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Abstract: Federico Commandino’s Latin editions of the mathematical works written by the ancient Greeks constituted an essential reference for the scientific research undertaken by the moderns. In his Latin editions, Commandino cleverly combined his philological and mathematical skills. Philology and mathematics, moreover, nurtured each other. In this article, I analyze the Greek and Latin manuscripts and the printed edition of Apollonius’ Conics to highlight in a specific case study the role of the editions of the classics in the renaissance of modern mathematics. PubDate: 2023-03-20
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Abstract: Much of the mathematics with which Felix Klein and Sophus Lie are now associated (Klein’s Erlangen Program and Lie’s theory of transformation groups) is rooted in ideas they developed in their early work: the consideration of geometric objects or properties preserved by systems of transformations. As early as 1870, Lie studied particular examples of what he later called contact transformations, which preserve tangency and which came to play a crucial role in his systematic study of transformation groups and differential equations. This note examines Klein’s efforts in the 1870s to interpret contact transformations in terms of connexes and traces that interpretation (which included a false assumption) over the decades that follow. The analysis passes from Klein’s letters to Lie through Lindemann’s edition of Clebsch’s lectures on geometry in 1876, Lie’s criticism of it in his treatise on transformation groups in 1893, and the careful development of that interpretation by Dohmen, a student of Engel, in his 1905 dissertation. The now-obscure notion of connexes and its relation to Lie’s line elements and surface elements are discussed here in some detail. PubDate: 2023-03-09 DOI: 10.1007/s00407-023-00305-1
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Abstract: The eighth-century Latin manuscript Milan, Veneranda Biblioteca Ambrosiana, L 99 Sup. contains fifteen palimpsest leaves previously used for three Greek scientific texts: a text of unknown authorship on mathematical mechanics and catoptrics, known as the Fragmentum Mathematicum Bobiense (three leaves), Ptolemy's Analemma (six leaves), and an astronomical text that has hitherto remained unidentified and almost entirely unread (six leaves). We report here on the current state of our research on this last text, based on multispectral images. The text, incompletely preserved, is a treatise on the construction and uses of a nine-ringed armillary instrument, identifiable as the “meteoroscope” invented by Ptolemy and known to us from passages in Ptolemy's Geography and in writings of Pappus and Proclus. We further argue that the author of our text was Ptolemy himself. PubDate: 2023-03-09 DOI: 10.1007/s00407-022-00302-w
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Abstract: This essay traces the history of early molecular dynamics simulations, specifically exploring the development of SHAKE, a constraint-based technique devised in 1976 by Jean-Paul Ryckaert, Giovanni Ciccotti and the late Herman Berendsen at CECAM (Centre Européen de Calcul Atomique et Moléculaire). The work of the three scientists proved to be instrumental in giving impetus to the MD simulation of complex polymer systems and it currently underpins the work of thousands of researchers worldwide who are engaged in computational physics, chemistry and biology. Despite its impact and its role in bringing different scientific fields together, accurate historical studies on the birth of SHAKE are virtually absent. By collecting and elaborating on the accounts of Ryckaert and Ciccotti, this essay aims to fill this gap, while also commenting on the conceptual and computational difficulties faced by its developers. PubDate: 2023-02-21 DOI: 10.1007/s00407-023-00306-0
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Abstract: The idea of a canonical transformation emerged in 1837 in the course of Carl Jacobi's researches in analytical dynamics. To understand Jacobi's moment of discovery it is necessary to examine some background, especially the work of Joseph Lagrange and Siméon Poisson on the variation of arbitrary constants as well as some of the dynamical discoveries of William Rowan Hamilton. Significant figures following Jacobi in the middle of the century were Adolphe Desboves and William Donkin, while the delayed posthumous publication in 1866 of Jacobi's full dynamical corpus was a critical event. François Tisserand's doctoral dissertation of 1868 was devoted primarily to lunar and planetary theory but placed Hamilton–Jacobi mathematical methods at the forefront of the investigation. Henri Poincaré's writings on celestial mechanics in the period 1890–1910 succeeded in making canonical transformations a fundamental part of the dynamical theory. Poincaré offered a mathematical vision of the subject that differed from Jacobi's and would become influential in subsequent research. Two prominent researchers around 1900 were Carl Charlier and Edmund Whittaker, and their books included chapters devoted explicitly to transformation theory. In the first three decades of the twentieth century Hamilton–Jacobi theory in general and canonical transformations in particular would be embraced by a range of researchers in astronomy, physics and mathematics. PubDate: 2023-01-31 DOI: 10.1007/s00407-022-00303-9
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Abstract: Modern color science finds its birth in the middle of the nineteenth century. Among the chief architects of the new color theory, the name of the polymath Hermann von Helmholtz stands out. A keen experimenter and profound expert of the latest developments of the fields of physiological optics, psychophysics, and geometry, he exploited his transdisciplinary knowledge to define the first non-Euclidean line element in color space, i.e., a three-dimensional mathematical model used to describe color differences in terms of color distances. Considered as the first step toward a metrically significant model of color space, his work inaugurated researches on higher color metrics, which describes how distance in the color space translates into perceptual difference. This paper focuses on the development of Helmholtz’s mathematical derivation of the line element. Starting from the first experimental evidence which opened the door to his reflections about the geometry of color space, it will be highlighted the pivotal role played by the studies conducted by his assistants in Berlin, which provided precious material for the elaboration of the final model proposed by Helmholtz in three papers published between 1891 and 1892. Although fallen into oblivion for about three decades, Helmholtz’s masterful work was rediscovered by Schrödinger and, since the 1920s, it has provided the basis for all subsequent studies on the geometry of color spaces up to the present time. PubDate: 2023-01-17 DOI: 10.1007/s00407-023-00304-2
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Abstract: In this article, we report the discovery of a new type of astronomical almanac by Joseph Ibn Waqār (Córdoba, fourteenth century) that begins at second station for each of the planets and may have been intended to serve as a template for planetary positions beginning at any dated second station. For background, we discuss the Ptolemaic tradition of treating stations and retrograde motions as well as two tables in Arabic zijes for the anomalistic cycles of the planets in which the planets stay at first and second stations for a period of time (in contrast to the Ptolemaic tradition). Finally, we consider some medieval astrological texts where stations or retrograde motions are invoked. PubDate: 2023-01-06 DOI: 10.1007/s00407-022-00301-x
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Abstract: We examine one of the well-known mathematical works of Abraham bar Ḥiyya: Ḥibbur ha-Meshiḥah ve-ha-Tishboret, written between 1116 and 1145, which is one of the first extant mathematical manuscripts in Hebrew. In the secondary literature about this work, two main theses have been presented: the first is that one Urtext exists; the second is that two recensions were written—a shorter, more practical one, and a longer, more scientific one. Critically comparing the eight known copies of the Ḥibbur, we show that contrary to these two theses, one should adopt a fluid model of textual transmission for the various manuscripts of the Ḥibbur, because neither of these two theses can account fully for the changes among the various manuscripts. We hence offer to concentrate on the typology of the variations among the various manuscripts, dealing with macro-changes (such as omissions or additions of proofs, additional appendices or a reorganization of the text itself), and micro-changes (such as textual and pictorial variants). PubDate: 2022-10-20 DOI: 10.1007/s00407-022-00297-4
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Abstract: By 1933, the class of generally accepted elementary particles comprised the electron, the photon, the proton as well as newcomers in the shape of the neutron, the positron, and the neutrino. During the following decade, a new and poorly understood particle, the mesotron or meson, was added to the list. By paying close attention to the names of these and other particles and to the sometimes controversial proposals of names, a novel perspective on this well-researched line of development is offered. Part of the study investigates the circumstances around the coining of “positron” as an alternative to “positive electron.” Another and central part is concerned with the many names associated with the discovery of what in the late 1930s was generally called the “mesotron” but eventually became known as the “meson” and later again the muon and pion. The naming of particles in the period up to the early 1950s was more than just a matter of agreeing on convenient terms, it also reflected different conceptions of the particles and in some cases the uncertainty regarding their nature and relations to existing theories. Was the particle discovered in the cosmic rays the same as the one responsible for the nuclear forces' While two different names might just be synonymous referents, they might also refer to widely different conceptual images. PubDate: 2022-09-21 DOI: 10.1007/s00407-022-00299-2
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Abstract: The Jeffreys–Lindley paradox exposes a rift between Bayesian and frequentist hypothesis testing that strikes at the heart of statistical inference. Contrary to what most current literature suggests, the paradox was central to the Bayesian testing methodology developed by Sir Harold Jeffreys in the late 1930s. Jeffreys showed that the evidence for a point-null hypothesis \({\mathcal {H}}_0\) scales with \(\sqrt{n}\) and repeatedly argued that it would, therefore, be mistaken to set a threshold for rejecting \({\mathcal {H}}_0\) at a constant multiple of the standard error. Here, we summarize Jeffreys’s early work on the paradox and clarify his reasons for including the \(\sqrt{n}\) term. The prior distribution is seen to play a crucial role; by implicitly correcting for selection, small parameter values are identified as relatively surprising under \({\mathcal {H}}_1\) . We highlight the general nature of the paradox by presenting both a fully frequentist and a fully Bayesian version. We also demonstrate that the paradox does not depend on assigning prior mass to a point hypothesis, as is commonly believed. PubDate: 2022-08-26 DOI: 10.1007/s00407-022-00298-3
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Abstract: In a 1936 manuscript submitted to the Physical Review, Albert Einstein and Nathan Rosen famously claimed that gravitational waves do not exist. It has generally been assumed that there was a conceptual error underlying this fallacious claim. It will be shown, through a detailed study of the extant referee report, that this claim was probably only the result of a calculational error, the accidental use of a pathological coordinate transformation. PubDate: 2022-08-25 DOI: 10.1007/s00407-022-00295-6
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Abstract: In this paper, we try to understand what considerations and possible sources of inspiration Desargues used to formulate his concepts of involution and transversal, and to state the related theorems that are at the basis of his Brouillon project. To this end, we trace some clues which are found scattered throughout his works, we connect them together in the light of his experience and knowledge in the field of perspective, and we investigate what were his motivations within Mersenne’s academy. As a result of our research, we can safely say that were his great geometrical insight and his projective vision of space which, guided by some classical theorems, led him to these completely new concepts in the panorama of the geometry of that time that were destined to remain misunderstood for about two centuries. PubDate: 2022-08-10 DOI: 10.1007/s00407-022-00296-5
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Abstract: In 1927, Paul Dirac first explicitly introduced the idea that electrodynamical processes can be evaluated by decomposing them into virtual (modern terminology), energy non-conserving subprocesses. This mode of reasoning structured a lot of the perturbative evaluations of quantum electrodynamics during the 1930s. Although the physical picture connected to Feynman diagrams is no longer based on energy non-conserving transitions but on off-shell particles, emission and absorption subprocesses still remain their fundamental constituents. This article will access the introduction and the initial reception of this picture of subsequent transitions (PST) by conceiving of concepts, models, and their representations as tools for the practitioners. I will argue for a multi-factorial explanation of Dirac’s initial, verbally explicit introduction: the mathematical representation he had developed was highly suggestive and already partly conceptualized; Dirac was philosophical flexible enough to talk about transitions when no actual transitions, according to the general interpretation of quantum mechanics of the time, occurred; and, importantly, Dirac eventually used the verbal exposition in the same paper in which he introduced it. The direct impact of PST on the conception of quantum electrodynamical processes will be exemplified by its reflection in diagrammatical representations. The study of the diverging ontological commitments towards PST immediately after its introduction opens up the prehistory of a philosophical debate that stretches out into the present: the dispute about the representational and ontological status of the physical picture connected to the evaluation of the perturbative series of QED and QFT. PubDate: 2022-07-05 DOI: 10.1007/s00407-022-00293-8
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Abstract: Aristarchus’s De magnitudinis et distantiis solis et lunae was translated into Latin and printed by Federico Commandino in 1572. All subsequent editions of Aristarchus’ treatise, published by John Wallis (1688), Fortia d’ Urban (1823) and Thomas Heath (1913), followed Commandino’s work. In this article, through a philological approach to the geometric diagrams, I tracked down one of the Greek sources used by Commandino for preparing his Latin version. Commandino pays particular attention to drawing figures. This article sheds light on the interaction between mathematical skills and the drawing of geometric diagrams implemented in his Latin edition of Aristarchus’ book. PubDate: 2022-06-29 DOI: 10.1007/s00407-022-00294-7
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Abstract: The paper brings into light and discusses a concentric solar model briefly described in Chapter 5 of Section III of ‘Abd al-Raḥmān al-Khāzinī’s On experimental astronomy, a treatise embedded in the prolegomenon of his comprehensive Mu‘tabar zīj, completed about 1121 c.e. In it, the Sun is assumed to rotate on the circumference of a circle concentric with the Earth and coplanar with the ecliptic, but the motion of the vector joining the Earth and Sun is monitored by a small eccentric hypocycle. The ratio between the distance of the hypocycle’s center from the Earth and the hypocycle’s radius is equal to the solar eccentricity in the eccentric model. The model is to account for the constancy of the apparent diameter of the solar disk as held by Ptolemy. The source of the model is unknown. Since the mechanism employed in it clearly resembles the pin-and-slot device, whose use in mechanical astronomical instruments has a long history from the Antikythera Mechanism to the medieval solar, lunar, and planetary equatoria and dials, we argue that the solar model can be positioned within this long-standing tradition and considered the result of the correct understanding of some Byzantine prototype and thus a typical example of the transmission of astronomical ideas via media of the material culture. PubDate: 2022-06-11 DOI: 10.1007/s00407-022-00292-9
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Abstract: We consider the Geometria Practica of Christopher Clavius, S.J., a surprisingly eclectic and comprehensive practical geometry text, whose first edition appeared in 1604. Our focus is on four particular sections from Books IV and VI where Clavius has either used his sources in an interesting way or where he has been uncharacteristically reticent about them. These include the treatments of Heron’s Formula, Archimedes’ Measurement of the Circle, four methods for constructing two mean proportionals between two lines, and finally an algorithm for computing nth roots of numbers. PubDate: 2022-05-13 DOI: 10.1007/s00407-022-00288-5
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Abstract: The manuscript UCLA 170/624 (ff. 75–76) contains Galileo’s proof of the center of gravity of the frustum of a cone, which was ultimately published as Theoremata circa centrum gravitatis solidorum in Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Leiden 1638). The UCLA copy opens the possibility of giving a fuller account of Theoremata dating and development, and it can shed light on the origins of this research by the young Galileo. A comparison of the UCLA manuscript with the other extant copies is carried out to propose a new dating for the composition of the Theoremata. This dating will then be reconsidered in light of the mathematical content. The paper ends with a complete description of the content of the UCLA manuscript and the edition of Galileo’s text there contained. PubDate: 2022-04-28 DOI: 10.1007/s00407-022-00289-4
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Abstract: This paper addresses an article by Felix Klein of 1886, in which he generalized his theory of polynomial equations of degree 5—comprehensively discussed in his Lectures on the Icosahedron two years earlier—to equations of degree 6 and 7. To do so, Klein used results previously established in line geometry. I review Klein’s 1886 article, its diverse mathematical background, and its place within the broader history of mathematics. I argue that the program advanced by this article, although historically overlooked due to its eventual failure, offers a valuable insight into a time of crucial evolution of the subject. PubDate: 2022-04-25 DOI: 10.1007/s00407-022-00290-x
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Abstract: Estimating the length of the Greek stadion remains controversial. This paper highlights the pitfalls of a purely metrological approach to this problem and proposes a formal differentiation between metrologically defined ancient measuring units and other measures used to estimate long distances. The common-sense approach to the problem is strengthened by some cross-over documentary evidence for usage of the so-called itinerary stadion in antiquity. We discuss the possibility of using statistical analysis methods to estimate the length of the stadion by comparing ancient routes with the actual distances. Simple numerical examples explain the limits of this approach, caused by the low number of data and by their mixed character. A special case of distances which can be calculated with the help of coordinates given in Ptolemy’s Geography is discussed, and has been shown to lead unavoidably to ambiguous solutions. PubDate: 2022-03-03 DOI: 10.1007/s00407-022-00287-6
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