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.     

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).      

Writing tea’s empire

The Almagest of Ptolemy (mid-second century ad) contains eleven dated reports of observations of the positions of planets made during the third century bc in Babylon and Hellenistic Egypt. The present paper investigates the character, purpose, and conventions of the observational programmes from which these reports derive, the channels of their transmission to Ptolemy's time, and the fidelity of Ptolemy's presentation of them. Like the Babylonian observational programme, about which we have considerable knowledge through cuneiform documents, the Greco-Egyptian ones were not directed towards the deduction of mathematical models of celestial motion but appear to have investigated patterns, correlations, and periodicities of phenomena. Ptolemy's immediate sources most likely were not the original series of observational records, but treatises by various astronomers of the intervening four centuries, including Hipparchus. While Ptolemy does not appear to have tampered with the wording of the reports, he faced difficulties and uncertainties in interpreting them; critically, he lacked sufficiently detailed information about the ancient calendars to be able to convert the reported dates accurately into his own chronological framework based on the Egyptian calendar.     

Today’s and tomorrow’s imaging and circulating biomarkers for pulmonary arterial hypertension

The pathobiology of pulmonary arterial hypertension (PAH) involves a remodeling process in distal pulmonary arteries, as well as vasoconstriction and in situ thrombosis, leading to an increase in pulmonary vascular resistance, right heart failure and death. Its etiology may be idiopathic, but PAH is also frequently associated with underlying conditions such as connective tissue diseases. During the past decade, more than welcome novel therapies have been developed and are in development, including those increasingly targeting the remodeling process. These therapeutic options modestly increase the patients' long-term survival, now approaching 60% at 5 years. However, non-invasive tools for confirming PAH diagnosis, and assessing disease severity and response to therapy, are tragically lacking and would help to select the best treatment. After exclusion of other causes of pulmonary hypertension, a final diagnosis still relies on right heart catheterization, an invasive technique which cannot be repeated as often as an optimal follow-up might require. Similarly, other techniques and biomarkers used for assessing disease severity and response to treatment generally lack specificity and have significant limitations. In this review, imaging as well as current and future circulating biomarkers for diagnosis, prognosis, and follow-up are discussed.     

‘Heated’ Debates in Apoptosis

Hippocrates’ assertion that ‘what the lance does not heal, fire will’ underscores the fact that for thousands of years heat has been used to treat a variety of diseases, including cancer. Indeed, spontaneous tumor remission has been observed in patients following feverish infection [1], and expression of activated oncogenes, such as Ras, can render tumor cells sensitive to heat compared with normal cells [2, 3]. In the past, a primary drawback to the use of heat as a clinical therapy was the inability to selectively focus heat to tumors in situ. Of late, however, several approaches have been devised to deliver heat more precisely, including the use of heated nanoparticles, making hyperthermia a more clinically tractable treatment option [4, 5]. Despite these practical advances, the mechanisms responsible for heat shock-induced cell death remain controversial and ill-defined. In this Visions and Reflections we discuss recent findings surrounding the initiation of heat shock-induced apoptosis, and propose future areas of research. Received 17 March 2007; received after revision 25 April 2007; accepted 22 May 2007     

The potter’s wheel: the host’s role in sculpting its microbiota

Animals, ranging from basal metazoans to primates, are host to complex microbial ecosystems; engaged in a symbiotic relationship that is essential for host physiology and homeostasis. Epithelial surfaces vary in the composition of colonizing microbiota as one compares anatomic sites, developmental stages and species origin. Alterations of microbial composition likely contribute to susceptibility to several distinct diseases. The forces that shape the colonizing microbial composition are the focus of much current investigation, and it is evident that there are pressures exerted both by the host and the external environment to mold these ecosystems. The focus of this review is to discuss recent studies that demonstrate the critical importance of host factors in selecting for its microbiome. Greater insight into host–microbiome interactions will be essential for understanding homeostasis at mucosal surfaces, and developing useful interventions when homeostasis is disrupted.     

Adult neurogenesis in Parkinson’s disease

Parkinson’s disease (PD), the second most common neurodegenerative disorder, affects 1–2 % of humans aged 60 years and older. The diagnosis of PD is based on motor symptoms such as bradykinesia, rigidity, tremor, and postural instability associated with the striatal dopaminergic deficit that is linked to neurodegenerative processes in the substantia nigra (SN). In the past, cellular replacement strategies have been evaluated for their potential to alleviate these symptoms. Adult neurogenesis, the generation of new neurons within two proliferative niches in the adult brain, is being intensively studied as one potential mode for cell-based therapies. The subventricular zone provides new neurons for the olfactory bulb functionally contributing to olfaction. The subgranular zone of the hippocampus produces new granule neurons for the dentate gyrus, required for memory formation and proper processing of anxiety provoking stimuli. Recent years have revealed that PD is associated with non-motor symptoms such as hyposmia, anhedonia, lack of novelty seeking behavior, depression, and anxiety that are not directly associated with neurodegenerative processes in the SN. This broad spectrum of non-motor symptoms may partly rely on proper olfactorial processing and hippocampal function. Therefore, it is conceivable that some non-motor deficits in PD are related to defective adult neurogenesis. Accordingly, in animal models and postmortem studies of PD, adult neurogenesis is severely affected, although the exact mechanisms and effects of these changes are not yet fully understood or are under debate due to conflicting results. Here, we review the current concepts related to the dynamic interplay between endogenous cellular plasticity and PD-associated pathology.     

‘Pitfalls’ in immune electron microscopy

Zusammenfassung Es werden die Fehlerquellen diskutiert, die bei der Untersuchung von Antigenen auftreten, wenn zu deren elektronenmikroskopischer Darstellung Antikörper, markiert mit Ferritin, verwendet werden. Durch geeignete Wahl der Fixierung können einige dieser Schwierigkeiten beseitigt werden.     

Crystalline acetates from ‘Croton Resin’

Zusammenfassung Durch Hydrolyse des aus Krotonöl gewonnenen Krotonharzes wurde eine empfindliche, wasserlösliche Polyhydroxyverbindung gewonnen, die teilweise als das bereits bekannte Alkohol-Phorbol abgeschieden werden konnte. Von ihren 4 Hydroxylgruppen scheinen im Krotonharz 2 mit Essigsäure verestert zu sein. Nur eine der Acetylgruppen ist verseifbar. Die dritte Hydroxylgruppe ist mit einer höheren Fettsäure von 10–16 C-Atomen verestert, die vierte scheint frei vorzuliegen. Zwei der Hydroxylgruppen im Verseifungsprodukt sind acetylierbar. Im Grundskelett der Formel C₁₉H₂₆O₆ ist daher eine Sauerstoff-Funktion noch ungeklärt.     

‘Passie’ thinking,a hypothesis

Summary Based on observations and on analogical deductions from the organization of the cerebellum, a hypothesis of a passive process of thinking is described. According to it, we try to program the cerebrum for the handling of an idea, regarding its solution we are, however, exposed passively to the brain.     

‘Cystine monosulfoxide’ and related compounds

Zusammenfassung Die früher als «Cystinmonosulfoxyd» beschriebene Verbindung (Thiosulfinat II) verhält sich wie eine äquimolekulare Mischung von Cystin (I) und dem entsprechenden Thiosulfonat (III). Oxydation von Cystin mit Perameisensäure führt, besonders in Gegenwart von HCl, über die Zwischenprodukte Thiosulfonat (III) und Sulfinsäure (VI) zu Cysteinsäure (VII).     

The coming-to-be of Hansen’s method

This article by Curtis Wilson is an account of the origin of Hansen’s powerful systematic method for finding contributions of higher order perturbations in celestial mechanics. Hansen’s method was developed in the course of improving on Laplace’s treatment of the mutual perturbations of Jupiter and Saturn. This method, an entirely new way of doing celestial mechanics when it first appeared, later made possible the successful treatment of the complicated motions of our moon (see Wilson 2010). In this paper Wilson gives a brief historical introduction followed by an account of relevant technical details of the Laplacian background, an account illustrating technical details in Hansen’s initial development in his Disquisitions of 1829, and a treatment illustrating details contributing to the achievement of Hansen’s more refined development in his Untersuchung of 1831. These details include conditional equations Hansen provides for checking the accuracy of calculations. Wilson also includes a detailed assessment showing the extraordinary improvement in empirical accuracy of Hansen’s treatment over the best earlier treatment of the Jupiter-Saturn  interactions.     

Spectrin and calpain: a ‘target’ and a ‘sniper’ in the pathology of neuronal cells

It is well documented that activation of calpain, a calcium-sensitive cysteine protease, marks the pathology of naturally and experimentally occuring neurodegenerative conditions. Calpain-mediated proteolysis of major membrane-skeletal protein, αII-spectrin, results in the appearance of two unique and highly stable breakdown products, which is an early event in neural cell pathology. This review focuses on spectrin degradation by calpain within neurons induced by diverse conditions, emphasizing a current picture of multi-pattern neuronal death and a recent success in the development of spectrin-based biomarkers. The issue is presented in the context of the major structural and functional properties of the two proteins.Received 7 March 2005; received after revision 22 April 2005; accepted 13 May 2005     

Darier’s disease: a calcium-signaling perspective

Ca²⁺ influx evoked across the plasma membrane upon internal store depletion is essential for a myriad of cellular functions including gene expression, cell proliferation, differentiation and even apoptosis. Darier’s disease (DD), an autosomal dominant inherited disorder of the skin, arising due to mutations in the isoform 2 of the sarco (endo) plasmic reticulum Ca²⁺ ATPase (SERCA2), exemplifies an anomaly of Ca²⁺ signaling disturbances. Owing to loss of function mutations in SERCA2, keratinocytes in DD patients have a reduced pool of endoplasmic reticulum (ER) Ca²⁺. Importantly, the status of ER Ca²⁺ is critical for the activation of a class of plasma membrane Ca²⁺ channels referred to as store operated Ca²⁺ channels (SOCs). The widely expressed transient receptor potential (TRP) family of channels is proposed to be SOCs. In this review we discuss DD from the viewpoint of Ca²⁺ signaling and present a potential role for TRPC1 in the disease pathogenesis. Received 30 August 2007; received after revision 17 October 2007; accepted 6 November 2007     

Aging clock: the watchmaker’s masterpiece

The phenomenon of cellular senescence has been known for almost four decades. Yet, until very recently, the molecular mechanisms that lead to senescence have been poorly understood. However, substantial progress has been made in the last few years toward identifying the pathways executing senescence. This r view focuses on two major advances in this field, the telomere aging clock theory and the cell cycle regulatory mechanisms in senescent cells. These recent studies indicate that cellular senescence is a highly elaborate and active process, which presumably works as an anti-oncogenic mechanism.