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every problem from the proof-sheets, so that I hope and believe 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 affect 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 obscure 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 OR PAPERS READ OR REFERRED TO,

SO FAR AS THEY CONCERN OR ARE USEFUL

TO THE SUBJECT.

1. Books directly upon Diophantos.

XYLANDER, Diophanti Alexandrini Rerum Arithmeticarum Libri sex......

Item Liber de Numeris Polygonis. Opus incomparabile......Latine redditum et Commentariis explanatum......

Basileae, 1575. BACHET, Diophanti Alexandrini Arithmeticorum Libri sex, et de numeris multangulis liber unus.

Lutetiae Parisiorum, 1621. Diophanti Alexandrini Arithmeticorum libri sex, et de numeris multangulis liber unus.

Cum commentariis C. G. BACHETI V.C. et obseruationibus D. P. de FERMAT Senatoris Tolosani.

Tolosae, 1670. Schulz, Diophantus von Alexandria arithmetische Aufgaben nebst dessen

Schrift über die Polygon-zahlen. Aus dem Griechischen übersetzt und mit Anmerkungen begleitet.

Berlin, 1822. POSELGER, Diophantus von Alexandrien über die Polygon-Zahlen. Uebersetzt, mit Zusätzen.

Leipzig, 1810. CRIVELLI, Elementi di Fisica......ed i Problemi arithmetici di Diofanto Alessandrino analiticamente dimostrati.

In Venezia, 1744. P. GLIMSTEDT, Första Boken af Diophanti Arithmetica algebraisk Oefversättning

Lund, 1855. STEVIN and GIRARD, “Translation ” in Les Oeuvres mathématiques de Simon Stevin.

Leyde, 1684.

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BOMBELLI, L'Algebra......diuisa in tre Libri......

Bologna, 1579. FERMAT, Opera Varia mathematica.

Tolosae, 1679. BRASSINNE, Précis des Oeuvres mathématiques de P. Fermat et de l'Arithmétique de Diophante.

Paris, 1853. Cossali, Origine, trasporto in Italia, primi progressi in essa dell’Algebra ......Storia critica........

Parma, 1797. NESSELMANN, Die Algebra der Griechen.

Berlin, 1842. JOHN KERSEY, Elements of Algebra.

London, 1674. Wallis, Algebra (in Opera Mathematica.

Oxoniae, 1695-9). SAUNDERSON, N., Elements of Algebra.

1740.

3. Books which mention or give information about Diophuntos,

including histories of mathematics. COLEBROOKE, Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bháscara.

London, 1817. SUIDAS, Lexicon (ed. G. Bernhardy).

Halis et Brunsvigae, 1853. FABRICIUS, Bibliotheca Graeca (ed. Harless). ABULFARAJ, History of the Dynasties (tr. Pococke).

Oxon. 1663. Ch. Th. v. Murr, Memorabilia Bibliothecarum publicarum Norimbergensium et Universitatis Altdorfinae.

Norimbergae, 1786. DOPPELMAYR, Historische Nachricht von den Nürnbergischen Mathematicis und Künstlern.

(Nürnberg, 1730.) Vossius, De universae mathesios natura et constitutione......

Amstelaedami, 1660. HEILBRONNER, Historia matheseos universae.

Lipsiae, 1742. MONTUCLA, Histoire des Mathématiques.

Paris, An 7. KLUEGEL, Mathematisches Wörterbuch.

Leipzig, 1830. KAESTNER, Geschichte der Mathematik.

Göttingen, 1796. Bossur, Histoire Générale des Mathematiques.

Paris, 1810. HANKEL, Zur Geschichte der Mathematik in Alterthum und Mittelalter.

Leipzig, 1874. CANTOR, Vorlesungen über Geschichte der Mathematik, Band I.

Leipzig, 1880. Dr HEINRICH SUTER, Gesch. d. Mathematischen Wissenschaften.

Zürich, 1873. James Gow, A short History of Greek Mathematics.

Camb. Univ. Press, 1884.

4. Papers or Pamphlets read in connection with Diophantos. POSELGER, Beiträge zur Unbestimmten Analysis.

(Berlin Abhandlungen, 1832.) L. Rodet, L'Algèbre d’Al-Khārizmi les methodes indienne et grecque.

(Journal Asiatique, Janvier, 1878.) WOEPCKE, Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed

ben Alhaçan Alkarkhi, précédé d'un memoire sur l'algèbre indéterminée chez les Arabes.

Paris, 1853 WOEPCKE, Mathématiques chez les Orientaux.

1. Journal Asiatique, Février Mars, 1855.

2. Journal Asiatique, Avril, 1855. P. TANNERY, "À quelque époque vivait Diophante ?" (Bulletin des Sciences

Mathém. et Astronom. 1879.) P. TANNERY, L'Arithmétique dans Pappus...... (Bordeaux Vemoirs, 1880.) Rosen, The Algebra of Mohammed ben Musa.

London, 1831. HEIBERG, Quaestiones Archimedeae.

Hauniae, 1879.

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THE WORKS OF DIOPHANTOS; THEIR TITLES AND GENERAL

CONTENTS; THE PORTIONS WHICH SURVIVE.

18-23

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23-26

§ 1. Titles: no real evidence that 13 books of Arithmetics ever existed

corresponding to the title .
No trace of lost books to be restored from Arabia. Corruption

must have taken place before 11th cent. and probably before

950 A.D.

Porisms lost before 10th cent. A.D.
§ 2. What portion of the Arithmetics is lost? The contents of the

lost books. The Polygonal Numbers and Porisms may have
formed part of the complete Arithmetics. Objections to this

theory
Other views of the contents of the lost Books
Conclusion

26

.

26-35
35-37

.

37
CHAPTER IV.

CHAPTER III.

THE WRITERS UPON DIOPHANTOS.

§ 1. Greek
§ 2. Arabian
$ 3. European generally

38-39
39-42
42-56

NOTATION AND DEFINITIONS OF DIOPHANTOS.

PAGES

57
57-67
67-69

§ 1. Introduction
§ 2. Sign for the unknown quantity discussed .
$ 3. Notation for powers of the unknown
4. Objection that Diophantos loses generality by the want of

more algebraic symbols answered

Other questions of notation : operations, fractions, &c.
g 5. General remarks on the historical development of algebraic

notation : three stages exhibited
8 6. On the influence of Diophantos' notation on his work

69
69–76

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76-80
80—82

CHAPTER V.

DIOPHANTOS' METHODS OF SOLUTION.

83-88
88—114

88-90
90–93
93_-94

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95-98

95

§ 1. General remarks. Criticism of the positions of Hankel and

Nesselmann
§ 2. 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 the unknown
(2) Mixed quadratics .

(3) Cubic equation
(B) Indeterminate equations.
I. Indeterminate equations of the first and second degrees.
(1) Single equation (second degree)

1. Those which can always be rationally solved .
2. Those which can be rationally solved only

under certain conditions
(2) Double equations.
1. First general method (first degree) .

Second method (first degree).

2. Double equation of the second degree
II. Indeterminate equations of higher degrees.
(1) Single equations (first class)

95-98

99-105
105-107

107

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(second class)
(2) Double equations .
$ 3. Summary of the preceding investigation

4. Transition
§ 5. Method of limits
§ 6. Method of approximation to limits

108-111
111-112
112-113

.

113–114
114--115
115—117
117--120

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