234081

The Relativization of Centrifugal ForceAuthor(s): Domenico Bertoloni MeliSource: Isis, Vol. 81, No. 1 (Mar., 1990), pp. 23-43Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/234081 .Accessed: 27/02/2011 22:36

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The elativization of Centrifugal Force

By Domenico Bertoloni Meli*

T HE CONCEPT OF REFERENCE FRAME is related to the notions of space, time, and, in particular, force.' Although these notions have at-

tracted considerable interest-one needs only to think of the literature on absolute space, on time, and on force in Newton-historians of science have devoted virtually no attention to the connection between the concept of reference frame and the concept of force in classical mechanics.2

In this paper I concentrate on a specific aspect related to the general theme, namely, on the notion of centrifugal force.3 Current historiography tends to consider the notion of centrifugal force after Christiaan Huygens and Isaac Newton as unproblematic. This attitude seems to be based on the assumption that the Huygensian theory of centrifugal force was accepted almost immediately after it was first published. However, there was no agreement about the measure of centrifugal force when motion is not circular. Moreover, the formal identity of the mathematical expression in the case of circular motion hides a conceptual gulf that cannot be overlooked. This gulf concerns the interpretation and location of centrifugal force. For Huygens and Newton centrifugal force was the result of a curvilinear motion of a body; hence it was located in nature, in the object of investigation. According to a more recent formulation of classical mechanics, centrifugal force depends on the choice of how phenomena can be conveniently

* Jesus College, Cambridge CB5 8BL, United Kingdom. I thank Professor I. Bernard Cohen for having made available to me his "Concordance of the

Words in Newton's Principia." 1 In modern terms, a reference frame is defined by three perpendicular axes with a common origin.

If the axes do not rotate and their origin is not accelerated, the reference frame is inertial. Centrifugal forces arise from a rotating frame and are a special case of fictitious forces.

2 Istv'an Szabo, in Geschichte der mechanischen Prinzipien, 2nd ed. (Basel: Birkhauser, 1979), seems to ignore the topic. The same can be said for Thomas L. Hankins, Science and the Enlighten- ment (Cambridge: Cambridge Univ. Press, 1985), Ch. 2, and H. J. M. Bos, "Mathematics and Ratio- nal Mechanics," in The Ferment of Knowledge, ed. G. S. Rousseau and R. Porter (Cambridge: Cambridge Univ. Press, 1980). A brief mention can be found in Rend Dugas, A History of Mechanics (Neuchatel: Editions du Griffon, 1955), pp. 370-383. See also Clifford Truesdell, "A Program toward Rediscovering the Rational Mechanics of the Age of Reason," Archive for History of Exact Sciences, 1960, 1:3-36, sec. 14; and the classic Ernst Mach, Die Mechanik in ihrer Entwickelung historisch- kritisch dargestellt, 7th ed. (Leipzig, 1912), Ch. 2.

3 A rotating reference frame introduces for each successive derivative of a vector r an additional factor wr, w being the angular velocity. Thus (drIdt)i = (drldt), + wr, where the indices i and r stand for "inertial" and "rotating," respectively. Introducing the velocity variable v, we have va = v, + w * r. Taking the derivative again, we have ai = a, + 2w * v, + w * [w - r], or a, = ai - 2w - v, - w - [w - r], where a represents acceleration and w is, for simplicity, taken as constant. The second term at second member is the complementary or Coriolis acceleration, whereas the third term is centrifugal force. If the radius is perpendicular to w, the modulus of the last term can be written as w2r.

ISIS, 1990, 81 : 23-43 23

24 DOMENICO BERTOLONI MELI

represented. Hence it is not located in nature, but is the result of a choice by the observer. In the first case a mathematical formulation mirrors centrifugal force; in the second it creates it.

The primary aim of this paper is to show that in the eighteenth century centrifugal force was a problematic notion in many respects. Moreover, I intend to show that current views concerning the ideas on centrifugal force expressed by Newton in the Principia mathematica are severely affected by the projection of modern methods and ideas that are found neither in the Principia nor in works contemporary with it. I hope that my analysis will stimulate a fresh reflection on Newton's mechanics and its reception.

In the first section I introduce Huygens's theory of centrifugal force and com- ment upon his theory about the cause of gravity. His ideas are compared with Newton's views in the next section. I stress the radical change in the meaning of centrifugal force: for Huygens it was the result of the circular motion of a body, and gravity had to be explained on the basis of the centrifugal force of a rotating fluid; for Newton, centripetal force was a mathematically well-established notion, and centrifugal force was explained, at least in orbital motion, as a reaction to centripetal force according to the third law of motion. Since Newton explained his views most clearly in a critique he made of Leibniz during the priority dispute over the calculus, a portion of this section is devoted to an account of Leibniz's theory.

The diversity of views emerging from the first two sections sets the stage for an investigation of their reception. In the third section I select some figures occupying a wide range of positions in the priority dispute and study their views on centrifugal force. Analyses of curvilinear motion related to Newton's views prevailed in the Leibnizian camp as well, despite the priority dispute. I argue that one reason for this puzzling phenomenon is that Newton's interpretation of centrifugal force in terms of the third law of motion exerted considerable influence on Continental mathematicians. Although this is an area where further research is needed, I believe that my analysis clearly emphasizes the variety of interpretations involved and the difference between eighteenth-century and more modern formulations of classical mechanics.

The last section centers on a memoir by Daniel Bernoulli containing some calculations and observations on centrifugal force. This memoir was part of a collective attempt by Alexis-Claude Clairaut, Leonhard Euler, Johann Bernoulli, and Daniel Bernoulli to find the motion of a body constrained within a rotating tube. The mathematics involved in the calculations relevant to my task was very simple. The problem was not to find difficult differential equations, but to reflect in a new way on the implications of well-known expressions.

This work is neither a history of centrifugal force in the eighteen century nor a collection of completely unrelated case studies. I believe that its format, which is intermediate between these two extremes, allows us both to follow the historical developments and to grasp the ideas of some of the main mathematicians of the time.4

4 I do not deal here with a problem widely debated by Leibniz, Pierre Varignon, Abraham De- moivre, Johann Bernoulli, and Jakob Hermann and up to the time of Jean d'Alembert. Very briefly, if the curve has a continuous curvature, centrifugal force makes the body move with accelerated motion. If the curve is conceived as a polygon with first-order infinitesimal sides, tangents are the

CENTRIFUGAL FORCE 25

I. HUYGENS'S QUANTIFICATION OF CENTRIFUGAL FORCE

The idea that a rotating body tends to escape from its curvilinear path precedes a formulation of the law of inertia. In the debates on the Copernican system at the time of Galileo, for example, both camps referred to the possibility of bodies being projected into the air by the daily rotation of the earth or along the tangent by the rotation of a wheel on which they were placed and then released.5 In the Principia philosophiae Descartes studied the tendency of rotating bodies to escape along the tangent because of their rectilinear inertia, using the famous example of the sling. According to Descartes, if a stone attached to a rotating sling was suddenly released, it would continue along the tangent to the point of the circumference where it is released. The cause for this is the inertia of the stone. Moreover, he believed that the stone has a tendency to escape in the direction of the radius, and this is precisely the component of motion that is hindered by the resistance of the sling. For a given sling the effort to escape along the radius is the larger, the faster the motion; the effort can be known from the tension of the sling. This interpretation affected mechanics for at least a century.6

Descartes's Principia was extremely influential, and the example of the sling was frequently quoted in the literature. It was not until Huygens's De vi centrifuga and Horologium oscillatorium, however, that his qualitative remarks were given a quantitative form. When a body rotates along a circumference, it has a tendency to escape along the tangent BCD (see Figure 1). This tendency would make the body move in such a way that while it rotates along BE, BF, the spaces traversed with a uniformly accelerated motion due to centrifugal force would be EC, FD.7 In more detail, if the body moving along the arc of circumference prolongations of such infinitesimal segments and accelerations are not required. Although these different interpretations are equivalent, improper applications often generated mistakes by a factor of 2. On this topic see Thomas L. Hankins, Jean d'Alembert (Oxford: Oxford Univ. Press, 1970), pp. 225-227; Hankins, Science and the Enlightenment (cit. n. 2), pp. 22-25; E. J. Aiton, "The Celestial Mechanics of Leibniz," Annals of Science, 1960, 16:65-82, esp. pp. 75-82; and Isaac Newton, The Mathematical Papers of Isaac Newton, ed. D. T. Whiteside, 8 vols. (Cambridge: Cambridge Univ. Press, 1967-1981), Vol. VI, pp. 540-541n (hereafter Newton, Mathematical Papers).

S Galileo, Dialogo sopra i due massimi sistemi del mondo, tolemaico e copernicano, in Galileo, Opere, ed. Antonio Favaro, 20 vols. (Florence, 1890-1909), Vol. VII, pp. 188-203, 242.

6 Rene Descartes, Principia philosophiae (Amsterdam, 1644), Pt. 2, prop. 38, and Pt. 3, props. 57-59. The tendencies to escape along the tangent and along the radius are not two components in different directions. The former is detected by an observer in an inertial frame, the latter by an observer rotating with the radius. Each observer, however, detects only one tendency. Moreover, the reaction to centripetal force is not centrifugal force, but the outward force acting along the radius. The difference between centrifugal force and the reaction to centripetal force emerges very clearly if we consider an elastic sling that oscillates while rotating with a uniform angular velocity w on a horizontal plane. In a reference frame rotating with the radius, the equation of radial motion is d2rIdt2 = w2r - k(r - ro), where k is the elastic constant and ro is the length of the elastic sling at rest. If w2 is smaller than k, the body oscillates around the point r = rokl(k - w2). The first term at second member represents centrifugal force, which in general is not equal and opposite to centripetal force. On this topic see Mach, Mechanik (cit. n. 2), Ch. 2, par. 2; and Heinrich Hertz, Die Prinzipien der Mechanik in neuem Zusammenhang dargestellt (Leipzig, 1894), sec. 1 of the Introduction.

7 De vi centrifuga, composed in 1659, was published posthumously (Leiden, 1703); see Christiaan Huygens, Oeuvres, 21 vols. (The Hague, 1888-1950), Vol. XVI, pp. 255-311. Huygens appended to the Horologium oscillatorium (Paris, 1673) (Oeuvres, Vol. XVIII, pp. 366-368), without proof, a series of propositions on centrifugal force. In De vi centrifuga, pp. 260-261, Huygens introduces centrifugal force by discussing the case of an observer placed on a rotating wheel. However, this passage has to be interpreted as a rhetorical ploy inserted in order to make centrifugal force more understandable, rather than as a statement implying that centrifugal force is fictitious and that it


BEFM was released, it would move along the tangent BS. In the time in which it travels along BE it would reach K from B, and similarly, in the times in which it travels along BF and BM it would reach L and N, respectively. Rigorously, EK, FL, and MN are arcs of the involute or evolvent of the circumference BEFM; however, they can be approximated by their tangents in E, F, and M-namely, EC, FD, and MS respectively.8

In a series of ensuing propositions in De vi centrifuga Huygens proved that centrifugal force is proportional to the mass of the rotating body and to the square of its velocity and inversely proportional to the radius. From Figure 1 we have that EB2 is proportional to AB and CE, where EB is proportional to velocity and CE to centrifugal force (with a constant factor of 2 because motion along CE is uniformly accelerated). Huygens provided no explanation of how to measure

S N D L CK B

Figure 1 A centrifugal force when motion is not circular. This problem was given different solutions in the following decades.

Successively Huygens compared centrifugal force to gravity and showed that if a body rotates along a circumference with the same velocity that it would acquire if it fell from a height of one quarter of the diameter, then centrifugal force and gravity would be equal. This result was used in his paper on the cause of gravity read at the Academy of Sciences in Paris in 1669. There Huygens claimed that gravity is caused by a vortex rotating around the earth with a velocity seventeen times greater than that of a point on the equator, owing to the daily rotation of the earth. According to Huygens, this rotation explains why seconds pendulums have different lengths at different latitudes. In the Discours de la cause de la pesanteur, and especially in the Addition dealing with Newton's Principia mathematica, Huygens tried to generalize his theory of gravity to planetary motion, claiming that if the matter of the vortex were too subtle, it would be extremely difficult to explain gravity and especially the motion of light.9

depends on the motion of the observer as opposed to the motion of the object. Recently Joella Gerstmeyer Yoder has questioned the accuracy of Huygens's editors with respect to De vi centrifuga, in "Christiaan Huygens' Theory of Evolutes: The Background to the 'Horologium oscillatorium'" (Ph.D. diss., Univ. Wisconsin-Madison, 1985), Ch. 2, esp. p. 38.

8 The theory of evolutes and evolvents was developed by Huygens in the third part of the Horolo- gium.

9 Christiaan Huygens, Discours de la cause de la pesanteur, in Huygens, Oeuvres (cit. n. 7), Vol. XIX, pp. 631-640; and Huygens, Addition au discours .. ., ibid., Vol. XXI, pp. 443-499, esp. pp. 466-473.


In summary, Huygens believed that gravity had to be explained in terms of the motion of a fluid and was an effect of the centrifugal force of that fluid. Further- more, for Huygens as for Descartes, centrifugal force was related to inertia: it is because of its inertia that a body moving along a circumference tends to escape along the tangent, and this tendency is the cause of centrifugal force along the radius. This explains why pendulums of equal length have different periods at different latitudes and why, commenting on Newton's Principia mathematica, Huygens stated that Newton had explained planetary motion in terms of gravity and centrifugal force, which counterbalance each other. 10 In fact Newton mainly referred to rectilinear inertia and centripetal force, but for Huygens the former was inextricably related to his own centrifugal force.

II. NEWTON AND LEIBNIZ

Newton's early views on circular motion can be found in the Waste Book, dating from the mid 1660s.1I He considers a hollow sphere and a body moving inside it along the perimeter of an inscribed square. From this he establishes the following proportion:

Total sum of shock at 4 corners Sum of sides of the square Force of motion of ball Radius of circle

Generalizing this proportion to the case in which the sides of the inscribed polygon become infinitely small, he attains the following result:

Total sum of shocks _ Sum of all sides 12 Force of motion of ball Radius of circle'

Further references occur in other sections of the Waste Book and in the Vel- lum Manuscript, dating from approximately the same time, where Newton shows that the ratio between the endeavor to escape from the earth at the equator and gravity is nearly as 1 to 300.13 In a manuscript on circular motion dating from the

10 Christiaan Huygens to G. W. Leibniz, 8 Feb. 1690, in Huygens, Oeuvres, Vol. IX, pp. 366-368. 11 University Library, Cambridge (ULC), MS Add. 4004. See John W. Herivel, The Background to

Newton's "Principia": A Study of Newton's Dynamical Researches in the Years 1664-1684 (Oxford: Clarendon Press, 1965), pp. 7-13, 45-48, 127-132; and the essay review by D. T. Whiteside, "New- tonian Dynamics," History of Science, 1966, 5:104-117.

12 For a circumference xTlmv = 2-r, where v is the velocity, x the total sum of shocks per unit time, T the period, and m the mass of the body. Therefore x = 2-rmvlT, and since v = 2rrR/T, we have x = mv2/R. In the Principia Newton referred to this result in the scholium following prop. 4. This is the expression of the force exerted by the surface on the ball and is equal and opposite to the action exerted by the ball because of its tendency to escape along the tangent. Therefore this expression is conceptually different from that of centrifugal force, which requires a rotating frame. It is only in the symmetric case of the circumference, or when the curve is perpendicular to the radius, that centripetal force is equal and opposite to the centrifugal force induced by a frame centered in the center of force and rotating with the body. This equality is broken as soon as we move from the circumference to the ellipse.

13 ULC MS Add. 3958, fol. 45; see Herivel, Background to Newton's "Principia" (cit. n. 11), sec. Ild, pp. 145-150, 184-186. See also Herivel, "Interpretation of an Early Newton Manuscript," Isis, 1961, 52:410-416. The same concept can be found in a letter of January 1681 to Thomas Burnet, where Newton refers to centrifugal force as the cause that flattens planets at the poles, as is apparent with Jupiter, and in a letter of 14 July 1686 to Edmond Halley, on the diminution of gravity at the equator. See The Correspondence of Isaac Newton, ed. H. W. Turnbull et al., 7 vols. (Cambridge: Cambridge Univ. Press, 1959-1977), Vol. II, pp. 329 (Burnett), 444-445 (Halley) (hereafter Newton, Correspondence).


late 1660s, Newton shows that the endeavor of a body moving in a circle to escape from the center can be calculated from the proportion BE:BA::BA:DB (see Figure 2), or, neglecting infinitely small differences, DE:DA::DA:DB. In the same manuscript we find that the endeavor to recede from the sun is inversely proportional to the squared distance, a result attained using Kepler's third law.14

Prior to the essay on motion of 1684 there are several references to centrifugal force in Newton's correspondence. They include a letter to the secretary of the Royal Society, Henry Oldenburg, and, most important, a reply to a letter from Robert Hooke. While Hooke had outlined his "theory of circular motions com- pounded by a direct motion and an attractive one to a centre," Newton referred in his reply to a body that would "circulate with an alternate ascent and descent made by its vis centrifuga and gravity alternately overballancing one another." 15

B A

D

Figure 2

In April 1681, writing to James Crompton at Cambridge, Newton referred to the same explanation, claiming that for a comet in perihelion centrifugal force would overpower attraction and make the comet recede from the sun.16

In summary, then, in Newton's early view centrifugal force is a real endeavor due to the inertia of a body moving along a curved path, just as it was for Huy- gens. In the case of a planet rotating around its axis, gravity greatly overpowers centrifugal force; for the motion of a ball in a hollow sphere, centrifugal force is equal and opposite to the force exerted by the container on the ball; last, in the case of orbital motion, Newton believed that centrifugal force and gravity alternately overpower each other.

14 ULC MS Add. 3958(5), fols. 87, 89; see Herivel, Background to Newton's "Principia," pp. 192-198. See also E. J. Aiton, The Vortex Theory of Planetary Motion (London: Macdonald; New York: American Elsevier, 1972), pp. 115-118; and R. S. Westfall, Force in Newton's Physics (Lon- don: Macdonald; New York: American Elsevier, 1971), pp. 350-360.

15 Isaac Newton to Henry Oldenburg, 23 June 1673, in Newton, Correspondence, Vol. I, p. 290; and Robert Hooke to Newton, 9 Dec. 1679, Newton to Hooke, 13 Dec. 1679, ibid., Vol. II, pp. 305-308. Newton referred to the letter to Oldenburg in 1686, in two letters to Halley, 20 June, 27 July, ibid., Vol. II, pp. 436, 446; he mentioned orbital motion, a conatus recedendi a centro, and a vis centrifuga. Newton's theory is very similar to those expressed in Descartes, Principia philosophiae (cit. n. 6), Pt. 3, prop. 120; and in Giovanni Alfonso Borelli, Theoricae medicearum planetarum (Florence, 1666). The latter work was in Newton's library-see J. Harrison, The Library of Isaac Newton (Cambridge: Cambridge Univ. Press, 1978)-and is referred to in Newton, Correspondence, Vol. II, p. 438.

16 Newton, Correspondence, Vol. II, p. 36. This may suggest that for a few years after Hooke's letter Newton still represented orbital motion as the resultant of the imbalance between gravity and centrifugal force. For this point and for Newton's theory of comets consult I. B. Cohen, The New- tonian Revolution (Cambridge: Cambridge Univ. Press, 1980), sec. 5.4.


With the various versions of the tract De motu, probably as a consequence of Hooke's suggestion, Newton changed his views on circular motion, which he eventually came to see as the resultant of rectilinear inertia and centripetal force.17 This new approach, however, did not automatically clarify the role and nature of centrifugal force; indeed, they remain virtually inexplicit in Newton's subsequent publications. What follows is a careful exegesis of how Newton rein- terpreted centrifugal force from 1684 onward.

In the last of a series of definitions that appear to be connected to De motu, we find the roots of Newton's later ideas:

The exercised force of a body is that by which it attempts to preserve that part of its state of rest or motion which it gives up instantaneously and it is proportional to the change of its state or to that portion of its state given up instantaneously, and not improperly is said to be the reluctance or resistance of the body, of which one species is the centrifugal force of rotating bodies.

Here centrifugal force is seen almost as a reaction proportional to the force that bends the body's orbit. According to the third law of motion, which Newton stated for the case of collision in the Waste Book and then in a series of laws in connection with De motu, centrifugal force would be equal and opposite to centripetal force, a view he explicitly endorsed several years later.18

In the Principia centrifugal forces do not constitute a problem and seem to play a negligible role. In Book I they are mentioned as mathematical examples on a number of occasions, but their nature is discussed only in the scholium following proposition 4.19 Newton refers to the Horologium, where Huygens had compared the force of gravity with the centrifugal force of revolving bodies. Immediately afterward he describes his Waste Book calculation of the endeavor to recede from the center, based on the polygon inscribed in a circumference. The scholium ends with the words: "This is the centrifugal force, with which the body impels the circle; and to which the contrary force, wherewith the circle continually repels the body towards the centre, is equal."20 This passage confirms the impression that Newton is using the third law: note in particular the words "huic aequalis est vis contraria." In a passage, omitted in the first edition, related to definition 5, Newton follows Descartes in setting together the tendencies to escape along the radius and along the tangent:

17 See D. T. Whiteside, "Newton's Early Thoughts on Planetary Motion: A Fresh Look," British Journal for the History of Science, 1964, 2:117-137; Newton, Mathematical Papers, Vol. VI, pp. 36-39, esp. n. 23; and Cohen, Newtonian Revolution, secs. 5.3, 5.5.

18 Herivel, Background to Newton's "Principia" (cit. n. 11), pp. 317, 320 (I have altered Herivel's translation slightly). For the view that centrifugal force is equal and opposite to centripetal force see ibid., pp. 31, 307, 312.

19 Isaac Newton's Philosophiae naturalis principia mathematica, the Third Edition (1726) with Vari- ant Readings, ed. A. Koyre, I. B. Cohen, and Anne Whitman (Cambridge: Cambridge Univ. Press, 1972); cf. the scholium at the end of prop. 10 (p. 54, 1. 10) and the end of prop. 12 (p. 58,1. 2), where Newton shows that if a body moves along a hyperbola, centripetal force toward the focus is inversely proportional to the squared distance; he adds that if the force was centrifugal, the body would move along the conjugate hyperbola. In cor. 3 to prop. 41 (pp. 127-128), Newton studies the curves described by a body acted upon by a centripetal or centrifugal force inversely proportional to the third power of the distance. In the third edition centrifugal force is mentioned in cor. 20 to prop. 66, in connection with tides.

20 The word centrifugal is used only in the second and third editions. Here I use the translation by A. Motte and F. Cajori (Berkeley/Los Angeles: Dover, 1934), p. 47. See also Newton, Mathematical Papers, Vol. VI, pp. 200-201.


A stone, whirled about in a sling, endeavors to recede from the hand that turns it; and by that endeavor, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as it is let go, flies away. That force contrary to this endeavor, and by which the sling continually draws back the stone towards the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits.2'

Newton seems to have the third law in mind in this passage as well. Moreover, he seems to indicate that his analysis is valid for all curves. Does Newton think that this result can be generalized to the case of planetary motion? In corollary 7 to proposition 4, for example, we find a generalization of the law of centripetal force to curves other than the circumference. Does this indicate that Newton is prepared to generalize the law of centrifugal force as well, using the third law? I discuss below two memoranda of Newton's that make this conjecture very plausible, and in Section IV we shall find that Euler's interpretation follows these lines.

In Book I Newton refers to the "vires recedendi ab axe motus circularis" in the famous scholium to definition 8, where he describes the rotating-vessel and the rotating-globes experiments. Newton's assumption is that true circular motion generates a tendency to escape from the axis of rotation. These tendencies make water in the bucket rise along the sides and cause a tension in the thread connecting the rotating globes. Probably the expression vis centrifuga does not occur explicitly both because centrifugal forces are formally introduced in the scholium to proposition 4 and because Newton thought that his reasoning was sufficiently clear.22 To rephrase Newton's assumptions: observers at rest with respect to absolute space see true circular motion and detect centrifugal forces; rotating observers do not see true circular motion and are unable to explain centrifugal forces. This is very different from the modern view, according to which centrifugal forces are detected in a rotating frame regardless of the motion that is being observed; we shall see that this motion can also be rectilinear uniform. With respect to rotations, for Newton there is only one possible representation of motion; different representations do not explain phenomena success- fully.23

In Book II of the Principia centrifugal forces occur mainly as mathematical examples. But in proposition 52, where Newton criticizes Cartesian vortices, he explicitly states that centrifugal forces arise from circular motion: "Caeterum cum motus circularis, et [ab] inde orta vis centrifuga."24

In Book III there are some relevant references in propositions 18 and 19 and in the corollary to proposition 36. The first and third cases refer to the axes of planets being shorter than the diameters perpendicular to those axes and are

21 Newton, Principia, def. 5. I have slightly improved the translation by Motte and Cajori (pp. 2-3). The crucial passage reads: "Vim conatui illi contrariam . .. centripetam appello."

22 There can be no doubt that the two experiments referred to in the scholium following def. 8 were understood in terms of centrifugal forces. Interpreters as diverse as George Berkeley and Colin Maclaurin, e.g., mentioned them explicitly. See George Berkeley, A Treatise concerning the Princi- ples of Human Knowledge (Dublin, 1710), sec. 114; Berkeley, De motu (London, 1721), sec. 62 (where he refers to a "conatus axifugus"); and Colin Maclaurin, An Account of Sir Isaac Newton's Philosophical Discoveries (London, 1748), pp. 101-102, 305ff.

23 Compare this to Huygens's defense of relative circular motion in Oeuvres (cit. n. 7), Vol. XVI, pp. 189-200, 209-233.

24 Newton, Principia (cit. n. 19), p. 379, 11. 9-10. Compare cor. 8 to prop. 22; prop. 23, on Boyle's law; prop. 33 (p. 319, 1. 35) and its cors. 3 and 6; and prop. 40, p. 783 (= p. 338, 1st ed.).


similar to the ideas in the letters to Burnet and Halley already discussed. Propo- sition 19 deals with the shape of the earth, which is a spheroid flattened at the poles. Centrifugal force at the equator is to gravity as 1 is to 2904/5. The principle, if not the numerical detail, is the same as in the Vellum Manuscript. In the third edition, in the scholium to proposition 4, Newton discusses the motion of a hypo- thetical little moon rotating very close to the surface of the earth: "Therefore if the same little moon should be deserted by all the motion which carries it through its orb, because of the lack of the centrifugal force with which it had endured in the orb, it would descend to the earth."25 This quotation is coherent with the interpretation that centrifugal force does play a role in orbital motion.

The passages from the Principia that I have just surveyed show that the change in Newton's views about centrifugal force-from his early manuscripts to his mature work-concerns exclusively the crucial problem of orbital motion, which is no longer seen as the result of two opposite tendencies overbalancing each other; after 1684 centrifugal force was related to the third law and to gravity. In the case we have just examined centrifugal force prevents an orbiting body from falling toward the center. In general, however, Newton neglects centrifugal force without explaining why this can be done.

Soon after the publication of the Principia, Leibniz expressed his views in the Tentamen de motuum coelestium causis. According to him, for all curves and in particular for the ellipse, centrifugal conatus or force is measured by the square of the velocity of rotation over the radius. The velocity of rotation is that component of orbital velocity perpendicular to the radius. Leibniz also refers to an outward conatus ("conatus excussorius"), which is measured by the distance from a point on a curve to the tangent from a point infinitely near. The outward conatus is the square of orbital velocity over the radius of the osculating circumference. Of course, if the curve is a circumference, the two coincide. If the curve is not a circumference, centrifugal force is obtained by fixing the radius and taking the square of the component of orbital velocity perpendicular to it; outward conatus is obtained by taking the square of orbital velocity over that radius perpendicular to it, namely, the osculating radius. The cause of both endeavors is the rotation of the body and its tendency to escape along the tangent. In his reading of the Principia, in an important passage of the so-called "zweite Bear- beitung" of the Tentamen, Leibniz claims that there are two different but equivalent ways to represent planetary motion (or, more generally, motion with central forces): either by rectilinear inertia and gravity alone, as if the body moved in a vacuum; or by a circular and a radial motion. Circular motion is due to a vortex rotating with a velocity inversely proportional to the radius, in order to account for Kepler's area law: this he calls circulatio harmonica; radial motion is due to the imbalance between gravity and centrifugal force: this he calls motus para- centricus. Thus for Leibniz the mathematical equivalence between the two representations is resolved on a physical level by the presence of the vortex, which was a pillar of his theory.26 His equation of paracentric motion is

25 Newton, Principia, p. 398, 11. 22-24, my translation; the translation by Motte and Cajori is not satisfactory. From the previous remarks, it appears that Newton became more explicit about centrifugal force in the second and especially in the third edition of the Principia.

26 See G. W. Leibniz, "Tentamen de motuum coelestium causis," Acta Eruditorum, February 1689, pp. 82-96; also (with some corrections) in Leibnizens mathematische Schriften, ed. C. I. Ger- hardt, 7 vols. (Berlin/Halle, 1849-1863), Vol. VI, pp. 144-161, secs. 10, 11, 19 (hereafter Leibniz,


ddr = 00aalr3 - 002a1r2,

or, in modern terms,

d2rldt2 = h21r3 - h221r2a,

where r is the radius and 0 = dt is the differential of time. For Leibniz a represents both the latus rectum of the ellipse and the proportionality constant between time and area; in the second equation a is the latus rectum and h the angular momentum.27

While criticizing Leibniz's views in the 1710s, Newton made his position on the matter clearer in two short memoranda.28 In the first text Newton explicitly states that centrifugal conatus (adopting Leibniz's vocabulary) and force of gravity are equal and opposite, as follows from the third law of motion. The context is planetary motion; therefore Newton's statement refers in particular to elliptical orbits. This crucial passage supports my previous conjecture that for Newton corollary 7 to proposition 4, Book I, read together with the following scholium, applies to centrifugal forces as well.

In the Principia the third law is used to prove that attraction must be mutual: if body 1 attracts body 2, the reaction is the equal and opposite force with which 2 attracts 1. This is made clear in particular in proposition 69, Book I. Therefore, in my interpretation the third law has a double role for Newton, as an explanation of both centrifugal force and the reciprocity of attraction.29 Contrary to his opponent's theory, Newton claims that orbital motion does not depend on the balance between gravity and centrifugal force, because the orbit is curved by the action of gravity alone.

In the second text Newton stresses his own views and makes a crucial obser- vation on Leibniz's equation of paracentric, or radial motion. Newton claims that if the curvature of the orbit is diminished in such a way that the orbit coincides with its tangent, there ought to be neither gravity nor centrifugal conatus. There

Mathematische Schriften). See also the "zweite Bearbeitung," ibid., pp. 161-187, esp. pp. 178, 184. On Leibniz see Aiton, Vortex Theory (cit. n. 14), Ch. 6. The "zweite Bearbeitung" was written between 1689 and 1690 and first published by Gerhardt. See Domenico Bertoloni Meli, "Leibniz on the Cen- sorship of the Copernican System," Studia Leibnitiana, 1988, 20:19-42, sec. 4.

27 This equation can be obtained by taking twice the derivative of the polar equation of an ellipse, r = b2l(q - e cos 0), where b and q are the minor and the major axes, respectively, e the eccentric- ity, and 0 the polar coordinate: a = b2:q; h = r2dOldt. The way in which Leibniz attained this equation is discussed in E. J. Aiton, "The Celestial Mechanics of Leibniz," Ann. Sci., 1960, 16:65- 82, esp. p. 76, n. 35. The term representing centrifugal force depends on the choice of a reference frame rotating with the radius. For Leibniz, however, centrifugal forces arise from curvilinear motion and are a real (as opposed to fictitious) tendency to motion. Originally he erroneously interpreted the first term at second member as twice the centrifugal conatus, but he corrected the mistake in Acta Eruditorum, October 1706, pp. 446-451, thanks to a letter from Varignon.

28 Newton's critiques are in The Correspondence of Sir Isaac Newton and Professor Cotes, ed. J. Edleston (London, 1850), pp. 310-313 (written around 1712); and Newton, Correspondence, Vol. VI, pp. 116-122 (written around 1714). See also Isaac Newton, "An Account of the Book Entituled Commercium epistolicum," Philosophical Transactions, 1715, 29:173-224, esp. pp. 208-209, where Newton accuses Leibniz of taking the center of the osculating circumference for the center of the circulation.

29 On this see E. J. Aiton, "The Mathematical Basis of Leibniz's Theory of Planetary Motion," in Leibniz Dynamica, ed. A. Heinekamp (Studia Leibnitiana, Sonderheft 13) (Wiesbaden: Steiner, 1984) pp. 209-225, esp. p. 222. Compare I. B. Cohen, "Newton's Third Law and Universal Gravity," Journal of the History of Ideas, 1987, 48:571-593.


is no doubt about the reason he has in mind: since motion would become uniform and rectilinear, force must disappear because of the first and second laws. How- ever, paracentric or radial motion would not be zero, nor would its differential, which dato tempore is proportional to acceleration. This is one reason why Newton believed that Leibniz was wrong. Here he seems to imply that, as far as force is concerned, there is only one possible representation of motion.30

Leibniz also dealt with the problem of whether one can have centrifugal force with rectilinear uniform motion. In a letter to Jakob Hermann he says that this is indeed possible, assuming an arbitrary point as a center; centrifugal force con- sists of how much the body has withdrawn from that point. However, he also seems to interpret the straight line as the tangent to a curve and to consider the body as moving along the curve and only instantaneously along the tangent.31 We shall come to other instances in which this problem was discussed.

In summary, before 1679 Newton-like Descartes, Borelli, and Leibniz-believed that orbital motion depended on the imbalance between gravity and centrifugal force; after 1684 he believed that centrifugal force was equal and opposite to gravity, from the third law of motion. In general, he explained curvilinear motion in terms of centripetal force and inertia alone, without centrifugal force; why in this case centrifugal force could be neglected, however, was not clear. In certain passages, such as definition 5, Newton seems to associate centrifugal force with inertia. In other passages, such as the scholium to proposition 4, Book III, centrifugal force prevents an orbiting body from falling toward the center. In cases different from orbital motion, such as a planet rotating around its axis, he believed that centrifugal force was different from gravity. Hence Newton's theory of centrifugal force followed a case-by-case pattern, and the solution to one particular problem could not be easily generalized.

Newton was not the only one to adopt different explanations in different con- texts. We have seen that Huygens thought that in the Principia orbital motion resulted from gravity and centrifugal force counterbalancing each other: for him gravity depended on the centrifugal force of a fluid. On the contrary, for Newton gravity was the cause of centrifugal force, the latter being only a reaction to the former. This diversity of views, both within Newton's thought and with respect to Huygens and Leibniz, sets the agenda for an investigation of the reception of the problem at the beginning of the eighteenth century.

III. THE RECEPTION OF CONTRASTING VIEWS

This section surveys the ways centrifugal force was perceived in the early decades of the eighteenth century. Centrifugal force is so common a notion, occur- ring in all treatises on mechanics as well as in the study of specific problems such

30 With hindsight, we see that the problem depends on a different choice of reference frame, which is inertial for Newton and rotating for Leibniz. In this case and in modern notation, the radial equation of motion for a body moving along a straight line at a distance R from the origin is r = R/cos 0. Taking twice the derivative with respect to time gives d2rldt2 = h2/r3, which is the expression for the centrifugal force along the rotating radius. Newton did not accept a representation in which (fictitious) forces occur in rectilinear inertial motion.

31 This passage deserves to be quoted: "Dici aliquomodo potest, vim centrifugam locum habere etiam, cum circularis motus non consideratur. Pro centro enim punctum quodcunque assumi potest, et concipi quantum continuato mobilis motu per tangentem curvae ab illo centro recedatur, et quantum mobile retrahendum sit ad curvam, in quo vis centrifuga consistit." G. W. Leibniz to Jakob Hermann, 21 Mar. 1709, in Leibniz, Mathematische Schriften, Vol. IV, p. 345.


as the shape of the earth, that the survey must be somewhat limited.32 Because of the striking difference between Newton and Leibniz on this issue, I have looked for a relation between particular perceptions of centrifugal force and given positions in the priority dispute over the invention of the differential calculus. I have selected five from among the mathematicians occupying the wide spectrum of positions in the dispute. These five can be set in three different camps: John Keill was the staunchest of Newton's champions; Johann Bernoulli and Christian Wolff were Leibniz's defendants in the opposite camp; and Pierre Varignon and Jakob Hermann occupied a more neutral position. Varignon actively tried to reconcile the two contenders. Hermann, in spite of his close contacts with Leib- niz, who had procured a post for him at the University of Padua, adopted several Newtonian tenets in a way that his mentor found distinctly unpleasant.33 All five knew not only Huygens's work and the Principia, but also Leibniz's essay on planetary motion.34 Any explanation of their different interpretations that relates it to the priority dispute is, however, surprisingly unsatisfactory.

John Keill, fed by Newton in the attack against Leibniz, predictably sided with his mentor: in an essay opposing Leibniz over second-order infinitesimals, he criticized the Tentamen and claimed that Leibniz had not explained the notion of centrifugal force, which in Keill's view would depend on the vis inertiae and would be equal and opposite to gravity. In "The Demonstrations of Monsieur Huygens's Theorems concerning the Centrifugal Force and Circular Motion," which was added to An Introduction to Natural Philosophy, he states his position very clearly: "A centrifugal force is the re-action or resistance which a moving body exerts to prevent its being turned out of its way, and whereby it endeavours to continue its motion in the same direction: and as re-action is always equal, and contrary to action, so in like manner is the centrifugal to the centripetal force. This centrifugal force arises from the vis inertiae of matter. "35 The same con- cepts, though less sharply defined, can be found in the Introductio ad veram physicam, which makes reference to proposition 4 of the Principia and to its corollaries.36 Concerning the shape of the earth, Keill once again sided with

32 A good survey of the exponents of vortex theories is in Aiton, Vortex Theory (cit. n. 14), Chs. 7-10. See also Isaac Todhunter, A History of the Mathematical Theories of Attraction and the Figure of the Earth, 2 vols. (London, 1873).

33 Leibniz to Hermann, 17 Sept. 1716, in Leibniz, Mathematische Schriften, Vol. IV, pp. 398-402. See also A. R. Hall, Philosophers at War (Cambridge: Cambridge Univ. Press, 1980), pp. 165-166; and Ernst Cassirer, Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit, 3rd ed. (Berlin: Bruno Cassirer, 1922), Vol. II, pp. 463-485, esp. p. 471. This is part of a chapter on space and time from Newton to Kant that devotes particular attention to Euler. On Varignon's role in the priority dispute see Hall, Philosophers at War, pp. 239-241.

34 John Keill discussed Leibniz's essay (the Tentamen, cit. n. 6) with Newton and considered it the most incomprehensible piece of philosophy ever written: "Reponse aux auteurs des remarques, sur le different entre M. de Leibnitz et M. Newton," Journal Lite'raire de la Haye, 1714, 4:319-358, esp. p. 348. Johann Bernoulli referred to it in a letter to Leibniz, 22 Feb. 1696, in Leibniz, Mathematische Schriften, Vol. III, p. 250. For Christian Wolff see Wolff to Leibniz, 24 Apr. 1715, in Newton, Correspondence, Vol. VI, pp. 216-218; and Wolff, Elementa matheseos universa, 5 vols., 2nd ed. (Geneva, 1732-1741), Vol. V, p. 84. Pierre Varignon referred to the Tentamen in "Des forces centrales," Me'moires de l'Acade'mie Royale des Sciences, 1700, pp. 218-237, on p. 224. Jakob Hermann mentioned it in "Metodo d'investigare l'orbite de' pianeti," Giornale de Letterati d'Italia, 1710, 2:447-467, on p. 450.

35 See Newton, Correspondence, Vol. VI, pp. 148-149; and Keill, "Reponse" (cit. n. 34), pp. 350-351. The quotation is from Keill, An Introduction to Natural Philosophy, ed. Willem Jakob s'Gravesande (London, 1745), p. 286; see also pp. 86-87. Notice the similarity to the passage from the definitions related to Newton's De motu quoted above.

36 John Keill, Introductio ad veram physicam, 2nd ed. (London, 1705), pp. 251-253.


Newton in claiming that centrifugal force is not equal and opposite to attraction. In the same passage he also repeated his view that with respect to planetary motion centrifugal force is equal and opposite to gravity, obviously perceiving no contradiction between the two cases.37 As I have already remarked about New- ton, Keill's observations too appear as a collection of case-by-case solutions that cannot be easily generalized.

The first surprise in these five cases comes from the French mathematician Pierre Varignon, who published a series of four memoirs on mechanics in 1700- 1701. The second paper contains a puzzling claim that central forces, both centrifugal and centripetal, are the foundation of Newton's "excellent book." The conclusion that Varignon interpreted the Principia as a book on centrifugal as well as centripetal forces, and that his own theorems on central forces must also be so interpreted, is very tempting. This suspicion is reinforced by the third memoir: referring to his previous piece, Varignon repeats the same definition of central forces, as if he wanted to explain that his theorems were valid for centripetal and centrifugal forces.38 Further evidence emerges from a letter-paradox- ically, it is addressed to Leibniz-in which Varignon explicitly states his views. The French mathematician thought the matter so unproblematic that he could simply write that centrifugal force is equal and opposite to centripetal force.39 From the context it appears that Varignon was referring to a curve in general, not to a circumference, and that centrifugal force would be equal and opposite to centripetal force because of the third law of motion. This sheds new light on his reading of the Principia and seems to indicate that Varignon-unlike Newton- did not perceive the existence of conflicting interpretations of centrifugal force.

For Jakob Hermann the situation at first sight appears simple. In the Phorono- mia he clearly states that centrifugal conatus arises from circular motion and that in an ellipse centrifugal endeavors are equal to the square of the velocity of rotation over the radius vector from the center to the orbit. Since the velocity of rotation is the component of the orbital velocity perpendicular to the radius, Hermann is using the same generalization of Huygens's theorems that Leibniz employed: centrifugal force and gravity are in general different.40 In another passage, however, Hermann expounds a different theory: he decomposes central solicitation, which corresponds to Newton's centripetal force, into a solicitatio tangentialis along the tangent to the curve and a solicitatio perpendicularis that is perpendicular to the tangent.41 The former would make the body move along the curve, whereas the latter would cancel the conatus recessorius of the body from the curve: this would be equal to the square of orbital velocity over the radius of the osculating circumference.42 This means that the component of the solicitation

37 John Keill, An Examination of Dr Burnet's Theory of the Earth, 2nd ed. (Oxford, 1734), pp. 91-93.

38 The second memoir: Pierre Varignon, "Du mouvement en general, par toutes fortes des courbes; et des forces centrales, tant centrifuges que centripetes, necessaires aux corps qui les decrivent," MWm. Acad. Roy. Sci., 1700 (1703), pp. 83-101, on p. 84. The full title is relevant to my point here. The third memoir: Varignon, "Des forces centrales" (cit. n. 34), p. 221. See also Varignon to Leibniz, 23 May 1702 and 6 Dec. 1704, in Leibniz, Mathematische Schriften, Vol. IV, pp. 99-104, on p. 101, and pp. 113-127, on p. 117.

39 Varignon to Leibniz, 29 Apr. 1706, in Leibniz, Mathematische Schriften, Vol. IV, p. 149. 40 Jakob Hermann, Phoronomia, sive de viribus et motibus corporum solidorum etfluidorum (Am-

sterdam, 1716), pp. 2, 97-98; see also pp. 91-92. 41 Ibid., p. 51. See also Jakob Hermann, "Modo facile di determinare la legge delle forze centrali,"

G. Lett. d'Italia, 1713, 13:321-362, esp. p. 323. 42 Hermann, Phoronomia (cit. n. 40), pp. 52-53, 68-69. Hermann's conatus recessorius corre-


of gravity perpendicular to the curve-namely, the square of orbital velocity over the osculating radius-is exactly equal and opposite to the conatus recessorius.43 In summary, whereas concerning centrifugal conatus Hermann adopts Leibniz's views, he partially follows Newton in claiming that the conatus recessorius is equal and opposite to the component of gravity perpendicular to the curve. In a letter to Leibniz Hermann also refers to the case of centrifugal force in rectilinear motion: it appears that he felt this was nonsense.44

In the Mathematisches Lexicon Christian Wolff is not very informative: under the headings "Vis centrifuga" and "Vis centripeta" he refers to Huygens's work and explains also that when a body moves along an ellipse around the sun, it will proceed along the tangent if unhindered. This force (of inertia) with which the planet tends to move, seen from the sun ("in Ansehung der Sonne"), is called vis centrifuga. The reference to the sun is meant to imply that centrifugal force acts along the radius from the sun to the planet; its nature is clearly related to inertia. In the Elementa matheseos universae Wolff defines centrifugal force as the tendency of a body revolving around a center to escape. Although he refers only to the case of a circumference, he claims, contrary to Leibniz and following New- ton, that vis centrifuga and vis centripeta are always equal and opposite.45

From the 1690s to the 1740s it is possible to trace several references to centrifugal force in Johann Bernoulli's published works and correspondence. These references concern the following problems: courbe centrifugue, falling bodies, and planetary motion.

The first case is mentioned in Bernoulli's correspondence with the Marquis de L'Hopital, whom Johann had instructed in the differential calculus in the early 1690s. The curve is described by a body attached to a thread unrolling from a further curve on a vertical plane in such a way that the tension of the thread is constant. To the marquis's request for explanations of centrifugal force, Johann answers that by centrifugal force he understands exactly what Huygens had explained in the Horologium. The curve described by the body is always perpendicular to the radius, which increases in length while the thread unrolls itself. After some uncertainties, Johann finds that centrifugal force is equal to the square of velocity over the radius. A few years later L'Hopital would publish the proofs of Huygens's theorem with a solution to the problem of the courbe centrifugue.46

sponds to Leibniz's conatus excussorius. Hermann, however, stressed the link between the former and a component of gravity, whereas Leibniz did not.

43 Ibid., p. 92. Centripetal force can be expressed as v2/p cos 0, where p is the osculating radius and 0 is the angle between the direction of gravity and the osculating radius (see Newton, Mathematical Papers, Vol. VI, pp. 548-549, n. 9); the component of centripetal force along the osculating radius is v2/p.

"Hermann to Leibniz, 21 Feb. 1709, in Leibniz, Mathematische Schriften, Vol. IV, p. 343. The discussion arose from the atrocious Essais et recherche de mathe'matique et de physique (Paris, 1705-1713), by Antoine Parent.

4 Christian Wolff, Mathematisches Lexicon (Leipzig, 1716), cols. 1459-1461; and Wolff, Elementa (cit. n. 34), Vol. II, pp. 159-160. Contrary to Hermann, but in accordance with Varignon, Wolff defines "vires centrales" as both centripetal and centrifugal forces (p. 160).

`6 Bernoulli states the problem in Acta Eruditorum (Suppl. 2), 1696, sec. 6, p. 291; in Johann Bernoulli, Opera, ed. G. Cramer (Geneva, 1745), Vol. I, pp. 141-142. See also L'H6pital to Johann, 19 Feb. 1695; and Johann to L'H6pital, 5 Mar. 1695, 26 Mar. 1695, and 21 Apr. 1696, where the term "courbe centrifugue" occurs; in Der Briefwechsel von Johann Bernoulli, ed. 0. Spiess, Vol. I (Basel: Birkhauser, 1955), pp. 263, 270-271, 276, 314-318; l'H6pital, "Solution d'un probleme physico-mathematique," Me'm. Acad. Roy. Sci., 1700 (1703), pp. 9-21; and L'H6pital to Leibniz, 25 Apr. 1695; in Leibniz, Mathematische Schriften, Vol. II, p. 201.


In this case gravity and centrifugal force have a component in the same direction. In his correspondence with Varignon, however, Johann discusses centrifugal force in the case of a body falling with an initial velocity LQ parallel to the horizon with a parabolic trajectory LMN (see Figure 3).47 After having con- structed the evolute RC of the parabola, Johann states that the centrifugal force at M is equal and opposite to the component of gravity along the osculating radius, namely, pMSIMO, where p represents gravity. We have seen that a similar theory was adopted by Hermann in the Phoronomia with respect to the conatus recessorius in the case of central forces: here, however, the impulsions of gravity are parallel among themselves. Johann defended a similar theory in the Discours sur les loix de la communication du mouvement and in the Nouvelles

L Q

M

R

C 0 N

Figure 3 pensees sur le systeme de M. Descartes; referring-in the first case explicitly-to circular and curvilinear motion, Johann explains centrifugal force in terms of action and reaction, which are always equal and opposite. Furthermore, in the Essai d'une nouvelle physique cedleste, one finds that centrifugal force depends on the curvature of the trajectory and on the speed of the body.48 This shlows that in spite of his early profession of loyalty to Huygens's views, on this topic Leib- niz's champion was closer to Newton's interpretation. However, Johann Ber- noulli seemed to apply the third law of motion only to the component of centripetal force perpendicular to the curve.

Euler claimed that Johann Bernoulli was the first to deal with the problem of motion of bodies relative to moving frames. Johann's first publication on this topic, however, concerns the motion of a body sliding along a movable inclined plane and is not relevant to our purpose here.49 We have to wait another decade

47 At present this correspondence is not in print. The relevant passage is quoted by Varignon in a letter to Leibniz, 9 Oct. 1705, in Leibniz, Mathematische Schriften, Vol. IV, pp. 136-138.

48 Johann Bernoulli, Discours sur les loix de la communication du mouvement (Paris, 1727) in Opera, Vol. III, pp. 1-107, esp. pp. 89-90; Bernoulli, Nouvelles pensees sur le systeme de M. Des- cartes (Paris, 1730), in Opera, Vol. III, pp. 131-173, esp. p. 137; and Bernoulli, Essai d'une nouvelle physique ceeste (Paris, 1735), in Opera, Vol. III, pp. 261-364, esp. p. 308. Applying the theory expressed in the letter to Varignon, centrifugal force would be directed along the osculating radius p and would be equal and opposite to the component of gravity along p.

49 This claim was made in Leonhard Euler, "Dissertation sur le mouvement des corps enferm6s dans un tube droit mobile autour d'une axe fixe," Opera postuma (St. Petersburg, 1862), Vol. II, pp.


to find his "De curva quam describit corpus inclusum in tubo circulante," dealing with a rotating frame. Johann finds the equation of motion of a body constrained in a tube rotating on a horizontal plane. Centrifugal force, however, is not mentioned explicitly.50

I have tried to show that in the first decades of the eighteenth century centrifugal force was interpreted in a number of ways. Some form of Newton's interpretation in terms of the third law prevailed not only with Keill, but also penetrated the "neutral ground" of Varignon and Hermann and the very stronghold of the Leibnizian camp with Wolff and Johann Bernoulli. Explanations relating the different interpretations to the priority dispute thus fail from the start here. With respect to this particular problem, moreover, it is difficult to invoke the general influence of the Principia mathematica, since mathematicians like Johann Ber- noulli were reading it with the specific intention of finding faults. On the Conti- nent a certain lack of perception of the controversial nature of centrifugal force possibly was fertile soil for Newton's interpretation. Second, its appeal to an indisputable principle such as the third law probably was an important point in favor of Newton's view; Leibniz's theory made no such reference. With respect to Johann Bernoulli and Jakob Hermann a further factor was important: only the component of gravity along the osculating radius was equal and opposite to centrifugal force-to the conatus recessorius, for Hermann-which was measured with respect to the center of the osculating circumference. In the mid 1700s the expression for centripetal force we saw above-v2/p cos 0-became widely known.51 The component along the osculating radius is V2Ip and this is equal and opposite to centrifugal force with respect to the center of the osculating circum-

85-113, in Leonhardi Euleri Opera omnia (Berlin/Gottingen, 1911-), Series II: Opera mechanica et astronomica, Vol. VII, pp. 266-307, on pp. 266-268; and in Euler, "De motu corporum in superficiebus mobilibus," Opuscula varii argumenti (Berlin, 1746), Vol. I, pp. 1-136, in Euler, Opera omnia, Series II, Vol. VI, pp. 75-174, on pp. 75-78. Johann's first publication on the topic was "Solutiones novorum quorundam problematum mechanicorum," Commentarii Academiae Scientiarum Petropoli- tanae, 1730/31 (1738), 5:11-25; in Bernoulli, Opera, Vol. III, pp. 365-375; it is based on the principle of conservation of living force. Daniel accused his father of plagiarism and claimed that he had solved this problem himself. See Daniel Bernoulli to Leonhard Euler, 20 Oct. 1742, in Correspondance mathematique et physique de quelques celebres geometres du XVIIIeme siecle, ed. P. H. Fuss, 2 vols. (St. Petersburg, 1843), Vol. II, p. 504; and Daniel's "De variatione motuum a percussione excentrica," Comm. Acad. Sci. Petropolitanae, 1737 (1744), 9:189-206.

50 Johann Bernoulli, "De curva quam describit corpus inclusum in tubo circulante," in Bernoulli, Opera, Vol. IV, pp. 248-252. This was also published in the following letters in Fuss, ed., Corre- spondance, Vol. II: Johann Bernoulli to Euler, 15 Mar. 1742 (pp. 67-71), 27 Aug. 1742 (pp. 73-81), 28 Dec. 1742 (pp. 144-145), Mar. 1743 (pp. 84-87) (Euler's replies are not extant). Johann says that the problem was posed to him by Samuel Koenig in 1734 when Clairaut and Pierre Louis Maupertuis were his guests in Basel. Andre-Marie Ampere and Gaspard Gustave Coriolis would refer to it some ninety years later in their memoirs on relative motion: A.-M. Ampere, "Solution d'un probleme de dynamique," Annales de Mathematiques, 1829/30, 20:37-58; and G. Coriolis, "Sur le principe des forces vives dans les mouvemens relatifs des machines," Journal de l'Ecole Polytechnique, 1831 (1832), 13:268-302, on p. 268.

51 Pierre Varignon, "Autre regle gen6rale des forces centrales," MWm. Acad. Roy. Sci., 1701 (1704), pp. 20-38, on pp. 21-22; and Abraham Demoivre to Johann Bernoulli, 27 July 1705, in K. Wollenschlaeger, "Der mathematische Briefwechsel zwischen Johann I Bernoulli und Abraham de Moivre," Verhandlungen der Naturforschenden Gesellschaft in Basel, 1933, 43:151-317, on pp. 213-214. In the text, which contains no proof, the expression for centripetal force is inverted. See also Johann Bernoulli to Demoivre, 16 Feb. 1706, ibid., pp. 224-225; Johann Bernoulli, "Extrait d'une lettre," Mem. Acad. Roy. Sci., 1710 (1713), pp. 521-533, on p. 529; and Bernoulli, "De motu corporum gravium," Acta Eruditorum, 1713, 2:77-95, 3:115-132, esp. p. 127. See also Newton, Math- ematical Papers, Vol. VI, pp. 548-549, n. 25; and E. J. Aiton, "The Inverse Problem of Central Forces," Ann. Sci., 1964, 20:81-99, esp. pp. 89-90.


ference. My conjecture is that the familiar use of this expression played a role in making more plausible the interpretation that Johann Bernoulli and Hermann adopted.

IV. A DEBATE OF THE MID 1740S

In the 1740s several mathematicians tried to solve the problem of motion of a body in a tube rotating on a horizontal plane. Among them were Johann Ber- noulli and his son Daniel, Leonhard Euler, and Alexis-Claude Clairaut.52 Here I am concerned, not with the original question, but with how the notion of centrifugal force was affected. A priori one might think that conceptual discussions on centrifugal force arose from some philosophical debate. Contrary to this reason- able presupposition, my readings suggest not only that the idea that centrifugal force is fictitious emerged from a mathematical context, but also that the mathematicians involved often needed material supports for thinking in terms of reference frames: sliding inclined planes, rotating tubes, and rotating turbines.53 For Johann Bernoulli, Euler, and Clairaut the origin of centrifugal force was not problematic; they did not doubt that its cause was the rotation of the tube and of the body in it. Daniel Bernoulli, however, began to see a problem in what his contemporaries were taking for granted.

In his Mechanica, Euler is not very explicit as far as our problem is concerned. In scholion 3 to proposition 77 he explains that all Huygens's theorems on centrifugal force in the Horologium are contained in the two preceding propositions. That which he had not made explicit could be "evidentissime" deduced.54 Propo- sitions 76 and 77 are similar to proposition 4 and to its corollary 7 in Book I of Newton's Principia. Probably for Euler these propositions were relevant to centrifugal force, but the expression vis centrifuga does not occur in them. Since they deal with the centripetal force required to keep a body in a circular and a curvilinear orbit, respectively, one is left in doubt as to whether centrifugal force is just equal and opposite to gravity or is measured in a different way-for example, with respect to the center of the osculating circumference. In Book II Euler clarifies his theory somewhat, explaining that the tendency of a body moving along a curved line to escape is called vis centrifuga because it is directed from the center of the osculating circumference. Indeed, in "De motu corporum in superficiebus mobilibus," Euler adopts exactly the same approach in discussing the problem of the body in the rotating tube.55 In the "Recherches sur l'origine

52 The problem is mentioned in Descartes, Principia philosophiae (cit. n. 6), Pt. 3, props. 58-59; and in G. W. Leibniz, "Specimen dynamicum," Acta Eruditorum, Apr. 1695, pp. 145-157, in Leib- niz, Mathematische Schriften, Vol. VI, pp. 234-246 (with a second part, pp. 246-254), on p. 238; also in Leibniz, Specimen dynamicum, ed. and trans. H. G. Dosch et al. (Hamburg: Meiner, 1982), pp. 10-12. See also Jean d'Alembert, Traite de dynamique (Paris, 1743), pp. 69-80.

53 See Truesdell, "Rediscovering the Rational Mechanics" (cit. n. 2), sec. 14. In the 1830s Gustave Coriolis discussed similar problems starting from the rotatory motion of machines, though in a more sophisticated way.

54 Leonhard Euler, Mechanica (St. Petersburg, 1736), in Euler, Opera omnia, Series II, Vols. I-II. ss Ibid., Vol. II, p. 15, def. 2; and Euler, "De motu corporum in superficiebus mobilibus" (cit. n.

49). This long paper is divided into three sections on motion in a tube moving parallel to itself, around a fixed axis, and without constraints. Compare pars. 16 and 41. See also Euler, "De motu corporum super superficies mobilibus," "De motu corporum in tubo rectilineo mobili circa axem fixum, per ipsum tubum transeuntem," and "De motu corporum in tubis circa punctum fixum mobilibus," Opera postuma, Vol. II, pp. 63-73, 74-84, and 114-124; in Euler, Opera omnia, Series II, Vol. VII, pp. 228-247, 248-265, and 308-326.


des forces" he claims that the origin of all forces is the impenetrability of bodies. With regard to the problem in question, he imagines a body moving along a curved surface; curvilinear motion due to the impenetrability of the surface causes in each point a centrifugal force, which is measured with respect to the center of the circumference osculating the curve in that point. It appears that for Euler this pattern was valid for all cases and that centrifugal force had to be measured along the osculating radius.56 Finally, in Theoria motus corporum solidorum seu rigidorum (1765), Euler clearly states that centrifugal force is due to the inertia of a body moving along a curvilinear path.57

The French historian Rene Dugas credited Clairaut with the discovery of the fictitious character of centrifugal force; however, this attribution needs to be revised and set in a wider context.58 Clairaut's essay on the topic opens with a short introduction stating that the problems dealt with had been posed by the Bernoullis and Euler. The remainder is divided into five articles, or sections. The first treats relative motion with respect to moving surfaces; sections 2-4 are devoted to the conservation of living force, mechanics of the rigid body, and motion of a system of interacting bodies; and the fifth contains a series of examples to elucidate the preceding principles.59 In the first section Clairaut considers a rectangle FGHI moving along the curves AB and CD, as in Figure 4. The problem is to determine the force acting on the body M as a result of the motion of the plane on which M is placed. Clairaut finds that the body on the plane FGHI experiences an acceleration MT equal and opposite to the acceleration MS required by the body supposed unconstrained to traverse the curve PQ, the curve traversed if M were fixed on FGHI. But this solution is not satisfactory. In fact, the incompleteness of Clairaut's attempt emerges very clearly from para- graphs 4 and 5, where he deals with the problem of the rotating tube. There he sets the force perpendicular to the direction of the radius equal to y ddrldt2, where y is the radius and r is proportional to the angle of rotation, and neglects the complementary or Coriolis acceleration. Centrifugal force is set equal to y dr2ldt2, but its cause seems to be related to the rotation of the body in the tube. This emerges further on as well, where Clairaut writes the equation with respect to an arbitrary point for a case that could be interpreted as a body moving in uniform rectilinear motion; however, he interprets it in terms of angles without any reference to motion and centrifugal force. For Clairaut, centrifugal force was still dependent on the true circular motion of a body.60

56 Leonhard Euler, "Recherches sur l'origine des forces," Histoire de l'Academie Royale des Sciences et Belles Lettres de Berlin, 1750 (1752), 6:419-447; in Euler, Opera omnia, Series II, Vol. V, pp. 109-131, esp. pp. 128-131. See also the introduction by Clifford Truesdell, ibid., Vol. XII, pp. xlii-xliv. On absolute space and time see Euler, "Reflexions sur l'espace et le tems," Hist. Acad. Roy. Sci. Belles Lettres Berlin, 1748 (1750), 3:324-333. On force see J. R. Ravetz, "The Representa- tion of Physical Quantities in Eighteenth-Century Mathematical Physics," Isis, 1961, 52:7-20; and Thomas L. Hankins, "The Reception of Newton's Second Law of Motion in the Eighteenth Cen- tury," Archives Internationales d'Histoire des Sciences, 1967, 78-79:43-65.

57 Leonhard Euler, Theoria motus corporum solidorum seu rigidorum (Rostock/Greifswald, 1765), in Euler, Opera omnia, Series II, Vol. III, esp. p. 97.

58 Rene Dugas, Histoire de la mechanique au XVIIe siecle (Neuchatel: Editions du Griffon, 1954), pp. 299-300; and Dugas, History of Mechanics (cit. n. 2), p. 370.

59 Alexis-Claude Clairaut, "Sur quelques principes qui donnent la solution d'un grand nombre de problemes de dynamique," MWm. Acad. Roy. Sci., 1742 (1745), pp. 1-52; and Clairaut to Euler, 28 Dec. 1742, 23 Apr. 1743, in Euler, Opera omnia, Series IV: Commercium epistolicum, Vol. V, pp. 144-148.

60 Clairaut, "Sur quelques principes," lemma 2, p. 25. A similar instance can be found in Johann


The idea that centrifugal force is fictitious emerges unmistakably in a memoir by Daniel Bernoulli on a problem posed by Euler: to find the motion of a tube rotating around a fixed point and containing a body freely moving in it.6' Daniel considers the problem of a straight tube rotating on a horizontal plane, general- izes it to the case of a tube filled with many bodies, and proves that his result is in accordance with the principle of conservation of living force. The reflection on centrifugal force is to be found in the first thirteen sections of the essay. In sections 14 and 15 he applies the preceding result to the case of a rotating tube; he also refers to correspondence he had on this matter with Clairaut. At the

Figure 4 B

A G

F

beginning Daniel defines three preliminary notions: circular motion as the motion along the arc of a circumference; centrifugal motion as motion perpendicular to circular motion; and momentum of circular motion, following Euler, as the prod- uct of velocity of rotation-which is the velocity of circular motion-radius, and mass. Equipped with these notions, he finds a series of results: the conservation of momentum of circular motion, or simply momentum, of a body moving along a straight line with a uniform velocity;62 the integral expression for the momentum of a rotating tube; and the conservation of the momentum for the tube-body system.

In section 12 Daniel attains the expression for the centrifugal velocity with respect to a point A for a body moving with constant velocity along a straight line:

dv = (VVIy)dt.

Bernoulli, "De curva quam describit corpus" (cit. n. 50). The last problem in Clairaut's essay is a generalization of that of the body in the tube; see the way in which the principle of relative motion is applied (pp. 48-52, esp. p. 49).

61 Daniel Bernoulli, "Noveau probleme de mechanique," Hist. Acad. Roy. Sci. Belles Lettres Ber- lin, 1745 (1746), 1:54-70. See also Daniel Bernoulli to Clairaut, in Correspondance, ed. Fuss (cit. n. 49), pp. 488, 497-504, 511, 525-526, 533, 539-540, 549.

62 An equivalent result concerning the area swept out by the radius from a fixed point to a body moving of inertial motion is proved at the beginning of prop. 1 in Newton's Principia.


V is the circular velocity CE perpendicular to the radius, and y is the distance from the fixed point A (see Figure 5).63 This result is relevant to us: centrifugal velocity has to be defined with respect to an arbitrary point, because a straight line has no center of force or osculating circumference. This clearly shows how centrifugal force results from a choice and is not determined by the data of the problem. Consequently, centrifugal motion must be calculated in an abstract "al- gorithmic" way. It is particularly interesting to see Daniel's interpretation of his own result in the following section. He explains that since the expression for centrifugal force is known, one could immediately attain from this his equation dv = dt V2/y. He says that while his reasoning retrospectively confirms the truth of his result, it was not at all clear at the beginning what the outcome would be. He thought the reason for this was that the short segment CE is not considered in the proof; in the measure of centrifugal force, however, CE must be seen as a small arc of a circumference with center A, and the reason for this was not immediately clear. In my opinion the cause of these worries is that the author was dealing with centrifugal motion without curvilinear motion, and this was not at all obvious.

In the same section Daniel claims that his proof can be useful in other cases, such as the trajectory around a center of force; no doubt he had planetary motion in mind here. Calling n the attractive force, he claims that the following equation can shorten the solution to some problems:

dv = (VVIy - 7t) dt.

The reader has certainly recognized the similarity to Leibniz's equation of paracentric motion, reproduced in Section II above. Leibniz, however, did not free himself completely from the intuitive relation between circular motion and centrifugal force, whereas Daniel Bernoulli reaches a considerable level of aware- ness. It is worth examining the vocabulary used in his essay: we have already seen that Daniel is obliged to talk of centrifugal force with respect to an arbitrary point; in section 13 he mentions the "utility" of his proof and says that it can shorten the solution of certain problems. Once again, this implies the existence of a plurality of possible descriptions of motion with different forces acting. The stress here is not on the discovery of a law of nature, but on the art of mathematical representation. Centrifugal force has changed location; disappearing from nature, it has become the result of a choice of the observer. The analogy with the problem analyzed by Newton in his attack against Leibniz emphasizes the gulf between Newton's and Daniel's ideas. Similar views can also be found in the main text on mechanics in the second half of the eighteenth century, the Me- chanique analytique. Joseph Louis Lagrange explicitly states that centrifugal force-with respect to the center of the osculating circumference-depends on the rotation of a system of perpendicular axes.64

63 The centrifugal velocity with respect to A is (BCIAC)c = c(yy - aa)Iy, where AC = y, AB = a, and c is the speed along BD; dv = aacdylyy(yy - aa); aly = Vlc. Substituting one has dv = VVdylc(- aa); dt = CD/c = ydylc(yy - aa) and c(yy - aa) = ydyldt; substituting again one has dv = (VVly)dt. (Notation has been altered from the original so as to run online.)

64 J. L. Lagrange, Mechanique analytique (Paris, 1788), pp. 162-165. Cf., however, ibid., 2nd ed. (Paris, 1811), pp. 225-226.


Priority issues or fundamental dates are not at stake here: the relativization of centrifugal force was a process rather than an isolated event. We have already seen how close to Daniel's views Leibniz came in his Tentamen and in his letter to Hermann. Johann Bernoulli, Euler, and Clairaut also considered force relative to the state of motion of the system and provided alternative mathematical representations of motion, although for them centrifugal force remained linked to the curvilinear motion of the body, as opposed to the motion of the observer. On the other hand, the issue was certainly not closed in 1746. In the Encyclope'die,

B C F D

A Figure 5

under the heading "Forces centrales et centrifuges," d'Alembert gives a Carte- sian account of centrifugal force. Referring to the example of the sling, the French mathematician claims that centrifugal force depends on the curvilinear motion of a body and on its inertia. Moreover, he gives an incorrect rule for measuring centrifugal force along radii drawn from different centers.65 D'Alem- bert's claim emphasizes yet again the problems related to the notion of centrifugal force.

In this essay I have emphasized the gulf between Newton's interpretation of centrifugal force in terms of the third law of motion and more modern interpretations, especially Daniel Bernoulli's. I have also argued that some version of Newton's analysis of curvilinear motion prevailed not despite the usage of the third law, but because of it. The reception of Newton's analysis is certainly a broader and more complex problem. I hope that my contribution may highlight some of the difficulties pertaining to these matters and stimulate further research in this area.

65 EncyclopMdie, Vol. VII (Paris, 1757). The transformation rule given by d'Alembert is based on the measure of centrifugal force along the osculating radius (velocity squared over the osculating radius). In his opinion a measure along a different radius would be the projection of the "standard" one, via the cosine rule.

Article Contentsp. 23p. 24p. 25p. 26p. 27p. 28p. 29p. 30p. 31p. 32p. 33p. 34p. 35p. 36p. 37p. 38p. 39p. 40p. 41p. 42p. 43

Issue Table of ContentsIsis, Vol. 81, No. 1 (Mar., 1990), pp. 1-180Front Matter [pp. 1-7]Gregory of Tours, Monastic Timekeeping, and Early Christian Attitudes to Astronomy [pp. 8-22]The Relativization of Centrifugal Force [pp. 23-43]Stereochemistry and the Nature of Life: Mechanist, Vitalist, and Evolutionary Perspectives [pp. 44-67]NotesSunspots, Galileo, and the Orbit of the Earth [pp. 68-74]

Letters to the Editor [pp. 75-76]Essay ReviewsReview: Science and Religion [pp. 77-80]Review: Technology and Society [pp. 80-83]

Book ReviewsGeneral WorksReview: untitled [pp. 84-85]Review: untitled [p. 85]Review: untitled [pp. 85-86]Review: untitled [pp. 86-87]Review: untitled [pp. 87-88]Review: untitled [pp. 88-89]Review: untitled [pp. 89-90]Review: untitled [pp. 90-91]Review: untitled [pp. 91-92]Review: untitled [pp. 92-93]Review: untitled [p. 93]Review: untitled [pp. 93-94]

AntiquityReview: untitled [pp. 94-95]Review: untitled [pp. 95-96]Review: untitled [pp. 96-97]Review: untitled [pp. 97-98]Review: untitled [pp. 98-99]

Middle Ages & RenaissanceReview: untitled [p. 99]Review: untitled [pp. 100-101]Review: untitled [pp. 101-102]Review: untitled [pp. 102-103]Review: untitled [pp. 103-104]

Early Modern PeriodReview: untitled [pp. 104-105]Review: untitled [pp. 105-107]Review: untitled [pp. 107-108]Review: untitled [p. 109]Review: untitled [pp. 109-110]Review: untitled [pp. 110-111]

Eighteenth CenturyReview: untitled [pp. 111-112]Review: untitled [p. 113]Review: untitled [pp. 113-114]Review: untitled [pp. 114-115]Review: untitled [pp. 115-116]Review: untitled [pp. 116-117]

Nineteenth CenturyReview: untitled [pp. 117-118]Review: untitled [pp. 118-119]Review: untitled [pp. 119-120]Review: untitled [pp. 120-122]Review: untitled [pp. 122-123]Review: untitled [p. 124]Review: untitled [pp. 124-125]Review: untitled [pp. 125-127]Review: untitled [pp. 127-128]

Twentieth CenturyReview: untitled [pp. 128-129]Review: untitled [pp. 129-130]Review: untitled [pp. 130-131]Review: untitled [pp. 131-132]Review: untitled [pp. 132-133]Review: untitled [pp. 133-134]Review: untitled [p. 134]Review: untitled [pp. 134-135]Review: untitled [pp. 135-136]Review: untitled [pp. 136-137]Review: untitled [pp. 137-138]Review: untitled [pp. 138-140]Review: untitled [p. 140]Review: untitled [p. 141]Review: untitled [pp. 141-142]Review: untitled [pp. 142-144]Review: untitled [pp. 144-145]Review: untitled [pp. 145-146]Review: untitled [pp. 146-147]Review: untitled [pp. 147-148]

Philosophy & Sociology of ScienceReview: untitled [pp. 148-149]Review: untitled [pp. 149-150]Review: untitled [pp. 150-151]Review: untitled [pp. 151-152]Review: untitled [pp. 152-153]Review: untitled [pp. 153-154]Review: untitled [pp. 154-155]Review: untitled [pp. 155-156]

Reference ToolsReview: untitled [pp. 156-157]Review: untitled [pp. 157-158]Review: untitled [pp. 158-159]Review: untitled [p. 159]Review: untitled [pp. 159-160]Review: untitled [pp. 160-161]Review: untitled [pp. 161-163]Review: untitled [p. 163]Review: untitled [p. 164]

Collections [pp. 164-173]

Back Matter [pp. 174-180]

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