The lecture courses given by Augustin–Louis Cauchy, published in 1820 s, contributed decisively to the reputation of the cours d’analyse at the Ecole Polytechnique in history. However, as B. Belhoste has pointed out, the teaching of analysis during the initial 5 year period (1794–1799) of the school was pivotal for the development of analysis in the nineteenth century. Joseph Fourier taught analysis from 1795 to 1798 at this institution, a period that coincided with the delivery of lectures on a more advanced level by Joseph–Louis Lagrange on the same mathematical topic from 1795 to 1799. However, the courses taught by Fourier have been overlooked, either because they were not published as the courses by Lagrange and Cauchy at the Ecole Polytechnique were, or because they were considered to be of lesser importance, being viewed merely as sources for teaching. This paper presents a comprehensive analysis of the two unpublished manuscripts that document Fourier’s pedagogical contributions at the Ecole Polytechnique. These manuscripts, Ms. 1852 and Ms. 2044, offer a unique insight into Fourier’s approach to teaching analysis at the Ecole Polytechnique. Each manuscript comprises notes taken during or produced after the courses taught by Fourier in a specific period, which will be elucidated in a greater detail in the article. The manuscripts are studied as material objects in order to gain insight into the content and textual forms of the documents, which in turn shed light on the teaching and learning activities of analysis at the end of the eighteenth century. The study of these manuscripts highlights that the lecture notes, produced and preserved in manuscript form, offer a unique perspective that published textbooks do not provide. By examining and comparing the manuscripts, this paper establishes that Fourier in fact presented the differential calculus twice to two different classes of students and it also highlights the evolution from one to the other. Furthermore, I argue that Fourier’s pedagogical approach was instrumental in shaping his methodology of the differential calculus. In addition, I bring to light Fourier’s epistemological concerns in dealing with this subject. In his methodology of analysis, Fourier shared with Lagrange an affinity for epistemological values, including simplicity, clarity and rigour. In conclusion, I suggest that an epistemological/mathematical culture took shape in this context, which cannot be dissociated from the enterprise of rigorizing analysis undertaken by Lagrange, then Fourier and subsequently Cauchy at the Ecole Polytechnique.
Archimedes’ treatise Measurement of a Circle is considered the oldest known text concerning circular sectors. We show that the areas of circular sectors and segments were calculated more than a thousand years earlier in ancient Mesopotamia.
Hipparchus based an estimate of the minimum possible value for the rate of precession on measurements of the distance of Spica from the Moon in mid-eclipse—an eclipse observed by himself on April 21, 146 BCE and another observed by Timocharis, on an unspecified date. This paper reaches three conclusions: (1) Hipparchus’s final adopted value for the longitude of Spica was determined by his wish to keep the latitude of Spica constant at 2° South; (2) Hipparchus carefully selected two eclipses for comparison so that the effects of the lunar parallax would nearly cancel out; and (3) Timocharis’s lunar eclipse was one of two possibilities: the eclipse of April 19, 295 BCE or that of April 8, 294 BCE.
Hankel used his principle of the permanence of formal laws (PFL) as a guide for the extension of number systems and as a necessary condition for the legitimacy of their formal theories. He acknowledged that these applications have important limitations, evidenced by the extension to hypercomplex numbers and by what he saw as the unavoidable inconsistency of a formal theory of irrational numbers. Yet, intriguingly enough, he remained fully committed to the PFL. I argue that this was due to his understanding it as an expression of a conservative strategy, inherited from Peacock and Hamilton, which permits the revision of the basic laws of arithmetic if there are reasons for revision that are found, upon deliberation, to outweigh the reasons for their preservation. Then I discuss criticisms by Schubert and Pringsheim, who reformulated the PFL to align it with their own anti-revisionary conservative strategy, at the cost of relinquishing parts of modern mathematics. I conclude by emphasizing the deep philosophical difference between these kinds of conservatism in mathematics.
In the first part of this study, we compare the different approaches Francesco Maurolico (1557) and Adriaan van Roomen took to develop a universal mathematics. Maurolico sought to ground practical numerical calculation of lines, surfaces, volumes, weights, times, etc., in Euclidean theory while van Roomen sought to establish a general theory of ratio and proportion that precedes arithmetic and geometry. One key difference is that Maurolico’s numbers measure the various species of quantity via a posited unit while van Roomen’s numbers measure their ratios. In the second part, we compare two unfinished drafts by van Roomen (ca. 1598–99) and Descartes (the Regulae, ca. 1628–31) which propose a universal algebra that can model all quantities, including numbers and geometric magnitudes. In fact, there can be no such algebra, since the arithmetical unit as a multiplicative identity is incompatible with the heterogeneous dimensions of Euclidean geometry. Thus, van Roomen abandoned his draft. Descartes, whose interest lay with numerical problem-solving, initially avoided the problem by positing a unit for each species of quantity. But once he became acquainted with theoretical, unitless geometry through the Pappus problem he, too, abandoned his draft and worked with two algebras in La géométrie (1637).
This paper investigates Muḥyī al‑Dīn al‑Maghribī’s measurements of Saturn at the Maragha observatory (northwestern Iran, 13th century), preserved in Chapters 2–5 of Book VIII of his Talkhīṣ al‑Majisṭī (The Compendium of the Almagest). Employing Ptolemy’s iterative three‑point method as described in Almagest X and XI, he measured Saturn’s orbital elements (eccentricity and the longitude of the apogee) from three observations conducted near the planet’s opposition to the mean Sun on 25 October 1263, 9 December 1266, and 27 February 1273 (with longitudinal errors not exceeding 1/6°). He obtained an eccentricity of 3;15 (radius of orbit = 60), a value already encountered in Ptolemy’s Canobic Inscription, and a longitude of the apogee of 258;38° in 1273. A calculational mistake prevented him from measuring the more accurate value of 3;23 for the eccentricity; nevertheless, his apogee longitude stands as the most precise determination in the Islamic astronomical tradition. The accuracy of his values for the Ptolemaic parameters of Saturn and the quantitative outcomes of his theory are examined comprehensively and comparatively with all other known theories established in the late Islamic period, especially Saturn’s theory laid out in the Īlkhānī zīj. As a technical case study, this investigation furnishes new insights into the nature of observational practice and theoretical innovation in the medieval Islamic period.
The motivation for undertaking this research came from reading Kossak’s 1872 publication about Weierstraß’ conception of numbers. It became clear that Weierstraß was developing a general notion of complex numbers that aligned closely with the concepts found in the eighteenth-century French arithmetic textbooks, an area already investigated by one of us. This prompted a deeper examination of the arithmetic framework that Weierstraß had introduced as the foundation for his lectures on analytic function theory. Our inquiry led to a comprehensive assessment of the lecture notes produced by students who attended these courses, which constitute the essential sources for understanding the development of Weierstraß’ foundational approach from the early 1860s to 1886. Rigorous notions of irrational numbers were developed in the second half of the nineteenth century by one French mathematician and three German mathematicians: Charles Méray, Georg Cantor, Richard Dedekind, and Carl Weierstraß. Nowadays, Dedekind’s approach to irrational numbers largely dominates the modern understanding of the real number system, while Weierstraß’ conception has been mostly forgotten. Weierstraß did not develop his theory with the aim of establishing a rigorous notion of real numbers, as is commonly reported by those who are aware of his achievements. Instead, his goal was to create a new notion of complex numbers—not merely the “standard” complex numbers introduced by Gauß in 1831, but a more general concept. To achieve this, he introduced his own method for constructing irrational numbers. This broader conception, which is little known today, provided the numerical foundation for Weierstraß’ preferred field of mathematical research: elliptic and Abelian functions. His primary concern, which he reformulated over time, came from Gauß’ 1831 observation about the impracticality of extending the number system to what Weierstraß called general complex numbers, later known as hypercomplex numbers.
The 1908 discovery of magnetic fields in sunspots raised the question of how such magnetic fields arise in astronomical bodies. The realization that the material in the Sun is in the plasma state made Larmor (1919) speculate that motions in the plasma may generate or sustain such magnetic fields. The first steps towards magnetohydrodynamics (MHD), which combines electrodynamics and hydrodynamics, were taken by Cowling (1933) and Ferraro (1937). These first results raised serious doubts whether MHD can explain the origin of the magnetic fields of the earth and of sunspots. The MHD equations first appeared in their full form in the works of Alfvén (1942a, 1943c), who used these equations to postulate what he called ‘electromagnetic-hydrodynamic waves’ (now known as Alfvén waves), which were discovered in the laboratory only about a decade later. Alfvén’s work was initially met with skepticism (and his theory of sunspots was rejected), and MHD was taken up very slowly at first. With the astrophysical interest generated by the discovery of magnetic fields in other astronomical systems from around 1950, the methods of MHD started spreading rapidly, leading to first solutions of the so-called ‘dynamo problem’—to explain how magnetic fields arise in astrophysical systems. The concept of ‘flux freezing’, first put forth by Alfvén (1943c), led to a new viewpoint that magnetic fields can be considered more fundamental than electric currents, which made MHD a more coherent and accessible theory. Curiously, Alfvén himself opposed this viewpoint and, although he was given the 1970 Nobel Prize “for fundamental work and discoveries in magnetohydrodynamics”, he became a harsh critic of MHD in later life.
We investigated the interpolation methods used in Chongxiu-Daming-li ((sic)(sic)(sic)(sic)(sic)), focusing on the principles behind the tables for sunrise times (Sunrise Parts (sic)(sic)(sic)) through the divided difference method. Our analysis compared these methods with modern polynomial interpolation techniques. Our findings show that the latitude derived from Chongxiu-Daming-li closely matches historical locations, though discrepancies suggest possible over- or underestimation of sunrise times. While the maximum interpolation error for sunrise times is approximately 15 s, discrepancies between computed and actual sunrise times can reach up to 4 min. Comparisons with earlier calendars, such as Xuanming-li ((sic)(sic)(sic)), reveal that fundamental observational techniques saw limited advancement. Despite the use of advanced interpolation methods, practical accuracy in traditional Chinese calendars did not significantly improve until the late 13th century with Guo Shoujing ((sic)(sic)(sic))'s Shoushi-li ((sic)(sic)(sic)).
Since the early twentieth century, numerous complete or partial Chinese nine-nines rhymes dating from the mid-Warring States period to the eleventh century ce have been excavated and published in China. This article surveys all of the as-yet known examples of such nine-nines rhymes, each of which is described, transcribed, collated, annotated, and translated into English, providing a comprehensive overview of what they have in common, as well as how they differ in certain important respects. As a result, the evolution of the nine-nines rhymes can be traced over a period of nearly a millennium-and-a-half down to the Tang (618–907 ce) and Southern Song (1127–1279 ce) dynasties. Thus, it will be possible for the first time to provide a general picture of how the nine-nines rhyme changed from pre-Qin times to survive in the form it usually assumes today.
We provide an interpretation of Husserl’s 1901 Doppelvortrag in Göttingen from the viewpoint of the modernist transformation of mathematics. We emphasise the dialectical aspects of the Doppelvortrag, and especially the underlying conflict between abstraction and intuition, which often resurfaces in Husserl’s philosophy. We focus on three key aspects: (1) the relation between Husserl’s idea of pure logic and Hilbert’s axiomatic method; (2) the concept of Definitheit and the early developments of model theory; (3) the nature of imaginary numbers and their justification in arithmetic and algebra. We stress that, in contrast to Hilbert’s Completeness Axiom from his Grundlagen, Husserl viewed the Definitheit as a statement requiring a proof, and he believed that one could be provided in the cases of arithmetic and geometry.
This note examines an apparently unpublished manuscript on special relativity written by Conrad Habicht in 1914 and made available online by the ETH-Bibliothek Zürich in December 2024. To the best of my knowledge, no study of its content has yet been published. Habicht was one of Einstein’s closest companions during the Bern years. Between February 1902 and mid-1904, he shared with Einstein many occasions for discussion and companionship in Bern. After leaving the city, he remained in close contact with Einstein through visits, reciprocal stays, and a substantial correspondence extending from the years immediately following 1905 to the eve of the First World War. The manuscript offers a clear and pedagogical presentation of special relativity. Its historical interest lies in the structure of the exposition and in the memory of the theory that the text preserves. Habicht does not present special relativity as an isolated creation beginning from Einstein’s 1905 paper alone. He devotes considerable space to the pre-Einsteinian problem situation: the classical principle of relativity, the ether, Fizeau’s experiment, Michelson–Morley, Lorentz’s theory, the contraction hypothesis, local time, and the privileged system of the stationary ether. Lorentz is treated as the central figure who brought the electrodynamics of moving bodies to its most acute form before Einstein’s intervention. This note provides a qualitative description of the manuscript, with particular attention to its structure, its treatment of the relation between classical mechanics and electrodynamics, and the respective roles assigned to Lorentz, Michelson–Morley, Einstein, and Minkowski. It also argues that Habicht’s exposition stands much closer to Einstein’s 1907 review article, Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen, than to the more compressed and self-contained presentation of Zur Elektrodynamik bewegter Körper in 1905. The manuscript thus offers an opportunity to examine a striking shift in Einstein’s own public presentation of special relativity: from the sparse, principle-based narrative of 1905 to the historically reconstructed and Lorentz-centered exposition of 1907. The document is therefore a historically significant witness to the early reception and narration of Einstein’s theory within the circle of one of his closest Bern companions. Its value lies in showing how, within Einstein’s extended milieu, special relativity could be understood not as an isolated act of conceptual creation, but as the principled resolution of a Lorentzian and electrodynamical problem situation.
This paper examines Kepler’s earliest applications of the distance law in a group of manuscripts from 1601 to 1602. It argues that these texts reveal not merely a technical innovation but a methodological transformation: Kepler assumed that a physically grounded hypothesis must hold across the entire celestial system. The study reconstructs several early extensions of the law—to the Earth, to lunar theory, and to the relation between terrestrial eccentricity and the length of the year—showing how rapidly he treated the principle as universally valid. It also offers a systematic presentation of these manuscript episodes and proposes refinements to the chronology of Pulkovo XIV.
The view that Peacock's principle of permanence has been invalidated by Hamilton's introduction of non-commutative algebras has always seemed rather odd, in light of Peacock's favorable reception of quaternions and the endorsement of his principle by Hamilton. But the view is not just odd; it is incorrect. In order to show this, I critically analyze Peacock's attempts to reject possible exceptions to his principle, like the factorial function and an infinite series due to Euler. Then I argue that the principle of permanence is best understood as an expression of a conservative strategy, philosophically grounded in Hume's conception of the laws of reasoning, which advocates their preservation to the furthest extent possible, thus allowing exceptions, i.e., violations of these laws. On this reading, non-commutative multiplication does not invalidate Peacock's principle, if the reasons for violating commutativity outweigh the reasons for its preservation. Finally, I show that Hamilton followed a conservative strategy of precisely this sort when he developed his quaternionic calculus.
In 1927, the eminent art historian Erwin Panofsky published an article about linear perspective, the history of its use in paintings, the principles underlying constructions and whether perspective could be regarded as a ‘symbolic form’, that is, as a subject in itself. Today his article, which is still widely read among historians of art, seems inadequate in several ways, notably in its treatment of what would now be regarded as the relevant mathematics and optics, and some matters of history of art; moreover, to today’s readers the article also raises questions not only about how the treatment of history of the sciences has developed in the last century but also about what we would now consider an acceptable approach to writing on perspective, a subject that does not wholly belong to either history of science or to history of art.
The prediction of planetary First/Last Visibility features prominently in the observational records and calendrical computations of many ancient cultures, although the ways in which these events were conceptualised and operationalised differ substantially. Focusing on the First/Last Visibility of the five classical planets (Venus, Jupiter, Mercury, Mars, and Saturn), this article reviews and compares the relevant textual evidence and computational procedures in Babylonian, Greek, Arabic, Indian, and Chinese traditions. A cross-cultural analysis shows that these visibility events were widely recognised as salient nodes within planetary synodic cycles across the traditions examined. Babylonian sources, however, tend to treat them as observationally significant without articulating a single, explicit and general criterion for judgement. In Greek and Arabic astronomy, shaped by geometric modelling, visibility was commonly handled through dedicated procedures adjunct to core positional computation, with careful attention to geographic latitude, seasonal geometry, and planetary ecliptic latitude. By contrast, Indian and Chinese approaches, largely procedure-based and algorithmic, typically treated First/Last Visibility as special cases within broader planetary computations and often did not incorporate planetary ecliptic latitude as an explicit parameter. Nevertheless, with the partial exception of the Babylonian material, the traditions surveyed most consistently relied on the Sun–planet separation—usually ecliptic elongation, and in some cases separation expressed in equatorial coordinates—to decide visibility. Comparison with modern visibility theory and our numerical simulations further indicates that elongation-based criteria are strongly conditioned by geographic latitude and seasonal observing geometry. Although such criteria are not highly precise, they are practical and intelligible, and in many historical contexts they were adequate for the predictive needs of working astronomers and calendar-makers. By clarifying both shared assumptions and systematic divergences in these traditions, this study contributes a comparative framework for understanding how visibility phenomena were incorporated into ancient astronomical theory and practice.
We study the origins and reception of the statistical mechanical theory of gas condensation put forward by the American scientist Joseph E. Mayer in 1937. Promising the first single and unified treatment of the gas and liquid phases of molecule assemblies on a solid theoretical footing, the theory was praised by Mayer’s contemporaries as a breakthrough. Mayer’s mathematical reasoning was, however, soon the subject of much debate, and while some of his peers accepted his reasoning, others questioned its validity on various grounds. With the intermission of World War II, it was first in the 1950s that the valid parts of the reasoning could be separated from the questionable ones, which meant that most physicists dropped the theory as a research object. The paper addresses Mayer’s mathematical reasoning and its reception, as well as the impact of the theory on physics. The theory helped put the problem of a statistical mechanical theory of phase transitions on physicists’ agenda as well as highlight the importance of mathematical sound reasoning in this area.
This note offers an overview of how Josiah Willard Gibbs’s Elementary Principles in Statistical Mechanics, published simultaneously in London and New York in 1902, spread through European university libraries. Information gathered through direct contacts with numerous academic libraries reveals an unexpectedly rapid material diffusion beginning on April 1, 1902. This early propagation can be explained by several channels: presentation copies sent by Yale University to leading universities, personal mailings by Gibbs himself to prominent scientists, and the distribution of copies by the American publisher to major scientific journals.
The half-century from 1910 to 1960, fractured by two world wars, was marked by profound changes in the French society and its scientific institutions. The mathematician Joseph Pérès, who played a major role in these changes, stands out as one of the most remarkable personalities of the time. A student of Émile Borel at the École normale supérieure and of Vito Volterra in Rome, then professor in 1921, Pérès became the first director, in 1930, of the institute for fluid mechanics of Marseille, a foundation of the Ministry of Air. Two years later, he joined the institute for mechanics of Paris where, with Lucien Malavard, he developed electrical analogies for solving hydrodynamic problems, work rapidly acknowledged by academics and the aeronautical industry. Elected in 1942 to the French Académie des Sciences, he helped found in 1946 the International Union of Theoretical and Applied Mechanics (IUTAM) and became its first president. A highly respected figure in aeronautical circles, he helped create the Office national des études et recherches aéronautiques (ONERA). Appointed deputy director of the CNRS, he promoted scientific computing, founding and directing the Institut Blaise Pascal, matrix for French laboratories in applied mathematics and computer science. Dean of the Paris faculty of sciences from 1954 and faced with a considerable increase in the number of students, he created two new campuses in Paris and Orsay. A radiant personality and member of numerous learned societies, he also contributed to general reflection on the relationship between science and society, notably at the International Academy of the History of Science. He died in 1962, leaving behind him a remarkable body of work and the memory of a “smiling wisdom”. His work would have a major influence on the subsequent development of French research and teaching in the 1960s and beyond.
