VOLUME 1: ARITHMETIC, ALGEBRA AND THEORY OF NUMBERS

Volume 1 consists of a series of articles published between 1972 and 2013. The volume deals with the evolution of arithmetic, algebra and the theory of numbers among Arabic-speaking mathematicians, all belonging to the era of the eighth to the fifteenth centuries, except for Al-Yazdī, who is a mathematician from the seventeenth century. Any mathematics historian interested in the contribution of Arabic mathematics during this period will find all their demands met in this first volume.

In his studies Rashed also focuses on the works of several Greek mathematicians such as Euclid and Diophantus, in addition to other Western mathematicians of the sixteenth and seventeenth centuries such as François Viète, Pierre de Fermat, Joseph-Louis Lagrange etc. Despite the commendable advantage of compiling all these articles in the same volume, some of them have become outdated due to the progress made in terms of critical editions of several mathematical texts, even though the order of these articles, as listed by Rashed, reflects the evolution of the critical edition of a batch of texts.

With high accuracy Rashed analyses the contribution of mathematicians such as al-Khwārizmī, al-Karajī, al-Samawʾal, Sharaf al-Dīn al-Ṭūsī, Ibn al-Haytham, al-Uqlīdīsī, al-Baghdādī etc., whether in relation to the Greeks who preceded them or in comparison to their contemporaries such as Fibonacci, as well as to those who succeeded them such as Viète, Fermat, Descartes or even Lagrange.

With regards to arithmetic, and through ten articles in this volume, Rashed traces the evolution of the conceptual traditions of this mathematical discipline: from induction, the definition of which was revisited by Rashed in the works of al-Karajī and Al-Samawʾal, through the contribution of Ibnal-Haytham to the resolution of a system of congruence (which is no more than a particular case of the Chinese remainder theorem) and the advent of Wilson’s theorem as a subsequent result during this resolution, to the entire Diophantine analysis of Al-Khāzin in the tenth century. Diophantus’ Arithmetica is also analysed in depth in this first volume, and Rashed demonstrates that there are two readings of the Arithmetica: an algebraic one and a Euclidean one. The latter led to the inception of a new tradition established by al-Khāzin, an endeavour that illustrates an entire Diophantine analysis whose privileged theme is the treatment of numerical triangles, a subject that has been well explored by Fibonacci, whose work is also explored by Rashed.

Rashed deals with the topic related to the theory of numbers in three articles. In the first he states that the primary use of combinatorial analysis in algebra dates back to the ninth century. One of the studies of the applications of combinatorial analysis dates back to the philologist Khalīl Ibn Aḥmad (died towards the end of the eighth century). His work replaced a lexicographical monograph with specific references to the lexicon of words in the Arabic language. The second article is related to amicable or figurative numbers. These numbers have been widely studied by several Arabic-speaking mathematicians such as Thābit Ibn Qūrra and al-Fārisī. This latter has provided various arguments that differ from Ibn Qūrra’s theorem. In the third article Rashed recommends the method employed by Ibn al-Haytham to identify amicable numbers.

The volume includes eleven articles by Rashed on algebra. They can be divided into two categories. The first category is related to the algebra of al-Khwārizmī, the school of al-Karajī and the algebraic reading of Diophantus’ Arithmetica. The second category relates to numerical analysis in the works of Sharaf al-Dīn al-Ṭūsī, al-Samawʾal and al-Bīrūnī, among others. After dealing with the beginning of algebra by al-Khwārizmī, Rashed presents the algebra of polynomials of al-Karajīas revisited by al-Samawʾal. As for numerical analysis, Rashed’s work includes the various mechanisms for approximating the nth root of an integer studied by several Arabic-speaking mathematicians, in addition to an excellent article on Sharaf al-Dīn al-Ṭūsī and his resolution of a third-degree equation. Rashed cannot be considered as a simple editor and translator of Sharaf al-Dīn al-Ṭūsī’s work. His first chapter firmly supports a thesis: Sharaf al-Dīn al-Ṭūsī laid down the first foundations for employing the first derivative phenomenon in a bid to seek the solution of a third-degree equation.

The volume includes two Arabic texts of al-Samawʾal on extracting roots and decimal fractions (pp. 269–70 and 270–3). In addition, several articles include the critical edition of an Arabic text accompanied by a French translation. Here is a list of these critical editions: demonstration with segments that it is impossible that a sum of two cubic numbers is a cubic one (al-Khāzin, pp. 305–8); on founding solutions of a numerical problem (Ibn al-Haytham, pp. 323–9); the seventh chapter of al-Samawʾal’s book The discovery of the errors of the astrologers in the most procedures and editions (al-Samawʾal, pp. 485–511); Al Yazdī’s text (pp. 545–63).

VOLUME 2: GEOMETRY

Volume 2 combines texts related to the history of geometry. As presented by Rashed in this volume, the authors write in Greek (Archimedes, Geminus, Apollonius, Theodosius, Menelaus), in Arabic (al-Kindī, Ibn Sahl, al-Qūhī, the Banū Mūsā brothers, Thābit ibn Qūrra, al-Sijzī, Abū al-Jūd, Ibn ʿIrāq, Ibn Hūd) and in French (Descartes, Fermat). Except for two biographical articles dedicated to Thābit ibn Qūrra and Abū Naṣr ibn ʿIrāq, the texts presented are primarily focused on mathematical concepts: conic sections, spherical geometry, projective methods, Archimedean ideas, curves and tangents. While studied over the long term, all these concepts are used to highlight both the continuities and the changes underlying the approaches employed.

Although from a different angle, three texts deal with the classification of curves. Similarly, the article published in Physis in 2005 provides a general overview of the history of the classification of curves, whereas the chapter published in 2004 displays the points of view of Géminus and al-Sijzī, and the chapter published in 1997 provides an analysis of the point of view of Descartes. Thus, and as far as this volume is concerned, it provides a juxtaposition of complementary elements of a section of the history of mathematics to which Rashed has contributed considerably.

From a historiographical point of view, three articles, published between 2010 and 2015 in Arabic Sciences and Philosophy, are dedicated to the Spheres of Menelaus. These articles shed light on the method being employed by Rashed to edit, translate and comment on the Arabic version of a major Greek text. The outcome of this endeavour resulted in the publication of Menelaus’ Spherics in 2017, of a reference work jointly authored with A. Papadopoulos.

Several articles include the critical edition of an Arabic text accompanied by a translation. Here is a list: On the approximation of the circumference to the diameter (al-Kindī, pp. 24–41); two fragments (al-Qūhī, quoted by al-Sijzī, pp. 60–1; 66–71); treatise on the construction of the perfect compass, which is the compass of conics (al-Sijzī, pp. 146–54); Al-Kāmil (al-Farghānī, extract, pp. 175–85); If two straight lines are drawn along two angles less than two right angles, they meet (Thābit ibn Qūrra, pp. 215–37); On the demonstration of Euclid’s famous postulate (Thābit ibn Qūrra, pp. 238–47); On the Euclidean definition of the straight line and on curved lines (Simplicius, quoted by al-Nayrīzī, pp. 327–8); Response of the eminent Sheikh Abū al-Jūd […] to what the eminent brother has requested […] al-Bīrūnī asked (Abū al-Jūd, pp. 422–55); excerpts from al-Ṭūsī’s commentary on Theodosius’ Spheres (al-Ṭūsī, pp. 476–83); excerpts from Ibn ʿIrāq’s commentary on the Spherics of Menelaus, book III (Ibn ʿIrāq, pp. 488–9); proposition of Theodosius taken up by Ibn ʿIrāq (Ibn ʿIrāq, pp. 491–5); excerpt from the book al-Istikmāl (Ibn Hūd, pp. 496–505); On the rectification of a proposition in the book of Menelaus on the Spherics whose difficulty was avoided by those who corrected that book (Ibn ʿIrāq, pp. 560–79); proposition III.5, from The Book of Menelaus on Spherical Propositions Rectified by the Emir Abū Naṣr Manṣūr Ibn ʿIrāq (Ibn ʿIrāq, pp. 580–91); the Text of Naṣīr al-Dīn al-Ṭūsī from his edition of Menelaus’ Spherics (al-Ṭūsī, pp. 604–13); commentary on Proposition III.5 of Menelaus’ Spherics (Ibn Abī Jarrāda, pp. 614–23).

VOLUME 3: OPTICS AND ASTRONOMY

Volume 3 brings together articles related to the history of optics and astronomy, which are arranged according to the chronological order of the date of their first publication. The division into two sections, one for optics and the other for astronomy, was probably avoided due to the imbalance between the two themes. Indeed, this volume will be of greater interest to historians of optics than to historians of astronomy, to which only seven articles are dedicated.

Two articles dealing with the history of astronomy, published in Arabic Sciences and Philosophy, are worth noting: ‘Al-Qūhī: From Meteorology to Astronomy’ (in English, pp. 359–407) and ‘Ibn al-Haytham et le movement d’enroulement’ (in French, pp. 559–651) published jointly with E. Penchèvre, a specialist in the area and editor of Ibn al-Šāṭir’s astronomical sum Nihāya al-sūlfītaṣḥīḥ al-ʾuṣūl. Based on two variations of the methods of historical epistemology, these two articles deal with the treatment of quantitative aspects, a topic that has been rarely analysed in the specialised literature. Yet, and in the field of the history of astronomy, the fundamental work of Rashed, to which we can safely recommend to the reader, is volume 5 of Les mathématiques infinitésimales du IX ^e au XI ^e siècle (2006). This work contains the edition, translation and notes of the astronomical treatises by Ibn Al-Haytham.

Optics and catoptrics represent the first topics in the history of Arab sciences ever to be explored by Rashed. The prelude to this work is conducted through a series of texts that have been used to analyse the epistemological content of the work of Ibn Al-Haytham in the area of optics. Having generated a great deal of analysis and commentary, all these works are of paramount importance for the history of optics, as well as the medieval Latin one, given their widespread dissemination, in addition to the name of the author that has been Latinised in Alhazen. Hence, the texts presented in the present volume are characterised by a twofold trend. The first one alludes to the sources of Ibn Al-Haytham and thus leads to that of Al-Quhī and of Al-Kindī. The second one tilts to the direction of mathematics in relation to construction and mirror features, thus joining another flagship area that is reminiscent of the works of Rashed, i.e. the history of conic sciences and geometric transformations.

Several articles in the volume contain the critical edition of an Arabic text accompanied by a translation. Here is a list of these critical editions: Fūthīṭos’s commentary on Ptolemy’s statement: ‘If the stars are on the horizon, their magnitudes appear greater to the eye’ (Fūthīṭos, pp. 200–5); The magnitudes of figures immersed in water (al-Kindī, pp. 210–11); Planetary Hypotheses (Ptolemy, excerpts, pp. 210–11 and 634–9); In a finite time, there is an infinite movement (al-Qūhī, pp. 318–21); Ibn Riḍwān’s refutations (Ibn Buṭlān, pp. 322–3); al-Muʿtabarfī al-ḥikma (Abū al-Barakāt al-Baghdādī, fragment, pp. 324–5); On the construction of the fiery mirror (Abū al-Wafāʾ, pp. 338–45); On the knowledge of the magnitude of the distance between the centre of the earth and the position of a shooting star in the night (al-Qūhī, pp. 378–87); On the knowledge of the magnitude of what is seen of the sky and of the sea from the top of an elevated thing (al-Qūhī, pp. 388–403); The resolution of doubts about the winding movement (Ibn al-Haytham, pp. 598–633); The ultimate understanding in the knowledge of orbs (Qūṭb al-Dīn al-Šīrāzī, excerpt, pp. 644–9); book of Mirrors (Euclid, pp. 692–707); book on Burning Mirrors (Thiasos, pp. 708–45); treatise on burning mirrors, by the <conic> sections (Ibn al-Haytham, pp. 764–87). In addition to this, there is a full French translation of Ibn al-Haytham’s discourse on light (pp. 11–27).

VOLUME 4: MATHEMATICS AND PHILOSOPHY

Volume 4 contains a collection of articles related to the philosophy of mathematics and historical methods. The first section, on probabilities and the social sciences, contains a series of four articles published between 1971 and 1994 in joint works. The first edition was conducted by G. Canguilhem, and the second one pertains to Gilles Gaston Granger. All in all, they refer to avenues that could be opened up by applying the methods of historical epistemology to the study of the long-term mathematisation of the entire section belonging to the area of social sciences.

The second section, on mathematics and philosophy, is much more extensive: it touches upon a field where Rashed has played a key role and been a major contributor for over 50 years, i.e. the philosophy of mathematics. The authors quoted here are Ibn al-Haytham, Avicenna, al-Ṭūsī, al-Ḥalabī, Maimonides, Descartes, al-Sijzī, Fermat and Menelaus. This long list of writers illustrates the historical scope of the framework that Rashed has embarked upon. Yet, more than the authors and the periods, it is the themes that lie at the heart of the articles gathered: analysis and synthesis; combinatory and metaphysics; the concept of demonstration; the mathematical study of the venue; the angle of contingency; the infinitely small; proof by contradiction or direct proof; equality or congruence. These are the objects of the philosophy of mathematics that Rashed helps to scrutinise and analyse by implementing the methods of historical epistemology. The whole process goes beyond periodisation as well as linguistic and cultural partitioning.

Consisting of three articles, the short third section, focused on the theme of the asymptote, is extremely rich from the point of view of the philosophy of mathematics. The articles highlight the spirit of deep analysis that is yielded both by the revisiting of texts and by exploring their prospects within the scholastic mindset. This section is devised as a proof of concept to reflect the fertility of the methods of historical epistemology that come into play in the field of the history of mathematics.

The fourth section, which contains thirteen articles published between 1978 and 2012 on the history of mathematics, problems and methods, is employed by Rashed to adopt a reflexive stance vis-à-vis the work and methods of the historian of mathematics. Hence, reading these texts, in conjunction with the introductions to these synthesis works, constitutes a rich source for future historians interested in this rich discipline and in the dynamics underlying the points of tension and points of convergence within historical epistemology. It is this trend that Rashed makes explicit through the use of these texts and the adoption of other approaches under way since the second half of the twentieth century.

Several articles in the volume include a critical edition of an Arabic text accompanied by a translation. Here is a list: glosses on certain propositions in the book of Conics of Apollonius (Maimonides [excerpts], pp. 131–9); The side is not commensurable with the diagonal (al-Sijzī, pp. 350–61); opuscule on the asymptote (al-Sijzī, pp. 416–33); epistle on the explanation of the asymptote (al-Qūmmī, pp. 454–67); treatise on the existence of the asymptote (anonymous, pp. 480–503). One should also mention the critical edition, without translation, of On the Demonstration of the Mode of Emanation of Things <in Infinite Number> from the Unique First Principle by al-Ṭūsī (pp. 110–17).