Heaviside's operational calculus and the attempts to rigorise it

At the end of the 19^th century Oliver Heaviside developed a formal calculus of differential operators in order to solve various physical problems. The pure mathematicians of his time would not deal with this unrigorous theory, but in the 20^th century several attempts were made to rigorise Heaviside's operational calculus. These attempts can be grouped in two classes. The one leading to an explanation of the operational calculus in terms of integral transformations (Bromwich, Carson, Vander Pol, Doetsch) and the other leading to an abstract algebraic formulation (Lévy, Mikusiski). Also Schwartz's creation of the theory of distributions was very much inspired by problems in the operational calculus.     

The early application of the calculus to the inverse square force problem

The translation of Newton’s geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which was a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem.     

J. L. Lagrange's changing approach to the foundations of the calculus of variations

Archive for History of Exact Sciences -     

Tetracalcium hydrogen triphosphate trihydrate,a constituent of dental calculus

Zusammenfassung Die überwiegenden Bestandteile des Zahnsteins sind Apatit, magnesiumhaltiger Whitlockit mit rund 7 Atomprozent Magnesium und (nur in der Mandibularregion) Brushit. Als untergeordneten Bestandteil haben wir in 48 von 124 untersuchten Zahnsteinproben auch Tetracalcium-hydrogen-triphosphat-trihydrat gefunden.      

Henry More and the development of absolute time

This paper explores the nature, development and influence of the first English account of absolute time, put forward in the mid-seventeenth century by the ‘Cambridge Platonist’ Henry More. Against claims in the literature that More does not have an account of time, this paper sets out More's evolving account and shows that it reveals the lasting influence of Plotinus. Further, this paper argues that More developed his views on time in response to his adoption of Descartes' vortex cosmology and cosmogony, providing new evidence of More's wider project to absorb Cartesian natural philosophy into his Platonic metaphysics. Finally, this paper argues that More should be added to the list of sources that later English thinkers – including Newton and Samuel Clarke – drew on in constructing their absolute accounts of time.     

The timber famine and the development of technology

In March 1912 A. N. Whitehead wrote a letter which sheds new and important light on his own view of his mathematical goals. In this article I publish the letter for the first time and relate its contents not only to his mathematical career but also to his scientific, philosophical and educational interests.     

Soluble factors and the development of rod photoreceptors

Photoreceptors are the most abundant cell type in the vertebrate neural retina. Like the other retinal neurons and the Müller glia, they arise from a population of precursor cells that are multipotent and intrinsic to the retina. Approximately 10 years ago, several studies demonstrated that retinal precursor cells (RPCs) are competent to respond to environmental factors that promote cell type determination and differentiation. Since those studies, significant effort has been directed at identifying the molecular nature of these environmental signals and understanding the precise mechanisms they employ to drive RPCs towards the different retinal fates. In this review, we describe the recent progress toward understanding how environmental factors influence the development of vertebrate rod photoreceptors.     

Some aspects of the early development and implantation of the mammalian egg

Summary The early development and implantation of the mammalian egg is described for various species and the differing and often contradictory solutions proposed by different authors for the many problems arising from their investigations are exposed, compared and discussed.     

Central adrenergic neurones and the initiation and development of experimental hypertension

Zusammenfassung Zerstörung zentraler adrenerger Neurone durch Injektion von 6-Hydroxydopamin in einen Seitenventrikel verursachte bei normotonen und genetisch hypertonen Ratten lediglich eine geringe und vorübergehende Blutdrucksenkung und beeinflusste die DOCA- und renale Hypertonie überhaupt nicht. Im Gegensatz dazu verhinderte 6-Hydroxydopamin die Entwicklung der DOCA- und renalen Hypertonie, wenn es 7–10 Tage vor deren Induktion in einen Seitenventrikel injiziert wurde. Bei genetisch hypertonen Ratten, bei denen eine allmähliche Blutdrucksteigerung schon bald nach der Geburt einsetzt, unterbrach intraventrikulär injiziertes 6-Hydroxydopamin die weitere Entwicklung der Hypertonie. Die Resultate weisen auf die Bedeutung zentraler adrenerger Neurone für die Pathogenese verschiedener experimenteller Hypertonieformen hin.     

High-fat diet and the development of obesity in albino rats

Zusammenfassung Männliche, geschlechtsreife Wistar-Ratten wurden 44 Wochen lang mit isokalorischen Diäten gefüttert, die entweder einen erhöhten Fett- oder Kohlehydrat- bzw. Eiweissanteil enthielten. Zwischen den eine fettreiche und eine kohlehydratreiche Diät erhaltenden Tieren zeigten sich keine statistisch bedeutsamen Unterschiede in bezug auf ihre Gewichtszunahme und den Gesamtanteil des Körperfettes. Diejenigen Tiere, die eine Diät mit erhöhtem Kaseingehalt erhielten, nahmen in Übereinstimmung mit den Literaturangaben weniger an Gewicht zu und lagerten auch weniger Körperfett ab. Wie aus unseren Ergebnissen hervorgeht, muss ein hoher Fettanteil in der Diät von sich aus noch nicht zur Fettsucht bei Ratten führen.     

Reactivation of the inactive X chromosome in development and reprogramming

In mammals, one of the two X chromosomes of female cells is inactivated for dosage compensation between the sexes. X chromosome inactivation is initiated in early embryos by the noncoding Xist RNA. Subsequent chromatin modifications on the inactive X chromosome (Xi) lead to a remarkable stability of gene repression in somatic cell lineages. In mice, reactivation of genes on the Xi accompanies the establishment of pluripotent cells of the female blastocyst and the development of primordial germ cells. Xi reactivation also occurs when pluripotency is established during the reprogramming of somatic cells to induced pluripotent stem cells. The mechanism of Xi reactivation has attracted increasing interest for studying changes in epigenetic patterns and for improving methods of cell reprogramming. Here, we review recent advances in the understanding of Xi reactivation during development and reprogramming and illustrate potential clinical applications.