Advances in Historical Studies 2014. Vol.3, No.1, 1221 Published Online February 2014 in SciRes (http://www.scirp.org/journal/ahs) http://dx.doi.org/10.4236/ahs.2014.31003 Open Access Euler, Reader of Newton: Mechanics and Algebraic Analysis Sébastien Maronn e1,2, Marco Panza3 1Institut de Mathématiques de Toulouse, Toul ous e, France 2Sciences Philosophie Histoire (SPHERE), Paris, France 3Institut d’Histoire et Philosophie des Sciences et des Techniques, Paris, France Email: Sebastien.Maronne@math.univtoulouse.fr, panzam10@gmail.com Received July 29th, 2013; revised September 1st , 2013; accepted September 10th, 2013 Copyright © 2014 Sébastien Maronne, Marco Panza. This is an open access article distributed under the Crea tive Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any me dium, provided the original work is properly cited. In accordance of th e Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intellectual property Sébastien Maronne, Marco Panza. All Copyright © 2014 are guarded by law and by SCIRP as a guardian. We follow two of the many paths leading from Newton’s to Euler’s scientific productions, and give an account of Euler’s role in the reception of some of Newton’s ideas, as regards two major topics: mechan ics and algebraic analysis. Euler contributed to a reappropriation of Newtonian science, though trans forming it in many relevant aspects. We study this reappropriation with respect to the mentioned topics and show that it is grounded on the development of Newton’s conceptions within a new conceptual frame also influenced by Descartes’s views sand Leibniz’s formal i sm . Keywords: Isaac Newton; Leonhard Euler; Newtonian Mechanics; Classical Mechanics; 18thCentury Algebraic Analysis Introduction The purpose of the present paper is to follow two of the many paths leading from Newton’s to Euler’s scientific produc tions1 and to give, at least partly, an account of Euler’s role in the reception of Newton as regards two major topics: mechan ics and algebraic analysis2. Euler contributed to a reappropria tion of Newtonian science. We will study this reappropriation with respect to the mentioned topics and show that it is ground ed on the development of Newton’s ideas within a new con ceptual frame also influenced by Cartesian ideas and Leibnizian formalism. From Newtonian Geometric Mechanics to Analytic Mechanics Euler’s works on mechanics concern different domains, some of which are not considered in Newton’s Principia (Newton, 1687). Beside his Mechanica (Euler, 1736)—a twovolume treatise on the motion of free or constrained punctual bodies— and a large number of papers on the same subject, Euler also much contributed to the mechanics of rigid and elastic bodies (Truesdell, 1960), the mechanics of fluids, the theory of ma chines and naval science. We shall limit ourselves to some of his contributions to the foundation of the mechanics of discrete systems of punctual bodies. We shall namely consider his views on the physical explanation of forces, his reformulation of the basic notions of Newtonian mechanics, and his works on the principle of least act io n . The Explanation of Forces The third book of Newton’s Principia offers an explanation of the motion of planets around the sun and of the satellites around them. It is based on the assumption that the celestial bodies act upon each other according to an attractive force act ing at a distance, the intensity of which depends on the mass of the attracting body and on its distance from the attracted one, and the effects of which are not influenced by a resistant me dium. This is a special central force—a force attracting bodies along a straight line directed towards a fixed or moving cen tre—characterised by the following wellknown equality where is the force by which the body attracts the body , and are, respectively, the masses of and , is the distance between (the centres of) them, and is an universal constant. In the first book of the Principia Newton provides a purely mathematical theory of central forces acting in absence of a resistance of the medium. In the third book, he is thus able to The quasitotality of Euler’s works is available online at the (Euler Ar ) , where one may also find many notices and references. Among other recent books devoted to Euler, cf. (Bradley & Sandifer, 2007) and (Backer ). 2For reaso ns of space, we w ill not addr ess in this p aper some related t opics like Euler ’s views on Newt on’s gravi tation theor y and his own cel estial me  chanics, p arti cular ly his lu nar th eory; Euler ’ s reflecti ons abou t ( abs olu te and relative) spac e and time; and Euler’s critical exposition of Newtonian sci Lettres à une princesse d’Allemagne (Euler, 17681772 ’s celestial mechanics, cf., for instance, (Schroeder, 2007) . On this matter, it is relevant to note that Euler had doubts in the 1740s about the validity of Newton ’s law of grav itat ion becau se of error s he ob served in cal  culations of planetary perturbat ions : cf. (Euler, 1743), Kleinert’s discus (Euler, OO, ser. II, vol. 31, appendix), and (Schroeder pp. 348379). On Euler’s views on space and t ime, cf. also (Cassirer , Book VII, Chapter II, section II) and (Maltese, 2000). Fi nally, on Euler’s Lettres à une princesse d’Allemagne, cf. (Cal inger, 1976).
S. MARONNE, M. PANZA Open Access give a mathematical theory of the world system by basing it only on the above assumption, that, according to him, is proved by empirical observations. Insofar as it allows the determina tion of the trajectories of the relevant bodies according to the mathematical theory of the first book, no supplementary hy pothesis about the nature of the relevant force is necessary. In Newton’s view, an hypothesis about forces is a conjecture con cerning their qualitative nature and causes. His “Hypot heses non fingo”—famously claimed in the General Scholium of the second edition of the Principia—is just intended to declare that he does not venture any such conjecture, since this is not nec essary to provide a satisfactory scientific explanation3. Apparently, this view is not shared by Euler. He seems to maintain that the notion of force cannot be primitive, and that a mathematical theory about forces cannot be separated from an account of their causes, even if this account depends more on “the province of metaphysics than of mathematics” and thus one cannot claim to undertake it with “absolute success”4. Euler’s account—on which, cf. (Gaukroger, 1982)—is based on a Cartesian representation of the world as a plenum of mat ter. Here is what he writes in his 55th letter to a German Prin cess5: As you see nothing that impels [small bits of iron and steel] toward the loadstone, we say that the loadstone at tracts them; and this phenomenon we call attraction. It cannot be doubted, however, that there is very subtle, though invisible, matter, which produces this effect by actually impelling the iron towards the lodestone [...]. Though this phenomenon be peculiar to the loadstone and iron, it is perfectly adapted to convey an idea of the signi fication of the word attraction, which philosophers so frequently employ. They allege then, that all bodies, in general, are endowed with a property similar to that of the loadstone, and that they mutually attract [...]. And then, in the 68th letter6: [...] as we know that the whole space which separates the heavenly bodies is filled with a subtle matter, called ether, it seems more reasonable to ascribe the mutual attraction of bodies to an action which the ether exercises upon them, though its manner of acting may be unknown to us, rather than to have recourse to an unintelligible property. The Cartesian vein of Euler’s account is even clearer in the Mechanica (Euler, 1736). It is very significant that in such a purely mathematical treatise, Euler devotes a scholium to dis cuss the causes and origins of forces. This is the scholium 2 of definition 10: the definition of forces, according to which, “a force [potentia] is the power [vis] that either makes a body pass from rest to motion or changes its motion.”7 This definition does not explain where the forces come from; Scholium 2 dis cusses the question. Euler begins by declaring that, among the real forces acting in the world, he only considers gravity. Then he argues that similar forces “are observed to exist in the mag netic and electric bodies” and adds: Some people think that all these [forces] arise from the motion of a somehow subtle matter; others attribute [them] to the power of attraction and repulsion of the bodies themselves. But, whatever it may be, we certainly see that forces of this kind can arise from elastic bodies and from vortices, and we shall inquire, at the appropriate occasion, whether these forces can be explained through these phe nomena8. One could hope that Euler’s Cartesian view be made more precise in a memoir presented in 1750 the title of which is quite promising: “Recherches sur l’origine des forces” (Eu ler, 1750)9. But the content of this memoir is somewhat sur prising. Euler begins by arguing that impenetrability is an essential property of bodies, and that it “comes with a force sufficient to prevent penetration”10. It follows—he says—that, when two Of course, we do not mean here that Newton had no views about the nature of forces, or never expressed them. In the same Principia , namely in the third of his Regulæ Phil os oph an di , opening t he third book (only added in the second and third edition), he argues, for example, that inertia universally belongs to all bodies. And in his third letter to Bentley, he explicitly writes the following (we quote from (Newton, LB, pp. 2526) ; a transcription of the original, kept at Trinity College Library, in Cambridge, is availabl e on lin e at the Newton Project website: www.newtonproject.sussex.ac.uk): “ That gravity should be innate, inherent and essential to Matter, so that one Body may act upon another at a Distance thro ’ a Vacuum , without the Mediation of any thing else, by and through which their Action or Force may be conveyed from one to another is to me so great an Absurdity, that I beleive no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constan tly according to certain Laws; but whether this Agent be material or immateri al I have left to the Consideration of my Readers. ” The point is that Newton does consider that the explanation of the nature of forces is neither essential to his theory of gravitation, nor a fortiori to hi s mathematical theory of motion. The last lines in the quoted passage of the third letter to Bentley is, among many others, an explicit expression of this attitude. cf. (Euler, LPAH, vol. I, p. 201). Here is Euler’s original (Euler, 1768 t. 68th, p. 265): “Il s’agit à présent d’approfon dir la véritable source de ces for ces att racti ves, ce qui appar tient p lutôt à la Metaphysique qu’ aux Mathematiques; & je ne saurois me flatter d ’y reussir aussi heureuse ” cf. (Euler, LPAH, vol. I, p. 165). Here is Euler’s original (Euler, 1768 1772 vol. I, lett. 55th, pp. 219220): “ Comme on ne voit rien, qui les [de petits morceaux de fer ou d ’acier] pousse vers l’aimant, on dit que l’ ’action même, se nomme attraction. On ne sauroit douter cepe ’il n’y ait quel que mati ere très s ubtil e, quoiqu ’ invisible, qui produise cet effet, en poussant effectivement le fer vers l ’ai man; [...] Quoique ce phénomene soit particulier à l ’ aimant, & au fer, il est tr ès propre à éclair cir le terme d’attraction, dont les Philosophes modernes se servent si freque m ment. Ils disent donc, qu ’une propriété semblable à celle de l’aimant, co n vient à tous les corps, en general, & que tous les corps au monde s ’ cf. (Euler, LPAH, vol. I, p. 203). Here is Euler’s original (Euler, 1768 1772 vol. I, lett. 68th, p. 268): “Puisque nous savons donc que tout l’ espace entr e les corps célestes est rempli d ’une manière subtile qu’on nomme l’ éther, il semble plus raisonnable d ’attribuer l’attraction mu tuelle des corps, â une action que l ’éther y exerce, quoique la maniere nous soit incon nue, que de recourir a une qualité inintelligible. ” On this same matter, cf. also the 75th letter: (Euler, 17681772, vol. I, pp. 297298). cf. (Euler, 1736, p. 39): “Potentia est vis corpus vel ex quiete in mo tum perducens vel motum eius alterans.” Here and later, we slightly modify I. Bruce’s translation available onl ine at http://www.17centurymaths.com . cf. (Euler, 1736, p. 40): “Similes etiam potentiae deprehenduntur in cor  poribus magneticis et electricis inesse, quae certa tantum corpora attr a hunt. Quas omnes a motu materiæ cuiusdam subtilis oriri alii putant, alii ipsis corporibus vim attrahendi et repellendi tribuunt. Quicquid autem sit, vid e  mus certe ex corporib us elastici s et vorticib us huiusmod i potentias origine m ducere posse, suoque loco inquiremus, num ex inde phaenomena haec p o  tentiarum explicari pos sint.” cf. also (Euler, 1746) and (Euler, 1765, Introduction) Romero , 2007) presents a det ail ed s tudy o f (Euler, 1746) and (Euler, (Gaukroger, 1982, pp. 134138) , an overview of the relevant parts of ( Euler, 1765) is offered. cf. (Euler, 1750, art. XIX, p. 428): “Aussi tot [...] qu’on reconnoit l’impénétrabilité des corps, on est obligé d’avoüer que l’impénétrabilité est accompagnée d’une force suffisante, pour empêcher la pénétration.”
S. MARONNE, M. PANZA Open Access bodies meet in such a way that they could not persist in their state of motion without penetrating each other, “from the im penetrability of both of them a force arises that, by acting on them, changes [...] [this] state11.” This being admitted, Euler shows how to derive from this only supposition the wellknown mathematical laws of the shock of bodies. This he considers enough to conclude that the changes in the state of motion due to a shock of two bodies “are produced only by the forces of impenetrability”12, so that, in this case, the origins (and cause) of forces are just the impenetrability of bodies. One would expect Euler to go on by describing a plausible mechanical model allowing him to argue that this is also the case of any other force, namely of central forces acting at a distance. But this is not so. He limits himself to considering the case of centrifugal forces which, not basing himself on any argument, he takes to being all reducible to the case where a body is deviated from its rectilinear motion because it meets a vaulted surface (Euler, 1750, art. LI, p. 443). Finally he writes13: If it were true, as Descartes and many other philosophers have maintained, that all the changes that bodies can suf fer come either from the shock of bodies or from the forces named “entrifugal”, we would now have clear ideas about the origins of forces producing all these changes [...]. I even believe that Descartes’ view would not be a little reinforced by those reflections, since, after having eliminated many imaginary forces with which philoso phers have jumbled the first principles of physics, it is very likely that the other forces of attractions, adherence, etc. are not better established. [...] For, though nobody has been able to establish mani festly the cause of gravity and forces acting upon heav enly bodies through the shock or some centrifugal forces, we should confess that neither has anybody proved the impossibility of it. [...] Now it seems as strange to reason since it is not proved by experience that two bodies sepa rated by a completely empty space mutually attract one another through some forces14. Hence, I conclude that, with the exception of forces whose spirits are perhaps able to act upon bodies, which are probably of a quite different nature, there is no other force in the world beside those originated in the impenetrability of bodies. Though advancing a nonNewtonian demand of explanation of the nature and causes of forces15 and sharing both the Carte sian requirement of deriving “basic concepts of mechanics from the essence of body” (Gaukroger, 1982, p. 139), and a Cartesian conception of the world as a plenum of matter allowing a re duction of all forces to contact ones, Euler reaches thus a quite Newtonian (and nonCartesian) attitude only disguised by rhe toric. His main point is finally clear, indeed: a mathematical science of motion is perfectly possible even in the ignorance of the actual causes of the forces of attraction, and the only way to ensure that there are reasons causing the forces are to show that the consideration of these reasons leads to the wellknown ma thematical laws of motion. These laws—rather than any possi ble mechanical model—are thus finally understood as the only sure expression of the reality of the universe. The Reformulation of Newton’s Me c hanics Usi ng Leibniz’s Differential Calculus of and the Introduction of Exte rnal Frames of Reference As is well known, Newton’s mechanics is essentially geo metric. Curves are used to represent trajectories of punctual bodies and a theorem is proved ensuring that nonpunctual bodies behave with respect to attractive forces as if their mass were concentrated in their centre of gravity. Instantaneous speeds are indirectly represented and measured by segments taken on the tangents of the curvestrajectories. They are taken to represent primarily the rectilinear space that a relevant point would cover in a given time, finite or infinitesimally small, if any force acting upon it ceased and the motion of this point were thus due only to its inertia. An analogous form of indirect representation and measure holds for any sort of force, or better for their accelerative punctual component. This provides a very simple way of composing forces and inertia, essentially based on the parallelogram law that is primarily conceived as holding for rectilinear uniform motions. When the consideration of time is relevant, this is typically represented and measured by ap propriate geometric entities, like appropriate areas: for example the areas that are supposed to be swept in that time by a vector radius, in the case of a trajectory complying with Kepler’s sec ond law. To solve mechanical problems, this fundamental geometric apparatus is of course not sufficient. Newton’s mechanics also includes two other fundamental ingredients. The first is a geometric method which allows to deal with punctual and/or instantaneous phenomena and to determine their macroscopic effects (like equilibrium configurations, ef fective trajectories, and continuously acting forces). It is pro vided by the method of prime and last ratios, together with a number of appropriate devices. The second ingredient is given by a number of fundamental laws expressing some basic relation between bodies (or better their masses), their motions, and the forces acting on them and because of them. It is provided by Newton’s well known laws of motion, occasionally supplemented by some principles—like the principle of maximal descent of the centre of gravity— cf. (Euler, 1750, ar t. XXV, p. 431): “ [...] à la rencontre de deux corps, qui se pénétr eroient s ’ils continu oient à demeurer dan s leur état, il nait de l’im ’un et de l’ autre à la fois une force qui en agissant sur les corps, change leur état. ” 2cf. (Euler, 1750, art. XLVI, p. 441): “ [...] dans le choc d es corps [. ..], il est clair que les changements, que les corps y souffrent, ne sont produits que leurs forces d ’impénetrabilité.” cf. also ibid. art. XLIV, p. 440. cf. (Euler, 1750, arts. LVIII and LIX, pp. 446447): “[...] s’ il étoit vrai, comme Descartes et quantité d ’autres Philosophes l’ont soutenu , que tous les changements, qui arrivent aux corps, proviennent ou du choc des corps, ou des forces nommées centrifuges; nous serions à present tout à fait éclaircis sur l ’ origine d es forces , q ui opérent to us ces changemens [ ... ]. J e crois même que le sentiment de Descartes ne sera pas médiocrement fortifié par ces réflexions; car ayant retranché tant de forces imaginaires, dont les Philos o  phes ont brouillé les premiers principes de la Physique, il est très pro b able que les au tr es f or ces d ’attraction , d ’a dhésion etc. ne sont pas mieux fondées. LIX. Car quoique personne n ’ait encore été en état de démontrer évide m ment la cause de la gravité et des forces dont les corps celestes sont sollicités par le ch oc ou quelque force centrifugue; il faut pourtant avouer que pe ’en a non plus démontré l’ impossibilité. [...] Or que deux corps éloignés entr ’eux par un espace entiérement vuide s’attirent mu tuellement par quelque force, semble aussi étrange à la raison, qu ’il n’ est prouvé par aucune expérience. A l ’exception donc des forces, dont les es être capables d’agir sur les corps, lesquelles sont sans doute d’ ature tout à fait différente, je conclus qu’il n’y a point d’ autres forces au monde que celles, qui tirent leur origine de l ’impénétrabilité des corps.” 14About Euler’s opposition to action at distance, cf. (Wilson, 1992). But we must keep in mind that Newton’ s rejection of any hypotheses about the natu re and t he caus es of the gr avitatio nal fo rce onl y concern s the limited domain of the mathematical principles of natural philosophy.
S. MARONNE, M. PANZA Open Access which are taken to follow from them. Though these laws are still considered as the more funda mental ingredients of classical mechanics, what we call today “Newtonian mechanics” is a quite different theory, reached through a deep transformation and reformulation of Newton’s original presentation. This transformation and reformulation mainly occurred during the 18th century and they were very much of Euler’s doing16. Giulio Maltese thus sums up the situa tion (Maltese, 2000, pp. 319320): In fact, it was Euler who built what we now call the “Newtonian tradition” in mechanics, grounded on the laws of linear and angular momentum (which Euler was the first to consider as principles general and applicable to each part of every macroscopic system), on the concept that forces are vectors, on the idea of reference frame and of rectangular Cartesian coordinates, and finally, on the notion of relativity of motion. This quotation emphasizes some basic ingredients of the Newtonian mechanics of today. We shall come back in a mo ment on some of them. We will then observe that the gradual emergence of these elements depends on a more basic trans formation (though perhaps, not so fundamental in itself). We refer to the replacement of Newton’s purely geometric forms of representation of motions, speeds and forces and of the con nected method of prime and last ratios by other forms of repre sentation and expression employing appropriate algebraic tech niques enriched by the formalism of Leibnitian differential calculus. This transformation is often described as a passage from a geometric to an analytic (understood as nongeometric) way of presenting Newton’s mechanics. This is only partially true, however. Though the use of algebraic and differential formal ism indeed allows the expression of the relation between the relevant mechanical quantities through equations involving the two inverse operators and submitted to a number of easily applicable rules of transformation, these equations are part of mechanics only if the symbols that occur in them take on a mechanical meaning. It is just the way in which this meaning is explained—and not the mere use of this formal ism—that decides whether the adopted presentation is geomet ric or not. For example, it is not enough to identify the punctual speed of a certain motion with the differential ratio to get a non geometric definition of speed: whether this definition is geo metric or not depends on the way in which this ratio, and namely the differential , are understood. If this differential is taken to be an infinitesimal difference in the length of a cer tain variable segment represented by an appropriate geometric diagram and indicating the direction of the speed in respect to another component of this diagram, the definition is still geo metric. This is exactly what happens in the first attempts to apply differential formalism to Newtonian mechanics, like those of Varignon, Johann Bernoulli, and Hermann17: the language of differential calculus is used to speak of mechanical configura tions represented by appropriate geometric diagrams and its rules are applied in order to get the relevant quantitative rela tions between the elements depicted in these diagrams (Panza, 2002). Like in Newton’s Principia, mechanical problems are thus, in these essays, distinguished from each other according to specific features manifested by the corresponding diagrams. Hence, differences in the problems depend on differences in the diagrams. This fragmentation of mechanics into several problems geo metrically different is still particularly evident in Hermann’s Phoronomia (Hermann, 1716), which Euler considers as the main treatise on dynamical matters written after the Principia. This is just what Euler wants to avoid in his Mechanica. Here is what he writes in the preface18: [...] what distracts the reader the most [in Hermann’s Phoronomia] is that everything is carried out [...] with oldfashioned geometrical demonstrations [...]. Newton’s Principia Mathematica Philosophiae are composed in a scarcely different way [...]. But what happens with all the works composed without analysis is particularly true with those which pertain to mechanics. In fact, the reader, even when he is persuaded of the truth of the things that are demonstrated, nonetheless cannot reach a sufficiently clear and distinct knowledge of them. So he is hardly able to solve the same problems with his own strengths, when they are changed just a little, if he does not research into the analysis and if he does not develop the same proposi tions with the analytical method. This is exactly what of ten happened to me, when I began to examine Newton’s Principia and Hermann’s Phoronomia. In fact, even though I thought that I could understand the solutions to numerous problems well enough, I could not solve prob lems that were slightly different. Therefore, in those years, I strove, as much as I could, to arrive at the analysis be hind those synthetic methods, and to deal with those propositions in terms of analysis for my own purposes. Thanks to this procedure I perceived a remarkable im provement of my knowledge. A major purpose of Euler’s Mechanica is to use Leibnitian differential formalism (which is what he calls “analytic me thod”) in order to generalise some of Newton’s results. Euler aims to arrive at some general procedures which allow him to solve large families of problems. He also looks for some rules for use in appropriate circumstances to determine, in a some what automatic way, appropriate expressions for relevant me 6As N. Guicciardini has remarked (Guicciardini, 1999, p. 6), “ Principia’ s mathematical methods belong definitely to what is past and obsolete .” 7On these essays, cf. (Aiton, 1989), (Blay, 1992), (Guicciar dini, 1995) Guicciardini, 1996), (Mazzone & Roero , 1997). 8 A large part of this passage is quoted and translated by N. Guicciardini in ( Guicciardini, 2004, p. 245). We quote his translation by adding a transla t he does not quote that slightly differs from Bruce’ 7]. Here is Euler’s original (Euler, 1736, Præfatio): “ [...] quod lectorem maxime distinet, omnia more veterum [...] geometricis demonstr a  tionibus est persecutus [...]. Non multum dissimili quoque modo conscripta sunt Neutoni Principia Mathematica Philosophiae [...]. Sed quod omnibus scriptis, quæ sine analysi sunt composita, id potissimum Mechanicis obti n git ut Lector, etiamsi de veritate eorum, quæ proferuntur, convi n catur, tamen non sati s claram et distinct am eorum cognitio nem assequatur, i ta ut easdem quaestiones, si tantillum immutentur, proprio marte vix r e solvere v aleat, n isi ipse in analysin inquirat easdemque propositiones an a lytica met h o do evolvat Idem omnino mihi, cum Neutoni Principia et Hermanni Phoronomiam per  lustrare cœ o ep i ssem, usu v eni t , ut, quamvis plurium pr ob l em at u m sol u tiones satis percepisse mihi viderer, tamen parum tantum discrepantia pro b lemata resolvere non potuerim. Illo igitur iam tempore, quantum potui, conatus sum analysin ex synthetica illa methodo elicere easdemque propos i tiones ad meam util itatem analytice p ertractare, quo negotio in signe cogn i tionis mea e augmentum percepi”.
S. MARONNE, M. PANZA Open Access chanical quantities. In order to reach this aim, Euler identifies punctual speeds and accelerations with first and second differential ratios, re spectively, and introduces an universal measure of a punctual speed given by the altitude from which a free falling body has to fall in order to reach such a speed. The basic elements of Newton’s mechanics appear in Euler’s treatise under a new form, quite different from the original. Nevertheless, in this treatise, mechanical problems are still tackled by relying on intrinsic coordinates systems: speeds and forces are composed and decomposed according to directions that are dictated by the intrinsic nature of the problem, for ex ample so as to calculate the total tangential and normal forces with respect to a given trajectory. This approach is quite natural, but limits the generality of possible common rules and prin ciples. A new fundamental change occurs when extrinsic reference frames, typically constituted by triplets of orthogonal fixed Cartesian coordinates, are introduced and when the relativity of motion is conceived to be the invariance of its laws with respect to different frames submitted to uniform retailer motions. Though this change was in fact a collective and gradual trans formation (Meli, 1993) and (Maltese, 2000, p. 6), Euler played a crucial role in it. Among other important contributions con nected with this change—described and discussed in (Maltese, 2000)—, it is important to consider his introduction of today’s usual form of Newton’s second law of motion. This is the ob ject of a memoir presented in 1750: “Découverte d’un nou veau principe de mécanique” (Euler, 1750). The argument that Euler offers in this memoir in order to jus tify the introduction of his “new principle” is so clear and apt to elucidate the crucial importance of this new achievement that it deserves to be mentioned. The starting point of this argument is the insufficiency of the tools provided both by Newton’s Pri n cipia and by Euler’s own Mechanica for studying the rotation of a solid body around an axis continuously changing its posi tion with respect to the elements of the body itself. To study this motion, Euler argues, new principles are needed and they have to be deduced from the “first principle or axioms” of me chanics, which, he says, cannot but concern the rectilinear mo tion of punctual bodies (Euler, 1750, art. XVIII, p. 194). The problem is precisely that of formulating these axioms in the most appropriate way to allow an easy deduction of all the oth er principles that are needed to study the different kinds of mo tion of the different kinds of bodies. According to Euler, these axioms are reduced to an unique principle, and his new one is just that. This principle is expressed by a triplet of equations that ex press Newton’s second law with respect to the three orthogonal directions of a reference frame independent of the motions to be studied (Euler, 1750, art. XXII, p. 196): 2Md 2 x=Pdt 2 ; 2Md 2 y=Qdt 2 ; 2Md 2 z=Rdt 2 , where is the mass of the relevant punctual body, , , and are the total forces acting along the directions of the threes axes and 2 is a factor of normalization. To understand the fundamental role that Euler assigns to this principle, a simple example is sufficient (Euler, 1750, art. XXIII, p. 196): from , one gets, by inte grating, Mdx=Adt;Mdy =Bdt;Mdz =Cdt, where , , are integration constants. It is thus proved that, in this case, the speed is constant in any direction so that the motion of any body on which no force acts is rectilinear and uniform, as Newton’s first law asserts19. A New Sort of Principles: Euler’s Program for Founding Newton’s Mechanics on Variational Principles Though fairly powerful, Euler’s new principle only directly deals with single punctual bodies. Let’s consider a system of several punctual bodies mutually attracting each other and pos sibly submitted to some external forces and internal constraints. In order to get the conditions of equilibrium or the equations of motions of such a system by relying on Euler’s principle, a detailed and geometric analysis of all the forces operating in this system is necessary. A fortiori, this is also the case of any other principle dealing with single punctual bodies set in New ton’s Principia and in Euler’s Mechanica. Hence, the study of any particular system of several punctual bodies through these principles requires a geometrical analysis of forces that differs from system to system. Consequently, only fairly simple sys tems can be studied in such a way. This is the reason why the need of a new sort of mechanical principles—directly concerned with whatever system of several punctual bodies—arose quite early. A similar principle, that would be later known as the principle of virtual velocities, was suggested by Johann Bernoulli in a letter to Varignon of Janu ary, 26th 1711 (Varignon, 1725, vol. II, pp. 174176). But a clear statement of the difference between these two kinds of principles only appears in a memoir presented by Maupertuis in 174020: If Sciences are grounded in certain immediately easy and clear principles, from which all their truths depend, they also include other principles, less simple indeed, and often difficult to discover, but that, once discovered, are very useful. They are to some extent the Laws that Nature fol lows in certain combinations of circumstances, and they teach us what it will do in similar occasions. The former principles need no proof, because they become obvious as soon as the mind examines them; the latter could not have, 9To appreciate the crucial difference between Euler’ s new principle and Newton ’s second law of motion, asserting that “A change in motion is pro  portional to the motive force impressed and takes place along the straight line in wh ich th e force is ex press ed [Mu tation em mot us propo rtiona lem ess e vi motrici impressæ , et fieri secundum lineam rectam qua vis illa imprimi ]” (Newton, PMCW, p. 12); (Newton, 1687, p. 416), remark that, by su pposing the motive force to be null, one can only deduce from this la s t l aw that the relevant motion is not changed, that is, it is inertial, but not that it is rectilinear uniform. To reach this conclusion, one has also to rely on Ne w ’s first law, which just fixes the nature of inertial motion: “ Every body preserves in its state of being at rest or of moving uniformly straight fo r ward except insofar as it is compelled to change its state by forces i m Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum , nisi quatenus a viribus impressis cogitur statum illum mutare] ibid.). 0cf. (Maupertuis, 1740, p. 170): “Si les Sciences sont fondées sur cer tains principes simples et clairs dès le premier aspect, d ’où dépendent toutes les vérités qui en son t l ’objet, ell es ont en cor e d’ autres princi pes, moi ns s imples à la vérité, etsouvent difficiles à découvrir, mais qui étant une fois déco u ’une trèsgrande utilité. Ceux ci sont en qu elque façon les Loix que la Nature su it dans certaines combinaisons de circon stances, et nous apprennent ce qu ’ elle fera dans de semblables occasions. Les premiers pri ncipes n’ont guére besoin de Démonstration, par l’ évidence do nt ils sont dès que l ’esprit les examine; les derniers ne sçauroient avoir de Démonstr a tion physique à la rig ueur, parce qu ’il est impossible de parco urir général e ment tous les cas où ils ont lieu. ”
S. MARONNE, M. PANZA Open Access strictly speaking, a physical proof, since it is impossible to consider in general all cases to which they apply. The aim of Maupertuis’s memoir is to suggest a new princi ple of the second kind, asserting that the equilibrium of any system of punctual bodies is obtained if an appropriate sum is maximal or minimal. This sum is M i i=1 n ∑P i dp i ∫+L+W i dw i ∫ where is the mass of the th body, , ... , are the forces acting upon it, and , ..., are the distances of this body from the centres of these forces, respectively. This is the first, static, version of the principle of least action. Maupertuis’s memoir originated quite an important program concerned with the foundation of mechanics, leading—through d’Alembert, Euler, and Lagrange, among others—to Hamil ton’s wellknown version of Newton’s mechanics (Fraser, 1983), (Fraser, 1985), (Szabó, 1987), (Pulte, 1989), (Panza, 1995), (Panza, 2003). Euler’s contributions to this program were essential and con cerned three major aspects: the elaboration of an appropriate mathematical tool for dealing with extremality conditions relative to integral forms including unknown functions; the generalisation of Maupertuis’s principle so as to get a general principle apt to provide the equations of motion of any system of several punctual bodies and also applicable, mutatis mutandis, to the solution of other mechanical prob lems; the justification of such a principle. In Euler’s view, these three aspects are intimately connected. His first contribution to this matter comes from his Methodus inveniendi (Euler, 1744): a treatise providing the first systema tisation of what is known, after Lagrange, as the calculus of variations21. The two appendixes to this treatise are solely de voted to enquiring the possibility of studying, respectively, the behaviour of an elastic band and the motion of an isolated body when they are submitted to forces, by relying on a general prin ciple asserting that “absolutely nothing happens in the world, in which a condition of maximum or minimum does not reveal itself”22. Euler’s main aim is not to find new results concerned with these problems, but to show how some already known results can be derived from a condition of maximum or mini mum for an integral form. The particular nature of this condi tion in the cases considered is taken to clarify the way in which a principle, which is analogous to Maupertuis’s one, can be stated in these cases and then, if possible, generalised. This same approach also governs Euler’s other works on the prince ple of least action: c f. in particular (Euler, 1748), (Euler, 1748), (Euler, 1751), (Euler, 1751), and (Euler, 1751). Euler’s and Maupertuis’s approaches are contrastive. Mau pertuis is mainly interested in looking for metaphysical and theological arguments (Maupertuis, 1744), (M aup ertuis, 1746), (Maupertuis, 1750), (Maupertuis, 1756). Indeed, he aims to support his claim to have found the very quantity in which Na ture is thrifty, and thus the real final cause acting in it. Euler is looking for mathematical invariances of the form emerging in different conditions and from which already known results regarding different mechanical problems can be drawn. Euler’s main idea is thus that of looking for an appropriate mathematical way to state a new principle that, being in agree ment with several results obtained through Newton’s original method of analysis of forces, could be generalised so as to get a principle of a new sort, namely a general variational principle. This research constituted a major event in the history of me chanics for it allowed to pass from a geometricalbased study of a particular concrete system to an analytical treatment of any sort of system based on an unique and general equation. In our view, these are the most fundamental origins of analytical me chanics (Panza, 2002). Lagrange’s first general formulation of the principle of least action (Lagrange, 1761) essentially de pends on the results obtained by Euler in this way. Algebraic Analysis Among the many wellknown differences between Newton’s and Leibniz’s approaches to calculus, a fairly relevant one deals with their opposite conceptions about its relation with the whole corpus of mathematics. Whereas Leibniz often stressed the novelty of his differential calculus, notably because of its special concern with infinity. Newton always conceived his results on tangents, quadratures, punctual speeds and connected topics as natural extensions of previous mathematics. Newton’s first research on these matters was explicitly based on the framework of Descartes’ geometry and geometrical al gebra provided in La Géométrie (Descartes, 1637). It mainly dates back to the years 16641666 (Panza, 2005), but culmi nates with the composition of the De analysis in 1669 (Newton, MWP, vol. II, pp. 206247) and of the De methodis in 1671 (Newton, MWP, vol. III, pp. 32353), where the new theory of fluxions is exposed. Later, Newton famously changed his mind about the respec tive merits of Descartes’ new way of making geometry and the classical (usually considered as synthetic) approach, especially identified with the style of Apollonius’ Conics (Galuzzi, 1990; Guicciardini, 2004), and based the Principia on the method of first and ultimate ratios, which he took to be perfectly compati ble with this last approach23. Finally, in the more mature pres entation of the theory of fluxions, the De quadratura curvarum (Newton, 1704)24, Ne wton stresses explicitely25: [...] To institute analysis in this way and to investigate the first or last ratios of nascent or vanishing finites is in harmony with the geometry of the ancients, and I wanted to show that in the method of fluxions there should be no need to introduce infinitely small figures into geometry. The De methodis begins as follows26: 1On Euler’s version of the calculus of variations, cf. (Fraser, 199 4). C. Fraser has also devoted many works to the history of the calculus of varia tion. A general survey of his result is offered in (Fraser, 2003 ). cf. (Euler , 1744, p. 245): “nihil omnino in mundo contingint, in quo non maximi minimive ratio q uæ piam e l uceat.” In the concl uding lemma of th e section I of book I of the Principia, Ne w ton claims that he proved the lemmas to which his method pertains “ in order to avoid the tedium of working out lengthy proofs by reductio ad absurdum in the ma nner of the Ancient geome te rs ” (Newton, PMCW, p. 441). See also (Newton, MWP, Vol. VIII, pp. 92168). cf. (Newton, MWP, VIII, p. 129): “[...] Analysin sic instituere, & finite  rum nascentium vel evanescentium rationes primas v el ultimas inves tigare, consonum est Geometriæ Veterum: & volui ostendere quo in Met h odo Fluxionum non opus sit figuras infinite parvas i n Geomet ri am introdu cere.” cf. (Newton, 16701671, pp. 3233): “Animadvertenti plerosque Geo excolendæ plurimum incumbere, et ejus ope tot tan  tasque d ifficultates superasse [ ...]: placu it sequenti a quibus campi analytici term inos expandere juxta ac curvarum doctrinam promovere
S. MARONNE, M. PANZA Open Access Observing that the majority of geometers [...] now for the most part apply themselves to the cultivation of analysis and with its aid have overcome so many formidable diffi culties [...] I found it not amiss [...] to draw up the fol lowing short tract in which I might at once widen the boundaries of the field of analysis and advance the doc trine of curves. In the earlymodern age, the term “analysis” and its cognates were used in mathematics in different, though strictly con nected, senses. Two of them were dominant in the middle of 17th century. Analysis, in the first sense, refers to the first part of a twofold method—the method of analysis and synthesis— paradigmatically expounded in the 7th book of Pappus Mathe matical Collection (Pappus, CMH). In the second sense, it re fers to a new domain of mathematics the introduction of which was typically ascribed to Viète, who had explicitly identified it with a “new algebra” (Viete, 1591). In our view, this discipline should be considered more as a family of techniques for making both arithmetic and geometry (Panza, 2007a), than as a separate theory somehow opposed to arithmetic and geometry. In the previous passage, Newton is undoubtedly referring to this discipline and he is claiming that his treatise aims to extend it so as to make it appropriate for studying curves. However, this extension crucially depends not only on the addition of new techniques, based on Descartes’ algebraic for malism, but a lso on a new conception about quantities, accord ing to which mathematics should deal not only with particular sorts of quantities, such as numbers, segments, etc., but also with quantities purely conceived, that is, with fluents (Panza, 2012). Hence, in the De methodis, the extended “field of analysis” no longer presents itself as a family of powerful techniques, but it rather takes the form of a new theory dealing with quantities purely conceived. These quantities are supposed to belong to a net of operational relations expressed through Descartes’ alge braic formalism appropriately extended so as to include infini tary expressions like series. Euler’s Theory of Functions Newton’s later opposition to Descartes’ way of doing ge ometry and the independence of the mathematical method of first and ultimate ratios from the analytic formalism of the the ory of fluxions—in the presence of the wellknown Newton Leibniz priority quarrel and its consequences—lead, in the 18th century, to a polarisation between two ways of understanding calculus. A Newtonian way, based on a classic conception of geometry, conceives fluxions as ratios of vanishing quantities. A Leibnitian way, based on the introduction of an appropriate new formalism, deals with infinitesimals. Maclaurin’s Treatise of fluxions (Maclaurin, 1742) is usually pointed out as the major example of the Newtonian view. “Flu xional ‘computations’ are not presented as a blind manipulation of symbols, but rather as meaningful language that could al ways be translated into the terminology of [...] [a] kinematic geometric model” (Guicciardini, 2004, pp. 239240). In contrast, Euler’s trilogy composed by the Introductio in analysin infinitorum, the Institutiones calculi differentialis, and the Institutionum calculi integralis (Euler, 1748), (Euler, 1755), (Euler, 1768) is indicated as the major example of the Leibni tian view. Nonetheless the two traditions were not as opposed, and the respective scientific communities were not as separated as it has been too often claimed. A clear example of this—which is mainly relevant here—is provided by Euler’s approach. Though there is no doubt that the theory expounded by Euler in the Institutiones and in the Institutionum uses Leibniz’s differ ential and integral formalism, some of the basic conceptions it is founded on derive from Newton’s views. Some of these conceptions are strictly internal to the organi sation of the theory. An example is provided by Euler’s idea that the main objects of differential calculus are not differen tials of variables quantities but differential ratios of functions conceived as ratios of vanishing differences (Ferraro, 2004). As he writes in the preface of the Institutiones27: Differential calculus [...] is a method for determining the ratio of the vanishing increments that any functions take on when the variable, of which they are functions, is given a vanishing increment [...] Therefore, differential calculus is concerned not so much with vanishing increments, which indeed are nothing, but with their mutual ratio and proportion. Since these ratios are expressed as finite quan tities, we must think of calculus as being concerned with finite quantities. In this way, Euler unclothes Newton’s notion of prime or ul timate ratio of its classically geometric apparel and transfers it to a purely formal domain using the language of Leibniz’s dif ferential calculus. Another strictly connected example comes from Euler’s de fi nition of integrals as antidifferentials and of the integral calcu lus as the “me tho d” to be applied for passing “from a certain relation among differentials to the relation of their quantities”, that is, in the simplest case (Euler, 1768), definitions 2 and 1, respectively, from to These definitions—which contrast with Leibniz’s conception of the integral as a sum of differentials—are instead clearly in agreement with the second problem of Newton’s De methodis: “when an equation involving the fluxions of quantities is exhib ited, to determine the relation of the quantities one to an oth er”28. The closeness of Euler’s and Newton’s views in both those examples depends on a more fundamental concern: the idea that both differential and integral calculus are part of a more general theory of functions (Fraser, 1989). This theory is exposed by Euler in the first volume of the In 7We slightly modify Blanton’s translation (Euler, ICDB , p. vii ’s original (Euler, 1755, p. VIII): “[...] calculi Differentialis [... terminandi rationem incrementorum evanescentium, quæ func  tiones qæ cunque accipiunt, dum quantitati variabili, cuius sunt functiones, increme n tum evanescens tribuitur [...]. Calculus igitur differentialis non tam in his ipsis incrementis evanescentibus, quippe quæ sunt nulla, exquirendis, quam in eorum ratione ac proportione mutua scrutanda occupatur: et cun hæ r a tiones finitis quantitabus exprimantur, etiam hic calculus circa quantitates finitas versari est cens endus.” cf. (Newton, 16701671, pp. 8283): “Exposita Æquatione fluxiones quantitatum involvente, invenire relationem quantitatum inter se”. On Ne wton’s notion of primitive, cf. (Panza, 2005, pp. 284293, 323 460).
S. MARONNE, M. PANZA Open Access troductio (Euler, 1748). According to him, it is not merely a mathematical theory among others. It is rather the fundamental framework of the whole of mathematics. Differential calculus is thus not conceived by Euler as a separated theory characterised by its special concern with infinity, as in Leibniz’s conception, but rather as a crucial part of an unitary building the founda tions of which consist of a theory of functions (Panza, 1992). As a matter of fact, this theory comes in turn from a large and ordered development of the results that Newton had pre sented in his De methodis before attaching the two main prob lems of the theory of fluxions, and that provided for him the base on which its extended “field of ana lysis” was grounded. A function is identified with an expression indicating the opera tional relations about two or more quantities and expressing a quantity purely conceived (Panza, 2007b). And the fundamental part of the theory concerns the power series expansions of functions. Though the language and the formalism that are used in Eu ler’s trilogy openly come back to the Leibnitian tradition, such a trilogy should thus be viewed, for many and fundamen tal reasons, as a realisation of the unification program that Newton had foreseen in the De methodis, a realisation that re lies, moreover, on the basic idea underlying Newton’s method of prime and last ratios. The Classification of Cubics and Algebraic Curves The second volume of the Introductio (Euler, 1748) is de voted to algebraic curves: the curves expressed by a polynomial equation in two variables when referred to a system of rectilin ear coordinates. Euler relies on some results obtained in the first volume to show that algebraic curves can be studied and classified without making use of calculus: as a matter of fact, this marks the birth of algebraic geometry. The problem of the classification of curves is quite ancient (Rashed, 2005). However, in his Géométrie (Descartes, 1637), Descartes returns to it in a new form: he concentrates only on algebraic curves (that he calls “geometric”, whereas he recom mends to reject from geometry other curves, termed “mechani cal”) and bases his classification on the degree of the corre sponding equation (Bos, 2001, pp. 356357), (Rashed, 2005, pp. 3250). This is, in fact, quite a broad classification, since equa tions of the same degree can express curves which look very different from each other. The classification of (nondegenerate) conics (the algebraic curves whose equations are the irre ducible ones of degree 2) is well known: they split up into ellipses (including circles), parabolas and hyperbolas. But what about curves expressed by equations of higher de grees? Newton answers the question for cubics (the algebraic curves whose equations are the nonreducible ones of degree 3), in a tract appeared in 1704 as an appendix to the Opticks (Newton, 1704), but the different stages of composition of which pre sumably date back to 16671695 (Newton, MWP, vol. VII, p. 565655): the Enumeratio Linearum Tertii Ordinis (Newton, 1704). In the second volume of the Introductio, Euler tackles the same problem using a quite different method, and shows that Newton’s classification is incomplete (Euler, 1748, vol. II, Ch. 9). He also provides an analogous classification of quartics (the algebraic curves whose equations are the nonreducible ones of degree 4), and explains how the method used for classifying cubics and quartics can, in principle, be applied to algebraic curves of any order (Euler, 1748, Vol. II, ch. 11 and 1214, respectively). Whereas Newton’s classification of cubics is based on their figure in a limited (i.e. finite) region of the plane and depends on the occurrences of points or line singularities as for instance nodes, cusps, double tangents, Euler suggests classifying alge braic curves of any order by relying on the number and on the nature of their infinite branches. Here is what he writes29: Hence, we reduced all third order lines to sixteen species, in which, therefore, all those of the seventytwo species in which Newton divided the third order lines are con tained30. It is not odd, in fact, that there is such a differ ence between our classification and Newton’s, since we obtained the difference of species only from the nature of branches going to infinity, while Newton considered also the shape of curves within a bounded region, and estab lished the different species on the basis of their diversity. Although this criterion may seem arbitrary, however, by following his criteria Newton could have derived many more species, whereas using my method I am able to draw neither more nor less species. Euler’s last remark alludes to much more than what it says. He rejects Newton’s criterion because of the impossibility of applying it as the order of curves increases, since so great a variety of shapes arise, as witnessed by the mere case of cubics. Indeed, a potentially general criterion has to deal with some properties of curves that can be systematically and as exhaus tively as possible explored in any order, like those of infinite branches, according to Euler. This task could be difficult, however, if these curves were studied through their equations taken as such, since the com plexity of a polynomial equation in two variables increases very quickly with its order. Insofar as a polynomial equation in two variables cannot be transformed by changing its global degree so that it continues to express the same curve, Euler shows how to determine the number and nature of infinite branches of a curve by considering its equation for some appropriate trans formations that lower its degree in one variable, namely “both by choosing the most convenient axis and the most apt inclina tion of the coordinates”, and attributing to a variable a conven ient value31. 9We slightly modify Blanton’s translation (Euler, IAIB, Vol. II, p. ’s original ( cf. Euler, 1748, Vol. II, 236 , p. 123): “ Omnes ergo Lineas tertii ordinis reduximus ad Sedeciem Species, in quibus pro p terea omnes illæ Species Septuagin ta dua, in quas Newtonus Lineas tertii ordinis divisit, continentur. Quod vero inter hanc nostram divisionem ac Newtonia nam tantum intercedat discrimen mirum non est; hic enim tantum ex ramo rum in infinitum excurrentium indole Specierum diversitatem desu m Newtonus quoque ad statum Curvarum in spatio finito spec tasset, atque ex hujus varietate diversas Species constituisset. Quanquam autem hæ c divisionis ratio arbitraria videtur, tamen Newtonus suam tandem rationem sequens multo plures Species producere potuisset, cum equidem mea m e thodo utens neque plures neque pauciores Species eruere queam. ” 0This is the reason why Euler prefers to use the term “genre” species” in order to indicate his classes of curves. The latter terms is re  served to distinguish curves of a given genre according to their shape in a limited region of the plane (Euler , 1748, vol. II, 238, p. 126). Once again, we slig htly modify Blanton’s translation (Euler, IAIB, Vol. 176). Here is Euler’s original (Euler , 1748, Vol. II , 272, p. 150): “Neg o tium autem hoc per reductionem æ quationis ad formam simpliciorem, dum et Axis commodissimus, et inclinatio Coordinatarum aptissima ass u mitur, valde sublevari potest: tum etiam, quia perinde est, utra Coordinata rum pro Absciss a accipiatur, labo r maxime dimin uetur, si ea Coordinata rum, cujus paucissimæ dimensiones in æ quatione occurrunt, pro Applica ta ass
S. MARONNE, M. PANZA Open Access Conclusion Euler’s results of which we have given an account, both in the case of foundation of mechanics and in that of algebraic analysis, depend on the effort to carry out or to extend a New tonian program. But in both cases, this is done by relying on Cartesian and Leibnitian conceptions and tools. C. A. Truesdell has summed up the situation about mechanics by saying that Euler inaugurated the tradition of Newtonian mechanics be cause he “put most of mechanics in their modern form”32. Mu tatis mutandis, the same also holds for Euler’s theory of func tions. In both cases the following question naturally arises: what does remain, then, of Newton’s conceptions in Euler’s theories? This is too difficult a question to hope to offer a com plete answer in a single paper. We merely hope to have pro vided some elements for such an answer. Acknowledgemen ts We thank Frédéric Voilley for his precious linguistic support. REFERENCES Aiton, E. J. (1989). 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