In 19th century Italy there was a reluctance to accept the idea of force as primitive concept (as proposed by Newton and Euler); this position was held by Piola too.
The preferred approach remained Jean D’Alembert’s, according to whom force is a
8After some reticence Piola would appreciate the new mathematical conceptions of Cauchy, but he did not get to share them. For a short note on the common religious ideas of Cauchy and Piola, cfr. [4], note (40) p. 29.
9eqs. (3) and (3bis) in Sect. I, pp. 513–517.
10See for instance [34,49].
derived concept,f =ma, or simply a definition and dynamics comes before statics.
This said, among others, Giambattista Magistrini (1777–1849), to whom Piola made reference [54]11:
Elements of the former [statics] cannot be but a particular determination of the elements of the latter [dynamics], and the equations of it [dynamics] could not be fine and general unless they did not include equilibrium with all its accidents. The practice itself of reasoning used to put statics prior to dynamics let us feel this truth by means of irregularity and contradiction […].
Indeed, [this practice] is compelled to use the expedient of a certain infinitesimal mechanical motion [44].12(A.2.2)
Piola’s epistemological vision as exposed in his early paper, with which he won the prize of the Regio istituto lombardo di scienze in 1824 remained virtually unchanged in subsequent works. Piola’s metaphysics was that of Lagrange: all the mechanics can be expressed by means of the differential calculus. It is not appropriate to resort to other branches of mathematics that use intuition (e.g. the Euclidean geometry) as they can mislead. Piola believed that there exists a ‘supreme equation’, which he called the
“equazione genaralissima”, a key instrument of his treatments. This coincides with what today we would call the virtual work equation based on Lagrangian calculus of variations. However, such an equation cannot be considered as obviousfor itself;
even Lagrange expressed some doubts on it:
It must be said that it is no evident in itself to be assumed as a primitive principle […] [41].13 (A.2.3)
In line with the epistemology of his time, Piola could not explicitly assume the equation of virtual work as a true principle and felt compelled to derive it from first principles which must be absolutely evident, at least in a purely empiric sense, that is experienced in everyday life. In this, Piola abandoned d’Alembert’s position [25],14 who considered mechanics as a purely rational science just like geometry, and linked himself to the ‘empiric’ epistemology of Newton, even though he did not accept Newton’s fundamental concept of force:
It is necessary to cut our claims and, by following the great precept of Newton, to look in the nature for those principles by means of which it is possible to explain other natural phenomena […]. These thoughts persuade us that he would be a bad philosopher who will persist to wish to know the truth about the fundamental principle of mechanics in the way he clearly understands axioms. […] But, if the fundamental principle of mechanics cannot be evident in itself, it should at least be a truth easy to be understood and to be convinced of [54].15(A.2.4)
The empiric first principle introduced by Piola is the superposition of motions: the motion due to the action of two causes is the sum, in the modern sense of vector sum, of
11pp. IX–X.
12p. 450. Our translation.
13p. 23. Our translation. For discussions on the logical and epistemological status of the principle of virtual work in the early nineteenth century see [11].
14p. XXIX.
15p. XVI. Our translation.
the motions due to each single cause.16Along with d’Alembert’s definition of force, this principle leads to the property of superposition of forces. These superpositions of motions and forces are not sufficient to study the mechanics of extended bodies, and the idea of mass must be introduced. Piola followed the norm of his time, by identifying mass with the quantity of matter: he believed that the substance of a given material could be considered to be formed by very small atoms which are all equal.
These can be arranged in space in various different ways and constitute bodies with apparently different densities; the mechanical behavior of a body depends only on the number of atoms it contains. In a scholion Piola clearly expressed his ideas on atoms, or infinitesimal components in mathematics and in physics, rejecting their existence in the former and accepting them in the latter:
I, educated by Brunacci in the school of Lagrange, have always avoided metaphysical infin- itesimal, by assuming that in analysis and geometry (if we want to have clear ideas) we must always substitute them [metaphysical infinitesimals] with an indeterminate quantity, as small as we need: but I accept what could be called a physical infinitesimal, about which the idea is quite clear. It is not an absolute zero, rather, it is a quantity that could be noticeable by other beings, but it is zero relative to our senses [58].17(A.2.5)
Piola ‘proved’ the equation of virtual work, believing to have eliminated all the mechanical and mathematical uncertainties which were in Lagrange’s formulation.
Indeed, Piola had no need to use the somewhat obscure concept of 18th century infinitesimal and used the calculus of variations established rigorously by Lagrange [42]. The equation of virtual work for a system of constrained material points is provided by Piola in the following form:
δL +λδC = 0, (2.1)
whereδL is the first-order variation of the work of all the active forces (including inertia),δCrepresents the first-order variation of the constraint equations andλis a Lagrange multiplier. Hence, the virtual displacements to take into account are free from any constraint and do not need to be infinitesimal.
Actually there was a weak point in Piola’s proof of the Eq. (2.1), that is the vanishing of the work of constraint reactions, which was implicitly assumed but not proved [11]. However, even if Piola had been conscious of the weakness of his reasoning, he would probably not have been severely worried. He had no doubt that the equation of virtual work was right and its rigorous proof was only a question of style, which did not modify the development of the mechanical theory.
By means of the “equazione generalissima”, the undisputed general equation of motion, Piola’s empiric and positivist strategy could be applied in a convincing and interesting way to the mechanics of extended bodies. In his papers, Piola questioned the need to introduce uncertain hypotheses on the constitution of matter by adopting a model of particles and forces among them, as the French mechanicians did. Piola stated that it was sufficient to refer to evident and certain phenomena: for instance,
16The same principle, using a similar vocabulary, was assumed in [48].
17 p. 14. Our translation.
in rigid bodies, the shape of the body remains unaltered. Then, one may use the undisputed equation of virtual work; only after one has found a model and equations based exclusively on phenomena, Piola said, it was reasonable to look for deeper analyses:
Here is the great benefit of Analytical Mechanics. It allows us to put the facts about which we have clear ideas into equation, without forcing us to consider unclear ideas […]. The action of active or passive forces (according to a well known distinction by Lagrange) is such that we can sometimes have some ideas about them; but more often there remain […]
all doubts that the course of nature is different […]. But in the Analytical Mechanics the effects of internal forces are contemplated, not the forces themselves; namely, the constraint equations which must be satisfied […] and in this way, bypassed all difficulties about the action of forces, we have the same certain and exact equations as if those would result from the thorough knowledge of these actions [55].18(A.2.6)
Piola’s approach to mechanics appears astonishingly modern to us; it can be found unchanged in many modern textbooks on rational mechanics:
It frequently happens that certain kinematical conditions exist between the particles of a moving system which can be stated a priori. For example, the particles of a solid body may move as if the body were rigid […]. Such kinematical conditions do not actually exist on a priori grounds. They are maintained by strong forces. It is of great advantage, however, that the analytical treatment does not require the knowledge of these forces, but can take the given kinematical conditions for granted. We can develop the dynamical equations of a rigid body without knowing what forces produce the rigidity of the body [39].19