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)     

Geometry of Light and Shadow: Francesco Maurolyco (1494–1575) and the Pinhole Camera

In his Theoremata de lumine, et umbre (1521), Francesco Maurolyco (1494–1575) discussed, inter alia, the problem of the pinhole camera. Maurolyco outlined a framework based on Euclidean geometry in which he applied the rectilinear propagation of light to the casting of shadow on a screen behind a pinhole. We limit our discussion to the problem of how the image behind an aperture is formed, and follow the way Maurolyco combined theory with instrument to solve the problem of the projection of light through small apertures. We show that Maurolyco not only reformed the classical sources which, he thought, were no longer the authoritative code of textual knowledge, but also established with the dioptra a novel linkage of method, theory, and instrument. He thereby demonstrated the importance of optics to the science of astronomy.     

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.     

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.     

The conceptual import of Carnot's theorem to the discovery of the entropy

Summary 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.     

The ascendancy of the Laplace transform and how it came about

The modern Laplace transform is relatively recent. It was first used by Bateman in 1910, explored and codified by Doetsch in the 1920s and was first the subject of a textbook as late as 1937. In the 1920s and 1930s it was seen as a topic of front-line research; the applications that call upon it today were then treated by an older technique — the Heaviside operational calculus. This, however, was rapidly displaced by the Laplace transform and by 1950 the exchange was virtually complete. No other recent development in mathematics has achieved such ready popularisation and acceptance among the users of mathematics and the designers of undergraduate curricula. Communicated by C. Truesdell     

P.S. Laplace's work on probability

Taken together with my previous articles [77], [80] devoted to the history of finite random sums and to Laplace's theory of errors, this paper sheds sufficient light on the whole work of Laplace in probability. Laplace's theory of probability is subdivided into theory of probability proper, limit theorems and mathematical statistics (not yet distinguished as a separate entity). I maintain that in its very design Laplace's theory of probability is a discipline pertaining to natural science rather than to mathematics. I maintain also the idea that the so-called Laplacian determinism was no hindrance to applications of his theory of probability to natural science and that one of his utterances in this connection could have well been made by Maxwell's contemporaries.Two possible reasons why the theory of probability stagnated after Laplace's work are singled out: the absence of new fields of application and, also, the insufficient level of mathematical abstraction used by Laplace. For all his achievements, I reach the general conclusion that he did not originate the theory of probability as it is now known. Dedicated to the memory of my Father, Boris A. Sheynin (1898–1975), the first generation of the Russian revolution Cette inégalité [Lunaire] quoique indiquée par les observations, était négligée par le plus grand nombre des astronomes, parce qu'elle ne paraissait pas résulter de la théorie de la pesanteur universelle. Mais, ayant soumis son existence au Calcul des Probabilités, elle me parut indiqués avec une probabilité si forte, que je crus devoir en rechercher la cause.(P. S. Laplace (Théor. anal. prob., p. 361))     

Laser diffraction used to monitor strain in mechanoreceptors ofJasus verreauxi

Summary Laser diffraction patterns from crayfish abdominal mechanoreceptors have been observed and the corresponding sarcomere lengths calculated and then correlated with sensory nerve discharge frequencies.Our thanks are due to the Australian Research Grants Committee for provision of the lasers and camera, to Dr.K. S. Imrie for the design of the interfacing cards, to ProfessorP. Mason and ProfessorR. E. Aitchison for their encouragement and advice, and toI. Paterson andC. March for constructing the equipement.     

Lipoperoxidation rates and drug-oxidizing enzyme activities in the liver and placenta of some mammal species during the perinatal period

Summary Lipoperoxidation and drug-metabolizing enzymes were measured in livers and placentas of different mammal species during the perinatal period. In placentas and fetal livers of rat, rabbit and guinea-pig, cofactor-supported lipoperoxidation was negligible, as were the activities of drug-oxidizing enzymes. Human fetal liver contained an intact drug-oxidizing electron transport chain, and lipoperoxidation activity was accordingly observed. It is suggested that lesions mediated by lipoperoxidation may be possible in human fetus, but they are less probable in animal fetuses.The skillful technical assistance of Ms.Liisa Tuhkanen and Ms.Vuokko Väisänen is gratefully acknowledged.     

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.     

The size distribution ofTetrahymena in relation to its position in the cell cycle

Summary Tetrahymena size distribution during the cell cycle was analyzed by means of radioautography with the aid of a sonic-digitizer, and a computer. The study demonstrates that as the organism ages and passes through the various cell cycle phases the volume distribution of the organisms in each phase remains lognormal.Acknowledgments. This work was supported by a research grant from Gtiftung Volkswagenwerk No. 112273 toA. Ron. The authors wish to acknowledge the technical help of Mrs.O. Horovitz and MissS. Urieli, as well as the expert photomicrography of Mrs.E. Salomon.     

Metabolism of the neuromodulator d-serine

Over the past years, accumulating evidence has indicated that d-serine is the endogenous ligand for the glycine-modulatory binding site on the NR1 subunit of N-methyl-d-aspartate receptors in various brain areas. d-Serine is synthesized in glial cells and neurons by the pyridoxal-5′ phosphate-dependent enzyme serine racemase, and it is released upon activation of glutamate receptors. The cellular concentration of this novel messenger is regulated by both serine racemase isomerization and elimination reactions, as well as by its selective degradation catalyzed by the flavin adenine dinucleotide-containing flavoenzyme d-amino acid oxidase. Here, we present an overview of the current knowledge of the metabolism of d-serine in human brain at the molecular and cellular levels, with a specific emphasis on the brain localization and regulatory pathways of d-serine, serine racemase, and d-amino acid oxidase. Furthermore, we discuss how d-serine is involved with specific pathological conditions related to N-methyl-d-aspartate receptors over- or down-regulation.     

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.     

Interactions between mechanics and differential geometry in the 19th century

Conclusion 79. This study of the interaction between mechanics and differential geometry does not pretend to be exhaustive. In particular, there is probably more to be said about the mathematical side of the history from Darboux to Ricci and Levi Civita and beyond. Statistical mechanics may also be of interest and there is definitely more to be said about Hertz (I plan to continue in this direction) and about Poincaré's geometric and topological reasonings for example about the three body problem [Poincaré 1890] (cf. also [Poincaré 1993], [Andersson 1994] and [Barrow-Green 1994]). Moreover, it would be interesting to find out how the 19^th century ideas discussed here influenced the developments in the 20^th century. Einstein himself is a hotly debated case.Yet, despite these shortcommings, I hope that this paper has shown that the interactions between mechanics and differential geometry is not a 20^th century invention. Klein's view (see my Introduction) that Riemannian geometry grew out of mechanics, more specifically the principle of least action, cannot be maintained. On the other hand, when Riemannian geometry became known around 1870 it was immediately used in mechanics by Lipschitz. He began a continued tradition in this field, which had several elements in common with the new view of mechanics conceived by the physicists and explicitly carried out by Hertz.Before 1870 we found only scattered interactions between differential geometry and mechanics and only direct ones for systems of two or three degrees of freedom. For more degrees of freedom the geometrical ideas were in some interesting cases taken over by analogy, but these analogies did not lead to formal introduction of geometries of more than three dimensions.     

Zur Diskussion über die Homogenität des Erdinnern

Summary According to considerations put forwards byW. Kuhn andA. Rittmann some years ago, it follows that the chemical composition of the interior of the earth must be rather homogeneous; the well-known discontinuity which occurs at 2900 km with respect to the propagation of longitudinal and transverse waves should not be due to a discontinuity of the material composition (not to an iron core). It is due to a continous decrease of the viscosity and thereby of the relaxation time; transverse waves of a period of e.g. 30 seconds will no more be propagated in a material whose relaxation time for tangential stress is below 30 seconds, while the longitudinal waves will suffer a decrease of the velocity at the same time.A criticism put forward byA. Eucken consists in the argument that a material in which the time of relaxation for tangential stress becomes equal to the period of the vibration will exhibit a considerable absorption coefficient for longitudinal waves too. It is now shown that the distance on which the period of vibration and the relaxation time are approximately equal is small compared with the wave length of the seismic waves in question, from which it follows that the resulting absorption of the longitudinal waves too will only be small.A further consideration shows that a mixture of 99 atomic % hydrogen and 1 atomic % of iron is most probably supercritical at a temperature of 5000° abs.A survey of the solubilities in question shows further that the hydrogen present in a mixture of 90% hydrogen and 10% iron should on the strength of the absorption coefficient be completely absorbed by the iron at 5000° abs. and at a total pressure of 2.10⁶ atmospheres.The main argument why the assumption of an iron core inside the earth must be dismissed remains the fact that the present state must be the result of an asymptotic processus which at least in its final phase has occurred under conditions similar to the present conditions of temperature, pressure and viscosity; these latter conditions are far from permitting the processus of sedimentation etc. which would be required.     

Aristotele e l'incommensurabilità

Riassunto Aristotele accenna a due diverse dimostrazioni della incommensurabilità tra lato e diagonale di uno stesso quadrato, una esatta e l'altra, avverte, errata. Nel presente lavoro viene presa in esame quest'ultima proponendone una ricostruzione in connessione, come é accennato da Aristotele, con le argomentazioni di Zenone.Nel corso del lavoro vengono esaminati due altri brani di Aristotele attraverso i quali é possibile osservare un momento dello sviluppo matematico dell'algoritmo euclideo e il cosiddetto postulato di Eudosso-Archimede nel più vasto ambito dei primi contatti con problemi di analisi infinitesimale.Nell'Appendice, poi, vengono elencati tutti i brani di Aristotele raccolti dall'autore sulla detta incommensurabilità.

---

Summary Aristotle mentions two different demonstrations of the incommensurability of side with diagonal of the same square, one of which is correct, and the other, as he points out, incorrect. This work examines the latter suggesting a reconstruction of it in connection, as Aristotle mentioned, with Zeno's argumentations.In the course of this work, two other passages by Aristotle are examined, by which it is possible to observe a moment of the mathematical development of the Euclidean algorithm and the so called Axiom of Eudoxus-Archimedes in the broader ambit of early contacts with problems of infinitesimal analysis.In the Appendix, secondly, all the passages by Aristotle regarding incommensurability gathered by the author are listed.

---

Résumé Aristote parle de deux différentes démonstrations de l'incommensurabilité entre le côté et la diagonale d'un même carré, une exacte et l'autre, nous prévient-il, inexacte. Dans ce travail on examine cette dernière démonstration en connexion, comme Aristote lui-même le suggère, avec les argumentations de Zénon.Au cours du travail l'auteur examine deux autres morceaux d'Aristote, à travers lesquels il est possible d'observer un moment du développement mathématique de l'algorithme euclidien et ce qu'on appelle le postulat d'Eudoxe-Archimède dans le domaine bien plus vaste des premiers contacts avec des problèmes d'analyse infinitésimale.Dans l'appendice on peut trouver la liste de tous les morceaux d'Aristote que l'auteur a rassemblés sur la susdite incommensurabilité.

---

---

Memoria presentata da C. Truesdell     

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.     

Über die Reizung des Ferntastsinns bei Fischen und Amphibien

Summary Fishes and aquatic amphibians are able to detect and locate moving bodies and even obstacles at some distance by means of their lateral-line sense organs (Ferntastsinn). As was shown experimentally, the main physical process involved in these reactions are certain local damming phenomena in front of the moving object (rise of pressure, displacement of water particles). Obstacles cause an alteration of the damming phenomena produced by the moving animal's body itself (increase of water resistance).Kramer's different view is rejected; he overlooked the existence of damming phenomena and was not aware of the fact thatXenopus laevis reacts to surface waves even when the animal is totally submerged.     

Geometrical constructions equivalent to non-linear algebraic transformations of the plane in Newton's early papers

Summary 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.Abbreviations MPN The Mathematical Papers of Isaac Newton, edited by D. T. Whiteside, Vols. 1–3, Cambridge, 1967–1969 - Hudson H. Hudson, Cremona Transformations in Plane and Space, Cambridge, 1927 - PT (abridged) Philosophical Transactions of the Royal Society 1665–1800 (abridged), London, 1809 - Andreev ₁ K. A. Andreev, On geometrical correspondences ... (in Russian), Moscow, 1879 - Andreev ₂ K. A. Andreev, On the Geometrical Formation of Plane Curves (in Russian), Kharkov, 1875     

Die Notwendigkeit einer Lichtreaktion für die Induktion photosynthetischer Aktivität

Summary Beans grown under a flash regime never show a PS-II activity at their first illumination with photosynthetically actinic light. But immediately when light is on, an induction period takes place for some 6 min. In this paper we show, by oxygen measurements and different light conditions, that the induction of PS-II activity is due to a photosynthetically independent light reaction.

---

---

Die Arbeiten wurden mittels eines Forschungsstipendiums anR. Strasser des Patrimoine de l'Université de Liège unterstützt. Frau.F. Hayet sei für die Kultivierung der Organismen undJ. F. Ohn für technische Konstruktionen gedankt.