore the product of the sides = 14 = - . 14.u
Thus the hypotenuse = 12 — ; — 1 4x'.
Therefore from the right-angled triangle
1 24 1
172 + 4 + 19Ga;'' - 336x - — = -^ + 196a;^
a;- X X
or, 172 - 336a;- ^- = 0.
' X
This equation gives no rational solution, unless 86"- 24. 336
IS a square.
But 172 :^ (perimeter)- + 4 times area, 24 . 336 = 8 times area
multiplied by (perimeter)".
(2) Let now the area = x, the perimeter = any number which
is both a square and a cube, say 64.
Therefore ( — a ] - 8 . 64" . a; must be a square,
or, 4a;' - 2 4 5 7 6a; + 4 1 9 4 3 4 ^ a square.
AUITMMETICS. I500K VI. li.ST
Therefore x' - GlU.c + 1048570 is a square.)
Also X + 04 is a square./
To solve this double equation, multiply the second equation
by such a square as will make the absolute t<.'rm the same
as in the first. Then, taking the difference and factors,
itc, the equations are solved.
[In the text we find i$i(Tw(r$o} aoi ol dpiO/JLoi, which, besides
being ungrammatical, would seem to be wrong, in that
dpiOfjiOL is used in an unprecedented manner for /loraoe?.]
25. To find a right-angled triangle such that the square of its
hypotenuse = the sum of a square and its side, <and the quotient
obtained by dividing the (hypotenuse)^ by one side of the triangle =
the sum of a cube and its side.
Let one of the sides be x, the other x'.
Therefore (hypotenuse)" = the sum of a square and its side,
and = a cube + its side,
X
Lastly, X* + x' must be a square.
Therefore of + 1 = a square = {x - 2)' say.
3
Therefore a; = -j , and the triangle is found.
26. To find a right-angled triangle such that one side is a cube,
the other = the diflerence between a cube and its side, the hypotenuse
= the sum of a cube and its side.
Let the hypotenuse = a;' + x, one side = x^-x.
Therefore the other side = 2x* = a cube.
Therefore x = 2,
and the triangle is (6, 8, 10).
TRACT ON TOLYGONAL NUMBERS.
1. All numbers, from 3 onwards in order, are polygonal, con-
taining as many sides as units, e.g. 3, 4, 5, &c.
" As a square is formed from the multiplication of a number
by itself, so it was proved that any polygonal multiplied
by a number in proportion to the number of its sides,
with the addition to the product of a square also in pro-
portion to the number of the sides, became a square.
This we shall prove, first showing how a polygonal num-
ber may be found from its side or the side from a given
polygonal number."
2. If there are three numbers equally distant from each other,
then 8 times the 2>i'odicct of the greatest and the middle + tlie square of
the least = a square whose side is (greatest + twice middle number).
Let the numbei-s be AB, BG, BD (in fig.) we have to prove
8 {AB){BG) + {BDy- = [AB + 2BGy.
E A B..D...G
Now AB = BG+GI).
Therefore
SAB . BG - 8 (BG' + BG . GD) = iAB . BG + iBG' + 4BG . GD.
and iBG . GD +DB'^ AB' [for AB=BG + GD, DB = BG-GD\
and we have to seek how AB" + iAB. BG -\- iBG^ can be made
a square.
Take AE^ BG.
Therefore iAB . BG = iAB . AE.
This together witli ABG' or iAE' makes iBE.EA, and this
together with AB' = [BE+EA)- = (AB + 2BGy.
3. If there are any numbers in A. p. the difference of the greatest
and the least > the common difference in the ratio of the number of
terms dimiuiahcd by 1.
POLYGONAL NUMBERS. 239
Let AB, BG, BD, BE... he in a. p.
B.A..G..D.. E
Therefore we must have, difference of AB, BE^ (difference of
AB, BG) X (number of terms- 1).
AG, GD, DE arc all equal. Therefore EA = AG >i (number
of the terms AG, GD, DE) ^ AG x (number of term.s in
series- 1).
Therefore (kc.
4. If there are any numbers in a.p. {greatest + least) x number of
terms = double the sum of all. [2s = 7i{l + a).]
Let the numbers be ^, 2^, C, D, E, F.
(A +F) X the number of them shall be twice the sum.
A.B.C.D.E.F
H.L.M.K...G
The number of terms is either even or odd ; and let their
number be the number of units in IIG.
First, let the number be even. Divide IIG into two equal
parts at A'.
Now the difference of i^, Z) = the difference of C, A.
Therefore F+ A =C + I),h\it F + A = {F+ A) HL.
Hence C + D = {F+A)LM, E+B = {F+A)MK.
Therefore A + B + ... = (F+A) UK.
And {F + A) IIG ^twice (A +B +...).
5. Secondly, let the number of terms be odd, A, B, C, D, E,
and let there be as many units in FU as there are terms, «J:c.
A.B.C.D.E
F.G.K.II
6. If titer e are a series of numbers beginning loilh 1 and increas-
ing in A. p., then the sicm of all x eight times the common difference
+ the square of {common difference - 2) = a square, whose side dimin-
ished by 2^ the common difference multijilied by a number, which
increased by 1 is double of the number of terms.
[Let the a.p. be 1, \ + a, ... 1 + n - 1 . a.
Therefore we have to prove
s.8a + {a-2y = {a{2n-\) + 2y,
i, e. 8as = 4a V - 4 (a - 2) na,
or 2» - an' - (o - 2) « = n (2 + n- la)].
240 DIOPHANTOS OF ALEXANDRIA.
Proof. Let AB, GD, EZ be numbers in A. p. starting from 1.
A.K..N...B G D E.L Z
H.M X—T
Let HT contain as many units as there are terms including \.
Difference between EZ and 1 = (difference between -4 5 and 1)
X a number 1 less than IIT [Prop. 3].
Put AK, EL, HM each equal to unity.
Therefore LZ=MT.KB.
Take KN = 2 and inquire whether the sum of all x eight times
KB + square on NB makes a square whose side diminished
by 2 = KB X sura of HT, TM.
Sum of all = I product {ZE + EL) .IIT=\ {LZ + 2EL) HT,
and LZ= AIT . KB from above.
Therefore the sum = \ (KB . iVT . TH+ 2TH),
or, bisecting MT at X, the sum = KB . TH . TX+ HT.
Thus we inquire lohether
KB. TH. TX. SKB + 8KB . HT + square on KB
is a square.
Now SHT . TX . KB' = iHT . TM . KB',
and SKB . HT = AHM. KB + i (HT + TM) KB.
Therefore toe must see lohether
i.HT. TM. KB' + iHM. KB + 4 (HT + TM) KB + NB"-
is a square.
But 4/7.1/ . KB = 2KB . NK,
and 2KB.NK+NB-=KB- + KN%
and again /JA'^ = HM' . BK\
and HM\BK"- + UlT . TM . BK'= {HT+ TMf BK\
Hence our expression becomes
{HT+ TMf Bid + 4 {HT + TM) KB + A'iV^^
A.K..N ...B R
H.M A'— T
or, putting {HT + TM) BK= NR, NR' + iNR + KN'
and 4.NR ^ 2NR . NK.
Therefore the given expression is a square whose side is RK,
and RK -2 = NR, which is KB {HT + TM),
and HT+ TM+ 1 = twice the number of terms.
Thus th(! proposition is proved.
7. Let
POLYGONAL NUMBERS.
HT+TM^A, KB=B.
K 1j
241
Therefore square on .1 x square on B = square ou G, where
G = {HT+TM)KB.
Let DE = A, EZ =^ B, in a straight line.
Complete squares DT, EL, and complete TZ.
Then DE : EZ^DT : TZ, and TE ■ EK=TZ : EL.
Therefore TZ is a mean proportional between the two squares.
Hence the product of the squares = the square of TZ, and
DT^ {IIT -h TMf, ZK = square on KB.
Thus the product (HT + T2If. KB' = NB^.
8. If there are any number of terms heginning from 1 in a. p.
the svm is a jiolygonal number, for it has as many angles as the common
difference increased by 2 contains units, ami its side = the number
of terms inclibding the term 1.
The numbers being represented in the figure, (sum of series
multiplied by ^KB) + NB- - RK\
O.A.K..N...B RG-
II . M A'-
-D /•; . L-
-T
Therefore, taking another unit AO, KO - 2, KN -- 2, and
OB, BK, BN are in arithmetical progression, so that
S.OB.BK + BN' = {OB + 2BKy,
[Prop. 2], and OB + 2BK- OK - ZKB an.l 3+12.2,
or 3 is one less than the double of the common difference
of OB, BK, BN.
Now as the sum of the terms of the j)rogressiou, including unity,
It;
H. D.
242 DIOPHANTOS OF ALEXANDRIA.
is subject to the same laws as Oi? ', while OB is any number
and OB always a polygonal (the first term being AO [1]
and AB the term next after it) whose side is 2, it follows
that the sum of all terms in the progression is a polygonal
equiangular to OB, and having as many angles as there
are units in the number which exceeds by OK, or 2, the
difference KB, and the side of it is HT which = number
of terms, including 1.
And thus is demonstrated what is said in Hypsikles^ definition.
If there are any numbers increasing from unity by equal
intervals, when the interval is 1, the sum of all is a tri-
angular number : wlien 2, a square: when 3, aj)entagon and
so on. And the number of angles = 2 + common difference,
the side = number of terms including 1.
So that, since we have triangles when the diffei'ence = 1, the
sides of them will be the greatest term in each case, and
the product of the greatest term and the greatest term
increased by 1 - twice the triangle.
And, since OB is a polygonal and has as many angles as
units, and when multiplied by 8 times (itself - 2) and
increased after multiplication by the square of (itself — 4)
[i.e. NB-] it becomes a square, the definition of polygonal
numbers will be :
Every polygonal multiplied 8 times into (number of angles
— 2) + square of (number of angles — 4) = a squax'e.
The Hypsiklean definition being proved, it remains to show
how, given the sides, we may find the numbers.
Now having the side HT and the number of angles we know
also KB, therefore we have {IIT + TM) KB = NR.
Hence KR is given [NK^1\
* This result Nesselmann exhibits thus. Take the aiithmetical progression
1, 6 + 1, 2& + l...(K-l)t + l.
If s is the sum, Qsh + (l) - 2)'^=[h (2k - 1) + 2p,
If now we take the three terms 6-2, h, h + 2, also in a. p.,
8b(?;+2) + (?i-2)' = [(6 + 2) + 26]'
= (3!> + 2)2,
Now 6 + 2 is the sum of the first two terms of first series; and 3 = 2.2-1,
therefore 3 corresponds to 2h - 1.
Hence s and h + 2 are subject to the same law.
POLYGONAL NUMBERS. 243
Therefore we know also the square of KR. Subtracting from
it the square of NB, we have tlie remaining term
which = number x '^KB.
Similarly given the number we can find the side.
9. Rule. To Jhid the number from the side.
Take the side, double it, subtract 1, and multiply the remamder
by (number of angles - 2). Add 2 to the product, and
from the square of the number subtract the square of
(number of angles — 4). Dividing the remainder by 8
times (number of angles - 2), we find the required
polygonal.
To filed the side from the numher. Multiply it l>y 8 times
(number of angles - 2), add to the product the square of
(number of angles - 4). We thus get a square. Subtract
2 from the side of this square and divide remainder by
(number of angles - 2). Add 1 to quotient and half the
result is the side required.
10. [A fragment.]
Given a numher, to find in how viany loays it can he a polygcmal.
Let AB be the given number, BG the number of angles, and
in BG take GD = 2, GU - 4.
A . T B E..D..G K
Z H
Therefore, since the polygonal AB has BG angles,
%AB . BD + BE- = a square = ZIP say.
Take in AB the length AT=\.
Therefore MB . BI)= iAT . BD + i (AB + TB) BD.
Take DK=i{AB+TB),
and for AAT.BD put 2BD . DE.
Therefore ZIP = KD . BD + 1BD .DE + BE*,
but 2BD . DE + BE' = BD' + DE\
Hence ZU ' ^ KD . BD + BD* + DE\
and KD . BD + BD^ - KB . BD.
Thus Zir=KB.BD+DE\
and, since DK = 4 {AB + TB), DK> 4 J T > 4, and half 4 - DG,
GK>GD.
244 DIOPHANTOS OF ALEXANDRIA.
Therefore, if DK is bisected at L, L will fall between G and K,
and the sqiiare on LB = LD' + KB . BD.
A . T B E..D..G L K
Z H N M
Therefore ZE' = BU - LD' + DE\
or ZH' + DL' = BU- + DE\
and LD"-~DE' = LB'~ZH\
Again since ED = DG and DG is produced to L,
EL.LG + GD'=DL\
Therefore DL' - DG' = DL' - DE' = EL . LG.
Hence EL . LG = LB' ~ ZIP.
Put ZM = BL {BL being > ZII).
Therefore ZM' - ZH' = EL . LG ; but DK is bisected in L,
so that DL = 2 (AB + BT) ; and DG = 2 AT.
Therefore GL = iB T, and BT-^^GL,
but also
AT {ov l) = ^^6-'(or 4).
Therefore AB = \ EL, but TB also = \ GL.
4 4
Hence AB.TB=^EL. LG,
or EL.LG=1(JAB.BT.
Thus UAB.BT = MZ' - ZII ' = 21 H ' + 2ZH . II M.
Therefore IIM is eve^i.
Let it be bisected in JV
[Here the fragment ends.]
INDEX
[The references are to pages.]
Ab-kismet, 41 u.
Abu'lfaraj, 2, 3, 12, 13, 41
Abu'l-Waffi Al-Biizjfmi, 13, 25—20,
40—42, 148, 155, 157
Abu Ja'far Mohammed ibn AUiusain,
156
Addition, how expressed by Diophantos,
69 ; Bombelli's sign for, 45 ; Vieta's,
78 «.
Algebraic notation, three stages of, 77
—SO
aljabr, 40, 92, 149—150, 158
Alkarkhi, 24—25, 71 ?;., 156—159
Al-Kharizmi, see Mohammed ibn Mfisfi
almuktibahi, 92, 149—150, 158
Al-Nadim, 39, 40 ii.
Al-Shahrastani, 41
Alsirfij, 24 u., 159
avaipopiKos of Hj-psikles, 5
6x)pl(TTu%, iv doplarcf), 140
ApoUonios, 4, 8, 9, 23
Approximations, 117—120, 147
Apukius, 15
Arabian scale of powers compared with
that of Diophantos, 70—71, 150—
151
Arabic translations, Ac, 23, 24, 25,
39—42, 148—159
Archimedes, 7, 142, 143, 144, 146, 147
Aristoxenos, 14, 15
Arithmetic and Geometry, 31, 141—
142
'ApiOfiriTLKo. of Diophantos, 33 and pas-
sim
apidfiijriKri and XoyiaTiK-f), IH, 136,
145
dpidfioi, 6 ; Diophantos' technical use
of the word, 57, 150; his sj-mbol for
it, 57- 66, 137—138, 160
apidfiOffTov, 74
Ars rei et census, 21 h.
Auria, Joseph, 51, 56
Autolykos, 5
Bacchios 6 y^pwv, 14, 15, 16
Bachet, 49 — 53 and passim
"Back-reckoning," H5 — 86, 114; ex-
amples of, 110, 111, and in the ap-
pendix passim
Bhaskara, 153
Billy, Jacobus de, 3, 54
Blancauus, 3
Bombelli, 13, 14, 15, 23, 35, 36. 42—
45, 52, 134—135; his algebraic no-
tation, 45, 68
Bossut, 32, 38, 90 n., 138—139 n.
Brahmagupta, 153
Brassinne, 221 n.
Camcrarius, Joachim, 2, 42
Cantor, 55 h., 58, 59. 67, 141 n., 151.
152, 156, 157
Cardan, 43, 46, 70
Casiri, 41 n.
Cattle-problem, the, 7, 142—117
Censo, 70
Coefficient, 93 «.
Colebrooke, 12, 19 n., 33, 133. 136.
137 n.
Cosa, 45, 70
Cossali, 1, 3, 10, 12, 31, 36, 41 n., 43 n..
49, 51, 70, 71. 107 n.. 133. 136. HO.
169 ;i.. 220 ;i.
Tridhara, 153
Cubes : transformation of a Bom of
two cube-s into the difference of two
others, and vice rer$ii, 123—125
Cubic equation. 30, 93—91. 114
246
Data of Euclid, 140
Dedication to Dionysios, 136
Definitions of Diophantos, 28, 29, 57,
67, 7-4, 137, 138, 163
Determinate equations : see contents ;
reduction of, 29, 149—150
Diagonal numbers, 142
Didymos, 14, 15, 16
Digby, 23
Dioi^hautos, see contents
s , . - J 35, 98
Division, how represented by Diophan-
tos, 73
Double-equations of the first and second
degrees, 98 — 107 ; of higher degrees,
112—113
Svva/jLis and the sign for it, 58 n., 62,
63, 66 7i., 67, 68, 140, 151 ; dvfa/xis
and Terpaywvoi, 67 — 68
dwa/xoduva/jus and the sign for it, 67 — 68
dvvainoOvvafJ.oa'Toi', 74
dvva/jMKv^os and the sign for it, 65 ii.,
67—68
Swa/JLOKV^offTov, 74
dwafioarov, 74
er5os = power, 29 7i.
Elements of EucUd, 4, 5, 142, 158
Epanthema of Thymaridas, 140
Epigrams, 2, 6, 7, 9, 142—147, 223
Equality, Diophantos' expression of,
75—76 ; Xylander's sign for, 76
Equations, classes of, see contents;
reduction of determinate equations,
29, 149-150
Eratosthenes, 5
Euclid, Elements, 4, 5, 142, 158; Data,
140
Eudemos, 67
Eunapios, 13
Fabricius, 1, 5, 14
Fakhn, the, 24—25, 71 n., 156—159
Fermat, 13, 23, 53, 54, 68, 123, 124,
125, 126, 128, 129, 130, 131, 221 n.
Fihrist, the, 39, 40, 41, 42
Fractious, representation of, 73 — 75
Gardthausen, 60, 64
Geminos, 18, 145—146
Geometry and algebra, 140 — 141, 151
—153, 156, 158
Geometry and arithmetic, 31, 141 —
142
Girard, Albert, 3 n., 55
Gow on Diophantos, 64 — 66 7i., 137 n.,
160
Hankel, 83—85, 129 n.
Harmonics of Diophantos, 14 ; of Pto-
lemy, 15
Harriot, 78 n.
Heiberg, 146—147, 160
Heilbronner, 3
Herakleides Ponticus, 16
Heron of Alexandria, 141, 153
Hipparchos, 5, 141
Hippokrates, 67
History of the Dynasties, see Abu'Ifaraj
Holzmann, Wilhelm, see Xylander
Hultsch, 146 n.
Hypatia, 1, 8, 9, 10, 11, 17, 38, 39 n.
Hypsikles, 4, 5, 6, 135, 242
Z for tffos, 75
lambHchos, 78, 79, 140
Identical formulae, 125
Indeterminate equations, 94 — 113, 144,
146, 147, 157, 158, 159
Irrationality, Diophantos' view of, 82
Isidoros, 5
Italian scale of powers, 70, 71
jabr, 40, 92, 149—150, 158
jiclr, 150
John of Damascus, 8
John of Jerusalem, 8
ka'b, 71 n., 157, 158
Kitab AljUtrist, 39
Kliigel, 11, 90 n., 144
Kostfi ibn Luk:l, 40
KvfioKvpos and the sign for it, 67—68
KvjBos and the sign for it, 58 n., 62, 63,
66h., 67— 68
Kuster, 8
INDEX.
247
Lato, 70
Lehmann, 60
Xfr^ty, and the symbol for it, 66 ;i., TI-
TS, 137, 163
Xei^tj iirl \e'i\pLv ■jroWairXaffiaffOuaa
TTOtet virap^LV, 13T n.
\i6((>avTos or Aew^aj'Tos, 14
Lessing, 142, 143, 144, 146 h.
Limits, jnethod of, 86, 8T, 115— IIT ;
approximation to, IIT — 120
\(yyi.<jTLKr] and apidfiy^TiKT], 18, 136, 145 —
146
Lousada, Miss Abigail, 56
Lnca Pacioli, 43, TO n.
Lucilius, 9
" Majuskelcursive " writing, 64, T2 71.
mal, Tin., 157, 158
Manuscripts of Diophantos, 19, 61
Maximus Planudes, 23, 38, 39, 51, 135
Meibomius, 14
Metrodoros, 10
7n(n!«, Diophantos' sign for, 66 /;., 71 —
73 ; Bombelli's, 45 ; Tartaglia'6,78 h.;
Mohammed ibn Miisfi's expression
for, 151
Minus vniltipUed bij minus gives plus,
137, 163
" Minuskclcursive " writing, 64
Mohammed ibn Miisa Al-Kliarizmi, 3,
40 n., 59, 92, 134, 148, 149—155,
156, 158
fxovaSis, 69 ; the symbol for, 69
Montucla, 3, 11, 53, 71, 136
imtfasxirln, 40 u.
mukdbala, 92, 149—150, 158
vifda, 150
Multiplication, modern signs for, 78 ;i.
lutqis, 151 n.
Nessehnann, 5, 10, 20, 21, 22, 23, 27,
81, 33, 34, 35, 36, 37, 44 n., 49, 51 h.,
54, 55, 58, 59, TT, T8, T9, 85, 8s, 91 h.,
92, lOS, 110, 114, 121, 125, 129 «.,
133, 142, 143 n., 144, 145, 140, 147,
169 n., 212 7!., 242 71.
Nikomachos, 6, 14, 15, 10, 38, 05 «.,
135, 151
Notation, algebraic: three stages, 77 —
80 ; drawbacks of Diophantos' nota-
tion, 80—82
Numbers which are the sum of two
squares, 127—130
Numbers wliich are the sum of three
squares, 130—131
Numbers as the sum of four squares,
131—132
dpyavQaai, 136 — 137
wpiafiivoi apidfjLol, 140
Oughtred, 78 77.
Pappos, 11, 12, 17, 65 71., 139
Trapio-OTTji or irapiffOTrrros ayoyy-ri, 117 —
120
Pcletarius, James, 2, 43
Pell, John, 56
Perron, Cardinal, 20
Phaidros, 14, 15
TrXaffficLTiKOv, 169 n.
Plato, 18, 141—142, 145
TrXij^os, coefficient, 93 71.
2)lus, Diophantos' expression of, 71,
137 71. ; Bombelli's s^-mbol for, 45 ;
Vieta's, 78 n.
Pococke, 2, 12, 41 ;i.
Polygonal Numbers, {31 — 35 and pas-
sim
Porisms, 18, 32—35, 37, 121—125, 210,
218
Poselger, 55, 120, 124 71.
Powers, additive and multiplicative
evolution of, 70—71, 150—161
Proclus, 142
Progression, arithmetical, summation
of, 239—240
irporacris and xp6fi\rjfi.a, 34
Ptolemy, Claudius, 9
Pythagoras, 141
Quadratic equation, solution of, 90—
93, 140—141, 151—155; tbo two
roots of, 92, 153—155
Radix, 68
Uumu-s, Peter, 10. \\, 15
248
INDEX.
Reduction of determinate equations,
29, 149—150
Eegiomontanus, Joannes, 2, 20, 21,
22, 23, 42, 46, 78
Eeimer, 32
Eelati, 71
Res, 68
Eiccati, Vincenzo, 27 n.
Eight-angled triangle: formation of,
in rational numbers, 115, 141, 142 ;
use of, 115, 127, 128, 155, 156 ; ex-
amples, APPENDIX, especially Book
VI.
pi^rj of Nikomachos, 151
Eodet, L., 29 n., 59, 60, 61, 62, 75—76,
91 n., 92, 134, 151, 155
Rosen, editor of Mohammed ibn Musa,
q. V.
Sursolides, 71
Suter, Dr Heinrich, 28 n., 50, 53 h.
Symbols, algebraic : see plus, minus,
&c.
tafsir on Diophantos, 40
Tannery, Paul, 6, 7, 9 n., 10, 13, 14,
15, 16, 133, 139, 142—146
tanto, Bombelli's use of, 45
Ta'rlkh Hokoma, 41
Tartaglia, 43, 78 n.
Theon of Alexandria, 8 n., 10, 11, 12,
13,38
Theon of Smyrna, 6, 135, 142
Thrasyllos, 15
Thymaridas, 140
Translations of Diophantos, see Chap-
ter III.
Salmasius, Claudius, 19 n., 224
Saunderson, Nicholas, 52 n., 133
Scholia on Diophantos, 38, 39, 135
Schulz, 55 and iwssiMi
Series, arithmetical; summation of,
239—240
shai, 150
" Side-numbers," 142
Simultaneous equations, how treated
by Diophantos, 80, 89, 113, 140
Sirmondus, Jacobus, 19 n., 20
Square root, how expressed by Dio-
phantos, 93 n.
Stevin, 3, 55
Struve, Dr J. and Dr K L., 142 n.
Subsidiary problems, 81, 86 ; examples
of, 97, 110, 111
Subtraction, Diophantos' symbol for,
66 71., 71—73; TartagUa's, 78 n.;
Bombelli's, 45
Suidas, 1, 8, 9, 10, 11, 12, 13, 45
Supersolida, 71
Unknown quantity and its powers in
Diophantos, 57—69, 139—140; in
other writers, 45, 68, 70, 71 n., 150,
151, 157, 158 ; Dioi^hantos' devices
for remedying the want of more than
one sign for, 80—82, 89, 179
ilnap^is, 29 n., 71, 137 n.
Usener, Hermann, 12 n.
Variable, devices for remedying the
want of more than one symbol for a,
80—82, 89, 179
Vieta, 52, 68, 78 n., 123—124
Vossius, 3, 21 71., 56
Wallis, 70 »„ 71, 138
Wopcke, 24, 25, 26, 155
Xylander, 45 — 51 and passim
Zcmus, 68
Zetetica of Victa, 52
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