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

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

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

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.

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

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

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

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

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

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

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

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

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

BM 76829, a fragment from the mid-section of a small tablet from Sippar in Late Babylonian script, preserves what remains of two new unparalleled pieces from the cuneiform astronomical repertoire relating to the zodiac. The text on the obverse assigns numerical values to sectors assigned to zodiacal signs, while the text on the reverse seems to relate zodiacal signs with specific days or intervals

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

Unfortunately, only after online first article publication, it was noticed that the first four sentences in footnote two were incorrect.

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

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

Pipe flow has been a challenge that gave rise to investigations on turbulence—long before turbulence was discerned as a research problem in its own right. The discharge of water from elevated reservoirs through long conduits such as for the fountains at Versailles suggested investigations about the resistance in relation to the different diameters and lengths of the pipes as well as the speed of flow

Among the most influential and well-known experiments of the 19th century was the generation and detection of electromagnetic radiation by Heinrich Hertz in 1887–1888, work that bears favorable comparison for experimental ingenuity and influence with that by Michael Faraday in the 1830s and 1840s. In what follows, we pursue issues raised by what Hertz did in his experimental space to produce and to

In my 2018 article in this journal, I described 15th-century Italian astronomer Giovanni Bianchini’s treatment of the problem of stellar coordinate conversion in his Tabulae primi mobilis , the first correct European solution. In this treatise Bianchini refers to a book he had written previously, containing the same error that had plagued his predecessors’ work on the problem. In this article, we announce

From 1873 to 1878, the American physicist Josiah Willard Gibbs offered to the scientific community three great articles that proved to be a milestone for the science of thermodynamics. On the other hand, between 1886 and 1896, the French physicist Pierre Maurice Marie Duhem translated thermodynamics into the language of Lagrange’s analytical mechanics. At the same time, he expanded its scope to include

We report on an unpublished and previously unknown manuscript of John von Neumann and contextualize it within the development of the theory of shock waves and detonations during the nineteenth and twentieth centuries. Von Neumann studies bombs comprising a primary explosive charge along with explosive booster material. His goal is to calculate the minimal amount of booster needed to create a sustainable

This article describes the emergence of formal methods in theory of partial differential equations (PDE) in the French school of mathematics through Janet’s work in the period 1913–1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems.

The most discussed and mysterious column within the Babylonian astronomy is column Φ. It is closely connected to the lunar velocity and to the duration of the Saros. This paper presents new ideas for the development and interpretation of column Φ. It combines the excellent Goal-Year method (for the prediction of Lunar Six time intervals) with old ideas and practices from the “schematic astronomy”.

The assumption that Ptolemy adopted star coordinates from a star catalog by Hipparchus is investigated based on Hipparchus’ equatorial star coordinates in his Commentary on the phenomena of Aratus and Eudoxus . Since Hipparchus’ catalog was presumably based on an equatorial coordinate system, his star positions must have been converted into the ecliptical system of Ptolemy’s catalog in his Almagest

Tycho Brahe was not just an observer; he was a skilled theoretical astronomer, as his lunar and solar models show. Still, even if he is recognized for proposing the Geoheliocentric system, little do we know of the technical details of his planetary models, probably because he died before publishing the last two volumes of his Astronomiae Instaurandae Progymnasmata, which he planned to devote to the

In this paper, we discuss Babylonian observations of a “massing of the planets” reported in two Astronomical Diaries, BM 32562 and BM 46051. This extremely rare astronomical phenomenon was observed in Babylon between 20 and 30 March 185 BC shortly before sunrise when all five planets were simultaneously visible for about 10 to 15 min close to the horizon in the eastern morning sky. These two observational

Charles Sanders Peirce wrote the article ‘The probability of induction’ in 1878. It was the fourth article of the series ‘Illustrations of the Logic of Science’ which comprised a total of six articles. According to Peirce, to get a clear idea of the conception of probability, one has ‘to consider what real and sensible difference there is between one degree of probability and another.’ He endorsed

Through an in-depth analysis of the notes that Leibniz made while reading Pascal’s manuscript treatise on conic sections, we aim to show the real extension of what he called “hexagrammum mysticum”, and to highlight the main results he achieved in this field, as well as proposing plausible proofs of them according to the methods he seems to have developed.

Here, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary differential equations. This leads us to the work of Mihailo Petrovic on algebraic differential equations and in particular his geometric ideas captured in his polygon method from the last years of the XIXth century, which have been left completely unnoticed by

In Resolution des quatre principaux problemes d’architecture (1673) then in Cours d’architecture (1683), the architect–mathematician Nicolas-Francois Blondel addresses one of the most famous architectural problems of all times, that of the reduction in columns (entasis). The interest of the text lies in the variety of subjects that are linked to this issue. (1) The text is a response to the challenge

It is well known that one of Poincaré’s most important contributions to mathematics is the creation of algebraic topology. In this paper, we examine carefully the stated motivations of Poincaré and potential applications he had in mind for developing topology. Besides being an interesting historical problem, this study will also shed some light on the broad interest of Poincaré in mathematics in a

To solve the direct problem of central forces when the trajectory is an ellipse and the force is directed to its centre, Newton made use of the famous Lemma 12 (Principia, I, sect. II) that was later recognized equivalent to proposition 31 of book VII of Apollonius’s Conics. In this paper, in which we look for Newton’s possible sources for Lemma 12, we compare Apollonius’s original proof, as edited

Babylonian methods for predicting planetary phenomena using the so-called goal-year periods are well known. Texts known as Goal-Year Texts contain collections of the observational data needed to make predictions for a given year. The predictions were then recorded in Normal Star Almanacs and Almanacs. Large numbers of Goal-Year Texts, Normal Star Almanacs and Almanacs are attested from the early third

Newton certainly regarded his second law of motion in the Principia as a fundamental axiom of mechanics. Yet the works that came after the Principia, the major treatises on the foundations of mechanics in the eighteenth century—by Varignon, Hermann, Euler, Maclaurin, d’Alembert, Euler (again), Lagrange, and Laplace—do not record, cite, discuss, or even mention the Principia’s statement of the second

The Jalālī (or Malikī) Calendar is well known to Iranian and Western researchers. It was established by the order of Sulṭān Jalāl al-Dīn Malikshāh-i Saljūqī in the 5th c. A.H. (The dates which are designated with A.H. indicate the Hijrī Calendar.)/11th c. A.D. in Isfahan. After the death of Yazdigird III (the last king of the Sassanid dynasty), the Yazdigirdī Calendar, as a solar one, gradually lost

Dans deux articles publiés en 1869 et 1878, Leopold Kronecker construit une théorie qui prend sa source dans le travail de Sturm sur la détermination du nombre de solutions réelles d’une équation. La présentation de cette théorie des caractéristiques par Émile Picard va donner lieu à une controverse entre les deux mathématiciens sur la paternité d’une formule donnant le nombre de solutions de certains

This paper provides an analysis of the use of axioms in Banach’s Ph.D. and their role in the progression of Banach’s mathematical thought. In order to give a precise account of the role of Banach’s axioms, we distinguish two levels of activity. The first one is devoted to the overall process of creating a new theory able to answer some prescribed problems in functional analysis. The second one concentrates

As is well known, it was only in 1926 that a comprehensive mathematical theory of braids was published—that of Emil Artin. That said, braids had been researched mathematically before Artin’s treatment: Alexandre Theophile Vandermonde, Carl Friedrich Gauß and Peter Guthrie Tait had all attempted to introduce notations for braids. Nevertheless, it was only Artin’s approach that proved to be successful

This article analyzes the angular spacing of the degree marks on the zodiac scale of the Antikythera mechanism and demonstrates that over the entire preserved 88° of the zodiac, the marks are systematically placed too close together to be consistent with a uniform distribution over 360°. Thus, in some other part of the zodiac scale (not preserved), the degree marks have been spaced farther apart. By

Johannes Kepler dedicated much of his work to discover a law for the refraction of light. Unfortunately, he formulated an incorrect law. Nevertheless, it was useful for anticipating the behavior of light in some specific conditions. Some believe that Kepler did not have the elements to formulate the law that was later accepted by the scientific community, that is, the Snell–Descartes law. However,

In this article, we publish the critical edition of Andalo di Negro’s De compositione astrolabii, with English translation and commentary. The mathematician and astronomer Andalo di Negro (Genoa ca. 1260–Naples 1334) presumably redacted this treatise on the astrolabe in the 1330s, while residing at the court of King Robert of Naples. The present edition has three purposes: first, to make available

Tables 2 and 4 contained an incorrect set of values for the mean lunar latitudes associated with the tabulated eclipse possibilities.

In the 1870s, Carl Neumann proposed the so-called method of the arithmetic mean for solving the Dirichlet problem on convex domains. Neumann’s approach was considered at the time to be a reliable existence proof, following Weierstrass’s criticism of the Dirichlet principle. However, in 1937 H. Lebesgue pointed out a serious gap in Neumann’s proof. Curiously, the erroneous argument once again involved

Farīd al-Dīn Abu al-Ḥasan ‘Alī b. al-Fahhād’s astronomical tradition as represented in the prolegomenon to his Alā’ī zīj (1172 AD) shows his experimental examination of the theories of his predecessors and testing the circumstances of the synodic phenomena as derived from the theories developed in the classical period of medieval Middle Eastern astronomy against his own observations. This work was

This article examines the research of Louis J. Mordell on the Diophantine equation $$y^2-k=x^3$$ as it appeared in one of his first papers, published in 1914. After presenting a number of elements relating to Mordell’s mathematical youth and his (problematic) writing, we analyze the 1914 paper by following the three approaches he developed therein, respectively, based on the quadratic reciprocity law

This paper presents a new edition of the Back Plate Inscription (BPI) of the Antikythera Mechanism, a series of descriptions of circumstances associated with eclipses indicated cyclically by the inscriptions of the Mechanism’s Saros Dial Scale. Our edition features several significant new readings as well as the confirmation of a disputed reading pertaining to one of the index letters by which the

In French mechanical treatises of the nineteenth century, Newton’s second law of motion was frequently derived from a relativity principle. The origin of this trend is found in ingenious arguments by Huygens and Laplace, with intermediate contributions by Euler and d’Alembert. The derivations initially relied on Galilean relativity and impulsive forces. After Bélanger’s Cours de mécanique of 1847,

Thomas Harriot was the finest English mathematician before Isaac Newton, but his work on the coinage of his country is almost unknown, unlike Newton’s. In the early 1600s Harriot studied several aspects of the gold and silver coins of his time. He investigated the ratio between the values of gold and silver, using data derived from the official weights of the coins; he used hydrostatic weighing to

RésuméLe Brouillon Project de Girard Desargues sur les coniques développe, dans sa partie centrale, la notion de traversale, notion qui généralise celle de diamètre d’Apollonius et permet d’unifier le traitement des diverses espèces de coniques. Il est souvent écrit qu’il s’agit là d’un équivalent de la polaire, concept émergeant au début du $$\hbox {XIX}{}^{\mathrm{e}}$$XIXe siècle. Nous allons dans

AbstractIn 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. Paper I (de Jong in Arch Hist Exact Sci 73:1–37, 2019) was devoted to a study of system A theory of the outer planets. In this second paper, I will study system A theory of the planet Venus. All presently known ephemerides of Venus appear

AbstractAstronomers have always considered the motion of the Moon as highly complicated, and this motion is decisive in determining the circumstances of such critical celestial phenomena as eclipses. Table-makers devoted much ingenuity in trying to find ways to present it in tabular form. In the late Middle Ages, double argument tables provided a smart and compact solution to address this problem satisfactorily

AbstractA few years ago, a manuscript by Jost Bürgi (1552–1632) was brought to scholarly attention, which included an ingenious sine calculation method. The purpose of this paper is to discuss two aspects of this manuscript. First, we wish to improve the current understanding of Bürgi’s method of sine calculation, especially with respect to the calculation of sines at a resolution of 1 min. Second

The paper shows that, contrary to what has been held since the sixteenth-century mathematician Christoph Clavius, there is no application of consequentia mirabilis (CM) in Greek mathematical works. This is shown by means of a detailed discussion of the logical structure of the proofs where CM is allegedly employed. The point is further enlarged to a critical assessment of the unsound methodology applied

AbstractIn this paper, I propose the idea that the French mathematician Michel Chasles developed a foundational programme for geometry in the period 1827–1837. The basic concept behind the programme was to show that projective geometry is the foundation of the whole of geometry. In particular, the metric properties can be reduced to specific graphic properties. In the attempt to prove the validity

The Mesopotamian system of sexagesimal counting numbers was based on the progressive series of units 1, 10, 1·60, 10·60, …. It may have been in use already before the invention of writing, with the mentioned units represented by various kinds of small clay tokens. After the invention of proto-cuneiform writing, c. 3300 BC, it continued to be used, with the successive units of the system represented

The original version of this article unfortunately contained mistakes.