Mathematical recreations of Dénes König and his work on graph theory

Dénes König (1884–1944) is a Hungarian mathematician well known for his treatise on graph theory (König, 1936). When he was a student, he published two books on mathematical recreations ( and ). Does his work on mathematical recreations have any relation to his work on graph theory? If yes, how are they connected? To answer these questions, we will examine his books of 1902, 1905 and 1936, and compare them with each other. We will see that the books of 1905 and 1936 include many common topics, and that the treatment of these topics is different between 1905 and 1936.     

German mathematicians in exile in Turkey: Richard von Mises,William Prager,Hilda Geiringer,and their impact on Turkish mathematics

There is a sizable and growing literature on scholars who fled from the Nazi regime, a literature which often focuses on the periods before leaving Germany and after settling permanently in the USA, but relatively less work on the interim period in which many of them found temporary homes in countries such as Turkey. In this article we would like to discuss the scholarly work, activities and the impact of mathematicians Richard von Mises, William Prager and Hilda Geiringer during their stay in Turkey. We argue that the establishment and the development of applied mathematics and mechanics in Turkey owe much to them.     

Calculations of faith: mathematics, philosophy, and sanctity in 18th-century Italy (new work on Maria Gaetana Agnesi)

The recent publication of three books on Maria Gaetana Agnesi (1718-1799) offers an opportunity to reflect on how we have understood and misunderstood her legacy to the history of mathematics, as the author of an important vernacular textbook, Instituzioni analitiche ad uso della gioventú italiana (Milan, 1748), and one of the best-known women natural philosophers and mathematicians of her generation. This article discusses the work of Antonella Cupillari, Franco Minonzio, and Massimo Mazzotti in relation to earlier studies of Agnesi and reflects on the current state of this subject in light of the author’s own research on Agnesi.     

Straddling centuries: The struggles of a mathematician and his university to enter the ranks of research mathematics, 1870–1950

This paper weaves two interlocking histories together. One strand of the fabric traces the development of the American mathematician Joseph B. Reynolds from a peripheral player to an active contributor to mathematics, astronomy, and engineering and to the founding of a sectional association of mathematicians. The other piece describes the evolution of his institution, Lehigh University, from its founding in 1865 to a full-fledged research department that began producing doctorates in 1939. Both Reynolds and Lehigh straddled the line between the pre- and post-Chicago eras in American mathematics.     

From anomaly to fundament: Louis Poinsot's theories of the couple in mechanics

In 1803 Louis Poinsot published a textbook on statics, in which he made clear that the subject dealt not only with forces but also with ‘couples’ (his word), pairs of coplanar non-collinear forces equal in magnitude and direction but opposite in sense. His innovation was not understood or even welcomed by some contemporary mathematicians. Later he adapted his theory to put forward a new relationship between rectilinear and rotational motion in dynamics; its reception was more positive, although not always appreciative of the generality. After summarising the creation of these two theories and noting their respective receptions, this paper considers his advocacy of spatial and geometrical thinking in mechanics and the fact that, despite its importance, historians of statics who cover his period usually ignore his theory of couples.     

The Luigi Cremona Archive of the Mazzini Institute of Genoa

Luigi Cremona (1830-1903) is unanimously considered to be the man who laid the foundations of the prestigious Italian school of Algebraic Geometry. In this paper we draw attention to the “Legato Itala Cremona Cozzolino”, which was given to the library of the Mazzini Institute, Genoa, Italy, by Cremona’s daughter, Itala, probably in 1939. This legacy, which contains over 6000 documents, mainly consisting of Cremona’s correspondence with scientific and institutional Italian interlocutors, can help us to understand the connections between the development of Italian mathematics in the second half of the XIX century and the main political issues of Italian history.     

Non-Classical Stems from Classical: N. A. Vasiliev’s Approach to Logic and his Reassessment of the Square of Opposition

In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What reasons forced him to reassess the status of particular propositions and to replace the square of opposition by the triangle of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the new ‘imaginary’ logic, as he called it?      

God, king, and geometry: revisiting the introduction to Cauchy’s Cours d’analyse

This article offers a systematic reading of the introduction to Augustin-Louis Cauchy’s landmark 1821 mathematical textbook, the Cours d’analyse. Despite its emblematic status in the history of mathematical analysis and, indeed, of modern mathematics as a whole, Cauchy’s introduction has been more a source for suggestive quotations than an object of study in its own right. Cauchy’s short mathematical metatext offers a rich snapshot of a scholarly paradigm in transition. A close reading of Cauchy’s writing reveals the complex modalities of the author’s epistemic positioning, particularly with respect to the geometric study of quantities in space, as he struggles to refound the discipline on which he has staked his young career.     

Euler’s fixed point theorem: The axis of a rotation

We give an elementary proof of what is perhaps the earliest fixed point theorem; namely Leonhard Euler’s theorem of 1775 on the existence of an axis v for any three-dimensional rotation R. The proof is constructive and shows that no multiplications are required to compute v. Dedicated to the memory of Leonhard Euler, “The Master of us all”, on the occasion of the 300th anniversary of his birth     

The rank filtration and Robinson’s complex

For a functor from the category of finite sets to abelian groups, Robinson constructed a bicomplex in [A. Robinson, Gamma homology, Lie representations and E_∞ multiplications, Invent. Math. 152 (2) (2003) 331-348] which computes the stable derived invariants of the functor as defined by Dold-Puppe in [A. Dold, D. Puppe, Homologie nicht-additiver Funktoren. Anwendungen., Ann. Inst. Fourier (Grenoble) 11 (1961) 201-312]. We identify a subcomplex of Robinson’s bicomplex which is analogous to a normalization and also computes these invariants. We show that this new bicomplex arises from a natural filtration of the functor obtained by taking left Kan approximations on subcategories of bounded cardinality.     

Symbolism, combinations, and visual imagery in the mathematics of Thomas Harriot

The mathematical work of Thomas Harriot (c. 1560–1621) is distinguished by extensive use of symbolism and other forms of visual imagery and by systematic use of combinations. This paper argues that these characteristics of his mathematical writing were already observable in the mid-1580s, in the phonetic alphabet he devised to record the speech of American Indians. The paper presents several little-known examples of Harriot's mathematics, demonstrating his use of symbolism both as a means of expression and as an analytic tool, and assesses Harriot's work in relation to the broader 17th-century trend toward symbolization in mathematics.     

Geometry at Cambridge, 1863–1940

This paper traces the ebbs and flows of the history of geometry at Cambridge from the time of Cayley to 1940, and therefore the arrival of a branch of modern mathematics in Great Britain. Cayley had little immediate influence, but projective geometry blossomed and then declined during the reign of H.F. Baker, and was revived by Hodge at the end of the period. We also consider the implications these developments have for the concept of a school in the history of mathematics.     

Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics

This article deals with Leibniz's reception of Descartes' “geometry.” Leibnizian mathematics was based on five fundamental notions: calculus, characteristic, art of invention, method, and freedom. On the basis of methodological considerations Leibniz criticized Descartes' restriction of geometry to objects that could be given in terms of algebraic (i.e., finite) equations: “Descartes's mind was the limit of science.” The failure of algebra to solve equations of higher degree led Leibniz to develop linear algebra, and the failure of algebra to deal with transcendental problems led him to conceive of a science of the infinite. Hence Leibniz reconstructed the mathematical corpus, created new (transcendental) notions, and redefined known notions (equality, exactness, construction), thus establishing “a veritable complement of algebra for the transcendentals”: infinite equations, i.e., infinite series, became inestimable tools of mathematical research.     

A dialogue on the use of arithmetic in geometry: Van Ceulen’s and Snellius’s Fundamenta Arithmetica et Geometrica

Snellius’s Fundamenta Arithmetica et Geometrica (1615) is much more than a Latin translation of Ludolph van Ceulen’s Arithmetische en Geometrische Fondamenten. Willebrord Snellius both adapted and commented on the Dutch original in his Fundamenta, and thus his Latin version can be read as a dialogue between representatives of two different approaches to mathematics in the early modern period: Snellius’s humanist approach and Van Ceulen’s practitioner’s approach. This article considers the relationship between the Dutch and Latin versions of the text and, in particular, puts some of their statements on the use of numbers in geometry under the microscope. In addition, Snellius’s use of the Fundamenta as an instrument to further his career is explained.     

Cristoforo Borri and the epistemological status of mathematics in seventeenth-century Portugal

This paper argues that the epistemological promotion of mathematics by the Jesuit Cristoforo Borri, while he was teaching at the Coimbra Jesuit College in the late 1620s, played a decisive role in the updating of cosmological ideas in 17th-century Portugal. The paper focuses on Borri's position on the celebrated quaestio de certitudine mathematicarum and on his understanding of the classification of sciences. It argues that by conferring on mathematics the status of Aristotelian causal science, Borri made it possible to integrate mathematical data into the philosophical debate, particularly with regard to the new cosmology.