On the structure of Carnot's theory of heat

A simple connection is pointed out between the theory of heat formulated in Sadi Carnot's: Réflexions sur la puissance motrice du feu (1824) and the later Kelvin-Clausius thermodynamics. In both theories two well-defined quantities, a heat function and a work function, exist and can be calculated by integrating along a reversible path. In thermodynamics the work function (energy) is conserved, whereas the heat function (entropy) increases by irreversible processes. In Carnot's theory the heat function is conserved, whereas the work function decreases, so that in this theory the irreversible process is characterized by a loss of work.     

A history of the axiomatic formulation of probability from Borel to Kolmogorov: Part I

This paper, the first of two, traces the origins of the modern axiomatic formulation of Probability Theory, which was first given in definitive form by Kolmogorov in 1933. Even before that time, however, a sequence of developments, initiated by a landmark paper of E. Borel, were giving rise to problems, theorems, and reformulations that increasingly related probability to measure theory and, in particular, clarified the key role of countable additivity in Probability Theory.This paper describes the developments from Borel's work through F. Hausdorff's. The major accomplishments of the period were Borel's Zero-One Law (also known as the Borel-Cantelli Lemmas), his Strong Law of Large Numbers, and his Continued Fraction Theorem. What is new is a detailed analysis of Borel's original proofs, from which we try to account for the roots (psychological as well as mathematical) of the many flaws and inadequacies in Borel's reasoning. We also document the increasing realization of the link between the theories of measure and of probability in the period from G. Faber to F. Hausdorff. We indicate the misleading emphasis given to independence as a basic concept by Borel and his equally unfortunate association of a Heine-Borel lemma with countable additivity. Also original is the (possible) genesis we propose for each of the two examples chosen by Borel to exhibit his new theory; in each case we cite a now neglected precursor of Borel, one of them surely known to Borel, the other, probably so. The brief sketch of instances of the Cantelli lemma before Cantelli's publication is also original.We describe the interesting polemic between F. Bernstein and Borel concerning the Continued Fraction Theorem, which serves as a rare instance of a contemporary criticism of Borel's reasoning. We also discuss Hausdorff's proof of Borel's Strong Law (which seems to be the first valid proof of the theorem along the lines sketched by Borel).In retrospect, one may ask why problems of geometric (or continuous) probability did not give rise to the (Kolmogorov) view of probability as a form of measure, rather than the study of repeated independent trials, which was Borel's approach. This paper shows that questions of geometric probability were always the essential guide to the early development of the theory, despite the contrary viewpoint exhibited by Borel's preferred interpretation of his own results.     

Prelude to dimension theory: The geometrical investigations of Bernard Bolzano

This paper treats Bernard Bolzano's (1781–1848) investigations into a fundamental problem of geometry: the problem of adequately defining the concepts of line (or curve), surface, solid, and continuum. Bolzano's interest in this problem spanned most of his creative lifetime. In this paper a full discussion is given of the philosophical and mathematical motivation of Bolzano's problem as well as his two solutions to the problem. Bolzano's work on this part of geometry is relevant to the history of modern mathematics, because it forms a prelude to the more recent development of topological dimension theory.     

“Will someone say exactly what the H-theorem proves?” A study of Burbury's Condition A and Maxwell's Proposition II

Summary Many historians of science recognize that the outcome of the celebrated debate on Boltzmann's H-Theorem, which took place in the weekly scientific journal Nature, beginning at the end of 1894 and continuing throughout most of 1895, was the recognition of the statistical hypothesis in the proof of the theorem. This hypothesis is the Stosszahlansatz or hypothesis about the number of collisions. During the debate, the Stosszahlansatz was identified with another statistical hypothesis, which appeared in Proposition II of Maxwell's 1860 paper; Burbury called it Condition A. Later in the debate, Bryan gave a clear formulation of the Stosszahlansatz. However, the two hypotheses are prima facie different. Burbury interchanged them without justification or even warning his readers. This point deserves clarification, since it touches upon subtle questions related to the foundation of the theory of heat. A careful reading of the arguments presented by Burbury and Bryan in their various invocations of both hypotheses can clarify this technical point. The Stosszahlansatz can be understood in terms of geometrical invariances of the problem of a collision between two spheres. A byproduct of my analysis is a clarification of the debate itself, which is apparently obscure.     

Newton's early computational method for dynamics

On December 13, 1679Newton sent a letter toHooke on orbital motion for central forces, which contains a drawing showing an orbit for a constant value of the force. This letter is of great importance, because it reveals the state ofNewton's development of dynamics at that time. Since the first publication of this letter in 1929,Newton's method of constructing this orbit has remained a puzzle particularly because he apparently made a considerable error in the angle between successive apogees of this orbit. In fact, it is shown here thatNewton's implicitcomputation of this orbit is quite good, and that the error in the angle is due mainly toan error of drawing in joining two segments of the oribit, whichNewton related by areflection symmetry. In addition, in the letterNewton describes quite correctly the geometrical nature of orbits under the action of central forces (accelerations) which increase with decreasing distance from the center. An iterative computational method to evaluate orbits for central forces is described, which is based onNewton's mathematical development of the concept of curvature started in 1664. This method accounts very well for the orbit obtained byNewton for a constant central force, and it gives convergent results even for forces which diverge at the center, which are discussed correctly inNewton's letterwithout usingKepler's law of areas.Newton found the relation of this law to general central forces only after his correspondence withHooke. The curvature method leads to an equation of motion whichNewton could have solvedanalytically to find that motion on a conic section with a radial force directed towards a focus implies an inverse square force, and that motion on a logarithmic spiral implies an inverse cube force.     

Boltzmann's ergodic hypothesis

Summary Boltzmann's ergodic hypothesis is usually understood as the assumption that the trajectory of an isolated mechanical system runs through all states compatible with the total energy of the system. This understanding of Boltzmann stems from the Ehrenfests' review of the foundations of statistical mechanics in 1911. If Boltzmann's work is read with any attention, it becomes impossible to ascribe to him the claim that one single trajectory would fill the whole of state space. He admitted a continuous number of different possible mechanical trajectories. Ergodicity was formulated as the condition that only one integral of motion, the total energy, is preserved in time. The two reasons for this are external disturbing forces and collisions within the system. Boltzmann found it difficult to ascribe ergodic behavior to a single system where the theoretical dependence on initial conditions, though never observed, has to be admitted as possible. To circumvent the dependence, he invented the concept of a microcanonical ensemble.     

Recurrences and continuity in Einstein's research on radiation between 1905 and 1916

We analyse some aspects of Einstein's research on the light quantum between 1905 and 1916. The central subject of our paper is the discussion of the possible relationship between a little known paper of his on photochemical equivalence and his well-known 1916 derivation of Planck's formula, indicating the possibility of a deep continuity between these, at first sight, little correlated, investigations. We also re-examine another chapter of Einstein's research on the behavior of radiation: momentum fluctuations (MF). The recurring use he made of his formula for MF has already been analysed in the literature. We emphasize that although through shifts of meaning, it establishes a close relationship between two distant stages of his research. We indicate where one might discover Einstein's first dissatisfaction with the probabilistic character of quantum physics, in relation to some aspects of his works of 1916.     

J. Zaragosa's centrum minimum,an early version of barycentric geometry

Summary Using the properties of the Centre of Gravity to obtain geometrical results goes back to Archimedes, but the idea of associating weights to points in calculating ratios was introduced by Giovanni Ceva in De lineis rectis se invicem secantibus: statica constructio (Milan, 1678). Four years prior to the publication of Ceva's work, however, another publication, entitled Geometria Magna in Minimis (Toledo, 1674), 2 appeared stating a method similar to Ceva's, but using isomorphic procedures of a geometric nature. The author was a Spanish Jesuit by the name of Joseph Zaragoza.Endeavouring to demonstrate an Apollonius' geometrical locus, Zaragoza conceived his idea of centrum minimum — a point strictly defined in traditional geometrical terms — the properties of which are characteristic of the Centre of Gravity. From this new concept, Zaragoza developed a theory that can be considered an early draft of the barycentric theory that F. Mobius was to establish 150 years later in Der barycentrische Calcul (Leipzig, 1827).Now then, whereas Ceva's work was rediscovered and due credit was given him, to this day Zaragoza's work has remained virtually unnoticed.     

Einstein,Nordström and the early demise of scalar,Lorentz-covariant theories of gravitation

Conclusion The advent of the general theory of relativity was so entirely the work of just one person — Albert Einstein — that we cannot but wonder how long it would have taken without him for the connection between gravitation and spacetime curvature to be discovered. What would have happened if there were no Einstein? Few doubt that a theory much like special relativity would have emerged one way or another from the researchers of Lorentz, Poincaré and others. But where would the problem of relativizing gravitation have led? The saga told here shows how even the most conservative approach to relativizing gravitation theory still did lead out of Minkowski spacetime to connect gravitation to a curved spacetime. Unfortunately we still cannot know if this conclusion would have been drawn rapidly without Einstein's contribution. For what led Nordström to the gravitational field dependence of lengths and times was a very Einsteinian insistence on just the right version of the equality of inertial and gravitational mass. Unceasingly in Nordström's ear was the persistent and uncompromising voice of Einstein himself demanding that Nordström see the most distant consequences of his own theory.     

Heaviside's operational calculus and the attempts to rigorise it

At the end of the 19^th century Oliver Heaviside developed a formal calculus of differential operators in order to solve various physical problems. The pure mathematicians of his time would not deal with this unrigorous theory, but in the 20^th century several attempts were made to rigorise Heaviside's operational calculus. These attempts can be grouped in two classes. The one leading to an explanation of the operational calculus in terms of integral transformations (Bromwich, Carson, Vander Pol, Doetsch) and the other leading to an abstract algebraic formulation (Lévy, Mikusiski). Also Schwartz's creation of the theory of distributions was very much inspired by problems in the operational calculus.     

Becquerel and the choice of uranium compounds

Conclusion The common assumption that Becquerel had no special reason to study uranium compounds in his search for substances emitting penetrating radiation cannot explain (a) Becquerel's own accounts, which refer to his choice as due to the peculiar harmonic series of bands; (b) Becquerel's systematic test of all uranium compounds (and metallic uranium), in contrast to his neglect of other substances; and (c) Becquerel's belief in invisible phosphorescence as an explanation of the radiation emitted by uranium compounds, even after his discovery that non-luminescent and metallic uranium also emit penetrating radiation.By comparing Becquerel's older studies of uranium to his radioactivity research, this paper has presented a reconstruction that can explain all of these points above. According to the historical evidence presented here, it is likely that Becquerel concentrated his attention on uranium and its compounds because the mechanical theory of luminescence opened up the possibility that, precisely in the case of uranium and its compounds, a violation of Stokes's law could occur, and penetrating short-wavelength radiation could be emitted through a special type of phosphorescence.     

Francis Hauksbee's theory of electricity

Conclusion Historians of science have usually assumed that the science of electricity developed in the period prior to Franklin, or at least prior to Nollet, in what amounted to a theoretical vacuum. It has been my aim in this paper to demonstrate the falsity of that assumption. I have shown, I hope, that Hauksbee's important researches were guided throughout by strong theoretical considerations, and I have indicated that Dufay's even more important studies were guided by exactly the same considerations. Nor was their theory in any sense a stagnant one. As it was developed by Hauksbee, it could give a fairly adequate explanation of almost all the known electrical phenomena; it even enabled him to predict the outcome of experiments such as the one involving the rubbing of a globe while it was positioned near a second, exhausted, globe. With the discovery of so many new phenomena in the 1730's, the theory turned out to be no longer adequate, but it is not at all surprising that it was a few years before the full extent of its inadequacies was appreciated, nor is it surprising that a strong continuity is evident between it and the theory which eventually replaced it. In the meantime, the theory continued to serve a useful function by suggesting new lines of research to its adherents. The theory functioned, then, in the same way as any other scientific theory, and it deserves a more serious treatment than it has usually received. This paper, I hope, can serve as a beginning.     

Isaac Newton and the problem of the earth's shape

In this article I discuss the theory of the earth's shape presented by Isaac Newton in Book III of his Principia. I show that the theory struck even the most reputable continental mathematicians of the day as incomprehensible. I examine the many obstacles to understanding the theory which the reader faced — the gaps, the underived equations, the unproven assertions, the dependence upon corollaries to practically incomprehensible theorems in Book I of the Principia and the ambiguities of these corollaries, the conjectures without explanations of their bases, the inconsistencies, and so forth. I explain why these apparent drawbacks are, historically considered, strengths of Newton's theory of the earth's shape, not weaknesses.     

Leonardo da Vinci's solution to the problem of the pinhole camera

The longstanding challenge of the pinhole camera for medieval theorists was explaining why luminous bodies cast onto a screen different images at different distances from the screen.I argue that this problem was first solved not by Francesco Maurolico, as David Lindberg concludes in his influential series of articles on the camera, but by Leonardo da Vinci. In studies in the Codex Atlanticus dating c. 1508–14, Leonardo explains the changes in screen patterns with distance by applying a key perspective principle to two kinds of projection pyramids that figure into pinhole camera imaging.In contrast, Maurolico's later conclusions about the pinhole camera are only partly correct. Maurolico gives a mistaken account of why pinhole images change with distance. He also introduces the erroneous notion that similar superimposed parts of the camera image actually fuse as the screen withdraws.     

Rapporto fra “metodo” e “contenuti matematici” in P. Gesualdo Melacrinò da Reggio Calabria

Summary Father Gesualdo Melacrinò (1725–1803), from Reggio Calabria (Italy), is an unknown Capuchin philosopher and theologian, who produced several works at the time he was teaching (only five years, from 1748–53); these works contained an original approach to the foundations and philosophy of mathematics. His main purpose was to reconciliate the classical traditions with the reality of his time. For him, this included a critical examination of the scholastic curriculum and a new orientation towards the methodological relevance of mathematics for all other sciences, especially for philosophy. Concerning mathematics, he emphasized the necessity of a basic revision and logical reconstruction of its foundations. This paper provides a comparative examination of Melacrinò's work with reference to its cultural and historical environment.     

The problem of the invariance of dimension in the growth of modern topology,part II

Summary This work examines the historical origins of topological dimension theory with special reference to the problem of the invariance of dimension. Part I, comprising chapters 1–4, concerns problems and ideas about dimension from ancient times to about 1900. Chapter 1 deals with ancient Greek ideas about dimension and the origins of theories of hyperspaces and higher-dimensional geometries relating to the subsequent development of dimension theory. Chapter 2 treatsCantor's surprising discovery that continua of different dimension numbers can be put into one-one correspondence and his discussion withDedekind concerning the discovery. The problem of the invariance of dimension originates with this discovery. Chapter 3 deals with the early efforts of 1878–1879 to prove the invariance of dimension. Chapter 4 sketches the rise of point set topology with reference to the problem of proving dimensional invariance and the development of dimension theory. Part II, comprising chapters 5–8, concerns the development of dimension theory during the early part of the twentieth century. Chapter 5 deals with new approaches to the concept of dimension and the problem of dimensional invariance. Chapter 6 analyses the origins ofBrouwer's interest in topology and his breakthrough to the first general proof of the invariance of dimension. Chapter 7 treatsLebesgue's ideas about dimension and the invariance problem and the dispute that arose betweenBrouwer andLebesgue which led toBrouwer's further work on topology and dimension. Chapter 8 offers glimpses of the development of dimension theory afterBrouwer, especially the development of the dimension theory ofUrysohn andMenger during the twenties. Chapter 8 ends with some concluding remarks about the entire history covered. Dedicated to Hans Freudenthal     

Kepler,Tycho, and the ‘Optical Part of Astronomy’: the genesis of Kepler's theory of pinhole images

In the last half of the 16^th century, the method of casting a solar image through an aperture onto a screen for the purposes of observing the sun and its eclipses came into increasing use among professional astronomers. In particular, Tycho Brahe adapted most of his instruments to solar observations, both of positions and of apparent diameters, by fitting the upper pinnule of his diopters with an aperture and allowing the lower pinnule with an engraved centering circle to serve as a screen. In conjunction with these innovations a method of calculating apparent solar diameters on the basis of the measured size of the image was developed, but the method was almost entirely empirically based and developed without the assistance of an adequate theory of the formation of images behind small apertures. Thus resulted the unsuccessful extension of the method by Tycho to the quantitative observation of apparent lunar diameters during solar eclipses. Kepler's attention to the eclipse of July 1600, prompted by Tycho's anomalous results, gave him occasion to consider the relevant theory of measurement. The result was a fully articulated account of pinhole images. Dedicated to the memory of Ronald Cameron Riddell (29.1.1938–11.1.1981)     

Evaluations of the beta probability integral by bayes and price

Summary The contribution of Bayes to statistical inference has been much discussed, whereas his evaluations of the beta probability integral have received little attention, and Price's improvements of these results have never been analysed in detail. It is the purpose of the present paper to redress this state of affairs and to show that the Bayes-Price approximation to the two-sided beta probability integral is considerably better than the normal approximation, which became popular under the influence of Laplace, although it had been stated by Price.The Bayes-Price results are obtained by approximating the skew beta density by a symmetric beta density times a factor tending to unity for n , the two functions having the same maximum and the same points of inflection. Since the symmetric beta density converges to the normal density, all the results of Laplace based on the normal distribution can be obtained as simple limits of the results of Bayes and Price. This fact was not observed either by Laplace or by Todhunter.     

A note on the fowler's solution of the lane-emden equation of index 3

Zusammenfassung FürFowler's Lösung der Lane-Emden-Gleichung von Index 3 wird der Gültigkeitsbereich erweitert.