A pervasive understanding holds that the foundation of calculus-based mathematical physics was laid by Issac Newton in his epochal treatise Philosophiae naturalis principia mathematica of 1687. In this historical understanding, what we now call "classical Newtonian mechanics" was born from Newton's work accomplished in his Principia, and was disseminated and digested throughout Enlightenment Europe as a result of the reception of this book. Before Voltaire challenges this understanding by demonstrating the historical gap separating Newton's work in the Principia from the calculus-based mathematical physics that only later became associated with his name. It also shows the important role played by Continental mathematicians, especially in France, in building from Newton's work, but also that of others such as Gottfried Wilhelm von Leibniz and Nicholas Malebranche, the modern science of analytical mechanics. It further distances this history from the direct life and legacy of Newton by demonstrating the important role that the French Académie Royale des Sciences played in creating the institutional crucible from which this new and innovative science was forged. Treating calculus-based mathematical physics as a contingent historical outcome produced through a wide array of intellectual, cultural, social, and political dynamics, this book frees the history of modern mathematical physics from the Enlightenment mythistory of the so-called "Newtonian Revolution." It does so by narrating a fully contingent cultural history of the birth, contests over, and then establishment of analytical mechanics as a foundational French science in the two decades around 1700.