History and Philosophy in the Classroom

In this chapter, the teaching of a single topic, pendulum motion, will be used
to illustrate the claims being made about the benefits of a liberal or contextual
approach to science education. Pendulum motion is chosen in part because it
has a place in nearly all science programmes, and also because it is a relatively
pedestrian topic. In ‘hot’ topics, such as evolution, genetic engineering, nuclear
energy, climate change or acid rain, historical and philosophical considerations
are obviously useful. If the case for HPS can be made with a ‘boring’ topic,
then the usefulness of HPS for science teaching is better established. Furthermore, the science of pendulum motion illustrates important general topics
alluded to in this book, including:
• the interplay of mathematics, observation and experiment in the
development of modern science;
• the reciprocal impacts of science on culture and society;
• the interactions of philosophy and science;
• the distinction between material objects and these objects as treated by
• the ambiguous role of empirical evidence in the justification or falsification
of scientific claims;
• the contrast between mode
solar system; a dynamical proof for the rotation of the earth on its axis; the
equivalence of inertial and gravitational mass; an accurate measurement of
the density and, hence, mass of the earth; and much more. Pendulum motion
was central to the argument between Aristotelians and Galileo over the role
of experience in settling conflicting claims about the world, and it figured in
Newton’s major metaphysical dispute with the Cartesians, namely the dispute
concerning the existence of the ether (Westfall 1980, p. 376). Domenico
Bertoloni Meli observed that:
Starting with Galileo, the pendulum was taking a prominent place in the study
of motion and mechanics, both as a time-measuring device and as a tool for
studying motion, force, gravity, and collision.
(Meli 2006, p. 206)
With good reason, the historian Bertrand Hall attested:
In the history of physics the pendulum plays a role of singular importance. From
the early years of the seventeenth century, when Galileo announced his
formulation of the laws governing pendular motion, to the early years of this
century, when it was displaced by devices of superior accuracy, the pendulum was
either an object of study or a means to study questions in astronomy, gravitation
and mechanics.
(Hall 1978, p.4 41)
The importance of the pendulum in science and philosophy was exceeded
only by its importance to commerce, navigation, exploration and Western
expansion. A convenient and accurate measure of the passage of time was
crucial for the pressing commercial and military problem of determining
longitude at sea, as well as for everyday economic and social affairs. The
pendulum answered these problems. Unfortunately, the centrality and
importance of the pendulum for the development of modern science are not
reflected in textbooks and school curricula, where it appears as an ‘exceedingly
arid’ subject and is mostly, even in the best classes, dismissed with wellremembered formulae (T = 2π√(l/g)) and some routine mathematical exercises
and maybe some practical classes.
The Textbook Myth and Prehistory of the Pendulum
The standard textbook treatment of pendulum motion features the story of
Galileo’s discovery of the isochronous movement of the pendulum. One such
account is:
When he [Galileo] was barely seventeen years old, he made a passive observation
of a chandelier swinging like a pendulum in the church at Pisa where he grew up.
He noticed that it swung in the gentle breeze coming through the half-opened
212 History and Philosophy: Pendulum Motion
church door. Bored with the sermon, he watched the chandelier carefully, then
placed his fingertips on his wrist, and felt his pulse. He noticed an amazing thing.
. . . Sometimes the chandelier swings widely and sometimes it hardly swings at
all . . . [yet] it made the same number of swings every sixty pulse beats.
(Wolf 1981, p. 33)
This same story appears in the opening pages of the most widely used highschool physics text in the world – the PSSC’s Physics (PSSC 1960).2
If the textbook account is to be believed, then a basic question is why it
was that the supposed isochronism of the pendulum was only seen in the
sixteenth century, when countless thousands of people of genius and with
acute powers of observation had, for thousands of years, been pushing children
on swings, and looking at swinging lamps and swinging weights, and using
suspended bobs in tuning musical instruments, without seeing their
isochronism. For centuries, people had been concerned to find a reliable
measure of time, both for scientific purposes and also in everyday life, to
determine the duration of activities and events, and the vital navigational
matter of determining longitude at sea. As the isochronic pendulum was the
answer to all these questions, the widespread failure to recognise something
so apparently obvious is informative. It suggests that there is not just a problem
of perception, but a deeper problem is involved, a problem of epistemology,
or how things are seen.
Nicole Oresme (1320–1382), in the fourteenth century, discussed pendulum
motion. In his On the Book of the Heavens and the World of Aristotle, he
entertained the thought experiment of a body dropped into a well that had
been drilled from one side of the Earth, through the centre and out the other
side (a thought experiment repeated by Galileo in his 1635 Dialogues
Concerning the Two Chief World Systems). Oresme likens this imaginary
situation to that of a weight that hangs on a long cord and swings back and
forth, each time nearly regaining its initial position (Clagett 1959, p. 570).
Albert of Saxony, Tartaglia and Benedetti all discussed the same problem in
the context of impetus theory. Leonardo da Vinci, a most acute observer, dealt
on many occasions with pendulum motion and, in the late 1490s, sketched
two pendulums, one on a reciprocating pump; the other on what appears to
be a clock. He recognised that the descent along the arc of a circle is quicker
than that along the shorter corresponding chord, an anticipation of Galileo’s
later Law of Chords. In 1569, Jacques Besson published a book in Lyon
detailing the use of the pendulum in regulating mechanical saws, bellows,
pumps and polishing machines.3
Thomas Kuhn, in his Structure of Scientific Revolutions, famously used
Galileo’s account of the pendulum to mark the epistemological transformation
from the old to the new science. Kuhn wrote:
Since remote antiquity most people have seen one or another heavy body swinging
back and forth on a string or chain until it finally comes to rest. To the
History and Philosophy: Pendulum Motion 213
Aristotelians, who believed that a heavy body is moved by its own nature from
a higher position to a state of natural rest at a lower one, the swinging body was
simply falling with difficulty. . . . Galileo, on the other hand, looking at the
swinging body, saw a pendulum, a body that almost succeeded in repeating the
same motion over and over again ad infinitum. And having seen that much,
Galileo observed other properties of the pendulum as well and constructed many
of the most significant and original parts of his new dynamics around them.
(Kuhn 1970, p. 118–119)
The question of how Galileo came to recognise and prove the laws of
pendulum motion is germane to the teaching of the topic. Teachers want
students to recognise and prove the properties of pendulum motion – period
being independent of mass and amplitude, and varying inversely as the square
root of length. How these properties were initially discovered can throw light
on current attempts to teach and learn the topic; as with all other subjects in
science, their history has intrinsic as well as pedagogical value.
Galileo’s Account of Pendulum Motion
In a letter of 1632, 10 years before his death, Galileo surveyed his achievements
in physics and recorded his debt to the pendulum for enabling him to measure
the time of free fall, which, he said, ‘we shall obtain from the marvellous
property of the pendulum, which is that it makes all its vibrations, large or
small, in equal times’ (Drake 1978, p. 399). To use pendulum motion as a
measure of the passage of time was a momentous enough achievement, but
the pendulum is also central to Galileo’s treatment of free fall, the motion of
bodies through a resisting medium, the conservation of momentum, and the
rate of fall of heavy and light bodies. His analysis of pendulum motion is,
thus, central to his overthrow of Aristotelian physics and the development of
the modern science of motion, a development of which the historian Herbert
Butterfield has said:
Of all the intellectual hurdles which the human mind has confronted and has
overcome in the last fifteen hundred years, the one which seems to me to have
been the most amazing in character and the most stupendous in the scope of its
consequences is the one relating to the problem of motion.
(Butterfield 1949/1957, p. 3)
Galileo, in his final work, The Two New Sciences, written during the period
of house arrest after the trial that, for many, marked the beginning of the
Modern Age, wrote:
We come now to the other questions, relating to pendulums, a subject which may
appear to many exceedingly arid, especially to those philosophers who are
continually occupied with the more profound questions of nature. Nevertheless,
214 History and Philosophy: Pendulum Motion
the problem is one which I do not scorn. I am encouraged by the example of
Aristotle whom I admire especially because he did not fail to discuss every subject
which he thought in any degree worthy of consideration.
(Galileo 1638/1954, pp. 94–95)
Galileo’s comment that pendulum investigations appear ‘exceedingly arid’ has,
unfortunately, been echoed by science students over the following 400 years.
Galileo’s Early Pendulum Investigations
Galileo’s new science had engineering or practical roots: he worked with and
was familiar with machines, and saw that pendulums were utilised in these
machines; he did not have to await the breeze blowing the chandelier in the
cathedral for his first experience of pendulum motion. And it was not just
observing or looking at machines, or even reading about them – inclined
planes, screws, levers, pendulums – but working with machines that provided
the grounds for Galileo’s transformation of extant science.4
While the youthful Galileo was briefly a medical student at Pisa, he utilised
the pendulum to make a simple diagnostic instrument for measuring pulse
beats. This was the pulsilogium. Medical practitioners in Galileo’s day realised
that pulse rate was of great significance, but there was no objective, let alone
accurate, measurement of pulse beat. Galileo’s answer to the problem was
ingenious and simple: he suspended a lead weight on a short length of string,
mounted the string on a scaled board, set the pendulum in motion and then
moved his finger down the board from the point of suspension (thus effectively
shortening the pendulum) until the pendulum oscillated in time with the
patient’s pulse. As the period of oscillation depended only on the length of
the string, and not on the amplitude of swing or the weight of the bob, the
length of the string provided an objective and repeatable measure of pulse
speed that could be communicated between doctors and patients, and kept
as a record.
The pulsilogium provides a useful epistemological lesson (and it is easily
made by students). Initially, something subjective, the pulse, was used to
measure the passage of time – occurrences, especially in music, were spoken
of as taking so many pulse beats. With Galileo’s pulsilogium, this subjective
measure itself becomes subject to an external, objective, public measure – the
length of the pulsilogium’s string. This was a small step in the direction of
objective and precise measurement, upon which scientific advance in the
seventeenth and subsequent centuries would depend.
After his appointment to a lectureship in mathematics at the University of
Pisa in 1588, Galileo quickly became immersed in the mathematics and
mechanics of the ‘Superhuman Archimedes’, whom he never mentions
‘without a feeling of awe’ (Galileo 1590/1960, p. 67). Galileo’s major Pisan
work is his On Motion (1590/1960). In it, he deals with the full range of
problems being discussed among natural philosophers – free-fall, motion on
History and Philosophy: Pendulum Motion 215
balances, motion on inclined planes and circular motion. In these discussions,
the physical circumstances are depicted geometrically, and mathematical
reasoning is used to establish various conclusions in physics: Galileo here
begins the thorough mathematising of physics, which is entirely modern.5
Galileo’s genius was to see that all of the above motions could be dealt with
in one geometrical construction. That is, motions that appeared so different
in the world could all be depicted and dealt with mathematically in a common
Consider a pendulum suspended at B, moving through C, F, L, J. This is a
most fruitful construction. It allowed Galileo to analyse pendulum motion as
motion in a circular rim and as motion on a suspended string. By considering
initial, infinitesimal motions, he was able to consider pendulum motion as a
series of tangential motions down inclined planes. Two years later, he was to
write an important letter to his patron, Guidobaldo del Monte, about these
Galileo made use of the diagram in Figure 6.1 to prove properties of
pendulum motion. It is important to note that Galileo then qualifies this
proof, saying:
But this proof must be understood on the assumption that there is no accidental
resistance (occasioned by roughness of the moving body or of the inclined plane,
or by the shape of the body). We must assume that the plane is, so to speak,
incorporeal or, at least, that it is very carefully smoothed and perfectly hard . . .
and that the moving body must be perfectly smooth . . . and of the hardest
Galileo here introduced crucial idealising conditions. His new science was not
going to be simply about how the world behaves, but rather how it should
behave. Or, to put it another way, his science was to be about how the world
would behave if various conditions were fulfilled: for the pendulum, if the
string were weightless, if the bob occasioned no air resistance, if the fulcrum
were frictionless and so on. In controlled experiments, some of these conditions
can be fulfilled, but other conditions cannot be fulfilled, and yet they were
crucial to Galileo’s science.7
Guidobaldo del Monte: Galileo’s Patron and Critic
The most significant opponent of Galileo’s nascent views about the pendulum
was his own academic patron, the distinguished Aristotelian engineer
Guidobaldo del Monte (1545–1607). Del Monte was one of the great mathematicians and mechanics of the late sixteenth century. He was a translator of
the works of Archimedes, the author of a major book on mechanics (Monte
1581/1969), a book on geometry (Planispheriorum universalium theorica,
1579), a book on perspective techniques, Perspectiva (1600), and an unpublished book on timekeeping, De horologiis, that discussed the theory and
216 History and Philosophy: Pendulum Motion
construction of sundials. He was a highly competent mechanical engineer and
director of the Venice Arsenal, an accomplished artist, a minor noble and the
brother of a prominent cardinal. He was also a patron of Galileo who secured
for Galileo his first university position as a lecturer in mathematics at Pisa
University (1588–1592) and his second academic position as a lecturer in
mathematics at Padua University (1592–1610).
The crucial surviving document in the exchange between Galileo and his
patron is a letter dated 29 November 1602, where Galileo writes of his
discovery of the isochrony of the pendulum and conveys his mathematical
proofs of the ‘pendulum laws’.8 As the letter is a milestone in the history of
timekeeping and in the science of mechanics, and as it illustrates a number
of other things about the methodology of Galileo and his scientific style, it
warrants reproduction:
You must excuse my importunity if I persist in trying to persuade you of the truth
of the proposition that motions within the same quarter-circle are made in equal
times. For this having always appeared to me remarkable, it now seems even more
History and Philosophy: Pendulum Motion 217
Figure 6.1 Galileo’s 1600 Composite Diagram of Lever, Inclined Plane, Vertical Fall and
Source: Galileo 1590/1960, p. 173
remarkable that you have come to regard it as false. Hence I should deem it a
great error and fault in myself if I should permit this to be repudiated by your
theory as something false; if it does not deserve this censure, nor yet to be banished
from your mind – which better than any other will be able to keep it more readily
from exile by the minds of others. And since the experience by which the truth
has been made clear to me is so certain, however confusedly it may have been
explained in my other [letter], I shall repeat this more clearly so that you, too, by
making this [experiment], may be assured of this truth.
Therefore take two slender threads of equal length, each being two or three
braccia long [four to six feet]; let these be AB and EF. Hang A and E from two
nails, and at the other ends tie two equal balls (though it makes no difference if
they are unequal). Then moving both threads from the vertical, one of them very
much as through the arc CB, and the other very little as through the arc IF, set
them free at the same moment of time. One will begin to describe large arcs like
BCD while the other describes small ones like FIG. Yet in this way the moveable
[that is, movable body] B will not consume more time passing the whole arc BCD
than that used up by the other moveable F in passing the arc FIG (see Figure 6.2).
I am made quite certain of this as follows.
The moveable B passes through the large arc BCD and returns by the same
DCB and then goes back toward D, and it goes 500 or 1,000 times repeating its
oscillations. The other goes likewise from F to G and then returns to F, and will
similarly make many oscillations; and in the time that I count, say, the first 100
large oscillations BCD, DCB and so on, another observer counts 100 of the other
oscillations through FIG, very small, and he does not count even one more – a
most evident sign that one of these large arcs BCD consumes as much time as
each of the small ones FIG. Now, if all BCD is passed in as much time [as that]
218 History and Philosophy: Pendulum Motion
Figure 6.2 Large- and Small-Amplitude Pendulums
Source: Galileo 1602/1978, p. 69
in which FIG [is passed], though [FIG is] but one-half thereof, these being descents
through unequal arcs of the same quadrant, they will be made in equal times. But
even without troubling to count many, you will see that moveable F will not make
its small oscillations more frequently than B makes its larger ones; they will
always be together.
The experiment you tell me you made in the [rim of a vertical] sieve may be
very inconclusive, perhaps by reason of the surface not being perfectly circular,
and again because in a single passage one cannot well observe the precise beginning
of motion. But if you will take the same concave surface and let ball B go freely
from a great distance, as at point B, it will go through a large distance at the
beginning of its oscillations and a small one at the end of these, yet it will not on
that account make the latter more frequently than the former (see Figure 6.3).
Then as to its appearing unreasonable that given a quadrant 100 miles long,
one of two equal moveables might traverse the whole and [in the same time]
another but a single span, I say that it is true that this contains something of the
wonderful, but our wonder will cease if we consider that there could be a plane
as little tilted as that of the surface of a slowly running river, so that on this [plane]
a moveable will not have moved naturally more than a span in the time that on
another plane, steeply tilted (or given great impetus even on a gentle incline), it
will have moved 100 miles. Perhaps the proposition has inherently no greater
improbability than that triangles between the same parallels and on equal bases
are always equal [in area], though one may be quite short and the other 1,000
miles long. But keeping to our subject, I believe I have demonstrated that the one
conclusion is no less thinkable than the other.
Let BA be the diameter of circle BDA erect to the horizontal, and from point
A out to the circumference draw any lines AF, AE, AD, and AC. I show that equal
moveables fall in equal times, whether along the vertical BA or through the
inclined planes along lines CA, DA, EA and FA. Thus leaving at the same moment
from points B, C, D, E, and F, they arrive at the same moment at terminus A;
and line FA may be as short as you wish.
And perhaps even more surprising will this, also demonstrated by me, appear:
That line SA being not greater than the chord of a quadrant, and lines SI and IA
being any whatever, the same moveable leaving from S will make its journey SIA
History and Philosophy: Pendulum Motion 219
Figure 6.3 Ball in Hoop
Source: Galileo 1602/1978, p. 70
more swiftly than just the trip IA, starting from I. This much has been
demonstrated by me without transgressing the bounds of mechanics. But I cannot
manage to demonstrate that arcs SIA and IA are passed in equal times, which is
what I am seeking.
Do me the favor of conveying my greetings to Sig. Francesco and tell him that
when I have a little leisure I shall write to him of an experiment that has come
to my mind for measuring the force of percussion. And as to his question, I think
that what you say about it is well put, and that when we commence to deal with
matter, then by reason of its accidental properties the propositions abstractly
considered in geometry commence to be altered, from which, thus perturbed, no
certain science can be assigned – though the mathematician is so absolute about
them in theory. I have been too long and tedious with you; pardon me, and love
me as your most devoted servitor.
(Drake 1978, p. 71)
Thus, in 1602, Galileo is claiming two things about motion on chords within
a circle:
1 That in a circle, the time of descent of a body free-falling, along all chords
terminating at the nadir, is the same regardless of the length of the chord.
220 History and Philosophy: Pendulum Motion
Figure 6.4 Law of Chords
Source: Galileo 1602/1978, p. 71
2 In the same circle, the time of descent along a chord is longer than along
its composite chords, even though the former route is shorter than the
This gets him tantalisingly close to a claim about motion along the arcs of
the circle, the pendulum case, but not quite there. He does allude to the
pendulum situation and says that two 6-foot pendulums keep in synchrony
through 1,000 swings, one being displaced widely, the other barely displaced
from vertical. He is not prepared to make the leap, saying, ‘But I cannot
manage to demonstrate that arcs SIA and IA are passed in equal times, which
is what I am seeking’. Galileo ‘sees’ that they are passed in equal times; he
has empirical proof – if one can take hypothetical behaviour in ideal situations
as empirical proof, but he lacks a ‘demonstration’. This is something that he
believes only mathematics can provide.
Del Monte was not impressed by these proofs, claiming that Galileo was a
better mathematician than a physicist. Reasonably enough, del Monte could
not believe that one body would move through an arc of 10 or 20 metres in
the same time as another, suspended by the same length of chord, would move
through only 1 or 2 cm. Further, as a mechanic, he conducted experiments
on balls rolling within iron hoops and found that Galileo’s claims were indeed
false: balls released from different positions in the lower quarter of the hoop
reached their nadir at different times.
This was yet another case where experience and common sense were at odds
with science. And, given the centrality of the pendulum in the foundation of
modern science, this case goes some way to explaining why modern science
was so late in appearing in human history, and why, when it did appear,
it was so geographically localised; it was western European science that,
within a few short decades, became universal science. Pleasingly for subsequent
scientific and social history, Galileo was not moved by del Monte’s or by
common-sense objections.
Galileo’s Mature Pendulum Claims
Galileo’s physics, with his notion of circular inertia and including his
pendulum analysis, was complete by 1610, when he was in his mid forties,
but this physics was not widely published or disseminated; fatefully, he was
introduced, in 1608, to the telescope, which side-tracked him into astronomical observations and into decades-long defence of the Copernican world
system. Galileo’s pendulum investigations would appear in public in his later,
celebrated defences of Copernicanism: his 1633 Dialogue and his 1638
Discourse. The well-known claims about pendulums were:
• Law of Weight Independence: period is independent of weight.
• Law of Amplitude Independence: period is independent of amplitude.
• Law of Length: period varies directly as length; specifically the square
root of length.
History and Philosophy: Pendulum Motion 221
• Law of Isochrony: for any pendulum, all swings take the same time;
pendulum motion is isochronous.
Although now routine and repeated in textbooks and ‘replicated’ in school
practical classes, these claims, when made, were, with good reason, very
contentious and disputed. Much about science and the nature of science can
be learned from the disputes about the legitimacy of mathematisation and
idealisation in science that they occasioned.
The pendulum claims can be briefly documented as follows. In the First Day
of his 1638 Dialogue, Galileo expresses his law of weight independence as
Accordingly I took two balls, one of lead and one of cork, the former more than
a hundred times heavier than the latter, and suspended them by means of two
equal fine threads, each four or five cubits long. Pulling each ball aside from the
perpendicular, I let them go at the same instant, and they, falling along the
circumferences of circles having these equal strings for semi-diameters, passed
beyond the perpendicular and returned along the same path. This free vibration
repeated a hundred times showed clearly that the heavy body maintains so nearly
the period of the light body that neither in a hundred swings nor even in a
thousand will the former anticipate the latter by as much as a single moment, so
perfectly do they keep step.
(Galileo 1638/1954, p. 84)
In the Fourth Day of the 1633 Dialogue, Galileo states his law of amplitude
independence, saying:
[It is] truly remarkable . . . that the same pendulum makes its oscillations with
the same frequency, or very little different – almost imperceptibly – whether these
are made through large arcs or very small ones along a given circumference. I
mean that if we remove the pendulum from the perpendicular just one, two, or
three degrees, or on the other hand seventy degrees or eighty degrees, or even up
to a whole quadrant, it will make its vibrations when it is set free with the same
frequency in either case.
(Galileo 1633/1953, p. 450)
In his final great work in mechanics, Dialogues Concerning Two New
Sciences (1638), Galileo says that:
It must be remarked that one pendulum passes through its arcs of 180º, 160º, etc
in the same time as the other swings through its 10º, 8º, degrees. . . . If two people
start to count the vibrations, the one the large, the other the small, they will
discover that after counting tens and even hundreds they will not differ by a single
vibration, not even by a fraction of one.
(Galileo 1638/1954, p. 254)
222 History and Philosophy: Pendulum Motion
In the First Day of his 1638 Dialogues, Galileo states his law of length when,
in discussing the tuning of musical instruments, he says:
As to the times of vibration of bodies suspended by threads of different lengths,
they bear to each other the same proportion as the square roots of the lengths of
the thread; or one might say the lengths are to each other as the squares of the
times; so that if one wishes to make the vibration-time of one pendulum twice
that of another, he must make its suspension four times as long. In like manner,
if one pendulum has a suspension nine times as long as another, this second
pendulum will execute three vibrations during each one of the first; from which
it follows that the lengths of the suspending cords bear to each other the [inverse]
ratio of the squares of the number of vibrations performed in the same time.
(Galileo 1638/1954, p. 96)
His fourth pendulum ‘law’, isochronous motion, is of the greatest importance for the subsequent scientific and social utilisation of the pendulum. In
the late fifteenth century, the great observer Leonardo da Vinci extensively
examined, manipulated and drew pendulums, but, as one commentator
remarks: ‘He failed, however, to recognize the fundamental properties of the
pendulum, the isochronism of its oscillation, and the rules governing its period’
(Bedini 1991, p. 5).
In the Fourth Day of the 1633 Dialogue, Galileo approaches his law of
isochrony by saying:
Take an arc made of a very smooth and polished concave hoop bending along
the curvature of the circumference ADB [Figure 6.5], so that a well-rounded and
smooth ball can run freely in it (the rim of a sieve is well suited for this experiment).
Now I say that wherever you place the ball, whether near to or far from the
ultimate limit B . . . and let it go, it will arrive at the point B in equal times . . . a
truly remarkable phenomenon.
(Galileo 1633/1953 p. 451)
In the First Day of the 1638 Dialogues, Galileo writes of his law of isochrony
But observe this: having pulled aside the pendulum of lead, say through an arc
of fifty degrees, and set it free, it swings beyond the perpendicular almost fifty
degrees, thus describing an arc of nearly one hundred degrees; on the return
swing it describes a little smaller arc; and after a large number of such vibrations
it finally comes to rest. Each vibration, whether of ninety, fifty, twenty, ten or
four degrees occupies the same time: accordingly the speed of the moving body
keeps diminishing since in equal intervals of time, it traverses arcs which grow
smaller and smaller. . . . Precisely the same thing happens with the pendulum of
(Galileo 1638/1954, p. 84)
History and Philosophy: Pendulum Motion 223
Since Galileo’s time, there have been countless empirical demonstrations or
tests of these mathematical proofs. This is a staple item in school physics pro –
grammes. The diagram in Figure 6.6 is from an eighteenth-century physics
text (Guadagni 1764, p. 32), and the photo is of a modern reproduction
exhibited in the Pavia University museum (Falomo et al. 2014). Both the figure
and the photo derive from Galileo’s geometrical proof of his chords theorem,
shown in Figure 6.4. It is the crucial scientific sequence of moving from
thought to reality via artisan craft and technology. In the text picture, two
balls are simultaneously released from the apex A; one travels the chord AEF,
one free-falls ADB, and the bells at the end of each traverse sound at the same
time. If they do not so sound, then the question becomes one of either finding
material ‘accidents’, including experimenter inadequacies, that interfere with
the ‘ideal’ or ‘world on paper’ or deciding that the theorem does not apply
in the world.
This Law of Chords is close to a proof for isochrony of pendulum motion.
The law has shown that the time of descent down inclined planes (chords) is
the same, provided the planes are inscribed in a circle and originate at the
apex or terminate at the nadir. This means that amplitude does not affect
the time. This is highly suggestive of a Law of Arcs, where amplitude should
not affect the time of descent or time of swing. This law is proved later in
Theorem XXII of the Dialogues, which also demonstrates the counter-intuitive
224 History and Philosophy: Pendulum Motion
Figure 6.5 Ball in Hoop
Source: Galileo 1633/1953, p. 451
proposition that the quickest time of descent in free fall is not along the
shortest path (see Figure 6.7). He says:
From the preceding it is possible to infer that the path of quickest descent from
one point to another is not the shortest path, namely a straight line, but the arc
of a circle. In the quadrant BAEC, having the side BC vertical, divide the arc AC
into any number of equal parts, AD, DE, EF, FG, GC, and from C draw straight
lines to the points A, D, E, F, G; draw also the straight lines AD, DE, EF, FG,
GC. Evidently descent along the path ADC is quicker than along AC alone or
along DC from rest at D. . . . Therefore, along the five chords, ADEFGC, descent
will be more rapid than along the four, ADEFC. Consequently the nearer the
inscribed polygon approaches a circle, the shorter is the time required for descent
from A to C.
(Galileo 1638/1954 p. 239)
Galileo realised that any truly isochronous movement could be used as
a clock; one only has to calibrate the movement against the duration of a
History and Philosophy: Pendulum Motion 225
Figure 6.6 Law of Chords Demonstration; Eighteenth-Century Text and Modern
: d
tA. 8
2000). 2000).
sidereal day and create some mechanism and technology for displaying the
steady isochronous movement. He further recognised that the swinging
pendulum was the ideal of such movement. Late in his life, Galileo proposed
using the pendulum as a clock, and his son Vincenzio produced sketches of
the proposal.9
Problems with Galileo’s Account and the Limits of
These marvellous proofs of Galileo did not receive universal acclaim: on the
contrary, learned scholars were quick to point out considerable empirical and
philosophical problems with them. The empirical problems were examples
where the world did not ‘correspond punctually’ to the events demonstrated
mathematically by Galileo. In his more candid moments, Galileo acknowledged that events do not always correspond to his theory; that the material
world and his so-called ‘world on paper’, the theoretical world, did not
correspond. Immediately after mathematically establishing his famous law of
parabolic motion of projectiles, he remarks that:
I grant that these conclusions proved in the abstract will be different when applied
in the concrete and will be fallacious to this extent, that neither will the horizontal
226 History and Philosophy: Pendulum Motion
Figure 6.7 Galileo’s Law of Chords Proof
Source: Galileo 1638/1954 p. 239
motion be uniform nor the natural acceleration be in the ratio assumed, nor the
path of the projectile a parabola.
(Galileo 1638/1954, p. 251)
One can imagine the reaction of del Monte and other hardworking
Aristotelian natural philosophers and mechanicians when presented with such
a qualification. It confounded the basic Aristotelian and empiricist objective
of science, namely to tell us accurately about the world in which we live. The
law of parabolic motion was supposedly true, but not of the world we
experience: this was indeed as difficult to understand for del Monte as it is
for present-day students.
As early as 1636, the notable mathematician, theologian and ‘net worker’
of all contemporary natural philosophers, Marin Mersenne (1588–1648),
reproduced Galileo’s experiments and not only agreed with del Monte, but
doubted whether Galileo had ever conducted the experiments (Koyré 1968,
pp. 113–117). Modern researchers have duplicated the experimental conditions described by Galileo and have found that they do not give the results
that Galileo claimed (Ariotti 1968, Naylor 1974, 1980, 1989).
Del Monte and others repeatedly pointed out that pendulums do not behave
as Galileo maintained; Galileo never tired of saying that ideal pendulums would
obey the mathematically derived rules. Del Monte retorted that physics was
to be about this world, not an imaginary, mathematical world. Opposition to
the mathematising of physics was a deeply held, Aristotelian, and more generally empiricist, conviction (Lennox 1986). The British empiricist Hutchinson
would later say of the geometrical constructions of Newton’s Principia that
they were just ‘cobwebs of circles and lines to catch flies in’ (Cantor 1991,
p. 219).10
It is easy to appreciate the empirical reasons for opposition to Galileo’s law.
The overriding argument was that, if the law were true, pendulums would
be perpetual motion machines, which clearly they are not. An isochronic
pendulum is one in which the period of the first swing is equal to that of all
subsequent swings: this implies perpetual motion. We know that any
pendulum, when let swing, will very soon come to a halt: the period of the
last swing will be by no means the same as the first. Furthermore, it was plain
to see that cork and lead pendulums have a slightly different frequency, and
that large-amplitude swings do take somewhat longer than small-amplitude
swings for the same pendulum length. All of this was pointed out to Galileo,
and he was reminded of Aristotle’s basic methodological claim that the
evidence of the senses is to be preferred over other evidence in developing an
understanding of the world.
The fundamental laws of classical mechanics are not verified in experience;
further, their direct verification is fundamentally impossible. Herbert
Butterfield (1900–1979) conveys something of the problem that Galileo and
Newton had in forging their new science:11
They were discussing not real bodies as we actually observe them in the real world,
but geometrical bodies moving in a world without resistance and without gravity
History and Philosophy: Pendulum Motion 227
– moving in that boundless emptiness of Euclidean space which Aristotle had
regarded as unthinkable. In the long run, therefore, we have to recognise that here
was a problem of a fundamental nature, and it could not be solved by close
observation within the framework of the older system of ideas – it required a
transposition in the mind.
(Butterfield 1949/1957, p. 5)
An objectivist, non-empiricist account of science stresses that the transposition in the mind is really the creation of a new theoretical object or system.
Even for Galileo, the pendulum seemed to stop at the top of its swing; it was
only in his theory, not his perceptual mind, that it continued in smooth
The Pendulum and Timekeeping
It is useful to outline some of the later developments in the science of pendulum
motion. They show the interaction of mathematics and experiment in scientific
development, and the importance to science of the development of theoretical systems and of conceptual frameworks within which to interpret and
interrogate nature. Both of these points are important for the teaching of
pendulum motion.
The pendulum played more than a scientific role in the formation of the
modern world. The pendulum was central to the horological revolution that
was intimately part of the scientific revolution. Huygens, in 1673, following
Galileo’s epochal analysis of pendulum motion, utilised the pendulum in
clockwork and so provided the world’s first accurate measure of time (Yoder
1988). The accuracy of mechanical clocks went, in the space of a couple of
decades, from plus or minus half-an-hour per day to a few seconds per day.
This abrupt increase in accuracy of timing enabled hitherto unimagined
degrees of precision measurement in mechanics, navigation and astronomy.
It ushered in the world of precision characteristic of the scientific revolution
(Wise 1995). Time could then confidently be expressed as an independent
variable in the investigation of nature.
Christiaan Huygens (1629–1695) refined Galileo’s pendulum laws and was
the first to use these refined laws to create a pendulum clock. Huygens modified
Galileo’s analysis by showing, mathematically, that it was movement on the
cycloid, not the circle, that was isochronous. He provides the following
account of this discovery:
We have discovered a line whose curvature is marvellously and quite rationally
suited to give the required equality to the pendulum. . . . This line is the path traced
out in air by a nail which is fixed to the circumference of a rotating wheel which
revolves continuously. The geometers of the present age have called this line a
cycloid and have carefully investigated its many other properties. Of interest to
228 History and Philosophy: Pendulum Motion
us is what we have called the power of this line to measure time, which we found
not by expecting this but only by following in the footsteps of geometry.
(Huygens 1673/1986, p. 11)
The cycloid is the curve described by a point P rigidly attached to a circle
C that rolls, without sliding, on a fixed line AB. The full arc ADB has a length
equal to 8r (r = the radius of the generating circle). A heavy point that travels
along an arc of cycloid placed in a vertical position, with the concavity
pointing upwards, will always take the same amount of time to reach the
lowest point, independent of the point from which it is released.
Having shown mathematically that the cycloid was isochronous, Huygens
then devised a simple way of making a suspended pendulum swing in a
cycloidal path – he made two metal cycloidal cheeks and caused the pendulum
to swing between them. Huygens’ first pendulum clock (Figure 6.9) was
accurate to 1 minute per day; working with the best clockmakers, he soon
made clocks accurate to 1 second per day.
After showing that the period of a simple pendulum varied as the square
root of its length, Huygens (Huygens 1673/1986, pp. 169–170) then derived
the familiar equation for small-amplitude pendulums, that is, ones moving on
the arc of a circle:
T = 2π√(l/g)
This development by Huygens of the theory of the pendulum is an example
that fits objectivist accounts of scientific development. Despite Galileo’s
personal brilliance, he nevertheless did not appreciate or work out correctly
the implications of the theoretical object that he himself created. The theory
had unseen or unintended consequences that needed others to work out or
discover. Huygens discovered that the cycloid (Figure 6.8), not the circle, was
the vital tautochronous curve by ‘following in the footsteps of geometry’, a
guide that Aristotelian philosophy distrusted in physical affairs. This discovery
of the tautochronous curve had very little to do with sensory input. Its
justification had even less to do with sense data or other putative empirical
foundations for belief so extolled by positivists.
History and Philosophy: Pendulum Motion 229
Figure 6.8 Cycloid Generated by a Moving Circle
Huygens’ Proposal of an International Standard of
Huygens saw that, in his pendulum equation, T = 2π√(l/g), the only variable
was l, as π was a constant and, provided that the Earth was a sphere and one
stayed near to sea level, g was also constant, and mass did not figure in the
equation at all. So, all pendulums of a given length will have the same period,
whether they be in France, England, Russia, Latin America, China or
Australia. And given l and T, then g could be determined for any location.
Huygens was clever enough to see that the pendulum would solve, not only
230 History and Philosophy: Pendulum Motion
Figure 6.9 Huygens’ Pendulum Clock Mechanism
Source: Huygens 1673/1986, p. 14
r $ . i . ‘Tfyjsr.
B t * ‘
j o
j o
f 5
Ce/rfj’.’^osc. _
C e n ir .^ r . IV Ij
c (
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2000). 2000).
the timekeeping and longitude problems, but an additional vexing problem,
namely establishing an international length standard, and, in 1673, he
proposed the length of a seconds pendulum (a pendulum that beats in seconds;
that is, whose period is 2 seconds) to be the international unit.13 The length
of the seconds pendulum was experimentally determined by adjusting a
pendulum so that it oscillated 24 × 60 × 30 times in a sideral day (each
oscillation takes 2 seconds); that is, between successive transits of a fixed star
across the centre of a graduated telescope lens (the sideral day being slightly
longer than a solar day).
This seems like a daunting task, but it was not so overwhelming. Huygens
and others knew that the length l of a pendulum varied as the square of period
or T2. So the length of a seconds pendulum is to the length of any arbitrary
pendulum as 1/T2. However, (1/T2) ∝ n2, where n is the number of oscillations
of a pendulum in 1 hour. As Meli observes: ‘therefore, by counting the number
of oscillations and the length of his pendulum, Huygens could determine the
length of the seconds pendulum’ (Meli 2006, p. 205).
Having an international unit of length, or even a national unit, was a major
contribution to simplifying the chaotic state of measurement existing in science
and everyday life. Within France, as in other countries, the unit of length varied
from purpose to purpose (timber to cloth), city to city, and even within cities.
This was a significant problem for commerce, trade, construction, military
hardware and technology, to say nothing of science. Many attempts had been
made to simplify and unify the chaotic French system. One estimate is that,
in France alone, there were 250,000 different, local measures of length, weight
and volume (Alder 1995, p. 43), and each European state had comparable
confusion of abundance, as did, of course, all other nations and cultures.
In the standard formulas, using standard approximations for g and π, it is
easy to show that the length of a seconds pendulum will be 1 metre.
T = 2π√(l/g)
So T2 = 4π2 (l/g)
So l = T2g/4π2
Substitute T = 2s (each beat is a second), g = 9.8 ms–2, π2 = 9.8
Then l = 0.993577 m, or very approximately 1 m
This result can reliably be demonstrated with even the crudest 1 m pendulum
– a heavy nut on a piece of string suffices: ten complete swings will take
20 seconds; twenty complete swings will take 40 seconds. A great virtue of
the seconds pendulum as the international length standard was that it was a
fully ‘natural’ standard; it was something fixed by nature, unlike standards
based on the length of a king’s arm or foot. And, of course, an international
length standard would provide a related volume standard and, hence, a mass
standard when the unit volume was filled with rainwater. A kilogram is the
weight of 1 litre (1,000 cc) of water. All of this can be engagingly reproduced
with classes; they can be challenged to think through ways in which a mass
standard can be derived from a length standard.
History and Philosophy: Pendulum Motion 231
It is not accidental that, 200 years after Huygens, the General Conference
on Weights and Measures meeting in Paris defined the standard universal
metre as ‘the length of path travelled by light in vacuum during a time interval
of 1/299,792,458 of a second’.14 This seemingly bizarre and arbitrary figure
is within 1 millimetre of Huygens’ original and entirely natural length
standard, and it was so chosen precisely to replicate the length of the seconds
pendulum. Unfortunately, it is the former, not the latter, that students meet
in the opening pages of their science texts, which confirms their worst fears
about the ‘strangeness’ of science. In the definition of standards, science gets
off to a bad pedagogical start.
The Pendulum and Determination of the Shape of
the Earth
Huygens’ proposal depended on g being constant around the world (at least
at sea level); it depended on the Earth being spherical. This seemed a most
reasonable assumption. Indeed, to say that the Earth was not regular and
spherical was tantamount to casting aspersions on the Creator: surely God
the Almighty would not make a misshapen earth. But, in 1673, contrary to
all expectation, this assumption was brought into question by the behaviour
of the pendulum.
When Jean-Dominique Cassini (1625–1712) became director of the French
Académie Royale des Sciences in 1669, he sent expeditions into different
parts of the world to observe the longitudes of localities for the perfection of
geography and navigation. The second such voyage was Jean Richer’s to
Cayenne in 1672–1673 (Olmsted 1942). Cayenne was in French Guiana, at
a latitude of approximately 5º N. It was chosen as a site for astronomical
observations because equatorial observations were minimally affected by
refraction of light passing through the Earth’s atmosphere – the observer, the
sun and the planets were all in the same plane.
The primary purpose of Richer’s voyage was to ascertain the value of solar
parallax and to correct the tables of refraction used by navigators and
astronomers. A secondary consideration was checking the reliability of
Huygens’ marine pendulum clocks, which were being carried for the purpose
of establishing Cayenne’s exact longitude. Richer surprisingly found that a
pendulum, set to swing in seconds at Paris, had to be shortened in order to
swing in seconds at Cayenne. Not shortened by much – 2.8 mm, about the
thickness of a matchstick – but nevertheless shortened. Richer found that a
Paris seconds clock lost two and a half minutes daily at Cayenne; the time
from noon to noon was 23 hours, 57 min, 32 s. The only apparent explanation
for the slowing of the pendulum at the equator was that g is less at the equator
than at Paris and the poles; in other words, that the Earth is a ‘flattened’ sphere,
an oblate, with the equator being further from the centre and so having less
gravitational attraction than the poles. The methodological lessons of this
episode will be fleshed out below.15
232 History and Philosophy: Pendulum Motion
The Pendulum in Newton’s Mechanics
The pendulum played a comparable role in Newton’s work to what it had
for Galileo and Huygens. Newton used the pendulum to determine the
gravitational constant g, to improve timekeeping, to disprove the existence of
the mechanical philosophers’ ether presumption, to show the proportionality
of mass to weight, to determine the coefficient of elasticity of bodies, to
investigate the laws of impact and to determine the speed of sound. Richard
Westfall, a distinguished Newtonian scholar, wrote that: ‘the pendulum
became the most important instrument of seventeenth-century science . . .
Without it the seventeenth century could not have begot the world of precision’ (Westfall 1990, p. 67). Concerning the pendulum’s role in Newton’s
science, Westfall has said that, ‘It is not too much to assert that without
the pendulum there would have been no Principia’ (Westfall, 1990, p. 82).
No small claim to fame!
Determination of the Gravitational Constant
Marin Mersenne (1588–1648) used Galileo’s circular pendulum, and its
theoretical underpinnings, to ascertain the value of the gravitational constant g. Neither Galileo, nor other seventeenth-century scientists, used the
idea of g in its modern sense of acceleration. Their gravitational constant was
the distance that all bodies fall in the first second after their release – this is,
numerically, one-half of the modern constant of acceleration.
Mersenne’s earlier investigations pointed to 3 Parisian feet being the length
of the seconds pendulum, and so, in a famed experiment (1647), he held
a 3-horological-foot (≈ 39 inch) pendulum (a seconds pendulum) out from a
wall and released it along with a free-falling mass (see Figure 6.10). He
adjusted a platform under the mass until he heard just one sound (the
pendulum striking the wall and the mass striking the platform simultaneously).
He reasoned that this should give him the length of free fall in half-a-second
(a complete one-way swing of the pendulum taking 1 second), and so, by the
times-squared rule (four times the length so determined), he could calculate
the length of free fall in 1 second, the gravitational constant of the period.
Huygens and Mersenne thought that the ear could not separate sounds to
better than 6 inches of free fall, and so thought that inaccuracy was going
to be ‘built into’ mechanics.16 All realised that a seconds pendulum was the
only way of getting a duration of 1 second; this was to be the key to obtaining
a measurement of the gravitational constant, the length fallen in the first
second of free fall.
Demonstration of Newton’s Laws
Three properties made the pendulum an ideal vehicle for the demonstration
of Newton’s laws and the investigation of collisions: pendulums of the same
length reach their nadir at the same time, irrespective of where they are
History and Philosophy: Pendulum Motion 233
released; they reach their nadir simultaneously, regardless of their mass; and
their velocity at the nadir is proportional to the length of the chord joining
the nadir to the point of release. Additionally, the paths of the colliding bodies
could be constrained. Newton set up pendulum collision experiments to
demonstrate his laws of motion and to explicate his nascent conservation law.
He concluded one demonstration by writing:
By the meeting and collision of bodies, the quantity of motion, obtained from the
sum of the motions directed towards the same way, or from the difference of those
that were directed towards contrary ways, was never changed.
(Newton 1729/1934, pp. 22–24)
Newton does not label this the conservation of momentum. He speaks of
conservation of ‘motion’, but it is modern momentum, mv, a vector quantity,
that he is describing. In current terminology, his conclusion is given by the
m1v1 + m2v1 = m1v2 + m2v2
The pendulum has here made a significant contribution to the foundation
of classical mechanics. The law of conservation of momentum was true for
both elastic (where there is no energy absorbed in the collision itself) and
inelastic collisions (where energy is absorbed). For instance, if the above
pendulums were made of putty, then, when they collided, they would deform
and simply come to a halt; there would be no motion after the collision. In
234 History and Philosophy: Pendulum Motion
Figure 6.10 Mersenne’s Gravitational-Constant Determination
this situation, one could hardly talk of conservation of ‘motion’ – although
Descartes, for instance, resolutely maintained that the quantity mv was the
basic measure of ‘motion’, and mv was the basic entity that was conserved in
the world. A theory-protecting strategy was to claim that, although gross
motion (the putty balls) ceased, the motion of the invisible corpuscules of the
mechanical worldview did not cease; they were energised.
Newton’s third law, ‘action is equal to reaction’, was demonstrated by
Newton using two long (3–4 m) pendulums and having them collide. He used
a result of Galileo (that the speed of a pendulum at its lowest point is
proportional to the chord of its arc) and applied it to the collision by
comparing the quantities mass multiplied by chord length, before and after
collision (Gauld 1998). For centuries, Newton’s ‘cradle’ apparatus, which
wonderfully manifests this conservation law, has intrigued students and
citizens (Gauld 2006).
Unifying Terrestrial and Celestial Mechanics
The major question for Newton and natural philosophers was whether
Newton’s postulated attractive force between bodies was truly universal: that
is, did it apply, not only to bodies on Earth, but also between bodies in the
solar system? Aristotle, as with all ancient philosophers, made a clear
distinction between the heavenly and terrestrial (sub-lunar) realms: the
former being eternal, unchanging and perfect, the realm of the Gods; the latter
being changeable, imperfect and corruptible, the realm of man. It was thus
‘natural’ that the science of both realms would be different, and, to speak
anachronistically, laws applying to the terrestrial realm would not apply to
the celestial realm. This cosmic divide lasted for 2,000 years.
It was the analysis of pendulum motion that rendered untenable the
celestial/terrestrial distinction and enabled the move from ‘the closed world to
the infinite universe’ (Koyré 1957). The same laws governing the pendulum
were extended to the Moon, and then to the planets. The long-standing
celestial/terrestrial distinction in physics was dissolved. The same laws were
seen to apply in the heavens as on Earth: there was just one universe, a unitary
solar system.
At 22 years of age, while ensconced in Lincolnshire to avoid London’s
Great Plague, Newton began to speculate that the Moon’s orbit and an apple’s
fall might have a common cause (Herivel 1965, pp. 65–69). He was able to
calculate that, in 1 second, while travelling about 1 kilometre in its orbit, the
Moon deviates from a straight-line path by about one-twentieth of an inch. In
the same period of time, an object projected horizontally on the Earth would
fall about 16 feet. The ratio of the Moon’s ‘fall’ to the apple’s fall is then
about 1:3,700. This was very close to the ratio of the square of the apple’s
distance from the Earth’s centre (the Earth’s radius), to the square of the
Moon’s distance from the Earth’s centre, 1:3,600. Was this a cosmic
coincidence? Or did the Earth’s gravitational attraction apply equally to the
apple and the Moon?
History and Philosophy: Pendulum Motion 235
Following the dictates of his own method, Newton then experimentally
investigated whether the derived consequences are seen in reality. He deferred
to Huygens’ experimental measurement, saying:
And with this very force we actually find that bodies here upon earth do really
descend; for a pendulum oscillating seconds in the latitude of Paris will be 3 Paris
feet, and 8 lines 1
⁄2 in length, as Mr Huygens has observed. And the space which
a heavy body describes by falling in one second of time is to half the length of
this pendulum as the square of the ratio of the circumference of a circle to its
diameter (as Mr Huygens has also shown), and is therefore 15 Paris feet, 1 inch,
1 line 7/9. And therefore the force by which the moon is retained in its orbit
becomes, at the very surface of the earth, equal to the force of gravity which we
observe in heavy bodies there.
(Newton 1729/1934, p. 408)
Newton then draws his conclusion: ‘And therefore the force by which the
moon is retained in its orbit is that very same force which we commonly call
gravity.’ The pendulum had brought heaven down to Earth.17
Timekeeping as the Solution of the Longitude
From the fifteenth century, when traders and explorers began journeying
away from European shores, problems of navigation and position-finding,
especially the determination of longitude, became more and more acute.
Chinese and Polynesian seafarers had the same problem. Gemma Frisius
(1508–1555) identified accurate timekeeping as the way to solve the longitude
problem. In 1530, he advised:
In our times we have seen the appearance of various small clocks, capably
constructed, which, for their modest dimensions, provide no problem to those
who travel. These clocks operate for 24 hours, in fact when convenient, they
continue to operate with a perpetual movement. And it is with their help that the
longitude can be found. . . . When one is on course for 15 or 20 miles, and wishing to know how distant one is from the point of departure, it would be preferable
to wait until the clock is at an exact division of time, and at the same time, with
the assistance of the astrolabe, as well as of our globe, to seek the hour of the
place in which we find ourselves. . . . In this manner it is possible to find the
longitude, even in a distance of a thousand miles, even without knowing where
we have passed, and without knowing the distance travelled.
(Pogo 1935, p. 470)
Frisius understood that the Earth makes one revolution of 360º in 24 hours,
and thus, in 1 hour, it rotates through 15º, or 1º each 4 minutes. Thus,
although the line of zero longitude is arbitrary, there is an objective
relationship between time and longitude. However, the theoretical solution
236 History and Philosophy: Pendulum Motion
was considerably ahead of the technology available and the ability of sixteenthcentury clockmakers. The pendulum was to play a pivotal role in solving this
pressing problem.18
Nearly all the great scientists of the seventeenth century – including Galileo,
Huygens, Newton and Hooke – worked intimately with clockmakers and used
their analysis of pendulum motion, specifically its isochronism, in the creation
of more accurate pendulum clocks and then portable watches, especially the
marine chronometer, with a view to accurate longitude determination.
Dava Sobel (1995) provides a very readable, bestseller account of the British
efforts at solving the problem of longitude. Sadly, few of the hundreds of
thousands of readers across twenty languages would relate the longitude story
to their pendulum studies in school science, because the former was never
mentioned alongside the latter, and Sobel leaves out completely the crucial
methodological matters that lay at the heart of Galileo’s discoveries. For
Galileo and timekeeping, she disappointingly repeats the chandelier mythology
(Sobel 1995, p. 37).
Huygens was aware of the clock’s role in solving the problem of longitude,
saying, ‘they are especially well suited for celestial observations and for
measuring the longitudes of various locations by navigators’ (Huygens
1673/1986, p. 8). Captain Robert Holmes, in 1664, gave the clocks their most
extensive sea tests, tests that vindicated Huygens’ faith in their ability to solve
the longitude problem. Ironically, Holmes was on a transatlantic voyage of
pillage against Dutch possessions in Africa, New Amsterdam (New York) and
South America. Huygens, in his Horologium Oscillatorium, repeats Holmes’s
account of the scientific part of this voyage.
The craftsman who first solved the technical problems of an accurate,
reliable marine chronometer was John Harrison (1693–1776) – known as
‘Longitude Harrison’ – who was born in 1693, the son of a Yorkshire
carpenter, and who died as a celebrated clockmaker in London in 1776.
Harrison died 3 years after receiving the British Longitude Board’s reluctant
final payment of the £20,000 reward established by the Longitude Act of 1714.
The prize required the then-unheard-of accuracy of 2 minutes during the
8–10 week voyage to the West Indies: Harrison’s clock would keep within 30
seconds of correct time on a voyage out to the Indies and back! After this,
the far reaches of the globe, and all islands and reefs in between, could be
accurately mapped, and European commerce, colonisation and conquest could
proceed apace. Students can be profitably engaged with all aspects of this
marvellous story, thereby learning astronomy, physics, navigation and history
(Bensky 2010).
Foucault’s Pendulum and the Earth’s Rotation
The pendulum provided the first-ever tangible and dynamic ‘proof’ of the
rotation of the Earth. On Newton’s theory, a pendulum set swinging in a
particular plane should continue to swing indefinitely in that same plane, the
History and Philosophy: Pendulum Motion 237
only forces on the bob being the tension in the cord and its weight directed
vertically downwards. Léon Foucault (1819–1868) – described as ‘a mediocre
pupil at school, [but] a natural physicist and an incomparable experimenter’
(Dugas 1988, p. 380) – ‘saw’ that, if a pendulum were placed exactly at the
North Pole and suspended in such a way that the point of suspension were
free to rotate (that is, it did not constrain the pendulum’s movement by
applying torque), then:
If the oscillations can continue for twenty-four hours, in this time the plane will
execute a whole revolution about the vertical through the point of suspension
. . . at the pole, the experiment must succeed in all its purity.
(Dugas 1988, p. 380)
As the pendulum is moved from the pole to the equator, Foucault easily
showed that, if T1 is the time in which the plane of the pendulum rotates 360º,
and T is the period of rotation of the earth, and β is the latitude where the
experiment is being conducted, then:
T1 = T/sinβ
From the formula, it can be seen that, at the poles, T1 = T (as sinβ = sin 90º
= 1); whereas, at the equator, T1 = ∞ (or infinity, as sin 0º = 0), and, thus,
there is no rotation of the plane of oscillation at the equator.
On 2 February 1851, Foucault invited the French scientific community ‘to
come see the Earth turn, tomorrow, from three to five, at Meridian Hall of
the Paris Observatory’. Foucault’s long and massive pendulum provided an
experimental ‘proof’ of the Copernican theory, something that had eluded
Galileo, Newton and all the other mathematical and scientific luminaries who
sought it (Aczel 2003, 2004, Tobin 2003).
Until Foucault’s demonstration, all astronomical observations could be
fitted, with suitable adjustments such as those made by Tycho Brahe, to the
stationary Earth theory of the Christian tradition. The ‘legitimacy’ of such ad
hoc adjustments in order to preserve the geocentric model of the solar system
was exploited by the Catholic Church, which kept the works of Copernicus
and Galileo on the Index of Prohibited Books up until 1835 (Fantoli 1994,
p. 473). To most nineteenth-century physicists, the manifest rotation of
Foucault’s pendulum, shown in the successive knocking down of markers
placed in a circle, was a dramatic proof of the Earth’s rotation.
Some Features of Science
As mentioned in Chapter 1, and as will again be outlined in Chapter 11, the
method of this book, and its recommendation for science teachers, is to make
explicit different features of science, particularly philosophical ones, as
curriculum content is taught. When treated historically, the pendulum provides
238 History and Philosophy: Pendulum Motion
a rich source for such lessons about science. Galileo is an outstanding example
of the scientist–philosopher or philosophical scientist (to use current
terminology). He made substantial philosophical contributions in a variety of
areas: in ontology, with his distinction of primary and secondary qualities; in
epistemology, both with his criticism of authority as an arbiter of knowledge
claims and with his subordination of sensory evidence to mathematically
informed reason; in methodology, with his development of the mathematical–
experimental method; and in metaphysics, with his critique of the Aristotelian
causal categories and rejection of teleology as an explanatory principle.19 It
is unfortunate that, despite his important contributions to philosophy, and
despite his acknowledged influence on just about all philosophers of the
seventeenth century and on such subsequent philosophers as Kant and Husserl,
Galileo makes, at best, a cameo appearance in most histories of philosophy.
And, of course, his philosophical achievements are ignored in science texts.
However, historically and philosophically informed pendulum teaching can
redress this lamentable oversight.
Well-informed teachers should also be aware of the pitfalls associated with
drawing philosophical lessons from the life of Galileo; the pitfalls are akin to
those involved in drawing theological and/or philosophical lessons from the
life of Jesus (or any other of the great religious figures). Albert Schweitzer, in
his monumental The Quest of the Historical Jesus (Schweitzer 1910/1954),
keenly observed that:
Thus each successive epoch of theology found its own thoughts in Jesus; that was,
indeed, the only way in which it could make Him live. But it was not only each
epoch that founds its reflection in Jesus; each individual created Him in accordance
with his own character.
(Schweitzer 1910/1954, p. 4)
The lesson from Schweitzer is not to avoid drawing lessons from historical
figures and their work, but just be careful and considered when doing so: the
historical figure cannot be treated as a Rorschach inkblot.20 With this caveat,
the following are some of the features of science that can arise in such teaching
of the pendulum.
Scientific Methodology
The history of pendulum motion study shows the limitations of both inductivism and falsificationism as accounts of scientific method. Concerning
inductivism, clearly Galileo did not induce his ‘marvellous properties’,
or laws, of pendulum motion from looking at various pendulums – balls in
hoops, chandeliers, swings, mechanical regulators, cork and iron bobs on
strings and so on – and then generalise towards universal statements of what
he saw in the particular instances. One enthusiast for inductivism has
commented that:
History and Philosophy: Pendulum Motion 239
For him [Galileo], the facts based on them [observations] were treated as facts,
and not related to some preconceived idea. . . . The facts of observation might
or might not fit into an acknowledged scheme of the universe, but the important thing, in Galileo’s opinion, was to accept the facts and build the theory to fit
(H.D. Anthony in Chalmers 1976/2013, p. 2)
This was, on the contrary, the methodology of the Aristotelians, for whom
the facts of experience were the starting points for science and who
unsuccessfully urged Galileo to be true to the facts; it was also the methodology
of Descartes, who concluded that, because the facts were so messy and erratic,
no science of pendulum motion was possible. What was seen to happen in
the world was important, but it did not have the importance that inductivism
attributed to it. This is well recognised by I.E. Drabkin, who said of
Aristotelian mechanics that it was,
impeded not by insufficient observation and excessive speculation but by too close
an adherence to the data of observation . . . an adherence to the phenomena of
nature so close as to prevent the abstraction therefrom of the ideal case.
(Drabkin 1938, pp. 69, 82)
Falsificationism, the view that the essence of science is to reject theories that
are contradicted by the facts, which is the methodological position associated
with Karl Popper, fares not much better than inductivism when dealing with
the pendulum example. Inductivism and falsificationism are two sides of the
same empiricist coin, and so it is to be expected that, where the first fails, the
second will also fail.
The type of situation faced by Huygens – the revision of a theory in the
light of contrary evidence – recurs constantly in the development of science.
Richer’s claim that the pendulum clock slows in equatorial regions nicely
illustrates some key methodological matters about science and about theory
testing. The entrenched belief since Erastosthenes in the second century BC
was that the Earth was spherical (theory T), and, on the assumption that
gravity alone affects the period of a constant-length pendulum, the
observational implication was that the period at Paris and the period at
Cayenne of Huygens’ seconds pendulum would be the same (O). Thus,
T implies O:
T → O
However, Richer seemingly found that the period at Cayenne was longer
(~ O). Thus, on simple, falsificationist views of theory testing such as were
enunciated first by Huygens himself, and famously developed by the
philosopher Karl Popper early in the twentieth century (Popper 1934/1959),
we have:
240 History and Philosophy: Pendulum Motion
T → O
∴ ~T
But theory testing is never so simple – a matter that was recognised by
Popper and articulated by Thomas Kuhn (1970) and Imre Lakatos (1970).
In the seventeenth century, many upholders of T just denied the second
premise, ~O. The astronomer Jean Picard, for instance, did not accept Richer’s
findings. Rather than accept the message of varying gravitation, he doubted
the messenger. Similarly, Huygens did not think highly of Richer as an
experimentalist, especially as the sea captain had dropped and smashed one
of Huygens’ clocks.
Others saw that theories did not confront evidence on their own; there was
always an ‘other things being equal’ assumption made in theory test; there
were ceteris paribus clauses (C) that accompanied the theory into the
experiment. These clauses characteristically included statements about the
reliability of the instruments, the competence of the observer, the assumed
empirical state of affairs, theoretical and mathematical devices used in deriving
O, and so on. Thus:
T + C → O
∴ ~T or ~C
More and more evidence came in, and from other experimenters, including
Sir Edmund Halley, confirming Richer’s observations. Thus ~O became
established as a scientific fact, to use Fleck’s terminology (Fleck 1935/1979),
and upholders of T, the spherical Earth hypothesis, had to adjust to it.
People who maintained belief in T reasonably said that the assumption that
other things were equal was mistaken. These, in principle, were legitimate
concerns. This was not easy; giving up established theories in science is never
easy, especially as the alternative was to accept that the Earth was oblate in
shape, an ungainly shape for the all-powerful, all-knowing Creator to have
fashioned. There were a number of obvious items in C that could be pointed
to as the cause of the pendulum slowing:
• C1: The experimenter was incompetent.
• C2: Humidity in the tropics caused the pendulum to slow because the air
was denser.
• C3: Heat in the tropics caused the pendulum to expand, hence, it beat
• C4: The tropical environment caused increased friction in the moving parts
of the clock.
Each of these could account for the slowing and, hence, preserve the truth of
the spherical Earth theory. However, each of them was in turn ruled out by
History and Philosophy: Pendulum Motion 241
progressively better-controlled and better-conducted experiments. Many, of
course, would say that adjustment of the thickness of a match (3 mm) as a
proportion of 1 metre (1,000 mm) was so minimal that it could just be
attributed to experimental error, or simply ignored. And, if the theory is
important, then that is an understandable tendency, but, for tougher-minded
scientists, it seemed that the long-held, and religiously endorsed, theory of the
spherical Earth had to be rejected on account of a persistent 3-mm discrepancy.
However, Huygens could see a more sophisticated explanation for the
lessening of g at the equator, while still maintaining T, the theory of a spherical
Earth. He argued:
• C5: Objects at the equator rotated faster than at Paris, and, hence, the
centrifugal force at the equator was greater; this countered the centripetal
force of gravity, hence diminishing the net downwards force (gravity) at
the equator and, hence, decreasing the speed of oscillation of the
pendulum; that is, increasing its period.
This final explanation for the slowing of equatorial pendulums, while
maintaining the spherical Earth theory, was quite legitimate and appeared to
save the theory. Many would be happy to just pick up this ‘get out of jail free’
card and continue to believe that the Earth was spherical. Huygens did not
do so. He calculated the actual centripetal force at the equator and, hence,
its effect on gravitational attraction.
In modern terms, the calculation is as follows. The equatorial object follows
a circular path covering an angle of 2π radians per day (86,400 seconds). So
angular velocity, ω, equals 2π/86,400 rad/sec, or 7.3 × 10–5 rad/sec. As the
radius of the Earth is about 6.4 × 106 m, the centripetal acceleration of the
object ac = ω2r ≈ 0.034 m/sec2. So this is the amount that the gravitational
acceleration of equatorial objects was diminished in virtue of the spinning of
the Earth.21 This translated as a 1.5 mm shortening of the seconds pendulum,
but it still left 1.5 mm unaccounted for. This is less than the thickness of a
match, and yet, for such a minute discrepancy, Huygens and Newton were
prepared to abandon the spherical Earth theory and claim that the true shape
of the earth was an oblate. For the new, quantitative science, the ‘near enough
is good enough’ mantra could not be maintained, something that students can
be taught to appreciate.22
Indeed, the testing situation is even a bit more complicated. Metaphysics
plays a role in theory appraisal. So the situation is as follows:
If theory (T) and conditions (C) and background metaphysics (M) imply
observation (O) T.C.M → O
And, if O is not the case,
Therefore, T is false, or C is false, or M is false
~M v ~T v ~C
242 History and Philosophy: Pendulum Motion
Willard van Orman Quine, and before him Pierre Duhem, elaborated the
methodological point about theory appraisal that is so apparent in the shapeof-the-Earth debate:
The totality of our so-called knowledge or beliefs, from the most casual matters
of geography and history to the profoundest laws of atomic physics or even of
pure mathematics and logic, is a man-made fabric which impinges on experience
only along the edges. . . . But the total field is so underdetermined by its boundary
conditions, experience, that there is much latitude of choice as to what statements
to reevaluate in the light of any single contrary experience.
(Quine 1953, p. 42)
The shape-of-the-Earth controversy is a wonderful episode in the history
of science. A great pedagogical story can be made, even a drama. All the
elements are there: powerful and prestigious figures, ‘no name’ outsiders,
controversy over a big issue, mathematics and serious calculations, religion
and, finally, decision-making, with ample opportunity and reason to preserve
the status quo. Sadly, however, the episode is little known and hardly ever
taught. If history and philosophy are valued, then there is good justification
for teaching the episode, but if ‘everyday, applied, immediate usefulness’ is
the guiding principle for constructing a science curriculum, then it is unlikely
ever to be taught. As Noah Feinstein, an advocate of the ‘usefulness’ position,
has written:
It [usefulness theory of curriculum] seems to suggest that the curriculum should
be stripped of canonical content that students are unlikely to find relevant to their
daily lives – such as, for instance, the shape of the earth.
(Feinstein 2011, p. 183)23
Real Versus Theoretical Objects
One way to conceptualise the methodological revolution of the seventeenth
century is to recognise the difference between real objects and such objects
when described from the standpoint of some scientific theory. Newton’s theory
of mechanics, for example, provides definitions of key concepts – momentum,
acceleration, average speed, instantaneous speed, weight, impetus, force, point
mass and so on. These concepts were hard-won theoretical constructs and are
utilised in his account of pendulum motion. At the beginning of the scientific
revolution, these concepts were only dimly seen, and were refined with time.
Consider the example of acceleration, which was initially defined by Galileo,
and by all his predecessors in the 2,000 years of natural philosophy, as rate
of increase of speed with respect to distance traversed – a natural enough
definition, given that accelerating bodies increase speed over both time and
distance, and that the passage of distance was both more measurable and more
easily experienced by sight and feel. On a galloping horse, you see poles go
History and Philosophy: Pendulum Motion 243
past, not time ticking over. Distance was measurable and so given to
geometrical depiction and analysis, in the way that time was not measurable
and so not given to precise geometrical analysis. It was only in Galileo’s
middle age that he changed the definition to the modern one of rate of change
of speed with respect to time elapsed. By 1604, Galileo had realised that
distances covered by a uniformly accelerating body (free fall) increased as the
square of the time elapsed. However, at this time and at least for another 6
years, he believed that speed increased in direct proportion to distance fallen.
The two beliefs, also held by Descartes, are inconsistent. Sometime after 1610,
he moved to the modern definition of acceleration and made his beliefs
consistent.24 Thirty years later, he refers to this in Day Three of his New
Sciences, where he writes: ‘A motion is said to be uniformly accelerated, when
starting from rest, it acquires, during equal time-intervals, equal increments
of speed’ (Galileo 1638/1954, p. 162).
Without such a change of definition, the fundamental laws of free fall
would not have been discovered. A good deal of a child’s school life would
pass while waiting for him or her to construct the modern definition of
acceleration, and, hence, most teachers routinely commence mechanics classes
by giving students the modern definition. The alternative is not considered,
even by the strongest advocates of discovery learning or of minimally guided
instruction. Yet the two options, and the historical story, certainly repay
discussion and analysis; students can appreciate the centrality of conceptual
‘discovery’ in science. However, as with so much, this depends on teachers
knowing the history and philosophy of what they are teaching.
Although less clearly defined than with Newton, interlocking concepts
formed the conceptual structure of Galileo’s physics and provided the meaning
of key terms. This is Galileo’s ‘world on paper’, as he referred to it, or his
proto-theoretical system. This is the indispensable scaffolding of science that
children need to be provided with; the scaffolding needs to be, in some sense,
transmitted to them. For realists, there is also a world of material and other
objects that exists apart from Galileo’s theorising. We can see in Galileo’s
practice a most important intervening layer emerging between theory and the
real world – the realm of theorised objects. These are natural objects as
conceived and described by the relevant theoretical concepts. Planets and
falling apples have colour, texture, irregular surfaces, heat, solidity and any
number of other properties and relations, but, when they become the subject
matter of mechanics, they are merely point masses with specified accelerations.
When thus conceptualised, they are no longer natural objects, but theorised
objects. In a similar way, when apples are considered by economists, they
become theorised objects of a different sort – commodities, with specific
exchange values. When botanists consider apples, they create yet other
theoretical objects. For Galileo, a sphere of lead on the end of a length of
rope, swinging in air, became, in his mechanical theory, a pendulum conceived
as a point mass at the end of a weightless chord suspended from a frictionless fulcrum. The theorised objects are created or constructed by scientists;
244 History and Philosophy: Pendulum Motion
contrary to the claims of many constructivists and sociologists of scientific
knowledge, the real objects are not so created. Failure to recognise this is the
cause of much philosophical confusion.
Galileo did not just develop a system of rational mechanics in the same way
as the medieval scientists who constructed mathematical models of physical
systems and then proceeded no further. In contrast, Galileo’s theoretical
objects were the means for engaging with and working in the natural world.
For him, the theoretical object provided a plan for interfering with the material
world and, where need be, for making it in the image of the theoretical. When
del Monte told Galileo that he had done an experiment with balls in an iron
hoop and the balls did not behave as Galileo asserted, Galileo replied that
the hoop must not have been smooth enough, that the balls were not spherical
enough and so on. These suggestions for improving the experiment are driven
by the theoretical object that Galileo had already constructed. This told
Galileo the things that had to be corrected in the experiment. Without the
theoretical object, he would not have known whether to correct for the colour
of the ball, the material of the hoop, the diameter of the hoop, the mass of
the ball, the time of day or any of a hundred other factors. It is this aspect of
Galileo’s work that moved Immanuel Kant to say that, with Galileo, ‘a light
broke upon all students of nature’ because he demonstrated that:
Reason has insight only into that which it produces after a plan of its own . . . It
is thus that the study of nature has entered on the secure path of a science, after
having for so many centuries been nothing but a process of merely random
(Kant 1787/1933, p. 20)
Galileo was a technician and an experimentalist. He put great effort
into devising, making and popularising novel technical instruments. He was
responsible for creating the pulsilogium, the bilancetta, the compasso di
proporzione, the thermoscopium and the telescope, and he drew workable
plans for the pendulum clock (Bedini 1986). He also measured and made
calculations of pendulum swings. Stillman Drake has unearthed these in
the Galilean manuscripts at Florence (Drake 1990, Chapter 1). However,
Galileo’s measurements and experimentation were directed; they were meas –
urements of behaviour in circumstances dictated by his theoretical conceptualisations. Further, as we have seen in his debate with del Monte, the
theoretical conceptualisation enabled him to identify ‘accidental’ departures
from the ideal. Once this was done, allowances could be made, and the
experiment could be refined. As has been remarked, the whole history of
classical mechanics is a long attempt to force nature to match Newtonian
History and Philosophy: Pendulum Motion 245
Scientific Laws
The regularity account of scientific law has been popular ever since David
Hume’s 1739 Treatise on Human Nature, where he attempted to refute
the necessitarian view of law. For Hume, and those following him, scientific
laws state constant relations between observables; they state what is uniformly
seen to be the case. The pendulum laws present an overwhelming problem
for the Humean account: the regularities do not occur. Under very refined
experimental conditions – small oscillations, heavy weights, minimum air and
fulcrum resistance – they almost occur, but ‘almost occurring’ is not what
regularity accounts of law are about. Moreover, it was a commitment to the
truth of the apparently disproved laws that enabled the approximate
conditions for the law’s applicability to be devised. That is, the truth of the
law is a presupposition for identifying approximations to it, and for identifying
when some behaviour is to be seen as ‘almost law-like’. Russell Hanson has
expressed the matter well:
The great unifications of Newton, Clerk Maxwell, Einstein, Bohr and Schrödinger,
were pre-eminently discoveries of terse formulae from which descriptions and
explanations of diverse phenomena could be generated. They were not discoveries
of undetected regularities.
(Hanson 1959, p. 300)
Michael Scriven once wrote that, ‘The most interesting thing about laws
of nature is that they are virtually all known to be in error’ (Scriven 1961,
p. 91). Nancy Cartwright, in her How the Laws of Physics Lie (Cartwright
1983), makes a similar point: if the laws of physics are interpreted as empirical,
or phenomenal, generalisations, then the laws lie. As Cartwright states the
matter: ‘My basic view is that fundamental equations do not govern objects
in reality; they govern only objects in models’ (Cartwright 1983, p. 129). The
world does not behave as the fundamental equations dictate. This claim is
not so scandalous: the gentle and random fall of an autumn leaf obeys the
law of gravitational attraction, but its distance of fall is hardly described by
the equation s = ut + 1
⁄2 at2. This equation refers to idealised situations. A true
description, a phenomenological statement, of the falling autumn leaf would
be complex beyond measure. The law of fall states an idealisation, but one
that can be experimentally approached. These laws are usually stated with a
host of explicit ceteris paribus, or ‘other things being equal’, conditions.26 For
the laws of pendulum motion, the ceteris paribus conditions would be:
• the string is weightless (so no dampening occurs);
• the bob does not experience air resistance;
• there is no friction at the fulcrum;
• all the bob’s mass is concentrated at a point;
• the pendulum moves in a plane and does not experience any elliptical
• gravity and tension are the only forces operating on the bob.
246 History and Philosophy: Pendulum Motion
However, these conditions can only be approached, never realised – something
that Ronald Giere has been at pains to point out (Giere 1988, pp. 76–78,
1999, Chapter 5). Giere believes, not only that scientific laws are false, they
are also neither universal nor necessary (Giere 1999, p. 90). His account of
the laws of pendulum motion is close to what has been advanced above in
discussion of the real and theoretical objects of science. He says:
On my alternative interpretation, the relationship between the equations and the
world is indirect . . . the equations can then be used to construct a vast array of
abstract mechanical systems. . . . I call such an abstract system a model. By
stipulation, the equations of motion describe the behavior of the model with
perfect accuracy. We can say that the equations are exemplified by the model or,
if we wish, that the equations are true, even necessarily true, for the model.
(Giere 1999, p. 92)
The seventeenth century’s analysis of pendulum motion supports objectivist
views of scientific theory and of epistemology.27 These objectivist views are
in opposition to all empiricist epistemologies and theories of scientific
methodology. Empiricists, since Bacon, Locke and Berkeley in the seventeenth
and eighteenth centuries, through to Alfred Ayer in the twentieth century, have
dominated philosophical reflection on science. Both Ayer’s The Foundations of Empirical Knowledge (Ayer 1955) and The Problem of Knowledge
(Ayer 1956) provide quintessential empiricist accounts of epistemology. Both
are preoccupied with the problem of perception, which is a telltale marker
of empiricism; despite all the fuss and bother about theory dependence of
perception, how things look is not of great scientific moment. Enough has
been said above to indicate that a concentration on perception, to the exclusion
of experimentation and theorising, misses the main thrust of the scientific
revolution. The empiricist tradition maintains the following:
1 There is a distinction between basic, observational or intuited knowledge
and theoretical knowledge.
2 Basic, observational or intuited knowledge does not involve theory.
3 Basic, observational or intuited knowledge is available or given to
individual observers or thinkers (or knowers, or subjects).
4 Theoretical knowledge is derived from, or ultimately justified by reference
to, basic, un-theoretical knowledge.
Rationalism is the other side of the empiricist coin. Both empiricist and
rationalist epistemologies are foundationalist: they seek foundations for
knowledge in the experience of the individual, but, whereas traditional empiricism defines experience as the ‘outer’ experience of the senses, rationalism
extends the definition of experience to include the ‘inner’ experiences of the
individual’s mind or of reason. In both epistemologies, it is the experience of
History and Philosophy: Pendulum Motion 247
the individual knower that is primary. Once this empiricist problematic is
accepted, in either its empiricist or rationalist guise, then there is only a short,
and oft-taken, step to scepticism and relativism. Clearly, both ‘outer’ and
‘inner’ experience is affected by an individual’s circumstances, language,
culture, ideology and theory. The ‘theory dependence of observation’ has
been much written upon and, when coupled with empiricist assumptions
about knowledge, it often results in relativist or sceptical epistemologies. If
experience is the foundation of knowledge, then the foundation is on shaky
ground, and doubts about the soundness of the building are easy to induce.
One modern, and energetic, variant of this empiricist problematic is
constructivism, which, as will be elaborated in Chapter 8, embraces the faulty
epistemological trifecta of individualism, experience and, inevitably, scepticism. From Aristotle through to modern constructivism, a major mistake has
been the elevation of passive experience (looking and observing) at the expense
of intervention, instrumentation and measurement. The latter connects the
investigator with causal processes in the world and thus puts limits on
otherwise unboundable, Rorschach-like subjectivity.
Karl Popper signalled an objectivist break with the empiricist problematic
in his 1934 The Logic of Scientific Discovery, but the account waited 40 years
for its full development in Objective Knowledge, where he wrote:
Traditional epistemology has studied knowledge or thought in a subjective sense
– in the sense of the ordinary usage of the words ‘I know’ or ‘I am thinking’. This,
I assert, has led students of epistemology into irrelevances: while intending to study
scientific knowledge, they studied in fact something which is of no relevance to
scientific knowledge. For scientific knowledge simply is not knowledge in the sense
of the ordinary usage of the words ‘I know’. While knowledge in the sense of ‘I
know’ belongs to what I call the ‘second world’, the world of subjects, scientific
knowledge belongs to the third world, to the world of objective theories, objective
problems, and objective arguments.
(Popper 1972, p. 108)
Most objectivists in theory of knowledge share Popper’s convictions that,
first, the growth of scientific knowledge is the core subject matter of epistemology; and, second, knowledge is something other than beliefs or psychological
states; it transcends individual consciousness. Popper delineated ‘three worlds’:
the first world of objects, processes and events (material and otherwise); the
second world of subjective, individual, mental operations (the life of the mind
or private consciousness); and the third world of scientific, and other, theories,
constructions and problem situations, which are a part of culture and which,
although created by people, nevertheless exist independently of first- and
second-world events.28
The objectivist tradition emphasises: first, that there is a separation of
cognitive or theoretical discourse from the real world. The world is neither
created by the discourse (as in idealism), nor does it somehow create the
248 History and Philosophy: Pendulum Motion
discourse (as in various reflection, or imprinting, theories from Locke to
Lenin), nor does it anchor or provide foundations for the discourse (as in
empiricism and positivism). Theoretical discourse and the world are each
autonomous. In this sense, theory exists independently of individuals. Thus,
scientific knowledge is, contrary to the claims of many constructivists, external
to individuals.
Second, objectivist views distinguish within theory between:
(a) the conceptual foundations of the discourse, containing the definitions of
theoretical and observational terms (there is no decisive distinction made
between these kinds of term);
(b) the conceptual structure of the theory, which is the elaboration and mani –
pulation of the basic concepts by techniques (mathematical and logical)
that produce the structure of the theory (Galileo’s theorems and propo –
(c) the theorised objects of the theory, which are objects in the world as they
are conceived and described by the theory – the balance treated as a
uniform line with parallel weights suspended from it, the pendulum treated
as a point mass on a weightless string, billiard balls treated as colourless,
perfectly hard bodies and so forth.
Objectivism in Education
A source of much confusion arises from there being a second sense of
‘objectivism’, especially common in science-education literature. This is
objectivism in the sense of ‘objective truth’, ‘universalism’ or ‘certainty of
belief’. This objectivism is completely orthogonal to the Popperian sense
(indeed, Popper denied the possibility of absolute truth). In science education,
objectivism, as well as having these absolutist overtones, is, with some
confusion, frequently interpreted as a version of positivism. Two science
educators write that:
At present, most science teaching is based on an objectivist view of knowing and
learning. . . . Here, objectivism subsumes all those theories of knowledge that hold
that the truth value of propositions can be tested empirically in the natural world.
(Roth & Roychoudhury 1994, p. 6)
Popper completely denies such a view. The index entry for ‘objectivism’ in
Kenneth Tobin’s The Practice of Constructivism in Science Education (Tobin
1993) reads ‘see positivism’. Positivism is, of course, a philosophical position
that Popper spent his life arguing against. This ‘crossing of philosophical wires’
is a major impediment to scholarly progress in science education and to its
productive engagement with other disciplines: Time is needlessly spent in
attacking straw men or enjoying ‘feel good’ euphoria, while significant
literature and arguments are ignored.
History and Philosophy: Pendulum Motion 249
Observation is an important element of science education, as teaching children
to observe carefully is part of every science curriculum. For some time, there
has been educational recognition of the more philosophical or problematic
dimensions of observation.29 The topic of observation has received renewed
attention in recent time, with the much-repeated claim of the Lederman
research group that an ‘empirical base’ is the first defining feature of the
nature of science. As they state the matter:
Students should be able to distinguish between observation and inference. . . . An
understanding of the crucial distinction between observation and inference is a
precursor to making sense of a multitude of inferential and theoretical entities
and terms that inhabit the worlds of science.
(Lederman et al. 2002, p. 500)
For this aim to be realised, teachers must first understand and appreciate the
distinction. The thesis of this book is that HPS can contribute to science
teaching and science teacher development by assisting with a better and deeper
understanding of the many theoretical, curricular and pedagogical issues that
engage science teachers. The thesis can be tested by elaborating the topic of
The pendulum story is an occasion for raising one basic matter about this
process of observation (including vision, perception, looking, noticing and
seeing) that links personal psychology to the observer’s social life: this is
the distinction between ‘object perception’ and ‘propositional perception’.
Galileo constantly saw things that those around him did not see; even
‘the lynx’, Leonardo da Vinci, did not see what Galileo saw. Clearly, in one
sense, Leonardo did see what Galileo saw, but in another sense he did not.
Ludwig Wittgenstein made the distinction in terms of seeing and seeing as
(Wittgenstein 1958). Popper made the distinction in terms of perception and
observation. Object perception occurs whenever healthy eyes pass over
reasonably lit situations; they see what there is to be seen. Among the things
seen, some are noticed or attended to. However, this kind of object perception
has no epistemological or scientific import, nor even much personal import.
The latter begins when, among the things noticed, people begin to verbalise
what they see; this then is propositional perception, where people see ‘that
p’, where p is some proposition or statement. This perception depends entirely
on language, social embeddness and available theory. The proposition, p, has
to be verbalised or written; it requires a language. Someone looks at a field
and sees ‘that there are trees’; they see, but do not notice, the fence. Someone
looks at a room of people, but only notices the person they want to meet.
Consider the two inscriptions in Figures 6.11 and 6.12. Everyone, from child
to adult, who passes their eyes over them can have an object perception of
the marks, whether they are noticed or not noticed.
Once the marks are noticed, then propositional perception can begin. At
the least sophisticated level, someone might see ‘that there are marks on the
250 History and Philosophy: Pendulum Motion
paper’; at the next level, someone might see ‘that there are Asian characters
on the paper’; at the next level, someone might see ‘that there is a Chinese
character on the paper’, whereas someone else might see ‘that there is a
Japanese character on the paper’; at the next level, someone might see ‘that
there are both Chinese and Japanese characters on the paper’, but not know
what they mean; with more sophistication, someone might see ‘that the
word STOP in both Chinese and Japanese is on the paper’; finally, a more
knowledgable person will recognise that each character needs to be qualified
for a precise translation into STOP rather than PAUSE. The ascent up the levels
of propositional perception is entirely dependent on what the person knows,
on what is in their mind, on the language competencies of the individual. All
of the latter is something that needs to be taught or enculturated; without this,
there cannot be propositional perception, and it is the latter that is of relevance
to most human life, including science and scientific investigation.
Two people might have the same propositional perception ‘that there is a
man in the room’, but one of them might see ‘that the prime minister of
Australia is in the room’; this propositional perception requires being able to
draw on certain knowledge or information in the head. To look at a
chessboard and see ‘that the black rook is threatened by the white queen’
depends on having been taught the rules of chess. No amount of looking, even
careful looking, at the above inscriptions, people or chess pieces will reveal
the word STOP, the prime minister of Australia or the threatened rook; to
see that the latter is the case requires teaching, learning and acquisition of
sophisticated language. As the English ‘ordinary language’ philosopher J.L.
Austin (1911–1960) commented: ‘we are using a sharpened awareness of
words to sharpen our perception of, though not as the final arbiter, of the
phenomena’ (Austin 1961, p. 130).
For students, the sharpened awareness of words has to come from the
outside, from engagement in a community, and it is teachers who are the major
conduit for this engagement. Two moderately sophisticated students might
share the propositional perception ‘that the body is moving smoothly in a
circle’; they will only be able to see ‘that the body is accelerating’ if they have
been taught elementary Newtonian mechanics, and certainly not by ‘brain
History and Philosophy: Pendulum Motion 251
Figure 6.11 Chinese Character Figure 6.12 Japanese Character
storming’ or ‘negotiating’ what they see. Likewise, students might see, as
Aristotle did, ‘that the hot air is rising to the ceiling’; only with basic
Newtonian mechanics will they see ‘that the falling cold air is pushing the less
heavy hot air up’. The first description leads to a mechanics of levitas; the
second leads to a mechanics of gravitas. So explanations depend on how things
are described, on what propositional perception people have. The explanation
for ‘the British invaded Australia’ is different from the explanation of ‘the
British discovered Australia’. The response to ‘I see that Billy is exhibiting
Attention Deficit Disorder’ will be different from that to ‘I see that Billy is
exhibiting spoilt brat behaviour’. Clearly, propositional perception in politics,
morals and aesthetics depends entirely on language, and the ‘more sharpened’
the language available, then the sharper can be the observations. Of course,
language is necessary, but not sufficient, for propositional perception.
Propositional perception is not tightly tied to object perception. The former
is always inferential. Two people looking at the same thing can see that
completely different things are the case. In the Middle East, some people see
that the US is defending democratic processes, whereas others see that the US
is defending its strategic interests. And so on, ad infinitum. Effort has to go
into seeing the ‘facts of the matter’, which will always be an item of
propositional perception, as every statement of fact is a proposition, and the
issue is to see how far down such agreement can go. The British empiricists,
and more recently the positivists, thought that there were some basic sensedata statements that grounded all variants of propositional perception about
the world.30 For instance, ‘I see a red patch’ or ‘I have a red patch sensation’
is the common perceptual foundation for ‘I see that there is a tomato’, ‘I see
that there is a stop sign’, ‘I see that blood is on the floor’ and so on. Having
the red sensation is object perception; when it is described, propositional
perception is occurring, and this is inferential.
Recall that, in Chapter 2, it was mentioned that the positivist Philipp Frank
said it was quite wrong for textbooks to write of how, in the morning, we
see ‘the sun rising on the horizon’. He said: ‘Actually our sense observation
shows only that in the morning the distance between horizon and sun is
increasing, but it does not tell us whether the sun is ascending or the horizon
is descending’ (Frank 1947/1949, p. 231). The statement ‘that the sun is
moving’ is just one propositional perception grounded by the object perception
we have when looking at the sun in the morning.
For these sorts of reason, Plato, 2,500 years ago, said that, ‘we see through
the eye, not with the eye’, which could be taken as the beginning of the long
history of philosophy of perception.31 More recently, philosophy of perception
has connected with philosophy of mind, cognitive psychology and phenomenology. In 1934, Karl Popper presciently warned against misunderstanding
the place of observation in science:
The doctrine that the empirical sciences are reducible to sense-perceptions, and
thus to our experiences . . . is here rejected . . . there is hardly a problem in
252 History and Philosophy: Pendulum Motion
epistemology which has suffered more severely from the confusion of psychology
with logic.
(Popper 1934/1959, p. 93)
This was written against the commitment of early positivists to sense
impressions, sense data and other putative perceptual bed-rocks of science.
Wallis Suchting, in a paper on ‘The Nature of Scientific Thought’, has
commented on these matters, saying that:
Thus the key inadequacy of empiricism has really nothing to do with the centrality
it accords to sense-experience; in particular, the controversy over whether the
‘basic language’ of science should be ‘phenomenonalistic’ or ‘physicalistic’ is
irrelevant to the main question, a mere internal family dispute, as it were. The
central deficiency of empiricism is one that it shares with a wide variety of other
positions, namely, all those that see objects themselves, however they are
conceived, as having epistemic significance in themselves, as inherently determining
the ‘form’, as it were, of their own representation, rather than as determining the
degree of applicability of representations of a given ‘form’, and hence, conversely,
that the nature of what is represented can be more or less directly ‘read off’ its
(Suchting 1995, p. 13; original italics)
Data, Phenomena and Theory
The foregoing considerations concerning the real and theorised object
distinction, object and propositional perception, idealisations and nonHumean accounts of law point to the importance of distinguishing data from
phenomena, and both of these from theory. The data are observed; phenomena characteristically are not. Scientific laws and theories are about the
phenomena, not about the data.32 If this is understood, a number of things
about science, and especially studies of pendulum motion, become clearer.
Real objects (processes, events, occurrences, states) are observed either in
natural (Aristotle’s preference) or experimental (Galileo’s and Newton’s
preference) settings. The observation can be immediate (with eyes, microscopes, etc.) or inferred (meter readings, instrument displays, etc.). All of this
occurs in the realm of the real, not the realm of discourse. The observations
are then verbalised, described, written or tabulated. This has to be done in a
language (including mathematics) and according to some theoretical
standpoint. This is all done in the realm of discourse. These descriptions are
characteristically sifted, sorted and selected – lots of readings and descriptions are simply thrown away, or ignored. The result is scientific data. These,
then, are the raw representations of real objects (processes, events, occurrences,
states). This step is clearly theory dependent. A range of falling red apples, or
swinging weights on a string, are, in physics, represented as points on a graph,
as printouts on a tickertape, as lines on a screen. These representations are
History and Philosophy: Pendulum Motion 253
not meant to mirror, or copy, the real. They are precisely meant to represent
the real. And, as mentioned earlier, economists, artists and farmers have
different ways of representing the same real events. Adequacy of representation
simply does not mean correspondence of representation, in the sense of the
representation mirroring the object.
For pendulums, even highly refined experimental apparatus will give a
scatter of data points. In plotting period against length, the scatter is enormous;
in plotting period against the square root of length, the scatter contracts
markedly, and the phenomenon of direct relation is revealed. The laws of
pendulum motion are not meant to, and cannot, explain these data points;
they are too erratic. However, in science, from data come phenomena; and it
is the phenomena that are the subject of scientific laws and theories. Often,
a line of best fit is put through the data points, and the line is then taken to
represent the phenomenon being investigated. Thereafter, it is the phenomenon
that is discussed and debated, not the data. Any number of individual
telescopic observations, when corrected and selected, constitute astronomical
data. From these, we infer, construct, invent planetary phenomena: circular
orbits, elliptical orbits, heliocentric or geocentric orbits. The latter are not seen.
They are not observational. However, this is no scientific impediment. Once
we settle on the phenomenon, it becomes the subject matter of our scientific
theories. Newton, in Book II of his Principia, after laying out his Rules of
Reasoning in Philosophy (our science), has a section on Phenomena. Among
six phenomena that he believes his System of the World has to account for,
That the fixed stars being at rest, the periodic times of the five primary planets,
and (whether of the sun about the earth, or) of the earth about the sun, are as
the 3/2th power of their mean distances from the sun.
That the moon, by a radius drawn to the earth’s centre, describes an area
proportional to the time of description.
(Newton 1729/1934, pp. 404–405)
These are not observational statements, and they are not data in the terms
we are using. They are statements of the phenomena to be explained. As
Newton acknowledged, these phenomena came from the work of the giants
on whose shoulders he stood: Galileo, Kepler and Brahe.33
Kepler’s ‘elliptical planetary paths’ were, in turn, phenomena separate from,
and not necessarily implied by, his astronomical data and measurements. As
William Whewell noted in the nineteenth century, in his critique of Mill’s
inductivist account of science, the concept of an elliptical path was supplied
by Kepler’s mind, not by his data. There is usually no univocal inference from
data to phenomena. Phenomena are underdetermined by data, just as theory
is underdetermined by evidence. In the above case, the data are probably
consistent with periodic times of 5/4th power of mean distance.
254 History and Philosophy: Pendulum Motion
The situation might usefully be represented as shown in Table 6.1.
• Level 6 is constituted by objects, processes and events in the world.
• Level 5 is constituted by perceptions, observations and psychological
states occasioned by events in world.
• Level 4 is constituted by the articulation or statement of observational
• Level 3 is constituted by the representation (graphs, tables, counts) of
observations; this constitutes data.
• Level 2 is constituted by phenomena such as ‘isochronous motion’ or
‘elliptical paths’ that can be represented by models, or empirical laws,
or equations such as T = 2π√(l/g).
• Level 1 is constituted by fundamental scientific laws or high-level theory.
A chaotically moving, falling autumn leaf ‘obeys’ a number of fundamental
causal mechanisms – gravitation, air resistance, etc. – but its path (data points)
does not illustrate or confirm the appropriate laws. Contrary to Nancy
Cartwright’s claims, we need not believe that the fundamental laws of physics
lie; they might lie about appearances (Level 3 items above), but, if we abandon
the long-entrenched, Aristotelian-based conviction that the laws should be
about Level 3 items, then we can maintain their truthfulness. They are true
of phenomena, not of data. This was a point well made by James Brown, who
concluded his analysis of this subject:
Phenomena are to be distinguished from data, the stuff of observation and
experience. They are relatively abstract, but have a strongly visual character.
They are constructed out of data, but not just any construction will do. Phenomena
are natural kinds that we can picture. They show up in thought experiments and
History and Philosophy: Pendulum Motion 255
Table 6.1 Objects and Processes, Perception, Data, Phenomena and Laws
Level 1 Fundamental laws and mechanisms Gravitational attraction
Simple harmonic motion
Level 2 Phenomena, scientific models, Four pendulum laws (mass and amplitude
idealisations independence; period varies as square
root of length; isochronous oscillation)
Level 3 Data Individual period measurements for
different masses, amplitudes, lengths;
scattered points on a graph
Level 4 Propositional perception Articulating different facts of the matter
as seen
Level 5 Object perception Seeing or observing swinging pendulum
Level 6 Objects, events and processes Weight swinging on end of cord
in world
they play an indispensable role in scientific inference mediating between data and
theory. So let’s attend to them.
(Brown 1996, p. 128)
Other Features
Other features of science and philosophical themes in the history of pendulum
study could have been singled out. For example, debate over the very
meaning of time. Any investigation or discussion of the pendulum leads
quickly to the question of time: what is it, and how is it best measured?
Such questions have been entertained for at least 2,500 years in the West, by
such luminaries as Plato, Aristotle, St Augustine, Aquinas, Newton, Mach and
Einstein. Other cultures have their own long traditions of engagement with
time and its measurement. For all cultures, this engagement includes contributions from artists, poets, musicians, theologians, philosophers, playwrights and
scientists. Discussion of time connects to calendars, religious and cultural
ritual, cosmology and countless things about everyday life from personal
punctuality to railways and airlines running ‘on time’. Pendulum studies are
an opportunity for students to appreciate and learn from these traditions
of temporal engagement; they are traditions where science, philosophy,
technology and culture intertwine.34
The themes identified above can be seen in the history of most other scientific
fields: genetics, combustion, atomic structure, astronomy, evolution and so
on. In all cases, discussion with students about the interplay of these influences
and themes is profitable.
The Pendulum and Recent US Science Education
Attention to the place accorded to the pendulum in curricula provides a good
testament to the degree to which science education has engaged with and
utilised HPS. Jerrold Zacharias, the driving force behind the 1960s PSSC
physics course, did not think highly of the educative value of the simple
pendulum, saying that, for students, ‘it is not very interesting’ (Zacharias 1964,
p. 69). He commends beginning instruction with the coupled pendulum,
which gives ‘lovely phenomena’, and then students will want to know about
the single pendulum, ‘And this happens every time’ (ibid.). It is an empirical
question whether this approach works or does not work in classrooms, but,
putting that aside, what students can learn about the simple pendulum will
depend on teachers having some appreciation of the rich history of the subject.
The evidence is that there is not much such appreciation.
It is instructive, if sobering, to look at the utilisation of the pendulum in
the past three decades of intense efforts to improve US school science
programmes. These efforts have involved thousands of individuals in bodies
such as the AAAS, the NRC, the National Academy of Science, the National
Academy of Engineering, the NSF; peak disciplinary bodies in physics,
256 History and Philosophy: Pendulum Motion
chemistry, biology, earth science; and all major national and state scienceeducation organisations, including the NSTA and the National Association
for Research in Science Teaching (NARST). Despite such massive and luminous oversight, the pendulum barely appears, and, when it does, so little of its
potential for teaching scientific content, methodology and the ‘nature of
science’ is utilised. This, in part, is a consequence of the unhealthy separation
of the science-education and HPS communities, and not just in the US.
Scope, Sequence and Coordination
The large-scale and influential curriculum proposal of the US NSTA – Scope,
Sequence and Coordination (Aldridge 1992) – highlights the pendulum to
illustrate its claims for sequencing and coordination in science instruction. Yet
nowhere in its discussion of the pendulum is history, philosophy or technology
Project 2061
In 1989, the AAAS published its wonderfully comprehensive Science for All
Americans report (AAAS 1989). It acknowledged that: ‘schools do not need
to be asked to teach more and more content, but rather to focus on what is
essential for scientific literacy and to teach it more effectively’ (AAAS 1989,
p. 4). The report saw that students need to learn about ‘The Nature of Science’,
and, hence, that was the title of its first chapter. The report recognised the
importance of learning about the interrelationship of science and mathematics,
saying: ‘The alliance between science and mathematics has a long history
dating back many centuries. . . . Mathematics is the chief language of science’
(AAAS 1989, p. 34). It also acknowledged that some episodes in the history
of science should be appreciated because, ‘they are of surpassing significance
to our cultural heritage’ (AAAS 1989, p. 111).
Among the ten such episodes it picks out is Newton’s demonstration that,
‘the same laws apply to motion in the heavens and on earth’ (AAAS 1989,
p. 113). It provides a very rich elaboration of this episode and its scientific,
philosophical and cultural impacts. Unfortunately, there is no mention of what
enabled Newton to achieve this unification, namely the pendulum; had such
mention been made, this ‘big idea’ could have been connected to something
tangible in all students’ experience, the place of mathematics in science could
have again been underlined, and a wonderful case study in the nature of science
could have been built upon.
The US National Standards
The under-utilisation of the pendulum can be gauged from looking at the
recently adopted US National Science Education Standards (NRC 1996). The
Standards adopt the same liberal or expansive view of scientific literacy as
the NCEE did in 1983, saying that it ‘includes understanding the nature of
History and Philosophy: Pendulum Motion 257
science, the scientific enterprise, and the role of science in society and personal
life’ (NRC 1996, p. 21). The Standards devote two pages to the pendulum
(pp. 146–147). However, there is no mention of the history, philosophy or
cultural impact of pendulum motion studies; no mention of the pendulum’s
connection with timekeeping; no mention of the longitude problem; and no
mention of Foucault’s pendulum.
Astonishingly, in the suggested assessment exercise, the obvious opportunity
to connect standards of length (the metre) with standards of time (the second)
and with standards of weight is not taken. Rather, students are asked to
construct a pendulum that makes six swings in 15 seconds. This is a largely
pointless exercise, especially when they could have been asked to make one
that beats in seconds and then measure its length and enquire about the
coincidence between their seconds pendulum and the metre (Matthews 1998).
Depressingly, the Standards document was reviewed in draft form by tens
of thousands of teachers and educators. It is clear that, if even a few of the
readers had a little historical and philosophical knowledge about the pendu –
lum, this could have transformed the treatment of the subject in the Standards
and would have encouraged teachers to realise the liberal goals of the
document through their treatment of the pendulum. This would have resulted
in a much richer and more meaningful science education for US students. That
this historical and philosophical knowledge is not manifest in the Standards
indicates the amount of work that needs to be done in having science educators
become more familiar with the history and philosophy of the subject they
teach, and of having science-education communities more engaged with the
communities of historians and philosophers of science.
America’s Lab Report
The US NRC commissioned a large study on practical work in US schools
that was published as America’s Lab Report: Investigations in High School
Science (NRC 2006). The book has 236 pages, seven chapters and hundreds
of references. The pendulum has three entries in the index. On its first
appearance, it is said to be regrettable that teachers simplify pendulum
experiments and ignore the ‘host of variables that may affect its operation’
(NRC 2006, p. 117). Teachers are advised to recognise these ‘impediments’,
such as friction and air resistance, but the writers go on to say that this ‘can
quickly become overwhelming to the student and the instructor’ (NRC 2006,
p. 118). This is not very helpful. It could have been an occasion to say
something about the fundamental importance of idealisation and abstraction
to the very enterprise of science, of not letting the trees get in the way of seeing
the forest. This was the problem identified by Thomas Kuhn in his discussion
of the pendulum and faced by da Vinci; it is the heart of the debate between
Galileo and his patron Guidabaldo del Monte, but the Lab Report says
nothing about this fundamental scientific procedure, much less provide some
historical background to its resolution. The pendulum allows students very
tangibly to begin seeing the effect of ‘impediments’ and ‘accidents’ (Koertge
258 History and Philosophy: Pendulum Motion
1977), or ‘errors’ in contemporary language, on the manifestation or ‘visibility’
of core natural processes.
On the pendulum’s second appearance in the index, the ‘typical pendulum
experiment’ is criticised, because it is ‘cleaned up’ and used just to teach
science content – that the ‘period of a pendulum depends on the length of the
string and the force of gravity’ – and not scientific process skills (NRC 2006,
p. 126). In contrast to these ‘bad’ pendulum practical classes, on the
pendulum’s third appearance, a ‘good’ class is described occupying two pages
in a highlighted box. In this class, teachers are first advised to demonstrate
swinging pendulums, then, in a very guided fashion, to have students graph
the relationships between period and mass, period and amplitude, and period
and length, and finally it is suggested that the teacher discuss the importance
of obtaining an adequate amount of data over a range of the independent
variable (NRC 2006, pp. 128–129).
This does no harm and can do some good, but there is nothing noteworthy
here. Everything about the rich history of the pendulum has been stripped
out: no mention of Galileo, Huygens, Newton, Hooke, universal gravitation,
timekeeping, clocks, length standards, longitude, shape of the Earth or
conservation laws. No connection intimated between science, technology and
society; no sense of participation in a scientific tradition. Nothing. Teachers
are not even told to talk about these great scientists and their pendulum-based
And in this set piece, nationally distributed, ‘model pendulum lesson’
teachers and students are told to plot period against length on a graph. This
is a task with only minimally useful outcomes: such a graph provides a scatter
of points that merely establishes a trend. After having done this, 17–18-yearold students could have been so easily asked to plot period against the square
root of length. When this is done, a straight line is obtained from the data,
not a scatter of points. As discussed above, the physical phenomenon is
revealed by mathematical manipulation. Period is seen, as it was by Galileo
and Huygens, not just to vary with length, but to vary directly with the square
root of length; the conclusion from the data moves from inconclusive T ∝ L
to conclusive T = k√L. The model lesson tells teachers to ‘avoid introducing
the formal pendulum equation, because the laboratory activity is not designed
to verify this known relationship’ (NRC 2006, p. 129). Final-year students in
Japan, Korea, Singapore and a good deal of the rest of the world have no
such problem, and US students deserve better than a dumbed-down, HPS-free
The graph of period against square root of length shows, in a manageable
way, the dramatic impact of mathematics on physics; without the mathematical notion of square root, we see qualitative trends; utilising the square
root, we see a precise, quantitative relationship. Further, this precise
relationship will allow the pendulum to be connected with free fall, where
distance of fall varies as the square of time. All of this is missed in the Lab
Report, and also missed is the opportunity for richer pendulum-informed
teaching of physics. What appears to have happened is what the NRC
History and Philosophy: Pendulum Motion 259
recognises in another publication: ‘As educators, we are underestimating what
young children are capable of as students of science – the bar is almost always
set too low’ (NRC 2007, p. vii). A pity they did not follow their own advice.
The Next Generation Science Standards
The NRC gives three reasons for producing updated NGSS in the US, one of
which is that there is a ‘growing body of research on learning and teaching
in science’ that can be utilised. The history of supposed science curriculum
‘reforms’ suggests that caution should be exercised about such claims. In the
1950s and 1960s, the ‘growing body of research on learning and teaching’
gave us behaviourism and behavioural objectives, which have disappeared
without educational trace; in the 1970s, the ‘growing body of research’
gave us discovery learning and ‘scientist for a day’ teaching, both with minimal
if not deleterious effects; in the 1980s and 1990s, the ‘growing body of
research’ gave us constructivism, which swept all before it in university schools
of education, but, in more sober light, its substantial philosophical and
pedagogical failings have been recognised, as will be shown in Chapter 8.
Good understanding of teaching and learning is certainly needed, but the
improvement of curricula does not flow just from knowledge about how to
better teach and learn material, but rather it flows from knowledge of what
material to teach and learn, and where to place the topics and concepts in
state and national standards. This is where a richer understanding of the
history and philosophy of pendulum studies and utilisation (and, of course,
of all other topics) can well contribute to science education. It can make for
better curricula and for better connections between disciplinary strands in
The pendulum ‘ticks all of the NGSS boxes’, so to speak. Very young
children, as shown in Japan, Korea and numerous other countries, can
profitably and enjoyably engage with pendulum activities (Kwon et al. 2006,
Sumida 2004).35 It is not accidental that Jean Piaget used the pendulum for
his investigation of the progressive development of children’s scientific
reasoning ability, especially their identification and control of variables (Bond
2004). The sophistication of pendulum activities and their relation with other
areas and topics in science can be enhanced with progression through school;
obvious connections with music, mathematics, technology and engineering can
be made, and even connections with chemistry (De Berg 2006). The full range
of process skills (data collection and representation, hypothesis generation),
methodological skills (generating hypotheses, evaluating these against
evidence, theory testing and so on) and model construction can all be cultivated
using pendulum classes.36
It remains to be seen how the pendulum will feature in the final NGSS
document, but the signs are not good. In the current (2012) draft, the
pendulum is mentioned four times, and each time it is in connection with
the transformation of energy from potential to kinetic forms. This is a level
of abstraction way beyond what is needed or called for, it is beyond the life
260 History and Philosophy: Pendulum Motion
experience of the students and it reifies the role played by the pendulum in
the history of physics and in its social utilisation. The draft document mentions
Newton’s laws, his theory of gravitation and the conservation of momentum,
but no mention of the pendulum, which could so easily be used to make
manifest and experiential each of these learning goals.
The pendulum case has been introduced in this chapter as an example where
the HPS can contribute to even routine science education. It provides an
opportunity to learn about science at the same time as one is learning the
subject matter of science. With good HPS-informed teaching the pendulummotion case enables students to appreciate the transition from commonsense and empirical descriptions characteristic of Aristotelian science, to
the abstract, idealised and mathematical descriptions characteristic of the
scientific revolution. The pendulum provides a manageable, understandable
and straightforward way into scientific thinking and away from everyday and
empirical thinking; it shows, at the same time, how scientific, idealised thinking
nevertheless is connected with the world through controlled experiment.
The ‘contextual’ teaching of science, as suggested here, is not a retreat from
serious, or hard, science, but the reverse. To understand what happened in
the history of science takes effort. Further, it is appealing to students.
A frequent refrain from intelligent students who do not go on with study in
the sciences is that, ‘science is too boring; we only work out problems’.37 The
history of human efforts to understand pendulum motion is far from boring:
it is peopled by great minds, their debates are engaging, and the history
provides a storyline on which to hang the complex theoretical development
of science. As well as gaining an improved understanding of science, students
taught in a contextual way can better understand the nature of science, and
have something to remember long after the equation for the period of a
pendulum is forgotten.
The following diagram (Figure 6.13), where the columns represent
curriculum subjects and the circles topics within subjects, displays the
integrative curricular function of history and philosophy.38
The content of the school day, or at least year, can be more of a tapestry,
rather than a curtain of unconnected curricular beads. The latter is a welldocumented problem with US science education, with its fabled ‘mile wide
and one inch thick’ curricula (Kesidou & Roseman 2002). However,
pendulum motion, if taught from a historical and philosophical perspective,
allows connections to be made with topics in religion, history, mathematics,
philosophy, music and literature, as well as other topics in the science
programme. And such teaching promotes greater understanding of science,
its methodology and its contribution to society and culture. However, such
connections first need to be recognised by curriculum writers and by teachers
charged with implementing curricula or achieving standards; this raises the
History and Philosophy: Pendulum Motion 261
whole question of HPS in pre-service or in-service teacher education, but that
is a different subject, for a different chapter.
1 This chapter has benefited from research published in Matthews (2000, 2001, 2014).
2 See the account of the philosophy and international impact of PSSC physics given in
Chapter 3.
3 There is no adequate history of the medieval pendulum, but see Büttner (2008) and
Hall (1978).
4 See Büttner (2008), Lefèvre (2001), Machamer (1998) and contributions to Renn
5 Galileo surely had precursors. There were medieval natural philosophers – John Buridan,
Nicole Oresme, Thomas Bradwardine and his Merton College colleagues, and others
– who utilised mathematics, but it was not fully engaged with their physics (natural
philosophy) as mathematics was with Galileo. See Clagett (1959) and Moody (1975).
6 This will be a recurrent theme in the history of pendulum-related science, where it is
seen that many different mechanical, biological and chemical processes will manifest
the mathematical formulae for simple harmonic motion.
7 This vital idealising feature of Galileo’s physics is discussed in Koertge (1977) and
McMullin (1985).
262 History and Philosophy: Pendulum Motion
Religion History Science Philosophy Technology
Figure 6.13 HPS-Informed Curricular Linkage. A: design argument; B: European voyages
of discovery; C: Aristotelian physics and methodology; D: pendulum clock; E: idealisation
and theory testing; F: timekeeping and social regulation; G: geometry of the circle; H:
applied mathematics; I: measurement and standards; J: time; K: energy; L: geodesy
8 Ronald Naylor (1980, pp. 367–371) and W.C. Humphreys (Humphreys 1967,
pp. 232–234) discuss the letter in the context of Galileo’s work on the law of fall.
9 Discussion and references can be found in White (1966, p. 109).
10 A reaction shared by many to the mathematisation of psychology and the ‘boxes and
arrows’ of modern cognitive science.
11 This disjunct between ‘lived experience’ and scientific conceptualisation bears specifically
on teaching the law of inertia, something that has been discussed in Chapter 5.
12 The foregoing problems of medieval Aristotelians are the same for contemporary
‘discovery’, ‘enquiry’ or ‘minimal guidance’ teaching; without guidance or instruction,
there is no learning.
13 This episode is elaborated in Matthews (2001).
14 Accounts of the development of the standard metre can be found in Alder (1995, 2002)
and Kula (1986, Chapters 21–23).
15 On the history of debate about the shape of the Earth, see Chapin (1994).
16 See Martins (1993).
17 On the pendulum’s role in this unification, see especially Boulos (2006).
18 The history is dealt with in Matthews (2000, Chapters 6, 7).
19 Maurice Finocchiaro (1980, p. 149) has provided a table of fourteen philosophical topics
to be found in Galileo’s Dialogue on the Two Chief World Systems (1633/1953).
20 Jaroslav Pelikan provides a richly documented historical study of the same quest (Pelikan
21 For the physics and mathematics of these calculations, see Holton and Brush (2001,
pp. 128–129).
22 This is a wonderful episode in the history of science. A great story can be made, even
a drama. All the elements are there: powerful and prestigious figures, ‘no name’
outsiders, controversy over a big issue, mathematics and serious calculations, religion,
final decision-making with ample opportunity to preserve the status quo. Sadly,
however, the episode is little known and hardly ever taught.
23 Feinstein elaborates his personal position by writing:
I would argue that knowing the shape of the earth is part of being ‘well-read’ in
science – an excellent thing indeed, and one capable of giving us joy and satisfaction,
but not one that all of our students will find useful.
(Feinstein 2011, p. 183)
24 On this important conceptual advance, see Moody (1975, p. 403).
25 In recent decades, philosophers have returned to the Kantian high estimation of
experiment in science and have written illuminating work on the topic. See at least:
Hacking (1988) and contributions to Radder (2003) and Gooding et al. (1989).
26 An extensive study of the logic of idealisation in science is Nowak (1980). See also
Laymon (1985) and Portides (2007).
27 Such objectivist epistemology is articulated in Popper (1972, Chapters 3, 4), Althusser
and Balibar (1970), Baltas (1988, 1990), Chalmers (1976/2013), Mittelstrass (1972),
Sneed (1979) and Suchting (1986).
28 See Irzik (1995) and Musgrave (1974).
29 See especially Hodson (1986) and Norris (1985).
30 On this tradition, see Yolton (2000).
31 Norwood Russell Hanson echoed Plato when he wrote: ‘there is more to seeing than
meets the eyeball’ (Hanson 1958). For a wide range of twentieth-century readings in
philosophy of perception, see Swartz (1965).
32 The distinction between scientific data and scientific phenomena has been developed
at length by James Bogen and James Woodward (1988), James Woodward (1989),
Ronald Laymon (1982, 1984) and James Brown (1996). For an interpretation of
Galileo’s work in terms of the data/phenomena distinction, see Hemmendinger (1984).
The conceptualisation can considerably illuminate persistent issues in educational
research, but it is rarely so utilised; see discussion in Brian Haig (2014, Chapter 2) and
Matthews (2004).
History and Philosophy: Pendulum Motion 263
33 For an analysis of the meaning of ‘phenomena’ in Newton’s work, see Achinstein
34 See at least: Barnett (1998), Landes (1983), Turetzky (1998) and van Rossum (1996).
35 The front cover of an excellent pendulum booklet produced for Japanese elementary
students is fully adorned with images of Galileo and Huygens – a nice comment on the
universality of science and its ability to be embraced by cultures beyond its original
European home.
36 On classroom utilisation of pendulum investigations as a way of teaching physics and
cultivating and assessing scientific reasoning skills, see at least: Kanari and Millar
(2004), Kwon et al. (2006), Stafford (2004), Zachos (2004) and the hundred-plus
references in Gauld (2004).
37 This was the refrain repeatedly made some years ago when an Australian deans of
science study reported on the top 10 per cent of school science achievers, most of whom
did not go into tertiary science-related programmes.
38 The idea for this visual representation of the argument comes from an AAAS lecture
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