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This chapter examines Cadwallader Colden's colonial philosophy and how it served to integrate European and American scientific thought. The Anglo-Irish philosopher, Bishop George Berkeley, questioned the intellectual credibility of new scientific knowledge. He rejected calculus on the basis that it invoked confusing and philosophically unsustainable terms and claimed that matter was imperceptible and therefore unknowable. These arguments were rejected by Colden, insisting that they threatened the eighteenth century's historic opportunity to create an enlightened age of useful knowledge. This chapter discusses Colden's efforts to defeat Berkleyan philosophy by writing an essay on fluxions in 1743 and introducing a theory of active matter that was published in 1746. It also considers how Colden became embroiled in religious as well as philosophical controversy in his attempt to answer Berkeley. Finally, it explores how Colden combined his investigations into natural philosophy with a revived interest in medicine and physiology during the 1740s.

This chapter discusses Isaac Newton's contributions to algebra and mathematics, and particularly in terms of using symbols. It first examines Newton's idea of unknown variables as quantities flowing along a curve. Fluents, as he called them (from the Latin fluxus, which means “fluid”), were very close to the things that we now call dependent variables, our x's, but limited by their dependence on time. Newton thought of curves as “flows of points” that represented quantities. According to Newton, the fundamental task of calculus was to find the fluxions of given fluents and the fluents of given fluxions. The chapter also considers Newton's work on infinitesimals and how his invention of calculus advanced a wide range of fields such as architecture, astronomy, chemistry, optics, and thermodynamics. It also describes some of the major developments that occurred in the fifty years following Newton's death.

This chapter presents a survey of Isaac Newton’s mathematical work and the development of his theories on the mathematical method that began to mature. It explores Newton’s early influences in the field of mathematics, such as Isaac Barrow and Descartes, and then outlines the first steps in his quest for mathematical mastery. The chapter also details Newton’s encounter with the problem of drawing tangents to plane curves and explores his discovery of the fluxions in the mid-1600s. It also briefly explores Newton’s mathematical maturity when he was elected as the Lucasian Professor at Cambridge University, succeeding Isaac Barrow, in which he discussed and claimed that certainty in natural philosophy can be attained through the use of geometry.

This chapter explores the analytical method of fluxions, as stated in De Methodis. Newton’s method of fluxions can be divided into two parts: The direct and the inverse. Newton considered the techniques of the direct method to be perfected, as presented in his treatise De Methodis. After making his De Methodis treatise, he also sought to develop his inverse method algorithm, while also creating a better conceptual foundation to the direct method. The chapter notes that Newton continued in improving the two methods until he composed the De Quadratura, a work which explains the most advanced refinement of his method of fluxions.

This chapter argues that being flows and folds, but is also distributed by a field. Flows join into stable folds, and folds are conjoined into things, but things are also arranged or ordered together through a field of circulation. This is the third concept in the theory of motion. A field is a single continuous flow that has a kinetic vector for each period on its surface. If flows intersect and folds periodically cycle, fields organize them all in a continuous feedback loop. This chapter provides a kinetic theory of how conjoined flows become organized according to distinct regimes or fields of motion.