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One of the main changes in European Renaissance mathematics was the progressive development of algebra from practical arithmetic, in which equations and operations began to be written with abbreviations and symbols, rather than in the rhetorical way found in earlier arithmetical texts. In Spain, the introduction of algebraic procedures was mainly achieved through certain commercial or arithmetical texts, in which a section was devoted to algebra or the ‘Arte Mayor’. This paper deals with the contents of the first arithmetical texts containing sections on algebra. These allow us to determine how algebraic ideas were introduced into Spain and what their main sources were. The first printed arithmetical Spanish text containing algebra was the Libro primero de Arithmetica Algebratica (1552) by Marco Aurel. Therefore, the aim of this paper is to analyse the possible sources of this book and show the major influence of the German text Coss (1525) by Christoff Rudolff, on Aurel's work. [ABSTRACT FROM AUTHOR]

In memory of Jackie Stedall, friend and colleague As Jackie Stedall argued in her 2011 book,From Cardano's great art to Lagrange's reflections: filling a gap in the history of algebra, there was a ‘transition from the traditional algebra of equation-solving in the sixteenth and seventeenth centuries to the emergence of “modern” or “abstract” algebra in the mid nineteenth century’ (page vii). This paper traces the evolution from the thirteenth-century work of the Pisan mathematician, Leonardo Fibonacci, to the early seventeenth-century work of the German Jesuit Christoph Clavius of what came to be considered ‘traditional algebra’. It contends that rather than a single ‘traditional algebra’, in fact, a plurality of intimately related yet subtly different algebras emerged over the course of those four centuries in different yet interacting national settings. [ABSTRACT FROM PUBLISHER]

The 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 to Elem. 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. [ABSTRACT FROM PUBLISHER]

The geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that the geometrical algebra interpretation should be reinstated as a viable historical hypothesis. [ABSTRACT FROM AUTHOR]

This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on ‘algebraical geometry’ and ‘geometrical algebra’ in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the ‘abstract algebra’ and ‘abstract geometry’ of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to ‘algebraical geometry’ and ‘geometrical algebra’ of the second generation of ‘scientific’ symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s–1840s. [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]

„She [Noether] then appeared as the creator of a new direction in algebra and became the leader, the most consistent and brilliant representative, of a particular mathematical doctrine – of all that is characterized by the term ,Begriffliche Mathematik‘.“ The aim of this paper is to illuminate this “new direction”, which can be characterized as a conceptual [begriffliche] perspective in mathematics, and to comprehend its roots and trace its establishment. Field, ring, ideal, the core concepts of this new direction in mathematical images of knowledge, were conceptualized by Richard Dedekind (1831– 1916) within the scope of his number theory research and associated with an understanding of a formation of concepts as a “free creation of the human spirit”. They thus stand for an abstract perspective of mathematics in their entirety, described as ‘modern algebra’ in the 1920s and 1930s, leading to an understanding of mathematics as structural sciences. The establishment of this approach to mathematics, which is based on “general mathematical concepts” [allgemein-mathematische Begriffe], was the success of a cultural movement whose most important protagonists included Emmy Noether (1882– 1935) and her pupil Bartel L. van der Waerden (1903– 1996). With the use of the term ‘conceptual’, a perspective is taken in the analysis which allows for developing connections between the thinking of Dedekind, the “working and conceptual methods” [Arbeits- und Auffassungsmethoden] of Noether as well as the methodological approach, represented through the thought space of the Noether School as presented under the term “conceptual world” [Begriffswelt] in the Moderne Algebra of van der Waerden. This essay thus makes a contribution to the history of the introduction of a structural perspective in mathematics, a perspective that is inseparable from the mathematical impact of Noether, her reception of the work of Dedekind and the creative strength of the Noether School. [ABSTRACT FROM AUTHOR]

In Cardano's classification in the Ars Magna (1545, 1570), the cubic equations were arranged in thirteen families. This paper examines the well-known solution methods for the families $$x^3 + a_1x = a_0$$ and $$x^3 = a_1x + a_0$$ and then considers thoroughly the systematic interconnections between these two families and the remaining ones and provides a diagram to visualize the results clearly. In the analysis of these solution methods, we pay particular attention to the appearance of the square roots of negative numbers even when all the solutions are real-the so-called casus irreducibilis. The structure that comes to light enables us to fully appreciate the impact that the difficulty entailed by the casus irreducibilis had on Cardano's construction in the Ars Magna. Cardano tried to patch matters first in the Ars Magna itself and then in the De Regula Aliza (1570). We sketch the former briefly and analyze the latter in detail because Cardano considered it the ultimate solution. In particular, we examine one widespread technique that is based on what I have called splittings. [ABSTRACT FROM AUTHOR]

Though it was written in Greek in a center of ancient Greek learning, Diophantus's Arithmetica is a curious synthesis of Greek, Egyptian, and Mesopotamian mathematics. It was not only one of the first purely number-theoretic and algebraic texts, but the first to use the blend of rhetorical and symbolic exposition known as syncopated mathematics. The text was influential in the development of Arabic algebra and European number theory and notation, and its development of the theory of indeterminate, or Diophantine, equations inspired modern work in both abstract algebra and computer science. We present, in this article, a selection of problems from the Arithmetica, which have been rewritten for ease of reading, and consider Diophantus's advancements in mathematics and mathematical notation in the context of ancient Greek mathematics. In particular, we examine Diophantus's use of syncopated mathematics, most notably his use of generic solutions that present an algorithm for solving an entire class of equations through the application of that algorithm to a single representational example, and how these techniques suggest a more extensive use of concrete examples when approaching modern mathematics. [ABSTRACT FROM AUTHOR]

This paper deals with the ancient origin of matrices, and the system of linear equations. Included are algebraic properties of matrices, determinants, linear transformations, and Cramer’s Rule for solving the system of algebraic equations. Special attention is given to some special matrices, including matrices in graph theory and electrical networks. It contains a wide variety of important materials accessible to college and even high school students and teachers at all levels. [ABSTRACT FROM PUBLISHER]

This article argues that papermaking supported cryptanalysis's invention; that cryptology—meaning both cryptography and cryptanalysis—could not be practiced before the sustained production of paper; and that Medieval Arabic cryptology originated in tandem with algebra. Furthermore, this article posits that the regional Islamicate postal service, or thebarīd, was used to relay Medieval Arabic cryptograms and thereby shaped the substance of cryptology. These conclusions stem from examining ninth century to fourteenth century Arabic cryptology as a technology and relating Arabic cryptology to three other technological devices: papermaking in the Middle East, algebra, and thebarīd. Extant documents suggest that cryptology originated in ninth century Baghdad. This is because no cryptanalytical writings are known to exist before this period, and cryptology requires both cryptography and cryptanalysis. However, evidence of Medieval Arabic cryptology exists almost exclusively in practitioners' treatises, not in cryptograms or the working papers of cryptanalysts. To redress this barrier to historical research, I move from text to context, or from the ideas in the treatises to thinking about the technologies these ideas called for. Placing cryptological technology within the wider technological context that inspired, shaped, and confined its development suggests why cryptology originated in the Islamicate world, how Arabic cryptology may have been implemented in practice, and what enabled cryptology's start. I hope that this technological exercise encourages the study of the context in which Medieval Arabic cryptology developed, until further primary sources surface. Furthermore, I intend to demonstrate Medieval Arabic cryptology's relevance to cryptological history and Middle Eastern studies. [ABSTRACT FROM PUBLISHER]

This article examines how the theory of proportions was explained during the second half of the seventeenth century in the works of Andreas Tacquet, Antoine Arnauld, Ignace Gaston Pardies, Bernard Lamy, and Jacques Rohault. These five authors had very different conceptions of this subject, and on one hand, they show that this question was not forgotten, even after the Geometry of Descartes, and on the other hand, their work displays the progressive transformation of mathematical objects. While Tacquet deepened Euclidean thought, the others stopped taking the Euclidean model as paradigmatic and tried to change the order of the Elements and to establish book V of Euclid in new ways. We shall see that this multiplicity of the approaches highlights both the vitality of the reflections and the difficulty in developing a new ontology of mathematics. Some of them nevertheless opened new perspectives that were to bloom only much later. We shall also see the increasingly important place of the algebra as time went by. [ABSTRACT FROM AUTHOR]

The article presents a biography of Belgian historian of mathematics and Jesuit priest Henri Bosmans. Among other topics described are his career both academically and as a Roman Catholic priest, his education, and his research publications. His work on the history of algebra is particularly highlighted.

The article offers a report from a March 20-22, 2013 conference in Wuppertal, Germany on the historiography of mathematics. Topics of presentations given included 19th-century criticisms of geometric algebra, T. Heath's 1908 translation of Euclid's geometry textbook "Elements," and the history of mathematics in Mesopotamia.

In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's "principle of equivalent forms," which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. [ABSTRACT FROM AUTHOR]

This paper is the result of a comparative literature study aimed at identifying relevant historical mathematical contexts that may be applicable to present day mathematics education. Five Swedish textbooks in algebra from the period around the year 1800 are scrutinized. A content analysis and a comparison between seven categories shows that there exist several historical features that may be used in teaching. These have the potential to serve both teachers and learners of mathematics. [ABSTRACT FROM PUBLISHER]

The aim of this paper is to provide a comprehensive exposition of the early contributions to the so-called Campbell, Baker, Hausdorff, Dynkin Theorem during the years 1890-1950. Related works by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff, and Dynkin will be investigated and compared. For a full recovery of the original sources, many mathematical details will also be furnished. In particular, we rediscover and comment on a series of five notable papers by Pascal (Lomb Ist Rend, 1901-1902), which nowadays are almost forgotten. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Emmy Noether's many articles around the time that Felix Klein and David Hilbert were arranging her invitation to Göttingen include a short but brilliant note on invariants of finite groups highlighting her creativity and perspicacity in algebra. Contrary to the idea that Noether abandoned Paul Gordan's style of mathematics for Hilbert's, this note shows her combining them in a way she continued throughout her mature abstract algebra. [ABSTRACT FROM AUTHOR]

In the early 1930s W. O. Kermack and W. H. McCrea published three papers in which they attempted to prove a result of E. T. Whittaker on the solution of differential equations. In modern parlance, their key idea consisted in using quantized contact transformations over an algebra of differential operators. Although their papers do not seem to have had any impact, either then or at any later time, the same ideas were independently developed in the 1960-1980s in the framework of the theory of modules over rings of microdifferential operators. In this paper we describe the results of Kermack and McCrea and discuss possible reasons why such promising papers had no impact on the mathematics of the twentieth century. [ABSTRACT FROM AUTHOR]

The common view of the history of contingency tables is that it begins in 1900 with the work of Pearson and Yule, but in fact it extends back at least into the 19th century. Moreover, it remains an active area of research today. In this paper we give an overview of this history focussing on the development of log-linear models and their estimation via the method of maximum likelihood. Roy played a crucial role in this development with two papers co-authored with his students, Mitra and Marvin Kastenbaum, at roughly the mid-point temporally in this development. Then we describe a problem that eluded Roy and his students, that of the implications of sampling zeros for the existence of maximum likelihood estimates for log-linear models. Understanding the problem of non-existence is crucial to the analysis of large sparse contingency tables. We introduce some relevant results from the application of algebraic geometry to the study of this statistical problem. [Copyright &y& Elsevier]

The article discusses the history of algebraic language. Arithmetic expressions inscribed in various artifacts recovered from Russia, Babylon and Greece are presented. Diophantus of Alexandria, a third century Greek mathematician, introduced an algebraic symbolism using an abbreviation of the unknown variable. This method marked the development of algebra. Several mathematical symbols used during the early stage of algebra are presented, such as arithmetic operations, square root and integer. Writing mathematical expressions using ancient symbols is demonstrated.

Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and other symbols (e.g. “+”, “=”) made it possible to achieve what is considered one of the greatest cultural accomplishments in human history, namely, the constitution of a symbolic algebraic language and the concomitant rise of symbolic thinking. Because of their profound historical ties with natural language, the emerging syntax and meanings of symbolic algebraic language were marked in a definite way by the syntax and meanings of the former. However, at a certain point, these ties were loosened and algebraic symbolism became a language in its own right. Without alluding to the theory of recapitulation, in this paper, we travel back and forth from history to the present to explore key passages in the constitution of the syntax and meanings of symbolic algebraic language. A contextual semiotic analysis of the use of algebraic terms in 9th century Arabic as well as in contemporary students' mathematical activity, sheds some light on the conceptual challenges posed by the learning of algebra. [ABSTRACT FROM AUTHOR]

This paper offers an epistemological reconstruction of the historical development of algebra from al-Khwārizmī, Cardano, and Descartes to Euler, Lagrange, and Galois. In the reconstruction it interprets the algebraic formulas as a symbolic language and analyzes the changes of this language in the course of history. It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form (in the sense of Wittgenstein's Tractatus) of the symbolic language of algebra. Thus the paper develops further the method of reconstruction which the author introduced for the analysis of the development of geometry. [ABSTRACT FROM PUBLISHER]

Explores the history of the Metropolis-Hastings algorithm, highlighting key personalities and events in its development. Popular Markov chain Monte-Carlo technique among statisticians; Reasons cited for the delay in the acceptance of the algorithm and reasons for its popularity.

At the turn of the century it seems to be appropriate to pause and to try to envision future possibilities. We want to discuss the prospects of algebra. To look into the future requires an understanding of the past, the longstanding aims, but also the difficulties which have been encountered. We are going to review part of the history of algebra in order to outline its present state. It is important to notice the missed opportunities and to analyze the reasons. To recognize the present possibilities requires to be aware of the tools which now are available and which may not yet have been used in an optimal way. We urge the reader to focus attention to the need for algebraic considerations in all parts of mathematics but also outside of mathematics. Of course, a view back should also strengthen the interest in classical open problems which now may be feasible to attack. [ABSTRACT FROM AUTHOR]

This paper traces three major stages in the development of algebraic notation: rhetorical or prose, syncopated, and symbolic. The development of algebra began in Babylonia and Egypt around 1700 BC. Examples of rhetorical algebra by al-Khowarizmi are used to illustrate potential difficulties that arise when algebraic problems are worked using words without symbols. Greek mathematician Diophantus was one of the pioneers of syncopated algebra. In this stage of notation, some shorthand was used along with prose. Indian mathematicians developed a syncopated algebraic notation independently of Diophantus. Around 1500 BC, symbolic algebra began to develop. The process of the development of a standardized, efficient symbol system is illustrated by tracing the evolution of some common symbols, including the symbols for equals, addition, subtraction, multiplication, and division. [ABSTRACT FROM AUTHOR]

Traces the birth of literal algebra. Mathematics in the First Centuries AD; Reference to Diophantus' `Arithmetica'; Diophantus' methods; Algebra after Diophantus.

Discusses Computational Geology, which aims to promote the use of mathematics in the undergraduate geosciences curriculum. Details on unit conversions; History of algebra; Difference of school algebra from modern or abstract algebra.

Presents a critique of Jacob Klein's book `Greek Mathematical Thought and the Origin of Algebra.' Argument of Klein's book; Science of nature in its modern incarnation with respect to First Principles; Nerve of Klein's inquiries; Greek concept of number; Appeal to the language of Scholastics; Modern concept of number; Results of the fully developed mode of conceptualization; Importance of Klein's work.

Discusses the evolution of algebra in the 1800s. Cyclotomic equation; First nontrivial example of an infinite abelian group; Group theories; Development of linear algebra; Achievements; Ideas and methods.

RESUMO O ensino de Álgebra tem se restringido a questões técnicas e operacionais, deixando de lado, muitas vezes, o desenvolvimento de conceitos e do pensamento algébrico. Acreditamos que esse enfoque está por trás das deficiências diagnosticadas em várias pesquisas e avaliações governamentais. Neste texto, apresentamos como os conceitos que foram relevantes para o desenvolvimento da álgebra ao longo dos séculos podem e devem participar do processo de ensino da álgebra.