NEWTON’S USE OF THE PENDULUM TO INVESTIGATE FLUID RESISTANCE: A
CASE STUDY AND SOME IMPLICATIONS FOR TEACHING ABOUT THE NATURE OF
SCIENCE
COLIN F. GAULD
School of Education, University of New South Wales, 9 Michael Crescent, Kiama Downs, NSW,
2533, Australia
Email: cgauld@smartchat.net.au
Abstract. Books I and III of Newton’s Principia develop Newton’s dynamical theory and show how it explains a
number of celestial phenomena. Book II has received little attention from historians or educators because it does not
play a major role in Newton’s argument. However, it is in Book II that we see most clearly Newton both as a
theoretician and an experimenter. In this part of the Principia Newton dealt with terrestrial rather than with celestial
phenomena and described a number of experiments he carried out to establish the success of his theory in explaining the
properties of fluid resistance. It demonstrates most clearly the activities of a scientist working at the forefront of
knowledge and working with phenomena which he did not fully understand.
In this paper the first of Newton’s set of experiments into fluid resistance are described and the theory which
underlies his explanation is outlined. A number of issues arising from this portion of the Principia together with
implications for teaching about the nature of science are discussed.
Keywords. Newton, fluid resistance, pendulum, nature of science, history of science
INTRODUCTION
The Principia represented a new way of thinking about physical phenomena and in it Newton
presented a number of analyses of a variety of phenomena − terrestrial motion, fluid motion and
resistance, sound, light, and celestial motion. These were marshalled to provide one large argument
for his new approach. The ultimate test of this approach (presented in Book I) was its ability to
explain the motion of a variety of celestial objects such as planets, satellites and comets (presented
in Book III). In presenting an outline of Newton’s argument in the Principia many authors proceed
from the theory in Book I to the successful explanations of celestial phenomena in Book III largely
ignoring the contents of Book II (see, for example, Chandrasekhar 1995, 343-344; Densmore 1992).
However, one of Newton’s objectives was to demonstrate that the alternative system proposed
by Descartes for explaining this motion − his vortex model − was inadequate (Aiton, 1972; Koyré
1965, pp.79-138; Newton 1729/1960, Book II, Section IX; Book III, final General Scholium;
Westfall 1980, p.454)1. As part of this programme it was necessary to show the effect of fluids
such as air on the motion of objects since it was not possible in Newton’s day to carry out
substantial experiments on motion in a vacuum. Book II of the Principia is primarily concerned
with an investigation into the nature and properties of the resistance to motion produced by fluids
although other things are dealt with in that book as well. While in Book I Newton presented the
theoretical basis for his approach and in Book III he applied his results to the understanding of
celestial phenomena it is in Book II that we see clearly Newton as an experimenter. In two places
(in the General Scholium following Proposition 31 and in the discussion of Proposition 40) he
reported the results of many experiments in which he used his theory to explain motion through
fluids.
Truesdell (1968b) has pointed out that in Book II Newton was dealing with previously
unexplored territory and that the material in this part of the Principia “is almost entirely original”
(p.145) something which was not generally the case with Books I and III. Because of this it
contains “a bewildering alternation of mathematical proof, brilliant hypothesis, pure guessing, bluff,
and plain error” (p.149). However, Truesdell also argued that “the brilliantly ingenious but, in the
end, largely unsatisfactory Book II laid out the areas and defined the problems for many of the
mechanical researches of the next century” (1968a, p.91).
In this paper Newton’s approach to studying fluid resistance using pendulum motion is
outlined and his results are discussed. The simplicity of his experiments make them ideal
1
candidates for replication by present day students and his theory provides a clear example of the
way Newton thought about physical phenomena. The relationship between theory and experiment
in Newton’s thinking is also clearly illustrated.
The first twenty-nine propositions in Book II provide an analysis of one and two dimensional
motion under the influence of resistances with different dependences on the velocity of the moving
object. In Proposition 30 Newton presented the theory which would enable him to use the decay in
the swing of a pendulum to find out the general properties of fluid resistance.
NEWTON’S THEORY OF HIS FLUID RESISTANCE EXPERIMENTS
Stage 1 – Newton’s expectation
In Section I of Book II (Propositions 1 to 4) of his Principia Newton dealt with the properties of
resistance which varies directly with the velocity. In the Scholium at the end of that section Newton
wrote
However, that the resistance of bodies is in the ratio of the velocity, is more a mathematical hypothesis
than a physical one. In mediums void of all tenacity, the resistances made to bodies are as the square of
the velocities. (Newton 1729/1960, p.244)
In other words Newton did not believe that resistance was really proportional to velocity but, in the
ideal case, to the square of the velocity. He argued that the moving body collided with the
stationary medium (or particles of the medium), exerted a force on it and started it moving. If one
considers the medium to be a collection of individual particles the velocity, v, imparted to one of
these particles is proportional to the velocity, V, of the moving body. If the collision is perfectly
elastic the particle acquires almost twice the velocity of the moving body; if the collision is
perfectly inelastic the velocity of the particle is almost equal to that of the moving body (see
Newton’s discussion of Propositions 33 and 35). Thus a more rapidly moving body produces a
proportionally larger velocity in the particles of the medium than a more slowly moving body. The
time taken to transfer a given total amount of motion (momentum) is inversely proportional to the
velocity. Thus the resistance, which is proportional to the amount of motion lost by the moving
body per unit time is proportional to change in motion per unit time, that is to mv/(1/V) or V2 where
m is the mass of the medium set in motion.
-- FIGURE 1 -Reference to Figure 1 shows that in one second the body moves a distance V so that the
momentum transfer per second is equal to mv where m is the amount of medium contained in the
volume traversed by the moving body in one second. Thus, from Newton’s third law, the resistance
to the motion of the moving body is
R = mv ∝ (dAV)V ∝ dAV2
where d is the density of the medium and A is the cross-sectional area perpendicular to the direction
of motion.
Newton showed in Book II that he expected the major contribution to resistance to be
proportional to the square of the velocity, the area of the moving body perpendicular to the direction
of motion and the density of the medium (Scholium following Proposition 14; Proposition 33;
Proposition 35, Cases 1-3, Corollaries 1-5).
This of course is only in “mediums void of all tenacity”. In the Scholium at the end of Section
III of Book II Newton wrote
2
The resistance of spherical bodies in fluids arises partly from the tenacity, partly from the attrition, and
partly from the density of the medium. And that part of the resistance which arises from the density of the
fluid is, as I said, as the square of the velocity; the other part, which arises from the tenacity of the fluid, is
uniform. (Newton 1729/1960, p.280)
Newton treated uniform resistance in a similar fashion to the effect of gravity on an ascending
body and he suggested that “one might also go on to the motion of bodies which are resisted in part
uniformly, in part in the ratio of the velocity, and in part in the duplicate ratio of the same velocity.”
He provided examples of how this might be done in his discussion of Propositions 11 to 14.
Stage 2 – Newton’s construction
In discussing Proposition 30 Newton began his analysis of the problem of resistance to the motion
of a cycloidal pendulum by using the same framework he used when dealing with the velocity of
the pendulum bob in the case where there is no resistance (Book I, Proposition 52). In Figure 2 the
diameter, BA, of a semi-circle represents the position of the bob as it moves from its beginning
point, B, to its end point, A (see also Figure 6(a)). The centre of this path is C and D and d are
points somewhere on the path of the pendulum bob separated by a small time interval, Δt. Newton
showed, in Propositions 38 and 52 of Book I that the vertical line DE is proportional to the velocity
of the bob at the point D and de is proportional to the velocity of the bob at point d (see Gauld
2004). In Proposition 51 of Book I Newton had already shown that
CD Force on pendulum bob along its path
=
L
Weight of the bob
-- FIGURE 2 -Because of the resistance the end point of the first swing of the bob is the point a which falls
short of A by the distance Aa. At the point D the velocity, instead of being represented by DE is
represented by the smaller length DF and, if from this point on, the bob continued unresisted it
would reach the point M at the left side of its path. If resistance exists between D and d the velocity
at d is represented by the line dj so that the line fj represents the decrease in velocity between the
points D and d due to the resistance from the medium through which the bob is travelling. The
point N represents the position the bob would now reach if resistance were zero after the point d and
O is the centre of the motion of the bob when resisted for the whole of its movement from B to a.
The line DK is constructed so that
DK Force of resistance at D
=
L
Weight of the bob
Stage 3 – Newton’s analysis
Figure 3 shows more detail of the region of Figure 1 between C and D. CF and Cg are radial lines
from the centre, C, and g is the point where the arc Nj extended intersects the line DF. The radius
CF cuts the arc jg at h. Fm is parallel to CD and so is equal to Dd. Comparing Figures 2 and 3, CN
= Cg, MN = Fh, CM = CF and fm = df - DF.
-- FIGURE 3 --
3
It can be seen that, if the distance Dd is small, the three triangles, Fmf, Fhg and FDC are
similar. Because Fmf and FDC are similar
fm
fm CD
=
=
Fm Dd DF
Fg is proportional to the decrease in velocity due to the resistance and fm is proportional to the
increase in velocity due to gravity so that, from Newton’s first law
Decrease in velocity due to resistance
Force on bob due to resistance
=
Increase in velocity due to force of gravity
Force on bob due to gravity
Fg DK
=
fm CD
That is
Fg fm DK CD
.
=
.
fm Dd CD DF
Thus
Fg DK
=
Dd DF
so that
Because triangles Fhg and FDC are similar
Fh DF
=
fg CF
so that
Fh fg
DF DK
.
=
.
fg Dd CF DF
Thus
Fh DK
=
Dd CF
or
MN
DK
=
Dd
CM
When the whole motion is resisted ΣMN.CM = ΣDd.DK. The area of the shaded region DdkK
is equal to Dd.DK so that ΣDd.DK is the area of the whole region BCaVK. The area of the shaded
region MNRS, where CM = MR, is equal to MN.CM so that ΣMN.CM is equal to the area whole
trapezoidal region AaUW. The area of AaUW is equal to Aa.½(CB + Ca) = Aa. ½aB = Aa.OB =
area of BCaVK.
Stage 4 – Newton’s application
Case 1: R = constant
In the situation where R is a constant the region BCaVK is a rectangle with a length equal to Ba and
a height equal to DK so its area is Ba.DK. Thus
Aa. ½aB = DK.aB
4
So that
OV = DK = ½Aa.
where OV represents the resistance at the centre of the swing.
Thus
( 1 2 )Aa
Resistance at O
=
L
Weight of the bob
[1]
Case 2: R ∝ velocity
Newton argued that if the resistance were small the velocity between B and a could be represented
by the line DF in Figure 4 where BFa is a semicircle with diameter Ba. He had shown in the
Corollary to Proposition 25 that, for a constant resistance, the fastest velocity occurred at O, the
centre of the line Ba. If the resistance were proportional to the velocity then the height of the curve
BKkVa would be a constant fraction of the height of the semicircle BFa. In other words AKkVa is a
semi-ellipse with an area equal to ½πOV.OB = ½πOV. ½Ba.
-- FIGURE 4 -Thus
So that
In other words
½πOV. ½Ba = Aa. ½aB
OV =
2 Aa
!
"
7
11
Aa
( 711) Aa
Resistance at O
=
L
Weight of the bob
[2]
Case 3: R ∝ velocity2
If the resistance were proportional to the square of the velocity and the curve AFa is a semicircle
then the curve AKkVa is a parabola with an area 2 3 Ba.OV. Thus in this case OV = ¾Aa and
( 3 4 ) Aa
Resistance at O
=
L
Weight of the bob
[3]
Newton acknowledged that the curve BFa might not be a semicircle and so in cases 2 and 3 the
curves might not be exactly an ellipse and a parabola. However, he considered his conclusions [1],
[2] and [3] to be close enough for his practical purposes using the word “nearly” for “approximately
true”.
NEWTON’S EXPERIMENTS INTO FLUID RESISTANCE
In order to use his theory to determine the properties of fluid resistance as they are affected by the
velocity and size of the body which is moving and the density of the fluid Newton set up a long
pendulum and varied the properties of the bodies which made up the bob. He described the care
with which he designed the support towards the end of the Scholium and explained the need for a
support such as a knife edge which did not inhibit the pendulum swing. He was able to vary the
velocity at the lowest point by altering the distance the bob was pulled back from its lowest point
(Gauld 2004). His procedure was to pull back the bob a known distance and count the number of
5
swings (from one side to the other) for the amplitude to reduce by one eighth or one quarter. He
then repeated the experiment by doubling the initial amplitude until he had a set of results in which
the initial amplitudes formed a geometric series. Newton determined the size of Aa for the first
swing by subtracting the size of the final ascent from the size of the initial descent and dividing by
the number of swings. Using his expression [1], [2] or [3] enabled him to use his value of Aa to
find the resistance at the lowest point in that swing.
Newton argued that the error in treating the circular pendulum as a cycloidal pendulum was
negligible. For a cycloidal pendulum the velocity at the lowest point of the swing is proportional to
the distance along the arc BC while for a circular pendulum it is proportional to the length of the
chord BC (Gauld 1998; 1999). The path of the bob of a circular pendulum has a constant curvature
while that of a cycloidal pendulum of the same length begins at the bottom with the same curvature
as the circular pendulum but this curvature decreases as the angle of the string increases. The
oscillating string length decreases as the string moves against the cycloidal cheeks. Thus, if the
length of the arc from which the bob starts its downward journey is the same for the two pendulums
the starting point for the cycloidal pendulum is higher and the velocity of the cycloidal pendulum is
larger when the bob reaches the lowest point.
For a circular pendulum the starting height is given by
h=
(chord ) 2
2L
For a cycloidal pendulum the starting height is given by
(arc) 2
h=
2L
Since the starting height is proportional to the square of the velocity at the bottom the ratio of the
velocity for the circular pendulum to that of the cycloidal pendulum is equal to (chord/arc).
Newton claimed that the ratio of the periods for these pendulums is (arc/chord). However, if
one compares these ratios one finds they are only approximately equal2. For example, for a starting
angle of 300 the ratios differ by about 0.5% (Kidd & Fogg 2002). Newton did not justify his claim
but an argument he might have used is the following. The pendulum bobs both start at rest and they
travel the same distance along almost identical paths. (The height above the lowest point at the
beginning for the circular pendulum is 17.21 inch while that for the cycloidal pendulum is 17.62
inch a difference of only 0.41 inch.) Newton may have then simply concluded that the time taken is
inversely proportional to the final velocity.
Newton’s conclusion from his discussion of Proposition 30 was that the decrease in amplitude
after one swing of the pendulum due to resistance was proportional to the resistance x length of the
pendulum (equations [1], [2] and [3]). In the General Scholium he restated this in a form which
referred only to the properties of the motion of the bob and not to the pendulum. In this form the
decrease in amplitude is proportional to the resistance x (period)2. Assuming that the resistance at
the bottom was proportional to V2 this means that the decrease in amplitude was proportional to V2 x
T2. However, Newton had already stated that V(cycloid)/V(circle) = (arc/chord) and that
T(cycloid)/T(circle) = (chord/arc) so his conclusion was that the decreases in the amplitude for a
circular pendulum and a cycloidal pendulum with the same length and the same amplitude were
“nearly” equal. Thus he believed that his use of circular rather than cycloidal pendulums did not
introduce substantial errors into his experiments.
-- TABLE 1 -Newton’s data for the 43 experiments he reported in the Scholium following his discussion of
Proposition 31 are given in Table 1 and a graphical summary is presented in Figure 5. His unit of
6
distance was the English inch (to be distinguished from the Paris inch) 12 of which make up an
English foot which is equal to 0.305 metre. His unit of weight (or mass) was the troy ounce which
is equal to 28.35 grams. He carried out three main series of experiments to demonstrate
respectively the dependence of the resistance on (a) the velocity of the bob, (b) the diameter of the
bob and (c) the density of the fluid.
-- FIGURE 5 -DEPENDENCE ON THE VELOCITY OF THE BOB (Cases 1-12)
Newton used cases 1-6, in which he measured the number of swings required for the pendulum
amplitude to decrease by one eighth, to investigate the dependence of the resistance on velocity.
The pendulum length, L, was 126 inches and the pendulum bob had a diameter of 6.875 inches and
a mass of 57.318 ounces. Newton placed a mark on the wire supporting the bob at a point 121
inches (L′) from the support. Behind this mark he placed a ruler which was used to locate the
position of the mark as the pendulum oscillated. The values in the second and third columns of
Table 1 are the distances moved by this mark when descending and ascending. It can be seen from
the geometry of the situation that the ratio Aa′/L′ for the mark is equal to Aa/L for the bob. The
measure of the velocity is the distance travelled in the first descent. For a given cycloidal pendulum
the arc length travelled during the first half of the first swing is proportional to the velocity at the
bottom of the path (Gauld 2004).
Some of the values for the number of oscillations in column 5 of Table 1 contain fractions. In
an early working paper upon which this Scholium is based (Cohen 1971, p.103) Newton explained
how he arrived at such values. For example, in case 6 where the number of oscillations is 9 2 3 the
bob would be above its target of 56 inches after 9 swings but below the target after 10 swings.
Newton noted the amounts by which the knot was above or below the target and used these values
to estimate the number of swings when the target was reached.
Newton’s expectation that the resistance would be proportional to the square of the velocity
was confirmed for larger velocities by dividing the values for Aa′ (which are the basic measures of
resistance according to equations [1], [2] and [3]) by the square of the arc travelled (which is the
basic measure of velocity at the bottom of the swing) in the first descent for cases 1-6.
In his more refined analysis he assumed that this resistance was made up of three components
proportional, respectively, to V, V3/2 and V2. He gave no reason for this and it appears to be a purely
formal assignment. His previous theoretical discussion had only dealt with resistances proportional
to V0, V and V2. Using cases 2, 4 and 6 and an expectation that Aa′ = AV + BV3/2 + CV2 Newton
determined the values of A, B and C which enabled him to calculate the relative sizes of these three
components for different velocities. In order to determine the resistance from the above equation
Newton used equations [2] and [3] to arrive at
Resistance
=
Weight
7
3
11
AV + 710 BV 2 + 3 4 CV 2
Length
The factor 7/10 in the second term is the mean of 7/11 and 3/4 found in equations [2] and [3].
Newton presented the data for cases 7-12 in which the amplitude was reduced by one quarter
of the initial descent stating that these results would provide more accurate information than cases
1-6 but he took their analysis no further. He wrote: “I leave the calculation to such as are disposed
to make it”! It is interesting that the values of Aa′ for the cases 1-6 are consistently larger than those
for the corresponding cases 7-12. One reason for the difference is that, as the pendulum oscillates,
its velocity decreases, the resistance decreases and the value of Aa′ decreases with each swing.
Thus, dividing the difference between the first descent and the last ascent by the number of swings
7
gives the average decrease per swing and not the decrease for the first swing. Because the smaller
decreases for the later swings in cases 7-12 are omitted in cases 1-6 the average values of Aa′ for
cases 1-6 are larger than for the corresponding values for cases 7-12. If one assumes a particular
dependence on V (or on the amplitude) for the value of Aa′ in each swing it is possible to derive a
more accurate estimate of Aa′ for the first swing (using, for example, the Goal Seek tool in MS
Excel). The values for Aa′ using this method are much closer for the two sets of results.
DEPENDENCE ON THE DIAMETER OF THE BOB (Cases 1-26, 27)
In cases 13-19 Newton used a bob with a smaller diameter (2 inches) and weight (26.25 ounces)
than for cases 1-12 while maintaining the same length for the pendulum. For these cases he counted
the number of swings to bring about a reduction in the amplitude of one eighth. For cases 20-26 the
experiments were repeated for a reduction in amplitude of one quarter but these results were not
analysed in the Principia.
Newton analysed cases 15, 17 and 19 using the same form of relationship for Aa′ as for cases
2, 4 and 6 and found that the ratio of the velocity squared components of resistance derived from
the values of C for cases 2, 4 and 6 and cases 15, 17 and 19 (for which corresponding velocities
were the same) was 7.33/1 while the ratio of the squares of the diameters was 11.8/1. He argued
that if one took into account the resistance of the thread, which he estimated to be “greater than a
third part of the whole resistance of the lesser pendulum”, the first ratio becomes (7.33 – 0.33)/(1.00
– 0.33) or 10.5/1 a value which is closer to the ratio of squares of the diameters.
Newton realised that the effect of the thread would be relatively less for globes with larger
diameters and he carried out an experiment (case 27) with a globe of diameter 18.75 inches and
mass 208 ounces. The pendulum length was 122.5 inches. The results of this experiment were
compared to those from case 6 and, since the pendulum lengths were different, this necessitated the
transformation of pendulum parameters for case 27 to those of case 6 before a conclusion could be
reached. He found the ratio of resistances to be 7/1 and the ratio of the square of the diameters to be
7.4/1.
DEPENDENCE ON THE DENSITY OF THE FLUID (Cases 28-43)
Newton carried out a series of experiments in which the bob of a pendulum moved through air
(cases 28-31A) and then through water (cases 30W-36). From cases 28 and 31A in air Newton
determined the values of A and C in the expression Aa′ = AV + CV2. He realised that he could not
simply use the arc length as a measure of the velocity at the bottom in water because the velocity
was much less in water than in air for the same pendulum parameters. However, he argued as he
did when comparing the behaviour of circular and cycloidal pendulums. He reasoned that if the
velocity of the pendulum in water were to be increased so it was the same as that for the pendulum
in air the time the bob spent travelling to the bottom of its path would be decreased in the same
ratio. Thus, since Aa is proportional to resistance x (time)2 and if the resistance were proportional
to V2, Aa would be unchanged and no error would be introduced by assuming that the measure of
the velocity of the pendulum bob travelling through water was the length of the arc.
If this conclusion were true then, for cases 31A and 31W, R(water)/R(air) =
Aa′(water)/Aa′(air) = N(air)/N(water) = 535/1.2. Using his values of A and C found previously
Newton then argued that R(air[V2])/R(air) = 48.16/61.71 where R(air[V2]) is the component of the
resistance due to the air which is proportional to V2 from which he concluded that R(water)/R(air
[V2]) = 571/1. Taking into account the fact that the thread was totally immersed in air but not
totally immersed in water, Newton, without any further details, arrived at his conclusion that
R(water)/R(air[V2]) was “about 850 to 1, that is as, the density of the water to the density of the air,
nearly”.
Newton carried out further experiments for which he does not provide all the details. He
realised that, in some cases, his vessel was too small for the size of the bob and he wrote
8
I intended to have repeated these experiments with larger vessels, and in melted metals, and other liquors
both hot and cold; but I had not the leisure to try all: and besides, from what is already described, it
appears sufficiently that the resistance of bodies moving swiftly is nearly proportional to the densities of
the fluids in which they move.
The last experiment he described in this Scholium was one to determine the resistance due to
the aether. The aether was conceptualised as a fluid which not only filled empty space but also
pervaded the interior of solid objects. He suspended a wooden box on the end of a thread 11 feet
long then drew it aside about 6 feet, allowed it to oscillate and noted the position of the bob after
one, two and three swings. The box was then filled with metal. When full the weight of the box
was 78 times the combined weight of the empty box, the air in it and the thread around it. When
allowed to oscillate, the filled box took 77 swings to reach the first mark. Newton was interested to
see if this difference was due to the effect of the aether on surfaces inside the box as well as on the
outside surface. Using the relationship [1], [2] or [3], namely,
kAa k ( Initial descent ! Final ascent ) / N
R
=
=
L
L
W
Newton concluded that “the whole resistance of the box when full, had not a greater proportion
to the resistance of the box, when empty, than 78 to 77”. Separating these resistances into two
components – that due to the outer surfaces and that due to the inner surfaces - he concluded that
“the resistance of the empty box in its internal parts will be above 5000 times less than the
resistance on the external superficies [surfaces]” that is, the resistance of the aether, if any, was very
small.
Newton pointed out that “this experiment is related by memory, the paper being lost in which I
had described it; so that I have been obliged to omit some fractional parts, which are slipt out of my
memory; and I have no leisure to try it again”.
DISCUSSION
In many respects Newton’s treatment of the problem of fluid resistance is significantly different
from the way this problem would be tackled today especially in educational settings. A number of
features of Newton’s work in this area are discussed in what follows.
The geometrisation of nature
The theory presented above illustrates the approach to understanding physical phenomena which
Newton adopted – an approach described by Koyré as “the geometrisation of nature” (Koyré 1965,
pp.6-7; Westfall 1990). Physical relationships could be mapped by and analysed through
geometrical relationships. Today the emphasis in science teaching is on algebraic rather than
geometrical models. One feature of Newton’s treatment of Proposition 30 is the close relationship
between the geometrical diagram and the physical situation. The base line, BA in Figure 2,
represents the path of the pendulum bob and the velocity of the bob is represented by vertical lines
starting from the particular position of the bob on that line. The resistance at D is represented by
the line DK. In each case there was a direct relationship between a dynamical or kinematic variable
and a line on the diagram and the connection between diagram and situation is not difficult to grasp.
The resistance is directly related to the decrease in the amplitude of the pendulum as it moves
through the fluid.
-- FIGURE 6 -9
However, this is not always so in the Principia and, in some cases, while Newton provided a
geometrical solution to a problem sometimes the link between the diagram and the situation was not
easy to conceptualise. His treatment of Problem 4 (Proposition 29) illustrates this point well. This
problem is “Supposing that a body oscillating in a cycloid is resisted in a duplicate ratio of the
velocity: to find the resistance in each place” so, in part, in this problem Newton attempted to solve
the same problem he dealt with in Proposition 30. Newton looked for a mathematical analogue for
the physical system he was studying and for many of the Propositions in Book II this was based on
the hyperbola. In Figure 6(a) a pendulum bob would move from B to A if there were no resistance
while, as a result of resistance, it moved from B to a. C is the lowest point and D is any point on
this path. Newton set up the hyperbola in Figure 6(b) in order to find a way of determining the
resistance experienced by the bob as it moved. He stipulated the following in order to begin his
analysis:
AreaPIEQ BC
=
AreaPITS Ca
(this defined the positions of S, P and Q)
AreaIEF OQ
=
AreaILT OS
AreaPINM CZ
=
AreaPIEQ BC
(this defined the position of M)
AreaPIGR CD
=
AreaPIEQ BC
(this defined the position of R)
where CZ is equal to the length of the pendulum.
Following his analysis of the situation Newton concluded that
OR
) AreaIEF ! AreaIGH
Resistance at D
OQ
=
Force of gravity
AreaPINM
(
Although this formally solved Newton’s problem it can easily be seen that little of this analysis
allows for an intuitive understanding of the problem itself and Newton acknowledged this when he
admitted “However, by reason of the difficulty of the calculation by which the resistance and the
velocity are found by this Proposition, we have thought fit to subjoin the Proposition following”. In
Proposition 30 he provided the elegant and generalised analysis of the problem of fluid resistance as
it applied to a cycloidal pendulum and which is outlined above although the approximations made
in this solution rendered it less rigorous than the Euclidean model which Newton attempted to
emulate in the Principia.
Plausible approximations without rigorous justification
In a number of places Newton appears to lack the mathematical tools to proceed in his analysis and
makes approximations on the basis of their apparent plausibility. For example, while his theory is
based on his analysis of the motion of a cycloidal pendulum, in his experiments he used a simple
circular pendulum. He made a case (outlined previously in this paper) that nevertheless his theory
could be applied to his experiment even though this case lacked the rigour of other parts of his
10
analysis. Similarly, he claimed that, in Figure 4, the curve Bfja was a semicircle – or at least
“nearly” a semicircle – a plausible assumption for which he provided some justification.
When his pendulum bobs moved through water, Newton realised that the velocity at the lowest
point would be significantly less that that given by the relationship between arc (or chord) length
and velocity used for a pendulum in air. However, he argued (without detailed analysis but with
reference to features of his theory presented earlier in this paper) that if this same relationship were
assumed no significant error would be introduced into his conclusions.
These examples are indications that Newton, working with a theory that did not fully perform
the function which he intended it to, nevertheless proceeded on the basis of what might reasonably
be assumed even where he was not able to establish the validity of those assumptions.
The use of raw data
When carrying out his experiments Newton took great care to ensure that his measurements were as
accurate as possible. In these experiments he used long pendulums and refined his experimental
technique as he progressed in his investigations. Table 1 contains information about 36 cases for
which Newton provided sufficient detail for us to know most of the relevant parameters and so
check his calculations. For cases 37-43 and the other cases he reported in the General Scholium
following Proposition 31 the detail is insufficient for this to be done and we have to take Newton’s
word for his conclusions. In his discussion of the implications of his data Newton analysed only 10
of those cases listed in Table 1 and no mention is made of possible errors in the data. From cases 16 he used cases 2, 4 and 6 to determine the coefficients A, B and C in the relation Aa′ = AV + BV3/2
+ CV2 and ignored cases 1, 3 and 5. Cases 7-12 were left to the reader to analyse.
Although Newton developed his theory in Book II to deal with resistances which were
constant, proportional to V, or proportional to V2 he gives no reason for introducing here a
component proportional to V3/2.
In many cases (for example, cases 1, 3 and 5 or 7, 9 and 11), if one applies the same analysis to
the data which Newton ignored, negative values are obtained for A or B. It is possible that Newton
was aware of this and so omitted these results from his report. On the other hand, the cases he did
include were those with larger values of V for which the V2 component would be expected to
dominate.
Nowadays all the data would be used and errors estimated to find curves of best fit so that no
available information is ignored. The presence of negative values for A or B would most likely be
attributed to experimental error.
Reconciliation between results and expectations
In all of the experiments described above there are major discrepancies between the results which
Newton obtained from his experiments and his expectations. The dependence of resistance on the
square of the velocity breaks down for small values of V, and the proportional dependence of
resistance on the square of the diameter of the bob or the density of the fluid was far from exact. In
all these situations Newton sought a reason for the difference between his experiments and his
expectations and the reconciliation is not always convincing. For example, he introduced a rough
estimate of the resistance of the thread (with no account of how he obtained this) to reconcile
experimental results in cases 1-6 and 13-19 with the ratio of diameters which was 11.3/1. The ratio
of resistances before modification was 7.33/1 while after modification was 10.5/1.
In comparing resistances in water and air his experiments indicated that the ratio was 570/1.
He rightly pointed out that the thread was immersed in air which exerted a resistance on the thread
but the thread was not immersed in the water so all the irrelevant variables were not controlled
11
adequately. He stated without support that if the thread were fully immersed in water the ratio
would be 850/1, which is “nearly” the ratio of the densities of water and air. (At 200C the ratio is
about 830/1). However, if the information he gave about the resistance of the thread when
discussing cases 13-19 in the experiment using a 2 inch bob is used, the ratio seems to be only
656/1. It is not clear how he arrived at the ratio of 850/1. In the third edition of the Principia
Newton used three different values for the water to air density ration – 850 here, 860 in the
Scholium after Proposition 40 when he investigated the resistance to falling bodies, and 870 in the
Scholium after Proposition 50 in which he investigated the speed of sound in air. Westfall (1973)
pointed out that, in the various drafts of the first edition of the Principia in which he described his
investigations into the speed of sound, Newton temporarily “adopted” 900 and 950 before settling
on 870 for this ratio, apparently to find the one which best suited his data. These might be examples
of what Truesdell calls “fudge factors” (1968b, p.149). Westfall (1973) indicated a number of other
places in the Principia where Newton appears to have “massaged” his results to bring about a closer
fit between theoretical predictions and experimental results.
Truesdell (1968b, p.91) points out that in Book II of the Principia Newton, using the new
physics, was dealing with phenomena which had never before been treated in this way. Unlike
Book I in which he was following in the footsteps of other natural philosophers, in Book II he was
treading new ground. Truesdell indicates that most of what Newton wrote in Book II was false
(Truesdell 1968a, p.91) but that nevertheless what Newton achieved there was noteworthy and must
be judged, not according to what we know now but by what was known in the 17th century.
Newton’s attempts to reconcile his results with his expectations is typical of someone breaking
new ground and this is just the position of Newton when investigating the complex phenomena
discussed in Book II of the Principia. We see here something of an individual scientist at work
doing all he can to show the value of his position rather than the completed, fully institutionalised,
objective science of textbooks.
Difference between terrestrial and celestial physics
Terrestrial phenomena are complex with the outcomes being the result of many influences while
celestial phenomena, dealt with in Book III are more simple and less prone to unpredictable
influences such as friction, resistance, turbulence and so on. The fit between Newton’s theory of
central forces presented in Book I and the astronomical data presented in Book III is much closer
than for the phenomena discussed in Book II. As far as possible, Newton attempted to control
carefully irrelevant variables but, as he did not fully understanding the phenomena he was dealing
with, he was not always aware of the nature of these irrelevant variables.
Part of Newton’s purpose in writing Book II was to refute Descartes’ theory of planetary
motion in which the planets were carried around the Sun by fluid vortices (Newton 1729/1960,
Book II, Section IX; Book III, Final General Scholium; Westfall 1980, p.454). Newton discussed
circular motion in fluids in Propositions 51, 52 and 53 of Book II. At the beginning of the
following Scholium he concluded:
Hence it is manifest that the planets are not carried round in corporeal vortices; for, according to the
Copernican hypothesis, the planets going round the sun revolve in ellipses, having sun in their common
focus; and the radii drawn from the sun describe areas proportional to the times. But now the parts of a
vortex can never revolve with such a motion. (Newton 1729/1960, p. 395)
CONCLUSION
The material presented above gives insight into Newton’s geometrical procedure and experimental
technique. There is much that students can learn from what Newton did. For example, his
experiments can be replicated and the results analysed using his theory; alternative estimates of Aa′
which take into account the reduction in size of this difference for each successive swing can be
12
derived; curve-fitting techniques can be used to analyse his own results (presented in Table 1) and
to translate what he did into more modern terms; the assumptions he made to enable his data to fit
his expectations can be explored; and more recent understanding of the nature of viscous drag
(proportional to V) can be introduced into his analyses. All of this involves coming to an
understanding of his mode of thought – in other words placing oneself in Newton’s shoes – and
beginning to appreciate the differences between what Newton did and what we would do with his
problem today. The experiments are simple enough not to require sophisticated instrumentation and
to be accessible to students in many contexts. They also show the genius of Newton when one
begins to appreciate the thinking of a man at the forefront of the development of mechanics.
NOTE
1
Aiton (1972) pointed out that, in spite of Newton’s efforts, the vortex theory flourished on the
continent well into the 1700s. Many European scientists had trouble with Newton’s theory because
it seemed to be based on the existence of occult forces at a distance where there was no physical
contact between the interacting objects. “All the objections to the Cartesian vortices had been know
for a long time. On the other hand, as a cause, the attraction was just as unacceptable as it had been
for over a century” (Aiton, 1972, p.247).
2
I am indebted to Alan Emmerson and Michael Fowler for helpful communications about the
period of a circular pendulum for large amplitudes.
REFERENCES
Aiton, E.J.:1972, The Vortex Theory of Planetary Motion, Science History Publications, New York.
Chandrasekhar, S.: 1995, Newton’s Principia for the Common Reader, Oxford University Press,
Oxford.
Cohen, I.B. 1971, Introduction to Newton’s Principia, Cambridge University Press, Cambridge.
Densmore, D.: 1995. Newton’s Principia: The Central Argument. Green Lion Press, Santa Fe,
NM.
Gauld, C.F.: 1998, Colliding Pendulums, Conservation of Momentum and Newton's Third Law,
Australian Science Teachers Journal, 44(3), 37-38.
Gauld, C.F. 1999, Using Colliding Pendulums to Teach Newton's Third Law, The Physics Teacher,
37, 25-28.
Gauld, C.F.: 2004, The Treatment of Cycloidal Pendulum Motion in Newton's Principia, Science
and Education.13, 663-673.
Kidd, R.B. & Fogg, S.L:. 2002, A Simple Formula for the Large-angle Pendulum Period, The
Physics Teacher, 40, 81-83.
Koyré, A.: 1965, Newtonian Studies. Chapman & Hall, London.
Newton, I.: 1729/1960, The Mathematical Principles of Natural Philosophy, (translated from the
third edition by Andrew Motte and revised by Florian Cajori), University of California Press,
Berkeley, CA.
Truesdell, C.: 1968a, A Program toward Rediscovering the Rational Mechanics of the Age of
Reason, In C. Truesdell, Essays in the History of Mechanics, Springer-Verlag, Berlin, 85-137.
Truesdell, C.: 1968b, Reactions of late Baroque Mechanics to Success, Conjecture, Error, and
Failure in Newton’s Principia, In C. Truesdell, Essays in the History of Mechanics, SpringerVerlag, Berlin, 138-183.
Westfall, R.S.: 1973, Newton and the Fudge Factor, Science, 179, 751-758.
Westfall, R.S:. 1980, Never at Rest: A Biography of Isaac Newton, Cambridge University Press,
Cambridge
13
Westfall, R.S.: 1990, Making a World of Precision: Newton and the Construction of Quantitative
Physics, In F. Durham & R.D. Pennington (eds) Some Truer Methods: Reflections on the
Heritage of Newton. Columbia University Press, New York, 59-87.
14
TABLE 1: Newton’s pendulum data
Case
First
descent
Last
ascent
Difference
Number of
swings
Aa′
Comments
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30A
31A
30W
31W
32
33
34
35
36
37
38
39
40
41
42
43
2
4
8
16
32
64
2
4
8
16
32
64
1
2
4
8
16
32
64
1
2
4
8
16
32
64
32
64
32
16
8
16
8
4
2
1
0.5
0.25
16
8
4
2
1
0.5
0.25
1.75
3.5
7
14
28
56
1.5
3
6
12
24
48
0.875
1.75
3.5
7
14
28
56
0.75
1.5
3
6
12
24
48
28
48
24
12
6
12
6
3
1.5
0.75
0.325
0.1635
12
6
3
1.5
0.75
0.325
0.1625
0.25
.5
1
2
4
8
0.5
1
2
4
8
16
0.125
0.25
0.5
1
2
4
8
0.25
0.5
1
2
4
8
16
4
16
8
4
2
4
2
1
0.5
0.25
0.125
0.0625
4
2
1
0.5
0.25
0.125
0.0625
164
121
69
35.5
18.5
9
374
272
162.5
83.3
41.7
22.7
226
228
193
140
90.5
53
30
510
518
420
318
204
121
70
10
85.5
287
0.00154
0.00413
0.01449
0.05634
0.21622
0.88889
0.00134
0.00368
0.01223
0.04800
0.19200
0.70588
0.00055
0.00110
0.00259
0.00714
0.02210
0.07547
0.26667
0.00049
0.00097
0.00238
0.00629
0.01961
0.06612
0.22857
0.40000
0.18713
0.02787
Diameter of wooden bob =
6.875 London inches
Mass of bob = 57.318
troy ounces
Pendulum length =
126 inches
535
0.483
1.2
3
7
11.3
12.7
13.3
3.325
6.5
12.1
21.2
34
53
62.2
0.00374
8.27586
1.66667
0.33333
0.07143
0.02222
0.01382
0.00656
1.20301
0.30769
0.08276
0.02358
0.00735
0.00330
0.00100
# see note below
in water
Diameter of lead bob =
3.625 inches
Mass of bob =
166.125 troy ounces
Length of pendulum =
134.375 inches
Diameter of bob = 1 inch
as for cases 1-6
Diameter of lead bob =
2 inches
Mass of bob = 26.25
troy ounces
Pendulum length =
126 inches
as for cases 13-19
* see note below
in air
as for cases 30W-36
* diameter = 18.75 inches; mass = 208 troy ounces; pendulum length = 122.5 inches; Newton says that this case took 5
swings but his analysis indicates that this should be 10.
# Newton’s Table gives 535 swings for case 30A but his analysis indicates that this should be for case 31A.
15
V
A
M
V
FIGURE 1
W
e
R
E
S
f
V
A
M N
a
F
j
U
C
FIGURE 2
16
O
g
k
K
d
D
B
f
F
m
j
C
h
g
D
FIGURE 3
17
e
E
F
j
V
A
a
C
O
k
K
d
D
FIGURE 4
aA' as a function of Initial Arc Length
1.00
Cases1-6
aA'
0.80
Cases 7-12
0.60
Cases 13-19
0.40
Cases 20-26
31A, 29, 28
0.20
0.00
0.00
Cases 30W-36
20.00
40.00
Initial Arc Length
FIGURE 5
18
60.00
B
(a)
(b)
FIGURE 6
CAPTIONS TO FIGURES
Figure 1. Motion of a cylinder through a resisting medium.
Figure 2. Diagram for Newton’s analysis of the motion of a cycloidal pendulum through a resisting
medium (Principia, Book II, Proposition 30)
Figure 3. Detail of Figure 2.
Figure 4. Simplified version of Figure 2.
Figure 5. Graphical summary of Newton’s results for cycloidal pendulums moving through
resisting mediums (Cases 1-36)
Figure 6. Newton’s diagrams for his solution to Problem VI (Principia, Book II, Proposition 29).
19