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In 1682, Leibniz published an essay containing his solution to the classic problem of squaring the circle: the alternating converging series that now bears his name. Yet his attempts to disseminate his quadrature results began seven years earlier and included four distinct approaches: the conventional (journal article), the grand (treatise), the impostrous (pseudepigraphia), and the extravagant (medals). This paper examines Leibniz's various attempts to disseminate his series formula. By examining oft-ignored writings, as well as unpublished manuscripts, this paper answers the question of how one of the greatest mathematicians sought to introduce his first great geometrical discovery to the world. [ABSTRACT FROM AUTHOR]

The Saṅgīta-ratnākara (Ocean of Music) of Śārṅgadeva (c. 1225 CE) is a 13th century text on musicology in Sanskrit. The fifth chapter of the Saṅgīta-ratnākara deals with a general analysis of all possible rhythms (tāla s) which can be obtained by combining a set of basic rhythmic components known as tālāṅga s. Here, Śārṅgadeva describes the construction of the saṃyoga-meru (Table of Combinations), a tool that tabulates the total number of possible tāla s of a given duration which are obtained by combining elements of different subsets of tālāṅga s. In this paper, we discuss the mathematical rationale employed by Śārṅgadeva in the construction of the saṃyoga-meru. [ABSTRACT FROM AUTHOR]

Non-standard analysis has often been presented as the proper framework for expressing rigorously Leibniz's conception of infinitesimals. This paper intends to study this interpretation from an historical point of view and to dispel a series of misunderstandings on which it rests. In order to do so, we propose to go back to Leibniz's conception of quantity, number and magnitude, an approach which has not been developed yet in the literature. [ABSTRACT FROM AUTHOR]

The Mainardi-Codazzi equations (MCE) and the fundamental theorem of surface theory (FT) are regarded as crucial achievements in the development of surface theory. The paper offers an analysis of three papers by Bour, Codazzi and Bonnet, submitted on the occasion of the Grand Prix des Mathématiques (1859), in which the MCE and the FT were systematically employed to deal with applicability problems. Our analysis provides a new insight into the historical process leading to a recognition of the relevance of the MCE and the FT and helps explaining why previous contributions on the subject could go unnoticed for years. [ABSTRACT FROM AUTHOR]

The paper is dedicated to analysing the role of "denominate numbers" in textbooks for Polish schools, whose author or co-author was Stefan Banach. Banach used the concepts of "number" and "denominate number" in his textbooks. The ways of introducing these concepts, as well as the associated manner of understanding numeration, are analysed in the paper. The methods of manipulating "denominate numbers" through various procedures are also discussed. The analysis is carried out in the context of the Polish ministerial ordinances related to teaching mathematics and the content of other textbooks, especially those used by Banach as a secondary school student. [ABSTRACT FROM AUTHOR]

This paper presents an analysis of a hitherto barely known conceptual problem in the foundations of arithmetic. An unknown type of number, so-called complex numbers, first emerged in 18th century France and became connected with the claim of non-commutativity of their multiplication. These first French practitioners are here analysed, along with examination of the impact of the introduction of the metric decimal system. International dissemination of the notion is then analysed, in particular in the case of Brazil. The paper concludes by looking at the mathematical foundations of multiplying (physical) quantities. [ABSTRACT FROM AUTHOR]

This paper evaluates possible reasons and motivations for 19th century geometer Julius Plücker's change in direction from his purely mathematical work to experimental physics. The author argues that this change did not happen suddenly in 1846 as is frequently suggested but rather, was a gradual change. This move took more than a decade and was triggered by Plücker's idea to apply his mathematical formalism to physical objects and phenomena, such as crystals and the trajectories of light in crystalline materials, a move which eventually led him to the newly discovered phenomena of diamagnetism. [ABSTRACT FROM AUTHOR]

Leibniz's manuscripts on perspective geometry remained unpublished and unknown until very recently. Among them, Scientia perspectiva stands out as the most complex and the most original. In this paper, we offer a thorough analysis of this manuscript, showing how Leibniz moves from perspective concepts fairly common at that time to a completely new idea of the practice that could have affected its entire history. This new science represents not only Leibniz's unique contribution to the development of perspective but also casts a new light on his own notion of space and geometry and their philosophical grounding. [ABSTRACT FROM AUTHOR]

The paper discusses Peano's argument for preserving familiar notations. The argument reinforces the principle of permanence, articulated in the early 19th century by Peacock, then adjusted by Hankel and adopted by many others. Typically regarded as a principle of theoretical rationality, permanence was understood by Peano, following Mach, and against Schubert, as a principle of practical rationality. The paper considers how permanence, thus understood, was used in justifying Burali-Forti and Marcolongo's notation for vectorial calculus, and in rejecting Frege's logical notation, and closes by considering Hahn's revival of Peano's argument against Pringsheim's reading of permanence as a logically necessary principle. [ABSTRACT FROM AUTHOR]

Pension funds, namely survivors' pension funds were established in Portugal in the late 18th century, in mutual benefit societies. The first Portuguese life assurance companies were created in the first half of the 19th century but their activity was not very extensive. Only in the 20th century did both institutions become actuarially-based. In this paper, we provide an overview of the development of actuarial calculus in Portugal until the late 19th century, in order to contextualise the scientific progress of those institutions, mainly the former. This paper lays the groundwork for further research on the history of actuarial calculus in Portugal. [ABSTRACT FROM AUTHOR]

We present a historical commentary together with an English translation of a mathematical-philosophical paper by the Czech differential geometer and later proponent of a geometrized unified field theory Václav Hlavatý (1894–1969). The paper was published in 1924 at the height of interpretational debates about recent advancements in differential geometry triggered by the advent of Einstein's general theory of relativity. In the paper he argued against a naive generalization of analogical reasoning valid for curves and surfaces in three-dimensional Euclidean space to the case of higher-dimensional curved Riemannian spaces. Instead, he claimed, the only secure ground to arrive at results is analytical calculation. We briefly discuss the biographical circumstances of the composition of the paper and characterize its publication venue the journal Ruch filosofický. We also give a discussion of the mathematical background for Hlavatý's argument. [ABSTRACT FROM AUTHOR]

In the preamble of the 1818 sangaku tablet of Sugino'o Shrine, the proposers acknowledged the help of an unnamed teacher/master in understanding and solving certain mathematical problems. Endō Tadashi argued that this unnamed teacher could be Saitō Naonaka (1773-1844). In this paper, we examine the famous travel diary of Yamaguchi Kanzan (?-1850) especially on his second trip to the Northeast. We compare the content of Yamaguchi's diary with the three problems of Sugino'o's tablet. Together with the timing of Yamaguchi's travel, we conclude that Yamaguchi Kanzan was likely the unnamed master mentioned in the preface of the Sugino'o Shrine sangaku. [ABSTRACT FROM AUTHOR]

Stefan Kempisty was a Polish mathematician, working on the theory of real functions, set theory, integrals, interval functions and the theory of surface area. In 1919 he defended his Ph.D. thesis, On semi-continuous functions , at the Jagiellonian University in Cracow under the supervision of Kazimierz Żorawski. In December 1924 he did his habilitation at the University of Warsaw and continued his work at the Stefan Batory University in Vilnius. Kempisty published over forty scientific papers, three textbooks and one monograph. Kempisty's name in mathematics appears in connection with the definition of quasi-continuous functions, different kinds of continuity of functions of several variables, the classification of Baire, Young and Sierpiński functions, interval functions, and Denjoy or Burkill integrals. This paper is prepared for a wide range of readers. It is an abridged version of the article written in Polish by the same authors (cf. Jóźwik et al., 2017), where can be found more detailed information. [ABSTRACT FROM AUTHOR]

Abstract This paper is meant as an introduction to a special issue on circulating mathematics via journals, from ca 1850 to 1950. It intends to study the practices, actors, places and institutions involved in mathematical circulation. Journals constitute one of the main carriers of mathematical knowledge and hence the main vehicle for its circulation. The focus of the paper is on 'mathematical' (understood very broadly) journals and the various forms of interplay between them. In order to understand and analyse the processes of circulation that are an important feature of mathematical production, the interactions that take place between journals are considered across different scales, from local practices and representations to more global appraisals. We shall use the notion of circulation space in order to break down the representations of the mathematical world in terms of the centres and peripheries suggested by the geographical area, with its massive publishing centres at different times – Paris, London, Berlin, Leipzig, New York, etc. Besides the geographical dimensions, the focus is on connections between journals, polarisations, competition, rivalries, etc. [ABSTRACT FROM AUTHOR]

At the turn of the 19th century, arithmetic underwent a reform. Lagrange and Delambre developed diverse reflections on this discipline and contributed to the revival of the study of the ancient Greeks' mathematics. Through analyzing Lagrange's lectures on arithmetic and the popular arithmetic treatises, I reestablish how Lagrange's innovative ideas were adopted in the later books. Moreover, this paper clarifies the role that "complex numbers" played in these treatises and investigates Delambre's memoir on the ancient Greek arithmetic, so as to reveal the way in which Delambre read the ancients' texts and reconstructed their arithmetic using "complex numbers". [ABSTRACT FROM AUTHOR]

In this paper we provide an analysis, both historical and mathematical, of two joint papers on the theory of connections by Élie Cartan and Jan Arnoldus Schouten that were published in 1926. These papers were the result of a fertile collaboration between the two eminent geometers that flourished in the two-year period 1925–1926. We describe the birth and the development of their scientific relationship especially in the light of unpublished sources that, on the one hand, offer valuable insight into their common research interests and, on the other hand, provide a vivid picture of Cartan's and Schouten's different technical choices. While the first part of this work is preeminently of a historical character, the second part offers a modern mathematical treatment of some contents of the two contributions. [ABSTRACT FROM AUTHOR]

Ada Lovelace is widely regarded as an early pioneer of computer science, due to an 1843 paper about Charles Babbage's Analytical Engine, which, had it been built, would have been a general-purpose computer. However, there has been considerable disagreement among scholars as to her mathematical proficiency. This paper presents the first account by historians of mathematics of the correspondence between Lovelace and the mathematician Augustus De Morgan from 1840–41. Detailed contextual analysis allows us to present a corrected ordering of the archive material, countering previous claims of Lovelace's mathematical inadequacies, and presenting a more nuanced assessment of her abilities. [ABSTRACT FROM AUTHOR]

Around the beginning of the year 1868, the first issue of the Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche , edited by Prince Baldassarre Boncompagni, was published. The Bullettino , which was the first European journal exclusively dedicated to the history of mathematics and physics, continued to be published for a total of twenty years in monthly issues. Officially only one-fourth of the two hundred copies of the journal were distributed in Italy, whilst the remaining copies reached the other main European countries. The Bullettino – which was in two parts, one for original papers, reviews and translations of memoirs published abroad, and one for bibliographical notices – had a range of aims: to establish links between historians of science all over Europe, to quickly spread Italian research in the field, and to provide the means to keep Italian scholars up to date with the latest developments abroad. The Bullettino was not just an academic journal to which researchers submitted their works for publication, but also the result of information, relationships, partnerships and debates, with Boncompagni as its chief editor at the centre. This paper has two aims: to describe some of the dynamics that led to the articles included in the Bullettino , and to provide an overview of the Bullettino 's content, focussing on historical contributions on Galileo, Copernicus, Domenico Maria Novara, on Lagrange's correspondence, on the correspondence between Sophie Germain and Gauss, and on contributions on astronomy and optics in Arab and Hebrew cultures by authors such Antonio Favaro, Gilberto Govi, Angelo Genocchi, Maximilian Curtze, Moritz Steinschneider and Enrico Narducci. [ABSTRACT FROM AUTHOR]

In this paper, I study Richard Dedekind and Heinrich Weber's 1882 Theorie der algebraischen Funktionen einer Veränderlichen , with a focus on the inherently arithmetical aspects of their work. I show that their paper provides an arithmetical rewriting of Riemannian function theory, i.e. a rewriting built on elementary arithmetical notions such as divisibility. I start with contextual elements concerning what is “arithmetical”, to put Dedekind and Weber's works into perspective from that viewpoint. Then, through a detailed analysis of the 1882 paper and using elements of their correspondence, I suggest that Dedekind and Weber deploy a strategy of rewriting parts of mathematics using arithmetic, and that this strategy is essentially related to Dedekind's specific conception of numbers and arithmetic as intrinsically linked to the human mind. [ABSTRACT FROM AUTHOR]

This paper is a historical account of the chords theorem, for conic sections from Apollonius to Boscovich. We comment the most significant proofs and applications, focusing on Newton's solution of the Pappus four lines problem. Newton's geometrical achievements drew L'Hospital's attention to the chords theorem as a fundamental one, and led him to search for a simple and direct proof, that he finally obtained by the method of projection. Stirling gave a very elegant algebraic proof; then Boscovich succeeded in finding an almost immediate geometrical proof, and showed how to develop the elements of conic sections starting from this theorem. [ABSTRACT FROM AUTHOR]

This paper discusses a theorem in birational geometry that J.L. Coolidge attributed to Alfred Clebsch. The background is reconstructed from letters Felix Klein exchanged with Max Noether in 1894, when Noether was completing work on a lengthy report with Alexander Brill on the history of algebraic functions. Noether was deeply troubled to learn that Klein had informed him back in 1869 about relevant results that Clebsch and Leopold Kronecker had discussed in Berlin. These exchanges with Klein led to revisions in the Brill-Noether report, made in part to ensure Noether's own priority rights and larger intellectual legacy. [ABSTRACT FROM AUTHOR]

This paper outlines the results of a study aimed at charting the spread and development of the mathematical sciences in Sardinia over the period from 1720, the year in which the island was awarded to the Savoy dynasty following the Treaty of The Hague, to 1848, the year in which King Charles Albert of Sardinia placed Sardinia on a level political and administrative footing with the Kingdom's mainland territory. In particular, the paper analyses the role played in the transmission and spread of mathematics by the Savoy educational institutions. [ABSTRACT FROM AUTHOR]

The paper ‘The chemical basis of morphogenesis’ [ Phil. Trans. R. Soc. Lond. B 237 , 37–72 (1952)] by Alan Turing remains hugely influential in the development of mathematical biology as a field of research and was his only published work in the area. In this paper I discuss the later development of his ideas as revealed by lesser-known archive material, in particular the draft notes for a paper with the title ‘Outline of development of the Daisy’. These notes show that, in his mathematical work on pattern formation, Turing developed substantial insights that go far beyond Turing (1952) . The model differential equations discussed in his notes are substantially different from those that are the subject of Turing (1952) and present a much more complex mathematical challenge. In taking on this challenge, Turing's work anticipates (i) the description of patterns in terms of modes in Fourier space and their nonlinear interactions, (ii) the construction of the well-known model equation usually ascribed to Swift and Hohenberg, published 23 years after Turing's death, and (iii) the use of symmetry to organise computations of the stability of symmetrical equilibria corresponding to spatial patterns. This paper focuses on Turing's mathematics rather than his intended applications of his theories to phyllotaxis, gastrulation, or the unicellular marine organisms Radiolaria . The paper argues that this archive material shows that Turing encountered and wrestled with many issues that became key mathematical research questions in subsequent decades, showing a level of technical skill that was clearly both ahead of contemporary work, and also independent of it. His legacy in recognising that the formation of patterns can be understood through mathematical models, and that this mathematics could have wide application, could have been far greater than just the single paper of 1952. A revised and substantially extended draft of ‘Outline of development of the Daisy’ is included in the Supplementary material . [ABSTRACT FROM AUTHOR]

König's paradox, that he presented for the first time in 1905, preserved the same structure in all his papers: there was a number that at the same time was and was not finitely definable. Still, he changed the way for forming it, and both its consequences and its solutions changed as well. In the present paper we are going to follow the story of König's paradox, that is an intriguing mix of labelling, solving, criticising an “object” from different viewpoints and for different aims. [ABSTRACT FROM AUTHOR]

The aim of this paper is to bring to light a previously unknown geometrical method for extracting the square root in seventh century China. In order to achieve this goal, a seventh century commentary by the scholar Jia Gongyan, 賈公彥, on a Confucian canon, the Rites of Zhou Dynasty [ Zhouli 周禮], is analysed. This is compared with the commentary by his contemporary Li Chunfeng, 李淳風, which is referred to in another mathematical book, the Mathematical Procedures of the Five Canons , [ Wujing Suanshu 五經筭術]. Although these two scholars probably knew each other, they used very different methods to solve the same problem in relation to square root extraction. It is argued that the differences mainly lie in two aspects: firstly, Jia Gongyan mostly made use of geometry while Li Chunfeng used counting rods; secondly, the two methods had different geometrical interpretations. Given the fact that the method of square root extraction Jia Gongyan uses is one among many other methods he employed in mathematics, and it has the same features as the others; moreover, other commentators on the Confucian Canons use similar mathematical methods, this paper closes with a general discussion on mathematical cultures. It is suggested that there were three elements to mathematical practice in seventh century China: geometry, counting rods, and written texts. The interplay and structure between the three elements is seen to influence mathematical practices. [ABSTRACT FROM AUTHOR]

This paper tackles the history of tactics, a field of investigation at the crossroads of algebra, combinatorics and recreational mathematics. Tactics was only taken up by mathematicians now and then between the 1850s and the 1900s, and its emergence was a process of mathematization of questions linked to the notions of “order” and “position”. To understand the long-term history of this field of investigation—one that became neither a theory nor a discipline—the paper analyzes the different historical configurations in which tactics took on its scientific meaning. It thus investigates how, under the banner of tactics, a continuity could be claimed by mathematicians that were, finally, working in very different scientific and historical context. [ABSTRACT FROM AUTHOR]

The German mathematician Olaus Henrici, 1 who was born in Denmark in 1840, studied engineering and mathematics in Germany before making his career in London. Initially, and for only a short time, he worked in an engineering business. He subsequently took on academic positions, first at University College London and then, from 1884, at the newly formed Central Institution (later Central Technical College) where he established a Laboratory of Mechanics. While at University College he became an active promoter of pure geometry and a producer of models of surfaces. In this paper I explore the geometrical side of Henrici's work, setting it into the context of his career and arguing that his interdisciplinary background was a key factor in his success as a creator of models. [ABSTRACT FROM AUTHOR]

During the first half of the 20th century the Danish geometer Johannes Hjelmslev developed what he called a geometry of reality. It was presented as an alternative to the idealized Euclidean paradigm that had recently been completed by Hilbert. Hjelmslev argued that his geometry of reality was superior to the Euclidean geometry both didactically, scientifically and in practice: Didactically, because it was closer to experience and intuition, in practice because it was in accordance with the real geometrical drawing practice of the engineer, and scientifically because it was based on a smaller axiomatic basis than Hilbertian Euclidean geometry but still included the important theorems of ordinary geometry. In this paper, I shall primarily analyze the scientific aspect of Hjelmslev's new approach to geometry that gave rise to the so-called Hjelmslev (incidence) geometry or ring geometry. [ABSTRACT FROM AUTHOR]

The Italian mathematician Antonio Bordoni is mainly known for his adherence to the Lagrangian approach to the foundations of calculus and for his role in creating an important school of mathematics. In this paper, I consider his less known work on the application of probability to design exams and analyze their outcomes. Within this framework, he obtained in 1837, as Mondésir and Poisson, the result that would lead Catalan to formulate his "new principle" of probability (Jongmans and Seneta, 1994). Moreover, in 1843, Bordoni also gave an early complete proof of the finite rule of succession. [ABSTRACT FROM AUTHOR]

This paper studies the Sanskrit and Malayalam versions of Citrabhānu's Twenty-One Problems: a discussion of quadratic and cubic problems from 16th-century Kerala. It reviews the differences in the approaches of the two versions, highlighting the distinction between the Sanskrit indeterminate integer-remainder arithmetical techniques and the Malayali fixed point iterations. The paper concludes with some speculations on the possible transmission of algebraic knowledge between Kerala and the west. [ABSTRACT FROM AUTHOR]

The Dutch introduced trigonometry to Japan in the middle of the 17th century, but the use of trigonometry was rarely seen until the 18th century, and its use was limited to practical purposes such as surveying and astronomy. It was rarely used to solve geometry problems currently called wasan , the reason for which is not yet fully explained. In this paper, I summarize a book by Aida Yasuaki (1747–1817) in which he strongly criticizes the use of trigonometry for wasan , and try to find the origins of his antipathy. [ABSTRACT FROM AUTHOR]

In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadratic transformations. In this sense, the work of Beltrami appears similar to the recent (after the anticipations of Magnus and Steiner) results of Schiaparelli and Cremona. Moreover, Beltrami's methods are closely related to the study of birational transformations, which in the same period were becoming one of the main topics of algebraic geometry. Finally, our work emphasises the role played by the nine-point conic problem in the studies of young Beltrami who, under Cremona's guidance, was then developing his mathematical skills. To this end, we make considerable use of the unedited correspondence Beltrami – Cremona, preserved in the Istituto Mazziniano, Genoa. [ABSTRACT FROM AUTHOR]

Though the central role of Gösta Mittag-Leffler in the promotion of specialized, research-oriented mathematics at Stockholms Högskola is widely acknowledged, the specific social and technical means by which he sought to cultivate a fledgling research community there during the early- to mid-1880s have received little attention. In particular, a detailed study of the relationship of his own research activity to that of his first Swedish students is absent from the existing literature. Through the juxtaposition of their research activities and unpublished correspondence, this paper explores Mittag-Leffler's active and deliberate efforts to engage his students Ivar Bendixson and Edvard Phragmén in open problems within his own research agenda, support them through his institutional connections, and instill within them norms concerning research ideologies, practices of communication and criticism, and frameworks for shared knowledge. It also illuminates the extent to which his teachings took root in at least one student to emerge from his program, who would perpetuate the mathematical practices set in place by his teacher and set forth on the international stage to promote his newly-acquired system of values. [ABSTRACT FROM AUTHOR]

This paper is devoted to the conception of mathematical objects and methods according to d'Alembert. We first recall his vision of the place of mathematics in the knowledge of nature, then the internal hierarchy of the various fields of this science, based on their degree of abstraction from sensations (§1 and 2). Then we come to the ideas of definitions, primitive ideas , simple ideas , and their generation as well as their generalization (§3 and 4). Then, having looked at what he means by quantities, numbers, quantities, as well as his conception of the objects and rules of algebra as abstract ideas by generalization (§5), we approach the question of the reality of mathematical objects with the example of the irrational (§6). The following paragraphs of the text are devoted to the difficulties encountered in various fields and the way d'Alembert tries to solve them: algebra and negative quantities (§7); principles of geometry (§8); the notion of limit as the basis of infinitesimal calculus (§9). His reflections, even if unfinished, were not without posterity (§10). [ABSTRACT FROM AUTHOR]

This paper is a study of Naṣīr al-Dīn al-Ṭūsī's comments on Euclid's Data. We produce a critical edition, translation and commentary of comments and marginal notes made by al-Ṭūsī in his Revision of the Euclidian text (Taḥrīr kitāb al-Muʿṭayāt li-Uqlīdis). The study results in some insight into what Ṭūsī thought was worth explaining from a mathematical perspective, some information about his manuscript sources, and, perhaps most importantly, some of his scholarly practices in producing his edition of this canonical mathematical text. Another result is that of two versions of this text that can be read in the manuscripts, one can be identified as the more polished draft. [ABSTRACT FROM AUTHOR]

The transmission and reception of the mathesis carried by Diophantus' Arithmetica has not attracted much attention among historians of Greek mathematics, who have devoted their scholarly activity almost exclusively to questions about the proper understanding of the character of the mathematical undertaking of the Alexandrian mathematician. As a result, the common belief is that Diophantus' Arithmetica is presented as an isolated, and thus uncontextualized phenomenon in the history of ancient Greek mathematics. The aim of this paper is to investigate testimonies and other piece of evidence suggesting that Diophantus' heritage was present in intellectual milieus of the Greek-speaking world during the late antique and early medieval times. Special emphasis is given to a number of scholia to the arithmetical epigrams of the Palatine Anthology which witness the persistence of the method of problem solving taught by Diophantus in the late antique world. [ABSTRACT FROM AUTHOR]

Mathematical commentaries in Arabic and Persian have come down to us in abundant numbers and manifoldness. There is no study of them as a literary genre, their distribution over time, space, or topic, nor of their forms or styles. The commentaries that have been scrutinized so far are few in number and focusing in topic mostly on geometry and in scope on those with innovative contributions. In this paper, I will try to survey more generally purposes, forms, and styles of mathematical commentaries based on a larger number of texts with different thematic content and range and produced in different contexts, periods, and localities. [ABSTRACT FROM AUTHOR]

Mathematical writers, above all, Euclid, tend to present their theorems as decontextualized, abstract propositions, which has become the standard modus of textual presentation in theoretical mathematics. Mathematical commentators, however, provide their readers with personal names and historical facts in order to elucidate problems, provide contexts of discovery, or construct doxographies, among other things. Modern readers have used such information for the construction of histories of science. When we look at these passages, however, we see that personal names and information about mathematicians of the past can serve quite a range of different objectives, such as the strategic self-positioning of the commentator vis-à-vis the past or present of mathematics, the education of the reader, mathematical or moral, the construction of the history of the field, etc. Not only does the commentator present a persona of himself to the reader, he can also turn colleagues and predecessors into personae. This paper attempts to elucidate the practice, by offering four examples of such plays of and with personae , in Pappus, Eutocius, al-Nayrīzī, and Proclus. [ABSTRACT FROM AUTHOR]

Abstract During the 20th century, impossibility theorems have become an important part of mathematics. Arrow's impossibility theorem (1950) stands out as one of the first impossibility theorems outside of pure mathematics. It states that it is impossible to design a welfare function (or a voting method) that satisfies some rather innocent looking requirements. Arrow's theorem became the starting point of social choice theory that has had a great impact on welfare economics. This paper will analyze the history of Arrow's impossibility theorem in its mathematical and economic contexts. It will be argued that Arrow made a radical change of the mathematical model of welfare economics by connecting it to the theory of voting and that this change was preconditioned by his deep knowledge of the modern axiomatic approach to mathematics and logic. [ABSTRACT FROM AUTHOR]