Newton’s Laws of Motion

An obstacle to reading the Principia is presuming that the Laws of Motion attributed to Newton in physics textbooks, and the concept of force in them, are the same as those in the book; they are not. This chapter provides an account of his Laws and his conception of force as his contemporaries would have understood them. It does so first by giving the historical background to them in works by Descartes, Christiaan Huygens, John Wallis, and Christopher Wren; and then by reviewing the history of Newton’s own reformulations of them not just in the sequence of manuscripts leading up to the first edition, but extending even to the second edition. In the Principia the Laws serve as the basis for deriving conclusions about forces governing the motions in our planetary system from the motions of the bodies within it “among themselves.” Crucial to their doing so are their six Corollaries, some of them initially formulated as Laws. The history of their development too is covered in parallel with that of the Laws, emphasizing their crucial role in licensing those inferences regarding forces from the observed motions.

The Indivisibles of the Continuum

This chapter considers some significant developments in seventeenth-century mathematics which are part of the pre-history of the infinitesimal calculus. In particular, I examine the “method of indivisibles” proposed by Bonaventura Cavalieri and various developments of this method by Evangelista Torricelli, Gilles Personne de Roberval, and John Wallis. From the beginning, the method of indivisibles faced objections that aimed to show that it was either conceptually ill-founded (in supposing that the continuum could be composed of dimensionless points) or that its application would lead to error. I show that Cavalieri’s original formulation of the method attempted to sidestep the question of whether a continuous magnitude could be composed of indivisibles, while Torricelli proposed to avoid paradox by taking indivisibles to have both non-zero (yet infinitesimal) magnitude and internal structure. In contrast, Roberval and Wallis showed significantly less interest in addressing foundational issues and were content to maintain that the method could (at least in principle) be reduced to Archimedean exhaustion techniques.

A história do problema das colisões na física do século XVII anterior a Newton

Resumo Este trabalho traça um panorama histórico do tratamento do problema das colisões ao longo do século XVII, até que se chegasse à grande síntese mecânica de Newton. Apresentamos primeiramente as abordagens pioneiras de Thomas Harriot e Isaac Beeckman. Em seguida, passamos à tentativa de solução do problema por Descartes, apresentando resumidamente os seus fundamentos filosóficos e a construção sistemática por ele buscada, que, enfim, se mostrou errada. Passamos, então, à análise das formulações propostas por Christopher Wren, John Wallis e Christiaan Huygens, todas elas atingindo resultados corretos, ainda que de aplicação restrita. Procuramos ressaltar as particularidades de cada uma dessas elaborações, destacando a introdução pioneira na física do uso de número negativos na determinação do sentido do movimento, efetuada por Wren e Wallis, e o caráter extremamente sistêmico do pensamento mecânico de Huygens, construindo sua solução a partir da aplicação de princípios físicos fundamentais, alguns já conhecidos e outros por ele esboçados.

Physical arguments and moral inducements: John Wallis on questions of antiquarianism and natural philosophy

In his posthumously published work Chartham News (1669), the antiquary William Somner tentatively sought to link the discovery of fossilized remains near Canterbury to the prehistoric existence of an isthmus connecting Britain and France, before calling on natural philosophers to pursue his explanation further. This call was eventually heeded by the Oxford mathematician John Wallis, but only after more than thirty years had elapsed. The arrival in England of a catalogue of questions concerning the geology of the Channel led to the republication of Chartham News in the Philosophical Transactions , prompting Wallis to develop a physical explanation based on his intimate knowledge of the Kent coastline. Unbeknown to Wallis at the time, that catalogue had been sent by G. W. Leibniz, who had in turn received it from G. D. Schmidt, the former Resident of Brunswick-Lüneburg in Sweden. Wallis's explanation, based on the principle of establishing physical causes both for the rupturing of the isthmus and for the origin of fossils, placed him in a camp opposed by Newtonian authors such as John Harris at a time when the priority dispute over the discovery of the calculus led to the severing of his ties with the German mathematician and philosopher Leibniz.

Polity and liturgy in the philosophy of John Wallis

John Wallis, a founding member of the Royal Society, theologian and churchman, participated in the leading ecclesiastical conferences in England from the beginning of the English Civil War to the Restoration. His allegiance across governments, both civil and ecclesiastical, has provoked criticism. Close investigation into his position on key church issues, however, reveals a deeper philosophical unity binding together his natural philosophy, mathematics and views on church polity and liturgy.

Geometry, religion and politics: context and consequences of the Hobbes–Wallis dispute

The dispute that raged between Thomas Hobbes and John Wallis from 1655 until Hobbes's death in 1679 was one of the most intense of the ‘battles of the books’ in seventeenth-century intellectual life. The dispute was principally centered on geometric questions (most notably Hobbes’s many failed attempts to square the circle), but it also involved questions of religion and politics. This paper investigates the origins of the dispute and argues that Wallis’s primary motivation was not so much to refute Hobbes’s geometry as to demolish his reputation as an authority in political, philosophical, and religious matters. It also highlights the very different conceptions of geometrical methodology employed by the two disputants. In the end, I argue that, although Wallis was successful in showing the inadequacies of Hobbes’s geometric endeavours, he failed in his quest to discredit the Hobbesian philosophy in toto .

ARITHMETICA INFINITORUM DE JOHN WALLIS

Este artigo apresenta uma pesquisa, ainda em andamento, que consiste em fazer um estudo da obra Arithmetica Infinitorum de John Wallis (1616-1703), publicada em 1655, na qual analisamos a aplicação da abordagem utilizada pelo autor na perspectiva de fazer uma conexão com o ensino nos dias de hoje, além de investigar sua efetividade em sala de aula para a aprendizagem de conceitos relacionados ao Cálculo Diferencial e Integral, procurando em sua metodologia elementos que possam contribuir para uma forma de ensino do Cálculo baseado mais na descoberta do que no rigor característico dessa disciplina. Desta forma procuramos compreender a forma inovadora de Wallis tratar algumas questões relacionadas ao Cálculo Diferencial e Integral, investigar como é realizada a abordagem de alguns de seus conceitos e relacioná-los com o ensino atual. Esse trabalho se fundamenta essencialmente nos estudos realizados nos últimos anos por pesquisadores interessados na área da História da Matemática que defendem o uso da História da Matemática como recurso a ser incorporado na sala de aula além de levar em consideração o que sugerem alguns documentos oficiais brasileiros em relação ao ensino da Matemática (PCN, DCN, OCM, etc.). Os resultados apurados apontam que esta obra foi de grande influência nas descobertas posteriores sobre os principais conceitos do Cálculo Diferencial e Integral. No nosso caso específico, até então, verificamos apenas a presença de comentários a respeito da obra Arithmetica Infinitorum e de seu autor em dois livros clássicos de História da Matemática e também em um artigo específico sobre este tema. Ao analisar estas obras, percebemos certas características específicas deste autor que podem ser de grande utilidade ao estudar questões relativas ao ensino nos dias atuais. Espera-se que esta pesquisa possa facilitar a compreensão dos conceitos de Cálculo Diferencial e Integral e assim contribuir para o seu ensino e, consequentemente para a sua aprendizagem.