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The introduction of logarithms into Spain
Abstract
During the first half of the 17th century, logarithms were taught by some professors in Spain, but knowledge of this subject remained scanty until the publication of Architectura civil by Juan Caramuel (1678) and especially of Trigonometria española by José Zaragoza (1672). Logarithms were considered only as an aid for computation up to the second half of the 18th century. Only when the infinitesimal calculus became more widely spread in Spanish mathematics, analytical interpretations of logarithms were also taken into account in books such as Elementos de matemáticas by Benito Bails (1776).ResumenDurante la primera mitad del siglo XVII, los logaritmos fueron explicados por algunos profesores en España, pero su conocimiento fue escaso hasta que se publicó la Architectura civil de Juan Caramuel (1678) y, sobre todo, la Trigonometria española de José Zaragoza (1672). Hasta la segunda mitad del siglo XVIII los logaritmos se consideraron únicamente como unos auxiliares que facilitaban el cálculo. Sólo cuando en las matemáticas españolas se generalizó el cálculo infinitesimal, se incluyeron también interpretaciones analíticas de los logaritmos en libros como los Elementos de matemáticas de Benito Bails (1776).
... Similar a la historia de los logaritmos (Navarro-Loidi & Llombart, 2008), Sörensen, en 1909, con el fin de evitar las potencias de 10 con exponente negativo con las que se expresa la concentración de iones de Hidrógeno, definió el concepto de pH (potencia del ion Hidrógeno): para una disolución neutra a 25°C, el pH sería igual a 7; para una disolución de ácido clorhídrico (1 mol por litro), el pH sería cero; y para una disolución de sosa de un mol por litro, el pH sería igual a 14. ¿En qué repercute la expresión matemática sobre la concepción química de las medidas de pH? Como éste es el 'exponente negativo de la concentración del ion hidrógeno', cuando la concentración es 0.01 (10 −2 ), su pH es 2. Esta conversión matemática genera mucha confusión entre los estudiantes: ¿por qué el pH sube cuando la acidez baja? ...
The integrated STEAM approach arises when a contextualized problem’s resolution requires different contributions (views) from each discipline. In this descriptive article we discuss the views needed from chemistry, technology, design skills, mathematical modelling and the artistic support necessary for constructing a scientific model that explains the pH of the mouth when chewing gum. To do this, we focus on describing the implementation of a short sequence of video-recorded inquiry-based contextualized activities (didactic innovation) in two classrooms with students in their third year of compulsory secondary education (14–15 years old). The scenes taken from the analysis of the implementation videos support the proposal of the connections needed in the STEAM contributions. The difficulties encountered from each discipline and the resolutions we propose shed light on how well-selected problems make STEAM views necessary, thus avoiding the usual summative multidisciplinary approach.
As Europeans ventured more frequently on transoceanic voyages in the 16th century, it became essential for navigators to be able to carry out mathematical computations. Iberian educators rose to the occasion, developing textbooks and curricula to transmit these abstract concepts. At the same time that they strove to present new mathematical techniques, these maritime authors also captured the tacit skills that had been practiced for centuries. Thus, even though few working sailors put pen to paper, it is possible to recover aspects of their epistemology from these texts. Because books traveled so easily across borders, these academic sources also had far-reaching effects, inspiring similar educational programs across maritime Europe. This article recommends adopting a comparative perspective, since much can be learned by tracing the evolution of these shared practices as they traveled from Spain and Portugal to the Netherlands, France, and England, and back again.
During the Spanish eighteenth century, a process of modernization took place in scientific knowledge, partly driven by the circulation and appropriation of new scientific ideas.
In this context, the Spanish mathematician Benito Bails (1731–1797) published his course Elementos de Matemática (Elements of Mathematics) consisting of ten volumes (1779–1799), in which, among other subjects, he presented one of the most complete mathematical developments of logarithmic calculation methods of his time, by using the infinity through infinite series.
The aim of our article is to demonstrate how algebraic analytical reasoning enabled Bails to obtain new and more efficient infinite algorithms that converge more quickly in the computation of logarithms in any system. We show how Euler's number ‘e’ is calculated, probably for the first time in Spanish teaching, in an eighteenth century mathematical text.
Our analysis concludes that Bails’ course constituted an innovation and provides evidence of its creativity, originality and ingenuity.
The four-figure of power of base numbers of tables needs to be created for easy synchronization
with the existing Logarithm table of power of base 10 which is working on four figures. Although the five-figure of power base numbers of tables are more accurate because it is having less approximation in its establishment. The four-figure table is easy to compute doing utilization
because it is having less digits to work with. This paper presents the four and five figures of Kifilideen (Power of base 11) and AntiKifilideen (Antipower of base 11) tables for the computation of mathematical problems. The four and five figures are both reliable to work with. However, there is a tradeoff between easy computation as related to the four-figure table and more accuracy as related to the five-figure table in their utilization.
This article analyzes Jost Bürgi's work (1620) and its place in the history of logarithms.
The discovery of hyperbolic logarithms has been variously attributed to Gregory of St-Vincent and to Alphonse Antonio de Sarasa. In this paper we describe the relationship between St-Vincent and de Sarasa, the challenge from Mersenne which provoked de Sarasa's publication, de Sarasa's understanding of logarithms, and the propositions of de Sarasa where the connection between hyperbolic areas and logarithms was first claimed. Since de Sarasa's hyperbolae were not defined analytically, he did not insist on a particular base for his logarithms, nor did he choose a value for log 1 which would have implied log AB=log A+log B, so we conclude that he did not focus on what Euler later called natural logarithms. An explanation of the different accounts of the origin of natural logarithms offered by twentieth century writers is tentatively proposed. Copyright 2001 Academic Press.Die Entdeckung hyperbolischer Logarithmen ist verschiedentlich Gregorius a S.Vincentio und Alphonse Antonio de Sarasa zugeschrieben worden. In diesem Artikel beschreiben wir die Beziehung zwischen St.Vincentius und de Sarasa, die Herausforderung von Mersenne, die de Sarasas Veröffentlichung provozierte, de Sarasas Verständnis von Logarithmen und die Sätze de Sarasas in denen der Zusammenhang zwischen Hyperbelflächen und Logarithmen zum ersten Mal behauptet worden ist. Da de Sarasas Hyperbeln nicht analytisch definiert waren, forderte er keine Basis fuer seine Logarithmen. Er wählte auch keinen Wert fuer log 1, der log AB=log A=log B impliziert hätte. So können wir schliessen, dass er sich nicht auf das konzentrierte, was Euler später natürliche Logarithmen nannte. Eine Erklärung der unterschiedlichen Darstellungen über den Ursprung der natürlichen Logarithmen, die Autoren des 20. Jahrhunderts geben, wird ansatzweise versucht. Copyright 2001 Academic Press.MSC 1991 subject classification: 01 A 45.
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