It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental propositions of (...) his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the definition of fluxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his definition of fluxion appeared too complex to some contemporaries. (shrink)

This article surveys measurements of celestial (chiefly solar) altitudes documented from twelfth- and thirteenth-century Latin Europe. It consists of four main parts providing (i) an overview of the instruments available for altitude measurements and described in contemporary sources, viz. astrolabes, quadrants, shadow sticks, and the torquetum; (ii) a survey of the role played by altitude measurements in the determination of geographic latitude, which takes into account more than 70 preserved estimates; (iii) case studies of four sets of measured solar altitudes (...) in twelfth-century Latin sources; (iv) an in-depth discussion of the evidence relating to altitude measurements performed in Paris in the period 1281–1290. The findings from the last part indicate that by the end of the thirteenth century Parisian astronomer had developed rigorous standards of observational practice in which altitudes were typically measured to a precision of minutes of arc and with a level of accuracy higher than ± 0;5°, and sometimes exceeding ± 0;1°. (shrink)

This study is a continuation of the authors’ previous work entitled “Helmholtz and the geometry of color space: gestation and development of Helmholtz’s line element” (Peruzzi and Roberti in Arch Hist Exact Sci. https://doi.org/10.1007/s00407-023-00304-2, 2023), which provides an account of the first metrically significant model of color space proposed by the German polymath Hermann von Helmholtz in 1891–1892. Helmholtz’s Riemannian line element for three-dimensional color space laid the foundation for all subsequent studies in the field of color metrics, although it (...) was largely forgotten for almost three decades from the time of its first publication. The rediscovery of Helmholtz’s masterful work was due to one of the founders of quantum mechanics, Erwin Schrödinger. He established his color metric in three extended papers submitted in 1920 to the Annalen der Physik. Two memoirs were devoted to the so-called lower color metric, which laid the basis for the development of his higher color metric, exposed in the last paper. Schrödinger’s approach to the geometry of color space has been taken as a starting point for future elaborations of color metrics and allows a close examination of the current assumptions about the analysis of color-matching data. This paper presents an overall picture of Schrödinger’s works on color. His color theory developed a tradition first inaugurated by Newton and Young, and which acquired strong scientific ground with Grassmann’s, Maxwell’s, and Helmholtz’s contributions in the 1850s. Special focus will be given to Schrödinger’s account of color metric, which responded directly to Helmholtz’s hypothesis of a Riemannian line element for color space. (shrink)

Power-law city-size distributions are a statistical regularity researched in many countries and urban systems. In this history of science treatise we reconsider Felix Auerbach’s paper published in 1913. We reviewed his analysis and found (i) that a constant absolute concentration, as introduced by him, is equivalent to a power-law distribution with exponent $$\approx 1$$ ≈ 1, (ii) that Auerbach describes this equivalence, and (iii) that Auerbach also pioneered the empirical analysis of city-size distributions across countries, regions, and time periods. We (...) further investigate his legacy as reflected in citations and find that important follow-up work, e.g. by Lotka (Elements of physical biology. Williams & Wilkins Company, Baltimore, 1925) and Zipf (Human behavior and the principle of least effort: an introduction to human ecology, Martino Publishing, Manfield Centre, CT (2012), 1949), does give proper reference to his discovery—but others do not. For example, only approximately 20% of city-related works citing Zipf (1949) also cite Auerbach (Petermanns Geogr Mitteilungen 59(74):74–76, 1913). To our best knowledge, Lotka (1925) was the first to describe the power-law rank-size rule as it is analyzed today. Saibante (Metron Rivista Internazionale di Statistica 7(2):53–99, 1928), building on Auerbach and Lotka, investigated the power-law rank-size rule across countries, regions, and time periods. Zipf’s achievement was to embed these findings in his monumental 1949 book. We suggest that the use of “Auerbach–Lotka–Zipf law” (or “ALZ-law”) is more appropriate than “Zipf’s law for cities”, which also avoids confusion with Zipf’s law for word frequency. We end the treatise with biographical notes on Auerbach. (shrink)

In his Metrica, Hero provides four procedures for finding the area of a circular segment (with b the base of the segment and h its height): an Ancient method for when the segment is smaller than a semicircle, $$(b + h)/2 \, \cdot \, h$$ ( b + h ) / 2 · h ; a Revision, $$(b + h)/2 \, \cdot \, h + (b/2)^{2} /14$$ ( b + h ) / 2 · h + ( b / 2 (...) ) 2 / 14 ; a quasi-Archimedean method (said to be inspired by the quadrature of the parabola) for cases where b is more than triple h, $${\raise0.5ex\hbox{$\scriptstyle 4$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 3$}}(h \, \cdot \, b/2)$$ 4 / 3 ( h · b / 2 ) ; and a method of Subtraction using the Revised method, for when it is larger than a semicircle. He gives superficial arguments that the Ancient method presumes $$\pi = 3$$ π = 3 and the Revision, $$\pi = {\raise0.5ex\hbox{$\scriptstyle {22}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 7$}}$$ π = 22 / 7. We are left with many questions. How ancient is the Ancient? Why did anyone think it worked? Why would anyone revise it in just this way? In addition, why did Hero think the Revised method did not work when $$b > 3\;h$$ b > 3 h? I show that a fifth century BCE Uruk tablet employs the Ancient method, but possibly with very strange consequences, and that a Ptolemaic Egyptian papyrus that checks this method by comparing the area of a circle calculated from the sum of a regular inscribed polygon and the areas of the segments on its sides as determined by the Ancient method with the area of the circle as calculated from its diameter correctly sees that the calculations do not quite gel in the case of a triangle but do in the case of a square. Both traditions probably could also calculate the area of a segment on an inscribed regular polygon by subtracting the area of the polygon from the area of the circle and dividing by the number of sides of the polygon. I then derive two theorems about pairs of segments, that the reviser of the Ancient method should have known, that explain each method, why they work when they do and do not when they do not, and which lead to a curious generalization of the Revised method. Hero’s comment is right, but not for the reasons he gives. I conclude with an exploration of Hero’s restrictions of the Revised method and Hero’s two alternative methods. (shrink)

In this paper we publish three fragments of a cuneiform tablet that, when complete, contained the dates and zodiacal positions of Saturn’s synodic phenomena for roughly 60 years. The text is unique in containing comparisons of computed data with observations. Through an analysis of the preserved data we propose that the dates and positions were computed by an otherwise unknown two-zone System A-type scheme and show that the computed data in the tablet can be dated to the fourth century BC. (...) This early date and the comparisons with observations suggest that the text was produced during the period of active development of the planetary systems. (shrink)

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. (shrink)

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. (shrink)

This Editorial reports an exchange in form of a comment and reply on the article “History and Nature of the Jeffreys–Lindley Paradox” (Arch Hist Exact Sci 77:25, 2023) by Eric-Jan Wagenmakers and Alexander Ly.

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. (shrink)

The article provides a reconstruction of Eudoxus' approach to simultaneous risings and settings in his two works dedicated to the issue: the Phaenomena and the Enoptron. This reconstruction is based on the analysis of Eudoxus’ fragments transmitted by Hipparchus. These fragments are difficult and problematic, but a close analysis and a comparison with the corresponding passages in Aratus suggests a possible solution.

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)