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.     

Die Bestimmung von Dibenamin im biologischen Material

Summary In order to identify the presence of dibenamine and dibenamine-like compounds in biological substances (urine, sera) bromine cresol purple was used as a formative element in the adduction of dyestuff (i.e. as Farbstoffadduktbildner). This method permits a more simplified and more sensitive process of work than that suggested byBrodie and his collaborators.     

Le lettere di Eugenio Beltrami nella corrispondenza di Ernesto Cesàro

Summary We publish seventeen letters by Eugenio Beltrami to Ernesto Cesàro, which are dated from 1883 until 1900. They are about academic and scientific questions. Beltrami communicated many of Cesàro's memoirs to the Accademia dei Lincei or the Istituto Lombarde di Scienze e Lettere, and often gave him suggestions for his books and papers in these letters. When Cesàro looked for a professorship in the United States, Beltrami gave him information.In some letters Beltrami discussed questions on geometry and mathematical physics. In particular, the third letter (dated December 1st, 1888) is devoted to the mechanical interpretation of Maxwell's equations. Here, Beltrami shows a new proof of the conditions when six given functions are the components of an elastic deformation.

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Memoria presentata da U. Bottazzini     

Anastácio da Cunha and the concept of convergent series

Summary Anastácio da Cunha's definition of convergent series (Principios Mathematicos, Lisboa 1790, p. 106) was analysed by the Portuguese mathematician J. Vicente Gonçalves (Análise do Livro VIII dos Principios Mathematicos de José Anastácio da Cunha, Congresso do Mundo Português Vol. XII, Tomo I, Lisboa 1940, 123–140) and more recently by the historian A. P. Youschkevitch (J. A. da Cunha et les fondements de l'analyse infinitésimale Revue d'Histoire des Sciences Tomme XXVI, N. 1, 1973, 9–22). This latter author contests Gonçalves' claim to the effect that Cunha anticipated in his definition the well known criteria of convergence commonly attributed to A. Cauchy. However, Youschkevitch's otherwise deeper analysis is based primarily on a French translation of Principios published in Bordeaux in 1811. In this paper, that translation is shown to be misleading at crucial places. Cunha's definition is further analysed, and an interpretation in terms of the potential infinity is proposed which results in a redress of Cunha's originality in this matter.     

Neue Gesichtspunkte zum 5. Buch Euklids

Summary The author's purpose is to read the main work of Euclid with modern eyes and to find out what knowledge a mathematician of today, familiar with the works of V. D. Waerden and Bourbaki, can gain by studying Euclid's theory of magnitudes, and what new insight into Greek mathematics occupation with this subject can provide.The task is to analyse and to axiomatize by modern means (i) in a narrower sense Book V. of the Elements, i.e. the theory of proportion of Eudoxus, (ii) in a wider sense the whole sphere of magnitudes which Euclid applies in his Elements. This procedure furnishes a clear picture of the inherent structure of his work, thereby making visible specific characteristics of Greek mathematics.After a clarification of the preconditions and a short survey of the historical development of the theory of proportions (Part I of this work), an exact analysis of the definitions and propositions of Book V. of the Elements is carried out in Part II. This is done word by word. The author applies his own system of axioms, set up in close accordance with Euclid, which permits one to deduce all definitions and propositions of Euclid's theory of magnitudes (especially those of Books V. and VI.).In this way gaps and tacit assumptions in the work become clearly visible; above all, the logical structure of the system of magnitudes given by Euclid becomes evident: not ratio — like something sui generis — is the governing concept of Book V., but magnitudes and their relation of having a ratio form the base of the theory of proportions. These magnitudes represent a well defined structure, a so-called Eudoxic Semigroup with the numbers as operators; it can easily be imbedded in a general theory of magnitudes equally applicable to geometry and physics.The transition to ratios — a step not executed by Euclid — is examined in Part III; it turns out to be particularly unwieldy. An elegant way opens up by interpreting proportion as a mapping of totally ordered semigroups. When closely examined, this mapping proves to be an isomorphism, thus suggesting the application of the modern theory of homomorphism. This theory permits a treatment of the theory of proportions as developed by Eudoxus and Euclid which is hardly surpassable in brevity and elegance in spite of its close affinity to Euclid. The generalization to a classically founded theory of magnitudes is now self-evident.

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Vorgelegt von J. E. Hofmann     

Der Eigenartige Einfluß Witelos auf die Entwicklung der Dioptrik

Summary Witelo's Perspectiva, which was printed three times in the sixteenth century, profoundly influenced the science of dioptrics until the Age of Newton. Above all, the optical authors were interested in the so-called Vitellian tables, which Witelo must have copied from the nearly forgotten optical Sermones of Claudius Ptolemy. Research work was often based on these tables. Thus Kepler relied on the Vitellian tables when he invented his law of refraction. Several later authors adopted Kepler's law, not always because they believed it to be true, but because they did not know of any better law. Also Harriot used the Vitellian tables until his own experiments convinced him that Witelo's angles were grossly inaccurate. Unfortunately Harriot kept his results and his sine law for himself and for a few friends. The sine law was not published until 1637, by Descartes, who gave an indirect proof of it. Although this proof consisted in the first correct calculation of both rainbows, accomplished by means of the sine law, the Jesuits Kircher (Ars Magna, 1646) and Schott (Magia Optica, 1656) did not mention the sine law. Marci (Thaumantias, 1648) did not know of it, and Fabri (Synopsis Opticæ, 1667) rejected it. It is true that the sine law was accepted by authors like Maignan (Perspectiva Horaria, 1648) and Grimaldi (Physico-Mathesis, 1665), but since they used the erroneous Vitellian angles for computing the refractive index, they discredited the sine law by inaccurate and even ludicrous results.That even experimental determinations might be unduly biased by the Vitellian angles is evident from the author's graphs of seventeenth century refractive angles. These graphs also show how difficult it was to measure such angles accurately, and how the Jesuit authors of the 1640's adapted their experimental angles to the traditional Vitellian ones. Witelo's famous angles, instead of furthering the progress of dioptrics, delayed it. Their disastrous influence may be traced for nearly thirty years after Descartes had published the correct law of refraction.

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Vorgelegt von C. Truesdell     

Further morphological and biochemical studies on normal and hybrid embryos of sea urchins

Zusammenfassung (1) Es wurde bei den SeeigelartenParacentrotus ,Arbacia undSphaerechinus sowie den drei BastardenPar ×Arb ,Arb ×Par undPar ×Sphaer für eine Reihe von Entwicklungsstadien die Anzahl der Kerne pro Keim bestimmt (Tabelle). Der morphogenetischen Hemmung der Bastarde entsprechen herabgesetzte Kernzahlen und parallel eine Herabsetzung der DNS-Synthese. (2) Es wurde bei den genannten reinen Arten und den Bastarden derUmsatz der RNS bestimmt. Er geht in allen Fällen, bei den reinen Arten wie bei den Bastarden, der DNS- Synthese parallel.

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Our work was supported by a grant from the Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung. We express our sincere thanks to the Nationalfonds and also to the authorities at the Zoological Station at Naples for generous help and supply of materials and facilities. — This paper is thankfully dedicated byF. Baltzer toF. E. Lehmann for his kind hospitality in the Bernese Institute.     

On the regulation of lactate dehydrogenase isoenzyme concentrations in mammalian cells

Zusammenfassung Fritz et al.³ haben in Nature New Biol. 230, 119 ein Modell publiziert, mit dem sie die Regulation der intrazellulären LDH-Isoenzym-Konzentration erklären. Die vorliegende Arbeit zeigt, dass die Resultate vonFritz et al. einfacher durch Heterogenität der Zellen des untersuchten Gewebes erklärt werden kann als durch ihr Modell mit non-random association der LDH-Untereinheiten und verschiedenen turnover-Raten der Isoenzyme in derselben Zelle.

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I thank Drs.Ursprung, D. Turner, D. Seybold andH. Eppenberger for helpful discussions. This work was supported by the Swiss National Science Foundation, Project No. 3.247.69.     

Molecular mechanism on the energy transfer and conversion in the photosynthetic system

Zusammenfassung Auf molekularer Basis wird die Energieübertragung und die Energieumsetzung in einem primären photosynthetischen System untersucht. Die Energieübertragung in den Pigmenten und im Chlorophyll-A wird mit der Energieübergangswahrscheinlichkeit und dem Diffusionsmodell eines intramolekularen «Exitons» mit Dipol-Dipol-Wechselwirkung behandelt. Bei einer Chlorophyllkonzentration von 0.1M/1 wird die Diffusionskonstante des «Exitons»D _e 1.1 × 10^–3 cm²/sec und seine Diffusionslängel _e 250 Å. Als Energieumsetzung wird die Anregung eines – -Tripletts des Chlorophyll-A mit Hilfe eines paramagnetischen Metallenzyms vorgeschlagen und diskutiert.

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This work was supported by grants from the Japanese Society for the Promote of Science. The author wishes to express his gratitude to ProfessorM. Kotani of Tokyo University and ProfessorK. Oomori and ProfessorG. Tomita for their encouragement.     

Hoplonemertine worms—a new source of pyridine neurotoxins

Summary Two pyridine bases were isolated from the marine hoplonemertineAmphiporus angulatus (Fabricius) and identified by mass and PMR-spectroscopy as 2,3-bipyridyl and 3,2; 3,2; 4,3'-tetrapyridyl (Nemertelline). Nemertelline, the first tetrapyridyl natural product to be reported, shows a structural resemblance to the tobacco constituent nicotelline. The crustacean toxicity of 2,3-bipyridyl is very similar to that of nicotine, but its mammalian toxicity is negligible.We thank DoctorsN. Meinkoth andN. Riser for collecting specimens, Dr.F. Kuffner for providing the nicotelline sample, Dr.C. Westerman for recording initial PMR-spectra, and Dr.R. King for furnishing the computer simulation program and recording the high resolution mass spectra, and Mr.J. Babashak for use of the Kontes densitometer. This research was supported by an NIH-GRS grant and by the Veterans Administration (No. MRIS-9416).     

“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.     

The Kepler Problem from Newton to Johann Bernoulli

Summary Newton solved what was called afterwards for a short time the directKepler problem (le problème direct): given a curve (e.g. an ellipse) and the center of attraction (e.g. the focus), what is the law of this attraction ifKepler's second law holds?The problème inverse (today: the problème direct) was attacked system-atically only later, first byJacob Hermann, then solved completely byJohann Bernoulli in 1710 and followingBernoulli byPierre Varignon. How didBernoulli solve the problem? What method did he use for this purpose and which of his accomplishments do we still follow today?In the second part various questions connected to the first part are dealt with from the point of view, Conflict and Cooperation, suggested byJ. van Maanen to the participants of the Groningen conference.     

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.     

Identification of a major cytokinin in coconut milk

A major cytokinin found in coconut milk was isolted by using the tobacco callus growth-promoting assay as a guide during purification. The structure of the factor was determined to be 14-O-{3-O-[-d-galactopyranosyl-(12)--d--galactopyranosyl-(13)--L-arabinofuranosyl]-4-O-(-L-arabinofuranosyl)--d-galactopyranosyl}-trans-zeatin riboside [G₃A₂-ZR] by various NMR techniques, including heteronuclear multiple bond connectivity by 2D multiple quantum NMR (HMBC), as well as mass spectroscopy and sugar analysis. The optimum concentration of G₃A₂-ZR for cytokinin activity in the tobacco callus assay was estimated to be 5×10^–6 M, so that G₃A₂-ZR is one order of magnitude more potent than 1,3-diphenylurea and one order less potent than zeatin riboside. At least 20% of the cytokinin activity of coconut milk could be attributed to G₃A₂-ZR.     

Synaptic frequency demodulation

Zusammenfassung Die synaptische Übertragung von Reizmustern wurde an der neuromuskularen Synapse beiRana ridibunda geprüft. Der Nerv wurde mit einer sinusoidal modulierten Frequenz gereizt und die postsynaptischen Potentiale intrazellulär abgeleitet. Daraus wird geschlossen, dass die Synapse Veränderungen in der Einsatzfrequenz überträgt, die einem Low-pass-Filter gleichkommt.

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We are grateful to Prof.S. Gitter for his help and encouragement.     

Wallis's product,Brouncker's continued fraction,and Leibniz's series

A historical sketch is given of Wallis's infinite product for 4/, and of the attempts which have been made, over more than three centuries, to find the method by which Brouncker obtained his equivalent continued fraction. A derivation of Brouncker's formula is given. Early results obtained by Indian mathematicians for the series for /4, later named for Leibniz, are reviewed and extended. A conjecture is made concerning Brouncker's method of obtaining close bounds for .     

Etude photophysique de la ptérobiline

Summary The first results of a photophysical study on pterobilin (biliverdin IX , a Lepidopter blue bile pigment) are presented. From the absorption and fluorescence spectra, it is deduced that the low yield of fluorescence indicates a desactivation of the excited singlet state occurring mainly by a non-radiative process. Analyses of the chemical compounds formed after irradiation of pterobilin in methanol show that it is rearranged into a series of new blue pigments among which phorcabilin and sarpedobilin (the two neobiliverdins IX isolated from Lepidopters) have been identified.

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Remerciements. Nous remercions le ProfesseurJ. Bergerard pour son intérêt. Ces recherches ont été réalisées dans le cadre de l'ATP CNRS No. 4699-10.     

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.