D'Alembert: Traité de dynamique (1743)

Jean le Rond d'Alembert is a name more often heard in a history of mathematics than in a history of philosophy. Yet, his connections to the world of philosophy in French Enlightenment are not non-existent, because he was one of the editors of the famous Encyclopédie, an encyclopedia meant to cover all human knowledge of the time. D'Alembert was particularly entrusted to care for all the articles engaged with natural sciences, but he also wrote a preliminary discourse explaining the purpose of the work and also engaging with some philosophical underpinnings of it and presenting a general framework for all sciences.

The book I am engaged with here belongs on the surface to d'Alembert's more scientific works, as even the name suggests. In effect, the purpose of d'Alembert in this treatise is, firstly, to present some fundamental principles for movement of material objects, such as inertia, and secondly, based on these principles, to solve some tricky problems, in which, for instance, the velocity of a body hanging from a thread is to be determined.

The topic of d'Alembert's treatise is then something that modern students of physics might find useful to consider. On the other hand, d'Alembert's method of presenting his result has some quaint elements, which show his work to belong to a transitional period between a geometric presentation of Greeks and algebraic presentation familiar in modern mathematics and physics. This is firstly shown by d'Alembert's assumption that everyone knows truths that have been demonstrated by Euclid in his Elements, as he without any explanation refers to a property of a circle that if two lines AB and AC are drawn from the same point A outside circle so that AB touches the circle on B and AC goes through a point D of the circle and then cuts circle also in the point C, then the square of the side AB equals a rectangle formed by AD and DC or AD x DC. While a modern student of mathematics might even have some difficulties following this description, let alone being convinced of it, any connoisseur of mathematics from d'Alembert's time would instantly recognise this statement as the proposition 36 in the third book of Euclid'sElements.

Even more clearly this transitional nature is seen in d'Alembert's habit of embodying quantities like velocity, time and others with lines, which for an eye unused to geometric way of presentation seem to have no connection with each other, although one acquainted with this method could see what mathematical relation the different lines are supposed to exhibit. Then again, in more difficult cases d'Alembert prefers algebraic denotations of mathematical relation. What makes it all even more confusing is that while d'Alembert clearly had an inkling that some quantities (e.g. forces) have a direction, neither the geometric presentations nor algebraic formulas clearly distinguish between scalar and vector quantities. Add to all this the rather muddled view of infinitesimals at the time, in which e.g. differentials of various levels were represented by lines that are supposed to be infinitely small, and a modern reader is bound to feel some puzzlement.

Philosophically most interesting part of the book is, as is often the case with such treatises, the preface. Here we find d'Alembert's take on the muddled question of ”living forces”, seemingly physical problem that still interested such philosophers as Descartes, Leibniz and Kant. The controversy itself started from the apparently obvious notion that something remains same in various interactions of bodies, such as collisions and that this same element would represent the quantity of motion itself. Descartes, among others, had suggested that this must be what is nowadays called momentum or mv, in which m means mass of the bodies involved and v their velocity. Some people, such as Leibniz, objected that what is conserved is actually a more intricate quantity, ½(mv^2). Leibniz also asserted that this quantity expressed a living force of the moving bodies themselves, while momentum was connected only with dead forces affecting bodies externally.

D'Alembert strategy, at least in the first edition of the treatise. is to deny that there is any true controversy. Momentum is conserved in momentanous collisions of bodies, while Leibnizian quantity of motion is conserved in cases where e.g. a medium continuously resists the movement of a body. Now, the case of a resisting medium can be regarded as an infinite sum of infinitely small collisions, which mathematically means that Leibnizian quantity of motion must be an integral of momentum, as it truly is. Which one to take as the more essential expression, d'Alembert concludes, is just a verbal quibble.

Although d'Alembert's solution is, as far as I know, quite correct, it also expresses a more general tendency of disregarding all philosophical problems as verbal quibbles – something quite common for modern scientists. This tendency is shown by d'Alembert also with a question concerning the status of physical laws : are they necessary or contingent? D'Alembert attempts once again to show that both sides are in a sense correct: laws of physics are contingent, because world could have worked according to different laws, but they are also necessary, if you take into account the empirical evidence about the motion of bodies. D'Alembert fails to notice that he has actually endorsed one side of the dilemma – contingency – and rephrased the other side, so as to fit with the first side. Furthermore, from a modern viewpoint, he has accepted a rather strong position . Not many would accept that empirical data would fit in with just one physical theory and thus necessitate it.

So much for d'Alembert, next time another philosopher from French Enlightenment.