PIERRE VARIGNON and the CALCULUS of MOTION Michael S

PIERRE VARIGNON AND THE CALCULUS OF MOTION Michael S. Mahoney Princeton University

The new currents of scientific thought that reshaped the Académie des Sciences in the 1690s flowed from sources outside France and met in Paris during the period 1689-92. From England came Newton's Principia, familiar in style perhaps to those who knew Huygens' work, but novel in substance surely to those schooled in Cartesian mechanics. From Germany1 by way of the Acta eruditorum came Leibniz' "Tentamen de motuum coelestium causis", familiar in its vortex cosmology but strange in its use of a new "differential calculus" to analyze the motion of bodies. From Basel came Johann Bernoulli, an unknown young mathematician skilled in that calculus and in the even newer and more mysterious "method of integrals", which his brother Jakob and he had recently be developing. Though the last of the new currents to arrive in Paris, the Bernoullis' infinitesimal analysis was the first to influence the direction of mathematics there. It opened the way to Leibniz' recent work on dynamics and thereby provided the tools with which a new generation of Academicians would recast and reorganize Newton's mechanics, integrating it into a program of research that would remain essentially Cartesian for another half-century. Hence, we shall consider the new currents in reverse order. Although Leibniz had worked out the essentials of his new calculus while still in Paris in the mid-1670s,2 he waited almost a decade before publishing an account of it in the May 1684 number of Acta eruditorum. The article, "Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractis, nec irrationales quantitates moratur, & singulare pro illis calculi, genus"3 set forth only the bare algorithms of differentiation, with no explanation or demonstration to speak of and with the sketchiest of illustrative examples. Articles that followed over the next few years shed a bit more light on the new method, especially on the ways in which it set the bases for a new, "transcendent" geometry, but they still offered little help or guidance to the reader who wanted to gain command over the calculus. Indeed, what constituted command remained itself unclear, as Leibniz failed to undertake a systematic exposition of the techniques of the calculus and of the range of its applications. 4 In particular, although he paid practically no attention at all to the inverse operation of integration and to its application to quadrature, rectification, and the inverse problem of tangents. It was instead Jakob Bernoulli and his younger brother Johann who assumed that responsibility in the late 1680s and who rapidly extended the purview of differential calculus to include all known curves and many new ones.

1More precisely, from Rome, where Leibniz was visiting when he first learned of the appearance of the Principia and hastened to put his own, similar thoughts to paper. See most recently Domenico Bertoloni Meli, Equivalence and Priority: The Newton-Leibniz Dispute on the World System (Oxford : Clarendon Press ; New York : Oxford University Press, 1993), together with the earlier studies of E.J. Aiton, "The Celestial Mechanics of Leibniz", Annals of Science 16(1960), 65-82; "The celestial mechanics of Leibniz in the light of Newtonian criticism", ibid. 18(1962), 31-41; and "The celestial mechanics of Leibniz: A new interpretation", ibid. 20(1964), 111-123.

2See J.E. Hofmann, Leibniz in Paris, 1672-1676: His Growth to Mathematical Maturity (London, New York: Cambridge University Press, 1974).

3pp.467-473; LMS.V.220-226

4See M.S. Mahoney, "Infinitesimals and Transcendent Relations", in D.C. Lindberg and R.S. Westfall (eds.), Reappraisals of the Scientific Revolution, Chap.12. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 2

The notes on the calculus made by the Marquis de l'Hôpital for Nicolas Malebranche in 1689 show by contrast the necessity and the originality of the Bernoullis' efforts.5 L'Hôpital failed to discern in Leibniz' notation the operational nature of d when prefixed to the symbol of a quantity. Even though Leibniz had earlier spoken of dx as "a certain modification" (quaedam modificatio) of x l'Hôpital treated it as simply a variation on the e that Barrow had used to denote an infinitesimal increment.6 The examples l'Hôpital offered Malebranche scarcely went beyond what Fermat and others had been doing fifty years earlier, either in substance or in style. Two years later, when Johann Bernoulli first appeared in Paris, the situation had changed little. L'Hôpital, who counted as the leading mathematician in Paris, had but a tenuous hold on the new infinitesimal methods. Quick to recognize Bernoulli's superior skills upon first meeting him at one of Malebranche's Thursday soirées, l'Hôpital contracted with the young Swiss to learn the calculus. Installed at the Marquis' summer home in Oucques, Bernoulli composed for l'Hôpital a short exposition of the differential calculus and a long treatise on the methods of the integral calculus.7 In a now notorious arrangement, Bernoulli ceded his rights of authorship to l'Hôpital, who published the work on the differential calculus, along with some material on the integral calculus, as his own Analyse des infiniment petits, pour l'intelligence des lignes courbes in 1696. Bernoulli had another eager student in Pierre Varignon, a friend of l'Hôpital and his sometime collaborator. At the time Professor of Mathematics at the Collège Mazarin, Varignon had established a reputation in the subject through his work in statics pursued in the traditional geometrical mode. Sharing in the lectures at Oucques, and more astute and talented as a mathematician than the Marquis, Varignon quickly made the calculus his own and was soon exchanging problems and solutions with Bernoulli, who recognized in him a colleague rather than a student and quasi-patron and who saw to it personally that Varignon received issues of the Acta eruditorum as they appeared, thus overcoming an embargo on trade with France.8 Hence, Varignon was able to make steady progress toward a command of the differential and integral calculus and to direct it to his own interests. In particular, Varignon strode out on his own precisely at the point at which l'Hôpital admitted having reached the limits of his understanding. Toward the end of the lectures on the integral calculus, Bernoulli offered a series of "physico-mechanical" problems, in which the conditions placed on quantities stemmed from the physics of static forces and of bodies moving under constraint. Bernoulli did not go all that far in his presentation, and he offered nothing like a system of analytic mechanics. His purpose was to illustrate applications of the integral calculus. Only two of the problems involved motion, namely the isochrone and the tautochrone, and both of them could be approached by means of Galileo's analysis of uniform motion and his laws of fall, in particular the law that ties the velocity of a falling body to the square root of the distance through which it has fallen from rest. To apply these

5Cf. Vol. 17,2 of Malebranche's Oeuvres|.

6"De geometria recondita et analysi indivisibilium atque infinitorum", Acta eruditorum (1686), pp. 292-300; LMS.V.226-33.

7/*references*/

8/*documentation*/ ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 3 laws, Bernoulli made an assumption that became fundamental to analytic mechanics: over infinitesimal distances ds traversed in infinitesimal times dt a motion may be considered as uniform, whatever its actual nature in the finite realm.9 That is, by Galileo's law of uniform motion, ds = vdt. This assumption is revealing in two ways of the conceptual structure of the differential calculus as it was originally developed and applied to mechanics. First, it treats the velocity as a primitive, rather than a derived quantity. If, that is, the relation ds = vdt defines anything, it defines the distance traversed by a body moving at a given speed. The relation stands as it is, rather than as a consequence of what has become a definition of velocity, namely v = ds/dt. Second, recall that, when Huygens moved from indivisibles to infinitesimal segments in his demonstration of the tautochrony of the cycloid, he worried about variations in speed over those segments. So he took pains first to try to show that the mean speed and the speeds at either end of the segment were close enough to be treated as equal and then to follow the laborious path of a double reductio ad absurdum. When Bernoulli first came to Paris, Varignon had been engaged in an effort to capture the science of weights and Galileo's laws of motion in algebraic form. It did not take much for Bernoulli to convince him that in the calculus he had a tool better suited to his purpose. Moreover, through Bernoulli's lectures Varignon gained access to the mathematical details of Leibniz' concise and confusing treatment of planetary orbits in his "Tentamen de motuum coelestium causis", hastily written upon learning of the publication of the Principia, which in turn was opened to for Varignon a new reading.10

Following a generalized account of isochronic curves in 1699, Varignon turned in 1700 to a general theory of motion determined by central forces and in a series of memoirs rendered into the language of the calculus the mechanical substance of Book I, Sections 2 - 10, of the Principia. His first memoir, "Manière générale de déterminer les forces, les vitesses, les espaces, & les temps, une seule de

9Cf. Euler, Mechanica (1736), Book I, Prop.3: In motu quantumvis inaequabili minima spatii elementa motu aequabili percurri concipi possunt. (In a motion that is non-uniform in whatever way, the smallest elements of distance can be thought of as traversed with a uniform motion.) The "demonstration" rested explicitly on the concepts of infinitesimal analysis: Quemadmodum enim in geometria curvarum linearum elementa ut lineolae rectae considerantur, ita enim simili modo in mechanica motus inaequabilis in infinitos aequabiles resolvitur. Vel enim revera elementa aequabili motu percurruntur, vel mutatio celeritatis per huiusmodi elementa est tantilla, ut incrementum aut decrementum sine errorem negligi possit. In utroque casu ergo apparet propositionis veritas. (For, just as in geometry the elements of curved lines are considered as straight linelets, so to in mechanics non- uniform motion is similarly resolved into an infinite number of uniform motions. For either the elements are in fact traversed with a uniform motion or the change of speed over elements of this sort is so small that the increase or decrease can be neglected without error. In either case, then, the truth of the proposition is clear.)

10Acta eruditorum (1689), pp.??? On what follows, cf. E.J. Aiton /*complete references*/ The relationship between Leibniz' and Varignon's analyses is both mathematically and historically interesting, especially given the sophistication of Leibniz' conceptualization of the problem. For these lectures, however, the comparison constitutes an unnecessary digression. The two treatments differ in essential ways, not the least of which is the level of technical detail presented to the reader and the clarity with which the methods of analysis are set out. Whatever Varignon's debt to Leibniz, it was Varignon's analysis that came to the attention of a wide audience. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 4

ces quatre choses étant donnée dans toutes sortes de mouvement rectilignes variés à discrétion",11 aimed at capturing Newton's theorems on rectilinear centripetal motion in two "general rules", from which all else followed by the techniques of ordinary and infinitesimal analysis. His modification of the configuration of Proposition 39 of the Principia reveals both the different form of mathematics Varignon was working with and the different ends to which he was applying it. All the rectilinear angles in the adjoined figure being right, let TD, VB, FM, VK, FN, FO be any six curves, of which the first three express through their common abscissa AH the distance traversed by some body moved arbitrarily along AC. Moreover, let the time taken to traverse it be expressed by the corresponding ordinate HT of the curve TC, the speed of that body at each point H by the two corresponding ordinates VH and VG of the curves VB and VK. The force toward C at each point H, independent of [the body's] speed (I shall henceforth call it central force owing to its tendency toward point C as center) will be expressed similarly by the corresponding ordinates FH, FG, FE of the curves FM, FN, FO.

Varignon's Curves Newton's Curves

The axis AC, with the center of force at C, stemmed from Newton. The six curves were inspired by

Leibniz. They represent graphically the various combinations of functional dependency12 among the parameters of motion: the "curve of times" TD represents time as a function of distance; the "curves of speed" VB and VX, the velocity as functions of distance and time respectively; and the "curves of force" FM, FN, and FO, the force as functions of distance, time, and velocity respectively. To translate those designations into defining mathematical relations, Varignon turned to algebraic symbolism. At any point H on AC set the distance AH = x, the time HT = AG = t, the speed (HV = AE = GV) = v, and the central force HF = EF = GF = y. "Whence," Varignon concluded from the perspective of the calculus, one will have dx for the distance traversed as if with a uniform speed (comme d'une vitesse uniforme) v at each instant, dv for the increase in speed that occurs there, ddx for the distance traversed by virtue of that increase in speed, and dt for that instant. Following Bernoulli, the first of the two general rules simply expressed symbolically the basic assumption of uniform motion over infinitesimal intervals. Since "speed consists only of a ratio of the distance traversed by a uniform motion to the time taken to traverse it", v = dx/dt, whence, by the rules of differentiation,13 dv = ddx/dt. The second rule took account of the change of speed and of the increment of distance that

12Note the close structural relation to the diagram in Leibniz's first account of the calculus in the Acta eruditorum (1684):

13Varignon takes dt as constant, i.e. ddt = 0. Doing so in Leibnizian calculus is equivalent in the modern form to choosing t as the independent variable, i.e. as the variable with respect to which one is differentiating. The Leibnizian version allows a greater flexibility in analyzing differential relations, as Bernoulli had shown in his lectures on the method of integrals and as Varignon shows in his next memoirs, to be discussed below. On the mathematical point, cf. H.J.M. Bos, "Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus", Archive for History of Exact Sciences (hereafter AHES), 14(1974), 3-90. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 6

results from it. Moreover, since the distances traversed by a body moved by a constant and continually applied force, such as one ordinarily thinks of weight, are in the compound ratio of that force and of the squares of the times taken to traverse them, ddx = y dt2, or y = ddx/dt2 = dv/dt. The rule appears to have stemmed in the first instance from the Principia. The first half of the measure expresses the second law, and the second half translates into the language of the calculus Lemma 10, which in turn sets out a principle also found in Huygens' analysis of centrifugal force.14 Varignon's version of the rule literally brings a new dimension to it, however, by capturing through the second differential dds that the effect is a second-order variation of the motion of a body. Those two rules, v = dx/dt and y = dv/dt sufficed, Varignon maintained, to give a full account of forced motion along straight lines. For, given any one of the six curves set out above, one can use the rules to carry out the transformations necessary to produce the other five. That central proposition reduces the mechanics in question to a matter of mathematics, and for the remainder of the memoir Varignon pursued an essentially mathematical point, echoing in the style and direction of his discussion

14/*Newton's expression; Huygens' use of the principle*/ Although Varignon used the principle effectively here, his later correspondence with Leibniz and Bernoulli shows that he did not fully understand it; see below, p.&fnref1.. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 7

two articles published by Leibniz in 1694.15 As Leibniz had there, so here Varignon struggled to express with words an emerging notion of function for which no symbolism yet existed. The proof of this proposition is easy; for, if one has any sort of variable or constant relation between, say, the speeds VH (v)16 and the distances traversed AH (x), that is, the equation of the curve of speeds BV, 1. the equation of which the variables are x and v will give the value of v in [i.e. in terms of] x and in constants, which value will form with dx/dt (Rule 1) another equation in which there will be only x's, t's and constants and which consequently will be that of the curve of times DT, of which the corresponding ordinates HT (t) will express the times required to go from A to H, [and] 2. the given equation of the curve VB will also give the value of dv in x (I am including there the dx as well) and in constants, which value will form with ydt (Rule 2) an equation in which there will be three sorts of variables, y, x, and t. But since the [equation] of the curve of times DT, which has just been found, gives the values both of x in t and of t in x, if one substitutes, one after the other, each of these values of x

15"Nova calculi differentialis applicatio & usus, ad multiplicem linearum constructionem, ex data tangentium condition", Acta eruditorum (1694), 311-316; cf. LMS.V.301-306. "Considérations sur la différence qu'il y a entre l'analyse ordinaire et le nouveau calcul des transcendantes", Journal des sçavans (1694), ???-???; cf. LMS.V.306-308. In the second, Leibniz set out as a challenge to those who still deemed "ordinary" analysis, i.e. algebra, adequate to the challenges of mathematics a general problem and in explicating it introduced a new term for an emerging concept (307-308): "Soit donné la raison, comme m à n, entre deux fonctions quelconques de la ligne ACC, trouver la ligne. J'appelle fonctions toutes les portions des lignes droites, qu'on fait en menant des droites indéfinies, qui répondent au point fixe et aux points de la courbe, comme sont AB ou A$ abscisse, BC ou $C ordonée, AC corde, CT ou Ch tangente, CP ou CB perpendiculaire, BT ou $h sous-tangente, BP ou $B sous-perpendiculaire, AT ou Ah resecta ou retranchée par la tangente, AP ou AB retranchée par la perpendiculaire, Th et PB sous-retranchées, sub-resectae a tangente vel perpendiculari, TP ou hB corresectae, et une infinité d'autres d'une construction plus composée, qu'on se peut figurer." Leibniz went on to note the necessity of the calculus for solving this general problem, a necessity he had built into it by extending the scope of mathematical relations to include tangents, subtangents, normals, subnormals, and similar properties of curves.

16Varignon's notation may be confusing here to the modern eye. He is not writing VH as a function of v but rather translating the geometrical segment into its algebraic form. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 8

and t in the equation of three variables, the substitution of the value of x, leaving only y and t as variables will make the equation that of the curve of forces NF, while the substitution of the value of t, leaving only y and x as variables will make the equation that of the curve of forces FM. That is, the solution of the differential equation v(x) = dx/dt yields the curve DT determined by HT = t(x), and, if VB = v(x), then v'(x)dx = dv = ydt will produce y(x,t), which can take two forms, depending on how the curve DT is expressed. Either FG = y(x(t),t) or FH = y(x,t(x)). The other curves emerge by similar transformations. As Varignon noted at the outset, the general claim rests on the dual assumption of complete solvability in the two Newton's Configuration realms of analysis: the resolution of any algebraic equation (i.e. getting x(t) from t(x)) and the integration of any differential equation. The limits of the mechanics in question were those of the calculus.17 Two examples would suffice, he thought, to illustrate the power of his method of analysis. The first stemmed from Galileo. On the assumption that v = /x = dx/dt, it follows directly that dt = dx//x, or t = 2/x, or x = t2/4. That, in shape at least, is the law of falling bodies. Newton provided the basis of the second example. From dt = dx/v = dv/y it follows immediately that ydx = vdv, or v2/2 = Iydx which, solved for v, is the first part of Proposition 39 of the Principia. Rewritten in the form

, it leads to the second part by way of the inverse curve

To anticipate the discussion to follow, it is worth noting here that, in all of the above discussion, Varignon's diagram played no more than an illustrative role, and an incomplete one at that. Nothing in the diagram exemplifies, much less conveys operationally, the relationships connecting the curves, since as differential relationships they are of a different dimension. Nor does either of the rules

17Cf. Leibniz in the second article cited above, p. 308: "Dans l'analyse ordinaire on peut toujours délivrer le calcul a vinculo et des racines par le moyen des puissances: mais le public n'a pas encore la méthode de le délivrer des puissances impliquées par le moyen des racines pures. [He had earlier noted the lack of a general solution of fifth-degree equations.] De même dans notre Analyse des transcendantes, on peut toujours délivrer le calculo a vinculo et des sommes par le moyen des différences: mais le public n'a pas encore la méthode de le délivrer des différences impliquées par le moyen des sommes pures ou quadratures: et comme il n'est pas toujours possible de tirer les racines effectivement pour parvenir aux grandeurs rationelles de l'Arithmétique commune, il n'est pas toujours possible non plus de donner effectivement les sommes ou quadratures, pour parvenir aux grandeurs ordinaires ou algébriques de l'analyse commune." ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 9

find graphic expression in the configuration, especially the second rule, which, resting on second-order differences, is two dimensions removed from the curves depicted. Varignon's second memoir of 1700, "Du mouvement en général par toutes sortes de courbes; & des forces centrales, tant centrifuges que centrepètes, nécessaires aux corps qui les décrivent", applied the two general rules of the first memoir to the motion of bodies along curves. The basic mathematical configuration remained the same; to it Varignon added a trajectory QL (which he called the "curve of paths", the "courbe des chemins") referred to the axis AC and to the center of force C via what may be termed "arc coordinates"; the point L is located by means of the abscissa AH (= x) and the arc-length HL (= z). If AC = a and CH = r, then L is also located by the radial distance r = a - x; in that case, dx = -dr. The coordinates point to Varignon's source: the diagram came from Propositions 40 and 41 of the Principia, though only in part; Varignon needed only the trajectory on the left of Newton's configuration, not the auxiliary curves on the right.18 As in the memoir on rectilinear motion, Varignon set out two "general rules" by means of which, given any two of the seven curves of his configuration, he could construct the remaining five. The first rule followed straightforwardly from the first memoir by replacing the axial interval dx by the infinitesimal increment ds of the distance QL (= s) along the trajectory. That is, v = ds/dt. (Rule 1) The crux of the memoir lies in Varignon's derivation of the second rule by analysis of the change of motion caused by a central force acting obliquely on the moving body over the infinitesimal distance Ll (= ds). Proposition 40 of the Principia provided the necessary hint, which Varignon developed in a manner peculiar to the

Varignon's Configuration

18Which were incorrect in any event; see the preceding lecture. Indeed, Newton may have reborrowed Varignon's versions for later editions of the Principia. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 10

infinitesimal analysis he was using.19 Thinking again of force in terms of weight, Varignon superimposed on the infinitesimal curvilinear triangle LRl the rectilinear configuration of an inclined plane and resolved the "absolute force" acting on the body at L (along the radius LR) into its normal and tangential components PR and LP. Concerned here with acceleration along the arc Ll, rather than with the measure of the centripetal force represented by PR, Varignon focused his attention on LP. By the law of the inclined plane,

Figure 5. absolute force:motive component = LR:LP = Ll:LR = ds/dx. If, that is, the absolute force is y, then the motive component is ydx/ds. That component acts as a constant force over the interval, producing an acceleration measured by the relation set out in the first memoir; that is

, or

(Rule 2)

Maintaining the parallel with his first memoir, Varignon changed his central proposition only to encompass the additional curve. As to how these two rules are to be used, I say for now that, being given any two of the seven curves noted above, that is to say, the equations of two taken at will, one will always be able to find the five others, supposing the required integrations and the solution of the equations that may be encountered. Focused on three curves in particular, the trajectory and the curves of time and force, the proposition incorporated several portions of the Principia into a single equation linking the three and solvable given any two. Given the trajectory and the curve of times, it yielded the central forces producing the motion. In particular, given the conic sections and the relation dt = rdz, that is, the relation of time and area established in Proposition 1, solving the resulting equations yielded Propositions 7-13. Given a force

19Si corpus, cogente vi quacunque centripeta, moveatur utcunque, & corpus aliud recta ascendat vel descendat, sintque eorum velocitates in aliquo aequalium altitudinum casu aequales, velocitates eorum in omnibus altitudinibus erunt aequales. /*diagram?*/ ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 11

and the same relation of time and area, it yielded Proposition 41. The two problems corresponded to the two components of the calculus: finding the forces called for differentiation, finding the trajectory called for integration. In this memoir and in one that followed, Varignon focused his attention on the determination of forces; indeed, he did not take up the second problem, the so-called "problem of inverse central forces" until 1710.20 Although he offered a general, theoretical justification along the lines of his first memoir, he knew that it would speak only to experts in the analysis. The general reader would need examples, though even then the lack of a suitable notation masked the generality of the techniques Varignon was applying and the way in which they took advantage of the methods of infinitesimal analysis. The particular form of the general rules lent itself well to the expression of the central forces determining a given trajectory. In Varignon's coordinate system, ds2 = dr2 + dz2, which may be rewritten in the form

By Kepler's hypothesis, r2dz2 = dt2, and, since dt is constant, differentiation of the left-hand side of the previous equation yields 2dsdds/dt2. Dividing that result by -2dr produces the value y of the central force. Assuming that dz can be expressed in terms of r and dr, one can rewrite the right-hand side of the equation in the form

to get an expression containing only r, dr, and constants. Differentiating that expression and dividing the result by -2dr will therefore give the value of y for the curve z = z(r). Conversely, an expression for y in terms of r will constitute a differential equation, the solution of which yields a trajectory z = z(r). As Varignon showed toward the end of the memoir, his method adapted easily to the case of parallel lines of force toward an infinitely distant center and thus incorporated the results of his 1699 studies on the motion of bodies according to various laws of fall.

20"Des forces centrales inverses", MARS (1710), 533-544. Varignon's article followed excerpts from an exchange of letters between Jakob Hermann and Johann Bernoulli concerning the same problem. As Fleckenstein suggests, in 1700 Varignon's command of the techniques of integration and hence of the solution of differential equations may simply not yet have sufficed for him to tackle the problem with any sophistication. He may also have been holding back out of deference to Bernoulli, who (for reasons not known to Varignon) had not yet published his lectures on the method of integration from which Varignon was working. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 12

Since in all of this the areas LCl (rdz/2) are as the dt, and the case of parallel directions renders LC (r) infinite and thus constant, one will have in that case the dz as the dt, or dz = dt. That is, the radial system r, z becomes the orthogonal system x, z, in which acceleration takes place in the direction of x and motion remains uniform in the direction of z. The pattern of analysis changes little; one differentiates ds2/dz2 holding dz constant and then divides by 2dx to get y = dsdds/dxdt2. Varignon's analysis of the motion of bodies under central forces effectively tied the development of mechanics to that of the calculus. Given the wholly algorithmic nature of differentiation, finding the forces had become a matter of straightforward computation. By contrast, finding the trajectory depended on the resources of the integral calculus and the theory of differential equations. As Varignon suggested in a subsequent memoir (see below), mathematics and the science of motion would henceforth march in tandem.21 Moreover, in addition to demonstrating his main proposition, Varignon's detailed analyses illustrated what Leibniz had always maintained was the deep significance of the calculus, namely that it opened a realm of mathematics closed to ordinary algebra, the realm of transcendental relations.22 Differentiation reduced those relations to quasi-algebraic equations in which dx as "a sort of modification of x" was added to the basic operations of finite algebra. Analysis of the differential equations, Leibniz argued, provides much information about the curves that cannot be elicited from their finite expression because of the opacity of transcendental relations. Varignon's analyses began with the differential form of the curves he was treating and hence obviated the question of whether they were transcendental. Thus he could consider the logarithmic spiral ds/dz = b/a and the inverted cycloid

where m = a - x, as directly as he could one of the conic sections. On the assumption of the general solvability of ordinary and differential equations, no trajectories lay beyond his method of determining the central forces that generate them. Again, the mechanics provided a vehicle for an essentially mathematical point. While Varignon took his own analytical path to the dynamics of particle motion, his approach lay closer to that of Leibniz' "Tentamen" than to that of the Principia. The essence of the derivation of expressions for the velocity and the force lies in algebraic relations and in the operations of the calculus, and the diagram reflects little of the structure of those expressions. The basic configuration represents the trajectory and an infinitesimal segment on it. As geometrical entities, they have no operational

21It is true that Newton's analyses of forces and trajectories also linked mechanics to mathematics, but the nature of his mathematics neither made the determination of forces algorithmic nor systematically related families of differential equations to classes of mechanical problems. In the first case, the geometrical reduction of QR/QT2.SP2 (see above, p.&newton.) is specific to each curve. In the second, the quadrature of curves is not coextensive with the solution of differential equations, especially if the parameters of the latter belong to mathematical spaces of differing dimension.

22Cf. "De geometria recondita ... (1686) ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 13 meaning. They acquire that only as algebraic entities linked to other algebraic quantities by means of dynamical relations expressed algebraically. At point L the body has a velocity v. By the fundamental assumption of infinitesimal analysis applied to dynamics, the infinitesimal increment of the trajectory, ds, is the product of its velocity at L and of the infinitesimal increment of time dt required to traverse it, neither of which has a representation in the diagram. Velocity remains a ratio of an element on the diagram, namely ds and of a quantity existing outside the space of the diagram, i.e. dt. The change in velocity, dv, results from the "d-modification"23 of that ratio and constitutes itself a ratio between two quantities, dds and dt, neither of which is represented on the diagram. The measure of force emerges then by appeal to a dynamical principle adapted to the special assumptions of infinitesimal analysis, namely that over dt the velocity dv = f dt can be considered uniform and hence that the distance traversed in that infinitesimal interval of time will be dds = dvdt = f dt2. None of these relations, nor any of the constituent quantities, has any place in the geometrical configuration, which contributes to the analysis in only one way, namely by suggesting the superposition of an inclined plane and thereby linking the components of force to the elements of the orbit. Once Varignon had derived his general expression for the force, the diagram of the trajectory ceased to serve any function in his reasoning.24 The various curves of time, velocity, and force retained from the graphical framework of the first paper play no role in the analyses, as indeed they had played none there. The reasoning proceeded symbolically, and as a result involved no special constructions or corollaries for the purposes of establishing measures of dynamical parameters in the geometrical configuration. Varignon's third paper read to the Académie in 1700, "Des forces centrales, ou des pesanteurs nécessaires aux planètes pour leur faire décrire les orbes qu'on leur a supposés jusqu'ici",25 pursued several lines of analysis. First, Varignon applied his rules to the specific case of motion on conic sections with the center of force at one focus, thus rendering Propositions 11-13 of the Principia into analytic form and confirming Leibniz's derivation of the same result in his "Tentamen de motuum coelestium causis". Second, he then explored other recent hypotheses about the shape of planetary orbits and the measure of time of motion, showing that none of them led to an inverse-square force.26 Third, he chased out analytically the consequences of assuming, as astronomers from Plato to Copernicus had done, that the planets move uniformly on their orbits. In this last undertaking he particularly revealed the mathematical purposes of his application of analysis to mechanics. At the beginning of the memoir, Varignon reviewed the principles of his analysis, commenting on the difference between his approach and Newton's. Now concerned directly with the analysis of orbits, he eliminated from his basic configuration the curves of the first paper and began simply with

23I shall use this term to capture the concept of differentiation Leibniz had in mind when he spoke of the process as quaedam modificatio of the variable to which it is applied.

24That is why it is not particularly disturbing that the diagram appears on a plate of diagrams at some distance in the volume from Varignon's article, rather than in the midst of it. It is hard to imagine trying to read the Principia if the diagrams were all located together at the end of the volume.

25MARS (1700), 218-237.

26Varignon was the first to subject to dynamical analysis Cassini's hypothesis that the orbits are ovals for which the product of the focal distances, rather than their sum, is a constant. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 14

the trajectory as a reference for several lemmas rehearsing the general rules of the second paper. In a scholium, he then added two curves, which he linked by construction to the trajectory. Curve TD is traced by the ordinate HT, which measures the time of motion corresponding to point L on the trajectory. Curve BO derives from drawing LN tangent to the trajectory at L and CN perpendicular to radius LC, and setting

where E is an

arbitrary constant. Constructing the subtangent HG, Varignon then asserted that the central force is measured by the expression

He offered no

demonstration, apparently assuming that readers would recognize the expression and its relationship to the analytic form previously derived.27 But Varignon suggested the construction only to reject it: But however simple this formula may appear for being in wholly finite magnitudes, and however easy its use may in fact be, one must say that it is much less so

27The result follows directly from setting LN/LC = ds/dx and HK/HT = subtan/t = dx/dt, whence

Then HG/HB = dx/dv, whence HG = E@v(x)dx/dv.

Since dv = dds/dt, and the geometric expression for the central force then

evaluates to or Here, as elsewhere, Varignon has dropped the constant

than that of the preceding [formula]. That is why we will not pause any longer at this one, and why the other on the contrary will always serve as general rule in what follows.28 Here Varignon consciously chose his style of analysis, rejecting the mixed analytic-geometric example of the Principia. The analysis of Newton's results and of recent alternatives to them was a matter of straightforward application of the method worked out in the second memoir. The question of uniform motion opened out into new directions of analysis. For by the analysis that led to the second general rule, acceleration along the arc ds was produced by the tangential component of the force; if force acted along the normal to the curve, no acceleration would occur. Hence, for a body to move uniformly on its orbit, the force at each point would have to be located on the radius of curvature for that point; for a smooth orbit, the center of force would have to move smoothly, which it would do if located on the locus of the centers of curvature. In the terms of analysis at the time, orbits of uniform motion belonged to the class of curves known as evolutes.29 They thereby forged a link between mechanics and the Bernoullis' proudest achievement in mathematics, the analytic derivation of the radius of curvature, the theorema aureum. 30 Varignon's next memoirs, read to the Académie in 1701, explored that link, and in doing so took account for the first time of Newton's geometrical analysis of the change of motion, tying the two approaches together. In "Autre règle générale des forces centrales, avec une manière d'en déduire & d'en trouver une infinité d'autres à la fois, dépendemment & indépendemment des rayons osculateurs qu'on va trouver aussi d'une manière infiniment générale", Varignon began with his familiar configuration with the infinitesimal curvilinear triangle LRl, where Ll = ds is an arc of the trajectory and RP is perpendicular to Ll. Taking that arc as the arc of the osculating circle of the trajectory at point L, Varignon now added to the configuration the radii of that circle drawn to L and l and meeting at the center of curvature

28It is not clear by what criteria Varignon was judging ease of use [usage] here, unless perhaps he was referring to Newton's methods of geometrical analysis.

29They would later shift to another class, namely that of orthogonal trajectories. /*references?*/

A. Drawing LH tangent to the curve at L and locating E on LH such that LE = Ll (= ds), Varignon drew El and turned to find an expression for it, for it is what has been traversed by virtue of the force along PR (= ydz/ds) during the instant dt in which the body traversed Ll instead of following the tangent LH as it would have done without that force, or y. That dynamical interpretation of El was intimately related to the mathematical interpretation by which Varignon derived an analytical expression for it. Following the lead of Bernoulli's lectures on the method of integrals,31 he took AL z LH, Al z Ll, and lE z LH, whence ª AlL ~ ª LlE, and AL 2 RP 2 lE. If AL = n, then n/ds = ds/lE, whence LE = ds2/n, which by the rule linking a constant force to the distance traveled in an instant is in turn equal to (ydz/ds)dt2; that is, y = ds3/ndzdt2 After a digression setting out the general expression for the radius of curvature,

where y and x are respectively r and z in the original configuration, Varignon made his point. That expression presupposed no differential as constant, that is, no variable as independent. Depending on that choice, the expression took on several different forms. Put another way, the expression determined a family of differential equations, each transformable into the others by a change of variable. Bernoulli had shown in his lectures how that set of transformations led to the solution of various integrals and differential equations. By linking it now to the mechanics of central forces, Varignon meant to extend its power even further, first by using it to free his mechanics from dependence on any particular choice of coordinates and second to bring mechanics to bear on mathematical problems. The details of his argument are of less concern here than the direction in which it takes him and his reader. Generalizing the expression of mechanical relations to the point of rendering them independent of the choice of independent variable brought the move from diagrams to dynamics to a culmination. Varignon's analyses proceeded by manipulation of symbols according to the rules of finite and infinitesimal algebra supplemented by those of kinematics and dynamics expressed symbolically, and the resulting combinations of variables took their meaning from those operations. Varignon, at least, had pointed mechanics toward Euler and Lagrange. Not everyone was prepared to follow his lead. However persuasive the results in mechanics, the mathematics at first spoke only to the converted. Throughout the papers discussed here, and in all those to follow, Varignon relied on the very techniques of infinitesimal analysis that its detractors found most troubling;32 Indeed, they might have said "tricks" instead of "techniques". In fact, Varignon had

31Chap.16, p.71 /*which edition?*/

32Not only the detractors were bothered. According to E.J. Aiton (The Vortex Theory of Cosmology, p.199) Guido Grandi (De infinitis infinitorum et infinite parvorum, Paris, 1710, p.34) objected that the two triangles in the derivation just described were not similar. Aiton notes that "Varignon justified his calculation by explaining that the tangent ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 17

been tricky, as had his mentor, Bernoulli. To begin with, he had treated a varying motion as a uniform one. Then he had treated a curvilinear triangle as a rectilinear one, two isosceles triangles (one with a curved base, the other with one side curved and the other straight) as right triangles, the differential of the central sector as rdz rather than as rdz + drdz, even though his derivation took second differences as significant, and so on. The tricks were not just shortcuts, the omission of steps that could be filled in if someone needed a more rigorous derivation. Rather, they rested on the skilled practitioner's sense of what was significant and what insignificant in an infinitesimal configuration. With no real foundations on which to build their analyses, the early practitioners of the calculus could perhaps best justify their actions by the results they achieved. The techniques worked often and they worked well. They produced the right answers. At least they worked most of the time. Sometimes, even the skilled went astray. Shortly after starting up correspondence with Leibniz in the early 1700s, Varignon wrote to announce that Huygens and all since had been incorrect in their measure of centrifugal force, the actual value of which was twice what they had derived. Varignon offered several versions of his argument, to which he held for some time before bowing to the combined arguments of Leibniz and Bernoulli. The versions show how infinitesimal analysis could reinforce errors of analysis. In the first version, Varignon compared the distance covered by acceleration during a finite time to that during an instant by appeal to Galileo's law of fall, i.e. that the distances are as the squares of the times, and the velocities as the times and hence as the square roots of the distances. But that comparison was invalid without the inclusion of a factor of two. For, in the case of finite intervals, the speed of motion is taken as the average speed over the interval, i.e. as half the final speed. In infinitesimal intervals, as has been seen, the speed of motion is assumed to be uniform over the entire interval. The constant factor of one half is essential in any comparison.33 Once on the wrong track, however, Varignon could find considerable support from infinitesimal analysis. Looking at the question by means of various configurations, he made what he thought were the same sorts of approximations that had guided his earlier investigations. All seemed to lead to the same conclusion. In fact, he was conflating configurations in incompatible combinations. For example, in one instance, he treated the tangent at the extension ofthe infinitesimal side of an infinitangular polygon at the same time that he considered the path of the body's motion as the true geometrical tangent drawn to the endpoint of the infinitesimal segment. The new methods were far from self-correcting. So muchy depended on the practitioner.34

was really a prolongation of a side of a polygon" and cites HARS for 1722. However, although Varignon clearly treats the arc ds as a straight line, he equally clearly treats the segment El as parallel to radius AL and hence as commonly perpendicular to the tangent LH. Moreover, the model in Bernoulli's lectures seems evident. Nonetheless, the reasoning is not transparent, even today, and that fact reinforces the complaint of contemporaries.

33Varignon's error here may be viewed as the mirror image of Huygens' misplaced concern about the mean speed of a body's fall over an infinitesimal segment of the cycloid.

34Curiously, Varignon had it right in 1700, for it followed from the analysis of uniform motion on orbits just discussed that on a circle of radius R the force pulling the body back from the tangent is ds2/rdt2 = v2/r. He had said so explicitly himself. MARS (1700), 233. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 18

Fontenelle, Secretary of the Académie and author of its annual Histoire, maintained there a running commentary on Varignon's efforts to further the cause of infinitesimal analysis by applying it to mechanics. Fontenelle's position required sensitivity to the needs of the group for explanations of new and unfamiliar concepts, and hence his remarks about Varignon's papers offer a guide both to what the general membership of the Académie found perplexing about them and to the terms in which proponents of the new mathematics thought about it themselves. For Fontenelle was an avid supporter of infinitesimal analysis, and he took pains both to explain its concepts and to place them in historical and philosophical context, thereby emphasizing the continuity of mathematical progress. Yet, for all his enthusiasm, he ultimately had to admit that infinitesimal analysis constituted a radical break with the past. As the man who first referred to developments in natural philosophy during the seventeenth century as a "revolution",35 he is the fitting person to sum up the impact of Varignon's analytic mechanics: Although the mathematical infinite is well understood, its principles quite unshakeable, its arguments fully coherent, most of its investigations a bit advanced, it does not cease still to cast us into the abyss of a profound darkness, or at the very least into realms where the daylight is extremely weak. ... a bizarre thing has happened in higher mathematics [haute géométrie]: certainty has undermined clarity. One always holds onto the thread of the calculus, the infallible guide; no matter where one arrives, one had to arrive, whatever shadows one finds there. Moreover, glory has always attached to great discoveries, to the solution of difficult problems, and not to the elucidation of ideas. Mathematics had come to be measured by new standards of effectiveness, and analytic mechanics represented one of those standards.