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.     

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.     

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.     

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.     

Heterogeneous responses of canine basilar and middle cerebral arteries to serotonin at normal and high CO2 tension

The responses of basilar arteries (BAs) to serotonin were attenuated by high \(P_{CO_2 } \) (86±1 mm Hg) and the pH matched acidotic solution ( \(P_{CO_2 } \) 37±1 mm Hg), whereas the responses of middle cerebral arteries (MCAs) were not. High \(P_{CO_2 } \) decreased the basal tone of both arteries, and the changes in basal tone due to high \(P_{CO_2 } \) were not influenced by 3×10^?7 M imipramine, 10^?5 M pargyline or 10^?4 M aspirin. The responses of BAs to serotonin were attenuated by high \(P_{CO_2 } \) in the presence of imipramine, pargyline and aspirin. The responses of MCAs to serotonin were not influenced by high \(P_{CO_2 } \) in the presence of pargyline and aspirin, but attenuated by high \(P_{CO_2 } \) in the presence of imipramine.     

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.     

Diffusion of d-glucose measured in the cytosol of a single astrocyte

Astrocytes interact with neurons and endothelial cells and may mediate exchange of metabolites between capillaries and nerve terminals. In the present study, we investigated intracellular glucose diffusion in purified astrocytes after local glucose uptake. We used a fluorescence resonance energy transfer (FRET)-based nano sensor to monitor the time dependence of the intracellular glucose concentration at specific positions within the cell. We observed a delay in onset and kinetics in regions away from the glucose uptake compared with the region where we locally super-fused astrocytes with the d-glucose-rich solution. We propose a mathematical model of glucose diffusion in astrocytes. The analysis showed that after gradual uptake of glucose, the locally increased intracellular glucose concentration is rapidly spread throughout the cytosol with an apparent diffusion coefficient (D _app) of (2.38 ± 0.41) × 10^?10 m² s^?1 (at 22–24 °C). Considering that the diffusion coefficient of d-glucose in water is D = 6.7 × 10^?10 m² s^?1 (at 24 °C), D _app determined in astrocytes indicates that the cytosolic tortuosity, which hinders glucose molecules, is approximately three times higher than in aqueous solution. We conclude that the value of D _app for glucose measured in purified rat astrocytes is consistent with the view that cytosolic diffusion may allow glucose and glucose metabolites to traverse from the endothelial cells at the blood–brain barrier to neurons and neighboring astrocytes.     

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.     

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.     

The ether lipid precursor hexadecylglycerol protects against Shiga toxins

Shiga toxin-producing Escherichia coli bacteria cause hemorrhagic colitis and hemolytic uremic syndrome in humans. Currently, only supportive treatment is available for diagnosed patients. We show here that 24-h pretreatment with an ether lipid precursor, the alkylglycerol sn-1-O-hexadecylglycerol (HG), protects HEp-2 cells against Shiga toxin and Shiga toxin 2. Also the endothelial cell lines HMEC-1 and HBMEC are protected against Shiga toxins after HG pretreatment. In contrast, the corresponding acylglycerol, dl-α-palmitin, has no effect on Shiga toxicity. Although HG treatment provides a strong protection (~30 times higher IC₅₀) against Shiga toxin, only a moderate reduction in toxin binding was observed, suggesting that retrograde transport of the toxin from the plasma membrane to the cytosol is perturbed. Furthermore, endocytosis of Shiga toxin and retrograde sorting from endosomes to the Golgi apparatus remain intact, but transport from the Golgi to the endoplasmic reticulum is inhibited by HG treatment. As previously described, HG reduces the total level of all quantified glycosphingolipids to 50–70 % of control, including the Shiga toxin receptor globotriaosylceramide (Gb3), in HEp-2 cells. In accordance with this, we find that interfering with Gb3 biosynthesis by siRNA-mediated knockdown of Gb3 synthase for 24 h causes a similar cytotoxic protection and only a moderate reduction in toxin binding (to 70 % of control cells). Alkylglycerols, including HG, have been administered to humans for investigation of therapeutic roles in disorders where ether lipid biosynthesis is deficient, as well as in cancer therapy. Further studies may reveal if HG can also have a therapeutic potential in Shiga toxin-producing E. coli infections.     

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.     

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.     

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.     

Leibniz’s syncategorematic infinitesimals

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. “If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83).