Authors:Bernard R. Goldstein; José Chabás Pages: 21 - 62 Abstract: In astronomy Abraham Zacut (1452–1514) is best known for the Latin version of his tables, the Almanach Perpetuum, first published in 1496, based on the original Hebrew version that he composed in 1478. These tables for Salamanca, Spain, were analyzed by the authors of this paper in 2000. We now present Zacut’s tables preserved in Latin and Hebrew manuscripts that have not been studied previously, with a concordance of his tables in different sources. Based on a hitherto unnoticed text in a Latin manuscript, we argue that Zacut is the author of the Tabule verificate which, in our publication of 2000, we took to be anonymous. We also discuss in detail Zacut’s tables for epoch 1513 for Jerusalem that are arranged for the Hebrew calendar, rather than the Julian calendar that he used elsewhere. We then consider a number of fragmentary texts that were found in the Cairo Geniza, now scattered in various European and American libraries. The new evidence is consistent with our earlier finding that Zacut depended both on the medieval Hebrew tradition in astronomy and on the Parisian Alfonsine Tables. PubDate: 2018-01-01 DOI: 10.1007/s00407-017-0199-2 Issue No:Vol. 72, No. 1 (2018)

Authors:Hamid-Reza Giahi Yazdi Pages: 89 - 98 Abstract: This article deals with an unstudied criterion for determining lunar crescent visibility, which appears in the Mufrad Z \(\bar{\iota }\) j, (compiled by Ḥāsib al-Ṭabar \(\bar{\upiota }\) , 5thc.A.H./11th c.A.D.). Al-Ṭabar \(\bar{\upiota }\) attributes this circular criterion to Al-B \(\bar{\upiota }\) rūn \(\bar{\upiota }\) . Initially, Prof. David King shed light on this criterion in 1987 and explained it briefly. We will examine this criterion by re-computing the underlying numerical values to reconstruct it, in order to demonstrate that it originates from Ḥabash’s simple criterion. PubDate: 2018-01-01 DOI: 10.1007/s00407-017-0201-z Issue No:Vol. 72, No. 1 (2018)

Authors:J. Climent Vidal; J. Soliveres Tur Abstract: The original version of this article unfortunately contained a mistake: The equation on page 34 was incorrect. The corrected equation is given below. PubDate: 2018-02-07 DOI: 10.1007/s00407-018-0203-5

Authors:J. Climent Vidal; J. Soliveres Tur Abstract: We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and set theory, which undoubtedly point towards the mathematics of twentieth and twenty-first centuries. PubDate: 2018-01-17 DOI: 10.1007/s00407-018-0202-6

Authors:C. Philipp E. Nothaft Abstract: This article examines an unstudied set of astronomical tables for the meridian of Cambridge, also known as the Opus secundum, which the English theologian and astronomer John Holbroke, Master of Peterhouse, composed in 1433. These tables stand out from other late medieval adaptations of the Alfonsine Tables in using a different set of parameters for planetary mean motions, which Holbroke can be shown to have derived from a tropical year of \(365\frac{1}{4} - \frac{1}{132}\) or \(365.\overline{24}\) days. Implicit in this year length was a 33-year cycle of repeating solar longitudes and equinox times, which has left traces in other astronomical tables from fifteenth-century England. An analysis of the manuscript evidence suggests that Holbroke owed his value for the “true length of the year” to a certain Richard Monke, capellanus de Anglia, who employed this parameter and the corresponding 33-year cycle in an attempt to construct a perfect and perpetual solar calendar, leading to his Kalendarium verum anni mundi of 1434. PubDate: 2018-01-04 DOI: 10.1007/s00407-017-0200-0

Authors:Christián C. Carman Abstract: It is well known that heliocentrism was proposed in ancient times, at least by Aristarchus of Samos. Given that ancient astronomers were perfectly capable of understanding the great advantages of heliocentrism over geocentrism—i.e., to offer a non-ad hoc explanation of the retrograde motion of the planets and to order unequivocally all the planets while even allowing one to know their relative distances—it seems difficult to explain why heliocentrism did not triumph over geocentrism or even compete significantly with it before Copernicus. Usually, scholars refer to explanations of sociological character. In this paper, I offer a different explanation: that the pre-Copernican heliocentrism was essentially different from the Copernican heliocentrism, in such a way that the adduced advantages of heliocentrism can only be attributed to Copernican heliocentrism, but not to pre-Copernican heliocentrism proposals. PubDate: 2017-12-23 DOI: 10.1007/s00407-017-0198-3

Authors:Marie Anglade; Jean-Yves Briend Abstract: Résumé Nous tentons dans cet article de proposer une thèse cohérente concernant la formation de la notion d’involution dans le Brouillon Project de Desargues. Pour cela, nous donnons une analyse détaillée des dix premières pages dudit Brouillon, comprenant les développements de cas particuliers qui aident à comprendre l’intention de Desargues. Nous mettons cette analyse en regard de la lecture qu’en fait Jean de Beaugrand et que l’on trouve dans les Advis Charitables. PubDate: 2017-11-08 DOI: 10.1007/s00407-017-0196-5

Authors:David Aubin Abstract: In this paper, we investigate the way in which French artillery engineers met the challenge of air drag in the nineteenth century. This problem was especially acute following the development of rifled barrels, when projectile initial velocities reached values much higher than the speed of sound in air. In these circumstances, the Newtonian approximation according to which the drag was a force proportional to the square of the velocity ( \(v^2\) ) was not nearly good enough to account for experimental results. This prompted a series of theoretical and experimental investigations aimed at determining the correct law of air resistance. Throughout the nineteenth century, contrary to what happened before or after, ballistician were—with very rare exceptions—alone in trying to tackle the problem of air resistance. This was a complex problem where theoretical considerations, experimental results, and computational algorithms intermingled with one another, as well as with the development of new materials and doctrine in artillery. By carefully studying the reasons why ballisticians finally opted for a complex empirical law at the end of the nineteenth century, we show that military procedures for evaluating materials became a yardstick for assessing the worth of mathematical theories as well. In conclusion, we try to assess why military specialists were not able to face the challenges posed by World War I and required the help of civilian scientists and mathematicians. PubDate: 2017-07-29 DOI: 10.1007/s00407-017-0195-6

Authors:Gerd Graßhoff; Florian Mittenhuber; Elisabeth Rinner Abstract: In his Geography, Ptolemy recorded the geographical coordinates of more than 6,300 toponyms of the known oikoumenē. This study presents the type of geographical information that was used by Ptolemy as well as the methods he applied to derive his geographical coordinates. A new methodological approach was developed in order to analyse the characteristic deviations (displacement vectors) of Ptolemy’s data from their reconstructed reference locations. The clusters of displacement vectors establish that Ptolemy did not obtain his coordinates from astronomical observations at each geographical location. The characteristic displacement vectors reveal how Ptolemy derived the coordinates: (1) he constructed locations on maps using a compass and ruler, for which he employed a small amount of astronomical reference data and geographical distance information; (2) he made schematic drawings of coastlines, based on textual descriptions of coastal formations; (3) and he situated additional locations within the established framework using reports of travel itineraries. PubDate: 2017-07-24 DOI: 10.1007/s00407-017-0194-7

Authors:Christopher D. Hollings Abstract: In the early years of the twentieth century, the so-called ‘postulate analysis’—the study of systems of axioms for mathematical objects for their own sake—was regarded by some as a vital part of the efforts to understand those objects. I consider the place of postulate analysis within early twentieth-century mathematics by focusing on the example of a group: I outline the axiomatic studies to which groups were subjected at this time and consider the changing attitudes towards such investigations. PubDate: 2017-07-20 DOI: 10.1007/s00407-017-0193-8

Authors:Dirk Grupe Abstract: Earlier than the Arabic-Latin transfer of Ptolemaic astronomy via the Iberian peninsula, a serious occupation with Arabic astronomy by Latin scholars took place in crusader Antioch in the first half of the twelfth century. One of the translators of Arabic science in the East was Stephen of Pisa, who produced a commented Latin version, entitled Liber Mamonis, of Ibn al-Haytham’s cosmography, On the Configuration of the World. Stephen’s considerations about the physical universe in relation to the doctrines of Ptolemaic astronomy have hitherto received but little attention. The present paper discusses Stephen of Pisa’s treatment of the planetary spheres in regard to Ptolemy’s theory of oscillating deferents. Emphasis is given to geometric arguments in Stephen’s criticism of Ibn al-Haytham’s spherical model of the inner planets and to Stephen’s own attempt at an improved theory based on additional spheres. The paper argues that astronomical studies in Antioch were of an advanced level, involving independent judgement as well as an influence of contemporary trends in Arabic astronomy. PubDate: 2017-06-23 DOI: 10.1007/s00407-017-0192-9

Authors:Nicola M. R. Oswald Abstract: Adolf Hurwitz’s estate contains a note from the early 1880s on the converse to Riemann’s proof of the functional equation for the zeta-function; this idea has later been elaborated by Hans Hamburger for a characterization of the zeta-function by its functional equation and by Eugène Cahen and Erich Hecke with respect to modular forms. In this note, we present Hurwitz’s reasoning and comment on the historical context. PubDate: 2017-04-27 DOI: 10.1007/s00407-017-0190-y

Authors:Călin Galeriu Abstract: The study of an electric charge in hyperbolic motion is an important aspect of Minkowski’s geometrical formulation of electrodynamics. In “Space and Time”, his last publication before his premature death, Minkowski gives a brief geometrical recipe for calculating the four-force with which an electric charge acts on another electric charge. The subsequent work of Born, Sommerfeld, Laue, and Pauli filled in the missing derivation details. Here, we bring together these early contributions, in an effort to provide a more modern, accessible, and unified exposition of the early history of the electric charge in hyperbolic motion. PubDate: 2017-04-09 DOI: 10.1007/s00407-017-0191-x

Authors:Steven Shnider Abstract: The following article has two parts. The first part recounts the history of a series of discoveries by Otto Neugebauer, Bartel van der Waerden, and Asger Aaboe which step by step uncovered the meaning of Column \(\varPhi \) , the mysterious leading column in Babylonian System A lunar tables. Their research revealed that Column \(\varPhi \) gives the length in days of the 223-month Saros eclipse cycle and explained the remarkable algebraic relations connecting Column \(\varPhi \) to other columns of the lunar tables describing the duration of 1, 6, or 12 synodic months. Part two presents John Britton’s theory of the genesis of Column \(\varPhi \) and the System A lunar theory starting from a fundamental equation relating the columns discovered by Asger Aaboe. This article is intended to explain and, hopefully, to clarify Britton’s original articles which many readers found difficult to follow. PubDate: 2017-03-04 DOI: 10.1007/s00407-017-0189-4

Authors:Gert Schubring Abstract: When did the concept of model begin to be used in mathematics? This question appears at first somewhat surprising since “model” is such a standard term now in the discourse on mathematics and “modelling” such a standard activity that it seems to be well established since long. The paper shows that the term— in the intended epistemological meaning—emerged rather recently and tries to reveal in which mathematical contexts it became established. The paper discusses various layers of argumentations and reflections in order to unravel and reach the pertinent kernel of the issue. The specific points of this paper are the difference in the epistemological concept to the usually discussed notions of model and the difference between conceptions implied in mathematical practices and their becoming conscious in proper reflections of mathematicians. PubDate: 2017-01-28 DOI: 10.1007/s00407-017-0188-5

Authors:Yaakov Zik; Giora Hon Abstract: The claim that Galileo Galilei (1564–1642) transformed the spyglass into an astronomical instrument has never been disputed and is considered a historical fact. However, the question what was the procedure which Galileo followed is moot, for he did not disclose his research method. On the traditional view, Galileo was guided by experience, more precisely, systematized experience, which was current among northern Italian artisans and men of science. In other words, it was a trial-and-error procedure—no theory was involved. A scientific analysis of the optical properties of Galileo’s first improved spyglass shows that his procedure could not have been an informed extension of the traditional optics of spectacles. We argue that most likely Galileo realized that the objective and the eyepiece form a system and proceeded accordingly. PubDate: 2017-01-20 DOI: 10.1007/s00407-016-0187-y

Authors:Athanase Papadopoulos Abstract: Nicolas-Auguste Tissot (1824–1897) published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the Tissot indicatrix. This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot indicatrix is a graphical representation of a field of ellipses on a map that describes its distortion. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane that are used in drawing geographical maps, and more generally he developed a theory for the distortion of mappings between general surfaces. His ideas are at the heart of the work on quasiconformal mappings that was developed several decades after him by Grötzsch, Lavrentieff, Ahlfors and Teichmüller. Grötzsch mentions the work of Tissot, and he uses the terminology related to his name (in particular, Grötzsch uses the Tissot indicatrix). Teichmüller mentions the name of Tissot in a historical section in one of his fundamental papers where he claims that quasiconformal mappings were used by geographers, but without giving any hint about the nature of Tissot’s work. The name of Tissot is missing from all the historical surveys on quasiconformal mappings. In the present paper, we report on this work of Tissot. We shall mention some related works on cartography, on the differential geometry of surfaces, and on the theory of quasiconformal mappings. This will place Tissot’s work in its proper context. PubDate: 2016-12-16 DOI: 10.1007/s00407-016-0186-z

Authors:C. Philipp E. Nothaft Abstract: A characteristic hallmark of medieval astronomy is the replacement of Ptolemy’s linear precession with so-called models of trepidation, which were deemed necessary to account for divergences between parameters and data transmitted by Ptolemy and those found by later astronomers. Trepidation is commonly thought to have dominated European astronomy from the twelfth century to the Copernican Revolution, meeting its demise only in the last quarter of the sixteenth century thanks to the observational work of Tycho Brahe. The present article seeks to challenge this picture by surveying the extent to which Latin astronomers of the late Middle Ages expressed criticisms of trepidation models or rejected their validity in favour of linear precession. It argues that a readiness to abandon trepidation was more widespread prior to Brahe than hitherto realized and that it frequently came as the result of empirical considerations. This critical attitude towards trepidation reached an early culmination point with the work of Agostino Ricci (De motu octavae spherae, 1513), who demonstrated the theory’s redundancy with a penetrating analysis of the role of observational error in Ptolemy’s Almagest. PubDate: 2016-11-11 DOI: 10.1007/s00407-016-0184-1

Authors:Christián C. Carman Abstract: The eighth book of Martianus Capella’s famous De Nuptiis Philologiae et Mercurii deserves a prominent place in the history of astronomy because it is the oldest source that came down to us unambiguously postulating the heliocentrism of the inner planets. Just after the paragraph in which Capella asserts that Mercury and Venus revolve around the Sun, he describes a method for calculating the size of the Moon, as well as the proportion between the size of its orbit and the size of the Earth. It is possible to find some descriptions of the argument in general histories of astronomy or in books dedicated to Capella’s work, but usually they do not try to make sense of the argument. Rather, they limit themselves to describe or paraphrase what Capella says. As far as I know, there is no single study of the argument itself. The explanation for this absence is simple: the calculation offers many difficulties in its interpretation, for it shows obvious inconsistencies in the steps of the argument and apparent arbitrariness in the selection of the data used. In this article, I offer an interpretation that tries to discover, behind Capella’s confusing presentation, a well-sound argument for calculating the Moon’s absolute size. Interestingly, we have no records of this argument in other sources, at least in the form described by Capella. PubDate: 2016-11-09 DOI: 10.1007/s00407-016-0185-0

Authors:Jip van Besouw Abstract: This article discusses the quest for the mechanical advantage of the wedge in the eighteenth century. As a case study, the wedge enlightens our understanding of eighteenth-century mechanics in general and the controversy over “force” or vis viva in particular. In this article, I show that the two different approaches to mechanics, the one that favoured force in terms of velocities and the one that primarily used displacements—known as the ‘Newtonian’ and ‘Leibnizian’ methods, respectively—were not at all on par in their ability to solve the problem of the wedge. In general, only those who used the Leibnizian concept of force or some related notion were able to get to the conventional results. This article thus rebuts the received view that the vis viva controversy was merely a semantic one. Instead, it shows that different understandings of “force” led to real and pragmatic differences in eighteenth-century mechanics. PubDate: 2016-10-17 DOI: 10.1007/s00407-016-0182-3