Rational mechanics was, indeed, the third field where the Newtonian legacy was of major importance. In the seventeenth century, the term mechanics had a double meaning. In his preface to the Principia, Newton made a clear-cut dichotomy between "practical mechanics" and "rational mechanics." The former referred to all manual arts people used to practice in varying degrees of exactness. Practical mechanics was closely related to geometry, for geometry "is nothing other than that part of universal mechanics which reduces the art of measuring to exact propositions and demonstrations." However, this was not the kind of mechanics Newton wanted to deal with. "Since the manual arts are applied especially to making bodies move, geometry is commonly used in reference to magnitude, and mechanics in reference to motion. In this sense, rational mechanics will be the science, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever" (Cohen and Whitman, p. 382). Half a century after the publication of the Principia, rational mechanics was a well-established branch of Newtonian physics, clearly distinguished from other aspects of natural investigation. A standard definition of the term implied three significant features:
- Rational mechanics was the mathematical study of motions generated from specific forces as distinguished from statics, which examined the forces of a system being in equilibrium.
- The mathematical analysis employed in rational mechanics should be able to represent the generation of the trajectories of moving bodies as distinguished from geometry, which sufficed only for the description of static curves.
- The current formulation of rational mechanics was based on the Principia, as opposed to practical mechanics, which originated in classical and Hellenistic antiquity.
The major contribution of Newton to the establishment of modern rational mechanics was threefold. First, he introduced the notion of attractive force as a dynamic factor of motion. He did so by mathematically constructing the modus operandi of a centripetal force acting as the inverse square of distance; subsequently, he assigned it a natural status by unifying terrestrial and celestial physics on the basis of attraction. His second contribution was that he clearly showed the limits of Euclidean geometry as far as the problems of motion were concerned. Although he himself did not totally reject Euclidean geometry when he composed the Principia, the modification of traditional geometry he suggested there, as well as his mathematical studies on "fluxions" and "fluents," indicated that the only proper mathematical way to treat the problems of motion was infinitesimal calculus. His third contribution was the comprehensive study of celestial mechanics and the explanation of a wide range of celestial phenomena on the basis of universal attraction.
Although the last contribution established Newton as a heroic figure throughout the eighteenth century, the two former did not have an equally straightforward effect on his philosophical profile. There is no doubt that Newtonian mechanics bridged the gap between astronomy and cosmology by presenting a concise physico-mathematical model for the operation of the Keplerian laws. However, the mathematical and ontological foundations of Newton's synthesis became the object of much discussion on the part of his successors. It is somewhat ironic that the transcription of Newtonian mechanics in the language of infinitesimal calculus was carried out on the basis of the mathematical notation suggested by Leibnitz, his major philosophical opponent. In fact, it was characteristic of Newtonian mechanics throughout the eighteenth century that many of the people who undertook the further advancement of Newtonian achievements combined the legacy of the Principia with the philosophical and mathematical ideas of Leibniz. The incorporation of the vis viva, or living force, theory in many Newtonian treatises that circulated widely on the Continent, along with various attempts aiming to render the laws of motion compatible with the metaphysical principles of Leibniz, were two other instances of this characteristic.
The thorn of Newtonianism, however, was the ontological status of attractive force. Thus, by the mid-eighteenth century quite a few significant mathematicians, like d'Alembert and Lazare Carnot (1753–1823), insisted that the notion of force should be expelled from mechanics. Others, like Johann Bernoulli (1667–1748) and Leonard Euler (1707–1783), suggested that a dynamic factor was, indeed, necessary in mechanics, but they also tried to keep a distance from the metaphysical consequences of such an assumption. In any case, the major pursuit of the time was the transformation of the Newtonian mechanics so that it might work exclusively on the basis of kinetic laws. This process culminated with the publication in 1788 of Méchanique analytique. Joseph Louis Lagrange's (1736–1813) work was entirely analytical in contrast to the method employed by Newton in the Principia, which was entirely geometrical. Lagrange was an admirer of Newton but he was also a disciple of d'Alembert. Thus, he shared with the latter the desire to develop a new science of mechanics that would not need the metaphysically laden concept of force. As a result, his Méchanique analytique drew upon d'Alembert's principle, the conservation of vis viva, and the principle of least action, none of which had a counterpart in Newton's work. Additionally, he applied his method to constrained systems of masses, rigid bodies, and continuous media, which was again a substantial departure from Newton's preoccupation with the legitimization of centripetal force acting at a distance.
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