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Euler, Reader of Newton: Mechanics and AlgebraicAnalysis

Marco Panza, Sebastien Maronne

To cite this version:Marco Panza, Sebastien Maronne. Euler, Reader of Newton: Mechanics and Algebraic Analysis.Advances in Historical Studies, Continuum, 2014, 3 (1), pp.12-21. <10.4236/ahs.2014.31003>. <hal-00415933v1>

Newton and Euler

Sebastien Maronne and Marco PanzaCNRS, REHSEIS

(CNRS and Universite Paris Diderot – Paris 7)

Preliminary version before publication in Scott Mandelbrot and Helmut Pulte, editors,

The reception of Isaac Newton in the European Enlightenment (2 vols.), Continuum,

London, forthcoming in 2010.

Contents

1 Introduction 2

2 From Newtonian Geometric Mechanics to Analytic Mecha-

nics 2

2.1 Explanation of forces . . . . . . . . . . . . . . . . . . . . . . . 32.2 The reformulation of Newton’s mechanics using Leibniz dif-

ferential calculus and the introduction of external frames ofreference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 A new sort of principles: Euler’s program for founding New-ton’s mechanics on variational principles . . . . . . . . . . . . 13

3 Algebraic analysis 16

3.1 Euler’s theory of functions . . . . . . . . . . . . . . . . . . . . 183.2 The classification of cubics and algebraic curves . . . . . . . . 21

4 Conclusions 23

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1 Introduction

To describe the totality of the links between Newton’s and Euler’s scientificproductions1 and to give a complete account of Euler’s role in Newton’sreception would be an herculean task. A large part of the scientific resultsof the former are indeed, either explicitly or not, based on or inspired bythose of the latter. Hence, we decided to leave apart a number of punctualconnections, and to concentrate on two major topics. Euler contributed to are-appropriation of Newtonian science that, among many other things, leadto the creation of analytic mechanics and of algebraic analysis or the theoryof functions. These are the two topics we shall deal with2.

2 From Newtonian Geometric Mechanics to

Analytic Mechanics

Euler’s works on mechanics concern different domains, some of which arenot considered in Newton’s Principia [Newton(1687)]. Beside his Mechan-ica [Euler(1736)]—a two volume treatise on the motion of free or constrainedpunctual bodies—and a large number of papers on the same subject, Euleralso highly contributed to the mechanics of rigid and elastic bodies ([Truesdell(1960)]),the mechanics of fluids, the theory of machines and the naval science. Weshall limit ourselves to some of his contributions to the foundation of themechanics of discrete systems of punctual bodies. We shall namely consider

1The quasi-totality of Euler’s works are available on line in [Eul()], where one mayalso find many notices and references. Among other recent books devoted to Euler,cf. [Bradley and Sandifer(2007)] and [Backer(2007)].

2We will not address, however, some related topics that other articles of the presentvolume deal with. Among them, there are: Euler’s views on Newton’s gravitation theoryand his own celestial mechanics, particularly his lunar theory (on which, cf. Nauenberg’sand Kollerstrom’s articles); Euler’s reflections about (absolute and relative) space andtime (on which, cf. Di Salle’s article); and Euler’s critical exposition of Newtonian sciencein his Lettres a une princesse d’Allemagne ([Euler(1768-1772)], on which, cf. Kleinert’sarticle). On Euler’s celestial mechanics, cf. also [Schroeder(2007)]. On this matter, itis relevant to note that Euler doubted in the 1740s about the validity of Newton’s lawof gravitation because of errors he observed in calculations of planetary perturbations(cf. [Euler(1743)], Kleinert’s discussion of this memoir, in [Euler(1911-...), ser. II, vol. 31,appendix] and [Schroeder(2007), pp. 348-379]. On Euler’s views on space and time, cf.also [Cassirer(1907), Book VII, Chapter II, section II] and [Maltese(2000)]. Finally, onEuler’s Lettres a une princesse d’Allemagne, cf. [Calinger(1976)].

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his views on the physical explanation of forces3, his reformulation of the ba-sic notions of Newtonian mechanics, and his works on the principle of leastaction.

2.1 Explanation of forces

The third book of Newton’s Principia offers an explanation of the motion ofplanets around the sun and of their satellites around them. It is based onthe assumption that the celestial bodies act upon each others according toan attractive force acting at distance whose intensity depends on the massof the attracting body and on its distance from the attracted one and whoseeffects are not influenced by a resistant medium. This is a special centralforce—a force attracting bodies along a straight line directed towards a fixedor moving centre—characterised by the following well-known equality

FC,c = mGM

r2

where FC,c is the force by which the body C attracts the body c, m andM are, respectively, the masses of c and C, r is the distance between (thecentres of) them, and G is an universal constant.

In the first two books of the Principia, Newton provides a purely math-ematical theory of central forces. In the third book, he is thus able to givea mathematical theory of the world-system by basing it only on the previ-ous assumption, that, according to him, is proved by empirical observations.Insofar as it allows to determine the trajectories of the relevant bodies ac-cording to the mathematical theory of the first two books, no supplementaryhypothesis about the nature of the relevant force is necessary. For Newton,an hypothesis about forces is a conjecture concerning their qualitative natureand causes. His “Hypotheses non fingo”—famously claimed in the GeneralScholium of the second edition of the Principia—is just intended to declarethat he does not adventure in any such conjecture, since this is not necessaryfor providing a satisfactory scientific explanation.

Apparently, this view is not shared by Euler. He seems to maintainthat the notion of force cannot be primitive, and that a mathematical theoryabout forces cannot be separated from an account of their causes, even if this

3On this matter, cf. also E. Watkins’ article in the present volume.

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account depends more on “the province of metaphysics than of mathematics”and thus one cannot pretend to afford it with “absolute success”4.

Euler’s account (on which, cf. [Gaukroger(1982)]) is based on a Cartesianrepresentation of the world as a plenum of matter. Here is what he writes inhis 55th letter to a German Princess5:

As you see nothing that impels [small bits of iron and steel] towardthe loadstone, we say that the loadstone attracts them; and thisphenomenon we call attraction. It cannot be doubted, however,that there is very subtle, though invisible, matter, which producesthis effect by actually impelling the iron towards the lodestone[. . . ]. Though this phenomenon be peculiar to the loadstone andiron, it is perfectly adapted to convey an idea of the significationof the word attraction, which philosophers so frequently employ.They allege then, that all bodies, in general, are endowed with aproperty similar to that of the loadstone, and that they mutuallyattract [. . . ].

And then, in the 68th letter6:

[. . . ] as we know that the whole space which separates the heav-enly bodies is filled with a subtle matter, called ether, it seems

4Cf. [Euler(1823), vol. I, p. 201]. Here is Euler’s original [Euler(1768-1772), vol. I, lett.68th, p. 265]: “Il s’agit a present d’approfondir la veritable source de ces forces attractives,ce qui appartient plutot a la Metaphysique qu’aux Mathematiques ; & je ne saurois meflatter d’y reussir aussi heureusement.”

5Cf. [Euler(1823), vol. I, p. 165]. Here is Euler’s original [Euler(1768-1772), vol. I, lett.55th, pp. 219-220]: “Comme on ne voit rien, qui les [de petits morceaux de fer ou d’acier]pousse vers l’aimant, on dit que l’aimant les attire, & l’action meme, se nomme attraction.On ne sauroit douter cependant qu’il n’y ait quelque matiere tres subtile, quoiqu’invisible,qui produise cet effet, en poussant effectivement le fer vers l’aiman ; [. . . ] Quoique cephenomene soit particulier a l’aimant, & au fer, il est tres propre a eclaircir le termed’attraction, dont les Philosophes modernes se servent si frequemment. Ils disent donc,qu’une propriete semblable a celle de l’aimant, convient a tous les corps, en general, & quetous les corps au monde s’attirent mutuellement”.

6Cf. [Euler(1823), 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 entre les corps celestes estrempli d’une maniere subtile qu’on nomme l’ether, il semble plus raisonnable d’attribuerl’attraction mutuelle des corps, a une action que l’ether y exerce, quoique la maniere noussoit inconnue, que de recourir a une qualite inintelligible.” On this same matter, cf. alsothe 75th letter: [Euler(1768-1772), vol. I, pp. 297-298].

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more reasonable to ascribe the mutual attraction of bodies to anaction which the ether exercises upon them, though its mannerof acting may be unknown to us, rather than to have recourse toan unintelligible property.

The Cartesian vein of Euler’s account is even clearer in the Mecha-nica [Euler(1736)]. It is highly significant that in such a purely mathematicaltreatise, Euler devotes a scholium to discuss the causes and origins of forces.This is the scholium 2 of definition 10: the definition of forces. Accordingto it, “a force [potentia] is the power [vis ] that either makes that a bodyto pass from rest to motion or changes its motion.”7 This definition doesnot explain what forces come from. Scholium 2 discusses the question. Eulerbegins by declaring that, among the real forces acting in the world, he onlyconsiders gravity. Then he argues that similar forces “are observed to existin the magnetic and electric bodies” and adds:

Some people think that all these [forces] arise from the motionof a somehow subtle matter; some others attribute [them] to thepower of attracting and repulsing of the bodies themselves. But,whatever it may be, we certainly see that forces of this kind canarise from elastic bodies and from vortices, and we shall inquire,at the appropriate occasion, whether these forces can be explainedthrough these phenomena8.

One could hope that Euler’s Cartesian view be made more precise in amemoir presented in 1750 whose title is quite promising: “Recherches surl’origine des forces” ([Euler(1750a)])9. But the content of this memoir isvery surprising.

7Cf. [Euler(1736), p. 39]: “Potentia est vis corpus vel ex quiete in motum perducens velmotum eius alterans.” Here and later, we slightly modify I. Bruce’s translation availableonline at http://www.17centurymaths.com.

8Cf. [Euler(1736), p. 40]: “Similes etiam potentiae deprehenduntur in corporibus mag-neticis et electricis inesse, quae certa tantum corpora attrahunt. Quas omnes a motu ma-teriæcuiusdam subtilis oriri alii putant, alii ipsis corporibus vim attrahendi et repellenditribuunt. Quicquid autem sit, videmus certe ex corporibus elasticis et vorticibus huius-modi potentias originem ducere posse, suoque loco inquiremus, num ex inde phaenomenahaec potentiarum explicari possint.”

9Cf. also [Euler(1746)] and [Euler(1765), Introduction]. The chapter 2 of[Romero(2007)] presents a detailed study of [Euler(1746)] and [Euler(1750a)]. In[Gaukroger(1982), pp. 134-138], an overview of the relevant parts of [Euler(1765)] is of-fered.

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Euler begins by arguing that impenetrability is an essential property ofbodies, and that it “comes with a force sufficient for preventing penetra-tion”10. It follows—he says—that, when two bodies meet themselves in sucha way that they could not persist in their state of motion without penetratingeach other, “from the impenetrability of both of them a force arises that, byacting on them, changes [. . . ][this] state11.” This being admitted, Euler showshow to derive from this only supposition the well-known mathematical lawsof the shock of bodies. This is enough, for him, to conclude that the changesin the state of motion due to a shock of two bodies “are produced only bythe forces of impenetrability”12, so that, in this case, the origins (and cause)of forces is just the impenetrability of bodies.

One would expect that Euler goes on by describing a plausible mechanicalmodel allowing to argue that this is also the case of any other force, namelyof central forces acting at distance. But this is not so. He limits himself toconsider the case of centrifugal forces that, basing on no argument, he takesto be all reducible to the case where a body is deviated from its rectilinearmotion because it meets a vaulted surface ([Euler(1750a), art. LI, p. 443]).Finally he writes13:

10Cf. [Euler(1750a)], art. XIX, p. 428: “Aussi-tot [. . . ] qu’on reconnoit l’impenetrabilitedes corps, on est oblige d’avouer que l’impenetrabilite est accompagnee d’une force suff-isante, pour empecher la penetration.”

11Cf. [Euler(1750a)], art. XXV, p. 431: “[. . . ] a la rencontre de deux corps, qui sepenetreroient s’ils continuoient a demeurer dans leur etat, il nait de l’impenetrabilite del’un et de l’autre a la fois une force qui en agissant sur les corps, change leur etat.”

12Cf. [Euler(1750a), art. XLVI, p. 441]: “[. . . ] dans le choc des corps [. . . ], il estclair que les changements, que les corps y souffrent, ne sont produits que leurs forcesd’impenetrabilite.” Cf. also ibid. art. XLIV, p. 440.”

13Cf. [Euler(1750a), arts. LVIII and LIX, pp. 446-447]: “[. . . ] s’il etoit vrai, commeDescartes et quantite d’autres Philosophes l’ont soutenu, que tous les changements, qui ar-rivent aux corps, proviennent ou du choc des corps, ou des forces nommees centrifuges; nousserions a present tout a fait eclaircis sur l’origine des forces, qui operent tous ces change-mens [. . . ]. Je crois meme que le sentiment de Descartes ne sera pas mediocrement fortifiepar ces reflexions; car ayant retranche tant de forces imaginaires, dont les Philosophes ontbrouille les premiers principes de la Physique, il est tres probable que les autres forcesd’attraction, d’adhesion etc. ne sont pas mieux fondees. LIX. Car quoique personnen’ait encore ete en etat de demontrer evidemment la cause de la gravite et des forcesdont les corps celestes sont sollicites, par le choc ou quelque force centrifugue ; il fautpourtant avouer que personne n’en a non plus demontre l’impossibilite. [. . . ] Or quedeux corps eloignes entr’eux par un espace entierement vuide s’attirent mutuellement parquelque force, semble aussi etrange a la raison, qu’il n’est prouve par aucune experience.A l’exception donc des forces, dont les esprits sont peut-etre capables d’agir sur les corps,

6

If it were true, as Descartes and many other philosophers havemaintained, that all the changes that bodies can suffer come ei-ther from the shock of bodies or from the forces named centrifu-gal, we would now have clear ideas about the origins of forces pro-ducing all these changes [. . . ]. Even I believe that Descartes’ viewwould not be scarcely reinforced by those reflections, since, afterhaving eliminated many imaginary forces with which philosophershave jumbled the first principles of physics, it is very likely thatthe other forces of attractions, adherence, etc. are not betterestablished.

[. . . ] For, though nobody has been able to establish manifestly thecause of gravity and forces acting upon heavenly bodies throughthe shock or some centrifugal forces, we should confess that any-body has neither proved the impossibility of it. [. . . ] Now itseems as much stranger to reason than it is not proved by ex-perience that two bodies separated by a completely empty spacemutually attract through some forces14. Hence, I conclude that,with the exception of forces whose spirits are perhaps able toact upon bodies, which are probably of a quite different nature,there is no other force in the world beside those originated by theimpenetrability of bodies.

Though advancing a non-Newtonian demand of explanation of the natureand causes of forces15 and sharing both the Cartesian 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 matterallowing a reduction of all forces to contact ones, Euler reaches thus a quiteNewtonian (and non-Cartesian) attitude only disguised by rhetoric. His mainpoint is finally clear, indeed: a mathematical science of motion is perfectlypossible also in the ignorance of the actual causes of attractive forces, and theonly way to ensure that some forces are caused by some reasons is to showthat the consideration of these reasons leads to the well-known mathematical

lesquelles sont sans doute d’une nature tout a fait differente, je conclus qu’il n’y a pointd’autres forces au monde que celles, qui tirent leur origine de l’impenetrabilite des corps.”

14About Euler’s opposition to action at distance, cf. [Wilson(1992)].15But remember that Newton’s rejection of any hypotheses about the nature and the

causes of the gravitational force only concerns the limited domain of the mathematicalprinciples of natural philosophy.

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laws of motion. These laws—rather than any possible mechanical model—are thus finally understood as the only sure expression of the reality of theuniverse.

2.2 The reformulation of Newton’s mechanics using

Leibniz differential calculus and the introduction

of external frames of reference

As it is well known, Newton’s mechanics is essentially geometric. Curves areused to represent trajectories of punctual bodies and a theorem is provedensuring that non-punctual bodies behave with respect to attractive forcesas if their mass were concentrated in their centre of gravity. Instantaneousspeeds are indirectly represented and measured by segments taken on thetangents of the curves-trajectories. They are taken to represent primarilythe rectilinear space that a relevant point would cover in a given time, finiteor infinitesimally small, if any force acting upon it ceased and the motion ofthis point were thus due only to its inertia. An analogous form of indirectrepresentation and measure holds for any sort of forces, or better for theiraccelerative punctual component. This provides a very simple way of com-posing forces and inertia, essentially based on the law of parallelogram thatis primarily conceived as holding for rectilinear uniform motions. When theconsideration of time is relevant, this is typically represented and measuredby appropriate geometric entities, like appropriate areas: for example theareas that are supposed to be swept out in that time by a vector radius, inthe case of a trajectory complying with Kepler’s second law.

To solve mechanical problems, this fundamental geometric apparatus isof course not sufficient. Newton’s mechanics also includes two other funda-mental ingredients.

The first is a geometric method which allows to deal with punctualand/or instantaneous phenomena and to determine their macroscopic effects(like equilibrium configurations, effective trajectories, and continuously act-ing forces). It is provided by the method of prime and last ratios, togetherwith a number of appropriate tricks.

The second ingredient is given by a number of fundamental laws express-ing some basic relation between bodies (or better their masses), their mo-tions, and the forces acting on them and because of them. It is provided bythe well-known Newton’s laws of motion, occasionally supplemented by some

8

principles—like the principle of maximal descent of the centre of gravity—which are taken to follow from them.

Though these laws are still considered as the more fundamental ingre-dients of classical mechanics, what we call today “Newtonian mechanics”is a quite different theory, reached through a deep transformation and re-formulation of Newton’s original presentation. These transformation andreformulation mainly occurred during the 18th century and Euler was themajor craftsman of them16. Here is as Giulio Maltese sums up the situation[Maltese(2000), pp. 319-320]:

In fact, it was Euler who built what we now call the “New-tonian tradition” in mechanics, grounded on the laws of linearand angular momentum (which Euler was the first to consideras principles general and applicable to each part of every macro-scopic system), on the concept that forces are vectors, on the ideaof reference frame and of rectangular Cartesian co-ordinates, andfinally, on the notion of relativity motion.

This quotation emphasises some basic ingredients of today’s Newtonianmechanics. We shall come back in a moment on some of them. Afterwardswe should observe that the gradual emergence of these elements depends ona more basic transformation (though perhaps, not so fundamental, as such).We refer to the replacement of Newton’s purely geometric forms of represen-tation of motions, speeds and forces and of the connected method of primeand last ratios by other forms of representation and expression employingappropriate algebraic techniques enriched by the formalism of Leibnitian dif-ferential calculus.

This transformation is often described as a passage from a geometricto an analytic (understood as non-geometric) way of presenting Newton’smechanics. This is only partially true, however. Though the appeal to thealgebraic and differential formalism allows, indeed, to express the relationbetween the relevant mechanical quantities through equations involving thetwo inverse operators d and

∫

submitted to a number of easily applicablerules of transformation, these equations are part of mechanics only if thesymbols that occur in them admit a mechanical meaning. It is just the way

16As N. Guicciardini has remarked [Guicciardini(1999), p. 6], “after Euler the Principia’smathematical methods belong definitely to what is past and obsolete.”

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as this meaning is explained—and not the mere use of this formalism—thatdecides whether the adopted presentation is geometric or not.

For example, it is not enough to identify the punctual speed of a certainmotion with the differential ratio ds

dtto get a non-geometric definition of

speed: whether this definition is geometric or not depends on the way asthis ratio, and namely the differential ds, are understood. If this differentialis taken to be an infinitesimal difference in the length of a certain variablesegment belonging to an appropriate geometric diagram and indicating thedirection of the speed with respect to other component of this diagram, thedefinition is still geometric.

This just happens in the first attempts to apply the differential formal-ism to Newtonian mechanics, like those of Varignon, Johann Bernoulli, andHermann17: the language of differential calculus is used to speak of mechan-ical configurations represented by appropriate geometric diagrams and itsrules are applied in order to get the relevant quantitative relations betweenthe elements depicted in these diagrams ([Panza(2002)]). Like in Newton’sPrincipia, mechanical problems are thus, in these essays, distinguished fromeach other according to specific features manifested by the correspondingdiagrams. Hence, differences of problems depends on differences of diagrams.

This fragmentation of mechanics in many problems geometrically differentis still particularly evident in Hermann’s Phoronomia ([Hermann(1716)]) thatEuler considers as the main treatise on dynamical matters written after thePrincipia. This is just what Euler wants to avoid in his Mechanica. Here iswhat he writes in the preface18:

17On these essays, cf., besides Mikhailov’s article in the present vol-ume, also [Aiton(1989)], [Blay(1992)], [Guicciardini(1995)], [Guicciardini(1996)],[Mazzone and Roero(1997)].

18A large part of this passage is quoted and translated by N. Guicciardini in[Guicciardini(2004a), p. 245]. We quote his translation by adding a translation of thepart he does not quote that slightly differs from Bruce’s one [cf. footnote (7)]. Here isEuler’s original [[Euler(1736), Praefatio]: “[. . . ] quod lectorem maxime distinet, omniamore veterum [. . . ] geometricis demonstrationibus est persecutus [. . . ]. Non multum dis-simili quoque modo conscripta sunt Neutoni Principia Mathematica Philosophiae [. . . ].Sed quod omnibus scriptis, quae sine analysi sunt composita, id potissimum Mechanicisobtingit, ut Lector, etiamsi de veritate eorum, quae proferuntur, convincatur, tamen nonsatis claram et distinctam eorum cognitionem assequatur, ita ut easdem quaestiones, sitantillum immutentur, proprio marte vix resolvere valeat, nisi ipse in analysin inquirat eas-demque propositiones analytica methodo evolvat. Idem omnino mihi, cum Neutoni Prin-

cipia et Hermanni Phoronomiam perlustrare coepissem, usu venit, ut, quamvis pluriumproblematum solutiones satis percepisse mihi viderer, tamen parum tantum discrepantia

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[. . . ] what distracts the reader the most [in Hermann’s Phorono-mia] is that everything is carried out [. . . ] with old-fashionedgeometrical demonstrations [. . . ]. Newton’s Principia Mathemat-ica Philosophiae are composed in a not-quite different way [. . . ].But what happens with all the works composed without analysisis particularly true with those which pertain to mechanics. Infact, the reader, even when he is persuaded of the truth of thethings that are demonstrated, nonetheless cannot reach a suffi-ciently clear and distinct knowledge of them. So he is hardly ableto solve the same problems with his own strengths, when they arechanged just a little, if he does not research into the analysis andif he does not develop the same propositions with the analyticalmethod. This is exactly what usually happened to me, when Ibegan to examine Newton’s Principia and Hermann’s Phorono-mia. In fact, even though I thought that I could understand thesolutions to numerous problems well enough, I could not solveproblems that were slightly different. Therefore, in those years, Istrove, as much as I could, to arrive at the analysis behind thosesynthetic methods, and to deal with those propositions in termsof analysis for my own purposes. Thanks to this procedure Iperceived a remarkable improvement of my knowledge.

The main purpose of Euler’s Mechanica is to use Leibnitian differentialformalism (which is what he calls “analytic method”) in order to generalisesome of Newton’s results. Euler aims to get some general procedures whichallow him to solve large families of problems. He also looks for some rules tobe used in appropriate circumstances to determine, in a somehow automaticway, appropriate expressions for relevant mechanical quantities.

In order to reach this aim, Euler identifies punctual speeds and acceler-ations with first and second differential ratios, respectively, and introducesan universal measure of a punctual speed given by the altitude from whicha 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 undera new form, quite different from the original. Still, in this treatise, mechanicalproblems are still tackled by relying on intrinsic coordinates systems: speeds

problemata resolvere non potuerim. Illo igitur iam tempore, quantum potui, conatus sumanalysin ex synthetica illa methodo elicere easdemque propositiones ad meam utilitatemanalytice pertractare, quo negotio insigne cognitionis meae augmentum percepi”.

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and forces are composed and decomposed according to directions that aredictated by the intrinsic nature of the problem, for example so as to calculatethe total tangential and normal forces with respect to a given trajectory. Thisapproach is quite natural, but limits the generality of possible common rulesand principles.

A new fundamental change occurs when extrinsic references frames, typ-ically constituted by triplets of orthogonal fixed Cartesian coordinates, areintroduced and when the relativity of motion is conceived as the invarianceof its laws with respect to different frames submitted to uniform retailer mo-tions. Though this change was in fact a collective and gradual transformation([Bertoloni Meli(1993)] and [Maltese(2000), p. 6]), Euler played a crucial rolein it. Among other important contributions connected with it (described anddiscussed in [Maltese(2000)]), it is important to consider his introduction ofthe today’s usual form of Newton’s second law of motion. This is the ob-ject of a memoir presented in 1750: “Decouverte d’un nouveau principe demecanique” ([Euler(1750b)]).

The argument that Euler offers in this memoir in order to justify the in-troduction of his “new principle” is so clear and apt to elucidate the crucialimportance of this new achievement that it deserves to be mentioned. Thestarting point of this argument is the insufficiency of the tools provided bothby Newton’s Principia and by Euler’s own Mechanica for studying the rota-tion of a solid body around an axis continuously changing its position withrespect to the elements of the body itself. To study this motion, Euler ar-gues, new principles are needed and they have to be deduced from the “firstprinciple or axioms” of mechanics, which—he says—cannot but concern, assuch, the rectilinear motion of punctual bodies [Euler(1750b), art. XVIII,p. 194]. The problem is just that of formulating these axioms in the mostappropriate way for allowing an easy deduction of all other principles thatare needed to study the different kinds of motion of the different kinds ofbodies. According to Euler, these axioms reduce to a unique principle whichis just his new one.

This principle is expressed by a triplet of equations that express Newton’ssecond law with respect to the three orthogonal directions of a reference frameindependent of the motions to be studied ([Euler(1750b), art. XXII, p. 196]):

2Md2x = Pdt2 ; 2Md2y = Qdt2 ; 2Md2z = Rdt2,

where M is the mass of the relevant punctual body, P , Q, and R are the

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total forces acting along the directions of the threes axes and 2 is a simplefactor of normalization.

To understand the fundamental role that Euler assigns to this princi-ple, a simple example is sufficient ([Euler(1750b), art. XXIII, p. 196]): fromP = Q = R = 0, one gets, by integrating,

Mdx = Adt ; Mdy = Bdt ; Mdz = Cdt,

where A, B, C are integration constants. It is thus proved that in this case thespeed is constant in any direction so that the motion of any body on whichno force acts is rectilinear and uniform, as the first Newton’s law asserts.

2.3 A new sort of principles: Euler’s program for foun-

ding Newton’s mechanics on variational principles

Though quite powerful, Euler’s new principle only deals directly with singlepunctual bodies. Let’s consider a system of several punctual bodies mutuallyattracting to each others and possibly submitted to some external forces andinternal constraints. In order to get the conditions of equilibrium or theequations of motions of such a system by relying on Euler’s principle, adetailed and geometric analysis of all the forces operating in this system isnecessary. A fortiori, this is also the case of any other principle dealing withsingle punctual bodies set in Newton’s Principia and in Euler’s Mechanica.Hence, the study of any particular system of several punctual bodies throughthese principles requires a geometrical analysis of forces that differs fromsystem to system. Consequently, only quite simple systems can be studiedin such a way.

This is the reason for the need of a new sort of mechanical principles—directly concerned with whatever system of several punctual bodies—arosequite early. A similar principle, that would be later known as the principle ofvirtual velocities, was suggested by Johann Bernoulli in a letter to Varignonof January, 26th 1711 ([Varignon(1725), vol. II, pp. 174-176]). But a clearstatement of the difference between these two kinds of principles only appearsin a Maupertuis’ memoir presented in 174019:

19Cf. [Maupertuis(1740), p. 170]: “Si les Sciences sont fondees sur certains principessimples et clairs des le premier aspect, d’ou dependent toutes les verites qui en sontl’objet, elles ont encore d’autres principes, moins simples a la verite, et souvent difficiles adecouvrir, mais qui etant une fois decouverts, sont d’une tres-grande utilite. Ceux-ci sont

13

If Sciences are grounded on certain immediately easy and clearprinciples, from which all their truths depend, they also includeother principles, less simple indeed, and often difficult to discover,but that, once discovered, are very useful. They are to some ex-tent the Laws that Nature follows in certain combinations of cir-cumstances, and they learn us what it will do in similar occasions.The former principles need no proof, because of their evidence;once the mind examines them; the latter could not have, strictlyspeaking, a physical proof, since it is impossible to consider ingeneral all cases to which they apply.

The aim of Maupertuis’ memoir is to suggest a new principle of the secondkind, asserting that the equilibrium of any system of n punctual bodies isobtained if an appropriate sum is maximal or minimal. This sum is

n∑

i=1

Mi

[∫

Pidpi + . . . +

∫

Widwi

]

where Mi is the mass of the i-th body, Pi, . . . , Wi are the forces acting uponit, and pi, . . . , wi are the distances of this body from the centres of theseforces, respectively. This is the first, statical, version of the principle of leastaction.

Maupertuis’ memoir originated a quite important program concernedwith the foundation of mechanics, leading—through d’Alembert, Euler, andLagrange, among other ones—to Hamilton’s well-known version of New-ton’s mechanics ([Fraser(1983)], [Fraser(1985)], [Szabo(1987)], [Pulte(1989)],[Panza(1995)], [Panza(2003)]).

Euler’s contributions to this program were essential and concerned threemajors aspects:

– the elaboration of an appropriate mathematical tool for dealing withextremality conditions relative to integral forms including unknownfunctions;

en quelque facon les Loix que la Nature suit dans certaines combinaisons de circonstances,et nous apprennent ce qu’elle fera dans de semblables occasions. Les premiers principesn’ont guere besoin de Demonstration, par l’evidence dont ils sont des que l’esprit lesexamine; les derniers ne scauroient avoir de Demonstration physique a la rigueur, parcequ’il est impossible de parcourir generalement tous les cas ou ils ont lieu.”

14

– the generalisation of Maupertuis’ principle so as to get a general prin-ciple apt to provide the equations of motion of any system of severalpunctual bodies and also applicable, mutatis mutandis, to the solutionof other mechanical problems;

– the justification of such a principle.

In Euler’s view, these three aspects are intimately connected. His firstcontribution to this matter comes from his Methodus inveniendi ([Euler(1744)]):a treatise providing the first systematisation of what is known, after La-grange, as the calculus of variations20. The two appendixes to this treatiseare just devoted to enquiring the possibility of studying, respectively, thebehaviour of an elastic band and the motion of an isolated body when theyare submitted to some forces, by relying on a general principle asserting that“absolutely nothing happens in the world, in which a condition of maximumor minimum does not reveal itself” 21.

Euler’s main aim is not to find new results concerned with these problems,but to show how some yet known results can be derived from a condition ofmaximum or minimum for an integral form. The particular nature of thiscondition in the cases considered is taken to clarify the way as a principle,which is analogous to Maupertuis’ one, can be stated in these cases and then,possibly, generalised. This same approach also governs Euler’s other works onthe principle of least action (cf., in particular [Euler(1748b)], [Euler(1748c)],[Euler(1751a)], [Euler(1751b)], and [Euler(1751c)]).

Euler’s and Maupertuis’ approaches are contrastive. Mauper-tuis is mainly interested in looking for metaphysical and theologicalarguments ([Maupertuis(1744)], [Maupertuis(1746)], [Maupertuis(1750)],[Maupertuis(1756)]). Indeed, he aims to support his claim to have foundthe very quantity that the nature is willing to save up, and thus the real finalcause acting in it. Euler is looking for mathematical invariances of the form

∫

Zdz = Max/Min

20On Euler’s version of the calculus of variations, cf. [Fraser(1994)]. C. Fraser has alsodevoted many works to the history of the calculus of variation. A general survey of hisresult is offered in [Fraser(2003)].

21Cf. [Euler(1744), p. 245]: “nihil omnino in mundo contingint, in quo non maximiminimive ratio quæpiam eluceat.”

15

emerging in different conditions and from which yet known results regardingdifferent mechanical problems can be drawn.

Euler’s main idea is thus that of looking for an appropriate mathematicalway to state a new principle that, being in agreement with several resultsobtained through Newton’s original method of analysis of forces, could begeneralised so as to get a principle of a new sort, namely a general variationalprinciple.

These researches constituted a major event in the history of mechanics forthey allowed to pass from a geometrical-based study of concrete particularsystem, to an analytical treatment of any sort of systems based on a uniqueand general equation. In our view, these are the more fundamental originsof analytical mechanics ([Panza(2002)]). Lagrange’s first general formulationof the principle of least action ([Lagrange(1761)]) essentially depends on theresults obtained by Euler in this way.

3 Algebraic analysis

Among the many well-known differences between Newton’s and Leibniz’s ap-proaches to the calculus, a quite relevant one deals with their opposite con-ceptions about its relation with the whole corpus of mathematics. WhereasLeibniz often stressed the novelty of his differential calculus, notably in rea-son of its special concern with infinity. Newton always conceived his resultson tangents, quadratures, punctual speeds and connected topics as naturalextensions of previous mathematics.

Newton’s first researches on these matters were explicitly based on theframework of Descartes’ geometry and geometrical algebra provided in LaGeometrie ([Descartes(1637)]). They date back overall to the years 1664-1666 ([Panza(2005)]), but culminate with the composition of the De analysisin 1669 ([Newton(1967-1981), vol. II, pp. 206-247]) and of the De methodisin 1671 ([Newton(1967-1981), vol. III, pp. 32-353]), where the new theory offluxions is exposed.

Later, Newton famously changed his mind about the respective meritsof Descartes’ new way of making geometry and the classical (usually consid-ered as synthetic) approach, especially identified with the style of Apollonius’Conics ([Galuzzi(1990)], [Guicciardini(2004b)]), and based the Principia onthe method of first and ultimate ratios, that he took to be perfectly compat-

16

ible with this last approach22.Finally, in the more mature presentation of the theory of fluxions, the De

quadratura curvarum ([Newton(1704c)])23, Newton stresses explicitely24:

[. . . ] To institute analysis in this way and to investigate the firstor last ratios of nascent or vanishing finites is in harmony withthe geometry of the ancients, and I wanted to show that in themethod of fluxions there should be no need to introduce infinitelysmall figures into geometry.

The De methodis begins as follows25:

Observing that the majority of geometers [. . . ] now for the mostpart apply themselves to the cultivation of analysis and with itsaid have overcome so many formidable difficulties [. . . ] I foundit not amiss [. . . ] to draw up the following short tract in whichI might at once widen the boundaries of the field of analysis andadvance the doctrine of curves.

In early-modern age, the term “analysis” and its cognates were used in math-ematics in different, though strictly connected, senses. Two of them weredominant in the middle of 17th century. According to the former, it refers tothe first part of a twofold method—the method of analysis and synthesis—paradigmatically expounded in the 7th book of Pappus Mathematical Col-lection ([Pappus(1876-1878)]). According to the former, it refers to a newdomain of mathematics whose introduction was typically ascribed to Viete,which had explicitly identified it with a “new algebra” ([Viete(1591)]).

22In the concluding lemma of the section I of book I of the Principia, Newton claimsthat he proved the lemmas that his method pertains to “in order to avoid the tediumof working out lengthy proofs by reductio ad absurdum in the manner of the Ancientgeometers” ([Newton(1999), p. 441]).

23See also [Newton(1967-1981), vol. VIII, pp. 92-168].24Cf. [Newton(1967-1981), VIII, p. 129]: “[. . . ] Analysin sic instituere, & finitarum

nascentium vel evanescentium rationes primas vel ultimas investigare, consonum est Ge-ometriæ Veterum: & volui ostendere quo in Methodo Fluxionum non opus sit figurasinfinite parvas in Geometriam introducere.”

25Cf.[Newton(1670-1671), pp. 32-33]:“Animadvertenti plerosque Geometras [. . . ] Ana-lyticæ excolendæ plurimum incumbere, et ejus ope tot tantasque difficultates superasse[. . . ]: placuit sequentia quibus campi analytici terminos expandere juxta ac curvarumdoctrinam promovere possem [. . . ].”

17

In our view, this domain should be considered more as a family of tech-niques for making both arithmetic and geometry ([Panza(2008)]), than asa separate theory somehow opposed to these latter ones. Undoubtedly, inthe previous passage, Newton is referring to this domain and he is claimingthat his treatise aims to extend it so as to make it appropriate for studyingcurves.

Still, this extension crucially depends not only on the addition of newtechniques, based on Descartes’ algebraic formalism. It also depends on a newconception about quantities, according to which mathematics should deal notonly with particular sorts of quantities, like numbers, segments, etc., but alsowith quantities purely conceived, that is, with fluents ([Panza(forthcoming)]).

Hence, in the De methodis, the extended field of analysis no more presentsitself as a family of powerful techniques; it rather takes the form of a new the-ory dealing with quantities purely conceived. These quantities are supposedto belong to a net of operational relations expressed through Descartes’ alge-braic formalism appropriately extended so as to include infinitary expressionslike series.

3.1 Euler’s theory of functions

Newton’s later opposition to Descartes’ way of making geometry and the in-dependence of the mathematical method of first and ultimate ratios from theanalytic formalism of the theory of fluxions—in presence of the well-knownNewton-Leibniz priority quarrel and its consequences—lead, in the 18th cen-tury, to a polarisation between two ways of understanding the calculus. ANewtonian way, based on a classic conception of geometry, conceives fluxionsas ratios of vanishing quantities. A Leibnitian way, based on the introductionof an appropriate new formalism, deals with infinitesimals.

Maclaurin’s Treatise of fluxions ([Maclaurin(1742)]) is usually pointed outas the major example of the Newtonian view. “Fluxional ‘computations’ arenot presented as blind manipulation of symbols, but rather as meaningfullanguage that could always be translated into the terminology of [. . . ][a]kinematic-geometric model” ([Guicciardini(2004a), pp. 239-240]).

Contrastively, Euler’s trilogy composed by the Introductio in analysin in-finitorum, the Institutiones calculi differentialis, and the Institutionum cal-culi integralis ([Euler(1748a)], [Euler(1755)], [Euler(1768)]) is indicated asthe major example of the Leibnitian view.

Guicciardini’s article in the present volume shows that the two traditions

18

were not so opposed, and that the respective scientific communities were notso separated as it has been too often claimed. A clear example of that—whichis mainly relevant here—is provided by Euler’s approach. Though there isno doubt that the theory expounded by Euler in the Institutiones and in theInstitutionum uses Leibniz’ differential and integral formalism, some of thebasic conceptions it is based on derive from Newton’s views.

Some of these conceptions are strictly internal to the organisation of thetheory. An example is provided by Euler’s idea that the main objects of differ-ential calculus are not differentials of variables quantities but differential ra-tios of functions conceived as ratios of vanishing differences ([Ferraro(2004)]).As he writes in the preface of the Institutiones26:

Differential calculus [. . . ] is a method for determining the ratioof the vanishing increments that any functions take on when thevariable, of which they are functions, is given a vanishing incre-ment [. . . ] Therefore, differential calculus is concerned not somuch with vanishing increments, which indeed are nothing, butwith their mutual ratio and proportion. Since these ratios areexpressed as finite quantities, we must think of calculus as beingconcerned with finite quantities.

In this way, Euler unclothes Newton’s notion of prime or ultimate ratio ofits classically geometric apparel and transfers it to a purely formal domainusing the language of Leibniz’s differential calculus.

Another, strictly connected, example comes from Euler’s definition of in-tegrals as anti-differentials and of the integral calculus as the “method” to beapplied for passing “from a certain relation among differentials to the relationof their quantities”, that is, in the simplest case ([Euler(1768)], definitions 2and 1, respectively), from

dy

dx= z

26We slightly modify Blanton’s translation ([Euler(2000), p. vii]). Here is Euler’s original[Euler(1755), p. LVII-LVIII]“[. . . ] calculi Differentialis [. . . ] est methodus determinandirationem incrementorum evanescentium, quæ functiones qæcunque accipiunt, dum quan-titati variabili, cuius sunt functiones, incrementum evanescens tribuitur [. . . ]. Calculusigitur differentialis non tam in his ipsis incrementis evanescentibus, quippe quæ sunt nulla,exquirendis, quam in eorum ratione ac proportione mutua scrutanda occupatur: et cunhæ rationes finitis quantitabus exprimantur, etiam hic calculus circa quantitates finitasversari est censendus.”

19

to

y = f(x) =

∫

zdx.

These definitions—which contrast with Leibniz’ conception of the integralas a sum of differentials—are clearly in agreement, instead, with the secondproblem of Newton’s De methodis : “when an equation involving the fluxionsof quantities is exhibited, to determine the relation of the quantities one toanother”27.

The closeness of Euler’s and Newton’s views in both those examplesdepends on a more fundamental concern: the idea that the differentialand the 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 Introduc-tio ([Euler(1748a)]). According to him, it is not merely a mathematicaltheory among others. It is rather the fundamental framework of the wholemathematics. Differential calculus is thus not conceived by Euler as a sepa-rated theory characterised by its special concern with infinity, as in Leibniz’sconception, but rather as a crucial part of an unitary building whose foun-dations consist of a theory of functions ([Panza(1992)]).

As a matter of fact, this theory comes in turn from a large and ordereddevelopment of the results that Newton had presented in his De methodisbefore attaching the two main problems of the theory of fluxions, and thatconstituted for him the base on which its extended “field of analysis” wasgrounded. A function is identified with an expression indicating the oper-ational relations about two or more quantities and expressing a quantitypurely conceived ([Panza(2007)]). And the fundamental part of the theoryconcerns the power series expansions of functions.

Though the language and the formalism that are used in Euler’s trilogyopenly come back to the Leibnitian tradition, such a trilogy should thus beviewed, for many and fundamental reasons, as a realisation of the unificationprogram that Newton had foreseen in the De methodis, a realisation thatrelies, moreover, on the basic idea underlying Newton’s method of prime andlast ratios.

27Cf.[Newton(1670-1671), pp. 82-83]: “Exposita Æquatione fluxiones quantitatum in-volvente, invenire relationem quantitatum inter se”. On Newton’s notion of primitive,cf. [Panza(2005), pp. 284-293, 323-325 and 439-460].

20

3.2 The classification of cubics and algebraic curves

The second volume of the Introductio ([Euler(1748a)]) is devoted to algebraiccurves: the curves expressed by a polynomial equation in two variables whenreferred to a system of rectilinear coordinates. Euler relies on some resultsobtained in the first volume to show that algebraic curves can be studied andclassified without appealing to the calculus: as a matter of fact, this marksthe birth of algebraic geometry.

The problem of classification of curves is quite ancient ([Rashed(2005)]).Still, in his Geometrie ([Descartes(1637)]), Descartes comes back on it in anew form: he concentrates only on algebraic curves (that he calls “geomet-ric”, whereas he recommends to reject from geometry other curves, termed“mechanical”) and bases his classification on the degree of the correspond-ing equation ([Bos(2001), pp. 356-357], [Rashed(2005), pp. 32–50]). This is,in fact, a quite broad classification, since equations of the same degree canexpress curves which look very different from each others. The classifica-tion of (non-degenerate) conics (the algebraic curves whose equations are theirreducible ones of degree 2) is well known: they split up into ellipses (in-cluding circles), parabolas and hyperbolas. But what about curves expressedby equations of higher degrees?

Newton answers the question for cubics (the algebraic curves whoseequations are the non-reducible ones of degree 3), in a tract ap-peared in 1704 as an appendix to the Opticks ([Newton(1704a)]), butwhose different stages of composition presumably date back to 1667-1695([Newton(1967-1981), vol. VII, p. 565-655]): the Enumeratio Linearum Ter-tii Ordinis [Newton(1704b)]28.

In the second volume of the Introductio, Euler tackles the same problemusing a quite different method, and shows that Newton’s classification isincomplete ([Euler(1748a), vol. II, Ch. 9]. He also provides an analogousclassification of quartics (the algebraic curves whose equations are the non-reducible ones of degree 4), and explains how the method used for classifyingcubics and quartics can, in principle, be applied to algebraic curves of anyorder ([Euler(1748a), vol. II, ch. 11 and 12-14, respectively]).

Whereas Newton’s classification of cubics is based on their figure in alimited (i.e. finite) region of the plane and depends on the occurrences ofpoints or line singularities like for instance nodes, cusps, double tangents,

28For the reception in Continental Europe of Newton’s Enumeratio, cf. Stedall’s articlein the present volume, which also deals shortly with Euler’s alternative classification.

21

Euler suggests to classify algebraic curves of any order by relying on thenumber and on the nature of their infinite branches. Here is what he writes29:

Hence, we reduced all third order lines to sixteen species, inwhich, therefore, all those of the seventy-two species in whichNewton divided the third order lines are contained30. It is notodd, in fact, that there is such a difference between our classifi-cation and Newton’s, since we obtained the difference of speciesonly from the nature of branches going to infinity, while Newtonconsidered also the shape of curves within a bounded region, andestablished the different species on the basis of their diversity.Although this criterion may seem arbitrary, still, by following hiscriteria Newton could have got many more species, whereas usingmy method I am able to draw neither more nor less species.

Euler’s last remark alludes to much more than what it says. He rejectsNewton’s criterion because on the impossibility of applying it as the orderof curves increases, since a so great variety of shapes arise, as witnesses themere case of cubics. Indeed, a potentially general criterion has to deal withsome properties of curves that can be systematically and as exhaustively aspossible explored for any order, like those of infinite branches, according toEuler.

This task could be difficult, however, if these curves were studied throughtheir equations taken as such, since the complexity of a polynomial equationin two variables increases very quickly with its order. Insofar as a polynomial

29We slightly modify Blanton’s translation ([Euler(1988-1990), vol. II, p. 147]). Hereis Euler’s original [cf. [Euler(1748a), Vol. II, § 236, p. 123]]: “Omnes ergo Lineas tertiiordinis reduximus ad Sedeciem Species, in quibus propterea omnes illæ Species Septuagintadua, in quas Newtonus Lineas tertii ordinis divisit, continentur. Quod vero inter hancnostram divisionem ac Newtonianam tantum intercedat discrimen mirum non est; hic enimtantum ex ramorum in infinitum excurrentium indole Specierum diversitatem desumsimus,cum Newtonus quoque ad statum Curvarum in spatio finito spectasset, atque ex hujusvarietate diversas Species constituisset. Quanquam autem hæc divisionis ratio arbitrariavidetur, tamen Newtonus suam tandem rationem sequens multo plures Species producerepotuisset, cum equidem mea methodo utens neque plures neque pauciores Species eruerequeam.”

30This is the reason why Euler prefers to use the term “genre” instead of “species” inorder to indicate his classes of curves. The latter terms is reserved to distinguish curvesof a given genre according to their shape in a limited region of the plane ([Euler(1748a)],vol. II, § 238, p. 126).

22

equation in two variables cannot be transformed in changing its global degreeso as it continues to express the same curve, Euler shows how to determinethe number and nature of infinite branches of a curve by considering itsequation for some appropriate transformations that lower its degree in onevariable, namely “both by choosing the most convenient axis and the mostapt inclination of the coordinates”, and attributing to a variable a convenientvalue31.

4 Conclusions

Euler’s results that we have accounted for, both in the case of foundation ofmechanics and in that of algebraic analysis, depend on the effort to carry outor to extend a Newtonian program. But in both cases, this is done by relyingon Cartesian and Leibnitian conceptions and tools. C. A. Truesdell hassummed up the situation about mechanics by saying that Euler inauguratedthe tradition of Newtonian mechanics because he “put most of mechanics inits modern form”32. But, then, what does remain of Newton’s conceptionsin Euler’s mechanics? Mutatis mutandis, a similar situation also holds forEuler’s theory of functions, and this suggests the same question: what doesremain of Newton’s conceptions in this last theory? These questions are toodifficult for hoping to afford a complete response in a single short paper.Still, we hope to have provided some elements of such a response.

31Once again, we slightly modify Blanton’s translation ([Euler(1988-1990), vol. II,p. 176]). Here is Euler’s original [Euler(1748a), vol. II, § 272, p. 150]: “Negotium autemhoc per reductionem æquationis ad formam simpliciorem, dum et Axis commodissimus,et inclinatio Coordinatarum aptissima assumitur, valde sublevari potest: tum etiam, quiaperinde est, utra Coordinatarum pro Abscissa accipiatur, labor maxime diminuetur, siea Coordinatarum, cujus paucissimæ dimensiones in æquatione occurrunt, pro Applicataassumatur.”

32Cf. [Truesdell(1968), p. 106]. See also [Truesdell(1970)] and [Maglo(2003), p. 139] thatresumes Truesdell’s views on this matter nicely and clearly.

23

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Index of names

Alembert, d’, 14Apollonius, 16

Bernoulli, Johann, 10, 13

Cartesian, 4, 5, 7, 9, 12, 23Cartesians, 23

Descartes, 7, 16, 18, 21

Euler, 1–7, 9–16, 18–23

Hamilton, 14Hermann, 10, 11

Kepler, 8

Lagrange, 14, 16Leibnitian, 9, 11, 18, 20Leibniz, 8, 16, 19, 20

Maclaurin, 18Maupertuis, 13–15

Newton, 1–3, 7–14, 16–23Newtonian, 2, 7, 9, 10, 18, 22, 23

Pappus, 17

Varignon, Pierre de, 10, 13Viete, 17

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