The Mathematics of the Area Law: Kepler's Successful Proof in Epitome Astronomiae Copernicanae (1621)

Epitome V (1621), and consisted of matching an element of area to an element of time, where each was mathematically determined. His treatment of the area depended solely on the geometry of Euclid's Elements, involving only straight-line and circle propositions – so we have to account for his deliberate avoidance of the sophisticated conic-geometry associated with Apollonius. We show also how his proof could have been made watertight according to modern standards, using methods that lay entirely within his power. The greatest innovation, however, occurred in Kepler's fresh formulation of the measure of time. We trace this concept in relation to early astronomy and conclude that Kepler's treatment unexpectedly entailed the assumption that time varied nonuniformly; meanwhile, a geometrical measure provided the independent variable. Even more surprisingly, this approach turns out to be entirely sound when assessed in present-day terms. Kepler himself attributed the cause of the motion of a single planet around the Sun to a set of `physical' suppositions which represented his religious as well as his Copernican convictions; and we have pared to a minimum – down to four – the number he actually required to achieve this. In the Appendix we use modern mathematics to emphasize the simplicity, both geometrical and kinematical, that objectively characterizes the Sun-focused ellipse as an orbit. Meanwhile we highlight the subjective simplicity of Kepler's own techniques (most of them extremely traditional, some newly created). These two approaches complement each other to account for his success. (Received April 19, 2002) Published online April 2, 2003 Communicated by N. M. Swerdlow     

Einstein’s quantum theory of the monatomic ideal gas: non-statistical arguments for a new statistics

In this article, we analyze the third of three papers, in which Einstein presented his quantum theory of the ideal gas of 1924–1925. Although it failed to attract the attention of Einstein’s contemporaries and although also today very few commentators refer to it, we argue for its significance in the context of Einstein’s quantum researches. It contains an attempt to extend and exhaust the characterization of the monatomic ideal gas without appealing to combinatorics. Its ambiguities illustrate Einstein’s confusion with his initial success in extending Bose’s results and in realizing the consequences of what later came to be called Bose–Einstein statistics. We discuss Einstein’s motivation for writing a non-combinatorial paper, partly in response to criticism by his friend Ehrenfest, and we paraphrase its content. Its arguments are based on Einstein’s belief in the complete analogy between the thermodynamics of light quanta and of material particles and invoke considerations of adiabatic transformations as well as of dimensional analysis. These techniques were well known to Einstein from earlier work on Wien’s displacement law, Planck’s radiation theory and the specific heat of solids. We also investigate the possible role of Ehrenfest in the gestation of the theory.     

GSK-3β is essential for physiological electric field-directed Golgi polarization and optimal electrotaxis

Endogenous electrical fields (EFs) at corneal and skin wounds send a powerful signal that directs cell migration during wound healing. This signal therefore may serve as a fundamental regulator directing cell polarization and migration. Very little is known of the intracellular and molecular mechanisms that mediate EF-induced cell polarization and migration. Here, we report that Chinese hamster ovary (CHO) cells show robust directional polarization and migration in a physiological EF (0.3–1 V/cm) in both dissociated cell culture and monolayer culture. An EF of 0.6 V/cm completely abolished cell migration into wounds in monolayer culture. An EF of higher strength (≥1 V/cm) is an overriding guidance cue for cell migration. Application of EF induced quick phosphorylation of glycogen synthase kinase 3β (GSK-3β) which reached a peak as early as 3 min in an EF. Inhibition of protein kinase C (PKC) significantly reduced EF-induced directedness of cell migration initially (in 1–2 h). Inhibition of GSK-3β completely abolished EF-induced GA polarization and significantly inhibited the directional cell migration, but at a later time (2–3 h in an EF). Those results suggest that GSK-3β is essential for physiological EF-induced Golgi apparatus (GA) polarization and optimal electrotactic cell migration.     

The platypus in Edinburgh: Robert Jameson,Robert Knox and the place of the Ornithorhynchus in nature, 1821–24

The duck-billed platypus, or Ornithorhynchus, was the subject of an intense debate among natural historians in the late eighteenth and early nineteenth centuries. Its paradoxical mixture of mammalian, avian and reptilian characteristics made it something of a taxonomic conundrum. In the early 1820s Robert Jameson (1774–1854), the professor of natural history at the University of Edinburgh and the curator of the University's natural history museum, was able to acquire three valuable specimens of this species. He passed one of these on to the anatomist Robert Knox (1791–1862), who dissected the animal and presented his results in a series of papers to the Wernerian Natural History Society, which later published them in its Memoirs. This paper takes Jameson's platypus as a case study on how natural history specimens were used to create and contest knowledge of the natural world in the early nineteenth century, at a time when interpretations of the relationships between animal taxa were in a state of flux. It shows how Jameson used his possession of this interesting specimen to provide a valuable opportunity for his protégé Knox while also helping to consolidate his own position as a key figure in early nineteenth-century natural history.     

The rise of geology and its influence on contemporary thought

Sous la Révolution, la conjoncture politique et militaire et l'échec de la poudre de muriate (chlorate) devaient conduire la France à privilégier la rationalisation des procédés de fabrication pour tenter d'accroître la portée des armes. En 1796, la poudre ronde de J. P. Champy (1744–1816), successeur de Lavoisier à la tête du service des Poudres et salpêtres, parut approcher de la poudre idéale: pour un prix de revient réduit, elle offrait une puissance suffisante et une sécurité optimale de fabrication et d'emploi. Malgré l'avis des experts scientifiques et militaires, sa production cessa pourtant à la suite d'un blocage politico-administratif. A la fin de l'Empire, J. S. Champy (1778–1845), perfectionna le procédé de son père pour en faire le premier système de production mécanisé, dont l'introduction devait modifier l'organisation du travail et la configuration des poudreries créées sous la Restauration. Le succès du système conduisit paradoxalement à son échec après une décennie de production qui resta toujours expérimentale: la poudre ronde remettait en cause les techniques métallurgiques en faisant éclater les canons. La publication inédite du premier rapport d'expertise de ce procédé (1813) fournit l'occasion de suivre les origines et l'évolution d'un système qui annonce la mécanisation de l'industrie poudrière au XIXe siècle, tout en se présentant clairement comme l'aboutissement des procédés révolutionnaires de l'an II. Elle permet aussi de s'interroger sur les critères et le statut de l'expertise, au moment même où l'analyse chimique de la poudre commence avec Proust, Gay-Lussac et Brianchon.     

Euler’s beta integral in Pietro Mengoli’s works

Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler in 1730, when he was trying to find the general term for the factorial function by means of an algebraic expression. Nevertheless, 70 years before, Pietro Mengoli (1626–1686) had computed such integrals for natural and half-integer exponents in his Geometriae Speciosae Elementa (1659) and Circolo(1672) and displayed the results in triangular tables. In particular, his new arithmetic–algebraic method allowed him to compute the quadrature of the circle. The aim of this article is to show how Mengoli calculated the values of these integrals as well as how he analysed the relation between these values and the exponents inside the integrals. This analysis provides new insights into Mengoli’s view of his algorithmic computation of quadratures.     

The current structural and functional understanding of APOBEC deaminases

The apolipoprotein B mRNA-editing enzyme catalytic polypeptide (APOBEC) family of cytidine deaminases has emerged as an intensively studied field as a result of their important biological functions. These enzymes are involved in lipid metabolism, antibody diversification, and the inhibition of retrotransposons, retroviruses, and some DNA viruses. The APOBEC proteins function in these roles by deaminating single-stranded (ss) DNA or RNA. There are two high-resolution crystal structures available for the APOBEC family, Apo2 and the C-terminal catalytic domain (CD2) of Apo3G or Apo3G-CD2 [Holden et al. (Nature 456:121–124, 2008); Prochnow et al. (Nature 445:447–451, 2007)]. Additionally, the structure of Apo3G-CD2 has also been determined using NMR [Chen et al. (Nature 452:116–119, 2008); Furukawa et al. (EMBO J 28:440–451, 2009); Harjes et al. (J Mol Biol, 2009)]. A detailed structural analysis of the APOBEC proteins and a comparison to other zinc-coordinating deaminases can facilitate our understanding of how APOBEC proteins bind nucleic acids, recognize substrates, and form oligomers. Here, we review the recent development of structural and functional studies that apply to Apo3G as well as the APOBEC deaminase family.     

The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin

The aim of this paper is to provide a comprehensive exposition of the early contributions to the so-called Campbell, Baker, Hausdorff, Dynkin Theorem during the years 1890–1950. Related works by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff, and Dynkin will be investigated and compared. For a full recovery of the original sources, many mathematical details will also be furnished. In particular, we rediscover and comment on a series of five notable papers by Pascal (Lomb Ist Rend, 1901–1902), which nowadays are almost forgotten.     

Newton's Propositions on Comets: Steps in Transition, 1681–84

Isaac Newton's closest approach to a system of the world in the critical period 1681–84 is provided in a set of untitled propositions concerning comets. They drastically revise his position maintained against Flamsteed in 1681 and may signal his adoption of a single comet solution for the appearances of 1680/1. Points of agreement and difference with the key pre-Principia texts of 1684–85 are analysed. He shows substantial control of the phenomena of tails which change very little in mechanical detail throughout his subsequent work. An emerging theory of gravitation brings planets, their satellites, and comets under the same laws of motion, yet retains a celestial vortex and includes a singular proposition in lieu of the usual formulation of Keplers area law. The analysis raises questions on a number of issues of recent Newtonian scholarship ranging from his achievement following his correspondence with Robert Hooke in 1679 to his veneration of the wisdom of the ancients. (Received September 7, 1999)     

Probability in 1919/20: the von Mises-Pólya-Controversy

The correspondence between Richard von Mises and George Pólya of 1919/20 contains reflections on two well-known articles by von Mises on the foundations of probability in the Mathematische Zeitschrift of 1919, and one paper from the Physikalische Zeitschrift of 1918. The topics touched on in the correspondence are: the proof of the central limit theorem of probability theory, von Mises' notion of randomness, and a statistical criterion for integer-valuedness of physical data. The investigation will hint at both the fruitfulness and the limits of several of von Mises' notions such as ``collective', ``distribution' and ``complex adjuncts' (characteristic functions) for further developments in probability theory and in ``directional statistics'. By pointing to the selectiveness of Pólya's criticism, the historical analysis shows the differing expectations of the two men with respect to the further development of the theory of probability and its applications. The paper thus gives a glimpse of the provisional state of the theory around 1920, before others such as P. Lévy (1886–1971) and A. N. Kolmogorov (1903–1987) stepped in and created a new paradigm for probability theory.     

Aristote et l'Arc-en-Ciel: Enjeux Philosophiques et Etude Scientifique

Résumé  L'objet de cet article est d'exposer et d'analyser l'explication proposée par Aristote pour l'arc-en-ciel dans les Météorologiques, livre III, ch. 2–5. Cette explication présente d'abord un intérêt au regard de l'histoire des sciences exactes, notamment parce qu'Aristote articule ici très concrètement une approche physique et une approche géométrique de ces phénomènes. Elle présente ensuite un intérêt et une énigme pour l'exégèse aristotélicienne, dans la mesure où ce texte des Météorologiques adopte l'idée de rayons visuels, plus exactement d'une émission hors de l'œil de quelque chose (laissée indéterminée par Aristote) rendant possible la vision, alors qu'Aristote avait longuement et rigoureusement combattu toute idée d'une telle émission dans deux traités précédents (De l'ame, et De la sensation et des sensibles). Elle présente enfin un intérêt philosophique dans la mesure où cette contradiction entre les Météorologiques et ces deux traités antérieurs trouve sa signification dans une exigence portant sur le statut ontologique de l'image spéculaire, exigence que l'on ne peut séparer de la visée scientifique qui guide Aristote. Manuscrit re?u le 31 juillet 2001     

From ultrasonic to frequency standards: Walter Cady’s discovery of the sharp resonance of crystals

In 1918–1919 Walter G. Cady was the first to recognize the significant electrical consequences of the fact that piezoelectric crystals resonate at very sharp, precise and stable frequencies. Cady was also the first to suggest the employment of these properties, first as frequency standards and then to control frequencies of electric circuits—an essential component in electronic technology. Cady’s discovery originated in the course of research on piezoelectric ultrasonic devices for submarine detection (sonar) during World War I. However, for the discovery Cady had to change his research programme to crystal resonance. This change followed Cady’s experimental findings and the scientific curiosity that they raised, and was helped by the termination of the war. Cady’s transition was also a move from “applied” research, aimed at improving a specific technology, to “pure” research lacking a clear practical aim. This article examines how Cady reached the discovery and his early ideas for its use. It shows that the discovery was not an instantaneous but a gradual achievement. It further suggests that disinterested “scientific” research (rather than “engineering” research) was needed in this process, while research aimed at design was required for the subsequent development of technological devices. I am very grateful to Chris McGahey for providing me with his research notes taken from Walter Cady’s diaries kept by the Rhode Island Historical Society, henceforth Diaries. I would like to thank Aharon (Arkee) Eviatar for linguistic comments, Ido Yavetz for our helpful discussion and Jed Buchwald for his thoughtful comments and editorial work. I thank the Lemelson Center in the National Museum for American History for a grant that enabled me to study Walter Guyton Cady Papers, 1903–1974, Archives Center, National Museum of American History (henceforth, ACNMAH) and the staff of the center, especially Alison Oswald, for their help. The following abbreviations are used: NB—Cady’s research notebooks kept at ACNMAH, AIP - Niels Bohr Library, American Institute of Physics, Cady’s dossier.     

Early nautical charts

Christian Wilhelm Blomstrand, Professor of Chemistry and Mineralogy at Lund University from 1862 to 1895, was one of the important chemists of the second half of the nineteenth century. His theoretical ideas and experimental accomplishments contributed to advances in several branches of chemistry. Living in Sweden during a transitional period between the older and newer chemistry and being a scientific as well as a political conservative, Blomstrand sought to reconcile Berzelius's dualistic theory with the unitary and type theories. He was opposed to Kekulé's dogma of constant valency, and he strove to establish a sound and complete theory of variable valency. This article briefly outlines Blomstrand's life and considers his best known book, Die Chemie der Jetztzeit (1869), as well as his work on mineralogy, inorganic chemistry (the earth acid elements, heteropoly acids, platinum complexes), and his theoretical views on valency, diazo compounds, and metal-ammines. His so-called ‘chain theory’, as developed and modified by his fellow Scandinavian chemist and close friend, Sophus Mads Jørgensen, was for more than three decades the most popular and successful of the numerous attempts to explain the constitution, properties, and reactions of coordination compounds.     

Oxidative stress and hypoxia-like injury cause Alzheimer-type molecular abnormalities in central nervous system neurons

Neuronal loss and neuritic/cytoskeletal lesions (synaptic disconnection and proliferation of dystrophic neurites) represent major dementia-associated abnormalities in Alzheimer’s disease (AD). This study examined the role of oxidative stress as a factor contributing to both the cell death and neuritic degeneration cascades in AD. Primary neuron cultures were treated with H₂O₂ (9–90 μM) or desferrioxamine (2–25 μM) for 24 h and then analyzed for viability, mitochondrial mass, mitochondrial function, and pro-apoptosis and sprouting gene expression. H₂O₂ treatment causes free-radical injury and desferrioxamine causes hypoxia-type injury without free radical generation. The H₂O₂-treated cells exhibited sustained viability but neurite retraction, impaired mitochondrial function, increased levels of the pro-apoptosis gene product CD95/Fas, reduced expression of N2J1-immunoreactive neuronal thread protein and synaptophysin, and reduced distribution of mitochondria in neuritic processes. Desferrioxamine treatment resulted in dose-dependent neuronal loss associated with impaired mitochondrial function, proliferation of neurites, and reduced expression of GAP-43, which has a role in path-finding during neurite outgrowth. The results suggest that oxidative stress can cause neurodegeneration associated with enhanced susceptibility to apoptosis due to activation of pro-apoptosis genes, neurite retraction (synaptic disconnection), and impaired transport of mitochondria to cell processes where they are likely required for synaptic function. In contrast, hypoxia-type injury causes neuronal loss with proliferation of neurites (sprouting), impaired mitochondrial function, and reduced expression of molecules required to form and maintain synaptic connections. Since similar abnormalities occur in AD, both oxidative stress and hypoxic injury can contribute to AD neurodegeneration. Received 24 May 2000; received after revision 7 July 2000; accepted 27 July 2000     

Continued fractions and the origins of the Perron–Frobenius theorem

The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory, numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s Theorem. This paper is primarily concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains. When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises and V.I. Romanovsky.     

A. M. Mayer's experiments with floating magnets and their use in the atomic theories of matter

In the years 1878 and 1879 the American physicist Alfred Marshall Mayer (1836–1897) published his experiments with floating magnets as a didactic illustration of molecular actions and forms. A number of physicists made use of this analogy of molecular structure. For William Thomson they were a mechanical illustration of the kinetic equilibrium of groups of columnar vortices revolving in circles round their common centre of gravity (1878). A number of modifications of Mayer's experiments were described, which gave configurations which were more or less analogous to Mayer's arrangements. It was Joseph John Thomson who, in publications between 1897 and 1907, used Mayer's results to obtain a good deal of insight into the general laws which govern the configuration of the electrons in his atomic model. This article is mainly concerned with Mayer's experiments with floating magnets and their use by a number of physicists. Through his experiments Mayer made a significant, although small, contribution to the theory of atomic structure.     

Synphilin-1 inhibits alpha-synuclein degradation by the proteasome

Intracellular deposits of aggregated alpha-synuclein are a hallmark of Parkinson’s disease. Protein–protein interactions are critical in the regulation of cell proteostasis. Synphilin-1 interacts both in vitro and in vivo with alpha-synuclein promoting its aggregation. We report here that synphilin-1 specifically inhibits the degradation of alpha-synuclein wild-type and its missense mutants by the 20S proteasome due at least in part by the interaction of the ankyrin and coiled-coil domains of synphilin-1 (amino acids 331–555) with the N-terminal region (amino acids 1–60) of alpha-synuclein. Co-expression of synphilin-1 and alpha-synuclein wild-type in HeLa and N2A cells produces a specific increase in the half-life of alpha-synuclein, as degradation of unstable fluorescent reporters is not affected. Synphilin-1 inhibition can be relieved by co-expression of Siah-1 that targets synphilin-1 to degradation. Synphilin-1 inhibition of the proteasomal pathway of degradation of alpha-synuclein may help to understand the pathophysiological changes occurring in PD and other synucleinopathies.