Pierre Varignon and the measurement of time/Pierre Varignon et la mesure du temps - article ; n°3 ; vol.50, pg 361-368
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Pierre Varignon and the measurement of time/Pierre Varignon et la mesure du temps - article ; n°3 ; vol.50, pg 361-368


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Revue d'histoire des sciences - Année 1997 - Volume 50 - Numéro 3 - Pages 361-368
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Source : Persée ; Ministère de la jeunesse, de l’éducation nationale et de la recherche, Direction de l’enseignement supérieur, Sous-direction des bibliothèques et de la documentation.


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Published 01 January 1997
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Pierre Varignon and the measurement of time/Pierre Varignon et
la mesure du temps
In: Revue d'histoire des sciences. 1997, Tome 50 n°3. pp. 361-368.
Citer ce document / Cite this document :
GOWING RONALD. Pierre Varignon and the measurement of time/Pierre Varignon et la mesure du temps. In: Revue d'histoire
des sciences. 1997, Tome 50 n°3. pp. 361-368.
doi : 10.3406/rhs.1997.1297
http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1997_num_50_3_1297Pierre Varignon
and the measurement of time
Ronald Gowing (*)
The work of Pierre Varignon (1654-1722), priest and scientist, géo
mètre in the Académie royale des sciences de Paris, correspondent of
Leibniz, Newton, the Bernoullis and many others, deserves wider atten
tion. He was one of those mathematicians who, in an age of slower
communication, were so important in the diffusion of new ideas. He
learned the differential and (some) integral calculus from Jean I Bernoulli,
initially during the latter's youthful visit to Paris in 1698, and thereafter
in the course of a lifelong and fruitful correspondence (1). Varignon' s
importance in the diffusion of the differential calculus in France has
been discussed by Pierre Costabel (2) and the importance of his influence
in the spread of Newtonian ideas on the continent of Europe has been
recognized by Henry Guerlac (3). He contributed papers, often of major
significance, to every volume of the Mémoires of the Paris Academy
from 1699 to 1721.
I have chosen to illustrate Varignon' s work by discussing what, on
the face of it, is a modest little paper dealing with a minor technical
problem, but which, like so much of Varignon's work, is related to a
major problem of the day. The major problem of the day is that of
finding longitude at sea, in particular by improving the accuracy of spring-
driven portable clocks; and the technical problem is to find the correct
shape for the fusee — part of the regulating mechanism. The mathemat
ical interest, is in deriving and solving differential equations with sepa
rated variables, and in the integration of binomial differentials.
The fusee is a device for compensating the declining power of a coiled
(*) Ronald Gowing, The Royal Institution Centre for the history of science and techno
logy, 21 Albemarle Street, London W1X4BS, Grande-Bretagne.
(1) Der Briefwechsel von Johann I Bernoulli, ed. by David Speiser, Band 2 : Der Brief
wechsel mit Pierre Varignon (Basel-Boston-Berlin : Birkhàuser Verlag, 1988).
(2) Pierre Costabel, Pierre Varignon et la diffusion en France du calcul différentiel
et intégral (Paris : Palais de la Découverte, 1965, publication D10).
(3) Henry Guerlac, Newton on the continent (Ithaca and London : Cornell Univ. Press,
Rev. Hist. Sri., 1997, 50/3, 361-368 Ronald Gowing 362
spring as it unwinds (fig. 1). The coiled spring S is housed in a barrel В
which turns as the spring uncoils. A thread T wound on the is
also wound on a conical sleeve F attached to the spindle P of the main
wheel of the going train of the clock. F is the fusee. As the spring uncoils,
the barrel is turned and the attached thread turns the fusee.
Fig. l.
When the spring is fully wound, the point of contact of the thread is
at X. As the uncoils it gets weaker, and the point of contact moves
up to Y. The declining tension in the thread is offset by the increasing
diameter of the fusee, thus ideally maintaining a constant moment about
the spindle, and so a constant motion of the clock. The problem is to
match the shape of the fusee to the law of variation in the tension of
the spring, transmitted by the string.
Fusees had been in use from as early as the fifteenth century, Leo
nardo Da Vinci drew suggested profiles. See e. g. Brusa (4). Robert Hooke
of course was aware of the problem, although not apparently of its diffi
« The Rule or Law of Nature in every springing body is that the force or
power thereof to restore it self to its natural position is always proportionate
to the distance or space it is removed therefrom...
« From this Principle also it will be easier to calculate the proportionate strength
(4) Guiseppe Brusa, Early mechanical horology in Italy, Antiquarian Horology, IS (1990),
495-513. Varignon and the measurement of time 363
of the Spring of a Watch upon the Fusey thereof and consequently of adjusting
the Fusey to the Spring so as to make it draw or move the Watch always with
an equal force (5). »
Varignon' s day fusee cutting machines which modelled the physical By
conditions of the problem were in general use. These still required great
skill of hand and eye, but the spring-driven portable clocks which resulted
were accurate enough for everyday purposes.
Varignon began his attack on the problem by assuming that the length
of string wound on the fusee would be a measure of the tension in the
spring, being as it were a measure of the extension (Hooke's tensio of
the spring). If the string were sufficiently flexible and closely wound,
the surface area covered by the string would be a measure of its length.
Taking coordinate axes x and y as shown (fig. 2) with dv as an element
of arc, we seek a curve such that y x tension = constant,
i. e. y \ - ydv = constant. (c = 2na, n not yet in use.)
x -*
Fig. 2.
(5) Robert Hooke, Lectures de potentia restitutiva (London, 1678), 4-5. 364 Ronald Gowing
The constant is chosen to be am1 where a is the initial radius and m2
the corresponding initial area of the fusee. Varignon follows the pract
ice, already widely going out of use, of keeping his quantities homogen
eous. Substituting
dv2 = dx2 + dy2
the resulting differential equation of the profile of the fusee is
Varignon noted that from this equation, y could not exceed *Ja2m2/c but
that from the defining property of the curve, as the area of the surface
approached zero, y would approach infinity.
Early in 1698, without mentioning the source of the problem he wrote
to Bernoulli about it (6). Bernoulli's response (7) was that the curve was
one of a type, Leibniz' isochrone (8) for example, being another, which
must be considered not at the origin, but at a little distance from it.
In the particular case of Varignon's curve, the logical difficulty could
be overcome by extending the curve AG by a straight line GH perpendi
cular to the axis, so that at G the surface does not become zero, but
is the annulus described by GH on rotation about the axis (fig. 3).
Fig. 3.
In his letter to Bernoulli, Varignon gave equation (A) without the minus
sign — as Costabel has pointed out — (9), but corrected this in later work.
(6) Op. cit. in n. 1, Bd 2, 159-160.
(7) Ibid., 166, 167.
(8)163, note 20.
(9) Ibid., 160, note 5. Varignon and the measurement of time 365
If the source of the problem had been mentioned to Bernoulli, it
is unlikely he would have let the assumption about the law of tension
pass without comment. Later in 1698 Varignon read a paper before the
Académie, giving the equation (A) in the form
and this is where he begins his published paper of 1702, « De la figure
ou curvité des fusées des horloges à ressort » (10).
Varignon begins by stating that so far as he knows, there is not yet
any satisfactory law of the way in which the tension in a coiled spring
varies as the spring uncoils. He has therefore looked at the problem
again, assuming the tension to vary as some power of the length of string
wound on the fusee, still taking area covered as a satisfactory measure
of length. Thus
УСт Г m
l^dv = constant (C)
is the defining equation (11), in which m is now any constant, and the
constant is chosen to be am . Replacing for dv as before, the
required differential equation is written as
Varignon immediately considers the conditions under which « the curve
will be geometric » /. e. in modern terms the conditions under which the
- In - 1
binomial can be rationalized. He gives these as m = , where n
4n + l -2/1-3 L . ..
is a positive integer, or where л is a positive integer or zero.
4/1 + 5
(10) Histoire et Mémoires de l'Académie royalendes ; science sciences, 1702 (Paris, 1704), 193-202.
(11) Modern writing of this equation : yl -\ydv v V I = constant.
VO / 366 Ronald Gowing
m = Varignon -2n-2 further and gives adds a that result the which curve he will attributes also to be Bernoulli, « quarrable namely » if
An + 5
-2л -3 -/2-1
m = or m = n being a positive integer or zero in
4л+4 In + 3
all three cases.
Substantial progress had been made in the solution of such different
ial equations involving trigonometric and logarithmic functions, by Vari-
gnon's younger contemporary Roger Cotes, first Plumian Professor of
experimental philosophy in the university of Cambridge, but this was
not published until 1722 (12), the year of Varignon's death. Varignon
is content to point out that there are conditions under which the expres
sion to be integrated can be rationalized, and to press on with his stated
intention — to construct the curve, i. e. the profile of the fusee. His
construction, a piece of fairly typical eighteenth century « geometrical
analysis » assumes that the integration can be done.
R E(t,y)
Fig. 4.
(12) Roger Cotes, Harmonia mensurarum, ed. by Robert Smith (Cambridge, 1722).
The Plumian Professor is the holder of the professorial chair in astronomy and experi
mental philosophy, founded by Dr Thomas Plume, archdeacon of Rochester, in 1707 and
tenable at Trinity College, Cambridge. Roger Cotes was the first Plumian Professor. The
appointment of course, continues to be made. Varignon and the measurement of time Ъ61
The first step is to rewrite equation (D) in the form
adx = - y/t2 - (P--áy
and construct the (t, y) curve QEO in which RE = t, AR = у (fig. 4).
By a simple construction, details of are not important, each point
E(t, y) is mapped on to a point N(Vř2 - a2, y) and the N curve GNO '
is drawn. The area GNR — the integral of the right hand side of equa
tion (D) — is, says Varignon, always « quarrable » when the sought curve
GBC is « geometric ».
The next step is to construct a rectangle AFMP, equal in area to
GRN and in which AP = a, AF = x. MF is drawn to meet NR in B.
The two areas GRN and AFMP being equal, their differentials are equal.
Therefore at В d(AFMP) = d(GRN),
i. e. d(ax) = d(\y/t2 - a2-dy),
a-dx = vr - cr-dy.
Thus В lies on the required curve.
Varignon is aware of the uncertainties in the result, but he is illustra
ting a process /. e. we could find the precise profile if only 1° we knew
the law of variation of tension, and 2° we could do the integrals. This
is a counsel of perfection. Even as late as the mid-nineteenth century,
when Swiss and other continental watchmakers had abandoned the fusee
(but British watchmakers persisted with it, to the detriment of their inter
national competitiveness), it was necessary, in very accurate work, to
make small adjustments to the fusee to accommodate imperfections and
inaccuracies in spring manufacture. But Varignon, as ever, is lured on
by the mathematics, although acknowledging this tendency.
Suppose, he says, that the volume of the fusee is a better measure
of spring tension, or what about the plane area under the fusee curve,
rather than surface area? He is able to derive the equations of the fusee
curve, in both cases a hyperbola, but adds :
« Neither area nor volume represents the length of the string. It is nearer
to the truth to say that surface area does represent the length of the string,
assumed to be very flexible and closely wound. If we are not to be in agreement
with mathematics rather than nature, we should retain the hypothesis that spring
tension is as some power of the length of string wound on to the fusee. [But
he cannot resist adding : Whilst waiting for experiment to determine the spring
law, here are two further examples, one geometric and one mechanical (13). »
(13) Op. cit. in n. 10, 200-201. 368 Ronald Gowing
The general law is that if y represents the radius of the fusee at a given
point, and z the tension in the string, then za - constant is the required
equation of the fusee. Suppose the tension in the string varies according
to a circular law, i. e. suppose z = lax - x2, then
laxy2 - x2y2 = a2b2 (E)
is the generating curve of the fusee. Suppose the tension varies as the ordinate
of a cycloid of which the circle above is the generating circle. From the defi
ning equation^ = ab, we have dz = (- ab/y2)dj\ Varignon uses the diffe
rential form of the equation of the cycloid (perhaps because the integrated form
contains an inverse cosine) giving
which Varignon again calls a generating curve. He does not solve the
In both of the above examples, since nothing is said to the contrary,
it must be assumed that x is distance measured along the axis of the
fusee, as in equations (A) and (B). A spring law giving the tension in
terms of such a variable seems most unlikely, and rather presupposes
knowledge of the shape of the fusee to start with.
A modern expression for the tension F in a coiled spring as it unwinds is
F = Ehe3nn/6 (14)
where F is the force or tension, E the length of the spring, h its height
or width, e and n have their usual meanings, n is the number of turns
initially — the number of turns currently. For a given spring, n is the
only variable, and this lends some support to Varignon.
As mentioned earlier, the fusee remained in use throughout the eigh
teenth century, and to some extent into the nineteenth. Perhaps it would
be suitable to finish with a remark attributed to Pierre Le Roy by Ferdi
nand Berthout : « La fusée est une des plus belles inventions de l'esprit
humain (15). »
(14) Leonard Weiss, Watchmaking in England, 1780-1820 (London : Hale, 1982).
(15) Ferdinand Berthout, Histoire de la mesure du temps par les horloges, 2 vol. (Paris,
1802), vol. 1, 77.