Mengoli's mathematical ideas in Leibniz's excerpts

In the seventeenth century many changes occurred in the practice of mathematics. An essential change was the establishment of a symbolic language, so that the new language of symbols and techniques could be used to obtain new results. Pietro Mengoli (1626/7–86), a pupil of Cavalieri, considered the use of symbolic language and algebraic procedures essential for solving all kinds of problems. Following the algebraic research of Viète, Mengoli constructed a geometry of species, Geometriae Speciosae Elementa (1659), which allowed him to use algebra in geometry in complementary ways to solve quadrature problems, and later to compute the quadrature of the circle in his Circolo (1672). In a letter to Oldenburg as early as 1673, Gottfried Wilhelm Leibniz (1646–1716) expressed an interest in Mengoli's works, and again later in 1676, when he wrote some excerpts from Mengoli's Circolo. The aim of this paper is to show how in these excerpts Leibniz dealt with Mengoli's ideas as well as to provide new insights into Leibniz's mathematical interpretations and commentsPeer ReviewedPostprint (author's final draft

Historical activities in the mathematics classroom: Tartaglia's Nova Scientia (1537)

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En record de Jacqueline (Jackie) Anne Stedall (1950-2014)

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Aspectos matemáticos del triángulo armónico de Gottfried Wilhelm Leibniz (1646-1716)

Las matemáticas del siglo XVII florecieron debido a su algebrización y a la introducción del infinito. En este artículo presentamos una aproximación a una de sus figuras, Gottfried Wilhelm Leibniz (Leipzig, 1646 - Hannover, 1716). Se analizan también algunos aspectos matemáticos del triángulo armónico, nuevo objeto creado por Leibniz a partir del triángulo aritmético de Pascal, que muestran que el infinito se convierte en un elemento más en los cálculos matemáticos de Leibniz. Cabe destacar la utilidad de estos textos de Leibniz sobre el triángulo aritmético y el triángulo armónico, para la enseñanza de las matemáticas.Postprint (author's final draft

La Reial Acadèmia de Matemàtiques de Barcelona (1720-1803). Matemàtiques per a enginyers

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Álgebra y geometría en el aula: la construcción geométrica de la solución de la ecuación de segundo grado

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The harmonic triangle in Mengoli’s and Leibniz’s works

The harmonic triangle was defined by Gottfried Wilhelm Leibniz (1646- 1716) in 1673, and its definition was related to the successive differences of the harmonic series. Leibniz studied it in many different texts throughout his life. Pietro Mengoli (1627-1686), rather at the same time, used the harmonic triangle as a triangular table to perform quadratures and also the interpola - ted harmonic triangle to calculate the quadrature of the circle. In this article we analyze and compare the independent treatment of harmonic triangle by Mengoli and Leibniz in their works, referring to their sources, their aims, and their uses. We show that, on the one hand, Mengoli uses triangular tables as a tool of calculus, and uses the harmonic triangle to perform quadratures through one procedure called by him “homology”. On the other hand, at the same time, Leibniz defines the harmonic triangle from the study on harmonic series, analyses its properties, and uses it to perform the summations of infi - nite series through one procedure called by him “sums of all the differences”. Harmonic triangle has an open visual structure in which the number of terms arranged in this way can be made infinite. The infinite therefore becomes one more element in the mathematical calculations of these authors, which in seventeenth century mathematics opened up a world of possibilities in the series and in their relations with infinitesimal calculusPostprint (author's final draft

Nous resultats i procediments en les matemàtiques del segle XVII: càlcul de màxims a Pietro Mengoli (1626/1627-1686)

La publicació, l’any 1591, de l’obra In artem analyticen isagoge de François Viète (1540–1603) va constituir un pas endavant important en el desenvolupament del llenguatge simbòlic. A començaments del segle xvii la difusió de l’obra de Viète va pro- vocar que altres autors, com ara Pietro Mengoli (1626/1627–1686), també consideressin la utilitat dels procediments algebraics per resoldre tot tipus de problemes. Mengoli va seguir el camí de Viète tot construint una geometria d’espècies, Geometriae speciosae elementa (1659), que li va permetre emprar conjuntament l’àlgebra i la geometria per resoldre problemes de quadratura. Mengoli, com Viète, va considerar la seva àlgebra una tècnica en la qual els símbols eren utilitzats no únicament per representar nombres sinó també valors de qualsevulla magnitud. Va tractar amb espècies, formes, taules triangulars, quasi raons i raons logarítmiques. Tanmateix, l’aspecte més innovador del seu treball va ser l’ús de les lletres per tractar directament les figures geomè- triques mitjançant les seves expressions algebraiques. En aquest article, analitzo la construcció algebraica d’aquestes figures geomètriques, l’ús de les taules triangulars i la demostració molt original que va fer Mengoli per trobar el màxim d’aquestes figures geomètriques abans del desenvolupament del càlcul de Newton i Leibniz. Aquestes anàlisis i l . l ustren les idees matemàtiques de Mengoli sobre la funció específica del llenguatge simbòlic com a mitjà d’expressió i com a eina analíticaPostprint (published version

The harmonic triangle in Mengoli 's and Leibniz's works

The harmonic triangle was defined by Gottfried Wilhelm Leibniz (1646- 1716) in 1673, and its definition was related to the successive differences of the harmonic series. Leibniz studied it in many different texts throughout his life. Pietro Mengoli (1627-1686), rather at the same time, used the harmonic triangle as a triangular table to perform quadratures and also the interpola - ted harmonic triangle to calculate the quadrature of the circle. In this article we analyze and compare the independent treatment of harmonic triangle by Mengoli and Leibniz in their works, referring to their sources, their aims, and their uses. We show that, on the one hand, Mengoli uses triangular tables as a tool of calculus, and uses the harmonic triangle to perform quadratures through one procedure called by him “homology”. On the other hand, at the same time, Leibniz defines the harmonic triangle from the study on harmonic series, analyses its properties, and uses it to perform the summations of infi - nite series through one procedure called by him “sums of all the differences”. Harmonic triangle has an open visual structure in which the number of terms arranged in this way can be made infinite. The infinite therefore becomes one more element in the mathematical calculations of these authors, which in seventeenth century mathematics opened up a world of possibilities in the series and in their relations with infinitesimal calculusPostprint (author's final draft