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Abstract: The Latin edition of the Mathematicae Collectiones was published in print in 1588, thirteen years after Federico Commandino’s demise. For his Latin version of Pappus’s work, Comandino used two Greek codices, formerly identified by Treweek. In this article, another Greek manuscript, revised and annotated by Commandino, is revealed. Two letters from Commandino to Ettore Ausonio shed new light on the edition of Pappus’s Collectio and show the partnership between the two mathematicians in elaborating supplementary proofs to include in the comments. Using these letters, we can date the first draft of the Latin version in the late 1560s. The posthumous edition of the Mathematicae Collectiones involved Commandino’s disciples and, in particular, Guidobaldo del Monte. The comparison between the manuscripts and the printed edition reveals an important role played by the disciple in revising the Latin translation of his master. PubDate: 2021-11-21 DOI: 10.1007/s00407-021-00284-1

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Abstract: Résumé Il est tentant de considérer l’œuvre mathématique de Girard Desargues, plus particulièrement son Brouillon Project sur les coniques, comme un travail de mathématiques appliquées à l’art de la perspective. Nous voudrions montrer dans cet article qu’il est sans doute plus pertinent de considérer que Desargues fait des mathématiques en praticien de la perspective ou, plus précisément, que son œuvre peut être lue comme un travail de perspective appliquée à la géométrie. Nous allons analyser quelques passages de l’œuvre du Lyonnais en adoptant ce point de vue perspectiviste afin de montrer comment ce parti pris permet d’éclairer les aspects novateurs d’un contenu mathématique parfois difficile à saisir dans le style touffu de l’auteur. Nous montrerons ensuite comment cette manière de faire de Desargues peut se retrouver chez Philippe de la Hire et Jacques-François le Poîvre, ce qui les a menés à l’idée nouvelle de considérer une transformation du plan dans lui-même comme objet explicite de la géométrie. PubDate: 2021-11-01 DOI: 10.1007/s00407-021-00275-2

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Abstract: This paper provides a detailed study of David Hilbert’s axiomatization of the theory of plane area, in the classical monograph Foundation of Geometry (1899). On the one hand, we offer a precise contextualization of this theory by considering it against its nineteenth-century geometrical background. Specifically, we examine some crucial steps in the emergence of the modern theory of geometrical equivalence. On the other hand, we analyze from a more conceptual perspective the significance of Hilbert’s theory of area for the foundational program pursued in Foundations. We argue that this theory played a fundamental role in the general attempt to provide a new independent basis for Euclidean geometry. Furthermore, we contend that our examination proves relevant for understanding the requirement of “purity of the method” in the tradition of modern synthetic geometry. PubDate: 2021-11-01 DOI: 10.1007/s00407-021-00278-z

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Abstract: This paper is concerned with the status of mathematical fictions in Leibniz’s work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, in turn, to three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectural principles. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz view, a presumptive or “conjectural” status. PubDate: 2021-11-01 DOI: 10.1007/s00407-021-00277-0

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Abstract: In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré’s works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not been addressed either. For example, one question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré’s works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré’s stated motivations for topology and the Poincaré conjecture. PubDate: 2021-10-13 DOI: 10.1007/s00407-021-00283-2

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Abstract: In the middle part of his Brouillon Project on conics, Girard Desargues develops the theory of the traversale, a notion that generalizes the Apollonian diameter and allows to give a unified treatment of the three kinds of conics. We showed elsewhere that it leads Desargues to a complete theory of projective polarity for conics. The present article, which shall close our study of the Brouillon Project, is devoted to the last part of the text, in which Desargues puts his theory of the traversal into practice by giving a very elegant tratment of the classical theory of parameters and foci. This will lead us to show that Desargues’ proofs can only be understood if one accepts that he reasons in a resolutely projective framework, completely assimilating elements at infinity to those at finite distance in his proofs. PubDate: 2021-09-16 DOI: 10.1007/s00407-021-00282-3

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Abstract: Biton’s Construction of Machines of War and Catapults describes six machines by five engineers or inventors; the fourth machine is a rolling elevatable scaling ladder, named sambukē, designed by one Damis of Kolophōn. The first sambukē was invented by Herakleides of Taras, in 214 BCE, for the Roman siege of Syracuse. Biton is often dismissed as incomprehensible or preposterous. I here argue that the account of Damis’ device is largely coherent and shows that Biton understood that Damis had built a machine that embodied Archimedean principles. The machine embodies three such principles: (1) the proportionate balancing of the torques on a lever (from Plane Equilibria, an early work); (2) the concept of specific gravity or density (from Floating Bodies, a late work); and (3) the κοχλίας, i.e., a worm drive (invented ca 240 BCE), with the toothed wheel functioning as the horizontal axis of rotation of the elevated ladder. Moreover, the stone-thrower of Isidoros of Abydos (the second machine in Biton) also embodies the κοχλίας. PubDate: 2021-09-08 DOI: 10.1007/s00407-021-00281-4

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Abstract: We discuss Einstein’s knowledge of projective geometry. We show that two pages of Einstein’s Scratch Notebook from around 1912 with geometrical sketches can directly be associated with similar sketches in manuscript pages dating from his Princeton years. By this correspondence, we show that the sketches are all related to a common theme, the discussion of involution in a projective geometry setting with particular emphasis on the infinite point. We offer a conjecture as to the probable purpose of these geometric considerations. PubDate: 2021-09-01 DOI: 10.1007/s00407-020-00270-z

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Abstract: The introduction of a new analytical method, due fundamentally to François Viète and René Descartes and the later dissemination of their works, resulted in a profound change in the way of thinking and doing mathematics. This change, known as process of algebrization, occurred during the seventeenth and early eighteenth centuries and led to a great transformation in mathematics. Among many other consequences, this process gave rise to the treatment of the results in the classic treatises with the new analytical method, which allowed new visions of such treatises and the obtaining of new results. Among those treatises is the Arithmetic of Diophantus of Alexandria (approx. 200–284) which was written, using the new algebraic language, by the French mathematician Jacques Ozanam (1640–1718), who in addition to profusely increasing the original problems of Diophantus, solved them in a general way, thus obtaining many geometric consequences. The work is handwritten, it has never been published, it has been lost for almost 300 years, and the known references show its importance. We will show that Ozanam’s manuscript was quoted as an important work on several occasions by others mathematicians of the time, among whom G. W. Leibniz stands out. Once the manuscript has been located, our aim in this article is to show and analyze this work of Ozanam, its content, its notation and its structure and how, through the new algebraic method, he not only solved and expanded the questions proposed by Diophantus, but also introduced a connection between the algebraic solutions and what he called geometric determinations by obtaining loci from the solutions. PubDate: 2021-09-01 DOI: 10.1007/s00407-021-00274-3

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Abstract: In this series of papers I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelve Ephemerides and seven Procedure Texts. Three computational systems of Mercury are known, all of system A. System A1 is represented by nine Ephemerides covering the years 190 BC to 100 BC and system A2 by two Ephemerides covering the years 310 to 290 BC. System A3 is known from a Procedure Text and from Text M, an Ephemeris of the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in the Astronomical Diaries and Planetary Texts we find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. To circumvent this difficulty I propose that the Babylonian scholars used an alternative more direct method to fit System A-type models to the observational data of Mercury. This alternative method is based on the fact that after three synodic intervals Mercury returns to a position in the sky which is on average only 17.4° less in longitude. Using reduced amplitudes of about 14°–25° but keeping the same zone boundaries, the computation of what I will call 3-synarc system A models of Mercury is significantly simplified. A full ephemeris of a synodic phase of Mercury can then be composed by combining three columns of longitudes computed with 3-synarc step functions, each column starting with a longitude of Mercury one synodic event apart. Confirmation that this method was indeed used by the Babylonian astronomers comes from Text M (BM 36551+), a very early ephemeris of the last appearances in the evening of Mercury from 424 to 403 BC, computed in three columns according to System A3. Based on an analysis of Text M I suggest that around 400 BC the initial approach in system A modelling of Mercury may have been directed towards choosing “nice” sexagesimal numbers for the amplitudes of the system A step functions while in the later final models, dating from around 300 BC onwards, more emphasis was put on selecting numerical values for the amplitudes such that they were related by simple ratios. The fact that different ephemeris periods were used for each of the four synodic phases of Mercury in the later models may be related to the selection of a best fitting set of System A step function amplitudes for each synodic phase. PubDate: 2021-09-01 DOI: 10.1007/s00407-020-00269-6

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Abstract: We examine a publication by Euler, De novo genere oscillationum, written in 1739 and published in 1750, in which he derived for the first time, the differential equation of the (undamped) simple harmonic oscillator under harmonic excitation, namely the motion of an object acted on by two forces, one proportional to the distance traveled, the other varying sinusoidally with time. He then developed a general solution, using two different methods of integration, making extensive use of direct and inverse sine and cosine functions. After much manipulation of the resulting equations, he proceeded to an analysis of the periodicity of the solutions by varying the relation between two parameters, \(a\) and \(b\) , eventually identifying the phenomenon of resonance in the case where \(2b=a\) . This is shown to be nothing more than the equality between the driving frequency and the natural frequency of the oscillator, which, indeed, characterizes the phenomenon of resonance. Graphical representations of the behavior of the oscillator for different relations between these parameters are given. Despite having been a brilliant discovery, Euler’s publication was not influential and has been neglected by scholars and by specialized publications alike. PubDate: 2021-07-10 DOI: 10.1007/s00407-021-00280-5

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Abstract: Late Babylonian astronomical texts contain records of the stationary points of the outer planets using three different notational formats: Type S where the position is given relative to a Normal Star and whether it is an eastern or western station is noted, Type I which is similar to Type S except that the Normal Star is replaced by a reference to a zodiacal sign, and Type Z the position is given by reference to a zodiacal sign, but no indication of whether the station is an eastern or western station is included. In these records, the date of the station is sometimes preceded by the terms in and/or EN. We have created a database of station records in order to determine whether there was any pattern in the use of these notation types over time or an association with any bias in the station date or the type of text the station was recorded in. Predictive texts, which include Almanacs and Normal Star Almanacs, almost always use Type Z notation, while the Diaries, compilations, and Goal-Year Texts use all three types. Type Z records almost never include in or EN, while other types seem to include these interchangeably. When compared with modern computed station dates, the records show bias toward earlier dates, suggesting that the Babylonians were observing dates when the planets appeared to stop moving rather than the true station. Overlapping reports, where a station on the same date was recorded in two or more texts, suggest that predicted station dates were used to guide observations, and that the planet’s position on the predicted stationary date was the true point of the observation rather than the specific date of the stationary point. PubDate: 2021-07-01 DOI: 10.1007/s00407-021-00272-5

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Abstract: The history of uniform convergence is typically focused on the contributions of Cauchy, Seidel, Stokes, and Björling. While the mathematical contributions of these individuals to the concept of uniform convergence have been much discussed, Weierstrass is considered to be the actual inventor of today’s concept. This view is often based on his well-known article from 1841. However, Weierstrass’s works on a rigorous foundation of analytic and elliptic functions date primarily from his lecture courses at the University of Berlin up to the mid-1880s. For the history of uniform convergence, these lectures open up an independent branch of development that is disconnected from the approaches of the previously mentioned authors; to my knowledge, Weierstraß never explicitly referred to Cauchy’s continuity theorem (1821 or 1853) or to Seidel’s or Stokes’s contributions (1847). In the present article, Weierstrass’s contributions to the development of uniform convergence will be discussed, mainly based on lecture notes made by Weierstrass’s students between 1861 and the mid-1880s. The emphasis is on the notation and the mathematical rigor of the introductions to the concept, leading to the proposal to re-date the famous 1841 article and thus Weierstrass’s first introduction of uniform convergence. PubDate: 2021-07-01 DOI: 10.1007/s00407-020-00266-9

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Abstract: In June 1888, Oliver Heaviside received by mail an officially unpublished pamphlet, which was written and printed by the American author Willard J. Gibbs around 1881–1884. This original document is preserved in the Dibner Library of the History of Science and Technology at the Smithsonian Institute in Washington DC. Heaviside studied Gibbs’s work very carefully and wrote some annotations in the margins of the booklet. He was a strong defender of Gibbs’s work on vector analysis against quaternionists, even if he criticised Gibbs’s notation system. The aim of our paper is to analyse Heaviside’s annotations and to investigate the role played by the American physicist in the development of Heaviside’s work. PubDate: 2021-07-01 DOI: 10.1007/s00407-020-00264-x

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Abstract: The gravitational influence of Jupiter on Saturn produces, among other things, non-negligible changes in the eccentricity of Saturn that affect the magnitude of error of Ptolemaic astronomy. The value that Ptolemy obtained for the eccentricity of Saturn is a good approximation of the real eccentricity—including the perturbation of Jupiter—that Saturn had during the time of Ptolemy's planetary observations or a bit earlier. Therefore, it seems more probable that the observations used for obtaining the eccentricity of Saturn were done near Ptolemy’s time, and rather unlikely earlier than the first century AD. Even if this is not quite a demonstration that Ptolemy used observations of his own, my argument increases its probability and practically discards the idea that Ptolemy borrowed values or observations from astronomers further back than the first century AD, such as Hipparchus or the Babylonians. PubDate: 2021-07-01 DOI: 10.1007/s00407-020-00271-y

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Abstract: In this paper we discuss in some depth the main theorems pertaining to Carnot’s theory of transversals, their initial reception by Servois, and the applications that Brianchon made of them to the theory of conic sections. The contributions of these authors brought the long-forgotten theorems of Desargues and Pascal fully to light, renewed the interest in synthetic geometry in France, and prepared the ground from which projective geometry later developed. PubDate: 2021-06-08 DOI: 10.1007/s00407-021-00276-1

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Abstract: The paper provides an analysis of Giuseppe Vitali’s contributions to differential geometry over the period 1923–1932. In particular, Vitali’s ambitious project of elaborating a generalized differential calculus regarded as an extension of Ricci-Curbastro tensor calculus is discussed in some detail. Special attention is paid to describing the origin of Vitali’s calculus within the context of Ernesto Pascal’s theory of forms and to providing an analysis of the process leading to a fully general notion of covariant derivative. Finally, the reception of Vitali’s theory is discussed in light of Enea Bortolotti and Enrico Bompiani’s subsequent works. PubDate: 2021-05-24 DOI: 10.1007/s00407-021-00273-4

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Abstract: In this paper I present an alternative reading and interpretation of the cuneiform tablet BM 76829. I suggest that the obverse of the tablet contains a simple astrological scheme linking the sign of the zodiac in which a child is born to the maximum length of life, and that the reverse contains a copy of a scheme relating parts of the body to the signs of the zodiac. PubDate: 2021-05-20 DOI: 10.1007/s00407-021-00279-y

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Abstract: Traditionally, “the operator calculus of Born and Wiener” has been considered one of the four formulations of quantum mechanics that existed in 1926. The present paper reviews the operator calculus as applied by Max Born and Norbert Wiener during the last months of 1925 and the early months of 1926 and its connections with the rise of the new quantum theory. Despite the relevance of this operator calculus, Born–Wiener’s joint contribution to the topic is generally bypassed in historical accounts of quantum mechanics. In this study, we analyse the paper that epitomises the contribution, and we explain the main reasons for the apparent lack of interest in Born and Wiener’s work. We argue that they did not solve the main problem for which the tool was intended, that of linear motion, because of their reluctance to use Dirac delta functions. PubDate: 2021-05-01 DOI: 10.1007/s00407-020-00262-z

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Abstract: This paper aims both to tackle the technical issue of deciphering Hobbes’s derivation of the sine law of refraction and to throw some light to the broader issue of Hobbes’s mechanical philosophy. I start by recapitulating the polemics between Hobbes and Descartes concerning Descartes’ optics. I argue that, first, Hobbes’s criticisms do expose certain shortcomings of Descartes’ optics which presupposes a twofold distinction between real motion and inclination to motion, and between motion itself and determination of motion; second, Hobbes’s optical theory presented in Tractatus Opticus I constitutes a more economical alternative, which eliminates the twofold distinction and only admits actual local motion, and Hobbes’s derivation of the sine law presented therein, which I call “the early model” and which was retained in Tractatus Opticus II and First Draught, is mathematically consistent and physically meaningful. These two points give Hobbes’s early optics some theoretical advantage over that of Descartes. However, an issue that has baffled commentators is that, in De Corpore Hobbes’s derivation of the sine law seems to be completely different from that presented in his earlier works, furthermore, it does not make any intuitive sense. I argue that the derivation of the sine law in De Corpore does make sense mathematically if we read it as a simplification of the early model, and Hobbes has already hinted toward it in the last proposition of Tractatus Opticus I. But now the question becomes, why does Hobbes take himself to be entitled to present this simplified, seemingly question-begging form without having presented all the previous results' My conjecture is that the switch from the early model to the late model is symptomatic of Hobbes’s changing views on the relation between physics and mathematics. PubDate: 2021-05-01 DOI: 10.1007/s00407-020-00265-w