The introduction of a new analytical method, due fundamentally to François Viète and René Descartes and the later dissemination of their works, resulted in a profound change in the way of thinking and doing mathematics. This change, known as process of algebrization, occurred during the seventeenth and early eighteenth centuries and led to a great transformation in mathematics. Among many other consequences, this process gave rise to the treatment of the results in the classic treatises with the new analytical method, (...) which allowed new visions of such treatises and the obtaining of new results. Among those treatises is the Arithmetic of Diophantus of Alexandria which was written, using the new algebraic language, by the French mathematician Jacques Ozanam, who in addition to profusely increasing the original problems of Diophantus, solved them in a general way, thus obtaining many geometric consequences. The work is handwritten, it has never been published, it has been lost for almost 300 years, and the known references show its importance. We will show that Ozanam’s manuscript was quoted as an important work on several occasions by others mathematicians of the time, among whom G. W. Leibniz stands out. Once the manuscript has been located, our aim in this article is to show and analyze this work of Ozanam, its content, its notation and its structure and how, through the new algebraic method, he not only solved and expanded the questions proposed by Diophantus, but also introduced a connection between the algebraic solutions and what he called geometric determinations by obtaining loci from the solutions. (shrink)

Three problems in book I of Diophantus’ Arithmetica contain the adjective plasmatikon, that appears to qualify an implicit reference to some theorems in Elements, book II. The translation and meaning of the adjective sparked a long-lasting controversy that has become a nonnegligible aspect of the debate about the possibility of interpreting Diophantus’ approach and, more generally, Greek mathematics in algebraic terms. The correct interpretation of the word, a technical term in the Greek rhetorical tradition that perfectly fits the (...) context in which it is inserted in the Arithmetica, entails that Diophantus’ text contained no (implicit) reference to Euclid’s Elements. The clause containing the adjective turns out to be a later interpolation, that cannot be used to support any algebraic interpretation of the Arithmetica. (shrink)

Adequality, or παρισóτης (parisotēs) in the original Greek of Diophantus 1 , is a crucial step in Fermat’s method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat’s collected works (1891, pp. 133–172). The first article, Methodus ad Disquirendam Maximam et Minimam 2 , opens with a summary of an algorithm for finding the maximum or minimum value of (...) an algebraic expression in a variable A. For convenience, we will write such an expression in modern functional notation as f (a). 3 The algorithm can be broken up into six steps in the following way:Introduce an auxiliary .. (shrink)

This study is the foundation of a new interpretation of the introduction and the three first books of Diophantus’s Arithmetica, one that opens the way to a historically correct contextualization of the work. Its purpose, as indicated in the title, is to renew the traditional discussion on the methods of problem-solving used by Diophantus, through the detailed exposition of a new analytical framework that aims to give an account of the coherence and progressive nature of the material included (...) in the three first books of the Arithmetica. One outcome of this new ‘toolbox’ is a new conspectus of the problems and solutions contained in the latter, which is presented in appendix. The first part of the article clarifies, as a necessary preliminary, the key notions and terminology underlying our analysis. Among these new concepts is the notion of “method of invention,” which accounts in general for any way, by which “positions” (hypostaseis) are used in the Arithmetica. The next part proposes a complete inventory of the various methods of invention found in the three first books. Finally the last part presents the above mentioned conspectus and proposes a series of preliminary conclusions that can be drawn from it. (shrink)

Over the past few decades, the question regarding the proper understanding of Diophantus’s method has attracted much scholarly attention. “Modern algebra”, “algebraic geometry”, “arithmetic”, “analysis and synthesis”, have been suggested by historians as suitable contexts for describing Diophantus’s resolutory procedures, while the category of “premodern algebra” has recently been proposed by other historians to this end. The aim of this paper is to provide arguments against the idea of contextualizing Diophantus’s modus operandi within the conceptual framework of (...) the ancient analysis and to examine the few instances, in the preserved books of the Arithmetica, which might be regarded as linked to practices belonging to the field of analysis. (shrink)

Lists the practical problems found in the manuscript of Book III of the Algebra which were not included in the printed text. The author believes that their omission reflects the influence of Bombelli's discovery of Diophantus.

ArgumentThe aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related toElem.II.5 as (...) a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. (shrink)

ExtractDuring the second half of the twentieth century an attention of historians of mathematics shifted to mathematics of the Late Antiquity and its subsequent development by mathematicians of the Arabic world. Many critical editions of works of mathematicians of the Hellenistic era have made their appearance, giving rise to a new, more detailed historical picture. Among these are the critical editions of the works of Diophantus, Apollonius, Archimedes, Pappus, Diocles, and others.Send article to KindleTo send this article to your (...) Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

En este ensayo presento un análisis comparativo entre los diversos procedimientos creados, respectivamente, por los matemáticos babilonios y Diofanto de Alejandría para resolver ecuaciones de segundo grado. Observaremos cómo los primeros recurrieron a la composición de diagramas mientras Diofanto aplicó un algoritmo abstracto que no consiguió generalizar.In this paper I present a comparative analysis among the diverse procedures invented respectively by the Babylonian mathematicians and Diophantus of Alexandria to solve quadratic equations. We will observe how the first ones appealed (...) to the composition of diagrams while Diophantus applied an abstract algorithm that he was not able to generalize. (shrink)

In this paper which consists of two parts (Teil I and Teil II) we champion Diophantus of Alexandria and Isabella BaÅ¡makova against Norbert Schappacher. In two publications ([Schappacher 1998a] and [Schappacher 1998b]) he puts forward inter alia two propositions: Questioning Diophantus’ originality he considers affirmatively the possibility that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls BaÅ¡makova’s claim (in [BaÅ¡makova 1972]) that Diophantus uses negative numbers, a nonsense , reproaching her (...) for her thoughtlessness . Teil I: First, we disprove Schappacher’s Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus’ keywords $ λvarepsilon tildeιψιζ and ὕπαρξιζ. Next, we discuss Schappacher’s epistemology of the history of mathematics and defend BaÅ¡makova’s methods. Finally we analyse in detail three problems from Diophantus’ Arithmetica (and their solutions) given by Thomas Heath and Helmuth Gericke as proof of the their claim that Diophantus did not use negative numbers. Teil II: In this Part, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a − b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis ( πλvarepsilonυρacuteα ) of a square number ( τvarepsilonτρacuteαγω ν o ζ ), which is afterwards computed by the formula (a - b) 2 = a 2 + b 2 - 2ab $ . Finally, we report how the topic Diophantus and the negative numbers has been dealt with by translators and commentators from Maximus Planudes onwards. Und er kommt zu dem Ergebnis: ≪Nur ein Traum war das Erlebnis. Weil≫, so schlieβt er messerscharf, ≪nicht sein k a n n, was nicht sein d a r f.≫ CHRISTIAN MORGENSTERN: Palmstrm. (shrink)

In this second part of “Negative Größen bei Diophant?” we start, as announced, by giving 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a − b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis $(\pi\lambda\varepsilon\upsilon\rho\acute{\alpha})$ of a square number $(\tau\varepsilon\tau\rho\acute{\alpha}\gamma\omega\nu o \zeta)$ , which is afterwards computed by the formula (a - b)2 = a 2 + b 2 - 2ab. Finally, we (...) report how the topic “Diophantus and the negative numbers” has been dealt with by translators and commentators from Maximus Planudes onwards. (shrink)

The volume D'al-Khwārizmī à Descartes is a monumental contribution to the world history of mathematical sciences, showing clearly that Arabic mathematics was an indispensable predecessor of early modern European mathematics. Roshdi Rashed is known, first of all, as an editor of classical mathematical writings in Arabic by such authors as al-Khwārizmī, Thābit ibn Qurra, Ibrāhīm ibn Sinān, Ibn al-Haytham, al-Khayyām, Sharaf al-Dīn al-Ṭūsī, as well as of the Arabic versions of Apollonius' Conics , Diophantus' Arithmetica , and Diocles' Burning (...) Mirrors . As the volume under review shows, he is also a historian of mathematics of the first class who has transformed historiography. This book is, in a sense, a manifesto of Prof. Rashed's entire œuvre. (shrink)

Teachers need to acquire technology and digital skills in order to be able to follow the rapid changes in society. Since 2003, a national project has been carried out in Greece concerning the certification of teachers in basic computer skills. During this project many teachers participated and certified through a Computer Based Assessment system that has been developed by the Computer Technology Institute & Press - Diophantus. In this paper a brief view of syllabus, item bank, and tools that (...) are used for preparing and delivering examinations for teachers’ certification on basic computer skills is given. Moreover, the participants profile based on teachers’ responses is explored. (shrink)

Essay Review of “Les Arithmétiques de Diophante. Lecture historique et mathématique” by Roshdi Rashed and Christian Houzel, and Histoire de l’analyse diophantienne classique : d’Abū Kamil à Fermat by Roshdi Rashed.