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Emulating the Greek geometers, Newton used synthetic demonstration to present the groundbreaking arguments of the Principia. This paper argues that we can better understand Newton's reasons for using geometry by considering John Wallis's interpretation of synthetic demonstration. Wallis condemned demonstration for failing to explain the mathematical truths it presented. He opposed to it a presentation that combined symbolic analysis with a documented account of discovery. In preferring symbols, Wallis was motivated both by the nascent tradition of symbolic analysis and by contemporary interest in artificial languages. Newton maintained Wallis's characterization of Greek demonstration as adapted to common understanding rather than as strictly elucidating, but he inverted the values Wallis associated with synthesis and analysis. In Newton's new account, synthetic demonstration was preferable precisely because it could address inexpert readers without exposing them to the complications of symbols-based analysis. Newton advanced his arguments on behalf of geometry through portraits of ancient mathematicians: Archimedes and Pythagoras. [ABSTRACT FROM AUTHOR]

The article focuses on the controversy between Isaac Newton and Gottfried Wilhelm von Leibniz in relation to the invention of calculus in the late 17th century. An evidence that Newton had some ideas about the calculus as early as his famous annus mirabilis in 1666, and hints of calculus in the work of his predecessors in England, in particular the writings of John Wallis and Isaac Barrow, is mentioned.

This paper explores Wallis's role as editor of Newton's mathematical work. My objective is to understand how two mathematicians who held different views concerning mathematical method could nonetheless cooperate with one another quite effectively. Most notably, Wallis and Newton pursued different policies as far as the printing of algebra is concerned. In the 1690s Newton held the view that algebra is a heuristic method 'not worthy of publication™. Wallis, instead, for all his life was keen on making algebraic methods explicit in print. As the analysis of the correspondence between Wallis, Collins and Newton reveals, the methodological tension between Wallis and Newton was resolved in such a way that Newton agreed to print his heuristic methods in Wallis's English Algebra (1685) and Latin Opera (1693-99). Newton wished to guarantee his priority rights on discoveries in algebra and calculus, yet he also sought to avoid any tight authorial commitment towards them. Wallis, in contrast, received from Newton material that turned out to be useful for the fulfilment of a nationalistic programme aimed at eulogizing British mathematicians as well as his own work. [ABSTRACT FROM AUTHOR]

In Galileo's opinion, his most important argument for the Earth's motion was based on his theory of the tides that combined the Earth's rotation with its orbital motion so that it alternately accelerates and decelerates the sea. His theory deliberately ignored the Moon's influence, which at that time was generally regarded as occult. Galileo's confidence in his theory was strongly reinforced by its providing a mechanical model. That a theory that ignored the Moon's influence could seem plausible is confirmed by comparison with the theories of Bacon and Wallis. That Galileo's theory could seem plausible despite encountering difficulties is confirmed by comparison with Newton's theory, which is deeply flawed by its inclusion of a vertical response to the total tidal force. That many people now regard Galileo's theory as wrong is due to our having absorbed as natural Newton's idea of lunar attraction. [ABSTRACT FROM AUTHOR]

The article focuses on the "General Scholium" of the "Principia Mathematica" by scientist Sir Isaac Newton. It explores the writings and ideas of his contemporary scientists and members of the Royal Society of London, England John Wallis, Christopher Wren, and Christiaan Huygens regarding the third law of motion from Book 1 of the "Principia Mathematica." The author argues that the "General Scholium," as added to the "Principia Mathematica" twenty-six years after it was first published, does not do justice to the remainder of the book's existing scientific and mathematical methodology. A discussion of Newton's use of logical reasoning and philosophical induction is also presented.