pinecone/core-2020-05-10-deduplication huggingface dataset:
Riemann and the Cauchy—Hadamard formula for the convergence of power series

The Cauchy-Hadamard formula for the radius of convergence of a power series was stated and proved by Riemann in his lectures of November 1856. This discovery revises the widespread opinion that, after Cauchy's publication in 1821, the formula was ignored until its rediscovery by Hadamard around 1890.Riemann hat die Cauchy-Hadamard-Formel fur den Konvergenzradius einer Potenzreihe in seinen Vorlesungen im November 1856 aufgestellt and bewiesen. Diese Entdeckung führt zur Revision der allgemeinen Meinung, die Formel sei nach ihrer Veröffentlichung durch Cauchy im Jahre 1821 bis zu ihrer Wiederemdeckung durch Hadamard um 1890 unbeachtet geblieben.Riemann a présenté et démontré dans ses cours, au mois de novembre 1856, la formule de Cauchy-Hadamard sur le rayon de convergence d'une strie entière. Cette découverte infirme l'opinion très répandue selon laquelle la formule serait restée dans t'oubli après sa publication par Cauchy en 1821 jusqu'à sa redécouverte par Hadamard vers 1890
Wine gauging in late 16th- and early 17th-century antwerp

Renaissance European cities imposed an excise on wine and beer. Wine gaugers were employed to measure the contents of barrels. Antwerp was no exception. It organized competitions to appoint new gaugers. Results of two such competitions are given and show that the results for full barrels were satisfactory. It is also shown that the gaugers' remuneration was not very high, since, most probably, they regarded gauging as a source of extra income. Despite the fact that competitions were held, doubts about the competence of gaugers remained. Some gaugers, Michiel Coignet among them, tried to improve measuring practices. In a second section it is demonstrated, using three approximations for barrels and using infinitesimal calculus, that the gaugers' formula for calculating the volume of a barrel yielded results which were presumably within the measuring errors. This is not the case for their formula for partly filled barrels
The visualization of quadratures in the mystery of corollary 3 to proposition 41 of Newton's Principia

When Newton gave his general solution to the inverse problem of central forces in Proposition 41 of Book I of his Principia he offered only one concrete example, the orbits for an inverse cube force with the initial velocity perpendicular to the line to the force center. This example is contained in Corollary 3 to Proposition 41. As it stands in the Principia, Corollary 3 is essentially inaccessible. An explanation of the corollary in a manuscript written by Newton seven years after the publication of the Principia made the corollary accessible to persons with a knowledge of Latin and with significant training in mathematics and physics. This explanation was “evidently written” at the request of the mathematician David Gregory, so Gregory qualifies as one person for whom the corollary became accessible. But the corollary remained inaccessible to most people interested in the history of science. An English translation and notes written by Turnbull in modern times made Corollary 3 much more accessible, but even this further explanation did not go into sufficient detail. The heart of the mystery of Corollary 3 concerns what we call “the visualization of quadratures.” In this paper we go beyond Turnbull's notes and provide a detailed analysis which makes minimal requirements on the technical background of readers. It is our hope that in this way this single important example of Newton's pathbreaking work on the inverse problem will be rendered accessible to the majority of historians of science
Ivory's treatment of pendulum observations

James Ivory (1765–1842) contributed to the mathematical theory of attraction. I describe his efforts (1826–1830) at determining the earth's ellipticity (e) through the adjustment of pendulum observations. At the time, several dozens of such observations had already been made in various latitudes, and their adjustment presented difficulties owing to the local anomalies of gravity. The very possibility of deducing a single ellipticity for the earth remained questionable. While achieving his goal, Ivory made methodical mistakes which he gradually corrected. His final result, 0.00333 < e < 0.00338, compares favorably with the value e = 0.00335 of the so-called Krasovsky ellipsoid. Ivory's work was forgotten mainly because new data, especially on meridian arc measurements, became available rather soon after its publication
Newton on the equiangular spiral: An addendum to Erlichson's account

In a recent article Herman Erlichson called attention to a flaw in Newton's proof of Proposition IX of Book I of the Principia. How did Newton fall into this error? A valid proof was near to hand, by an easy addition to Lemma III of Book II; but evidently Newton wished to attempt a different line of argument. The figure for Proposition IX in the first two editions of the Principia differs from that in the third edition, and does not involve the quadrilateral “given in kind” that Erlichson rightly objects to. But the basic error remains: the assumption without proof of the similarity of all segments of the spiral with the same central angle. By 1671 Newton had proved this assumption by an integration, establishing the logarithmic property of the equiangular spiral. In Proposition IX it would thus appear that he either simply presented as known an analytic result now familiar to himself, or hoped that his readers would consider it as valid for infinitesimal segments
On connecting socialism and mathematics: Dirk Struik, Jan Burgers, and Jan Tinbergen

The Dutch mathematicians, Dirk Struik, Jan Burgers, and Jan Tinbergen, each struggled to find ways to combine their political and scientific aspirations. Although they came from similar social backgrounds and shared much the same scientific training—all were pupils of Paul Ehrenfest—their later approaches to this dilemma varied markedly. Struik's views on mathematics were the most radical, asserting that mathematical conceptions can better be understood in conjunction with larger social and intellectual processes. By contrast, of the three his mathematics changed the least under the influence of external factors. The approaches taken by Burgers and Tinbergen illustrate ways in which the social context can affect a mathematician's work. Their novel ideas helped launch a new paradigm for using mathematics to address social problems, viz., the notion of mathematical modeling, which superseded the conventional approach to applied mathematics
“Scientific control” in mathematical reviewing and German-U.S.-American relations between the two World Wars

Reflections of American mathematicians on the German leadership in mathematical reviewing between the two World Wars reveal one aspect of the rise of American mathematics to worldwide importance. Singular political events such as the mass exodus of German mathematicians and the interference in the work of the then dominant German Zentralblatt led to the foundation of the American Mathematical Reviews in 1940. A look at German national competition in mathematical reviewing during the 1930s (Jahrbuch and Zentralblatt), however, reveals deeper disciplinary as well as ideological and political roots of the “control function” of scientific reviewing that are not necessarily connected to international competition
How probabilities came to be objective and subjective

Between 1837 and 1842 at least six mathematicians and philosophers, writing in French, English, and German, and working independently of one another, introduced distinctions between two kinds of probability. Although the grounds, contents, and implications of these distinctions differed significantly from author to author, all revolved around a philosophical distinction between “objective” and “subjective” which had emerged ca. 1840. It was this new philosophical distinction which permitted the revisionist probabilists to conceive of the possibility of “objective probabilities,” which would have been an oxymoron for classical probabilists such as Jakob Bernoulli and Pierre Simon Laplace. Without relinquishing the rigid determinism of the classical probabilists, the revisionists were nonetheless able to grant chance an objective status in the world by opposing it to the subjective variability of the mind
On mathematics in the history of Sub-Saharan Africa

This article presents an overview of research findings and of sources on or related to mathematics in the history of Africa south of the Sahara. Topics such as counting and numeration systems, mathematical games and puzzles, geometry, graphs, and continental and international connections are included. Attention is paid to the objectives of research in the history of mathematics in Africa, to methodology, to the relationship with ethnomathematical research, and to the uses of research findings in mathematics education. Some possible directions for further research are also identified
On Jacobi's remarkable curve theorem

One of the prettiest results in the global theory of curves is a theorem of Jacobi (1842): The spherical image of the normal directions along a closed differentiable curve in space divides the unit sphere into regions of equal area. The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper, Jacobi (1836). In this note the context for this theorem and its proof are presented as well as a discussion of the ‘error’ corrected by Clausen
Origins of the analysis of the Euclidean algorithm

The Euclidean algorithm for computing the greatest common divisor of two integers is, as D. E. Knuth has remarked, “the oldest nontrivial algorithm that has survived to the present day.” Credit for the first analysis of the running time of the algorithm is traditionally assigned to Gabriel Lamé, for his 1844 paper. This article explores the historical origins of the analysis of the Euclidean algorithm. A weak bound on the running time of this algorithm was given as early as 1811 by Antoine-André-Louis Reynaud. Furthermore, Lamé's basic result was known to Émile Léger in 1837, and a complete, valid proof along different lines was given by Pierre-Joseph-Étienne Finck in 1841
Carl Liebermeister's hypergeometric tails

In 1877 the medical doctor Carl Liebermeister proposed a Bayesian test procedure for homogeneity of two binomial populations. In its hypergeometric distribution structure, it is a forerunner of Fisher's exact test. After some historical remarks, a self-contained probabilistic derivation (based on the duality between the negative and positive hypergeometric distributions) of Liebermeister's procedure is given. The pattern of Liebermeister's own analytical derivation is filled out to show his early and remarkable mastery of the theory of the hypergeometric function
Al-Mu'taman ibn Hūd, 11th century king of Saragossa and brilliant mathematician

This article describes the discovery and the structure of the Kitāb al-Istikmāl, an extensive mathematical work by Yūsuf al-Mu'taman ibn Hūd, who was king of Saragossa between 1081 and 1085. By means of a specific example it is shown that al-Mu'taman was one of the leading mathematicians of 11th-century Islamic Spain. The Istikmāl contributes to our knowledge of Saragossa as a scientific center. This article is a revised version of a lecture given on the occasion of the XIXth International Conference on History of Science in the Palacio de Aljafería in Saragossa, the palace where al-Mu'taman lived
Halley, cotes, and the nautical meridian

The “true division of the nautical meridian,” that is, the correct spacing of the parallels of latitude on the Mercator chart, presented theoretical problems to 17th-century mathematicians. Edmond Halley solved the problem by showing that under stereographic projection a rhumb line (line of constant bearing) projected into a logarithmic spiral. Hence he developed a rule, simple to apply, for calculating the meridional distance corresponding to any given latitude interval. Halley's younger contemporary, Roger Cotes, showed how his own logometric methods yielded Halley's rule more economically, and further, that the use of the logarithmic spiral was not necessary
“What fermented in me for years”: Cantor's discovery of transfinite numbers

Transfinite (ordinal) numbers were a crucial step in the development of Cantor's set theory. The new concept depended in various ways on previous problems and results, and especially on two questions that were at the center of Cantor's attention in September 1882, when he was about to make his discovery. First, the proof of the Cantor-Bendixson theorem motivated the introduction of transfinite numbers, and at the same time suggested the “principle of limitation,” which is the key to the connection between transfinite numbers and infinite powers. Second, Dedekind's ideas, which Cantor discussed in September 1882, seem to have played an important heuristic role in his decision to consider the “symbols of infinity” that he had been using as true numbers, i.e., as autonomous objects; to this end Cantor introduced in his work, for the first time, ideas on (well) ordered sets
Euclidean geometry in the mathematical tradition of islamic India

This paper describes the importance of Euclidean geometry for the educational system in medieval Islamic India and surveys the kinds of sources available for study of this branch of Euclidean scholarship. It examines several types of source documents important for the study of Euclidean thought in India and its ties to other branches of the medieval Euclidean tradition. The major types of sources described are: (1) Arabic and Persian translations, (2) Recensions of these translations, (3) Summaries of the Euclidean corpus, (4) Encyclopedic works that include descriptions of Euclidean geometry, and (5) Rhymed prose (manzūmāt)
Outline of a scientific biography of Ernst Meissel (1826-1895)

A short analysis is given of the scientific work of Ernst Meissel (1826-1895), who worked for most of his active life as a secondary school director in Germany. In particular, Meissel performed many interesting numerical calculations. He is best known perhaps for his computation of 0(109), where, generally speaking, π(x) denotes the number of primes ≤x. Many of his results were announced without proofs, which makes the story more exciting. In passing, more sporadic biographical information is given about some other 19th-century mathematicians: Gram, F. Meisel (≢ Meissel), Oppermann (Gram's teacher), and Weihrauch, among others
The axiomatization of linear algebra: 1875-1940

Modern linear algebra is based on vector spaces, or more generally, on modules. The abstract notion of vector space was first isolated by Peano (1888) in geometry. It was not influential then, nor when Weyl rediscovered it in 1918. Around 1920 it was rediscovered again by three analysts—Banach, Hahn, and Wiener—and an algebraist, Noether. Then the notion developed quickly, but in two distinct areas: functional analysis, emphasizing infinitedimensional normed vector spaces, and ring theory, emphasizing finitely generated modules which were often not vector spaces. Even before Peano, a more limited notion of vector space over the reals was axiomatized by Darboux (1875).L'algèbre linéare moderne a pour concept fondamental l'espace vectoriel ou, plus généralement, le concept de module. Peano (1888) fut le premier à identifier la notion abstraite d'espace vectoriel dans le domaine de la géometric, alors qu'avant lui Darboux avait déjà axiomatisé une notion plus étroite. La notion telle que définie par Peano cut d'abord peu de répercussion, même quand Weyl la redécouvrit en 1918. C'est vers 1920 que les travaux d'analyse de Banach, Hahn, et Wiener ainsi que les recherches algébriques d'Emmy Noether mirent vraiment l'espace vectoriel à l'honneur. A partir de là, la notion se développa rapidement, mais dans deux domaines distincts: celui de l'analyse fonctionnelle où Yon utilisait surtout les espaces vectoriels de dimension infinie, et celui de la théorie des anneaux où les plus importants modules etaient ceux qui sont générés par un nombre fini d'éléments et qui, pour la plupart, ne sont pas des espaces vectoriels.Grundlage der modernen linearen Algebra sind Vektorraume oder, allgemeiner, Moduln. Der abstraiete Begriff des Vektorraumes wurde zuerst von Peano (1888) in der Geometric herausgearbeitet. Er gewann damals kaum Einfluβ, and auch dann noch nicht, als er 1918 von Weyl wiederentdeckt wurde. Um 1920 wurde er von drei Analytikern—Banach, Hahn, Wiener—und einer Algebraikerin, Noether, nochmals wiederentdeckt. Danach wurde der Begriff rasch weiterentwickelt, allerdings auf zwei unterschiedlichen Gebieten: in der Funktionalanalysis unter Betonung endlich-dimensionaler normierter Vektorräume and in der Ringtheorie unter Betonung endlich erzeugter Moduln, die oft keine Vektorräume Bind. Noch vor Peano wurde ein eingeschränkterer Begriff des Vektorraumes über den reellen Zahlen von Darboux (1875) axiomatisiert
L.E.J. Brouwer: Toward Intuitionistic Logic

The evolution of Brouwer's ideas on logic throughout his entire scientific life is practically unknown. Still, the study of this evolution is important, on the one hand, for grasping how Brouwer's notion of intuitionism allowed for the construction of logical systems and, on the other, in order to see how some statements of intuitionistic logic which are now considered obvious were only formulated in their correct form after much rethinking. In this paper, we will thus consider Brouwer's ideas on logic in general as well as some laws in particular while tracing their development (i.e., the laws of excluded middle, testability, and reciprocity of complementarity).Die Entwicklung von Brouwers Auffassungen uber Logik während seines Lebens ist nahezu unbekannt. Die Analyse dieser Entwicklung ist sehr wichtig, weil sie zeigt, erstens daβ Brouwers Intuitionismus die Grundlage fur die Konstruktion eines logischen Systems abgeben konnte; zweitens daβ einige Satze der intuitionistischen Logik, die heute als selbstverständlich angesehen werden, erst nach einem langen ProzeB von Überlegungen korrekt formuliert werden konnten. Deshalb wird in diesem Aufsatz Brouwers Auffassung uber Logik im allgemeinen and uber einige logische Sätze (d.h. den Satz vom ausgeschlossenen Dritten, den von der Alternative von Absurditat oder Absurdität der Absurditat, and den von der Reziprozitat der Komplementaritat) im besonderen betrachten and ihre Entwicklung umrissen.L'evoluzione delle idee di Brouwer sulla logica durante tutta la sua vita scientifica è praticamente sconosciuta. Tuttavia, to studio di questa evoluzione è importante, da un lato per comprendere come il concetto di intuizionismo di Brouwer conteneva già in sé le basi per la costruzione di un sistema logico, d'altro lato per vedere come asserti della logica intuizionista, the sono ora considerati ovvi, furono formulati correttamente solo dopo molti ripensamenti. In questo articolo considereremo le idee di Brouwer sulla logica in generale e su alcune leggi in particolare (la legge del terzo escluso, di provabilità, di reciprocity della complementarità), delineandone l'evoluzione
Branch Points of Algebraic Functions and the Beginnings of Modern Knot Theory

Many of the key ideas which formed modern topology grew out of “normal research” in one of the mainstream fields of 19th-century mathematical thinking, the theory of complex algebraic functions. These ideas were eventually divorced from their original context. The present study discusses an example illustrating this process. During the years 1895-1905, the Austrian mathematician, Wilhelm Wirtinger, tried to generalize Felix Klein's view of algebraic functions to the case of several variables. An investigation of the monodromy behavior of such functions in the neighborhood of singular points led to the first computation of a knot group. Modern knot theory was then formed after a shift in mathematical perspective took place regarding the types of problems investigated by Wirtinger, resulting in an elimination of the context of algebraic functions. This shift, clearly visible in Max Dehn's pioneering work on knot theory, was related to a deeper change in the normative horizon of mathematical practice which brought about mathematical modernity
Drawing the Boundaries: Mathematical Statistics in 20th-Century America

AbstractStatistics in Europe and in the United States acquired the status of a separate field only in the 20th century. Until then, it had provided a quantitative tool for analyzing data in various sciences. In the early 1930s, the founding of theAnnals of Mathematical Statisticsand of the Institute of Mathematical Statistics served to institutionalize the profession of mathematical statistics in the United States, and helped to distinguish it as a discipline independent from the social sciences which had previously supported it. This paper examines the American mathematical statisticians' attempts to delineate the boundaries of their profession, focusing on their association with the social sciences, on the founding of theAnnalsand of the IMS, and on the profession's continuing advancement during World War II.Il n'était qu'au 20èmesiècle que la statistique est devenue une discipline distincte. Antérieurement, elle a servi d'outil quantitatif pour l'analyse des données dans les sciences diverses. Dans les 1930s, l'établissement desAnnals of Mathematical Statisticset de l'Institute of Mathematical Statistics a amené l'institutionalisation d'une profession “statistique mathématique” et a servi à distinguer cette profession des sciences sociales. Ici, j'analyse les tentatives des statisticiens mathématiques américains d'établir les frontières de leur discipline en concentrant sur leurs associations avec les sciences sociales, le fondement desAnnalset de l'IMS, et le progrès de la profession pendant la deuxième guerre mondiale.Erst im 20. Jahrhundert bekam Statistik in Europa und in den Vereinigten Staaten von Amerika die Stellung einer eigenständigen Wissenschaft. Bis dahin hatte man es als ein quantitatives Hilfsmittel zur Datenanalyse in verschiedenen Wissenschaften verwendet. Die Einführung derAnnals of Mathematical Statisticsund die Gründung des “Institute of Mathematical Statistics” um 1930 verhalf, den Stand der mathematischen Statistik in den Vereinigten Staaten zu institutionalisieren und sich von den Sozialwissenschaften zu unterscheiden, in die es zuvor eingeordnet worden war. Diese Veröffentlichung untersucht den Versuch der mathematischen Statistiker in Amerika ihre Arbeit abzugrenzen, unter besonderer Berücksichtigung ihrer Verbindung zu den Sozialwissenschaften, denAnnals of Mathematical Statistics,des “Institute of Mathematical Statistics” und der Weiterentwicklung ihrer Arbeit während des zweiten Weltkrieges