Search

reviewer of Alexander's paper in [64, p. 672], is "of little practical use". In doing so, Alexander developed several classes of univalent functions and some tests guaranteeing univalence which involve Taylor coefficients and location of zeroes and critical points, consequently opening up new areas of investigation for researchers in geometric function theory. Some prominent mathematical analysts, among them J. Dieudonne, G. Szego, M. S. Robertson, R. Nevanlinna, L. Fejer, and L. Bieberbach picked up on topics introduced by Alexander, and work on them has continued throughout the century. It is the purpose of this paper to analyze the contents of Alexander's work, detailing his highly intuitive arguments and how they were developed by later workers, and to follow several lines of development of this material. It is believed by some [5] that his paper was not appreciated for a long time after its publication, and the informal nature of its arguments might have led to its being described as "much underestimated" [19, p. 103]. Yet the work does introduce the notions of starlike functions, close-to-convex functions, functions of bounded turning, as well as other ideas and theorems which were rediscovered as much as twenty to forty years after its appearance, in some cases apparently without knowledge of Alexander's pioneering work. We will first give some biographical background on Alexander, then summarize briefly his mathematical contributions outside of the work under discussion. We will turn next to the paper itself, analyzing its contents section by section, and include several histories of themes initiated by Alexander, and how they developed in the modern era. Our history will concentrate most heavily on the era from 1915 to the late fifties and early sixties with an occasional reference to later work.

It is this paper which will be discussed. In it, as the title indicates, he studied the motions (deplacements) of a rigid body (systeme solide) in three dimensional space independently of any dynamical considerations. He began by giving a complete description of motions in synthetic terms, establishing successively that a body is fixed in space once three non-collinear points have been determined; that if two points are fixed the motion is a rotation about an axis through those points; that translations when composed give a translation which is independent of the order of composition and can be found by the 'loi du polygone des translations' (p. 383) and that a translation is equal to an infinitesimal rotation about an axis perpendicular to the direction of the translation but situated at an infinite distance. RODRIGUES described the resultant as

In the paper here commented upon, its author discusses the various versions of the discovery of X-ray diffraction under the aspect of the formation and embellishment of myths of origin of the clan formed by crystallographers. To that effect, he endeavours to show that most details of the traditional descriptions of motives and obstacles encountered in the planning of the Laue-Friedrich-Knipping experiment are fictitious and fabricated for the purpose of giving the clan a separate identity. This interpretation by the author is possible only because he fails to appreciate the vagueness of the physical information confronting Lauebefore the experiment. To make his interpretation plausible, the author repeatedly aggrandizes statements taken from the literature, especially from Laue's Nobel Lecture and from the Festschrift for the semicentennial of the discovery. These unjustified accents, needed in support of his main thesis, show that his scheme is pre-conceived, artificial, and unnecessary.

The theory of constructive formation of plane algebraic curves in Newton's writings is discussed in § 1: the apparatus by which Newton forms the curves, Newton's theorems on forming unicursal curves, his theory of conics, and his theory of (m, n) correspondence. Special Cremona plane and space transformations obtained by Newton's organic method are dealt with in § 2. The article ends with § 3, which shows two different directions in the theory of the constructive formation of plane algebraic curves in the XVIII-XIXth centuries. A synopsis is appended.

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. 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 to have been written after 200 BC so that the development of system A theory of Venus may have been a late development. On the other hand, there are several earlier texts in which the motion of Venus, going from one synodic phenomenon to the next one, is parametrized in quite some detail. At least six computational systems of Venus are known of which only two, systems A0 and A3, are genuine type-A systems. Both are based on the hypothesis that in exactly 1151 years Venus experiences 720 synodic phenomena of the same kind and that after this period of 1151 years Venus returns to exactly the same position in the sky. This period relation was probably derived from the observational fact that after 8 years Venus returns to a position in the zodiac that falls on average 2.5° short of its previous position. The study of the Babylonian planetary theory of Venus presented here will be primarily based on the system A3 theory of Venus because it is the most complete of the two systems. The parameters of the four step functions characterizing the system A3 model of the first and last appearances of Venus are known from previous studies of tablet BM 32599 (ACT 1050). Based on a database of synthetic observations of the first and last appearances (morning first, morning last, evening first and evening last) and the two stationary points (morning station and evening station) of Venus between 315 BC and 50 BC, I first discuss the observational material from the point of view of a Babylonian astronomer. This involves deriving synodic time intervals and synodic arcs from the observed dates and longitudes of Venus. Both vary with the position of Venus in the zodiac. This variation shows up most purely in the synodic arcs and synodic time intervals of Venus at its stations. The variation pattern of the synodic arcs and of the synodic time intervals at the first and last appearances of Venus differ considerably because they are strongly affected by atmospheric extinction and by the ecliptic latitude of Venus. A comparison of predictions of the first and last appearances of Venus computed according to system A3 with observations in the synthetic database shows that the models provide fairly accurate fits to the observations of evening last and morning first with typical standard deviations of about 2° but rather poor fits to the observations of morning last and evening first with typical standard deviations of about 7° and 4°. Quite surprisingly, it turns out that the morning first A3 model fits the observed longitudes of Venus at its stations almost perfectly, with standard deviations of about 1°. This suggests that observations of the stations of Venus must have been available to the Babylonian astronomer(s) who constructed the system A3 model of Venus and who computed the longitudes of Venus at its first and last appearances preserved in tablet BM 32599. This is quite puzzling because observations of the stations of Venus were definitely not part of the standard observing program of the Babylonian astronomers as we know it from the Astronomical Diaries. To resolve this problem, I propose that observations could have been made by some individual astronomer for his own use but that these observations never became part of the tradition of what was regularly observed and recorded in the Astronomical Diaries. It turns out that 25 years of observations of Venus (here illustrated for the years 203 BC to 178 BC) is sufficient for the construction of a system A model that perfectly fits the variation of the synodic arcs of Venus at its stations. By combining observed time intervals and known velocities of Venus, observations over this limited timespan can also be used to construct longitudes of Venus at its first and last appearances from previous known positions of Venus based on Normal Stars passages or from known positions at its stations. This is the only way in which positions of Venus at its first and last appearances can be determined because at its first and last appearances the sky is too bright for Normal Stars to be visible. I propose that the Babylonian system A3 models of Venus were based on longitudes constructed in this way. I finally suggest that the tablet, of which BM 32599 is the remainder, was filled line by line starting with a set of initial values in the bottom left-hand corner of the tablet and that the choice of these initial values was based on an observation of Venus at its evening station when it was exceptionally close to the Normal Star η Piscium in February 179 BC. Based on this choice of initial conditions, the tablet covers the years 403 BC to 170 BC and it may have been composed around 170 BC. It was probably computed with the purpose of creating a raster of longitudes with mazes of about 2.5° that could be used to predict future positions of Venus at its first or last appearance once a position at a previous first or last appearance was available.

The theoretical solid-state physicist Walter Kohn was awarded one-half of the 1998 Nobel Prize in Chemistry for his mid-1960s creation of an approach to the many-particle problem in quantum mechanics called density functional theory (DFT). In its exact form, DFT establishes that the total charge density of any system of electrons and nuclei provides all the information needed for a complete description of that system. This was a breakthrough for the study of atoms, molecules, gases, liquids, and solids. Before DFT, it was thought that knowledge of the vastly more complicated many-electron wave function was essential for a complete description of such systems. Today, 50 years after its introduction, DFT (in one of its approximate forms) is the method of choice used by most scientists to calculate the physical properties of materials of all kinds. In this paper, I present a biographical essay of Kohn’s educational experiences and professional career up to and including the creation of DFT. My account begins with Kohn’s student years in Austria, England, and Canada during World War II and continues with his graduate and postgraduate training at Harvard University and Niels Bohr’s Institute for Theoretical Physics in Copenhagen. I then study the research choices he made during the first 10 years of his career (when he was a faculty member at the Carnegie Institute of Technology and a frequent visitor to the Bell Telephone Laboratories) in the context of the theoretical solid-state physics agenda of the late 1950s and early 1960s. Subsequent sections discuss his move to the University of California, San Diego, identify the research issue which led directly to DFT, and analyze the two foundational papers of the theory. The paper concludes with an explanation of how the chemists came to award “their” Nobel Prize to the physicist Kohn and a discussion of why he was unusually well suited to create the theory in the first place.

Some time in the late 1590s, the Welsh amateur mathematician John Bulkeley wrote to Thomas Harriot asking his opinion about the properties of a truly gargantuan (but totally imaginary) plano-spherical convex lens, 48 feet in diameter. While Bulkeley’s original letter is lost, Harriot devoted several pages to the optical properties of “Mr Bulkeley his Glasse” in his optical papers (now in British Library MS Add. 6789), paying particular attention to the place of its burning point. Harriot’s calculational methods in these papers are almost unique in Harriot’s optical remains, in that he uses both the sine law of refraction and interpolation from Witelo’s refraction tables in order to analyze the passage of light through the glass. For this and other reasons, it is very likely that Harriot wrote his papers on Bulkeley’s glass very shortly after his discovery of the law and while still working closely with Witelo’s great Optics; the papers represent, perhaps, his very first application of the law. His and Bulkeley’s interest in this giant glass conform to a long English tradition of curiosity about the optical and burning properties of large glasses, which grew more intense in late sixteenth-century England. In particular, Thomas Digges’s bold and widely known assertions about his father’s glasses that could see things several miles distant and could burn objects a half-mile or further away may have attracted Harriot and Bulkeley’s skeptical attention; for Harriot’s analysis of the burning distance and the intensity of Bulkeley’s fantastic lens, it shows that Digges’s claims could never have been true about any real lens (and this, I propose, was what Bulkeley had asked about in his original letter to Harriot). There was also a deeper, mathematical relevance to the problem that may have caught Harriot’s attention. His most recent source on refraction—Giambattista della Porta’s De refractione of 1593—identified a mathematical flaw in Witelo’s cursory suggestion about the optics of a lens (the only place that lenses appear, however fleetingly, in the writings of the thirteenth-century Perspectivist authors). In his early notes on optics, in a copy of Witelo’s optics, Harriot highlighted Witelo’s remarks on the lens and della Porta’s criticism (which he found unsatisfactory). The most significant problem with Witelo’s theorem would disappear as the radius of curvature of the lens approached infinity. Bulkeley’s gigantic glass, then, may have provided Harriot an opportunity to test out Witelo’s claims about a plano-spherical glass, at a time when he was still intensely concerned with the problems and methods of the Perspectivist school.

Douglas Hartree and Hilleth Thomas were graduate students together at Cambridge University in the mid-1920s. Each developed an important approximation method to calculate the electronic structure of atoms. Each went on to make significant contributions to numerical analysis and to the development of scientific computing. Their early efforts were fused in the mid-1960s with the development of an approach to the many-particle problem in quantum mechanics called density functional theory. This paper discusses the experiences which led Hartree and Thomas to their approximations, outlines the similarities in their subsequent careers, and highlights the essential role their work played in the foundational papers of modern density functional theory.

The aim of this paper is to provide a comprehensive exposition of the early contributions to the so-called Campbell, Baker, Hausdorff, Dynkin Theorem during the years 1890–1950. Related works by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff, and Dynkin will be investigated and compared. For a full recovery of the original sources, many mathematical details will also be furnished. In particular, we rediscover and comment on a series of five notable papers by Pascal (Lomb Ist Rend, 1901–1902), which nowadays are almost forgotten.

This is the last in a series of three papers on the history of the Lenz–Ising model from 1920 to the early 1970s. In the first paper, I studied the invention of the model in the 1920s, while in the second paper, I documented a quite sudden change in the perception of the model in the early 1960s when it was realized that the Lenz–Ising model is actually relevant for the understanding of phase transitions. In this article, which is self-contained, I study how this realization affected attempts to understand critical phenomena, which can be understood as limiting cases of (first-order) phase transitions, in the epoch from circa 1965 to 1970, where these phenomena were recognized as a research field in its own right. I focus on two questions: What kinds of insight into critical phenomena was the employment of the Lenz–Ising model thought to give? And how could a crude model, which the Lenz–Ising model was thought to be, provide this understanding? I document that the model played several roles: At first, it played a role analogous to experimental data: hypotheses about real systems, in particular relations between critical exponents and what is now called the hypothesis of scaling, which was advanced by Benjamin Widom and others, were confronted with numerical results for the model, in particular the model’s so-called critical exponents. A positive result of a confrontation was seen as positive evidence for this hypothesis. The model was also used to gain insight into specific aspects of critical phenomena, for example that diverse physical systems exhibit similar behavior close to a critical point. Later, a more systematic program of understanding critical phenomena emerged that involved an explicit formulation of what it means to understand critical phenomena, namely, the elucidation of what features of the Hamiltonian of models lead to what kinds of behavior close to critical points. Attempts to accomplish this program culminated with the so-called hypothesis of universality, put forward independently by Robert B. Griffiths and Leo P. Kadanoff in 1970. They divided critical phenomena into classes with similar critical behavior. I also study the crucial role of the Lenz–Ising model in the development and justification of these ideas.

The theory of exterior differential systems plays a crucial role in Cartan’s whole mathematical production. As he once recognized, all the germs of his subsequent work were contained there. Indeed, it provided him with powerful technical tools that turned out to be very useful in many different fields such as the theory of partial differential equations, the theory of infinite dimensional Lie groups (Lie pseudogroups) and differential geometry. Nevertheless, scarce attention has been paid to this area of historical research thus far. Although authoritative scholars have investigated the foundation of exterior differential calculus in Cartan’s early papers, no specific analysis of Cartan’s subsequent works laying the foundations of what nowadays is known as the Cartan–Kahler theory has been yet provided. This article represents a first attempt to remedy this unsatisfactory state of affairs by focusing on Cartan’s work on Pfaffian systems at the very beginning of the past century. The analysis of Cartan’s relevant papers is preceded by a description of the historical context in which such contributions were conceived. In this respect, special emphasis will be put on some works by Engel and von Weber on Pfaffian systems, which laid the basis for the subsequent geometrical developments of Cartan’s theory of exterior differential systems.

This paper is the third of a multi-part examination of the Babylonian mathematical lunar theories known as Systems A and B. Part I (Britton, AHES 61:83–145, 2007) addressed the development of the empirical elements needed to separate the effects of lunar and solar anomaly on the intervals between syzygies, accomplished in the construction of the System A lunar theory early in the fourth century B.C. Part II (Britton, AHES 63:357–431, 2009) examines the accomplishment of this separation by the construction of a successful theory depicting the variations due to lunar anomaly in System A and its subsequent adaptation in System B. The present paper examines the introduction of the uniform zodiac, necessary for any theory depicting variations depending on the position of syzygy. It addresses three questions: (1) In light of all available evidence, what is the magnitude of the constant term in the expression Δλ* = C − 1.3828°Y, describing the difference between the Babylonian sidereal longitudes and modern tropical longitudes? (2) What considerations governed the placement of the Babylonian sidereal zodiac relative to the fixed stars? (3) When was the uniform zodiac introduced? To the first question it finds C = 3.20° ± 0.1°, scarcely different from Huber’s (Centaurus 5:192–208, 1958) estimate of 3.08°, essentially confirming Huber’s result obtained from much less data. For the second it shows that accommodating the three asterisms comprising Taurus limited the placement of the zodiac to within 3°, while the prominence of half sign multiples among the measured intervals between prominent Normal Stars led irresistibly to the choice adopted. Finally, it finds that the zodiac was introduced between −408 and −397 and probably within a very few years of −400.

In the early 1930s W. O. Kermack and W. H. McCrea published three papers in which they attempted to prove a result of E. T. Whittaker on the solution of differential equations. In modern parlance, their key idea consisted in using quantized contact transformations over an algebra of differential operators. Although their papers do not seem to have had any impact, either then or at any later time, the same ideas were independently developed in the 1960–1980s in the framework of the theory of modules over rings of microdifferential operators. In this paper we describe the results of Kermack and McCrea and discuss possible reasons why such promising papers had no impact on the mathematics of the twentieth century.

This paper is the second of a multi-part examination of the creation of the Babylonian mathematical lunar theories known as Systems A and B. Part I (Britton 2007) addressed the development of the empirical elements needed to separate the effects of lunar and solar anomaly on the intervals between syzygies. This was accomplished in the construction of the System A lunar theory by an unknown author, almost certainly in the city of Babylon and probably early in the 4th century B.C. The present paper focuses mainly on System A and the likely process of its construction. The first three sections are largely descriptive – first of the basic concepts which underlie the theory; then of the component schemes comprising the theory; and finally of two distinctive texts which suggest how the theory was constructed. The crux of the paper is Sect. 4, which describes how the theory seems likely to have been constructed. Here the crucial insight in separating the effects of lunar and solar anomaly appears to have been recognizing that – of all the measurable intervals bounded by eclipses – only 235 months exhibits a variation due solely to lunar anomaly, and that by means of an elegant mathematical model the amplitudes of 223 and 12 months could be deduced from its amplitude. The rest of the section describes the likely details of the derivation of the Φ ~ Λ scheme, and the extension of the methodology to the other components of the theory. It concludes with a demonstration that Φ and its dependent schemes were anchored through the Φ ~ W scheme to the syzygy on –403 Aug 18 (GN 7391) which concluded the shortest 6-month interval in the first 24 saros cycles since –746 (and in fact in the 900 years separating Nabonassar and Ptolemy). The next three sections address a number of largely technical details and amplifications of the theory, beginning with the schemes describing the variation of lunar velocity (column F) in Sect. 5. Section 6 addresses issues concerning the interpretation of Φ and the so-call Saros Text (BM 36705), while Sect. 7 discusses System B’s corresponding treatment of the effects of lunar anomaly, illustrating both its derivative nature and mathematically less rigorous structure. Section 8 examines the accuracy of the two theories, showing that the System A theory was both remarkably accurate and superior to System B. The final section offers some brief remarks on the power and elegance of the mathematical treatment of the problem by the author of System A.

This is the second in a series of three papers that charts the history of the Lenz–Ising model (commonly called just the Ising model in the physics literature) in considerable detail, from its invention in the early 1920s to its recognition as an important tool in the study of phase transitions by the late 1960s. By focusing on the development in physicists’ perception of the model’s ability to yield physical insight—in contrast to the more technical perspective in previous historical accounts, for example, Brush (Rev Modern Phys 39: 883–893, 1967) and Hoddeson et al. (Out of the Crystal Maze. Chapters from the History of Solid-State Physics. Oxford University Press, New York, pp. 489–616, 1992)—the series aims to cover and explain in depth why this model went from relative obscurity to a prominent position in modern physics, and to examine the consequences of this change. In the present paper, which is self-contained, I deal with the development from the early 1950s to the 1960s and document that this period witnessed a major change in the perception of the model: In the 1950s it was not in the cards that the model was to become a pivotal tool of theoretical physics in the following decade. In fact, I show, based upon recollections and research papers, that many of the physicists in the 1950s interested in understanding phase transitions saw the model as irrelevant for this endeavor because it oversimplifies the nature of the microscopic constituents of the physical systems exhibiting phase transitions. However, one group, Cyril Domb’s in London, held a more positive view during this decade. To bring out the basis for their view, I analyze in detail their motivation and work. In the last part of the paper I document that the model was seen as much more physically relevant in the early 1960s and examine the development that led to this change in perception. I argue that the main factor behind the change was the realization of the surprising and striking agreement between aspects of the model, notably its critical behavior, and empirical features of the physical phenomena.

The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory, numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s Theorem. This paper is primarily concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains. When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises and V.I. Romanovsky.

The correspondence between Richard von Mises and George Polya of 1919/20 contains reflections on two well-known articles by von Mises on the foundations of probability in the Mathematische Zeitschrift of 1919, and one paper from the Physikalische Zeitschrift of 1918. The topics touched on in the correspondence are: the proof of the central limit theorem of probability theory, von Mises' notion of randomness, and a statistical criterion for integer-valuedness of physical data. The investigation will hint at both the fruitfulness and the limits of several of von Mises' notions such as ``collective'', ``distribution'' and ``complex adjuncts'' (characteristic functions) for further developments in probability theory and in ``directional statistics''. By pointing to the selectiveness of Polya's criticism, the historical analysis shows the differing expectations of the two men with respect to the further development of the theory of probability and its applications. The paper thus gives a glimpse of the provisional state of the theory around 1920, before others such as P. Levy (1886–1971) and A. N. Kolmogorov (1903–1987) stepped in and created a new paradigm for probability theory.

Our object in this paper is to study the antecedents, contents, implications, and impact of a not well-known or appreciated paper by EHRENFEST in 1911 on the essential nature of the different quantum hypotheses in radiation theory. After a careful analysis of EHRENFEST’s notebooks, correspondence, and publications, we conclude that the essential points of EHRENFEST’s paper were not perceived to a large extent, and hence that its implications were not considered thoroughly. Specifically, we show that EHRENFEST contributed significantly to the clarifications of the differences between PLANCK’s and EINSTEIN’s respective quantum hypotheses, as well as to those of the concepts ‘‘discontinuity,’’ ‘‘quantization,’’ and ‘‘corpuscularity.’’

Pavel Alekseevich Nekrasov (1853-1924) was an outstanding mathematician, who made important contributions to algebra, analysis, probability theory, and mechanics. He was, however, unable to present his work on probability satisfactorily and Markov and Liapunov simply rejected it. Seneta [Sen81] was the first to discover Nekrasov's pertinent work; he further explored this subject [Sen84], and Soloviev [Sol97] recently investigated Nekrasov's study of the central limit theorem. I myself [She89b, §7.2], also dealt with this theme; I discovered archival and newspaper materials concerning him (cf. [She89a], [She93], [She95], [She96], [She98a, pp. 11-16, 83-102], [SheOO, pp. 134-136], and [Chi94]), and I translated the entire debate of Markov and Liapunov with Nekrasov (in all, seven papers [She98b]), as well as the report of a commission denouncing Nekrasov's proposal on teaching probability theory in school [Marl 6], and one of Nekrasov's papers [Nekl2] on the method of least squares. My translations remain hardly known because they were published as microfiches; this is an additional reason for my attempt to provide here a general picture of Nekrasov.

This paper focuses on scientific practices and problems of modeling in the case of meteorology. This domain is especially interesting because of its crucial influence on the conception of mathematical models and its links with numerical instability, computers, and the science of chaos and dynamic systems. All of these questions involve for their solution a considerable mathematical apparatus which was developed by several mathematicians. An early example is von Neumann’s and Charney’s Princeton Meteorological Project in the period 1946–53 which ended with daily numerical prediction in less than 2 hours. After this stage, the questions of long-range forecasting and general circulation of the atmosphere became of greater importance. The late 1950s saw the emergence of an alternative: were atmospheric models used mainly for prediction or understanding? This controversial debate in particular occurred during an important colloquium in Tokyo in 1960 which gathered together J. Charney, E. Lorenz, A. Eliassen, and B. Saltzman, among others, and witnessed discussions on statistical methods for predictions and/or maximum simplification of dynamic equations. This phase ended in 1963 with Lorenz’s seminal paper on “Deterministic non periodic flows.”

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 Pascal’s theory of forms and to providing an analysis of the process leading to a fully general notion of covariant derivative. Finally, the reception of Vitali’s theory is discussed in light of Enea Bortolotti and Enrico Bompiani’s subsequent works.

Among many other things, Carnot stated a principle and proved a theorem. In 1850, Clausius corrected Carnot's theory, modifying it according to Joule's principle. He might have considered a corollary of the theorem as the mathematical formulation of Carnot's principle. We challenge the corollary: it is based on hidden assumptions, nor is it true for all cycles. Clausius realized the corollary's lack of generality, but on different grounds. In 1854, he generalized the theorem, and gave an (other) expression to Carnot's principle. We analyze Clapeyron's account of Carnot's theory, Thomson's account of 1849 and some of Clausius belated comments on his 1850 paper, as well Clausius' paper of 1854. We hope that they shed light on the corollary's tacit hypotheses and on the meaning of Carnot's principle. It is our contention: Clausius took seriously a contemporary meaning of the principle, and looked for a condition of integrability that could express recovery of the initial conditions of the reservoirs. Furthermore, he seems to have had some prior knowledge of the form the expression of the principle should take. Actually, this was the theory's natural candidate.

Niels Henrik Abel proved in 1824 that the general equation of degree 5 cannot be solved by rational operations and root extractions starting with the coefficients. Two years later, in a letter to Crelle, he gave explicit expressions of the roots of an equation of degree 5 which can be obtained in this way. He also announced that he had analogous results for any equation of prime degree. To substantiate this assertion, there is the posthumously published and unfinished manuscript (1828?) whose main part deals with solvable equations of prime degree. In addition, the introduction contains four statements about solvable equations in general of which indications of proofs are given at the end of the manuscript. After Abel's death, when Galois' papers had been understood, the theory of solvable equations was tied up with the theory of solvable groups. Using the results by Jordan (1870) in this area, Weber was able to prove Abel's statements (see Weber's Algebra II (1895)) about the zeros and factorizations of the polynomials of solvable equations. The purpose of this paper is to analyze Abel's manuscript in the light of modernized versions and extensions of his proofs. The main tool is a normalization (Theorem 1) of the radical extensions of a ground field K to the splitting field K(f) of an irreducible polynomial /(x) with coefficients in K whose zeros can be obtained by successive root extractions. The theorem says that there is a tower of Galois fields

Albert Einstein came very close to the final form of the equations describing the gravitational field during his collaboration with Marcel Grossmann a fecund period which began in August 1912 and lasted nine months. Nevertheless, the theory published in their joint paper1 showed an "ugly dark spot," as Einstein wrote to Lorentz2 : contrary to initial expectations, the field equations were not generally covariant. Einstein struggled hard to iron out what he considered a major flaw, and eventually succeeded in November 1915, when he presented the final version of his general theory of relativity in a number of communications to the Prussian Academy of Science. Accepted reconstructions of the genesis of the Entwurf paper generally take in the following steps: Einstein & Grossmann initially made the right choice, taking the Ricci tensor into account as the gravitational field tensor, but rejected it later in the paper due to the difficulties encountered in recovering the Newtonian limit in the case of a weak and static gravitational field, making do with equations which were covariant with respect to linear transformations of coordinates. Moreover, in a paragraph entitled "Remarks," added at the end of the Mathematical part (the paper has a "Physical part" and a "Mathematical part," written by Einstein and Grossmann, respectively), Einstein gave what he believed was a proof that all generally covariant equations for the gravitational field were physically unacceptable. Interpretation of the way Einstein accounted for his failure to obtain the correct Newtonian limit and for the limited covariance of the equations is still a moot point among historians of physics. Some believe that Einstein's difficulties were of an essentially mathematical nature, while others stress their physical nature, viewed in terms of Einstein's conception of static gravitational fields as it stood in 1912.

This paper provides new elements for reconstructing Heinrich Hertz's conversion from an electrodynamic concept to a concept of field theory, a process that took place between September 1887 and February 1888. First of all, it is argued that, contrary to what one could deduce from Hertz's own presentation in his Untersuchungen (or Electric Waves), paper N∘ 5 was actually written after the first publication of papers N∘ 6 and N∘ 7; this fact is illuminated by Hertz's unpublished correspondence (Section 1). Following this reordering, an approximative process of Hertz's conversion based only on his published papers is described (Section 2). Second, by complementing Hertz's published Erinnerungen (or Memoirs, Letters, Diaries) with his unpublished Versuchsprotokolle (Laboratory notes), a much finer description of this conversion process is discussed (Section 3). It is finally concluded that Hertz's idea of “air waves” belongs at the very end of this process, and was mainly attained through experimental hints found within Helmholtz's theoretical framework.

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”. Inspired by the old “TU11” rule for prediction of times of lunar eclipses, it proposes that column Φ, in a similar way, used the sum of the Lunar Four to predict times of lunar eclipses as well as the duration of one, 6, and 12 months by means of what usually is called “R–S” schemes. It also explains fully the structure and development of such schemes, a fact that strongly supports the new interpretation of column Φ.

Summary John Wallis's Arithmetica infinitorum (1655) was his finest piece of work and a major influence on the course of seventeenth-century English mathematics. This paper describes the background, content and methods of the book, and its reception by English and continental mathematicians; it also explores some of the mathematical ideas that arose in or from the Arithmetica infinitorum and suggests that the book played an important role in the seventeenth-century transition from geometrical to algebraic thinking. Right at the beginning of his mathematical career John Wallis, Savilian Professor of Geometry at Oxford from 1649 to 1703, embarked on the work that was to be published in 1655 as the Arithmetica infinitorum} It was his masterpiece and was to have a profound influence on the course of seventeenth-century English mathematics. The book contains the result for which Wallis is now best remembered, his infinite fraction for 4/π,2 but to him and his contemporaries the real importance of the Arithmetica infinitorum was in its methods: Wallis showed how classical problems of quadrature could be handled arithmetically and algebraically, and in doing so gave new impetus to the seventeenth-century shift from geometrical to algebraic thinking. Wallis himself was well aware of the book's importance and later devoted the final quarter of his lengthy Treatise of algebra both historical and practical to describing the contents and implications of the Arithmetica infinitorum, as worked out in the text itself and as developed by mathematicians from Brouncker to Newton during the ten years following publication.3 From a much longer perspective this paper revisits the Arithmetica infinitorum and reviews its significance afresh.4