Wallis on Indivisibles

International audienceThe present chapter is devoted, first, to discuss in detail the structure and results of Wallis's major and most influential mathematical work, the Arithmetica Infinitorum ([51]). Next we will revise Wallis's views on indivisibles as articulated in his answer to Hobbes's criticism in the early 1670s. Finally, we will turn to his discussion of the proper way to understand the angle of contingence in the first half of the 1680s. As we shall see, there are marked differences in the status that indivisibles seem to enjoy in Wallis's thought along his mathematical career. These differences correlate with the changing context of 17th-century mathematics from the 1650s through the 1680s, but also respond to the different uses Wallis gave to indivisibles in different kinds of texts—purely mathematical, openly polemical, or devoted to philosophical discussion of foundational matters

Influencia del granallado en los aceros inoxidables austeníticos metaestables

En el presente trabajo se muestra el efecto del granallado sobre la microestructura y las propiedades mecánicas de un acero inoxidable austenítico metaestable EN 1.4318 (AISI 301LN). Se han considerado dos condiciones de partida distintas: recocido (microestructura totalmente austenítica) y laminado en frío (con un porcentaje inicial de martensita del 38%). La granalla utilizada es de acero inoxidable S300, proyectada a una velocidad de 65 m/s con factores de cobertura del 200 y 400%. El análisis microestructural se ha llevado a cabo por difracción de rayos-X, microscopía láser confocal (CLSM) y microscopía electrónica de barrido (MEB). Para cada condición se han realizado ensayos de fatiga a alto número de ciclos. Los resultados ponen de manifiesto que el proceso de granallado permite aumentar en un 25% el límite a fatiga del acero laminado, mientras que no se observa ningún efecto sobre el acero recocido. Para esta condición, la creación de microgrietas en la superficie enmascara el efecto beneficioso del endurecimiento debido a la formación de martensita por deformación durante el proceso de granallado.The aim of this work is to study the microstructural changes and its effect on mechanical properties of a metastable austenitic stainless steel grade EN 1.4318 (AISI 301LN) subjected to shot peening process. Two different material conditions were considered: annealed (totally austenitic microstructure) and cold rolled (with an initial martensite content of 38%). Stainless steel shots type S300 impacted the surface at 65 m/s with two different coverage factors: 200 and 400%. Microstructural analysis was performed by x-ray diffraction, Confocal Laser Scanning Microscopy (CLSM) and Scanning Electron Microscopy (SEM). For each condition fatigue tests at high number of cycles were carried out. Results point out that only cold rolled samples displayed an improvement of 25% on the fatigue limit while no effect was detected for annealed steel. For this condition, the presence of surface microcracks was expected to offset the advantages introduced by the presence of deformation induced martensite formed during shot peening process.Peer Reviewe

Algebra as language: Wallis and Condillac on the nature of algebra

[EN] The present article focuses on a specific stage in the process through which algebraic notations became a powerful formal language. The article compares the views of Condillac and Wallis on the nature of algebra as a way to understand the differences between the all powerful algebraic language of the first decades of the 18th century and the limitations of 17th-century algebraic notations. In particular, it considers the role the graphical element of algebraic notations, and their symmetry in some specific cases, played in John Wallis's understanding of the demonstrative power of the algebraic notations, It shows that the weakness of Wallis's algebra qua formal language is fully consistent with his emphasis on algebra as a tool of discovery. It is also consistent with Wullis's view that algebra does not produce "new" demonstrations, or that algebra does not alter the substance of the classical demonstrations. Finally, and perhaps more significantly, such a weakness is also consistent with the notion that algebra was but a tachigraphy that "condensed" mathematical arguments and "offered them at one glance".[ES] El presente artículo estudia un episodio concreto en el proceso que transformó las notaciones algebraicas de los siglos XVI y XVII en un poderoso lenguaje formal. En el mismo se comparan las ideas de Wallis y de Condillac sobre la naturaleza del álgebra con el objetivo de entender las diferencias entre el todopoderoso lenguaje algebraico de las primeras décadas del siglo XVIII y las limitaciones inherentes a las notaciones algebraicas del siglo XVII . De forma particular, se valora aquí el papel desempeñado por el elemento gráfico de las notaciones algebraicas, así como su simetría en algunos casos particulares, en la manera como Wallis entendía el poder demostrativo de las notaciones algebraicas. El artículo arguye que la debilidad del álgebra qua lenguaje formal en Wallis es consistente con la importancia que él le acordaba como instrumento de descubrimiento. También es consistente con la opinión de Wallis que el álgebra no altera de forma substancial las demostraciones clásicas. Finalmente, y quizá de forma más importante, esta debilidad también es consistente con la idea que el álgebra sólo era una taquigrafía que "condensaba" los argumentos matemáticos y "los mostraba en una sola mirada".Partial support from the research project BHA2002-02648 of the Spanish Ministry of Science and Technology is gratefully acknowledged.Peer reviewe

Cold War geopolitics and French mathematicians: The Groupement des mathématiciens d’expression latine (1957-1985)

Introduction The Groupement des mathématiciens d’expression latine was born in Nice, 12-19 September 1957, in the first Réunion des mathématiciens d’expression latine. Many more meetings were to gather scores of mathematicians linked together by their use of “Latin languages”. While the rationale behind the “Union of Latin Mathematicians” is to say the least surprising for nowadays networks of international cooperation, however the first Réunions des Mathématiciens d’Expression Latine were ou..

Cambio de nociones sobre la proporcionalidad en las matemáticas pre-modernas

Defined as «a sort of relation in respect of size between two magnitudes of the same kind», a ratio was not a number nor a geometrical magnitude in Euclid's Elements. During the first half of the 18th century, however, ratios were identified with numerical magnitudes for all practical purposes. This paper argues that in order to understand the changing notions of ratio and proportionality in the early modern period two questions are to be answered separately. One concerns the numerical status of the objects compared through a ratio, or term of the ratio and the paper shows that by the turn of the 17th century this difficulty had been overcome. The second question concerns the status of ratios themselves, which was not solved until the 18th century. In studying this development particular attention is paid to the problem of evaluating the influence of the medieval notion of denomination of ratios. Thanks to Pedro Nunez's algebra book it is possible to show that this concept had become fossilized and lost its virtuallity as an «arithmetizing» agent by the mid 16th century. Furthermore, it is argued here that the changes overcoming ratios and proportionality in the 16th century are understandable only with reference to the social background, and particularly to the so-called abbaco books and to 16thcentury algebra.Una razón entre dos objetos matemáticos era definida en los Elementos de Euclides como «una clase de relación respecto del tamaño entre dos magnitudes del mismo tipo»; las razones no eran números ni magnitudes geométricas en los Elementos. Sin embargo, durante la primera mitad del siglo XVIII las razones fueron identificadas con magnitudes numéricas. Para entender los cambios que afectaron a las nociones de razón y de proporcionalidad durante los siglos XVI y XVII hay que distinguir, y responder separadamente, a dos cuestiones. La primera concierne al status numérico de los objetos comparados en una razón, o términos de la razón. Esta dificultad había sido superada a principios del siglo XVII, en el sentido de que en la práctica toda clase de términos había sido identificada con una magnitud numérica. La segunda dificultad concierne al status de las razones mismas; esta no fue superada hasta bien entrado el siglo XVIII. El presente artículo dedica particular atención al problema de la influencia de la noción medieval de denominación de una razón. Gracias al libro de álgebra de Pedro Núñez es posible demostrar que ya a mediados del siglo XVI este concepto se había fosilizado y había perdido todo su poder «aritmetizador». Se arguye, además, que sólo es posible entender los cambios sufridos por las nociones de razón y proporcionalidad haciendo referencia al contexto social, y particularmente a los llamados libros de abbaco