Der Mathematiker Abraham de Moivre (1667–1754)

Summary Before examining de Moivre's contributions to the science of mathematics, this article reviews the source materials, consisting of the printed works and the correspondence of de Moivre, and constructs his biography from them.The analytical part examines de Moivre's contributions and achievements in the study of equations, series, and the calculus of probability. De Moivre contributed to the continuing development from Viète to Abel and Galois of the theory of solving equations by means of constructing particular equations, the roots of which can be written in the form . He also discovered the reciprocal equations. In the course of this work de Moivre discovered an expression equivalent to (cos +i sin ) ⁿ =cos n +i sin n and, following Cotes, he succeeded in expressing the nth roots of unity in trigonometric form.In the theory of series, de Moivre developed a polynomial theorem encom-passing Newton's binomial theorem and, in particular, a theorem of recurrent series useful in the calculus of probability.The demands of the calculus of probability led de Moivre to an approximation for the binomial coefficients for large values of n. The interaction between de Moivre and James Stirling, particularly in regard to the asymptotic series for log (n!), is treated at length. This work supplied the foundation for de Moivre's limit theorem for the binomial distribution.The calculus of probability, which occupied him from 1708 onward, became in time ever more the center of de Moivre's inquiries. Proceeding from contemporary collections of gambling exercises, de Moivre, by introducing an explicit measure of probability for the so-called Laplace experiments, found the beginnings of a theory of probability. De Moivre expanded the classic application of probability calculus to games of chance by addressing himself to the problem of annuities and by adopting Halley's work with its conception of Probability of life. De Moivre was the first to publish a mathematically formulated law for the decrements of life derived from mortality tables.

---

Abkürzungen a.a.O. am angegebenen Ort = Verweis auf das nach Verfassern alphabetisch geordnete Literaturverzeichnis. Eine vor a.a.O. in runden Klammern angegebene Zahl kennzeichnet die entsprechende Nummer der im Literaturverzeichnis aufgeführten Arbeiten eines Autors. - AC Ars conjectandi = Jakob Bernoulli (4) a.a.O. - AE Acta Eruditorum - a.S. alter Stil⁴⁷ - BM Bibliotheca Mathematica - DMV Deutsche Mathematiker-Vereinigung - JL Journal Literaire - MA Miscellanea Analytica, London 1730 - n.S. neuer Stil⁴⁷ - PT Philosophical Transactions of the Royal Society of London - r.F. rekurrente Folge - r.R. rekurrente Reihe - SMA Miscellaneis Analyticis Supplementum, London 1730 - v. veröffentlicht (nur im Briefverzeichnis verwendet) Prof. Dr. Kurt Vogel zum 80. Geburtstag 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.

---

---

Vorgelegt von C. Truesdell     

Action de la capsicine sur la multiplication et la virulence de l'Agrobacterium tumefaciens

Summary It was established that the treatment of liquid cultures ofAgrobacterium tumefaciens with capsicine (in doses of 0.0025–0.01) in the course of 6 h, causes a more abundant and quicker multiplication of the micro-organism and an increase of its virulence, as measured by the method ofBernaerts andDe Ley.     

Newton and the classical theory of probability

Summary Probabilistic ideas and methods from Newton's writings are discussed in § 1: Newton's ideas pertaining to the definition of probability, his probabilistic method in chronology, his probabilistic ideas and method in the theory of errors and his probabilistic reasonings on the system of the world. Newton's predecessors and his influence upon subsequent scholars are dealt with in §2: beginning with his predecessors the discussion continues with his contemporaries Arbuthnot and De Moiver, then Bentley. The section ends with Laplace, whose determinism is seen as a development of the Newtonian determinism.An addendum is devoted to Lambert's reasoning on randomness and to the influence of Darwin on statistics. A synopsis is attached at the end of the article.Abbreviations PT abridged Philosophical Transactions of the Royal Society 1665–1800 abridged. London, 1809 - Todhunter I. Todhunter, History of the mathematical theory of probability, Cambridge, 1865 To the memory of my mother, Sophia Sheynin (1900–1970)     

The Lunar Theories of Tycho Brahe and Christian Longomontanus in the Progymnasmata and Astronomia Danica

Tycho Brahe's lunar theory, mostly the work of his assistant Christian Longomontanus, published in the Progymnasmata (1602), was the most advanced and accurate lunar theory yet developed. Its principal innovations are: the introduction of equant motion for the first inequality in order to separate the determination of direction and distance; a more accurate limit for the second inequality although requiring a more complex calculation; additional inequalities of the variation and, in place of the annual inequality in Tycho's earlier theory, a reduction in the equation of time; in the latitude theory a variation of the inclination of the orbital plane and an inequality of the motion of the nodes; a reduction in the range of variation of distance, parallax, and apparent diameter. Some of these were already present in Tycho's earlier lunar theory (1599), but all were changed in notable ways. Twenty years later Longomontanus published a modified version of the lunar theory in Astronomia Danica (1622), for the purpose of facilitating the calculation through new correction tables, and also explained his reasons for parts of the theory in the Progymnasmata. This paper is a technical study of both lunar theories.     

Mosaicism for sulfoiduronate sulfatase deficiency in carriers of Hunter's syndrome

Summary Using an assay for sulfoiduronate sulfatase based on the degradation of³⁵S mucopolysaccharides in a cellfree system, two clonal populations have been demonstrated in fibroblasts of heterozygotes for Hunter's syndrome. The locus responsible for sulfoiduronate sulfatase deficiency in thisX-linked mucopolysaccharidosis is therefore subjected to dosage compensation in females.Acknowledgments. The technical assistance of Mr.Antonio De Falco and Mrs.Carmela Salzano is gratefully acknowledged. The authors are indebted to Dr.H. Kresse (Münster, BRD) for helpful suggestions.     

Chu Shih-chieh's Suan-hsüeh ch'i-meng (introduction to mathematical studies)

Conclusion The multi-faceted content of the SHCM and its collection of rules and tables made it an important mathematical work not only in China, but also in Korea and Japan. This book clearly demonstrated Chu's predominant interest in the field of algebra and his contribution to the solution of numerical equations of higher degree, which was a prelude to his famous treatise the Ssu-yüan yü-chien.     

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.     

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)     

A novel tetracyclic sesquiterpene from the oil of orejuela ofCymbopetalum penduliforum (Dunal)

Résumé De l'éther de pétrole extrait de l'huile de l'Orejula (Cymbopetalum penduliforum Dunal, Baill) fut examiné pour son contenu en terpènes. On a isolé les sesquiterpènes tétracycliques (ishwarone). L'évidence chimique et spectral appuie la constitution chimique et sa conversion à isoishwarone.

---

---

The authors are grateful to Dr.F. Will, Mr.W. F. Kuhn and Mr.G. Vilcins for their valuable analytical data and to Dr.R. F. Dawson for the oil of orejuela and Dr.P. C. Parthasarathy for the authentic sample of isoishwarane. We wish to thank ProfessorsA. Burger andN. H. Cromwell for helpful discussions.     

La regola dei segni dall'enunciato di R. Descartes (1637) alla dimostrazione di C. F. Gauss (1828)

Conclusioni Le dimostrazioni algebriche della regola, sia che riguardino il caso a radici reali o quello generale si fondano tutte su una proposizione che descrive il comportamento dei segni di un'equazione quando la si moltiplica per x – a oppure per x + a.Le prime dimostrazioni algebriche relative ad equazioni a radici reali, cioé quelle diSegner (1728),Campbell, Stübner eDe Gua, benchè trovate indipendentemente, presentano notevoli somiglianze. Per esempio tutti gli autori menzionati dimostrano laproposizione di Leibniz ricorrendo al lemma che afferma che, in un'equazione completa a radici reali, il quadrato di ogni coefficiente è maggiore del prodotto dei due coefficienti adiacenti, anche se di esso forniscono dimostrazioni diverse.Nella dimostrazione algebrica relativa al caso generale laproposizione di Leibniz è invece dimostrata direttamente. Già nella dimostrazione diSegner (1756) vengono introdotti, anche se in forma oscura e prolissa, quegli elementi che verranno poi semplificati e chiariti nelle dimostrazioni diWaring, Fourier eGauss.Il punto essenziale delle dimostrazioni analitiche, sia che riguardino il caso a radici reali o quello generale, è una proposizione che trasporta la validità del teorema dall'equazione derivata di un'equazione, all'equazione stessa. La regola viene poi dimostrata per induzione. Nelle prime dimostrazioni analitiche, cioè quelle diDe Gua, Kaestner edAepinus, gli autori fanno diretto riferimento al grafico della curva che rappresenta l'equazione. Successivamente con il consolidarsi delle conoscenze dell'analisi infinitesimale,Euler, Milner, Lagrange eRuffini omettono ogni riferimento al grafico dell'equazione.Osserviamo infine che è proprio l'applicazione dell'analisi infinitesimale alla teoria delle equazioni a suggerire aFourier una generalizzazione della regola dei segni, che condurrà successivamente al più generaleteorema di Sturm.     

Solar and lunar observations at Istanbul in the 1570s

From the early ninth century until about eight centuries later, the Middle East witnessed a series of both simple and systematic astronomical observations for the purpose of testing contemporary astronomical tables and deriving the fundamental solar, lunar, and planetary parameters. Of them, the extensive observations of lunar eclipses available before 1000 AD for testing the ephemeredes computed from the astronomical tables are in a relatively sharp contrast to the twelve lunar observations that are pertained to the four extant accounts of the measurements of the basic parameters of Ptolemaic lunar model. The last of them are Taqī al-Dīn Mu?ammad b. Ma‘rūf’s (1526–1585) trio of lunar eclipses observed from Istanbul, Cairo, and Thessalonica in 1576–1577 and documented in chapter 2 of book 5 of his famous work, Sidrat muntaha al-afkar fī malakūt al-falak al-dawwār (The Lotus Tree in the Seventh Heaven of Reflection). In this article, we provide a detailed analysis of the accuracy of his solar (1577–1579) and lunar observations.     

Master Georg Dörffel and the rise of cometary astronomy

This paper examines the purport of epact tables encountered on scientific instruments, and explains their use. The epact is a valuable chronological aid for calculating the age of the moon. In handbooks of chronology, usually two types of epacts are distinguished: the epact used in medieval times, and the so-called Lilian epact used after 1582 in the Gregorian perpetual calendar. By examining the rules for calculating the age of the moon, it turns out that the Julian and Gregorian epacts encountered on instruments must be distinguished from the medieval and Lilian epacts. It is shown that the Julian epact was already in use in 1478, and that, by adjusting for the shift of ten days in the date of the vernal equinox, the Gregorian epact was derived from it in 1582. The common association of the latter with the Lilian epact employed in the Gregorian perpetual calendar is incorrect. It is further shown that in contrast to the medieval and Lilian epacts, which served purely ecclesiastical purposes, the Julian and Gregorian epacts were mainly used to calculate the true age and zodiacal position of the moon. This knowledge was applied to secular interests such as ‘lunar astrology’, tidal computations, and the conversion of lunar into solar time.     

Britton’s theory of the creation of Column $$\varPhi $$ in Babylonian System A lunar theory

The following article has two parts. The first part recounts the history of a series of discoveries by Otto Neugebauer, Bartel van der Waerden, and Asger Aaboe which step by step uncovered the meaning of Column \(\varPhi \), the mysterious leading column in Babylonian System A lunar tables. Their research revealed that Column \(\varPhi \) gives the length in days of the 223-month Saros eclipse cycle and explained the remarkable algebraic relations connecting Column \(\varPhi \) to other columns of the lunar tables describing the duration of 1, 6, or 12 synodic months. Part two presents John Britton’s theory of the genesis of Column \(\varPhi \) and the System A lunar theory starting from a fundamental equation relating the columns discovered by Asger Aaboe. This article is intended to explain and, hopefully, to clarify Britton’s original articles which many readers found difficult to follow.     

Das Parallelenproblem im Corpus Aristotelicum

Summary In the Corpus Aristotelicum are numerous items suggesting that the assertion of the fifth postulate in Euclid's Elements had been preceded by attempts to demonstrate this postulate itself, or some equivalent fundamental proposition, within the rigorous frame of Absolute Geometry in Bolyai's sense. Thus geometers contemporary with Aristotle tried to solve the problem which became known commonly in later centuries as the Problem of Parallels.Probably these geometers first attempted a direct solution. Only one text at our disposal supports this hypothesis: (1) Anal. Prior. 65 a 4–7. My analysis below in Chapter I shows that a mathematical meaning can be read from this somewhat obscure text only if it is interpreted as an allusion by Aristotle to those geometers who believe they are demonstrating, obviously in an absolute way, the proposition Elem. I 29, equivalent to the fifth postulate, but do not realize that in the process they are using lemmas which result themselves from the proposition to be demonstrated. Such a lemma would assert the uniqueness of the parallels, existence of which was shown in an absolute way in Elem. I 27. My conjecture and reconstruction afford a natural explanation for an inconsequence singular for Book I of the Elements, namely, the presence of the proposition Elem. I 31 in the purely Euclidean part of the book, in spite of the fact that the assertion merely repeats the absolute proposition Elem. I 27 without explicitly containing any Euclidean element.It is probable that the failure of these direct attempts led to an indirect approach to the problem through reductio ad absurdum of some hypothesis contrary to what was to become Postulate V or to some equivalent proposition. Numerous texts survive from which it is clear that geometers contemporary with Aristotle followed fairly far the consequences of an hypothesis contrary to the fifth postulate, obtaining important results which are partly identical with some theorems of Saccheri. Some of these texts attest first of all that what Saccheri called the Hypothesis of the Obtuse Angle had been stated in an independent and explicit way and that the fundamental result, identical with Prop. 14 of Saccheri's Euclides ab omni naevo vindicatus (1733), had been obtained, namely, that within Bolyai's Absolute Geometry this hypothesis leads to the remarkable formal contradiction that parallels intersect. This conclusion followed from two different formulations of the Obtuse Angle Hypothesis: (2) Anal. Prior. 66a 11–14, if the exterior angle (formed by a secant which intersects two parallel straight lines) is smaller than the interior angle (opposite and situated on the same side of the secant), and (3) 66a 14–15, if the sum of the angles in a triangle is greater than 2R. Finally, an item in (4) Ethica ad Eudemum 1222b 35–36 shows us that by investigating the Obtuse Angle Hypothesis, the Greek geometers also discovered the quadrilateral in which the sum of the angles is equal to 8R; this quadrilateral, which does not appear even in Saccheri's book, is the maximal quadrilateral of the Riemann geometry, a quadrilateral degenerated into a straight line closed upon itself (Chapter IV 20).Nowhere in the Corpus does the Hypothesis of the Acute Angle appear in an independent formulation. Nevertheless in (5) Anal. Poster. 90a 33–34 this Hypothesis is mentioned along with the other two: namely, Aristotle states that the essence of the triangle consists in the sum of its angles' being equal to, greater than or less than 2R (Chapter V 27). The formulation of the fifth postulate in the Elements allows greater probability to the conjecture of independent existence of the Acute Angle Hypothesis as well. Indeed, in its original formulation the fifth postulate is redundant, since it unnecessarily specifies in which of the half-planes (bounded by the secant) the intersection of the two straight lines occurs; this specification is itself a theorem. The Acute Angle Hypothesis must have been formulated not only symmetrically to (3) Anal. Prior. 66 a 14–15, that is, the sum of the angles of the triangle is less than 2R, as results from (5) Anal. Poster. 90 a 33–34, but also symmetrically to (2) Anal. Prior. 66 a 11–14. In the latter case the following final conclusion should have been reached in order to reduce to absurdity the Acute Angle Hypothesis: Two straight lines cut by a secant are incident if the sum of the interior angles (on the same side of the secant) is smaller than 2R, and the incidence occurs on that side of the secant where the sum of the angles is less than 2R. In the frame of the Acute Angle Hypothesis, this end conclusion is relevant only if this final specification (concerning the half-plane where the incidence occurs) is explicitly emphasised. According to my conjecture, it was precisely the practical impossibility of reaching this conclusion as a theorem of Absolute Geometry that later determined Euclid to transpose this decisive end conclusion from the Acute Angle Hypothesis, without changing its wording, and to include it among the postulates (Chapter II 13).A queer passage of Proklos (In primum Euclidis Elementorum, ed. Friedlein p. 368, 26–369, 1) in which the Acute Angle Hypothesis is presented in the form of a Zenonian paradox reinforces the conjecture that this hypothesis was studied independently by the ancient geometers (Chapter VI 33). Thus failure to solve the Problem of Parallels preceded not only the later Non-Euclidean geometry but also Euclidean geometry itself.The general undifferentiated Contra-Euclidean Hypothesis appears in the following form in all the other texts examined: The sum of the angles in the triangle is not equal to 2R. This hypothesis is nowhere qualified by Aristotle as being absurd or impossible: On the contrary, he takes it always as being just as much justified a priori as is the Euclidean theorem Elem. I 32 which contradicts it. For instance in (6) Anal. Poster. 93 a 33–35 Aristotle puts the problematical alternative: Which of the two propositions is right (or, which of the two constitutes the Logos, the raison d'être of the triangle), the one that states that the sum of the angles in the triangle is equal to 2R, or on the contrary, the one that states that the sum of the angles in the triangle is not equal to 2R (Chapter V 28)?In a number of texts the theorem Elem. I 32 itself and the general Contra-Euclidean Hypothesis are treated as being a sort of principle, and stress is laid on the idea that the logical consequences of each of these items invariantly preserve its specific (Euclidean or non-Euclidean) geometrical content [(7) 1187 a 35–38 (Chapter IV 18); (8) 1222 b 23–26 (Chapter IV 19); (9) 1187 b 1–2 (Chapter IV 18); (10) 1222 b 41–42 Chapter IV 21); (11) 1187 b 2–4 (Chapter IV 18)]; (12) Physica 200 a 29–30: If the sum of the angles in the triangle is not equal to 2R, then the principles of geometry cannot remain the same (Chapter V 25); (13) Metaph. 1052 a 6–7: It is impossible that the sum of the angles in the triangle be sometimes equal to 2R and sometimes not equal to 2R (Chapter V 24). Finally, the most important item of this sort is to be found in (14) De Caelo 281 b 5–7: If we accept as a starting hypothesis that it is impossible for the sum of the angles in the triangle to be equal to 2R, then the diagonal of the square is commensurable with its side (Chapter III).Another group of texts reveal Aristotle's attitude as regard these Contra-Euclidean theorems: (15) 1222 b 38–39 (Chapter IV 20); (16) 200 a 16–19 (Chapter VI 30); (17) 402 b 18–21 (Chapter VI 31); (18) 171 a 12–16 (Chapter VI 32); (19) 77 b 22–26 (Chapter V 26); (20) 101 a 15–17 (Chapter VI 31); (21) 76 b 39–77 a 3 (Chapter VI 31). These passages reveal Aristotle's conviction that these paradoxical Contra-Euclidean propositions (which cannot be annihilated by reductio ad absurdum) are nevertheless inacceptable as bad, probably because their graphical construction requires curved lines for representing the concept of straight lines.Finally, another group of texts show that Aristotle sensed in a way the necessity of adding to the foundations of Geometry a new postulate, from which the proposition Elem. I 32 should follow rigorously.

---

---

Aram Frenkian zum Gedächtnis

---

Vorgelegt von J. E. Hofmann     

Muḥyī al-Dīn al-Maghribī’s lunar measurements at the Maragha observatory

This paper is a technical study of the systematic observations and computations made by Mu?yī al-Dīn al-Maghribī (d. 1283) at the Maragha observatory (north-western Iran, c. 1259–1320) in order to newly determine the parameters of the Ptolemaic lunar model, as explained in his Talkhī? al-majis?ī, “Compendium of the Almagest.” He used three lunar eclipses on March 7, 1262, April 7, 1270, and January 24, 1274, in order to measure the lunar epicycle radius and mean motions; an observation on April 20, 1264, to determine the lunar eccentricity; an observation on August 29, 1264, to test the model; and another on March 15, 1262, for measuring the lunar parallax. In the second period of activity at the Maragha observatory, Shams al-Dīn Mu?ammad al-Wābkanawī (c. 1254–1320) adopted all of al-Maghribī’s parameter values in his Zīj, but decreased his value for the mean longitude of the moon at epoch by 0;13,11 $^{\circ }$ . By comparing the times of the new moons and lunar eclipses in the period of 1270–1320 as computed from the astronomical tables of the Maragha tradition with the true modern ones, it is argued that this correction was very probably the result of actual observations.     

“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.     

Heterogeneity in the fibre composition in the flight muscles ofPeriplaneta americana andBelostoma sp

Zusammenfassung In Flugmuskeln vonBelostoma sp. und vonPeriplaneta americana kommen neben den normalen Fasern auch abnorm grosse Fasern vor. Diese Riesenfasern zeigen gegenüber Sudanschwarz B das gleiche färberische Verhalten wie die Normalfasern.

---

---

The work was carried out during the year 1965/66 when the author was the recipient of a U.G.C scholarship under Dr. J. C.George.     

Die physikalische Kieme der Wasserinsekten

Summary The air stores carried by a number of aquatic insects have: a) a hydrostatic function (Brocher, Oortwijn-Botjes, Thorpe, andCrisp); b) the function of an oxygen store (Ege, de Ruiter et al.) and c) the function of a physical gill (Strauss-Durckheim, Ege, etc.). The fact that oxygen is taken up from water with the aid of an air bubble was demonstrated forNotonecta by comparing the life time of insects with and without physical gill (while replenishing the oxygen store from the air was prevented) byEge, and forCorixa byPopham, whileVlasblom determined the oxygen uptake from water with and without air bubble forNotonecta, Naucoris, Corixa, Sigara andNepa. Nepa andSigara can take up considerable quantities of oxygen by cutaneous respiration.During the summer, the gill function of the air store ofNotonecta andNaucoris is of importance only when a water current passes along the animal, caused by ventilation movements of the legs (de Ruiter et al.). At low temperatures, however, the metabolic rate is so low that in many instances the physical gill provides the oxygen required without ventilation movements.An apparatus for the simultaneous determination of oxygen uptake from air and water (Wolvekamp andVlasblom) gave results that provided a means of evaluating the importance of the physical gill function.In some cases, the air store, although in direct contact with the water, does not need to be replenished. InAphelocheirus andElmis, the negative pressure in the bubble, caused by oxygen consumption and the diffusing out of part of the nitrogen, is compensated for by the mechanical resistance of a feltwork of thin hairs and the surface tension of the boundary layer of the water (Thorpe andCrisp). In the African beetle,Potamodytes, the unprotected air bubble is permanent because the strong river currents produce a lowered pressure around the animal according toBernoulli's principle (Stride).     

Costaclavine fromPenicillium chermesinum

Zusammenfassung Es wurden inPenicillium Mutterkornalkaloide nachgewiesen, die bisher nur unter den Pilzen ausAspergillus undClaviceps bekannt waren.

---

---

For the identification of the species used in this investigation, I am indebted to Dr.J. R. Kinsley, Purdue University. The technical assistance of MissS. Roos and the financial support from the Swedish Natural Science Research Council was appreciated. The sample of authentic costaclavine was generously provided by Dr.M. Abe.