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A Cr^Onica Do C Alculo: II. a Epoca De Newton E Leibniz

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A Cr^Onica Do C Alculo: II. a Epoca De Newton E Leibniz o . Revista Brasileira de Ensino de Fsica, vol. 18, n 3, setembro, 1996 181 A Cr^onica do Calculo: II. A Ep o ca de Newton e Leibniz Jose Maria Filardo Bassalo Departamento de Fsica da UniversidadeFederal do Para 66075-900, Belem, Para, Brasil Trabalho recebido em 30 de outubro de 1995 Nesta Cr^onica vamos mostrar como se desenvolveu o que ho je conhecemos como Calculo Diferencial e Integral. Nesta segunda parte, estudaremos o desenvolvimento desse Calculo devido, principalmente, aos trabalhos de Newton e Leibniz. Abstract In this Chronicle weshowhowwas developp ed what to day means Integral and Di e- rential Calculus. In this second part, we study the development of this Calculus due, mainly, to the Newton's and Leibniz's works. 1. Intro duc~ao ritmos. Assim, entusiasmado com a descob erta, ele cal- culou diversos logaritmos das areas sob uma hiperb ole 1 Na primeira parte da Cr^onica, vimos como se de- equilatera ate 55 casas decimais. Na primavera de 1665, senvolveu o Calculo Diferencial e Integral desde a Newton demonstrou que o problema das tangentes (di- Antiguidade ate os trabalhos que antecederam aos de ferenciac~ao, na linguagem atual) e o das quadraturas 2 Newton e Leibniz. Nesta segunda parte, estudaremos (integrac~ao, tambem na linguagem atual) apresentavam os trabalhos de Newton e de Leibniz e de seus contem- entre si uma relac~ao inversa, relac~ao que constitui o Te- p or^aneos. orema Fundamental do Calculo. As principais contribuic~oes do fsico e matematico ingl^es Sir Isaac Newton (1642-1727) para o desenvolvi- Na primavera-ver~ao de 1665, Newton descobriu um mento do Calculo,aconteceram no p ero do entre 1664 novo asp ecto para tratar o problema das tangentes e e 1676, quando estudou os principais livros escritos, ate quadraturas. Ateent~ao, conforme vimos, ele havia con- 3 ent~ao, sobre Matematica. Vejamos essas contribuic~oes. siderado as areas como somas estaticas de in nitesi- Newton havia aprendido, no livro de Wallis, mais, no mesmo estilo de Wallis. No entanto, naquela como este realizavaocalculo de areas sob curvas, o casi~ao, passou a tratar as areas cinematicamente, considerando-as como soma estatica de in nitesimais. como se fossem varridas p or uma linha movel. No ou- Para esse calculo, Wallis utilizava-se de induc~oes, inter- tono de 1665, estendeu essa ab ordagem cinematica a p olac~oes, aproximac~oes e logaritmos, trabalhando no gerac~ao de curvas, e tratou-as como o lugar geometrico 4 domnio do in nito, isto e, com series in nitas. Ent~ao, (lo cus)deumponto que se deslo cava em condic~oes no inverno de 1664-1665, estendeu o meto do de Wallis determinadas. Com essa ideia de movimento, passou a 5 usando expans~oes binomiais em termos dessas series. considerar suas variaveis como geradas p or movimentos Com a expans~ao (teorema) binomial, Newton obser- contnuos de p ontos, linhas e planos, o que lhe p ermitiu vou que p o deria encontrar a area abaixo de pratica- derivar o termo uxional esuasvariantes. Assim, a mente qualquer curvaalgebrica, e expressa-la em loga- essas variaveis (p or exemplo, x e y) denominou-as de Este artigo e em homenagem ao meu amigo PAULO RENATO BENTIVEGNA, medico paulista, estudioso e entusiasta do Calculo. 182 J. M. Filardo Bassalo uentes eas suas velo cidades chamou-as de ux~oes, geral e inverso ao utilizado para se tracar tangentes a e denotadas p or p e q, resp ectivamente. Ainda nesse curvas. Com efeito, Newton demonstrou que a area z m + n m n n n ax estudo baseado no movimento, Newton a rmou que \as e dada p or .Para sob a curvay=ax m + n linhas in nitamente p equenas" descritas p elos corp os a fazer a demonstrac~ao, Newton considerou um acrescimo cada momento eram as velo cidades com que eles as des- in nitesimal da variavel x (denominado p or ele de mo- creviam, e que a raz~ao entre as velo cidades de y e x em mento de x e denotado p or o), com o corresp ondente 12 qualquer p onto de uma curva, de nia a tangente nesse aumento oy (momento da area) para a area z. Com 6 ponto. isso, obteveaarea aumentada na forma: z + oy= m + n n n a(x + o) . Em seguida, aplicou a expans~ao Ometo do baseado no movimento foi tambem ex- m + n binomial ao lado direito dessa equac~ao, cancelou termos plorado p or Newton em tr^es outras o casi~oes: 13 de No- semelhantes, dividiu tudo p or o, desprezou termos que vembro de 1665, 14 e 16 de Maio de 1666. Por m, m 13 n ainda continham o, e obteve nalmentequey=ax . em Outubro de 1666, em um manuscrito intitulado To Resolve Problems by Motion (ParaResolver Problemas Uma exp osic~ao mais completa sobre o meto do do pelo Movimento), Newton apresentou a exp osic~ao de calculo de tangentes e de quadraturas (p or intermedio 7 seu meto do das ux~oes, emb ora com uma notac~ao de uentes e ux~oes), desenvolvido entre 1665 e 1666, provisoria e ainda complicada. foi apresentada p or Newton no manuscrito Tractatus de Methodis Serierum et Fluxionum (Tratado dos Metodos A partir do nal de 1666, Newton afastou-se um das Series e Flux~oes), manuscrito iniciado no inverno p ouco da matematica e passou a se dedicar ao estudo 14 de 1670-1671 e que, contudo, nunca foi concludo. do movimento dos corp os, da luz e das cores, temas Ainda nesse manuscrito, Newton apresentou um grande que jahavia questionado (juntamente com alguns te- numero de aplicac~oes das ux~oes, tais como, diferen- mas matematicos), enquanto aluno do Trinity College 8 ciac~oes de func~oes implcitas, calculos de tangentes, de da Universidade de Cambridge, e que os havia regis- maximos e mnimos, de p ontos de in ex~ao, e de curva- trado em um caderno de ap ontamentos, com o ttulo de turas de curvas, chegando, nesteultimo caso, a calcular Quaestiones quaedam Philosophicae, escrito, provavel- 15 9 corretamente os seus raios de curvatura, dado p or (na mente, p or volta de 1664. 3 2 2 (1 +y _ ) 16 . notac~ao de ho je): R = A publicac~ao dos livros do matematico esco c^es Ja- y mes Gregory (1638-1675), De VeraCirculi et Hyper- Quando o matematico alem~ao Gottfried Wilhelm bolae Quadratura, em 1667, Geometriae Pars Univer- Leibniz (1646-1716) comecou a publicar seu trabalho salis e Exercitationes Geometricae,em1668,edoli- sobre o Calculo Diferencial, em Outubro de 1684, no vro Logarithmotechnia do matematico alem~ao Nikolaus Acta Eruditorum Lipsiensium (Ata dos Eruditos de Mercator (Kau mann) (1620-1687), tambem em 1668, Lpsia (Leipzig)), Newton considerou a hipotese de fez com que Newton voltasse novamente sua atenc~ao tambem publicar seus trabalhos sobre Matematica. As- para a Matematica. Nesses livros, hao calculo de qua- sim, em meados de 1691, resolveu dar continuidade 10 draturas p or intermedio de representac~oes em serie. aummanuscrito que comecara em 1676, intitulado Como, anteriormente, Newton jahavia obtido quadra- De Quadratura Curvarum (SobreaQuadraturadas 17 turas, usando um meto do que desenvolvera e envol- Curvas), escrevendo ent~ao um novomanuscrito, no vendo, tambem, representac~oes em series, conforme vi- qual apresentou um conjunto de problemas resolvidos mos acima, resolveu ent~ao exp or esse meto do no ma- pelo meto do das ux~oes, problemas semelhantes aos nuscrito intitulado De Analysi per Aequationes Numero que Leibniz havia resolvido com o seu proprio meto do. Terminorum In nitas (SobreaAnalise de Equac~oes Alem disso, considerou um sistema de notac~oes como 11 com Numer o Ilimitado de Termos), escrito em 1669. alternativa ao apresentado p or Leibniz. Portanto, foi no De Quadratura que adotou, em de nitivo, a notac~ao Ometo do usado p or Newton e descrito no De Anal- x_ para representar a ux~ao de x, assim como a notac~ao ysi, consistia em obter quadraturas p or um pro cesso o . Revista Brasileira de Ensino de Fsica, vol. 18, n 3, setembro, 1996 183 tentativa de de nir limite de uma func~ao. Por outro x' para representar o uente cuja ux~ao e x,alem de lado, no Lema VI I, ele demonstrou que a secante a um exp erimentar o Q (de quadratura) como substituto R arco transforma-se em uma tangente, quando se usam de usado p or Leibniz. E mais ainda, dobrando os 20 \asultimas raz~oes". pontos (.) e as linhas ('), Newton representou, resp ec- tivamente, ux~oes de ux~oes e uentes de uentes. Muito emb ora Newton ha ja utilizado o meto do das raz~oes na demonstrac~ao de alguns Lemas do Livro I, Ainda no De Quadratura, Newton abandonou a conforme registramos acima, somente no Livro I I do ideia de acrescimos in nitesimais e explicou o meto do Principia, apresentou os algortimos de seu meto do de das ux~oes em termos de raz~oes: primeira raz~ao 21 calculo. Por exemplo, no Lema I I do Livro, escreveu: de quantidade nita nascente e ultima raz~ao de - \O momento de qualquer genitum e igual aos momen- quantidade n ma.Vejamos como Newton intro du- tos de cada um dos lados geradores, multiplicado p elos ziu esses conceitos, ao calcular, p or exemplo, a ux~ao n 18 ndices das p ot^encias desses lados, e p or seus co e cien- da func~aoy=x . Assim, para encontrar a ux~ao de tes continuamente".
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