Comte’s first volume of his positive philosophy ends with a study of mechanics. Although mechanics is for Comte even more concrete than geometry, it still falls within mathematics. Indeed, Comte is against all readings of mechanics, where some elements of analysis are interpreted as real forces, although they would be nothing but means for making calculations. What forces are real can be decided only by observation and experience. Furthermore, Comte adds, mechanics does not investigate what is the nature of these forces, but merely the movements caused by them.
The abstract nature of mechanics, characteristic to mathematics in general, Comte notes, is seen in the fact that mechanical calculations are simplified by assuming bodies to be passive or inert, although in reality they are in many ways active. This idealisation of bodies as inert in mechanics, Comte explains, is not to be confused with inertia in the sense expressed in one of the basic laws of mechanics - the fact that bodies tend to move in straight lines and retain their state of movement. Together with the other two basic laws of mechanics - one being Newton’s law of action and reaction, the other being Galilei’s discovery that forces are independent of one another and can thus be composed with the parallelogram law - the law of inertia is, according to Comte, based on observation, not on any a priori deduction. Particularly, Comte adds, the law of inertia cannot be deduced from the law of sufficient reason.
Comte divides mechanics, expectedly, into statics dealing with instantaneous forces and uniform movement or equilibrium arising from them and dynamics dealing with continuous forces and varied movement arising from them. Within both statics and dynamics, he then differentiates a part examining solid bodies from a more complex part examining fluids. An important problem in this classification in Comte’s opinion concerns the relative status of statics and dynamics. Statics is clearly the older discipline, studied by ancient mathematicians long before dynamical questions. But as Comte has said earlier, the historical order of disciplines does not necessarily correspond to the order of the disciplines in a completed science. Indeed, when dynamics was finally introduced at the start of the Modern Age, statics was regarded as a mere abstract limit case of dynamics.
Yet, as Comte’s favourite mathematician, Lagrange, had argued, the notion of virtual displacement - an application of his calculus of variations to mechanics - could be used to reduce all of dynamics to statics. In effect, Comte is referring to the central idea of the so-called d’Alembert principle that all apparently dynamical systems can be regarded as being in equilibrium. Comte further links Lagrange’s idea with Poinsot’s notion of a force couple, which Comte thinks is a modification of a notion of force from translation to rotation.
Just like with other parts of mathematics, Comte is especially interested in the applications of mechanics. Thus, he points out that statics is used for finding mass centres of bodies, while in dynamics we are trying to calculate movement of a particle from forces affecting it, or the other way around, to find forces creating a known movement. We need not go in great detail to theorems that Comte lists as consequences of the three basic mechanical principles. I will just point out that Comte speaks against interpreting Maupertuis’ principle of least action in a theological or metaphysical manner suggesting that bodies would somehow choose to move in accordance with the principle.
