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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Marking out a disciplinary common ground: The role of chemical pedagogy in establishing the doctrine of affinity at the heart of British chemistry

This paper presents a case study that contributes to the current debate among historians of chemistry concerning the role and influence of pedagogy in science. Recently, Bernadette Bensaude-Vincent and her colleagues concluded that in nineteenth-century France, ‘textbooks played an important role in discipline building and in creating theories’.1 ¹A. Garcia-Belmar, B. Bensaude-Vincent and J.R. Bertomeu-Sánchez, ‘The Power of Didactic Writings: French Chemistry Textbooks of the Nineteenth Century’, in Pedagogy and the Practice of Science: Historical and Contemporary Perspectives, edited by D. Kaiser (Cambridge, MA, 2005), 243. Developing this idea further, this paper examines the dissemination of knowledge through face-to-face chemical lectures, showing that the influence of pedagogical strategy on theoretical content of the science is far from negligible. The pedagogy of William Cullen was essentially responsible for the prevalence of the doctrine of affinity in British chemistry from the 1760s onwards. Cullen used his affinity theory as a pedagogical tool that to a large extent defined his discipline, and the pedagogical pyramid that he headed similarly ensured that the doctrine would remain at the heart of British chemistry. From a pedagogical tool, the doctrine of affinity was transformed over time into a chemical tool, offering British chemists a disciplinary common ground that both set and reinforced the boundaries to their discipline.     

Delineating multiple functions of VEGF-A in the adult brain

Vascular endothelial growth factor-A (abbreviated throughout this review as VEGF) is mostly known for its angiogenic activity, for its activity as a vascular permeability factor, and for its vascular survival activity [1]. There is a growing body of evidence, however, that VEGF fulfills additional less ‘traditional’ functions in multiple organs, both during development, as well as homeostatic functions in fully developed organs. This review focuses on the multiple roles of VEGF in the adult brain and is less concerned with the roles played by VEGF during brain development, functions described elsewhere in this review series. Most functions of VEGF that are essential for proper brain development are, in fact, dispensable in the adult brain as was clearly demonstrated using a conditional brain-specific VEGF loss-of-function (LOF) approach. Thus, in contrast to VEGF LOF in the developing brain, a process which is detrimental for the growth and survival of blood vessels and leads to massive neuronal apoptosis [2–4], continued signaling by VEGF in the mature brain is no longer required for maintaining already established cerebral vasculature and its inhibition does not cause appreciable vessel regression, hypoxia or apoptosis [4–7]. Yet, VEGF continues to be expressed in the adult brain in a constitutive manner. Moreover, VEGF is expressed in the adult brain in a region-specific manner and in distinctive spatial patterns incompatible with an angiogenic role (see below), strongly suggesting angiogenesis-independent and possibly also perfusion-independent functions. Here we review current knowledge on some of these ‘non-traditional’, often unexpected homeostatic VEGF functions, including those unrelated to its effects on the brain vasculature. These effects could be mediated directly (on non-vascular cells expressing cognate VEGF receptors) or indirectly (via the endothelium). Experimental approaches aimed at distinguishing between these possibilities for each particular VEGF function will be described. This review is only concerned with homeostatic functions of VEGF in the normal, non-injured brain. The reader is referred elsewhere in this series for a review on VEGF actions in response to various forms of brain injury and/or brain pathology.     

Mechanisms regulating immune surveillance of cellular stress in cancer

The purpose of this review is to explore immune-mediated mechanisms of stress surveillance in cancer, with particular emphasis on the idea that all cancers have classical hallmarks (Hanahan and Weinberg in Cell 100:57–70, 67; Cell 144:646–674, 68) that could be interrelated. We postulate that hallmarks of cancer associated with cellular stress pathways (Luo et al. in Cell 136:823–837, 101) including oxidative stress, proteotoxic stress, mitotic stress, DNA damage, and metabolic stress could define and modulate the inflammatory component of cancer. As such, the overarching goal of this review is to define the types of cellular stress that cancer cells undergo, and then to explore mechanisms by which immune cells recognize, respond to, and are affected by each stress response.