O processo de integração em Blaise Pascal
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Data
2007-08-10
Autores
Venturin, Jamur Andre [UNESP]
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
O propósito desse trabalho é identificar de que maneira Pascal solucionava problemas matemáticos de integração, fazendo uso dos indivisíveis. Para isso, consultamos em particular três de suas obras: Postestatum Numericarum Summa1, LETTRE DE M. DETTONVILLE A M. DE CARCAVI2 e o Tratado do Triângulo Aritmético. Vimos a estreita relação que existe entre essas obras, a saber, na primeira delas é exibida a regra geral para encontrar áreas sob curvas do tipo y = xn , bem como mostra a relação entre a soma de potências numéricas com grandezas contínuas. Faz a integração das curvas segundo a abordagem dos indivisíveis. Já na segunda, que é associada diretamente com a terceira obra, é apresentada tanto a compreensão do indivisível e do infinitamente pequeno na constituição do contínuo, quanto à relação de soma simples, triangular e piramidal (encontradas a partir do triângulo aritmético) com suas respectivas integrais. Desse modo, a fim de entendermos o que aconteceu no século XVII, e sabermos quais as possíveis influências matemáticas de Pascal, buscamos outros métodos de integração como aqueles utilizados pelos gregos, isto é, o método de exaustão e as técnicas do século XVI. Sendo assim, foi possível observar que seus procedimentos de integração podem ser contemplados em dois aspectos: no primeiro, como contribuição para a história do desenvolvimento do cálculo, em um período em que ele estava na eminência de ser estruturado. No segundo, destacamos a relação existente com o cálculo moderno, contudo seu campo teórico é fundamentado nos indivisíveis.
The intention of this work is to identify of which way Pascal solved mathematical problems of integration, making use of the indivisibles. For this we consulted in particular three of his works: Postestatum Numericarum Summa, Lettre De M. Dettonville A. M. de Carcavi and the Treatise on the Arithmetical Triangule. We saw the close connection that exist between these works, namely, the first of them is showed the general rule to find areas under curves y = x2 , as well as it shows the relation between the sums of numerical potences with continuous magnitudes He makes the integration of curves according to approach of indivisibles. Already on the second, itþs associate directly to the third work, is presented the understanding of indivisible and the infinitive little in the constitution of the continuous, and also in the relation of simple, triangular and piramidal sums (found from the arithmetical triangule) with their respective integrals. In this way, in order to understand what happened in the 17th century and to know the possibles mathematical influences of Pascal, we search others methods of integration like those used by the Greeks, that is to say, the method of exhaustion and the techniques of the 16th century. Such being the case, it was possible to observe that its procedures of integration can be contemplated in two aspects: the first one, as a contribution for the history development calculation, in a period that it was the eminence of being structured. The second one detach the relation existent with the modern calculation, however, its theoretical field is based on indivisibles ones.
The intention of this work is to identify of which way Pascal solved mathematical problems of integration, making use of the indivisibles. For this we consulted in particular three of his works: Postestatum Numericarum Summa, Lettre De M. Dettonville A. M. de Carcavi and the Treatise on the Arithmetical Triangule. We saw the close connection that exist between these works, namely, the first of them is showed the general rule to find areas under curves y = x2 , as well as it shows the relation between the sums of numerical potences with continuous magnitudes He makes the integration of curves according to approach of indivisibles. Already on the second, itþs associate directly to the third work, is presented the understanding of indivisible and the infinitive little in the constitution of the continuous, and also in the relation of simple, triangular and piramidal sums (found from the arithmetical triangule) with their respective integrals. In this way, in order to understand what happened in the 17th century and to know the possibles mathematical influences of Pascal, we search others methods of integration like those used by the Greeks, that is to say, the method of exhaustion and the techniques of the 16th century. Such being the case, it was possible to observe that its procedures of integration can be contemplated in two aspects: the first one, as a contribution for the history development calculation, in a period that it was the eminence of being structured. The second one detach the relation existent with the modern calculation, however, its theoretical field is based on indivisibles ones.
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Matemática - História, Educação matemática, Mathematics history, Mathematics education
Como citar
VENTURIN, Jamur Andre. O processo de integração em Blaise Pascal. 2007. iii, 118 f. Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas, 2007.