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Despite what one might infer from its main title, this book does not undertake to help the modern reader with the task of following Newton's exposition. What it does do, richly, importantly, and usefully, is to trace the implications of Newton's approach in Principia and the debate among the mathematicians and natural philosophers of Newton's time about his methods and conclusions.Guicciardini explores the plurality of mathematical methods used in Principia and traces how they were interpreted and understood by Newton, his contemporaries, and those immediately after him. "The combatants debated issues whose content is worth considering", Guicciardini says. "The cluster of problems addressed which concern the Principia's mathematical methods is particularly interesting. How should natural philosophy be mathematized? Is it legitimate to use in this context uninterpreted symbols? Can we depart from the established Archimedian or GalileanlHuygenian tradition of geometrizing Nature? What is the value of elegance and conciseness? What is the relation between Newton's geometrical methods of the Principia and calculus? What is the advantage of using the latter?"The book examines reactions and discussions from the publication of Principia (1687) to the publication of Euler's Mechanica (1736), after which the author considers Newton's geometrical methods of presentation "past and obsolete", fully replaced by analytical representations of Newton's results.In Part I, Guicciardini reviews the mathematical style and methods of Principia. He looks at how and why Newton chose to use a geometric calculus in Principia instead of an analytic (algebraic) calculus of the sort he himself had championed and helped develop earlier. The author examines the places where Newton refers to his calculus of quadratures as having provided the proof needed, and identifies the places where Newton actually uses his analytical calculus. In Part 2, he devotes Ja chapter each to Newton, Huygens, and Leibniz, tracing their responses to Principia. He shows how Newton's commitment to that approach to solving mathematical problems actually solidified in the years after publication of Principia, while Leibniz and his followers were following a different path with the differential calculus. In Part 3, he studies the two schools that emerged during the period of this study, the British Newtonian school and the Continental Leibnizian.Guicciardini's discussions of the differences between the synthetic and the analytic approaches to solving problems of dynamics and his explanation of theory of proportions and its use by Newton are particularly illuminating. Newton's