Authors:Jeffrey A. Oaks Pages: 245 - 302 Abstract: Françios Viète (1540–1603) was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek geometric equalities. His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”. PubDate: 2018-05-01 DOI: 10.1007/s00407-018-0208-0 Issue No:Vol. 72, No. 3 (2018)

Authors:Sara Confalonieri Pages: 303 - 352 Abstract: In the framework of the De Regula Aliza (1570), Cardano paid much attention to the so-called splittings for the family of equations \(x^3 = a_1x + a_0\) ; my previous article (Confalonieri in Arch Hist Exact Sci 69:257–289, 2015a) deals at length with them and, especially, with their role in the Ars Magna in relation to the solution methods for cubic equations. Significantly, the method of the splittings in the De Regula Aliza helps to account for how Cardano dealt with equations, which cannot be inferred from his other algebraic treatises. In the present paper, this topic is further developed, the focus now being directed to the origins of the splittings. First, we investigate Cardano’s research in the Ars Magna Arithmeticae on the shapes for irrational solutions of cubic equations with rational coefficients and on the general shapes for the solutions of any cubic equation. It turns out that these inquiries pre-exist Cardano’s research on substitutions and cubic formulae, which will later be the privileged methods for dealing with cubic equations; at an earlier time, Cardano had hoped to gather information on the general case by exploiting analogies with the particular case of irrational solutions. Accordingly, the Ars Magna Arithmeticae is revealed to be truly a treatise on the shapes of solutions of cubic equations. Afterwards, we consider the temporary patch given by Cardano in the Ars Magna to overcome the problem entailed by the casus irreducibilis as it emerges once the complete picture of the solution methods for all families of cubic equations has been outlined. When Cardano had to face the difficulty that appears if one deals with cubic equations using the brand-new methods of substitutions and cubic formulae, he reverted back to the well-known inquiries on the shape of solutions. In this way, the relation between the splittings and the older inquiries on the shape of solutions comes to light; furthermore, this enables the splittings to be dated 1542 or later. The last section of the present paper then expounds the passages from the Aliza that allow us to trace back the origins of the substitution \(x = y + z\) , which is fundamental not only to the method of the splittings but also to the discovery of the cubic formulae. In this way, an insight into Cardano’s way of dealing with equations using quadratic irrational numbers and other selected kinds of binomials and trinomials will be provided; moreover, this will display the role of the analysis of the shapes of solutions in the framework of Cardano’s algebraic works. PubDate: 2018-05-01 DOI: 10.1007/s00407-018-0209-z Issue No:Vol. 72, No. 3 (2018)

Authors:Mathieu Ossendrijver Pages: 145 - 189 Abstract: Between ca. 400 and 50 BCE, Babylonian astronomers used mathematical methods for predicting ecliptical positions, times and other phenomena of the moon and the planets. Until recently these methods were thought to be of a purely arithmetic nature. A new interpretation of four Babylonian astronomical procedure texts with geometric computations has challenged this view. On these tablets, Jupiter’s total distance travelled along the ecliptic during a certain interval of time is computed from the area of a trapezoidal figure representing the planet’s changing daily displacement along the ecliptic. Moreover, the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area. In the present paper these procedures are traced back to precursors from Old Babylonian mathematics (1900–1700 BCE). Some implications of the use of geometric methods by Babylonian astronomers are also explored. PubDate: 2018-03-01 DOI: 10.1007/s00407-018-0204-4 Issue No:Vol. 72, No. 2 (2018)

Authors:S. Mohammad Mozaffari Pages: 191 - 243 Abstract: From Antiquity through the early modern period, the apparent motion of the Sun in longitude was simulated by the eccentric model set forth in Ptolemy’s Almagest III, with the fundamental parameters including the two orbital elements, the eccentricity e and the longitude of the apogee λA, the mean motion ω, and the radix of the mean longitude \( \bar{\lambda }_{0} \) . In this article we investigate the accuracy of 11 solar theories established across the Middle East from 800 to 1600 as well as Ptolemy’s and Tycho Brahe’s, with respect to the precision of the parameter values and of the solar longitudes λ that they produce. The theoretical deviation due to the mismatch between the eccentric model with uniform motion and the elliptical model with Keplerian motion is taken into account in order to determine the precision of e and λA in the theories whose observational basis is available. The smallest errors in the eccentricity are found in these theories: the Mumtaḥan (830): − 0.1 × 10−4, Bīrūnī (1016): + 0.4 × 10−4, Ulugh Beg (1437): − 0.9 × 10−4, and Taqī al-Dīn (1579): − 1.1 × 10−4. Except for al-Khāzinī (1100, error of ~ + 21.9 × 10−4, comparable to Ptolemy’s error of ~ + 33.8 × 10−4), the errors in the medieval determinations of the solar eccentricity do not exceed 7.7 × 10−4 in absolute value (Ibn al-Shāṭir, 1331), with a mean error μ = + 2.57 × 10−4 and standard deviation σ = 3.02 × 10−4. Their precision is remarkable not only in comparison with the errors of Copernicus (− 7.8 × 10−4) and Tycho (+ 10.2 × 10−4), but also with the seventeenth-century measurements by Cassini–Flamsteed (− 2.4 × 10−4) and Riccioli (+ 5.5 × 10−4). The absolute error in λA varies from 0.1° (Taqī al-Dīn) to 1.9° (al-Khāzinī) with the mean absolute error MAE = 0.87°, μ = −0.71° and σ = 0.65°. The errors in λ for the 13,000-day ephemerides show MAE < 6′ and the periodic variations mostly remaining within ± 10′ (except for al-Khāzinī), closely correlated with the accuracy of e and λA. PubDate: 2018-03-01 DOI: 10.1007/s00407-018-0207-1 Issue No:Vol. 72, No. 2 (2018)

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:Robert Goulding Abstract: Kepler’s 1604 Optics (Ad Vitellionem Paralipomena) proposed among many other things a new way of locating the place of the image under reflection or refraction. He rejected the “perspectivist” method that had been used through antiquity and the Middle Ages, whereby the image was located on the perpendicular between the object and the mirror (the “cathetus”). Kepler faulted the method for requiring a metaphysical commitment to the action of final causes in optics: the notion that the image was at that place because it was best or appropriate for it to be there, and for no other discernible reason. Kepler’s new theory relied on binocular vision and depth perception to determine the location of the image. No final causes were required, and he showed that the image would in general not be found on the cathetus. According to modern scholarship, Kepler’s theory was part of his revolutionary transformation of the science of optics, and his abandonment of perspectivist optics; as a consequence, the theory of binocular vision is also thought to be original with him. This article demonstrates that the very same theory of binocular image location was set out by Giovanni Battista Benedetti some twenty years earlier, and his writings on this subject may have been Kepler’s unacknowledged source for his own theory. Furthermore, another mathematician, Simon Stevin, developed much the same theory at the same time as Kepler and, it seems, independently of either Benedetti or Kepler. The discovery of these other binocular theories, especially Benedetti’s, requires us to recognize that Kepler’s revolution (if it can be called that) emerged out of a wider dissatisfaction with the foundations of perspectivist optics, which other lesser-known opticians resolved in much the same way that Kepler did. PubDate: 2018-05-16 DOI: 10.1007/s00407-018-0210-6

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: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