-
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
Your use of the JSTOR archive indicates your acceptance of
JSTOR's Terms and Conditions of Use, available at
.http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's
Terms and Conditions of Use provides, in part, that unlessyou have
obtained prior permission, you may not download an entire issue of
a journal or multiple copies of articles, and youmay use content in
the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this
work. Publisher contact information may be obtained at
.http://www.jstor.org/action/showPublisher?publisherCode=ucpress.
.
Each copy of any part of a JSTOR transmission must contain the
same copyright notice that appears on the screen or printedpage of
such transmission.
JSTOR is a not-for-profit service that helps scholars,
researchers, and students discover, use, and build upon a wide
range ofcontent in a trusted digital archive. We use information
technology and tools to increase productivity and facilitate new
formsof scholarship. For more information about JSTOR, please
contact [email protected].
The University of Chicago Press and The History of Science
Society are collaborating with JSTOR to digitize,preserve and
extend access to Isis.
http://www.jstor.org
-
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 abso- lute 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 con- sider 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 centrif- ugal 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 no- tion, 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 cen- trifugal 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 interpreta- tions 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 mo- tion. 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 centri- fuga 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 dif- ferent 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
-
26 DOMENICO BERTOLONI MELI
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 ma- thematica, 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 centri- fuga,
in "Christiaan Huygens' Theory of Evolutes: The Background to the
'Horologium oscillator- ium'" (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.
-
CENTRIFUGAL FORCE 27
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 poly- gon 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 expres- sion 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).
-
28 DOMENICO BERTOLONI MELI
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 alter- nately 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.
-
CENTRIFUGAL FORCE 29
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 cen- tripetal
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 scho- lium 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 contin- ually 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 de- scribed 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.
-
30 DOMENICO BERTOLONI MELI
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 plau- sible, 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 mo- tion 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
uni- form. With respect to rotations, for Newton there is only one
possible represen- tation 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.).
-
CENTRIFUGAL FORCE 31
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 compo-
nent 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 circum- ference.
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; out- ward 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 equiva-
lent 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 repre- sentations 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
centrif- ugal 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,
-
32 DOMENICO BERTOLONI MELI
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 repre- sents both the latus rectum of the ellipse and
the proportionality constant be- tween 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 grav- ity 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
explana- tion 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.
-
CENTRIFUGAL FORCE 33
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-be- lieved that orbital motion depended on the imbalance
between gravity and cen- trifugal 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 de- cades 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 equa- tion 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 (ficti- tious)
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 quan- tum 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.
-
34 DOMENICO BERTOLONI MELI
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 posi- tions
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 cen- trales," 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.
-
CENTRIFUGAL FORCE 35
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 centrip- etal 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 pas- sage, 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-
-
36 DOMENICO BERTOLONI MELI
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
ten- dency 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 centrif- ugal 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 ex- plained in the Horologium. The curve
described by the body is always perpendic- ular 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-math- ematique," 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.
-
CENTRIFUGAL FORCE 37
In this case gravity and centrifugal force have a component in
the same direc- tion. In his correspondence with Varignon, however,
Johann discusses centrifu- gal 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 simi- lar theory was
adopted by Hermann in the Phoronomia with respect to the co- natus
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
centripe- tal 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.
-
38 DOMENICO BERTOLONI MELI
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 men- tioned
explicitly.50
I have tried to show that in the first decades of the eighteenth
century centrifu- gal force was interpreted in a number of ways.
Some form of Newton's interpre- tation 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 dif- ferent 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 cen- trifugal 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
superfi- ciebus 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.
-
CENTRIFUGAL FORCE 39
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 centrif- ugal 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 mathe- maticians involved
often needed material supports for thinking in terms of refer- ence
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 cen- trifugal 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 cen- trifugal 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 exam- ple, 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.
-
40 DOMENICO BERTOLONI MELI
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 soli- dorum 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 exam- ples 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
-
CENTRIFUGAL FORCE 41
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.
-
42 DOMENICO BERTOLONI MELI
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 para- centric motion, reproduced in Section II above.
Leibniz, however, did not free himself completely from the
intuitive relation between circular motion and cen- trifugal 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 mathemati- cal representation.
Centrifugal force has changed location; disappearing from na- ture,
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.
-
CENTRIFUGAL FORCE 43
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 repre- sentations 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 centrifu- gal 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 interpreta- tions, 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]