## Abstract and Keywords

This article examines the mutual influences between mathematics and the new sciences that emerged in the long seventeenth century, whereby new scientific enterprises fostered the development of new mathematical methods and mathematical developments in turn paved the way for new scientific research. It begins with an overview of the revolutions in mathematics in the long seventeenth century and the status of the mathematical sciences in the Late Renaissance, followed by a discussion on the work of mathematicians in the late sixteenth century including Galileo, René Descartes, Gottfried Wilhelm Leibniz, Jacob Bernoulli, Gerhard Mercator, and Edmond Halley. It also describes the work done in the areas of organic geometry and mechanical curves, infinitesimals, and mechanics.

Keywords: mathematics, new sciences, mathematicians, Galileo, René Descartes, Gottfried Wilhelm Leibniz, organic geometry, mechanical curves, infinitesimals, mechanics

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