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Über dieses Buch

This edition of Books IV to VII of Diophantus' Arithmetica, which are extant only in a recently discovered Arabic translation, is the outgrowth of a doctoral dissertation submitted to the Brown University Department of the History of Mathematics in May 1975. Early in 1973, my thesis adviser, Gerald Toomer, learned of the existence of this manuscript in A. Gulchln-i Macanl's just-published catalogue of the mathematical manuscripts in the Mashhad Shrine Library, and secured a photographic copy of it. In Sep­ tember 1973, he proposed that the study of it be the subject of my dissertation. Since limitations of time compelled us to decide on priorities, the first objective was to establish a critical text and to translate it. For this reason, the Arabic text and the English translation appear here virtually as they did in my thesis. Major changes, however, are found in the mathematical com­ mentary and, even more so, in the Arabic index. The discussion of Greek and Arabic interpolations is entirely new, as is the reconstruction of the history of the Arithmetica from Diophantine to Arabic times. It is with the deepest gratitude that I acknowledge my great debt to Gerald Toomer for his constant encouragement and invaluable assistance.

Inhaltsverzeichnis

Frontmatter

Introduction

Frontmatter

Chapter I. The Four Arabic Books and the Arithmetica

Abstract
The Greek mathematician Diophantus of Alexandria is known with certainty to have lived between 150 B.C. and A.D. 350, as we infer from his having mentioned Hypsicles and from his having been mentioned by The on of Alexandria; it seems fairly probable, though, that he flourished about A.D. 250.1 We can be sure that he wrote at least two treatises: one dealing with problems in indeterminate analysis, the Arithmetica, and another, smaller, tract on polygonal numbers, both of which are only partially extant today.
Jacques Sesiano

Chapter II. The Extant Arabic Text

Abstract
Books IV to VII of Diophantus’ Arithmetica are found in a codex, apparently a unicum, which is described under the number 295 in the eighth volume of the catalogue of the manuscripts kept in the library attached to the shrine of Imam Rezā at Mashhad (cf. Gulchīn-i Macānī, Fihrist, pp. 235-36). This codex is said to have come to the Shrine Library as the result of an endowment (waqf) made in 1932 by a certain Mirza Reza Khan from Nā’īn (Mīrzā Ridā Ḫān Nā’īnī).1 The manuscript is protected by a cardboard cover bound with and reinforced on the corners by leather. In recent times its eighty reddish-brown leaves (175 x 130 mm) have been numbered as pages.2 On each of these—except for the title-page and the last page—figure twenty lines of text (128 x 92 mm).3
Jacques Sesiano

Chapter III. Tentative Reconstruction of the History of the Arithmetica

Abstract
In the beginning of the seventh Book of his Collection, Pappus mentions two types of analyses and syntheses distinguished by the Greeks.1 The first, ποριστικóν, type is commonly used by geometers in connection with the demonstration of a proposition, or of an (already known) solution. In the corresponding analysis, what is to be proved is supposed to be true (or known), and must be reduced by passing through its successive consequences, either to an identity or to a known proposition. The synthesis then reverses the process. The second kind of analysis, of the ζητητικóν type, is used in the finding of a solution to a problem. Supposing the problem solved, the mathematician establishes between the known and the unknown magnitudes some relation, which is then reduced, by elimination, to a final relation containing the smallest number of unknowns possible (one for a determinate problem). This is the analysis. The synthesis simply verifies the exactness of the solution found.
Jacques Sesiano

Translation

Frontmatter

Fourth Book of the Treatise of Diophantus on Squares and Cubes

Abstract
I have presented in detail, in the preceding part of this treatise on arithmetical problems, many problems in which we ultimately, after the restoration and the reduction1, arrived at one term equal to one term, (namely) 10 those (problems) involving (either of) the two species of linear and plane number and also those which are composite. I have done that according to categories which beginners can memorize and grasp the nature of.
Jacques Sesiano

Fifth Book of the Treatise of Diophantus the Alexandrian on Arithmetical Problems

Abstract
We wish to find two numbers, one square and the other cubic, such that when we add to the square of the square a given multiple of the cubic number, 1620 the result is a square number, and when we subtract from the same another given multiple1 of the cubic number, the remainder is a square number.
Jacques Sesiano

Sixth Book of the Treatise of Diophantus

Abstract
We wish to find two numbers, one cubic and the other square, having their sides in a given ratio, such that when their squares are added, the result is a square number.
Jacques Sesiano

Seventh Book of the Treatise of Diophantus

Abstract
Our intention is to expound in the present Book many arithmetical problems without their departing from the type of problems seen previously in the fourth and fifth Books—even if they are different in species1—in order that it be an opportunity for (acquiring) proficiency and an increase in ex-2925 perience and skill.
Jacques Sesiano

Mathematical Commentary

Frontmatter

Book IV

Abstract
The introduction to Book IV can be divided into three distinct parts.
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Book V

Without Abstract
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Book VI

Without Abstract
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Book VII

Abstract
The introduction to Book VII consists of a single sentence, which is to all appearances genuine—although the elucidation of its meaning poses some difficulty. In it the following three points are made.
Jacques Sesiano

Text

Frontmatter

Book IV

Without Abstract
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Book V

Without Abstract
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Book VI

Without Abstract
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Book VII

Without Abstract
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Backmatter

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