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1 Introduction

Leonhard Euler (1707–1783), born in Basel, the first child of Paul Euler and Margaretha (née Brucker), spent his beginning youth not in Basel but in the nearby Swiss countryside Riehen. (Over time the Euler family name has had a variety of spellings, with Öwler and Äweler in the fourteenth and fifteenth centuries and Öwbler, Ewbler, and Ouwler in the sixteenth.) His parents were his first teachers. Margaretha is thought to have instructed him in beginning reading. His father gave an elementary education in mathematics. Like his teacher Jakob Bernoulli, Paul Euler taught his young son mathematics not as an isolated discipline but as underlying all natural knowledge, interrelated with other fields.

Euler, who required instructional preparation completed, was sent by his parents to Basel, perhaps as early as the age of eight. By the second decade of the 1700s, Basel was no longer in its golden period; nonetheless the presence of Jakob and Johann Bernoulli made it a center for scientific and mathematical research.

In Euler’s day schoolboys were sometimes beaten or knocked about until they bled. At other times the teacher might find himself dragged by the hair by an enraged father making an unannounced visit to the class. There is no reason to suppose that Euler’s intelligence saved him from routine indignities and brutalities.

In 1715 Paul Euler hired as a private tutor Johannes Burckhardt (1691–1743), a young theologian with a tolerable background in mathematics. At the time Burckhardt was supporting Johann Bernoulli in arguments with Brook Taylor and other members of the Royal society of London over which was superior, Leibniz’s differential calculus or Newton’s.

After completing his gymnasium education in 1720, Leonhard registered at the University of Basel for courses in the philosophical faculty, which covered fields of learning outside recognized professions. It was the equivalent of the modern secondary school. The philosophical faculty imparted a general education before a student chose a specialty for a higher degree. Through hard work and an astonishing memory, Euler mastered all of his subjects. During his first 2 years, he was enrolled in Johann Bernoulli’s class for beginners in geometry as well as practical and theoretical arithmetic.

Euler spent 2 years earning his prima laurea, roughly equivalent to a bachelor’s degree, receiving it in 1722 at the age of 15 after presenting the ad lectiones publicas, his undergraduate thesis, De temperantia, extolling moderation. In conversations and intellectual debates, Euler began to display to a wider public his substantial learning and his command of Latin. While he advanced rapidly in lower level mathematics courses, Euler did not however quickly gain the attention of his teacher Johann Bernoulli.

In the autumn of 1723, Euler passed the examination of the philosophical faculty for the Master of Arts degree and officially received it on 8 June 1724 at the age of 17. At the graduation session on that day, Euler gave a public lecture in Latin comparing the natural philosophy of René Descartes with that of Isaac Newton and indicating the consequences of each. It was probably in consultation with Burckhardt and Johann Bernoulli that he had chosen this important and timely topic.

Euler’s increasing attention toward mathematics and natural philosophy did not please his father, who obliged him to register in the theology faculty in 1723 in preparation for taking holy orders. He felt fortunate to continue Saturday meetings with the stern Johann Bernoulli. As Leibniz was dead and Newton old and less active, Bernoulli was since 1707 the leading mathematical preceptor in all of Europe. While likely concerned about employment for his genial son, Paul Euler accepted a Johann Bernoulli’s request for Euler’s shift out of theology.

Euler’s friend and classmate Johann II, the youngest Bernoulli son, had tried gain him access to a possible tutorial with the father. But Johann Bernoulli was busy, and flatly refused to give Euler private lessons. Instead he advised him to start reading some more difficult mathematical books and work through them as diligently as he could, and if he came across some obstacle or difficulty, he could visit him every Saturday afternoon.

All suggests that Euler examined such classics as the second edition of Copernicus’s De revolutionibus orbium coelestium from 1566 as well as Kepler’s Astronomia nova of 1610 and Galileo’s Dialogo sopra i due massimi sistemi del mondo published in 1632. Euler’s master’s lecture shows that he was studying Descartes’s Principia philosophiae and La géométrie. He possibly also read Rohault’s Cartesian masterful Traité de physique, a major physics text of the late seventeenth century, which appeared in 1671.

The interests of Euler’s teacher make it probable that he read a range of works relating to the new calculus and its applications, beginning with the Analyse des infiniment petits pour l’intelligence des lignes courbes published in 1696 in Paris, attributed to de L’Hôpital. Euler possibly examined two texts of Varignon, a correspondent, disciple, and close friend of Johann Bernoulli: the Projet d’une nouvelle méchanique, published in 1687, and Nouvelles conjectures sur la pesanteur of 1690. Varignon’s two-volume Nouvelle mécanique ou statique and Éclaircissemens sur l’analyse des infiniment petits, both from 1725, may also have been available. Euler must have studied Jakob Bernoulli’s articles on the theory of infinite series, published from 1682 to 1704 and reprinted in 1713; his Ars conjectandi on probability, with a preface by Nikolaus I, published posthumously in 1713; and Jakob Hermann’s Phoronomia, sive de viribus et motibus corporum solidorum et fluidorum of 1716. Other books to which Euler referred to are John Wallis’s Arithmetica infinitorum (1656) and Brook Taylor’s Methodus incrementorum directa et inversa (1715). No confirming evidence exists on whether Euler saw Johann Bernoulli’s pioneering articles for the Académie des sciences of Paris in 1718 on what would become the calculus of variations. (Most of the bibliographic information is drawn from Calinger (2016).)

Euler was called to the Academia scientiarum imperialis petropolitanae of St. Petersburg by his friend Daniel Bernoulli in 1727. He remained there until 1741 to reach the Académie royales des sciences et belles lettres de Berlin. Here he had some problems with Frederick II that finally pushed him back to St. Petersburg in 1766 where he remained until his death in 1783, at the age of 78 and almost completely blind. One of the most prolific writers, among the greatest mathematicians of all time, Euler treated all the themes of physics, with an approach, however, more as a “mathematician” than an experimental physicist, from astronomy to optics, from electricity to magnetism, and from hydraulics to mechanics to music, leaving an indelible mark in all sectors. He also wrote many pages of philosophy of nature, still of interest today because written with the sober language of the mathematician. His writings, more than 20 books and pamphlets and about 800 papers, are collected in his Opera omnia (Euler 1911–2018). It remains to complete the correspondence, written in several languages. Most are in Latin and French and some in German and Russian (Euler also had competence in Italian, Spanish, Chinese, and Japanese). Despite his strong religiousness, Euler was an illuminist philosopher-scientist, and his faith in reason transpires in all his writings.

2 Philosophy of Nature

Since entered the University of Basel, Euler, as discussed just above, had been deeply interested in the studies of theology and natural philosophy, and only 17 he gave a lecture for his master’s degree comparing the natural philosophies of Descartes and Newton.

Certainly he was not a professional philosopher, but he reflected at length on the classical themes of philosophy of nature. He was not interested in discussions of a metaphysical character, and even for that, he was generally hostile to the approach to the natural philosophy and mechanics of Leibniz’s school, represented in his time by his colleague Christian Wolf (1679–1754). Studying Euler’s philosophy is challenging not so much to understand his works on mechanics or physics (modern sense) but rather to see how natural philosophy was being transformed into the hands of mathematicians. It is indeed his mathematics that influenced his philosophy and not vice versa; in particular his research on Calculus influenced his conception of matter and space.

Euler’s ideas about philosophy of nature are scattered everywhere. The natural reference is however to the Anleitung zur naturlehre (Euler 1862) of the 1750s, referred in the following as the Anleitung, and Lettres à une princesse d’Allemagne (Euler 1770–1774) of 1760s, referred in the following as Lettres. The Anleitung did not spread greatly, even because of his late publication. The Letters had instead a different fate. Michael Faraday (1791–1867), for instance, read the letters (Calinger 2016, p. 565, note 35) but not the Anleitung which was probably read by Bernhard Riemann (1826–1866) (Speiser 2010, p. 46).

Euler’s philosophy of nature was generally well received, praised by Voltaire, for instance. According to Alexandre Koyré, the Lettres may be included among prominent Newtonian popularizations (Koyré 1965, p. 18). But also by Leibniz, even though here and there, Euler criticized his philosophy of nature. Euler had no particular reason to stand on one side or the other in the philosophical debates of the period though he had a profound estimate of Newton referring to him as the Summus Neutonus; not being a professional philosopher, he could limit himself to accepting those ideas that seemed closer to his sensitivity as a mathematician without looking deeply into their consistency.

It should be cited however a strong criticism by two scholars well connected with Euler, d’Alembert, and Lagrange. They judged him severely. Lagrange wrote to d’Alembert, “Our friend is a great analyst but quite poor a philosopher” (Lagrange 1867–1892, Vol. 13, p. 135). (Letter of Lagrange to d’Alembert, 16th June 1769.) D’Alembert replied indignant: “It is incredible that such a great genius as him on geometry and analysis is in metaphysics so inferior to the smallest schoolboy, not to say so flat and so absurd, and it is the case to say: Non omnia eidem Dii dedere” (Lagrange 1867–1892, Vol. 13, p. 148). (Letter of d’Alembert to Lagrange, 7th August 1769.)

2.1 Anleitung zur Naturlehre

The Anleitung has been received incomplete. The end of Chap. 5 and the beginning of Chap. 6 are missing (which probably occupied about seven pages). There are reasons to think that, even with these pages, the work could not be not complete and we can therefore ask why Euler did not finished it (Speiser 2010). In its current form, the book consists of 21 chapters that can be grouped: Chaps. 1–5, the role of natural philosophy and the main properties of matter; Chaps. 7–9, general introduction to the principles of mechanics; Chaps. 10–11, apparent motion, general rules of motions; Chaps. 12–18, the property of matter; Chaps. 15–18, possible forms of matter; Chap. 19, gravity; and Chaps. 20–21, principles of hydrostatics and hydrodynamics.

The opening of the Anleitung suggests a traditional treatise on natural philosophy or natural science, even of Aristotelian mold. According to Euler, natural science is a science that aims to explain the causes of changes that occur on material bodies. Wherever there is such a science, he continued, it is very incomplete, since it is able to state with certainty the causes of only very few changes. Moreover, changes are restricted to those occurring on inorganic or material bodies, thus distinguishing natural science from the science of the mind, which aims to explain mental changes. Nothing is said about living beings (Euler 1862, Vol. 2, p. 449). All changes involving material bodies must arise from the essence and from the properties of the bodies themselves. What is common to all material bodies without exception is called a property of the bodies, and therefore all things not sharing this property are excluded from the domain of material bodies. The general properties of material bodies are those shared without exception. The essence of material bodies is a property that is not only shared by all material bodies but which is such that all things having this property must of necessity be considered to be material bodies. The best way for natural science to proceed is first to investigate the general properties, that is to say the essence of material bodies, and subsequently to subject all particular types of bodies to a similar examination (Euler 1862, Vol. 2, pp. 450–453).

Following the reading of the Anleitung, it is understood however that the causes and the properties of which Euler spook are those useful for a mathematical treatment of the philosophy of the nature, adopting a mechanistic approach in the broad sense; they are the extension (that is, the position and or a configuration), the mobility (velocity and acceleration), the persistence or inertia (mass), and the impenetrability (force, because forces are due to impenetrability). Euler wanted to found a coherent system, able to describe all the inorganic nature, starting only from some principles of mechanics and a few other hypotheses. Even Newton, another “mathematician” who wondered about the relationship between the philosophy of nature and mathematics, in some of his Queries (28–31) had sketched some ideas in this direction, but they remained only queries, that is, questions.

The first general property of material bodies is the extension, and anything that has no extension cannot be regarded as a material body. We are not only convinced by our experience that all material bodies that we know possess extension, but our concept incorporates extension in such a way that we can exclude all things without extension from the category of material bodies. It follows then that whatever can be said of extension per se can without exception also be said of material bodies. Everything with extension is divisible, and divisibility can be continued ad infinitum; therefore all material bodies are infinitely divisible (Euler 1862, Vol. 2, pp. 453–455). However Euler did not enter the question of the actual corpuscular (atoms) or continuum constitution of matter. What he could say is that one knows from experience that the actual subdivision of many material bodies can be carried out to an astonishing degree and that our tools and senses are too blunt to permit this subdivision to be carried even further. However he was not discussing what can actually be done but rather the merely the possibility of taking the subdivision even further.

For liquid matter Euler however denied the existence of atoms.

A liquid matter cannot be formed from a number of small particles that are solid and hard, for whatever the shape and arrangement of the particles might be, for it is not possible that a pressure that acts at one location will propagate in all directions with equal force. (Euler 1862, Vol. 2, p. 526; Translation by Hirsch E.) (A1)

In the following a verbatim exposition of Euler argument is referred to. If one imagines these particles in the first place to have the shape of a cube and imagines them to be arranged in an orderly manner on one another, then one sees easily that if the uppermost is pressed down by a force, the lowest will press on the ground with equal force, but sideways no force will be exerted; therefore if many such rows fill a vessel and a force presses on them from above, then the bottom of the vessel will experience an equal force, but the sides will experience none at all. If the particles were lying together in a disordered manner, then the pressure could also be propagated sideways, but it would not be the same in all directions. But what can be said of cubic particles is equally valid for all other shapes that involve corners, for which reason most natural scientists ascribe to the particles a spherical shape; but it is easy to show that from these, if they are assumed to be solid and hard, the main property of liquids cannot be obtained. It is only necessary to imagine a heap of spheres, arranged as is usual for cannon balls, to see that if they are pressed from above, they exert no force sideways, or that at least such a force would not be the same in all directions. In addition spheres that fill a vessel cannot lie together in such a regular manner that a great dissimilarity in their arrangement would not develop, that also would prevent a uniform pressure. If one assumes the spheres to be in uniform motion, then the pressure can change but cannot be maintained the same in all directions; and one would have to regard as extremely rare a case where the pressure was the same, remembering that this constitutes the essence of a liquid. One need only to consider that if three spheres lie in a straight line, the middle one can always be pushed sideways, whatever forces may be acting on the outer ones. On the various types of bodies (Euler 1862, Vol. 2, p. 526).

An important concept that Euler considered worthy to be exposed in his foundational text is what today is called potential energy or possibly work, to which he referred to as the effectiveness and to which he also assigned the name effort des forces sollicitantes (Euler 1757b, p. 287). (It must be said that Daniel Bernoulli in his work of 1750, Remarques sur le principe de la conservation des forces vives pris dans un sens général, introduced something like the effectiveness (Bernoulli 1750, p. 359).) The effectiveness of a force is the integral quantity that is found if one multiplies the force with the differential of the distance to which it pushes the body and then integrates. This concept is the more of the utmost importance, because (1) the sum of the effectiveness of all forces ∫Pdx +  ∫ Qdy +  ∫ Rdz always has the same magnitude – given by the increase of living force – if one had assumed three different planes, that is, in modern term, it is invariant with the change of coordinates. (2) The whole theory of equilibrium is based on it. For it can be shown that equilibrium cannot occur, if the sum of the effectivenesses is not a minimum or occasionally a maximum: “This marvelous theorem was first derived by the world renowned President de Maupertuis and is closely connected with the other general principle of frugality. From this we see at the very least that the effectiveness has a major influence on all motions that can be produced by forces, and that it deserves to be given a special name [emphasis added]” (Euler 1862, Vol. 2, p. 493; Translation by Hirsch E.). It also must be remarked however that Euler seems not to be conscious that he had to limit the nature of forces in his definition; only for what are now called conservative forces – for which the integrals appearing in the definition of effectiveness are independent of the path – Euler considerations has any meaning.

One more concept Euler maintained worth to be underlined is that of pressure into a fluid with its modern meaning, in English stress that is internal force per unit of surface. Until then pressure was simply a force of contact, the greater, the greater the surface. Euler introduced this new concept, maintaining ambiguously the same name pressure both for the global force and for the force per unit of surface. Because the pressure (traditional meaning), said Euler, depends on the magnitude of the area on which it acts, it is best if one represents it by a magnitude, the pressure (new meaning), which, when multiplied by an area, expresses the magnitude of the force that acts on the area, and the constancy of the pressure rests on the fact that this magnitude is constant. If we express by aa × k the force p, which acts on the base area S = aa, then the pressure on any area ff is:

$$ \frac{ff}{aa}k\times aa=k\times ff $$

Therefore when k remains the same, then the pressure on the liquid matter is also the same (Euler 1862, Vol. 2, pp. 523–524). To picture this completely, Euler regarded k as a height and expressed the force acting on the area ff by the volume of a cylinder of base area ff and height k, since the volume of such a cylinder is ff × k. This picture is very convenient, according to Euler, because forces are best represented by weights; for this one selects a uniform matter and says that the force, by which the area ff is pressed, is just the weight of such matter that fills the volume of the cylinder of volume ff × k. Therefore such a height k gives us a clear understanding of the force that presses on the inner wall of the vessel filled with liquid matter, and that at the same time presses on all the parts within. For the larger or smaller this height is, the larger or smaller, in the same proportion, will be the force of the pressure (Euler 1862, Vol. 2, p. 524). A similar statement can also been found in the Principes generaux the l’état d’équilibre des fluides (Euler 1757a, pp. 5–7). Here he introduced the symbol p, still used today though limited to a pressure of external loads, instead of k to indicate stress. Euler also used his concept of stress for solids bodies (Capecchi 2001, p. 27).

It may seem strange that in a treatise dealing with general aspects, Euler had also found ample space to exquisitely technical aspects concerning the laws of hydrostatics and hydrodynamics (Chaps. 20–21). It must be kept in mind however that for Euler fluids were not only the ordinary ones, for example, water and air, but also the aether (see next sections) was a fluid and as such subjected to the laws of hydraulics. David Speiser, with a bit of whiggism, suggests that the advantage of Euler over Newton, in order to provide a unified theory of the various branches of physics, is the concept of the partial derivative equations, the mathematical basis of any field theory, a new tool, created by Euler himself and d’Alembert – and others – to formulate the mechanics of fluids. And it is thanks to this new mathematical tool that Euler was the first to imagine a unified field theory, an idea, and a hope that are at the center of the ambitions of physicists (Speiser 2010, p. 45).

In fact, according to Euler, all physical phenomena, with the possible exclusion of gravity (see below), can be reduced to the interactions of four scalar fields (the densities and pressures of gross matter and aether) and two vector fields (the velocities of matter and aether), interconnected and governed by partial derivative equations. Seen as an attempt, Euler’s way of proceeding with hydrodynamics is not very different from that used in modern physics to unify field theories, despite the present more in-depth knowledge of the structure of matter and the huge amount of accumulated empirical data. Even today, the basic differential equations of a field of physics, whose theory is known, are also considered as fundamental for other sectors, in which a theory is still lacking. In the case of Euler, the basic differential equations are those of hydrodynamics, and the radical hypothesis is that of an unique aether (Speiser 2010, p. 46). Despite the considerations of Speiser, which seem convincing, in the Anititung there is however no mention of the aethereal fluid, neither for what concerns electricity nor for what concerns magnetism. These topics that will be treated with plenty in the Letters.

2.2 Lettres à une princesse d’Allemagne

At the time of the publication of the Letters, Euler stood at the peak of his career. He was the director of the mathematics class (division) of the St. Petersburg Academy and a well-known mathematician. Although he sent the individual letters between 1760 and 1762, he had derived them from articles dating back to the 1720s. Thus, notwithstanding the didactic character, they offer an outline of his scientific ideas and the modifications occurring in them.

The letters, 234 in number (a thousand pages), originated from lessons delivered to the princess Charlotte Ludovica Luisa, a relative of Frederik II of Prussia, but were conceived more generally for young students. They met with prodigious success; by the turn of the nineteenth century, they had been translated from the original French into eight other languages: Russian, German, Dutch, Swedish, Italian, English, Spanish, and Danish. The numbering is different from edition to edition, here that of 1770–1774 edition is considered (Euler 1770–1774).

Euler’s letters essentially concern philosophy of nature considered from different points of view, that of the traditional philosopher, with considerations of physica generalis, and that of the philosophers emerging at the time. It is not easy to group them by themes; it can be said that the first letters (about 140) have a more general character; the others are more technical and deal with current topics in physical research, among them at least 17 letters are devoted to electricity and nineteenth to magnetism.

Among the first letters, there are five dedicated to music (4–8). We are in a period in which the musical theory was undergoing somehow a revolution with Jean-Philippe Rameau (1683–1764) among the protagonists. Euler, only 24, had written an interesting paper on music, the Tentamen novae theoriae musicae of 1730–1731 (Euler 1739b), which was criticized by Rameau. The two “musicians” exchanged some letters about consonances in the 1750s (Calinger 2016, pp. 362–363).

Letters (17–44) follow that deal with optics. The first four of the theory of propagation of light with particular reference to Newton; only a nod is made at Descartes; the others face problems of geometrical optics, of brightness and color of bodies, and also of the structure of the human eye. Euler compared the “emanation” (projectile) theory attributed to Newton with his vibrational theory, reported in particular in the Nova theoria lucis of 1744 (Euler 1746b). He did so without mentioning Huygens who had proposed a theory in which light spread in a medium as pulses. (About the difference between the conceptions of Euler and Newton and for a justification of the absence of the name of Huygens in Euler’s writings of optics, see Hakfoort (1995).) According to Euler, to explain the nature of light and color, it must be admitted celestial spaces filled with a subtle matter, he called aether, a fluid matter like air but much thinner, nevertheless, with much greater elasticity (using a modern term, stiffer than air). And as in the air, the sound is propagated, so the light propagates in the aether (Euler 1770–1774, Vol. 1, Letter 19).

In the letters from 45 to 79, there are general considerations on mechanics. They speak of gravity in a general sense, of the law of universal gravitation, to conclude with the explanation of gravity that is associated with the elasticity of the aether, without however going into details. It is here that one sees that a mature Euler is not fully satisfied of his mechanical explanation of gravity. Indeed, in the letters 46 and 54, for instance, he wrote that “philosophers have warmly disputed, whether there actually exists a power which acts in an invisible manner upon bodies; or whether it be an internal quality inherent in the very nature of the bodies, and, like a natural instinct, constraining them to descend” (Euler 1770–1774, Vol. 1, Letter 46, p. 195), “On this question philosophers are divided. Same are of opinion that this phenomenon is analogous to impulsion; others maintain, with Newton, and the English in general, that is it consists in attraction” (Euler 1770–1774, Vol. 1, Letter 54, p. 230). Thus declaring his incapacity of a definitive choice.

Letters from 80 to 87 concern aspects of natural philosophy, bodies, and spirits. In the letter 76 Euler criticized Wolf’s conceptions of dynamics. He did this by referring substantially to what he had already written in his Gedancken von den Elementen der Cörper of 1746 (Euler 1746b) (see below).

Letters 88 to 114 address issues of ethic, psychology, and logic. Particularly in the letters 103–105, the diagrams of Euler-Venn are used to explain some logical relations. It is true that the idea of representing graphically logical relationships was not completely new; this had happened in some treatises of the eighteenth century. But it was to Euler that the mathematician John Venn (1834–1923) referred to in his studies on logic more or less a century later. Some letters about epistemology follow, from which the influence of Descartes and Locke and perhaps even of Condillac transpire – without explicit reference to them – and letters of metaphysics in which Euler repeated the criticism toward Leibniz’s monad system. From letter 133 onward, physical (modern sense) problems are addressed. In particular letters from 138 to 154 deal with electricity and letters from 169 to 186 with magnetism.

Apart from a summary of a paper by Aepinus (Euler 1761), the only left written works on the theory of electricity are on the Letters. This notwithstanding he considered electricity a very interesting field to be explored:

The subject which I am now going to recommend to your attention almost terrifies me. The variety it presents is immense, and the enumeration of facts serves rather to confound than to inform. The subject I mean is electricity, which for some time past has become an object of such importance in physics that everyone is supposed to be acquainted with its effects. (Euler 1770–1774, Letter 138, Vol. 3, p. 277; Translation in Speiser (2008)) (A2)

Euler was primarily interested in the electric field, not in the charge (he was among those who rejected the idea that there were two types of electricity). For him electricity is due to the presence of one electric fluid, the aether which in normal circumstances fills the pores which travers all bodies in all directions. The electric qualities of a body depend on how easy or how difficult the pores are to open. In conductors (modern term) – metals and such – the aether can easily flow in and out; in this case, the pores are always open, so to speak. With other materials a certain strength is required to open the pores through friction: these are the insulators (modern term).

To magnetism instead Euler dedicated some papers and letters, see, for instance, Radelet de Grave and Speiser (2004) and Euler (1748a, b). Magnetism also is associated to a fluid. Originally – in the Anleitung, for instance – in Euler’s opinion, the same aether was responsible for gravity, magnetism, electricity, and light. In the Lettres, in particular in Letter 176, instead he was of the opinion that the aether associated to magnetic forces was a special kind of subtle matter and not simply the subtlest part of a single all-pervading aether (Euler 1770–1774, Letter 176, Vol. 3, pp. 118–119). He seems never to have had any doubt that the light aether is responsible for the electric phenomena as well.

2.3 Matter and Its Properties

Euler dealt at length with bodies, matter, and related properties here and there in his scientific works. However, there are some papers in which he focused specifically on the subject. Among them: Dissertatio de igne, in qua ejus natura et proprietates explicantur written in 1737, Gedancken von den Elementen der Körper of 1746, Recherches physiques sur la nature des moindres parties de la matiere of 1746, Réflexions sur l’espace et le tems of 1748, Recherches sur l’origines des force of 1750, Lettres à une princesse d’Allemagne sur divers sujets de physique & de philosophie of 1760–1761, and Anleitung zur Naturlehre of 1750s. (Euler published anonymously Gedancken von den Elementen der Körper, an anti-Wolffian work. It was immediately recognized as bearing his signature however.)

According to Euler, bodies are characterized by the following general properties: extension, mobility, inertia, and impenetrability. Among them the fundamental is impenetrability which as such should be considered the very essential property of the bodies.

Whatever is impenetrable belongs to the category of bodies, and therefore the essence of bodies is their impenetrability, on which therefore all the other properties must be founded (Euler 1862, Vol. 2, p. 472). (A3)

A body has extension in common with space and mobility with moving images projected on the wall, both of which however do not possess the property of impenetrability.

According to Euler, from impenetrability, extension and mobility follow. His argumentation on these points has more a rhetorical than demonstrative value but is equally interesting. From the very convincing thesis (a) no-extension → no-impenetrability, from the rule of classical logic, it follows (b) impenetrability → extension. More difficult is to argue (c) impenetrability → mobility; indeed a thing can be at rest though impenetrable; thus impenetrability is not necessary for actual motion; what can be said is that impenetrability is not in contrast with the possibility of motion, that is, mobility, but this makes weak the implication (c).

Inertia, called persistence (Standhaftigkeit) in the Anleitung, is also associated by Euler with impenetrability being linked to mobility “because when a thing is mobile, it should also have persistence, since otherwise any change would occur without sufficient reason” (Euler 1862, Vol. 2, p. 472), and mobility implies impenetrability. But the derivation (d) impenetrability → persistence has more problems than derivations discussed previously. Euler’s strategy is to prove the implication (e) mobility → persistence that with the implication (c) would give (d). The derivation of the implication (e) implies the principle of inertia that Euler, like d’Alembert, based on the principle of sufficient reason, which has been the object of criticism by many scholars. Because implication (d) would derive from two weak implications (c) and (e), it would consequently be two times weak. For a different view on this point, see Gaukroger (1982, pp. 141–142).

According to (Gaukroger 1982), the idea that impenetrability constitutes the essence of body depends upon two Euler’s claims: it is unique and necessary to a body, and it is irreducible, in the sense that conceiving a body as impenetrable is a primitive clear notion. Both Boscovich in the Theoria philosophiae naturalis of 1758 and Kant in the Metaphysische Anfangsgründe der Naturwissenschaft of 1786 thought impenetrability of bodies an obscure idea requiring elucidation in terms of a more basic notion that of repulsive force. For them, repulsive force was the physically primitive notion not requiring further elucidation, just as attractive force was physically primitive (Gaukroger 1982, p. 140). Considering that Euler was deriving force from impenetrability (see below), his is exactly the opposite path from that of Boscovich and Kant.

Bodies are portions of matter; thus Euler devoted a lot of attention to the properties of matter. The Recherches physiques sur la nature des moindres parties de la matiere, a relatively early work, began with the following question: “It is an important question in physics and metaphysics to establish whether or not the smallest part of matter are similar to each other” (Euler 1746c, p. 289). Euler tried to give a physical answer to this metaphysical question coming to the conclusion that all the smallest parts of bodies, referred to as the smallest molecules of matter, are made of the same stuff. In order to speak about the smallest molecules, it is necessary to distinguish between the properly said matter (groben Materie), gross matter, characterizing the inertia of bodies to which only the term molecules is referred to, from another type of matter, which is usually not recognized as such, called subtle matter, interposed among molecules. Notice that notwithstanding Euler used the term molecules, he said nothing about a corpuscular conception of matter, that is, the molecules may be not corpuscles.

Following these premises, repeated some years later in the Anleitung, Euler argued that there are only two kinds of matter, a subtle matter and a gross matter. The different apparent nature and density of bodies depend on a different combination of subtle matter and gross matter, with the subtle matter that fills the pores of the gross matter that may be more or less numerous and great. The molecules of gross matter have no pores, and even if they cannot be divided in act, being extended, they can be divided with the imagination. The division of matter remained a theme dear to Euler, who maintained, if not an ambiguity, at least a lack of explanation. For liquids, with fairly simple reasoning, as already shown in the previous section, he had “succeeded” in proving that there are no indivisible finite molecules; for solids Euler escaped the problem of the actual existence of molecules; what he said concerned the extension which is always indefinitely divisible.

The thesis that there are only two kinds of matter is carried out by Euler with both physical and metaphysical arguments. Assuming a metaphysical point of view, he invoked a principle of economics, which however is not so rigid because at least two kinds are allowed. While for the gross matter Euler arrived at a more or less convincing argument on the fact that it is only of one kind, for the subtle one, he hypothesized that there may be more than one, perhaps only two. For what the uniqueness of gross matter is concerned, after making considerations based on daily experience that lead to believe that there is a single type of (gross) matter and that the differences found among the various bodies depend on the nature and quantity of the pores in it present, Euler passed to the demonstration of this affirmation, on the basis of two laws of empirical nature. The first law concerns the fact that the force of gravity varies with the inverse of the square of the distance, as Newton has shown (Euler’s words), and thus the weight of bodies decreases with the square of their distance from the center of the earth. The second empirical law concerns the fact that bodies of different weight fall with the same temporal law (in vacuum).

From the law of the inverse of square, Euler proved that the density and thus the pressure of the aether toward the center of the earth which causes gravity (see below) must decrease with inverse proportion of the distance. Thus, if x is the distance from the center of the earth, the pressure of the aether should vary as h − A/x, being A a constant of proportionality and h the pressure of the aether when at rest, as it is for x = ∞. From this expression of the pressure, it is easy for Euler to prove that the weight of a body is proportional to the volume of the gross matter it contains. Then, from the independence of the temporal law of fall for bodies of different weight, and from the law of mechanics (force and mass are proportional), it easy to prove that weight and mass are proportional. Which implies that the mass is proportional to volume, and because mass for Euler is the same as quantity of gross matter, it means that the quantity of gross matter of body is proportional to its volume, and thus the density of gross matter is constant, and thus the gross matter is of one kind only (Euler 1862, pp. 542–545).

A first draft of the hypothesis of the existence of a subtle matter or aether can be found in the Dissertatio de igne, in qua ejus natura et proprietates explicantur that Euler wrote in 1737 for the award of 1738 proposed by the Académie des sciences of Paris, which was assigned to him (Euler 1752a). To explain the phenomenon of combustion, Euler claimed that the flame arises from the sudden explosion of molecules of “materiae subtilis igneae compressae” and to illustrate his conjecture used a corpuscular model: “Let’s imagine a great quantity of glass spheres, full of air strongly compressed, and this mass of spheres is the material we want. Suppose that only a sufficient force intervenes to break a single sphere; and it is clear that both the impetus of the air and the projection of the fragments of glass will produce a similar effect in the nearby spheres, and then from these in all the others, until all are broken, emitting with immense roar the air that they enclosed” (Euler 1752c, p. 10).

Euler went on to argue that the properties of fire, due to a specific igneous matter, are not reducible to the ordinary laws of mechanics, like the principle of inertia or the proportionality of action and reaction. The heat given off by the fire and the rapid spread of the flame in all directions violate these principles and must be explained in another way. It is necessary to resort to a further postulate: the existence of an aether, a substance distinct from the igneous particles. The flame does not instantly disperse in the air due to the resistance of this aether, much more elastic and rarefied of the igneous matter, which surrounds and holds the flame in an unstable equilibrium. The continuous shocks that are created at the limit between the aether and flame give place to vibrations that generate light and transmit it in a straight line, like it occurs for the waves in the air (Euler 1752c, p. 19). The postulate of the ether was certainly not new in the eighteenth century. Many variants of it were formulated that referred above all to the conjectures that Newton had advanced, albeit in a temporary problematic form. For a discussion of the various hypotheses discussed in the period 1740–1750 by physicists, see Cantor and Hodge (1981).

The properties of subtle matter had been discussed more in depth in the Recherches physiques sur la nature des moindres parties de la matiere and in the Gedancken von den Elementen der Körper, to be taken again in the Anleitung. In the second part of the former of the two papers, Euler first denied the possibility of the existence of vacuum with metaphysical considerations, scarcely convincing indeed while he did not mention a physical more convincing (for us) motivation, namely, that the presence of vacuum would not justify the oscillatory theory of light proposed in his Nova theoria lucis, where propagation is conceived, in analogy to that of sound, as elastic waves propagating in the medium. With regard to the nature of the subtle matter, he limited to assert that it is very different from the gross matter; it is not gross matter made of very tiny corpuscles separated by vacuum but rather a continuous substance or, with a technical modern word, a continuum.

There will be thus at least two main kinds of matter; one [the gross matter] which gives the fabric of sensible bodies, whose the particles have all an unchangeable degree of density, which is even greater than the apparent density of gold; the other [the subtle matter] kind of matter will be that of which the subtle fluid which cause the gravity is composed, and we call the aether. (The density does not vary in time, that is, gross matter is incompressible (Euler 1862, Vol. 2, p. 510).) (Euler 1746c, p. 300) (A4)

In the Anleitung, Euler spook at length about his continuous aether and did not feel embarrassed to introduce a concept which was alien to the dominating mechanistic philosophy where the aether had usually a corpuscular nature. Euler’s aether fills the whole space, leaving no emptiness. Air consists essentially of aether with small corpuscles of gross matter. While the gross matter is incompressible, the aether is compressible:

Gross matter is therefore not capable of any change other than in the appearance of its shape, which, if appropriate forces are available, can be changed in arbitrary ways.

[…] It does not appear to be true that subtle matter also has always and everywhere the same density, such that it could through no force be driven into a smaller space. Instead an important difference between gross and subtle matter seems to be that the latter can be compressed. (Euler 1862, Vol. 2, pp. 511–512; Translation adapted from Hirsch E.) (A5)

The aether although has the property to expand or shrink and behaves like an elastic medium and has a certain density that is proper to it. However, in no case it is possible to reduce a portion of aether to a point, that is, to effectively annihilate it (Euler 1862, Vol. 2, p. 513). Euler clearly stated that the compressibility of subtle matter does not imply the violation of its property of impenetrability.

Euler did not propose a convincing, at least for a modern, explanation of the force/pressure caused by the elasticity of aether which is not deducible from any of the typical properties of matter, extension, inertia, impenetrability, and mobility and therefore should be treated as a primitive (essential?) concept. Forces on gross matter are explained in Euler’s mechanics by means of impenetrability and inertia. For instance, suppose an elastic body in motion that impacts with a rigid surface, it receives a force and rebounds because of the impenetrability of the surface and the resistance to change its motion from the inertia of the ball. The pressure of the aether against a rigid surface cannot be explained in the same way because the inertia of the aether is assumed to be negligible: “because experience shows that the celestial bodies do not suffer effects with their impact with the aether that fills the universe” (Euler 1862, Vol. 2, p. 542). Certainly there is an analogy with gases; but in this case, one (Euler himself) can resort to a mechanistic explanation; this was what Daniel Bernoulli did in his Hydrodynamica of 1738: the pressure of a gas is determined by the impact of the particles on the walls of the vessel that contains it (Bernoulli 1738, pp. 200–203). But for Euler the aether is not formed by particles.

Euler also doubted that there is on kind only of subtle matter: “Whether there are several kinds of this subtle matter, some of which are denser than others, we shall not be discussing here, but if indeed there are several kinds, we shall refer to them collectively as subtle matter. As long as the explanation of what occurs in nature does not require several such kinds, it would be bold and against the rules of a sound science of nature if, merely following our imagination we were to increase the number of kinds of subtle matter” (Euler 1862, p. 509).

According to him, the aether that is found in the universe is compressed far beyond its natural state and consequently exerts a great elastic force on the bodies formed by the gross matter (Euler 1862, Vol. 2, p. 517), able to explain the resistance of the materials to ropture, their elasticity, and the force of gravity.

The resistance of solid bodies is associated with the hydrostatic pressure that the subtle compressed matter exerts on the various particles of ordinary gross matter, an explanation similar to that suggested by Galileo with the use of horror vacui. When two smooth grains of gross matter come into contact with each other, the elastic force of the aether makes them adhere strongly, and the breaking strength is maximum. If, on the other hand, the contact between the grains is not complete, then the breaking strength will be the minor the minor the contact surface.

To show this more clearly, let the two bodies of Fig. 1, ABCD and ABEF, be joined at the surface AB such that between them the cavities ab, cd, and ef are filled with the aether. The body ABCD will then be forced against the other body by the aether that presses on the planes CD and EF; on the other hand, it will be pushed back by the aether in the cavities ab, cd, and ef. Therefore the force with which the body ABCD is pressed against the other body ABEF is the result of the two constraining forces.

Fig. 1
figure 1

Strength of bodies (Euler 1862, Vol. 2, Table XX, Fig. 235)

Full size image

The lake of resistance in fluid is explained by the fact that here gross matter is so suffused by the aether that the particles have nowhere immediate contact. To see how this could happen, imagine that every particle of gross matter is surrounded by subtle matter, and the particles consequently never approach each other so closely that there would not remain some subtle matter between them. If every particle were at rest, such a mixing would be hard to understand; but if the subtle matter is in motion so that it continuously flows between the gross particles, then in this way immediate contact can be impeded (Euler 1862, Vol. 2, p. 535).

The elasticity is explained by admitting that inside bodies there are cavities (pores) filled with subtle matter that do not communicate with the outside. When a body is deformed, either compressed into a smaller space or expanded into a larger one, then there will be change in the size of its pores, some being expanded, but others being compressed.

To explain this more clearly let us assume in the first place that a body is compressed into a smaller space. Since gross matter itself is incompressible, this cannot happen other than by making the pores smaller. In that case the apparent density of the body must increase, because the whole matter of which the body consists, or at least the gross matter, since the subtle matter can be neglected here, must have been brought into a smaller space. Let a3 be that part of the body that is filled with gross matter, and e3 the remaining part that only contains subtle matter, the sum of all pores taken together, then a3 + e3 is the size of the body, a is its mass and a3/(a3 + e3) is its density. But a3 cannot be changed, therefore when the body is brought into a smaller space only e3 is diminished. […] If after the change in shape of a body the closed pores are neither larger nor smaller, then the body retains this changed shape. But if the closed pores become wider or narrower, then a force will develop in the body tending to restore the previous shape. (Euler 1862, Vol. 2, pp. 537–538; Translation by Hirsch E.) (A6)

The elasticity of commonly experienced bodies is thus explained by means of the elasticity of the aether, which is clearly a vicious circle. Thus what Euler actually explained was not the elasticity itself but rather why bodies made by an incompressible matter exhibit elasticity and why the elasticity differs from body to body. The difference in elasticity lies in the diversity of the dimensions and arrangement of the pores.

The account above referred about elasticity explains the behavior of the various elastic bodies, for example, the loss of elasticity as a result of heating. Indeed, the subtle matter contained in the pores is set into motion by heat, opening up access to previously closed pores; at the end the matter has less and larger pores than before. Therefore even though before heating there were large forces, these can vanish after heating.

The weight of the bodies, that is, gravity, is explained by admitting that the density, and therefore also the pressure of the aether surrounding the earth, increases by increasing the distance from its center. Euler is very precise in his explanation. Gravity should be associated to the gradient of pressure in the aether around the earth that determines the movement of the ethereal fluid downward with the consequent downward drag of the gross matter, determining the heaviness of the bodies (Euler 1862, Vol. 2, p. 542).

The section devoted to the heaviness of bodies ends with the consideration that even if one does not know why the pressure decreases in proximity to the planets, to stay with this doubt is always better than not knowing anything as when one says that the heaviness is due to attraction.

Although we have to stop here and hardly can hope ever to find the cause of the diminution of the elastic force of the aether, it is easier to resign to this than to merely maintain that all bodies are by their nature endowed with a force to attract each other. For since one can not even form an understandable concept of this attraction, one can by way of contrast at least understand how it is possible that the elastic force of a liquid matter is reduced, and one also understands that this can occur in a way that is in accordance with the laws of Nature. (Euler 1862, Vol. 2, p. 547; Translation by Hirsch E.) (A7)

2.4 The Origin of Forces

Force in the eighteenth century was the fundamental magnitude of mechanics and was introduced in various ways, each with its own problems. In his youthful treatise of 1736, the Mechanica sive de motu analytice exposita, Euler introduced the concept of force in a classical and rather generic way. The force [potentia] is “an action on a free body that either leads to the motion of the body at rest, or changes the motion of that body” (Euler 1736, p. 78). If in this treatise Euler payed no care to explain the nature of force, he did in subsequent works. It is worth noting that Euler had two Latin terms to indicate force: potentia and vis. To potentia he gave the technical meaning of cause of the variation of the motion of a body; to vis a generic meaning, either technical or common, for example, he can speak of vis gravitatis as an example of technical use, as well as of vis inertiae and the vis of impenetrability, as an example of common use.

Many years later, when he wrote the Theoria motus corporum solidorum seu rigidorum, with a much clearer understanding of mechanics, Euler proposed the definition:

Definition 12. What induces to change the absolute state of bodies is named a force [vis]; it should be due to external causes, since the body will remain in its own state due to internal causes (Euler 1765c, p. 44). (A8)

Is there any change in the status of force (in the ontology); from a being has it became a name and instead of a potentia a vis? The question mark is justified also because in the Anleitung, preceding of some years the Theoria, Euler in a succinct proof of the laws of motion had written that force is measured by the effect it produces: Here he wrote: “A force twice as big must in the same time produce twice as big an effect, because just in view of that do we consider it twice as big” (Euler 1862, Vol. 2, p. 476). A similar position is also reiterated a few pages after Definition 12 in the Theoria, as will be commented in a subsequent section. In any case the above definition hides some ambiguities, always present in the scholars of the eighteenth century when they deal with force.

In the following in the attempt to decipher the complex argumentations on the concept of force, I will assume that force is for Euler a primitive concept, which in particular is reduced to the anthropomorphic idea of pressure. What Euler tried to explain is not what is force or pressure but what is its origin. The fact then that the Definition 12 defines force as the name given to the external cause of the change of motion does not regard ontology but simply serves to delimit the broader concept that force has in the natural philosophy of the period, somehow a hypostatization of any cause, also internal.

In the Recherches sur l’origine des forces of 1750, he had paused on the role of impenetrability only to explain forces. Euler observed that despite the inertia or the general property of bodies by virtue of which each body tends to preserve its state, we see that bodies that fall under our observation continually change their state. So these bodies are subject to forces that must necessarily be external (Euler 1752d, p. 423).

To illustrate the role of impenetrability to generate forces Euler considered two bodies A and B. A is at rest, while B is moving toward A. After the impact, in order to avoid its own penetration, A acts to change the state of B, and in turn B acts on A. In other words, A applies a force to B (due the impenetrability of A, which is an external cause) and B to A. Euler concluded that these forces see their source in the impenetrability only (Euler 1752d, p. 427). That is, the forces due to impenetrability are passive forces (modern term); they are not determined neither for their quantity nor for their direction; they only are “obliged” to act when the impenetrability of a body is threatened; otherwise they do not produce any effect.

It is true that a greater action would prevent penetration as well, but it is more natural to assume a minimal value; that is forces have at all times value and direction strictly necessary to prevent penetration, so that if a minimum force is sufficient for the purpose only it acts (Euler 1752d, pp. 430–431). In this criterion of “least action,” Euler took inspiration from Maupertuis, and in it, one can find a justification of the principle of least action. To Maupertuis he made explicit reference in the letter 78 of the Lettres (Euler 1775, Vol. 1, p. 304).

Euler argued that the forces that two bodies exchange are equal and opposite to each other. For Euler this equality of forces, which is commonly known as the principle of action and reaction, is a necessary consequence of the penetration. So, for him, it is a rational and not an empirical law.

This equality of forces, hence the great principle of equality between action and reaction, is a necessary consequence of the nature of penetration. For, if it would be possible for the A body to penetrate the body B, body A would be equally as much penetrated by the body B; therefore, since the damage that these bodies that penetrate, is equal, these two bodies must also employ equal forces to resist penetration. So, as much as the body B is solicited by the body A, that will be solicited equally by this, both deploying exactly as much force as necessary to prevent penetration. But these two bodies acting one on the other by any force, will be in the same state as if they were compressed together by the same force (Euler 1752d, p. 434). (A9)

Euler was also able to analyze the evolution of the force due to impenetrability in the case of impact. Differently from many scholars of the period and in agreement with Johann Bernoulli (and Leibniz), he assumed that impact is not an instantaneous process: “The instantaneous impact would not be in accordance with the always respected law of nature, for which nothing can occur instantaneously, and as for a leap. According to this law a so great change, as sometimes is that occurring in the impact of bodies cannot occur without corresponding to a some interval of time” (Euler 1746a, p. 36).

In the hypothesis that the impacting bodies are linear elastic Euler furnished an explicit expression of the force of impact during the process of compression following the impact. If P is such a force, it is given by the expression P = F/kz, where F is the force necessary to get a penetration k, while z is the current value of the penetration. For reasons of space and also because it is not relevant for the present text, I do not repeat the mathematical steps involved in the proof. I limit myself to say that the approach and the result are essentially the same Euler had obtained in his youthful work De communicatione motus in collisione corporum of 1731 (Euler 1738). The difference is that now Euler had a clear concept of mass, whereas in 1731 he had not. But the equations are the same, demonstrating how mathematics may be more powerful than physics.

In subsequent works Euler associated force not only with impenetrability but with inertia also. In fact if there were no inertia, the state of bodies could be changed, and in principle it could be argued that as impenetrability causes a change in state, it also determines forces. But it would only be a meaningless word game. Imagine, for example, stopping a moving ball by opposing it a hand; if inertia would have no role in the explication of force, one could be free to assume the ball without inertia. In this case the ball would be rejected or stopped, but the hand would not notice anything; which is contrary to what happen in the real world. To obtain an effect, it is necessary that the ball is equipped with the ability to resist changes of motion, that is, that it has not zero mass (inertia). In the Theoria Euler limited himself to say that forces originate from impact of bodies because of their impenetrability. He wondered if all the forces have this origin. Without excluding the existence of other forces, he was content to say that a very important class of forces has this origin.

Here’s how Euler describes the role of inertia: “The cause of those forces by which the state of a body is changed may be agreed to lie not in inertia alone but in inertia coupled with impenetrability. Indeed, seeing that only bodies can be said to be impenetrable, and since bodies are necessarily endowed with inertia, impenetrability as such involves inertia, so that impenetrability alone is rightly considered the source of all forces by which the state of bodies is changed. It will therefore be proper to consider this property more exactly as being the origin of all forces” (Euler 1765c, p. 46). He thus can still pretend that only impenetrability counts because inertia is given when impenetrability is given.

The account of the force of impact between the two bodies A and B considered above may be completed as follows: when B comes into contact with A, it experiences a force which we would normally term A’s force of resistance to change of state. Note that the force is not in any sense in A: what is in A is its inertia, which is not a force (according to Euler’s definition) because it only maintains A’s state. But this internal principle is experienced by B as a force. There is, therefore, an external force acting on B which is not internal to A. Nor does it act at a distance because it is a prior condition of there actually being a force that A and B be impenetrable and that they be in contact (Gaukroger 1982, p. 147).

Euler underlined in several occasions that the forces of impact are external to the bodies and that inertia is not a force in the sense that it is not responsible for the change in the motion of a body; it has only the role of preventing it. In his work of 1746 Gedancken von den Elementen der Körper (Euler 1746b), Euler criticized Leibniz’s, or rather Wolf’s, philosophy of forces and monads. Euler’s main criticism concerned Wolf’s thesis that the monads, which are the ultimate components of reality, are endowed with an internal force that intends to continually change their state. Euler believed that there was indeed an internal strength, which however tended to maintain the state and not to change it and concluded: “One must therefore stipulate that two particular well differentiated classes of things exist in the world, to one of which belong the corporeal things, whose essence consists in the force [ability] steadily to maintain their state. The other however comprises the souls and ghosts, which are endowed with a force to change their state” (Euler 1746b, p. 15).

2.5 Concepts of Time and Space

Euler discussed the concepts of space in his more technical works such as the Mechanica sive de motu analytice exposita and Theoria motus corporum solidorum seu rigidorum and in some more philosophical ones such as the Lettres and Anleitung. But the text in which it did so in a systematic way is Réflexions sur l’espace et le tems of 1748 (Euler 1750). Euler’s discussion is interesting because it shows how a mathematical-oriented mind addressed some issues of philosophy of nature and metaphysics; he had perhaps less philosophical culture than a professor of philosophy, but he knew much more directly the topic he is dealing with. Euler’s paper did not escape to Immanuel Kant (1724–1804) who read it 15 years after its publication. Kant observed that the mathematical considerations of motion and space furnished many data to guide the metaphysical speculation in the track of the truth, by thus considering mathematics and physics as a tool to avoid the void speculations of the philosophers of his time (Cassirer 1953, p. 351).

Euler defended his thesis of absolute space against the theses of “metaphysicians” who believe that only the concept of relative space makes sense. The metaphysicians in question are naturally the followers of Leibniz and Wolf, not explicitly named however. That he referred to them is evident when he stated “I strongly doubt […] that the equality of the spaces should be judged by the number of monads that fill it” (Euler 1750, p. 331).

He addressed his thesis of absolute space with metaphysical and physical reasons. The metaphysical reason consists in affirming that a fertile concept in physics cannot be empty: “One should instead assert that both the absolute space and the absolute time, such as mathematicians look at, are real things which submit outside our imagination, because it would be absurd to sustain that pure imaginary objects could be assumed as foundation of the real principles of mechanics” (Euler 1750, p. 326). The physical reason is that only by conceiving absolute space can one explain the inertia of bodies.

Euler discussed inertia by referring it to a body at rest. He affirmed that to explain the difficulty to put it in motion from rest, its place (position) must be thought of as absolute – although in fact it is not possible to individuate any absolute system he added. Euler argued with a simple example that inertia cannot depend on the presence of nearby bodies to which refer motion considering a body that floats on still water, remaining at rest in its turn. If the water begins to flow, an observer, rigidly linked with it, sees the body move without any force being applied to it. Or if he sees the body remaining at rest with respect to the water, a careful examination shows him that this is due to the effect of water dragging. So Euler can conclude: “I strongly doubt that metaphysicians dare to sustain that bodies maintain their position with respect to other bodies thanks to their inertia, because it could be easy to show the falsehood of this explanation because of the consideration I discussed on bodies close to their neighbor” (Euler 1750, p. 328).

Euler then dealt with the case in which the position of a body is referred to fixed stars. Being them at rest – a possibility that is not certain for Euler (Euler 1765c, p. 38)) – the absolute space of the mathematicians would coincide with that of the metaphysicians. But, he wondered, how the fixed stars so far can determine inertia? This is impossible according to Euler.

This thesis, rejected by Euler, is endorsed by some modern scientists, who also refer to the reflections on the subject by the epistemologist Ernst Mach.

As Ernst Mach has pointed out, it cannot be a coincidence that the fixed stars appear indeed fixed relative to inertial frames, and hence that it is reasonable to consider inertia as a force exerted on local objects by the totality of the objects in the entire universe. Thus, Newton’s law may best be interpreted as a consequence of the basic axiom that the sum of the forces, including the inertial force, acting on a particle should be zero and of the constitutive law of inertia, which states that this inertial force should be given by −ma, where a is the acceleration relative to an inertial frame (Noll 2004, p. 9).

In essence Euler concluded that the preservation of the state (motion or rest) of bodies is explained only by conceiving place according to the criterion of mathematicians in an absolute space and not in relation to other bodies. And nobody can say that the principle of inertia is based only on something that exists in our head.

The reality of space is confirmed by the principle of preservation of uniform motion in a certain direction. If space was a relative concept, what sense would it have to speak of direction? It must be a direction of an absolute space that can be conceived in a natural way by abstraction.

Euler ended his considerations on space by reiterating the metaphysical argument that one cannot say that a principle accepted as true in physics by nearly all scholars (the principle of inertia), is founded on a thing that exists in our imagination only (the absolute space), and from this it must be concluded that the mathematical idea of absolute space cannot be in any way imaginary, but that there is something of real in the world which corresponds to this idea (Euler 1750, p. 329).

According to Ernst Cassirer, the Réflexions sur l’espace et le tems set up in fact not only a program for the construction of mechanics but a general program for the epistemology of the natural sciences. It sought to define the concept of truth of mathematical physics and contrasted it with the concept of truth of the metaphysicians. The considerations of Euler rested entirely on the foundations on which Newton had erected the classical system of mechanics. His concepts of absolute space (and absolute time, see below) were revealed not only as the necessary fundamental concepts of mathematical-physical knowledge of nature but as true physical realities. To deny these realities on philosophical grounds means to deprive the fundamental laws of dynamics – above all the law of inertia – of any real physical significance. In such an alternative, the outcome cannot be questioned: the philosopher must withdraw his suspicions concerning the “possibility” of an absolute space and an absolute time as soon as the reality of both can be shown to be an immediate consequence of the validity of the fundamental laws of motion (Cassirer 1953, p. 351).

Once established the reality of space, Euler went on to examine time. Beginning with the observation that the ideas of space and time always go together (Euler 1750, p. 331). Arguing on the reality of time is however more complicated. Euler distinguished between the idea of time and time itself. He had no objection to the idea that time is conceived as a succession of changes. But this idea is not sufficient to assert that two time intervals are the same, as one has to say in mechanics. Thus, according to Euler, time cannot be reduced to the idea of succession of changes; there must be something else: absolute time. Euler concluded by saying that he realized that his considerations could only be taken on by philosophers who are willing to give some sense of reality to time and motion, while they will not make the slightest impression to those who consider everything as relative (Euler 1750, p. 333).

3 Mechanics and Mathematics

Euler was lucky enough to be born in a period when there was to build all the mechanics and all the Calculus starting from the foundations of the seventeenth century. His precursors and heirs have been sought. He has often been seen as the successor of Newton and Leibniz. It must be said that his role in mechanics has been overshadowed by the historians of mechanics such as Mach, Dugas, Montucla, and Dühring, very careful to the fundamentals, who see him more as a mathematician than a mechanician. After all, more than 60% of his work deals with pure mathematics, and even those whose object is mechanics and astronomy contain many sections that can be classified as mathematics. Today Euler’s role, even for what concerns the fundamentals of mechanics, is re-evaluated. Some see him as the founder of modern classical mechanics as well as the one who threw the germs of the theory of relativity and quantum mechanics (Suisky 2009).

Regarding the legacy left by Euler, we must distinguish between the actual one, that is, the influence that he had on his contemporaries and the influence he could have had if all of his writings had been published and understood and also the influence it could have on modern mathematicians and physicists if they were still studied. On this last point, I limit myself to note that many of Euler’s scientific writings could still well understood by a modern reader and that it is still possible to identify promising lines of research now abandoned.

In any case the genesis of Euler’s thought, although interesting, is not a priority of this book. The reference to predecessors and contemporaries in mechanics is considered only on specific points, the main objective being to expose and explain his ideas. In this chapter I will focus almost exclusively on the fundamentals of Euler’s mechanics, largely neglecting his elaborations in the areas of rigid body dynamics, theoretical astronomy, theory of elasticity, and fluid dynamics, even though they represent the greatest contribution of Euler to mechanics.

In my analysis, I will undergo a critical examination of the current points of view, due in large part to Truesdell’s studies who edited (Vols. 12 and 13) or co-edited six volumes of the collected works of Euler and wrote appreciated articles and essays on him. He was Euler’s greatest advertising agent, making him an eighteenth-century hero, a genius like Newton and perhaps superior to him. The current point of view can be summarized as follows:

  1. 1.

    Euler translated Newton’s mechanics into Leibnizian language.

  2. 2.

    He introduced the use of vector calculus in mechanics.

  3. 3.

    He defined precisely the mechanics of the mass point, starting from the uncertain Newtonian mechanics.

  4. 4.

    He defined precisely the mechanics of the rigid body.

  5. 5.

    He made fundamental contributions to the systems of deformable bodies.

  6. 6.

    He founded modern hydrodynamics.

  7. 7.

    He introduced the concept of observer.

I will not stop to question these theses, limiting myself to say that it is one of the many myths of historiography that sees the evolution of science due to the work of isolated geniuses. It is hard to imagine that one man, not even one with exceptional memory and great workmanship and intelligence like Euler, could have done so much on his own. However, I believe that it is true that Euler was able to gather and summarize the ideas of his time, and for this reason, when one does not care about the genesis of the various ideas but wants to take stock of a certain era, the study of his research must be considered as essential.

The reconstruction of the basis of Euler’s mechanics is carried out here by studying his published works – articles and books – and some letters. The main text which has been considered is Theoria motus corporum solidorum seu rigidorum (hereinafter Theoria), written in 1760 and published in 1765 (Euler 1765c). That is, I will start from a text in which the foundations of mechanics have already been laid. Of course, I will also refer to the Mechanica sive de motu analytice exposita (herein after Mechanica), published in 1736, and to articles and books that show both his researches and conceptions of natural philosophy, among which the most important is the Anleitung.

Euler had his own clear idea of how to develop a mechanical theory able to solve the problems that were then still waiting for a solution. Already in his early treatise, the Mechanica, he presented the work program that engaged him throughout his life: developing all the mechanics starting from the laws of motion of the mass point, a concept that he was among the first to specify. The mass point is the model with which the finite or infinitesimal bodies are studied when there is no interest in the extended objects. The mass of the whole body is thus imagined concentrated in one point, and the laws of kinematics and dynamics are applied to it. Here is Euler’s program as reported in the general scholium of Chap. 1 of the Mechanica:

  1. 1.

    In the first place, very small bodies which can be considered as points are referred to.

  2. 2.

    Then the approach follows for these bodies of finite magnitudes which are rigid and are not allowed to change their shape.

  3. 3.

    In the third case, flexible bodies will be considered.

  4. 4.

    Fourthly, those which allow extension and contraction.

  5. 5.

    Fifthly the motion of bodies, constrained by others.

  6. 6.

    In the sixth case, the motion of fluids is in the agenda (Euler 1736, p. 37).

Never was Euler explicit about his metaphysical conceptions about mechanics. In particular, although in practice he treated mechanics as a purely rational discipline, widely described with the language of Calculus, nowhere made his position explicit and never equated the status of mechanics with that of mathematics. Probably this silence was not due to an oversight but to some form of uncertainty. The only explicit statement he made about the role of mathematics in mechanics was his declaration of the preference given to the analytic treatment over the geometric one. For Euler only the former allows a systematic approach to all problems, while the latter involves the search for new routes for every problem. But this is of course true not only for mechanics but for mathematics in general, and it was for this reason that in the eighteenth century in every field of mathematics and physics, geometry was replaced by algebra and calculus, bringing the discipline of geometry to a state of decadence, from which it will recover only in the nineteenth century. Here is what Euler wrote about this point in the preface of the Mechanica:

Thus, I always have the same trouble, when I might chance to glance through Newton’s Principia or Hermann’s Phoronomia, that comes about in using these [synthetic methods], that whenever the solutions of problems seem to be sufficiently well understood by me, that yet by making only a small change, I might not be able to solve the new problem using this method. Thus I have endeavored a long time now, to use the old synthetic method to elicit the same propositions that are more readily handled by my own analytical method, and so by working with this latter method I have gained a perceptible increase in my understanding. Then in like manner also, everything regarding the writings about this science that I have pursued, is scattered everywhere, whereas I have set out my own method in a plain and well-ordered manner, and with everything arranged in a suitable order. Being engaged in this business, not only have I fallen upon many questions not to be found in previous tracts, to which I have been happy to provide solutions, but also I have increased our knowledge of the science by providing it with many unusual methods, by which it must be admitted that both mechanics and analysis are evidently augmented more than just a little. (Euler 1736, Preface; Translation by Bruce I.) (A10)

The last sentence of the passage quoted above, in which Euler declared that the developments of mechanics and mathematics are closely linked, is worthy of note; mathematics allows to solve problems of mechanics; mechanics suggests cues to mathematics to treat unresolved problems.

3.1 Theoria Motus Corporum Solidorum Seu Rigidorum: The Motion of Mass Points

The first part of the Theoria, the Introductio (103 pages over a total of 527, preface excluded) reports a summary of the main results on the dynamics of the mass point, largely taking up from Mechanica, with many clarifications and updatings.

What a modern reader notices first, in reading the Theoria, though to a less extent to what happens for the Mechanica, is the verbosity of the exposition that in many cases instead of helping confuses and makes it difficult a reading because it does not allow to distinguish what is strictly necessary from what is superfluous. The sense of frustration is increased by the fact that in what is presented as a formal and generally axiomatic exposition, there is little or no distinction between theorems and principles. Principles are almost never declared; the main exception concerns the principle of inertia.

Here is what Euler wrote about this point in the Lettres, a text following the Theoria, albeit slightly:

This principle [of inertia] is commonly expressed in the two following proposition: A body once at rest will remain eternally at rest, unless it be put in motion by some external or foreign cause: Secondly, A body once in motion will preserve it eternally, in the same direction, and with the same velocity; or will proceed with an uniform motion, in a straight line, unless it is disturbed by some external, or foreign cause. In these two propositions consists the foundation of the whole science of motion, called mechanics. (Euler 1775, Vol. 1, p. 309; Translation in Euler (1802)) (A11)

While in the Mechanica the principle of inertia is not explicitly referred to as such, in the Theoria it is widely introduced with two propositions, qualified as axioms (see below). At no point is it referred to as a principle what a modern would treat as such, that is, the metaphysical principle of sufficient reason that plays a fundamental role in Euler’s proofs.

The way of introducing and using principles becomes less mysterious if one reflects that the eighteenth-century mathematicians understood the term principle differently from us. They referred to a principle as a proposition placed at the foundation of a theory, requiring nothing else. Then the principle can be either first, that is, primitive, evident in itself or true empirically, or it can be a second principle if it can be demonstrated by first principles. Among the principles of mechanics, there are, for example, the principle of minimum action and that of the conservation of living forces. In Euler’s time the logical status of these principles was not exactly defined; it was thought that they could be proved by first principles (and Euler will in part do so), but they still were not. Here is what Euler wrote about principles:

XVlll. Although the principles in question are new, as they are not yet known or spread by the authors who have treated of Mechanics, it is understood, however, that the foundation of these principles cannot be new, but that is absolutely necessary that these principles should be deduced from first principles, or rather axioms, over which the doctrine of motion is established (Euler 1752a, p. 194). (A12)

The term axiom is used by Euler in the Introductio only three times to extend the validity of the properties from relative to absolute motion and to introduce the concept of inertia, with the classical meaning of a self-evident proposition. The term theorem is used eight times; there are then 15 definitions and 19 propositions qualified as problems. These propositions could generally be formulated as theorems because they provide the solution of the problems also. Euler’s operation, perhaps similar to that of Euclid, serves to make the discussion less abstract.

The Theoria opens with the definitions of rest and motion:

Definition I. Just as a body at rest remains perpetually in the same place, so a body in motion continues to change its position. Clearly a body that is observed to adhere always to the same place is said to be at rest: but that body which advances by gliding from place to place in time is said to be moving (Euler 1765c, p. 1). (A13)

The primitive concepts necessary to understand this definition, those of space and time, are introduced soon after, at a discursive level into two Explications and one Scholium. Space is introduced as an absolute entity “Now we can only conceive a notion about this space itself by abstraction, by considering all the bodies to be removed, and what is left we decide to call space” (Euler 1765c, p. 1), “a concept which was argued at length by philosophers (and uselessly).”

But the idea of absolute space, admitted Euler, cannot be used in practice neither to define the position of a body nor its motion. They must be referred to nearby bodies which must be able to maintain invariable in time positions relative to one another in order to fulfill the purpose. To determine the location of a point P, it is necessary to know its distance from at least four non-coplanar points A, B, C, and D (Euler 1765c, p. 2).

After introducing the reference system or using a modern term the observer, Euler went on to define the notions of relative motion and rest. A body is at rest with respect to A, B, C, D if its distances from them do not vary, otherwise it is in motion (Defs. 2, 3) (Euler 1765c, pp. 3–5). Note that Euler did not use the term relative, but “with respect to.”

In a few definitions Euler introduced the concepts of path (Def. 4) and uniform and difform motions (Def. 5) and velocity (Def. 5), limited to uniform motion. Some comments are dedicated to the definition of velocity as the relationship between two heterogeneous quantities. The difficulty, inherent in the geometric concept of proportion and magnitude that requires comparison between uniform magnitudes, is solved by introducing units of measurement for space and time; in this way it is possible to define (the numerical value of) the velocity as the ratio between the numerical values that represent the measurements of space and time. Def. 7 concerns the direction of motion in the rectilinear and curvilinear cases, assumed as that straight line in which the motion is occurring or, if it is along a curve, as the tangent to the curve.

After these definitions Problem 1 follows, which deals with the use of the differential calculus to evaluate the space traveled by a point moving on a straight line with a velocity v assigned as function of time. Euler assumed that for a small dt (the element of time), the passed space ds, an elementary space indeed, is traversed with constant v velocity defined by v = ds/dt, in which v and s are explicitly thought of as generic functions of time, without particular restrictions on their regularity. (This is referred to as Proposition 3 in the Mechanica. “As indeed in geometry the elements of curved lines are considered to be the elements not-uniform motion is resolved into an infinite number of uniform motions. This is not the case, but it can be ignored without error. In either case the truth of the proposition is therefore apparent” (Euler 1736, p. 12; Translation by Bruce I.).)

The idea of considering constant any magnitude (in the present case the velocity) on condition of referring to a very small spatial or temporal intervals is what gives calculus a great heuristic power. If the velocity of the body v is given at individual instants, then the distances s passed in time t can be obtained with the help of the relation ds = vdt, the integral of which gives s =  ∫ vdt. Similarly if the speed v corresponding to a distance s is known, then the time t, in which the distance s is passed, is given by the differential equation dt = ds/v, which gives t =  ∫ ds/v.

More than one historian concedes that this approach to motion contains elements of novelty. Meanwhile, velocity is defined as a derivative (modern term, actually for Euler it was a ratio of differentials) of a function s(t) which is considered as a generic mathematical function and therefore an abstract concept not necessarily connected to a geometric curve. Then time is treated in the same way as any other physical quantity, and it does not have a particular role as it happens, for example, in Newton, with his idea of fluxions (Suisky 2009, p. 132).

I do not pronounce on this judgment about Euler’s originality. Certainly a similar treatment was not possible before the introduction of Calculus; furthermore it was established that Euler contributed a lot to the generalization of the concept of function. If one compares the way of treating velocity in the Theoria with that of the Mechanica, written 20 years earlier, he notices greater ease due to greater confidence both with the Calculus and the concept of function. It should be kept in mind that Euler’s text also had a didactic function and the maturation of the concepts referred to was not only his but mainly of the contemporary readers who did not need to be reassured about the validity of the procedure.

Problem 2 considers the motion of a mass point on a plane curve. It is reduced to the study of two monodimensional motions projecting the velocity into two directions not necessarily orthogonal, even if, added Euler, it would be better if they were such, for reasons of simplicity of calculations. For the motion in space (Problem 3), the projection is on to three axes. It should be noted that this is only a kinematic problem where the problem of the legitimacy of decomposition does not arise.

Studying motion with respect to orthogonal coordinate axes instead of intrinsic or natural coordinates is considered a fundamental turning point in Euler’s mechanics, a choice that appears first in his work Recherches sur le mouvement des corps célestes en général of 1747 (Euler 1749). The use of intrinsic coordinates, the standard approach at the turn of the eighteenth century, had two drawbacks. On the one hand, it required skill to find the right frame of reference. On the other hand, the approach was too difficult for bodies moving into a three-dimensional space. To be carried out consistently, it needed concepts of differential geometry such as curvature, torsion, and so on, fully developed only in the nineteenth century. The use of Cartesian coordinates does not present these problems. Although the equations may be complicated, they can be written in a standardized way. Even in this case, however, it must be said that Euler was not the first to project the equations of motion on two or three axes. According to Lagrange the first was Colin Maclaurin (1698–1746) in his A treatise on fluxions of 1742 (Lagrange 1811, p. 243).

If the curve FM (Fig. 319) be described by powers directed in any manner whatsoever, and the force at any point M, resulting from the composition of these powers, act in the direction MK, and be measured by MK; let MK be resolved into the force MO in the direction of the ordinates MP (=y), and the force OK parallel to the base AP (=x); then, the time being supposed to flow uniformly, or the velocity at M being represented by the fluxion of the curve FM, the force MO will be measured by \( \ddot{y} \) and the force OK by \( \ddot{x} \) (Maclaurin 1801, Vol. 1, p. 298).

Johann Bernoulli also had done it according to (Maltese 2003, p. 210; Truesdell 1960, pp. 184–236, 1968, p. 112). But only with Euler, the use became systematic; indeed it became the only one.

The use of decomposing the motion into two or three directions can be seen as a step toward modern vector algebra. A same magnitude, a geometric vector, can give rise to different pairs (or triads if in space) of components as the coordinate system varies. The choice of the coordinate system can be arbitrary. To the decomposition of the motion in several directions, Euler gave the name resolution. The motion is said to be resolved, provided that the small interval traversed in the element of time is considered as the diagonal of a parallelogram or parallelepiped (Euler 1765c, Def. 8, p. 21). Euler also considered other types of coordinates and therefore of decomposition. In Problems 5 and 6, the polar coordinates are introduced in the plane and in the space, respectively.

In Chap. 1 of the Introductio, Euler carried out only kinematic considerations. In the following chapters, he went on to discuss the causes of motion. They are classified internal and external:

1. Internal causes. Responsible for the reason either for rest or motion of a body, with the exclusion of all external causes. They are able to contribute anything to change the state of motion or rest.

2. External causes. Responsible of the change of the state of motion and rest of a body.

To explain the nature and characteristics of internal causes, Euler devoted the entire Chap. 2. To achieve his purpose, he is forced to introduce the then controversial concept of absolute motion. In essence Euler said that, if there is the absolute space, of anybody one can say if it is at rest or in motion with respect to the absolute space. To these types of situation, he referred to as absolute motion or rest. This obvious fact can be expressed through an axiom:

Axiom 1. Every body, even without being relative to other bodies, either remains at rest or moves, that is, it is either at absolute rest or in absolute motion (Euler 1765c, p. 30). (A14)

Two more axioms concern absolute motion; they express the principle of inertia, a term which Euler has not yet introduced.

Axiom 2. A body, which is absolutely at rest, if subjected to no external actions, will persist indefinitely in the state of rest.

Axiom 3. A body, which is absolutely in motion, if subjected to no external actions, continues to move uniformly along the same direction (Euler 1765c, pp. 32, 33). (A15)

These axioms, which Euler called the principles of internal motion, are justified in a simple way, basing on the principle of sufficient reason. For the state of rest, Euler argued, for instance, that because all the external causes of motion have been withdrawn, no reason is present that a body should begin to move in one direction rather than in any other: “This truth depends on the principle of sufficient reason” (Euler 1765c, p. 32).

A somewhat more sophisticated reasoning concerns the permanence in the state of uniform motion. No change in the direction can indeed occur, since there is no reason it should be deflected from that, in one rather than in all the other directions; “clearly it surely maintains the same direction, for the principle of sufficient reason.” About speed [the modulus of velocity] it can be said that “unless it always remains the same, either it increases or decreases, of which neither absurdity can be said; for if it is either being increased or decreased, it must happen to follow a certain law; but what this law may be cannot be conceived in any way, since nothing surely will be agreed upon” (Euler 1765c, p. 34). The reader is asked to reflect on the validity of these demonstrations.

After the introduction of the three axioms, the first theorem (Theorem 1), which asserts that the axioms valid for absolute motion also apply to relative motion, provided that it refers to a body (to an observer) that is at (absolute) rest or in uniform rectilinear motion (Euler 1765c, p. 37), appears very simple in reality.

In Definition 11 the internal cause of motion is given a name, inertia (the persistence of the Anleitung):

That quality of bodies, the reason for persisting in the same state present within a body itself, is called inertia, and also sometimes the force of inertia (Euler 1765c, p. 35, translation from Bruce I). (A16)

with the concept of state (absolute) introduced as follows: “While a body is either absolutely at rest or moving uniformly in a direction, it is said to persist in the same state” (Euler 1765c, p. 35).

Inertia indicates that property of bodies, whereby being at rest, means that they will continue to be at rest, therefore as if they oppose to motion; but since bodies set up in motion themselves equally oppose all to be changed, either on account of the speed or direction; this name seems a good choice. Sometime, said Euler, it is called force of inertia, because the body is resistant to change the state; but because often force is defined as the (external) cause which is changing the state of the body, force of inertia is not acceptable with this meaning – though Euler occasionally did. Whereby, as confusion should arise, it is better to omit the name force and refer to it by the simpler name of inertia (Euler 1765c, p. 36).

Chapter 3 is devoted to the external cause of motion; to it also is given the name force, with Definition 12. Though it has already been introduced before, I rewrite it for the sake of clarity:

What induces to change the absolute state of bodies is named a force [vis]; it could be due to external causes, since the body will remain in its own state due to internal causes. (Euler 1765c, p. 44) (A17)

After the introduction of external and internal causes of motion, Euler could go on to demonstrate what is now known as the Newtonian equation of motion, for the one-dimensional case. In essence Euler considered as a necessary truth, that is, rational, the law that Newton considered contingent that is empirical. The demonstration takes place into two steps.

Theorem 2a. The small space [dω], through which a given body at rest is advanced in the assigned small interval of time dt by different forces, is proportional to the forces. (Euler 1765c, p. 55) (A18)

Theorem 3. If equal forces act on unequal bodies at rest, the effect [dω] produced in the same small time intervals [dt] is inversely proportional to the inertias of the bodies. (Euler 1765c, p. 56, translation from Bruce I) (A19)

The proof of Theorem 2a is very simple, based on the assumption of the additivity of the effect of different forces. Basically, said Euler, if a small body, a corpuscle, is pulled forward by a force equal to p in the short time interval dt through the small space equal to dω and if another force equal to p is acting on the same body along the same direction, the body progresses through another equal small interval equal to dω. Thus this corpuscle acted on by a force equal to 2p in the same interval of time dt is pulled through the small distance equal to 2dω. Similarly if n forces – or equivalently a force np – act on the corpuscle at rest for the same time interval dt, they move the body through the interval equal to ndω.

The demonstration is not convincing however. Euler assumed indeed that the effect of the sum of two forces (two causes) is the sum of their effects, that is, he is assuming a principle of superposition and this is not granted. (The principle of superposition presupposes for its validity that the two cause do not interact to give a cause different from their sum.) And even if this is conceded, the proof would be only a trivial theorem of arithmetics (at Euler’s time) for which if two quantities increase of the same amount they are proportional. Moreover the demonstration follows a reasoning different from that carried out in the Anleitung. Here (Theoria), in the proof of Theorem 2a, force is considered as measured a priori; in the sense that it is independent of the formulation of dynamics, there (Anleitung) force was measured from effect, and the proof of the theorem becomes a simple truism.

Euler’s writing is substantially contemporary to one of Daniel Bernoulli, the Examen principiorum mechanicae, et demonstrationes geometricae de compositione et resolutione virium of 1726 (Bernoulli 1728), in which the latter posed the problem of the logic status of the law that links acceleration to force, assuming force as defined a priori concluding that the law is empirical and not rational. Bernoulli said that the part of mechanics that deals with the balance of forces can be deduced from the principle of composition of forces, as Varignon has shown. If another principle is added to this principle, namely, that according to which the increases in velocity are proportional to the increments of time multiplied by the force, mechanics is completed, relative to the motion. Galileo used this principle. Bernoulli believed that the principle of composition of forces is a necessary truth, while that of Galileo is a contingent truth: “Nature could have made the increases in velocity in the bodies proportional to the increments of time multiplied by any function of the pressure, so that said t the time, p the pressure and v the velocity, it was not dv = pdt, but for example dv = p2dt or dv = p3dt” (Bernoulli 1728, pp. 126–127).

The proof of Theorem 3 is analogous. Basically it is said that if one joins two bodies of equal mass under a given force, the effect is halved. With some details, following Euler’s reasoning: Let consider a corpuscle having an inertia A, which at rest is moved by a force equal to p for the short interval of time dt through the small space dω; if another corpuscle B equal to A is acted on by a force also equal to p along the same direction, it moves in the same dt by the same small space dω. If the two corpuscles are joined together into one resulting in a body with an inertia 2A (it is an implicit assumption by Euler), it acted by the force equal to 2p, in the time equal dt moves still of the small space dω. Thus a force 2p on the body of inertia 2A produces the displacement dω. Similarly a force np applied to a body n A produces the same displacement dω. Thus for the Theorem 2a, the force p on a body n A produces a displacement dω/n.

Note that inertia is not equivalent to the Newtonian quantity of matter; it is rather a property of a body whose measurement is defined in an operational way as the constant of proportionality between force and displacement dω. Euler can thus introduce the concept of mass in the following way, with Definition 15:

The mass or the quantity of matter of a body is the name given to the amount of the inertia which is present in that body, by which just as it tries to continue in its own state so it tries to resist all changes (Euler 1765c, p. 57). (A20)

where also the term quantity of matter is redefined by means of the concept of mass, inverting the usual approach where the mass is defined by the quantity of matter. The property of additivity is implicitly attributed to the mass as clear from the proof of Theorem 3. That is, if one joins two bodies with mass m1 and m2, he gets a body with mass m1 + m2.

The operational definition of mass in the eighteenth century used by Euler is not common; almost all mathematicians treat mass and inertia as proportional to the quantity of matter, taking for granted the meaning of quantity of matter (for instance, the number of atoms). The operational definition of mass will be resumed in the nineteenth century by Mach in his Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt written at the end of the nineteenth century (Mach 1919).

In the Mechanica Euler had given a different formulation of mass, and it is after a theorem – and not a definition – (Proposition 17) that the amount of inertia (vis inertia) is proportional to the amount of matter or mass (Euler 1736, p. 57). The amount of matter, however, was defined in a vague way, like the set of points (atoms?) that make up a body. Only that not all the points have the same mass. More precisely the points can be taken having the same mass when the same force exerts an equal effect on them (that is they have the same inertia). Thus the demonstration of the theorem on the equality between inertia and mass of the Mechanica is the result of a vicious circle of the type: the amount of matter is proportional to the amount of inertia, so the amount of inertia is proportional to the amount of matter.

At this point, by putting together Theorem 2a and Theorem 3 and using the definition of mass, Euler could enunciate Theorem 4, according to which:

Theorem 4. If corpuscles at rest with masses in an unequal ratio, are acted on by some singular forces, the small intervals through which they are thrust forwards in the same short time intervals will be composed in the direct ratio of the forces and the inverse of the masses. (Euler 1765c, p. 59; Translation adapted from Bruce I.) (A21)

That is, for a force p and a mass A, it can be written dω ∝ p/A.

In Problem 9, Euler started to replace the elementary displacement dω measured from the rest with the variation dv of velocity. Coming to the relation (with his symbols):

$$ dv=\frac{\uplambda pdt}{A} $$

where λ is a proportionality constant necessary to move from a proportion to a formula, which is the natural way to work with algebra and Calculus. (Incidentally if A is the weight, as Euler always assumes, then λ coincides with the acceleration of gravity (modern meaning).) Euler basically said that the increment dω, for a given interval of time dt, is proportional to the element of velocity dv, and therefore for Theorem 4, the above relation follows.

A modern reader sees in the previous formula the second law of motion of Newtonian mechanics in the monodimensional case. To discuss whether Euler’s was the first formulation of the equation of motion – it was not indeed – is not relevant here; certainly it was one of the first times the second law of motion was written in a very clear and unequivocal form. The only thing missing is the meaning of the constant λ; but this concerns only the choice of units of measurement and Euler made a choice later.

What leaves a little “surprised,” especially if reading the Theoria immediately after the Mechanica, is the fact that dv is placed directly proportional to the time interval dt; Theorem 4 authorizes only to say that dv is proportional to p and inversely proportional to A. The proportionality between dv and dt had been treated as a theorem in the Mechanica (Proposition 15), attributing its discovery to Galileo, who actually saw it as a plausible hypothesis to be verified experimentally, or if not it, its consequence, that is, the law according to which spaces vary with the square of times.

135. Galileo was the first to use this theorem in the investigation of falling heavy bodies. Indeed he did not give a demonstration of it, however because of the strong agreement with phenomena, nobody doubted it anymore. (Euler 1736, p. 53) (A22)

Euler’s demonstration is very simple but circular. This is how it works: consider n infinitesimal time intervals dτ. The increase in velocity du is the same in each interval of time dτ, assumed equal to each other, because by hypothesis the force, that is, the cause of motion is the same. So after n equal intervals dτ, the velocity becomes dv = ndu, from which it follows that dv is proportional to dt = ndτ (Euler 1736, p. 51). The circularity is to admit the constancy of the increase in velocity over time, which is the same as saying that the increase in velocity is proportional to time.

The reasons for omitting the proof of proportionality of dv with dt in the Theoria are not clear, at least to me. It is possible that everything derives from Euler’s, and others, studies on finite differences, in fashion in the early eighteenth century. These studies had shown that the variation of a function y of x, in a certain small interval dx, is necessarily proportional to dx itself, that is, Δy ∼ ψdx, width ψ a proportionality constant, generally dependent on x. Euler assumed it natural to consider the variation of velocity dv that in a given interval of time, once force and mass are fixed, cannot but vary linearly with the infinitesimal interval of time dt, and therefore dv = ψdt. The fact that ψ can only vary with p and A (and not also on the time t) depends on the physical nature of the problem that tells us that the variations of velocity in a certain interval are always the same (ψ= const.), or it varies as ψ = ψ(p, A), but the case ψ = ψ(t) is not discussed.

The motion of a body moving along a plane curve (Problem 13) following a force acting on the plane is treated by resolving the force into two (orthogonal) components and then studying two separate one-dimensional motions. It should be noted that unlike the case presented in the Problem 2, which was purely kinematic in nature, the idea of resolution or projection tends not to show the underlying physical principle, the parallelogram rule, for which the projections of forces are forces themselves that act independently of each other (without interaction). This is one of the cases in which the power of the mathematical instrument hides the physical nature of the problem. Indeed the use of Cartesian coordinates has deeper advantages than that of mere simplicity; the addition of vectors located at different points is so natural as to become customary at once, because it is reduced to the ordinary algebraic summation, giving for granted that this addiction has a physical meaning. Somehow Euler was aware of the problem and said that a force is traced back to three forces P, Q and R acting in the (orthogonal) directions x, y and z, with the “static resolution criterion” (Euler 1765c, p. 77).

The equations of motion eventually take the form (Euler 1765c, p. 85):

$$ ddx=\frac{2{gPdt}^2}{A};\quad ddy=\frac{2{gQdt}^2}{A};\quad ddz=\frac{2{gRdt}^2}{A} $$

in which A is the mass, while the symbol g denotes the height through which a heavy body drops in the time of a second, in a specified region of the globe, in case mass and weight are assumed to have the same numerical value: “Let the mass of the corpuscle be equal to A, which clearly indicates as well the weight, if the corpuscle is situated in a region of the earth chosen to evaluate an absolute measure” (Euler 1765c, p. 77). (Note that g is one half the value of the acceleration of gravity G, this follows from the relation s = 1/2Gt2: By assuming t = 1 it follows g = 1/2G.)

In these equations the constant λ has been replaced by 2g. That is, a coefficient that had only the role to transform a proportion in an algebraic equation is given a mechanical meaning, the space traveled in one second. Still in his work of 1749, Euler had assumed λ = 1/2, which, in modern terms, corresponds to assume an acceleration of gravity G = 1/2 and thus g = 1/4. The choice had derived from the desire of giving a simple expression to the speed of fall from the height h, which is simply given by v2 = h, instead of v2 = 2Gh. No inconsistency is at play; it is just a different choice of units of measurement.

To note that the equations of motion Euler wrote are in accord with the rule of calculus of the eighteenth century, where the concept of differential was fundamental. Using modern notation, Euler normally wrote the equation of motion as mdv = fdt, and not ma = f, where the symbols are the usual. However, he also used the acceleration a in a technical way, as the ratios dv/dt or dds/dt2. This is what he wrote when integrating the equation of motion: [In dynamics] the effect should be measured by the acceleration or the change in the speed, that is impressed on the body in a given time: this is proportional to that force divided by the mass of the body. […] dv is equal to the product of the acceleration and the element of distance travelled (Euler 1736, p. 84).

3.2 Measurement of Forces

The explanation of the origin of forces given in Euler’s writings on natural philosophy, in the Researches sur l’origine des forces in particular, does not provide any criterion for their measurement. The force due to impenetrability is not measurable in itself as it exists only at the moment of impact. What, at least in principle, is measurable is its effect, which, for Euler according to the Newtonian approach, is expressed by the variation in velocity in an assigned small interval of time, in the case of the impact much less than the duration of the contact between the bodies that collide with each other.

In the writings specific on mechanics, such as the Mechanica and the Theoria, the problem is discussed a little more. In the first treatise, Euler showed no particular difficulty to introduce the measurement of forces. It could be obtained by the rules of statics, a discipline at the time considered well founded. These rules indeed allow to measure a force as function of a sample force, for instance, a weight. It is enough to evaluate how many unities of the sample force should be summed (the additivity of forces is granted, for instance, by the rule of parallelogram) to reach the equilibrium when they act in the opposite direction with respect to the force to be measured. This means that the measure of a force is a priori and independent of dynamics that is independent of the dynamical effect it produces.

Euler is more careful in the Theoria; here he distinguished between a static and a dynamic measure; of course when two forces measured with the two approaches give the same numerical value for Euler, the forces are equal under all the aspect, that is, there are not dynamic and static forces but only different ways to measure them; the dynamic way is used only when static measurements are not possible, as, for instance, it occurs for gravitational forces of astronomy.

Euler had established his equations of motion, Theorems 2a and 3, referring to the increment of the displacement of a body due a force starting from the rest, because only in this case he had an independent criterion to measure force, the static one. In the case of motion Euler said that nothing is known concerning the measurement of forces and the way to measure them is left to us: “Since in statics, from which we draw the measurement of forces, the bodies to which the measurements are applied, may be considered in a state of rest, and thus nothing is defined concerning the measurement of these forces when they act on the body in motion” (Euler 1765c, p. 53).

In dynamics the only way left for the measure of forces should be searched in the measure of the effects, that is, they can measured a posteriori only: “Therefore the magnitude of these forces is determined not by the impenetrability, which clearly cannot be quantified, but from the change of the state which must be effected lest the body penetrate each other” (Euler 1765c, p. 50).

Euler, however, did not dwell at length on the problem of measuring forces and did not relate the dynamic and a posteriori measure of force, based on motions, with the static and a priori measure, based on the equilibrium. In such a way he avoided to make clear if the foundation of his mechanics is dynamic or static. Euler left comments on the dynamic measure of force only in some Explicationes of the Theoria, as he would not to compromise himself with strong declarations. Here he proposed the convention that if a force acting on a body in motion causes a displacement shift σ in the direction in which it acts, that force is assumed to be equal to the force that would cause the same displacement σ from the rest, evaluated, for instance, according to Theorem 2a (that is, if σ = kp is the law derived from Theorem 2a and the shift of displacement measured for a body in motion is \( \overline{\upsigma} \), the force associated to this variation of motion is given by \( \overline{p}=\overline{\upsigma}/k \)).

144. For the forces, then, by which bodies already in motion are acted upon, we set up this ground of measuring, so that we shall judge these equal to those which would have executed the same effect on the same bodies at rest in the same time. This ground, however, does not require proving, because it rests upon a definition and thus it was open to us to establish it. For if for any motion the small space sσ should be equal to small space Sσ, through which the same small body at rest is brought forward in that same little time by force p, we also call those forces equal. (Euler 1765c, p. 53; Translation in Pourciau (2016)) (A23)

Of course if it is admitted to measuring force through its effect, the Newtonian law of motion could be considered as a simple definition, and the concept of force could become superfluous. Only a weak ontological substratum would remain which allows to attribute some reality to force; but it is a link that can be broken without substantial changes in the formal development of mechanics. This link will be broken, shortly after the publication of the Mechanica, for example, by d’Alembert.

One could say that Euler preferred pragmatism to rigor. His alleged apodictic foundation of mechanics based on principles, self-evident for him, as those of impenetrability and inertia of bodies, works only if one remains at a superficial level, but above all it is not able to explain all the mechanics, for example, the link between statics and dynamics. Of course one could also criticize the evidence of the ideas of impenetrability and inertia or the lawfulness of the use of the principle of sufficient reason, but these criticisms should apply to the whole period in which Euler lived.

4 The Apparent Motion and the Observer

Although Euler believed in the existence of an absolute space, he nevertheless believed that motion, as a matter of fact, could only be studied in a relative space, defined by a certain conventional reference frame. It will then be the task of the physicist to establish what relationships exist between the relative space chosen to study motion and the absolute space. This is, for instance, what Euler wrote in his Mechanica, and in other occasions:

Because of the immense nature of space and of its unbounded nature […] we are unable to form a fixed idea of this. Thus, in place of this immense space and of the boundaries of this, we are accustomed to defining a finite space and the limits within which bodies can move, from which we can indicate the states of motion and of rest of bodies. Thus, we are accustomed to say that a body that keeps the same situation with respect to its boundaries is at rest, and truly that which changes with respect to the same, to be in a state of motion. (Euler 1736, Vol. 1, p. 2; Translation by Bruce I.) (A24)

In the Anleitung Euler posed himself the problem to see what happens by changing the reference frame, in particular by considering two reference frames in motion with respect to each other, a problem which had ancient origins. Galileo founded his mechanics referring to thought experiments imagining to be in the cabin of a moving ship. Huygens also set out on a uniform motion to study the problem of collision. Euler is part of this tradition, also followed by some mathematicians at the beginning of the eighteenth century, but as usual, he specified and enlarged it. A fairly detailed analysis of his modus operandi can be found in (Maltese 2000; Bertoloni 1993).

Already in one of his works of 1739 in which he addressed the problem of the influence of the speed of light propagation in astronomical observations, Euler faced the purely kinematic problem to correct the observational data to take into account the fact that the observers (spectatores in Latin) based on the earth mobile with respect to the fixed stars (Euler 1739a).

In Chap. 10 of the Anleitung, Euler systematically introduced the concept of observer, to which he referred to with the German term Zuschauer, which is properly translated as spectator, basically a fixed or mobile coordinate frame (without clock). There are some important results that clarify what today is known as the Galilean principle of relativity. Thanks to the use of Cartesian coordinates and calculus, the mathematical passages that Euler had to face appears very simple to a modern.

The first proposition states:

If the observer moves at constant speed along a straight line, and estimates directions correctly, then all bodies that are either at rest or are moving at constant speed in a straight line will appear to him to remain in the same state. (Euler 1862, Vol. 2, p. 497; Translation by Hirsch E.)

Notice that it is not specified very clearly the observer is considered to be moving with respect to the absolute space. The second proposition basically states that the equations of motion are the same for an absolute system and for a system that moves uniformly with respect to it:

If the observer moves uniformly in a straight line, and if he judges directions correctly, that is according to parallel running lines, then the maintenance of the apparent movement requires the same forces as the true movement, however much the apparent movement may differ from the true movement. (Euler 1862, Vol. 2, p. 497; Translation by Hirsch E.) (A25)

In the two propositions, the statement asserting that the direction is judged correctly means in modern term that the observer is moving without rotation.

In the following I refer with some details Euler’s proof. Let’s assume three coordinate axes in the absolute space, OA, OB, and OC and call the components of the motion (velocities) along the axes u, v, w. The motion of the observer is defined by velocities α, β, γ that by assumption are constant. To the observer, the motion of the body as seen from the axis OA will appear the smaller, the faster his own motions. The apparent motion in the direction of the axes OA, OB, and OC will be \( \underline{u}=u-\alpha, \underline{v}=v-\upbeta, \underline{w}=w-\gamma \), respectively. From the equations of motion that Euler had already developed, it follows that for the maintenance of the true (absolute) motion, one requires three forces, MP = P, MQ = Q, MR = R, so that:

$$ P=\frac{Mdu}{ndt};\quad Q=\frac{Mdv}{ndt};\quad R=\frac{Mdw}{ndt} $$

where M is the mass and n a normalizing factor. Now replace u, v and w, by \( \underline{u},\underline{v},\underline{w} \), in these equations. Since α, β and γ are constants, the differentials remain unchanged, so that it is apparent that for the maintenance of the apparent motion the same forces P, Q and R are required as for the true motion.

In a third proposition, perhaps the newest one, Euler considered an observer which moves in a translational way (without turning) unevenly. In this case, he concluded that fictitious forces must be considered alongside real forces.

But if the observer does not move uniformly in a straight line, but does estimate directions correctly [emphasis added], then to maintain the apparent motion of all bodies, in addition to the forces that are actually acting on the bodies, further forces are required that will at every instant and in every body produce the change that takes place at the location of the observer, but acting in the opposite direction. (Euler 1862, Vol. 2, p. 498; Translation by Hirsch E.) (A26)

In this case also the proof is very simple. However arbitrarily the motion of the observer may change, with reference to the three assumed axes; it can always be represented by the three motions (velocities) α, β and γ by taking these quantities to be variable. Now if the true movement requires the forces:

$$ P=\frac{Mdu}{ndt};\quad Q=\frac{Mdv}{ndt};\quad Q=\frac{Mdw}{ndt} $$

By replacing u, v, w by u − α, v − β, w − γ, the maintenance of the apparent motion will require the following three forces:

$$ {\displaystyle \begin{array}{l}\mathrm{Kraft}\quad MP=\frac{Md u}{ndt}-\frac{Md\upalpha}{ndt}=P-\frac{Md\upalpha}{ndt}\\ {}\mathrm{Kraft}\quad MQ=\frac{Md v}{ndt}-\frac{Md\upbeta}{ndt}=Q-\frac{Md\upbeta}{ndt}\\ {}\mathrm{Kraft}\quad MR=\frac{Md w}{ndt}-\frac{Md\gamma}{ndt}=R-\frac{Md\alpha}{ndt}\end{array}} $$

Therefore, apart from the forces P, Q, R that actually act on each body, three additional forces are needed that in each body produce the same change that occurs at the location of the observer, but in the opposite direction.

The above proposition of Euler finds its applications on several occasions. In the Theoria, for example, he finally applied it to astronomy, in order to eliminate the non-inertial motion of the observer’s frame based on the earth (Euler 1765c, pp. 120–121). In his important work on water wheels, Euler studied the action of a jet of water coming out of a rotating pipe (Euler 1756). To determine this action, the observer is located in a frame rigidly linked with the pipe (Maltese 2000, p. 342; Bertoloni 1993, pp. 322–326). An approach that will be followed by the nineteenth-century engineers involved the study of turbines. In a more general study of 1755, Euler arrived to formulate relations that have analogies, a part from some mistakes in calculation, with the famous relations that Gaspard-Gustave de Coriolis (1792–1843) found almost a century later (Bertoloni 1993, p. 326; Maltese 2000, pp. 342–342), with reference to a problem similar to that faced by him, that is a rotating wheel.

5 Dynamics of Rigid Bodies

The problem of the motion of rigid bodies, particularly of their rotation, has been the subject of careful studies since the middle of the seventeenth century. It was then presented as the problem of the composite pendulum and its reduction to a simple pendulum. Descartes, Roberval, and Huygens had dealt with the problem (Capecchi 2014) and then the Bernoullis. In 1745 Daniel Bernoulli wrote to his friend Euler at the end of 1745 of the problem as a very difficult and almost impossible to solve for anyone (Truesdell 1968, p. 257).

Euler succeeded in the enterprise shortly after Bernoulli’s letter with his famous paper Decouverte d’un nouveau principe de mecanique (Euler 1752b), presented at the Académie royale de science et belles lettres de Berlin in 1750. The locution “noveau principe” referred to in the title of the paper at a first reading appears to coincide with those that today go under the name of equations of motion of the mass point in a system of three coordinated axes, which Euler had already presented in another of his works, the Recherches sur le mouvement des corps célestes en général of 1749, without referring to it as a new principle. Historians have often questioned in which sense this principle was new. Generally, in the past, it was considered that the title adopted by Euler was improper; after all it was not Newton who had written the equations of motion? And in any case Euler himself had rewritten them a year before. More recently, however, a great originality has been sustained, thus passing from one extreme to another (Truesdell 1968; Maltese 2001).

There are some novelties with respect to the paper of 1749. The mass to which “Newton’s equations” refer are not necessarily of isolated bodies; they may also be of infinitesimal bodies, that is, the elementary mass dM of the corpuscles which are – true or imaginary – part of an extended body, for the moment considered as rigid. And the forces acting on these corpuscles are not only the external forces acting on the whole body but also the constraint reactions that come from the surrounding corpuscles to maintain the constraint of rigidity. So the novelty of Euler would be of method, in the sense that the motion of an extended body is studied using the same laws that apply to a mass point, thus reducing mechanics to a single principle, as Euler had always declared he wanted to do, and of merit because in these equations a broader concept of force is used.

The historians who praise Euler’s role, however, are silent on the fact that d’Alembert in 1749, a year before Euler presented his work, wrote Recherches sur la precession des equinoxes that although starting from different principles of mechanics presented an approach that was equivalent to that of Euler from a mathematical point of view. It equally based on infinitesimal elements of the extended body, the earth, by calculating the necessary force to keep it in motion, as a product of the infinitesimal masses for their acceleration due to gravitation forces, according to relations derived from the study of the kinematics of rigid bodies and by imposing the equivalence of the (static) moments of these forces with those of the external forces. Among other things, with his principles, d’Alembert was not obliged to call for the constraint reactions, which somehow caused embarrassment to Euler (see below). However, regardless of whether Euler’s work was original or not, it was certainly fundamental for the evolution of his thought and also of the mechanics of eighteenth century.

Here is as Euler presented his new principle: Let consider an infinitely small body, whose mass M (dM) is assumed to be contained in a single point. Suppose that that body of mass M has received an arbitrary motion along the direction x and that it is acted on by arbitrary forces P, we will have:

$$ 2 Mddx={Pdt}^2 $$

with the symbols having the usual meaning. “And it is this formula alone, which contains all the principles of mechanics” (Euler 1752b, p. 195). A similar reasoning can be repeated for the direction y and z and forces Q, R arriving to the equations (Euler 1752b, p. 196):

$$ 2 Mddx={Pdt}^2;\quad 2 Mddy={Rdt}^2;\quad 2 Mddx={Rdt}^2 $$

which are exactly the same as those reported in the paper of 1749.

It is thus from this same great principle that it will be necessary to derive all the rules which we need in order to determine the motion of a solid body whose axis of rotation does not remain immobile. For this purpose, it will be necessary to consider not only all the elements of the body, but also their mutual connection, in virtue of which all the elements maintain among themselves the same order and the same distances. For the motion of the whole body is composed of the motions of all its elements, and the motion of each of these must follow the principle which I have just explained, given that each element participates in the forces [f] which act on the body, and, in addition, is affected by certain forces [r] which prevent it from losing its connection with the others [emphasis added]. But before we can determine the effect of the forces to which the elements are subjected, we must consider in general the motion to which such a body is susceptible. (Euler 1752b, p. 197; Translated by Langton SG.) (A27)

The motion of a rigid body subject to generic forces is decomposed into the motion of the center of gravity and the motion around it, that is, a rotation. For the first motion, Euler merely stated that it obeys the laws of motion valid for the mass point by considering the mass of the whole body concentrated in its center of gravity and subjected to a force equal to the resultant of all the forces acting on the body.

To study the rotating motion, Euler considered a frame fixed with the origin in the center of gravity of the body and three non-rotating axes AO (z), BO (y), and CO (x) and studied the motion, displacement, velocity, and acceleration, of a mass corpuscle dm, which occupies the position x, y, z, when the body rotates around an axis passing through O. Rotation is defined by the components λ, μ, and ν of the angular velocity vector with respect to the axes x, y, z.

Once found the expression of ddx, ddy, ddz, starting by the kinematic relations of x, y, z and the angles of rotation, he equated the resultant moment of the masses multiplied by acceleration (ma) to the resultant moment of the forces (f + r) acting on the body with respect to the fixed axes. (In modern term, given the equation of motion ma = f + r, Euler made equal the moments of the two members of the equality.) In the analysis of the forces, Euler believed he could ignore the constraint forces (r). For which “it is to be remarked that the internal forces mutually cancel one another, so that the continuation of the motion requires only the external forces, to the extent that those forces do not mutually cancel” (Euler 1752b, p. 206). It is not clear here whether Euler refers to the principle of action and reaction or, more simply, to the shared axiom of mechanics that a body cannot modify its motion due to internal forces. Truesdell thinks that Euler drew his idea from Daniel Bernoulli (Truesdell 1968, pp. 252–256).

The result of Euler’s calculations is shown in Fig. 2. In the Eqs. I, II, and III, M is the mass of the whole body; ff, gg, and hh are in the order the axial moments of inertia (Euler term) with respect to the axes z, y, x, while ll, mm, and nn are the mixed moments of inertia with the pairs of axes xy, xz, yz, respectively, and λ, μ, ν are the components of the vector of angular rotation according to the axes z, y, x. Pa, Qa, Ra are the (static) moments of the external forces with respect to the axes z, y, x.

Fig. 2
figure 2

Motion of a rigid bodies about a movable axis (Euler 1752b, p. 213)

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Euler referred to the Eqs. I, II, and III as “three formulas which contain the new principles of Mechanics which we need in order to determine the motion of solid bodies.” They are quite complex, especially because the moments of inertia are referred to fixed axes, thus: “it is necessary to calculate anew for each instant the values ll, mm, nn, ff, gg, hh, because they will vary as a result of the change in the position of the body with respect to the three [fixed] axes” (Euler 1736, p. 214).

Euler took up the subject of the rigid body motion in later works, the results of which are summarized in his Theoria of 1760 where there are no substantial new elements compared to what Euler had written in the previous 10 years. Besides writing the equations of motion of the rigid body, Euler also dealt with complex problems, including gyroscopic motions.

A superficial reading of this “applicative” part makes it clear the perspectives that calculus has opened to the mathematical investigation of motion. It also makes manifest the theoretical and practical difficulties that arise in the solution of the differential equations of motion which requires the development of new fields of mathematics. To confirm what was already clear to Euler since 1736, that mechanics and mathematics proceed together, because a part of mechanics has now become mathematics itself.

However, the way in which Euler reached the equations of motion is also interesting from a foundational point of view. He did so by making explicit what was implicit in his article of 1750. He started from the idea of elementary (disturbing) forces, that is, those forces that applied to each element of the body separately may produce the same change in their state as likewise actually found (Euler 1765c, p. 117). The elementary forces, if applied in the opposite direction to the body, reduce it to rest. This is, as shown in the next chapter, what goes under the name of d’Alembert principle (introduced in 1743). Euler did not mention d’Alembert, and with a little reason because he had already used this approach in his previous works, but without giving him the relief that he is giving it here. To distinguish the elementary disturbing forces from the “ordinary forces,” he will refer to the latter as the actual forces. Euler referred to the principle of d’Alembert as a “well known circumstances which can be deduced from the principles of metaphysics, always fully effective according to reason. But this principle should be correctly understood; and indeed it is usually proposed generally in a vague manner, so that its conclusions cannot be reached without any risk” (Euler 1765c, pp. 119–120).

Moreover here the actual forces acting bear the causes in turn, that we designate by the letter V; then the effect is twofold: on the one hand, by which the motion of the body is affected, in place of this the elementary forces are to be assumed immediately effecting a change in the motion, which forces likewise we denote by the letter T. Now the other effect consists in trying to change the structure of the body, in place of this the forces affecting the bonds must be taken, which we denote by the letter S. Therefore since from the cause V there is produced the effect equal to T + S, it must be considered that V = T + S, thus on collecting terms together, S = V − T, in short as we have found above. (Euler 1765c, pp. 119–120; Translation by Bruce I.) (A)

The equations of motion to translation and rotation of a rigid body are obtained by equating the resultant and (static) moment of the elementary and external forces, according to the rules of statics.

Euler devoted a particular care to constraint forces, that is, the forces, to which the constraints of body must resist, which can be obtained as the resultant of the forces actually disturbing the body and the elementary forces applied in the opposite direction. The actual forces with the elementary forces taken away must give the forces affecting the constraints. How a body resists to these forces depends on its structure and on the way the parts of the body are connected together, which refers to physics rather than to mechanics. However, Euler said we only assume bodies, of which we consider the motion, to be provided with a sufficient level of stiffness, so that no change in the shapes be apparent. Of course it is sufficiently plausible that the connection of the parts is not so strong to not give way a little bit, although this could escape our senses. If it is true, then clearly no body is able to be taken as rigid, unless those where in general there are no forces trying to disturb the bond, since even by the smallest forces a certain change in the shape is being produced. “Now whatever such rigid bodies, assumed here, are present in the world or not, this question does not touch on the present discussion, since in all disciplines it is not needed that the objects are in existence to be contemplated” (Euler 1765c, p. 120).

Studies carried out by Euler on the motion of a rigid body in the period 1750–1760 are very well summarized in (Maltese 2000, 2003). In 1751 Euler employed axes rigidly linked to the body (in the following body axes) and noticed the existence of axes of free rotation (Euler 1767). After János András Segner (1704–1777) had discovered in 1755 the existence of at least three of such axes named principal axes of inertia, Euler explored their properties in Du mouvement de rotation des corps solides autour d’un axe variable (Euler 1765b) and arrived in 1758 to the three equations of motion of a rigid body, which are named after him in the paper Du mouvement de rotation des corps solides autour d’un axe variable of 1758 (Euler 1765a) and which are much simpler that those reported in Fig. 2.

They are in the form (Euler 1765a, p. 165) (Note that the two papers of 1751 and 1758 have very similar title and sometimes are confused. This has occurred also in the Euler archive.):

$$ {\displaystyle \begin{array}{l}P= Maa\frac{d\propto \cos }{2 gdt}+M\left( cc- bb\right)\frac{\propto \propto \cos \, \upbeta \cos \, \gamma }{2g}\\ {}Q= Mbb\frac{d\propto \cos \, \upbeta}{2 gdt}+M\left( aa- cc\right)\frac{\propto \propto \cos \, \gamma \cos \, \alpha }{2g}\\ {}R= Mcc\frac{d\propto \cos \, \gamma }{2 gdt}+M\left( bb- aa\right)\frac{\propto \propto \cos \, \alpha \cos \, \upbeta}{2g}\end{array}} $$

where ∝ is the modulus of the angular velocity; α, β, γ are the angles the axis of rotation makes with the body axes a, b, c, the principal axes of inertia passing from the center of gravity; aa, bb, cc are the moment of inertia with respect to the principal axes; P, Q, R are the component of the (static) moment of the external forces with respect to the body axes; M is the total mass of the body (actually the weight); and g is one half of the acceleration of gravity. These equations are much simpler than those referred to in Fig. 2.

After 1760, Euler came back to the study of the motion of rigid bodies. In a work of 1775, Nova methodus motum corporum rigidorum determinandi (Euler 1776), Euler now 70 years old and with impaired sight rewrote the equations of motion of the rigid body in the form shown in Fig. 3, where x, y, z are the coordinates of the mass point dM with respect to a fixed coordinate system; P, Q, R and S, T, U are, respectively, the components with respect to x, y, z of the resultant of forces and (static) moments with respect to the center of gravity of external forces; i is the acceleration of gravity. The integral must be extended to the body of which the motion is studied.

Fig. 3
figure 3

Equations of motion of a rigid body (Euler 1776, p. 224)

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Truesdell (Truesdell 1968) gives much importance to the Eqs. I–VI of Fig. 3. In his opinion:

  1. 1.

    It is the first time that the equations of motion appear altogether, rotation and translation.

  2. 2.

    They have general validity, that is, they apply to all types of bodies, even those that are not rigid.

  3. 3.

    The equations apply not only to the whole body but also to each of its finite or infinitesimal portions.

  4. 4.

    The Equations of moments IV, V, and VI are treated as if they were independent of the Eqs. I, I, and III.

Truesdell’s theses deserve respect, considering the long study he has dedicated to Euler, but they are ideologized, in the sense that Truesdell having made Euler his champion tends to attribute him the authorship of the principles used in modern continuum mechanics.

I will comment on Truesdell’s thesis point by point.

  1. 1.

    It is true that the equations of equilibrium to translation and those to rotation appear together for the first time in Euler’s texts and therefore also in the history of mechanics. But this is not particularly significant. Euler had in fact already written since 1750 equations similar to those presented in the work of 1775. Any compiler by reading Euler’s texts would effortlessly summarize his mechanics with the equations of Fig. 3.

  2. 2.

    It is true that the equations of Fig. 3 have general validity. They represent the equilibrium equations of mechanics regardless of the nature of the bodies to which they refer, hence independently of what today goes under the name of constitutive relationship, that is, the fact that the body be rigid, or perfectly flexible as a chain or elastic (the elastica, for instance), or fluid body. For example, in Euler’s text of 1775, it can be assumed that Euler had consciously defined the constraint of rigidity starting from coordinate transformation expressed by the relations:

    $$ x=f+ FX+{F}^{\prime }Y+{F}^{\prime \prime }Z $$
    $$ y=g+ GX+{G}^{\prime }Y+{G}^{\prime \prime }Z $$
    $$ x=h+ HX+{H}^{\prime }Y+{H}^{\prime \prime }Z $$

    where x, y, z are the coordinates of the element dm, in the fixed coordinate system; X, Y, Z are the coordinates of the same element with respect to the body axes, whose axes which, for instance, coincide with the barycentric principal axes of inertia; f, g, h are the coordinates of the center of gravity, in the fixed coordinate system. F, F′, F′′, G, G′, G′′, H, H′, H′′ are the cosines directors of the body axes with respect to the fixed coordinate system. Substituting these relations in the equations of Fig. 3, one obtains the equations of motion of the rigid body. But it is not at all certain that Euler was aware of the generality of his results. At least this does not result from the reading of the work of 1775, in which explicit reference is made to the rigid body only.

  3. 3.

    In the work of 1775, the integrals are extended to the whole body only.

  4. 4.

    Reading the work, it seems that Euler proceeded in the same way as in 1750; that is, he started from the forces on the mass element dm and then evaluated the resultant for translation and rotation. However, compared to 1750 or even 1760, he forgot the constraint reactions and their role. It is obviously difficult that this was a forgetfulness. It remains to be seen whether Euler ignored constraint reactions for simplicity of treatment, considering that he intended essentially to present equations of the motion of the rigid body in a different form from what has been done in previous works. Or instead he chooses of adopting two independent principles instead of just one. This however would completely transform the axiomatic of Euler and would void his ideal of unification and rationalization. A discipline based on a single principle demonstrated a priori, and therefore purely rational, would pass to a physical-mathematical discipline based on two principles, of which no explanation can be given, which confer hypothetical nature to the theory, leaving its verification to the agreement with the experimental results, perhaps too modern for Euler.

According to Truesdell his thesis would also be confirmed by the approach followed by Euler in the study of deformable continua. An in-depth study of the study in this direction is not of interest for the following writing; moreover, most likely it would not lead to a conclusive result.

6 Quotations

A1:

Eine flüssige Materie muss zu allererst diese Eigenschaft haben, dass ihre Theilchen nicht aneinander befestigt sind, so dass ein jegliches Teilchen ohne einigen Widerstand von den übrigen abgesondert und in Bewegung gesetzt werden kann

A2:

La matière, sur laquelle je voudrois à présent entretenir V.A. me fait presque peur. La variété en est surprenante, & le dénombrement des faits sert plutôt a nous éblouir qu’à nous éclairer. C’est de l’Electricité dont je parle, & qui depuis quelque tems en devenüe un article si important dans la Physique, qu’il n’est presque plus permis à personne d’en ignorer les effets.

A3:

Alles was undurchdringlich ist, gehört in das Geschlecht der Körper, und daher besteht das Wesen der Körper in der Undurchdringlichkeit, in welcher folglich alle, übrigen Eigenschaften ihren Grund haben müssen.

A4:

II y aura donc deux especes de matiere, l’une qui fournit l’étoffe à tous les corps sensibles et dont toutes les particules ont la même densité, qui est très considerable et qui surpasse meme de plusieurs fois celle de l’or; l’autre espece de matiere sera celle dont ce fluide subtil qui cause la gravite est compose, et que nous nommons l’ether.

A5:

Die grobe Materie ist also an sick selbst keiner anderen Veränderung fähig als in Ansehung ihrer Figur, welche, wenn hinlängliche Kräfte vorhanden, auf alle mögliche Arten verändert werden kann.

[…] Dass die subtile Materie auch allezeit und allenthalben eine, beständige Dichtigkeit haben sollte, dergestalt, dass dieselbe durch keine Kräfte in einen kleineren Raum getrieben werden kannte, scheint der Wahrheit nicht gemäss zu sein. Vielmehr mochte auch hierin ein Hauptunterschied zwischen der groben und subtilen Materie bestehen, dass sick diese zusammendrücken liesse.

A6:

Um dieses deutlicher darzuthun, so wollen wir erstlich setzen, ein Körper werde in einen kleinern Raum zusammengepresst. Weil sich nun die grobe Materie fur sich nicht zusammenpressen lasst, so kann dieses nicht anders geschehen, als wenn die Poren kleiner gemacht werden. In diesem Falle muss demnach die scheinbare Dichtigkeit des Körpers wachsen, weil die ganze Materie, woraus der Körper besteht, oder zum wenigsten die grobe, da die subtile in Ansehung derselben für nichts zu achten, in einen kleinern Raum gebracht worden. Es sei a3 der Theil des vom Körper eingenommenen Raumes, welcher mit grober Materie angefüllt ist, e3 aber der ülbrige Theil, so nur subtile Materie in sich enthalt, oder die Summe von allen Poren zusammen genommen, so wird a3 + e3 die Grosse des Körpers, as seine Masse und a3/(a3 + e3) seine Dichtigkeit aüsdrucken. Nun aber kann a3 nicht verändert werden, daher, wenn der Körper in einen kleinern Raum gebracht wird, so wird nur e3 verringert oder […] Wenn nach geschehener Veränderung der Figur eines Körpers die verschlossenen Poren weder grosser noch kleiner werden, so behalt der Körper diese veränderte Figur. Wenn aber die verschlossenen Poren weiter oder enger werden, so wird sick in dem Körper eine Kraft äussern, sick wieder in seine vorige Figurherzustellen.

A7:

Ungeachtet wir aber hier stehn bleiben müssen und kaum hoffen können, jemals die wahre Ursache dieser Verminderung der elastischen Kraft des Aethers zu ergründen, so kann man sich doch damit leichter begnügen; als wenn man blosserdings vorgiebt, alle Körper seien von Natur mit einer Kraft begabt, einander anzuziehen. Denn da man sich von diesem Anziehn nicht einmal einen verständlichen Begriff mach en kann, so kann man im Gegentheil zum wenigsten überhaupt einsehn, wie es möglich sei, dass die elastische Kraft einer flüssigen Materie vermindert werde, und man begreift auch, dass dieses auf eine den Gesetzen der Natur gemässe Art geschehen könne.

A8:

Qiucquid statum corporum absolutum mutare valet, id vis vocatur; quae ergo, cum corpus ob causas internas in statu suo esset permansurum, pro causa externa est habenda.

A9:

XXXL. Cette égalité des forces, d’ou depend le grand principe de l’égalité entre l’action & reaction, est une suite necessaire de la nature de la penetration. Car, s’il étoit possible que le corps A pénétrat le corps B, le corps A seroit précisément autant péenétre par le corps B; donc, puisque le danger que ces corps se pénètrent, est égal de part & d’autre, il faut aussi que ces deux corps employent des forces égales pour resister a la penetration. Ainsi, autant que le corps B est sollicité par le corps A, précisément autant sera celui-cy sollicité par celui-là, l’un & l’autre déployant exactement autant de force qu’il faut pour prevenir Ia penetration. Or ces deux corps agissant l’un sur l’autre par une force quelconque, se trouveront dans le même état que s’ils étoient comprimes ensemble par Ia même force.

A10:

Idem omnino mihi,cum Neutoni Principia et Hermann Phoronomiam perlustrare coepissem, usu venit, ut, quamvis plurium problematum solutiones satis percepisse mihi viderere, tamen parum tantum discrepantia problemata resolvere non potuerim. Illo igitur iam tempore, quantum potui, conatus sum ex synthetica illa methodo elicere easdemque propositiones ad meam utilitatem analytice pertractare, quo negotio insigne cognitionis meae augmentum percepi. Simili deinde modo alia quoque passim dispersa ad hanc scientiam spectantia scripta sum persecutus, quae omnia ad meum usum methodo plana et aequabili exposui atque in ordinem idoneum digessi. Hoc in negotio occupatus non solum in plurimas antea nondum tractatas incidi quaestiones, quas feliciter solutas dedi:sed etiam complures peculiares methodos sum adeptus, quibus tam mechanica quam ipsa analysis non parum augmenti accepisse videantur.

A11:

Or on énonce communément ce principe par deux propositions, dont l’une porte, qu’un corps étant une fois en repos demeure éternellement en repos, à moins qu’il ne soit mis en mouvement par quelque cause externe ou étrangere. L’autre proposition porte, qu’un corps étant une fois en mouvement, conservera toujours éternellement ce mouvement avec la même direction & la même vitesse, ou bien sera porté d’un mouvement uniforme suivant une ligne droite, à moins qu’il ne soit troublé par quelque cause externe ou étrangere.

A12:

Quoique les principe dont il s’agit ici soient nouveaux, entant qu’ils ne sont pas encore connus ou étalés par les Auteurs, qui on traité la Mécanique, on comprend néanmoins, que le fondement de ces principes ne saurait être nouveau, mais qu’il est absolument nécessaire, que ces principes soient déduits des première principes, ou plutôt des axiomes, sur le quels toute la doctrine du mouvement est établie.

A13:

Definitio I. Quemadmodum Quies est perpetua in eodem loco permanentia, ita Motus est continua loci mutatio. Corpus scilicet, quod semper in eodem loco haerere observatur, quiescere dicetur:quod autem labente tempore in alia atque alia loca succedit, id moveri dicitur.

A14:

Axioma I. Omne corpus, etiam sine respectu ad alia corpora, vel quiescit vel movetur, hoc est vel absolute quiescit vel absolute movetur.

A15:

Axioma 2. Corpus, quod absolute quiescit, si nulli externae actioni fuerit subiectum, perpetuo in quiete perseverabit.

Axioma 3. Corpus, quod absolute movetur, si nulli externae actioni subiiciatur, secundum eandem directionem motu aequabili progredi perget.

A16:

Definitio 11. Proprietas illa corporum, quae rationem perseverationis in eodem statu in se continet, inertia appellatur, quandoque etiam vis inertiae.

A17:

Definitio 12. Qiucquid statum corporum absolutum mutare valet, id vis vocatur; quae ergo, cum corpus ob causas internas in statu suo esset permansurum, pro causa externa est habenda.

A18:

Theorema 2a. Spatiola, per quae idem corpusuclum quiescens eodem tempusculo dt a diveris viribus promovetur, sunt ipsis viribus proportionalia.

A19:

Theorema 3. Si aequales vires corpuscula inaequalia quiescentia sollicitent, effectus eodem tempusculo producti erunt reciproce inertiae corpusculorum proportionales.

A20:

Definitio 15. Massa corporis vel quantitas materiae vocatur quantitas inertiae, quae in eo corpore inest, qua tam in statu suo perseverare quam omni mutationi reluctari conatur.

A21:

Theorema 4. Si corpuscula ratione massae inaequalia quiescant atque a viribus quibuscunque singula sollicitentur, erunt spatiola, per quae eodem tempusculo protrudentur, in ratione composita ex directa virium in inversa massarum.

A22:

Theoremate hoc ex solutione problematis invento primus est usus GALILAEUS ad motum gravium delabentium investigandum. Eius quidem demonstrationem non dedit, sed tamen propter insignem eius cum phaenomenis congruentiam de eo amplis dubitari noluit.

A23:

Pro viribus ergo, quibus corpora iam mota sollicitantur, hanc dimetiendi rationem stabilimus, ut eas aequales iudicemus iis, quae in iisdem corporibus quiescentibus eodem tempore eundem effectum essent praestaturae. Haec autem ratio non indiget probatione, quia definitioni innititur nobisque adhuc liberum fuerat eam constituere. Si enim promotu quocunque spatiolas sσ aequalia fuerint spatiola Sσ, per quod idem corpusculum quiescens tempusculo eodem profertur a vi p, huic etiam illas vires aequales appellamus

A24:

Quoniam autem immensi illius spatii cuiusque terminorum, […] nullam nobis certam formare possimus ideam; loco huius immensi spatii eiusque terminorum considerare solemus spatium finitum, limitesque corporeos, ex quibus de corporum motu et quiete indicamus. Sic dicere solemus, corpus, quod respect horum limitum situm eundem conservat, quiescere, id vero, quod situm eodem respectu mutat, moveri.

A25:

Wenn der Zuschauer gleichgeschwind in einer graden Linie fortrückt und die Gegenden richtig, das ist, nach gleichlaufenden Linien schätzet, so werden zur Unterhaltung der scheinbaren Bewegung, wie sehr dieselbe auch von der wahren unterschieden sein mag, eben diejenigen Kräfte erfordert, als zur Unterhaltung der wahren Bewegung.

A26:

Wenn aber der Zuschauer sich nicht gleichförmig in einer graden Linie bewegt, dennoch aber die Gegenden richtig schätzet, so werden, um die scheinbare Bewegung aller Körper zu bewerkstelligen, noch ausser den Kräften, welche wirklich auf dieselben wirken, solche Kräfte erfordert, welche in einem jeden Körper alle Augenblicke eben die Veränderung hervorbringen, welche in dem Orte des Zuschauers vorgeht, aber nach einer umgekehrten Richtung.

A27:

XXIV. C’est done de ce même grand principe, d’où il faudra dériver les regles, donc no us avons besoin pour determiner le mouvement d’un corps solide, lorsque l’axe de rotation ne demeure pas immobile. Pour cet effet il faudra considerer non seulement tous les élémens du corps, mais assi leur liaison mutuelle, en vertu de laquelle tous les élémens conservent entr’eux le meme ordre & les mêmes distances. Car le mouvement du corps entier est composé des mouvemens de tous ses élémens, & le mouvement de chacun doit suivre le principe, que je viens d’expliquer, entant que chaque élément participe des forces qui agissent fur le corps, & qu’il est outre cela sollicité par de certaines forces, qui l’empêchent, qu’il n’abandonne la connéxion avec les autres. Or avant que de déterminer cet effet des forces, auxquelles les élémens sont assujettis, il faut considerer en general le mouvement, donc un eel corps en susceptible.

A28:

Hic autem vires actu sollicitantes vicem causae gerunt, quam littera V designemus; deinde effectus est duplex: alter, quo motus corporis afficitur, cuius loco assumi debent vires elementares mutationem motus immediate efficientes, quas vires simul littera T denotemus. Alter vero effectus in conatu structuram corporis turbandi consistit, cuius loco sumi debent vires compagem afficientes, quas littera S notemus. Cum igitur a causa V producatur effectus = T + S, censeri debet V = T + S, unde colligitur S = V − T, prorsus uti invenimus. Verum in tanta rerum metaphysicarum caligine malim demonstrationem allatam adhibere ad principium metaphysicum illustrandum.

7 Conclusions

The justification by the scholars of the eighteenth century of the foundations of mechanics required substantial involvement in metaphysics and epistemology to introduce fundamental notions: the nature of space, time, force, constitution and properties of bodies, nature of motion. This effort was pursued by Euler not with the classical and organic approach of canonical philosophy but with the pragmatism of a mathematical philosopher. Euler’s role in mechanics and philosophy has been overshadowed by the historians such as Mach, Dugas, Montucla, Dühring, very careful to the fundaments, who see him more as a mathematician than a mechanician or a philosopher. After all, more than 60% of his work deals with pure mathematics, and even those whose object is mechanics and astronomy contain many sections that can be classified as mathematics. Today Euler’s role, even for what concerns the fundaments of mechanics, is reevaluated. Euler was lucky enough to be born in a period when all the mechanics and all the calculus were to build starting from the foundations of the seventeenth century. The section has showed that he used his luckiness however as much as fruitful it was possible.

8 Cross-References