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 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.     

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.     

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.     

Detection and substitution-pattern determination of guanidines

Summary The guanidine function can be detected and its substitution pattern determined taking into account the¹H-NMR signals of the N–H and N–C–H groups. Satisfactory results were obtained with mono-to penta-substituted guanidines (as picrate salts).Part XX ofStudies on Plants; preceding part,R. A. Corral, O. O. Orazi andI. A. Benages, Tetrahedron29, 205 (1973).     

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.     

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.     

Transparency in organisms

Summary The occurrence in animal phyla of species having a relatively transparent body is noted and measurements of the transmittance of medusae made in a spectrophotometer are reported, but the approximate nature of the results obtained with a commercial instrument and the importance of the correct physical design of the measuring apparatus are emphasized. The application to invertebrates of the structural explanation of the predominant transmission of incident light by the vertebrate cornea is discussed and the role of other factors considered. Destructive interference of the scattered rays, sufficient to account for the transparency of the cornea, has been shown not to demand a completely regular arrangement of collagen fibres. The small diameter and regularity of the fibrillar components in the muscles ofSagitta may be adequate to account for their transparency.I am grateful to Dr.D. M. Maurice for the encouragement of this interest, to Dr.E. G. Jordan for electron micrographs ofSagitta and to Mr.G. Ross (Department of Physics, Queen Elizabeth College) for helpful and critical discussion.     

Numerical solving of equations in the work of José Mariano Vallejo

The progress of Mathematics during the nineteenth century was characterised both by an enormous acquisition of new knowledge and by the attempts to introduce rigour in reasoning patterns and mathematical writing. Cauchy’s presentation of Mathematical Analysis was not immediately accepted, and many writers, though aware of that new style, did not use it in their own mathematical production. This paper is devoted to an episode of this sort that took place in Spain during the first half of the century: It deals with the presentation of a method for numerically solving algebraic equations by José Mariano Vallejo, a late Spanish follower of the Enlightenment ideas, politician, writer, and mathematician who published it in the fourth (1840) edition of his book Compendio de Matemáticas Puras y Mistas, claiming to have discovered it on his own. Vallejo’s main achievement was to write down the whole procedure in a very careful way taking into account the different types of roots, although he paid little attention to questions such as convergence checks and the fulfilment of the hypotheses of Rolle’s Theorem. For sure this lack of mathematical care prevented Vallejo to occupy a place among the forerunners of Computational Algebra.     

The mensuration of quadrilaterals and the generation of Pythagorean triads: A mathematical,heuristical and historical study with special reference to Brahmagupta's rules

Summary The ancient rule for the area of a quadrilateral, , is examined to show its inaccuracy and its arbitrariness, and in order to see how those using it might have become aware of its shortcomings. Consideration ofBrahmagupta's vastly more sophisticated rule, , and of the Hindu schedule of recognized quadrilateral types, leads to the question of the proper assessment to be made of this strand of Indian mathematics. The work on quadrilaterals was intimately connected with the generation of Pythagorean triangles out ofbja (=seed) numbers by the same method as had probably been used by the mathematicians of Old Babylonia. Ways in which this procedure could have arisen by induction from particular numerical calculation are shown and the rest of the study consists primarily of an investigation as to whether the same kind of approach might not also have been used to obtain the remarkable rules on quadrilaterals given byBrahmagupta. It is found that there is no compelling need to adopt the common assumption thatBrahmagupta must have had access to Alexandrian mathematics. But even if he did chance to learn of some helpful items from the works ofHero orPtolemy, it still seems necessary for a proper appreciation of his understanding of the mensuration of quadrilaterals to suppose that he would have worked such items into the contemporary Indian mathematical context in some such way as is indicated here. In particular, it is argued thatBrahmagupta, far from indulging in reckless generalization, could well have proceeded with great caution, from one amenable type of quadrilateral to another, verifying his inductions by comparing them with results given by alternative calculation methods.     

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.     

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 .     

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.     

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     

Untersuchungen über die Tumorbildung beim amphidiploiden Bastard vonNicotiana langsdorffii undNicotiana glauca

Summary The occurrence of tumours of the amphidiploid hybrid ofNicotiana glauca andN. langsdorfii can — as was found in organculture — be connected with the specific formation of callus ofN. glauca. Those factors ofN. glauca which hinder callus growth, are obviously disturbed by the hybridization withN. langsdorffii, so thatN. glauca's potency in forming callus can manifest itself uninhibitedly and thus produce tumours.

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Den Herren Prof. Dr.G. de Lattin (Zoologisches Institut der Universität des Saarlandes) und Prof. Dr.F. Anders (Genetisches Institut der Justus Liebig-Universität, Giessen) bin ich für die Förderung dieser Arbeit zu grösstem Dank verpflichtet.     

Essais de solubilisation et de détoxification de la Tyrothricine

Summary As Tyrothricin —Dubos's discovery of the antibiotic agent extracted from cultures ofB. brevis — does not give true aqueous solutions and retains, in the colloïdal state of the usually employed solutions, toxic properties for polymorphonuclear blood cells, we have attempted to effect the solubilization and detoxification of this compound. Treatment with formalin in given proportions results in making it water-soluble. The action of the formalin-treated compound on blood cells suspended in Ringer's solution seems to show a loss of toxicity. Subcutaneous and intramuscular injections in mice do not give visible general toxic effects, and the experiments incite to further investigations in this direction.     

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)     

Zur Wirkung des Calciums auf die Myofibrillen-ATPase des Herzmuskels

Summary The inhibition of cardiac myofibrillar ATPase activity by EDTA can be completely reversed by Ca⁺⁺ under certain experimental conditions. This shows that there is no fundamental difference between the reaction of cardiac myofibrillar ATPase and the tension development of glycerinated cardiac fibres, as was supposed byBriggs andHannah on account of the results ofParker andBerger.     

Bolzano,Cauchy and the “new analysis” of the early nineteenth century

Summary In this paper I discuss the development of mathematical analysis during the second and third decades of the nineteenth century; and in particular I assert that the well-known correspondence of new ideas to be found in the writings of Bolzano and Cauchy is not a coincidence, but that Cauchy had read one particular paper of Bolzano and drew on its results without acknowledgement. The reasons for this conjecture involve not only the texts in question but also the state of development of mathematical analysis itself, Cauchy both as personality and as mathematician, and the rivalries which were prevalent in Paris at that time.     

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