Cyclotomie et formes quadratiques dans l’œuvre arithmétique d’Augustin-Louis Cauchy (1829–1840)

Augustin-Louis Cauchy publie une majorité de ses recherches arithmétiques entre 1829 et 1840. Celles-ci ne sont pourtant qu’évoquées dans certaines histoires de la théorie des nombres centrées sur les lois de réciprocité ou sur la théorie des nombres algébriques. Elles y sont décrites comme contenant quelques résultats similaires à ceux de Gauss, Jacobi ou Dirichlet mais de manière incomplète et désordonnée. L’objectif de cet article est de présenter une analyse des textes arithmétiques de Cauchy publiés entre 1829 et 1840 pour montrer qu’ils contiennent au contraire un ensemble cohérent de résultats en lien avec les formes quadratiques $4p^{\mu }=x^2+ny^2$ , où $p$ est un nombre premier et $n$ un diviseur de $p-1$ . Nous discuterons également la forme particulière de ce corpus et la stratégie utilisée pour retrouver les lignes directrices du travail de Cauchy. Augustin-Louis Cauchy published most of his arithmetical research between 1829 and 1840. These are however only mentioned in some number theory history centered on reciprocity laws or on theory of algebraic numbers. They are described as containing some results similar to those of Gauss, Jacobi and Dirichlet but in a incomplete and disorganized way. The objective of this paper is to present an analysis of Cauchy’s arithmetical texts published between 1829 and 1840 to show that they contain a rather consistent set of results related to quadratic forms $4p^{\mu } = x ^2 + ny ^2 $ , where $p$ is a prime and $n$ a divisor of $ p-1 $ . We will also discuss the particular form of this body of texts and the strategy we used to find the guidelines of the work of Cauchy.     

Investigations of the coordinates in Ptolemy’s Geographike Hyphegesis Book 8

In Book 8 of his Geographike Hyphegesis Ptolemy gives coordinates for ca. 360 so-called noteworthy cities. These coordinates are the time difference to Alexandria, the length of the longest day, and partly the ecliptic distance from the summer solstice. The supposable original conversions between the coordinates in Book 8 and the geographical coordinates in the location catalogue of Books 2–7 including the underlying parameters and tabulations are here reconstructed. The results document the differences between the ${\Omega}$ - and ${\Xi}$ -recension. The known difference in the longitude of Alexandria underlying the conversion of the longitudes is examined more closely. For the ecliptic distances from the summer solstice of the ${\Omega}$ -recension, it is revealed that they were originally computed by means of a so far undiscovered approximate, linear conversion. Further it is shown that the lengths of the longest day could be based on a linear interpolation of the data in the Mathematike Syntaxis 2.6.     

Isaac Newton’s ‘Of Quadrature by Ordinates’

In Of Quadrature by Ordinates (1695), Isaac Newton tried two methods for obtaining the Newton–Cotes formulae. The first method is extrapolation and the second one is the method of undetermined coefficients using the quadrature of monomials. The first method provides $n$ -ordinate Newton–Cotes formulae only for cases in which $n=3,4$ and 5. However this method provides another important formulae if the ratios of errors are corrected. It is proved that the second method is correct and provides the Newton–Cotes formulae. Present significance of each of the methods is given.     

On Jacobi’s transformation theory of elliptic functions

The main interpretative challenge set by the Fundamenta Nova Theoriae Functionum Ellipticarum lies in Jacobi’s transformation theory upon which the entire theoretical edifice of the treatise depends. Unfortunately, Jacobi did not convey any indication of how he attained his general formulae for rational transformations of elliptic functions. He limited himself to providing a posteriori verification of the validity of his claims. The aim of this paper is precisely to describe the heuristic path by which in 1827 Jacobi succeeded in finding these transformation formulae. The proposed historical reconstruction will hopefully shed new light upon the emergence in Jacobi’s work of the inversion process of elliptic integrals of the first kind and thus of the elliptic function sinam \(u\) itself.     

Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929)

We analyze the paper “The wave mechanics of $\alpha $ -ray tracks” Mott (Proc R Soc Lond A 126:79–84, 1929), published in 1929 by N. F. Mott. In particular, we discuss the theoretical context in which the paper appeared and give a detailed account of the approach used by the author and the main result attained. Moreover, we comment on the relevance of the work not only as far as foundations of Quantum Mechanics are concerned but also as the earliest pioneering contribution in decoherence theory.     

Two problems in Aristarchus’s treatise on the sizes and distances of the sun and moon

The book of Aristarchus of Samos, On the distances and sizes of the sun and moon, is one of the few pre-Ptolemaic astronomical works that have come down to us in complete or nearly complete form. The simplicity and cleverness of the basic ideas behind the calculations are often obscured in the reading of the treatise by the complexity of the calculations and reasoning. Part of the complexity could be explained by the lack of trigonometry and part by the fact that Aristarchus appears unwilling to make some simplifications that could be simply taken for granted. But an important part of the complexities is due to some unnecessary inconsistencies, as recently discovered by Berggren and Sidoli (Arch Hist Exact Sci 61:213–254, 2007). In the first part of this paper, I will try to show that some of these inconsistencies are just apparent. But the complexity of the calculations and reasoning is not the only reason that could disturb a reader of the treatise. The great inaccuracy—even for the measurement methods and instruments available at those times—of one of the three input values of the treatise is really astonishing. In the sixth and last hypothesis, Aristarchus states that the moon’s apparent size is equal to 2 $^{\circ }$ , while the correct value is one-fourth of that. Some attempts have been made in order to explain such a big value, but all of them have problems. In the second part of this paper, I will propose a new speculative but plausible explanation of the origin of this value.     

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.     

Wnt–Notch signalling crosstalk in development and disease

The Notch and Wnt pathways are two of only a handful of highly conserved signalling pathways that control cell-fate decisions during animal development (Pires-daSilva and Sommer in Nat Rev Genet 4: 39–49, 2003). These two pathways are required together to regulate many aspects of metazoan development, ranging from germ layer patterning in sea urchins (Peter and Davidson in Nature 474: 635–639, 2011) to the formation and patterning of the fly wing (Axelrod et al in Science 271:1826–1832, 1996; Micchelli et al in Development 124:1485–1495, 1997; Rulifson et al in Nature 384:72–74, 1996), the spacing of the ciliated cells in the epidermis of frog embryos (Collu et al in Development 139:4405–4415, 2012) and the maintenance and turnover of the skin, gut lining and mammary gland in mammals (Clayton et al in Nature 446:185–189, 2007; Clevers in Cell 154:274–284, 2013; Doupe et al in Dev Cell 18:317–323, 2010; Lim et al in Science 342:1226–1230, 2013; Lowell et al in Curr Biol 10:491–500, 2000; van et al in Nature 435:959–963, 2005; Yin et al in Nat Methods 11:106–112, 2013). In addition, many diseases, including several cancers, are caused by aberrant signalling through the two pathways (Bolós et al in Endocr Rev 28: 339–363, 2007; Clevers in Cell 127: 469–480, 2006). In this review, we will outline the two signalling pathways, describe the different points of interaction between them, and cover how these interactions influence development and disease.     

The development of the Laplace transform, 1737–1937

This paper, the first of two, follows the development of theLaplace Transform from its earliest beginnings withEuler, usually dated at 1737, to the year 1880, whenSpitzer was its major, if himself relatively minor, protagonist. The coverage aims at completeness, and shows the state which the technique reached in the hands of its greatest exponent to that time,Petzval. A sequel will trace the development of the modern theory from its beginnings withPoincaré to its present form, due toDoetsch.     

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.     

Pre-Euclidean geometry and Aeginetan coin design: some further remarks

Some ancient Greek coins from the island state of Aegina depict peculiar geometric designs. Hitherto they have been interpreted as anticipations of some Euclidean propositions. But this paper proposes geometrical constructions which establish connections to pre-Euclidean treatments of incommensurability. The earlier Aeginetan coin design from about 500 bc onwards appears as an attempt not only to deal with incommensurability but also to conceal it. It might be related to Plato’s dialogue Timaeus. The newer design from 404 bc onwards reveals incommensurability, namely in the context of ‘doubling the square’. It thereby covers the same topic but a different geometry as passages in Plato’s dialogue Meno (385 bc). This coin design incorporates important elements of ancient Greek geometrical analysis of the fifth century bc like the gnomon, Hippocrates’ squaring of the lunule (ca. 430 bc), and a geometrical version of monetary equivalence. Through this venue, the design’s conceptual lineage might be traced as far back as Heraclitus’ cosmology of about 500 bc.     

Christian Huygens’ Lost and Forgotten Pamphlet of his Pendulum Invention

Until recently it was believed that Christian Huygens’ earliest publication of his pendulum invention was Horologium of 1658. He published the more famous general treatise, Horologium Oscillatorium, fifteen years later in 1673. Two years ago, an article1 ¹Whitestone, Sebastian, ‘The Identification and Attribution of Christiaan Huygens’ First Pendulum Clock', Antiquarian Horology, December (2008), 201–222. suggesting an unknown collaboration in developing the clock pendulum between Huygens and the Paris clockmaker Isaac Thuret, presented the evidence of Benjamin Martin, an 18th century educationalist and retailer of scientific material. Martin described a Huygens publication of 1657 and reproduced the illustration it contained. This illustration shows a different clock from the one drawn in Horologium and different also from those previously considered as Huygens’ earliest surviving examples. However, the illustration is similar to part of a plate in Horologium Oscillatorium and this similarity caused one historian to cast doubt on the existence of the 1657 publication.2 ²Plomp, R., ‘Letter', Antiquarian Horology, December (2009), 714–17. See also author's reply, ibid, 717–19. This article, with information presented for the first time, seeks to prove the existence of that work and thereby establish it in the canon of Huygens’ writings while re-examining the invention in the light that it casts.     

eIF2A,an initiator tRNA carrier refractory to eIF2α kinases,functions synergistically with eIF5B

The initiator tRNA (Met-tRNA _i ^Met ) at the P site of the small ribosomal subunit plays an important role in the recognition of an mRNA start codon. In bacteria, the initiator tRNA carrier, IF2, facilitates the positioning of Met-tRNA _i ^Met on the small ribosomal subunit. Eukarya contain the Met-tRNA _i ^Met carrier, eIF2 (unrelated to IF2), whose carrier activity is inhibited under stress conditions by the phosphorylation of its α-subunit by stress-activated eIF2α kinases. The stress-resistant initiator tRNA carrier, eIF2A, was recently uncovered and shown to load Met-tRNA _i ^Met on the 40S ribosomal subunit associated with a stress-resistant mRNA under stress conditions. Here, we report that eIF2A interacts and functionally cooperates with eIF5B (a homolog of IF2), and we describe the functional domains of eIF2A that are required for its binding of Met-tRNA _i ^Met , eIF5B, and a stress-resistant mRNA. The results indicate that the eukaryotic eIF5B–eIF2A complex functionally mimics the bacterial IF2 containing ribosome-, GTP-, and initiator tRNA-binding domains in a single polypeptide.     

Fructose transport-deficient Staphylococcus aureus reveals important role of epithelial glucose transporters in limiting sugar-driven bacterial growth in airway surface liquid

Hyperglycaemia as a result of diabetes mellitus or acute illness is associated with increased susceptibility to respiratory infection with Staphylococcus aureus. Hyperglycaemia increases the concentration of glucose in airway surface liquid (ASL) and promotes the growth of S. aureus in vitro and in vivo. Whether elevation of other sugars in the blood, such as fructose, also results in increased concentrations in ASL is unknown and whether sugars in ASL are directly utilised by S. aureus for growth has not been investigated. We obtained mutant S. aureus JE2 strains with transposon disrupted sugar transport genes. NE768(fruA) exhibited restricted growth in 10 mM fructose. In H441 airway epithelial-bacterial co-culture, elevation of basolateral sugar concentration (5–20 mM) increased the apical growth of JE2. However, sugar-induced growth of NE768(fruA) was significantly less when basolateral fructose rather than glucose was elevated. This is the first experimental evidence to show that S. aureus directly utilises sugars present in the ASL for growth. Interestingly, JE2 growth was promoted less by glucose than fructose. Net transepithelial flux of d-glucose was lower than d-fructose. However, uptake of d-glucose was higher than d-fructose across both apical and basolateral membranes consistent with the presence of GLUT1/10 in the airway epithelium. Therefore, we propose that the preferential uptake of glucose (compared to fructose) limits its accumulation in ASL. Pre-treatment with metformin increased transepithelial resistance and reduced the sugar-dependent growth of S. aureus. Thus, epithelial paracellular permeability and glucose transport mechanisms are vital to maintain low glucose concentration in ASL and limit bacterial nutrient sources as a defence against infection.