About the Pendulum Project

The International Pendulum Project: An Overview

by Michael Matthews

The International Pendulum Project (IPP) is a collaborative research project examining the history of the study of pendulum motion, and its scientific, horological, philosophical, cultural and educational ramifications. The Project demonstrates how understanding the pendulum story can assist teachers to improve science education by developing pendulum-related curricular content, by showing connections between pendulum studies and other parts of the school programme especially mathematics and social studies. The Project involves about thirty or so researchers in sixteen countries plus a large number of 'just interested' participants.[1] As the pendulum is a universal topic in university mechanics courses, high school science programmes, and a common topic in elementary school science, an enriched approach to its study can result in enhanced science literacy for all, a literacy that includes appreciating the part played by science in the development of society and culture.

The Pendulum in Western Science

The pendulum has played a significant role in the development of Western science, culture and society. The pendulum was studied by Galileo, Huygens, Newton, Hooke and all the leading figures of seventeenth-century science. The pendulum was crucial for, among other things, establishing the collision laws, the conservation laws, the value of the acceleration due to gravity g, ascertaining the variation in g from equatorial to polar regions and hence discovering the oblate shape of the earth, and, perhaps most importantly, it provided the crucial evidence for Newton's synthesis of terrestrial and celestial mechanics.

The pendulum was important for the Galileo's new science, and it had a central place in Newton's physics, with the historian Richard Westfall remarking that 'without the pendulum, there would be no Principia' (Westfall 1990, p.82). Subsequently the pendulum was at the core of classical mechanics as it developed through the eighteenth, nineteenth and early twentieth centuries, with the work of Stokes, Atwood and Eotvos being especially notable. Foucault's pendulum, as well as providing dynamical evidence for the rotation of the earth, also played a role in the popularisation of science in the late nineteenth and early twentieth centuries (Conlin 1999). Pendulum measurements enabled the shape of the earth to be determined, and were pivotal for the science of geodesy (Heiskanen & Vening Meinesz 1958).

The simple pendulum, when displaced through a small amplitude (<10°) oscillates with a natural frequency that depends solely upon its length. The pendulum manifests simple harmonic motion, whereby the restoring force on the bob (the tangential vector component of the pull of gravity) varies linearly with displacement. This is a marvellous physical system and is emblematic of a wide range of other such oscillating natural and perhaps social systems. The ideal, non-damped, simple pendulum is a conservative system in which the energy created by displacement is retained in the system when it swings. Galileo had an understanding of this, and demonstrated it so simply by showing how the pendulum, once released, retained its initial height, but did not exceed it. Low-level mathematical models can 'capture' the motion of simple pendulums. With more complicated pendulums - when the mass of the string, air disturbance, and fulcrum resistances are taken into account Ð more sophisticated mathematics and differential equations are required in order to 'capture' the behaviour. With double and triple pendulums chaotic motion can be induced which in turn requires still more sophisticated mathematics in order to be properly modelled. The whole pendulum system becomes more complex when a torque is delivered to the point of support; as the torque increases relative to the gravitational force, then the pendulum's behaviour becomes more complex and consequently more resistant to mathematical capture. In recent decades mathematicians and physicists have jointly worked on this problem. [2]

The IPP will indicate how the pendulum can anchor a long pedagogical journey in which this interplay of mathematics, experiment, observation and modelling can be explored, discovered, demonstrated and appreciated. It is a manageable case study that can be elaborated all the way from elementary school to postgraduate studies.

The Pendulum and Timekeeping

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 tied to 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 quantum increase in accuracy of timing enabled hitherto unimagined degrees of precision measurement in mechanics 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.

Accurate time measurement was long seen as the solution to the problem of longitude determination which had vexed European maritime nations in their efforts to sail beyond Europe's shores. [3] If an accurate and reliable clock was carried on voyages from London, Lisbon, Genoa, or any other port, then by comparing its time with local noon (as determined by the sun's shadow), the longitude of any place in the journey could be ascertained. As latitude could already be determined, this enabled the world to be mapped. In turn, this provided a firm base on which European trade and colonisation could proceed. The perils of being lost at sea were hugely decreased.

The clock transformed social life and customs: patterns of daily life could be 'liberated' from natural chronology (the seasonally varying rising and setting of the sun) and subjected to artificial chronology; labour could be regulated by clockwork and, because time duration could be measured, there could be debate and struggle about the length of the working day and the wages that were due to agricultural and urban workers; timetables for stage and later train and ship transport could be enacted; the starting time for religious and cultural events could be specified; punctuality could become a virtue; and so on. The transition from 'natural' to 'artificial' hours was of great social and psychological consequence: technology, a human creation, begins to govern its creator.[4]

The clock did duty in philosophy. It was a metaphor for the new mechanical worldview that was challenging the entrenched Aristotelian, organismic and teleological, view of the world that has sustained so much of intellectual and religious life. In theology, the clock was appealed to in the influential argument from design for God's existence Ð if the world functions regularly like a clock, as Newton and the Newtonians maintained, then there must be a cosmic clockmaker.[5]

Horology

The IPP is concerned to make the oft-ignored link between the seventeenth century's revolution in timekeeping and developments in physics and methodology. Despite there being scores of excellent books, and hundreds of research articles, on the technical, social and comparative history of timekeeping, there are few studies that connect the pendulum clock to Galileo and Huygens' discoveries of the physics of the pendulum, and even less studies that connect the pendulum clock to the Galilean revolution in scientific methodology. Galileo's law of isochronous motion, and hence his directions for using the pendulum in timekeeping, could not be accepted until he threw off the straight-jacket placed on science by the epistemological primacy given by Aristotelians to experience and the evidence of the senses. As long as scientific claims were judged by what could be seen, and as long as mathematics and physics were kept separate, then Galileo's pendulum claims could not be substantiated. Their substantiation required not just a new science, but a new way of judging scientific claims, a new methodology of science.

The Seconds Pendulum as a Universal Length Standard

The IPP will elaborate the little told story of how Huygens, in the process of elaborating his theory of pendulum motion and clockwork design argued in 1673 that the seconds pendulum could provide a new international standard of length (its length is effectively one modern metre). Undoubtedly this would have been a major contribution to simplifying the chaotic state of measurement existing in science and everyday life. He thought that this standard was dependent only upon the force of gravity, which he took to be constant all over the earth, and thus the length standard would not change with change of location. The standard was to be portable over space and time. Alas, Jean Richer's Cayenne voyage of 1672 suggested that the Paris seconds pendulum had to be very slightly shortened to beat seconds in tropical Cayenne (Matthews 2000, pp.144-146). Still, if a specific latitude were agreed upon (Paris? London? Berlin? Madrid?) then Huygens' proposal would answer to the pressing need of a natural, invariant length unit. Once a subsidiary volume standard was created, by filling this volume with rain water, an international mass unit would also be created. How Huygens' 1673 proposal of the seconds pendulum as a universal length standard was related to the century later (1793) decree of the French Revolutionary Assembly establishing the metre length standard as one 40th million part of the circumference of the earth, is an intriguing story with rich methodological, social and political overtones.[6]

Philosophy of Science and Pendulum Studies

The IPP adopts the view that philosophy of science should be informed by history of science. This is one of the important contributions of Thomas Kuhn's legacy. Historical study of the pendulum case shows how Galileo initiated the methodological transition which was to culminate in the Galilean-Newtonian Paradigm (GNP) which quickly came to characterise the Scientific Revolution, and the subsequent centuries of modern science. The Aristotelian epistemological taboo on manipulating nature, or experimenting, was lifted, as was the Aristotelian hesitancy to mix mathematics with science. The long entrenched conviction that only undisturbed or 'natural' states-of-affairs would reveal their essence was slowly replaced by the view that nature has to be simplified, that variables had to be controlled, that 'inputs' and 'outputs' needed to be measured and represented mathematically, and that scientific understanding was something other than grasping the essence or nature of things and ascertaining their final causes or teleological purposes. [7]

There are, admittedly, problems with 'historicised' philosophy of science. One is that history can be mined merely to find support for antecedently arrived at epistemological positions. History is then 'reconstructed' to suit whatever philosophical position is being advocated. Albert Schweitzer, in his monumental 1910 work on The Quest of the Historical Jesus that traced the history of Christian interpretation of Jesus, remarked that 'each successive epoch of theology found its own thoughts in Jesus É But it was not only each epoch that found its reflection in Jesus; each individual created Him in accordance with his own character' (Schweitzer 1910, p.4). Schweitzer could equally have been talking of Galileo. It is notorious that Galileo has been made out to be a shining example of the full range of epistemological positions: from rationalist, through empiricist and experimentalist, to positivist, and to methodological anarchist (Crombie 1981). The common thread is that the epistemology attributed to Galileo is usually the one favoured by the biographer or interpreter.

There is a chicken-and-egg problem with the Kuhnian stance. If philosophy of science emerges from history of science, how is the history first demarcated? Independently of a philosophical, normative, position what will count as the subject matter of history from which our methodological lesson is to be drawn? Do we draw lessons equally from Christian Science, National Socialist Science, Lysenkoism, Astrological Science, Islamic Science, Hindu Science, New Age Science as well as classical mechanics, thermodynamics, and quantum mechanics?

The two standard ways around these problems are essentialist approaches on the one hand, and nominalist approaches on the other. For essentialists, history is ignored and science is characterised on a priori grounds Ð usually philosophical, political or sometimes religious. For nominalists, philosophy is ignored, and science is taken to be hatever people claiming to do science actually do. This option is popular among cultural historians of science and sociologists of science. It is better to steer a path between these two alternatives by focussing on an episode that all can agree upon as being good science, and then teasing out some methodological lessons from that. If the achievements of Galileo and Newton are not considered good, or at least, representative, science, then the very question of the epistemology of science loses its cogency. This is a version of the common 'paradigm case' argument in philosophy: to understand something, first find an exemplary instance of it, and examine its features and ramifications. [8]

Galileo's Methodological Revolution

The seventeenth century's analysis of pendulum motion is a particularly apt window through which to view the methodological heart of the scientific revolution. More particularly, the debate between the Aristotelian Guidobaldo del Monte and Galileo over the latter's pendular claims, represents, in microcosm, the larger methodological struggle between Aristotelianism and the new science. This struggle is about the legitimacy of idealisation in science, and the utilisation of mathematics in the construction and interpretation of experiments. Del Monte was a prominent mathematician, engineer and patron of Galileo (Renn et al. 2000, Matthews 2000, pp.100-108). He kept indicating how the behaviour of pendulums contradicted Galileo's claims about them. Galileo kept maintaining that refined and ideal pendulums would behave according to his theory. Del Monte said that Galileo was a great mathematician, but a hopeless physicist. This is the methodological kernel of the scientific revolution. The development of pendular analyses by Huygens, and then Newton, beautifully illustrates the interplay between mathematics and experiment so characteristic of the emerging Galilean-Newtonian Paradigm. If students can be made familiar through their own investigations with some highlights of this nascent history of the pendulum, then they will have learnt something important about the origins and nature of modern science.

It is acknowledged that science has moved on, and that it can be claimed that understanding seventeenth century debates about the pendulum is irrelevant to understanding modern techno-industrial science and its methodology. This is a complex issue but, in brief, understanding origins, and development, is important for understanding and judging the present. This is true in just about all spheres Ð political, religious, social and personal - and no less so in conceptual matters. [9] Further modern science has not so outgrown its methodological roots as to make irrelevant an examination of central seventeenth century epistemological debates. Even if it could be shown that modern science is methodologically different from its origins, nevertheless understanding where modern science has come from and, consequently, what occasioned the change, is still important.

In education it is sensible to begin with simple or idealised cases. Presenting students with the full story Ð the truth, the whole truth, and nothing but the truth Ð is rarely a good idea. Concentrating on just some key aspects of a topic, be it in history, economics, biology, or what ever, makes pedagogical sense. The Galileo Ð del Monte debate does capture in comprehensible form some of the core issues of epistemology Ð the distinction between observation and experiment, the relationship of evidence to knowledge claims, the role of theory in guiding experiment, and so on - and this gives an educational justification for its presentation. Provided students are made aware that the complete picture, or the modern picture, might be more complex, and provided they are encouraged to examine how science may have changed, then dealing with the seventeenth century is educationally and philosophically justified.

'Big Picture' History of Science

The IPP fits into the 'big picture' or 'grand narrative' genre of history of science: it is dealing with the interrelatedness of timekeeping, pendulum science, philosophy and social forces; and it endeavours to draw methodological lessons from all this. Big Pictures in the history of science need not be painted with broad brush strokes. The IPP endeavours to compose a big picture but does so with fairly fine brushes. The IPP deals with both internal matters concerning the development and refinement of scientific concepts, and external matters concerning the social and cultural context in which the development of science occurs. This distinction needs detailed attention, and ultimately it is somewhat conventional. For instance, a change in epistemology was fundamental to Galileo's achievements in understanding pendulum motion. Is then epistemology internal or external to science? Huygens' recognition of the isochronous nature of cycloidal motion rested upon the new geometrical analysis of the cycloid curve. Is then mathematics internal or external to science? Neither Galileo's or Huygens' proposals for utilising the pendulum in timekeeping could be experimentally tested until technological advances in metallurgy, gear-cutting and escapement design were made. Is then technology internal or external to science? Once science is recognised as part of the intellectual culture of a society then the separation of 'internal' and 'external' elements borders on being conventional.

That the distinction is blurred, does not mean that it cannot be made in some form. It is clear that the longitude problem played a major role in the development of clockwork. Solving longitude was one of the major preoccupations of European nations from the fifteenth to the eighteenth centuries. Kings' ransoms were offered for its solution. Despite all the external financial and political pressure, a solution had to wait on scientific, methodological and mathematical progress. The world was the judge of putative solutions, not political or ideological interests. This is an important point to be appreciated at a time when many maintain that science simply dances to the tune of the last patron who paid the fiddler. In science, paying the fiddler and getting a good dance, are two different things.

The Pendulum and Piagetian Research

The pendulum entered into educational research and cognitive psychology with the publication in 1958 of the English translation of Barbel Inhelder and Jean Piaget's The Growth of Logical Thinking from Childhood to Adolescence (Inhelder & Piaget 1958). Chapter Four of the book describes the pendulum tasks that Piaget and Inhelder gave to children to ascertain the extent to which they could isolate and manipulate potential variables (length, amplitude, weight, impetus) that affected the periodicity of the pendulum. The chapter is titled 'Operations of Exclusion of Variables' because only one of the four potential variables impact upon the duration of swing. Performing the task of isolating and uncoupling (controlling) the variables was seen as a window onto the child's cognitive structures or capacities and their developmental sequencing. The tasks subsequently became a commonplace in diagnostic testing, being labelled 'Piagetian Reasoning Tasks' (PRT); as they involved extensive engagement with the child, the test procedure was called 'Methode Clinique'. Successful completion of the tasks was seen as indicative of the change from concrete to formal operational thinking. The subheadings of the chapter indicate the cognitive sequencing:

• Stage I: Indifferentiation between the subject's own actions and the motion of the pendulum.
• Stage II: Appearance of serial ordering and correspondence, but without separation of variables.
• Stage IIIa: Possible but not spontaneous separation of variables.
• Stage IIIb: The separation of variables and the exclusion of inoperant links.

The pendulum did for reasoning and formal thinking tests what it centuries earlier had done for timekeeping. Subsequently Piaget's cognitive theory, and his test protocols, have been extensively scrutinised. [10] The IPP will appraise this research tradition, commenting on its strengths, weaknesses and seeing how pendulum investigations might still be used to assess higher order mental capacities and children's ability to reason proportionally, to control variables, to make inferences, to draw conclusions about the truth of hypotheses given certain evidence Ð in brief, to think scientifically.

Enriched Scientific Literacy

The IPP supports efforts to widen the meaning of science literacy so that literacy is seen as involving an understanding and appreciation the nature of science, including its history, methodology and interrelations with culture. This is a demanding objective, but given the centrality of science to the development of society, culture and self-understanding, it is one that is worthy of pursuit by educationalists. In the USA, the National Standards for Science Education, and the AAAS's Project 2061 both endorse this wider, liberal idea of scientific literacy. So also does the National Curriculum in the UK, a number of provincial science curricula in Canada, the Norwegian science curriculum, the Danish science curriculum, and the New South Wales state syllabus in Australia. Most science programmes aspire to having students know more than just a certain amount of science content, and having a certain level of competence in scientific method and scientific thinking. Most programmes want students to have some sense of the 'big picture' of science: its history, philosophy and relationship to social ideologies, institutions and practices (McComas &Olson 1998). In most countries, science education has dual goals: promoting learning of science, and also learning about science. Or, as it has been stated, science education has both disciplinary and cultural goals (Gauld 1977). Teaching the history and philosophy of pendulum motion is an ideal vehicle for realising some of these more ambitious aspirations for scientific literacy.

Liberal Education and Pendulum Teaching

The contextual, intellectualist, cross-disciplinary proposals of the IPP find their natural home in the liberal education tradition, whose core commitment is that education is concerned with the development of a range of knowledge and a depth of understanding, and with the cultivation of intellectual and moral virtues. [11] The intellectual virtues certainly include developing capacities for clear, logical and critical thought. These liberal goals are contrasted with goals such as professional training, job preparation, promotion of self-esteem, social engineering, entertainment, or countless other putative purposes of schooling that are enunciated by politicians and administrators.

On this liberal view, science education is seen as contributing to the overall education of students, and thus considerations about aims and purposes of education constrain decisions about science education. The development of an educated person is the telos of school science teaching; this is the 'prize' that teachers' eyes need to be kept on. Liberal educational theory is a normative theory about good education, not a social scientific theory about actual schools and their function in society. A theory of education is not the same as a theory of schooling, although they are frequently confused. The situation is quite comparable to philosophy of philosophy which is concerned to identify the salient methodological and other features of good science, not provide a description or theory of any old science that happens to be practised in the Soviet Union in the 1940s, Nazi Germany in the 1930s, or in current day Institutes of Creation Science, Islamic Science or Hindu Science.

Curriculum Considerations

The educational importance of the IPP can be gauged from a glance at the recently adopted US National Science Education Standards (NRC 1996). In the Standards, two pages are devoted to the pendulum but there is no mention of the history, philosophy, or cultural impact of pendulum motion; no mention of the pendulum's connection with timekeeping; no mention of the longitude problem; and in the suggested assessment exercise, the obvious opportunity to connect standards of length with standards of time, is not taken. The Standards document was reviewed by tens of thousands of teachers and educators, and putatively represents current best practice in science education. It is clear that a little historical and philosophical knowledge about the pendulum could have transformed the treatment of the subject in the Standards, and consequently could 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. This problem is not confined to the US: it is an international problem. Hopefully the research publications and classroom materials generated by the IPP will do something to ameliorate this problem.

References

• (NRC) National Research Council: 1996, National Science Education Standards, National Academy Press, Washington.
• Adler, M.J.: 1939/1988, Reforming Education, (G. van Doren ed.), Macmillan, New York.
• Alder, K.: 1995, 'A Revolution to Measure: The Political Economy of the Metric System in France'. In M.N. Wise (ed.), The Values of Precision, Princeton University Press, Princeton, NJ, pp. 39-71.
• Andrewes, W.J.H. (ed.): 1998, The Quest for Longitude: The Proceedings of the Longitude Symposium, Harvard University, Cambridge, Massachusetts, November 4-6, 1993, 2nd Edition, Collection of Historical Scientific Instruments, Harvard University, Cambridge, MA.
• Bantock, G.H.:1981, The Parochialism of the Present, Routledge &Kegan Paul, London.
• Barger, V. &Olsson, M.: 1973, Classical Mechanics: A Modern Perspective, McGraw-Hill, NY.
• Berriman, A.E.: 1953, Historical Metrology: A New Analysis of the Archaelogical and Historical Evidence Relating to Weights and Measures, J.M. Dent &Sons, London.
• Bond, T.G. &Bunting, E.: 1995, Archives de Psychologie 63, 'Piaget and Measurement III: Reassessing the Methode Clinique', 231-255.
• Cipolla, C.: 1967, Clocks and Culture: 1300-1700, Collins, London.
• Conlin, M.F.: 1999, 'The Popular and Scientific Reception of the Foucault Pendulum in the United States', Isis 90(2), 181-204.
• Crombie, A.C.: 1981, 'Philosophical Presuppositions and the Shifting Interpretations of Galileo'. In J. Hintikka et al. (eds.), Theory Change, Ancient Axiomatics, and Galileo's Methodology, Reidel, Boston, pp. 271-286. Reproduced in A.C. Crombie, Science, Optics and Music in Medieval and Early Modern Thought, The Hambledon Press, London, 1990, pp. 345-362.
• Eisner, E.W.: 1992, 'Curriculum Ideologies'. In P.W. Jackson (ed.), Handbook of Research on Curriculum, Macmillan, New York, pp. 302-326.
• Gauld, C.F.: 1977, 'The Role of History in the Teaching of Science', Australian Science Teachers Journal 23(3), 47-52.
• Gould, R. T.: 1923, The Marine Chronometer, Its History and Development, J.D. Potter, London. Reprinted by The Holland Press, London, 1978.
• Heilbron, J.L.: 1989, 'The Politics of the Meter Stick', American Journal of Physics 57, 988-992.
• Heiskanen, W.A. &Vening Meinesz, F.A.: 1958, The Earth and its Gravity Field, McGraw, NY.
• Hirst, P.H.: 1974, 'Liberal Education and the Nature of Knowledge'. In his Knowledge and the Curriculum, Routledge &Kegan Paul, London, pp. 16-29.
• Howse, D.: 1980, Greenwich Time and the Discovery of Longitude, Oxford University Press, Oxford.
• Inhelder, B. &Piaget, J.: 1958, The Growth of Logical Thinking, Basic Books, New York.
• Kimball, B.: 1986, A History of the Idea of Liberal Education, Teachers College Press, New York.
• Kline, H.A.: 1988, The Science of Measurement: A Historical Survey, Dover Publications, New York.
• Kuhn, D. &Brannock, J.: 1977, 'Development of the Isolation of Variables Scheme in Experimental and ÒNatural ExperimentÓ Contexts', Developmental Psychology 13(1), 9-14.
• Kula, W.: 1986, Measures and Man, Princeton University Press, Princeton NJ.
• Landes, D.S.: 1983, Revolution in Time. Clocks and the Making of the Modern World, Harvard University Press, Cambridge, MA.
• Macey, S.L.: 1980, Clocks and Cosmos: Time in Western Life and Thought, Archon Books, Hamden, CT.
• Machamer, P.: 1998, 'Galileo's Machines, His Mathematics, and His Experiments'. In P. Machamer (ed.) The Cambridge Companion to Galileo, Cambridge University Press, pp.53-79.
• Matthews, M.R.: 2000, Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion can Contribute to Science Literacy, Kluwer Academic Publishers, New York.
• Mayr, E.: 1982, The Growth of Biological Thought, Harvard University Press, Cambridge MA.
• McComas, W.F. &Olson, J.K.: 1998, 'The Nature of Science in International Science Education Standards Documents'. In W.F. McComas (ed.), The Nature of Science in Science Education:Rationales and Strategies, Kluwer Academic Publishers, Dordrecht, pp. 41-52.
• McKeon, R.: 1994, On Knowing. The Natural Sciences, compiled by D.B Owen, edited by D.B. Owen &Z.K. McKeon, University of Chicago Press, Chicago.
• McMullin, E.: 1978, 'The Conception of Science in Galileo's Work'. In R.E. Butts &J.C. Pitt (eds.) New Perspectives on Galileo, Reidel Publishing Company, Dordrecht, pp. 209-258.
• McMullin, E.: 1990, 'Conceptions of Science in the Scientific Revolution'. In D.C. Lindberg &R.S. Westman (eds.) Reappraisals of the Scientific Revolution, Cambridge University Press, Cambridge.
• Mittelstrass, J.: 1972, 'The Galilean Revolution: The Historical Fate of a Methodological Insight', Studies in the History and Philosophy of Science 2, 297-328.
• Orrill, R. (ed.): 1995, The Condition of American Liberal Education: Pragmatism and a Changing Tradition, College Entrance Examination Board, New York.
• Peters, R.S.: 1966, Ethics and Education, George Allen and Unwin, London.
• Renn, J., Damerow, P. &Rieger, S.: 2000, 'Hunting the White Elephant: When and How did Galileo Discover the Law of Fall', Science in Context 13(3-4)
• Rogers, E.M.: 1960, Physics for the Inquiring Mind, Princeton University Press, Princeton.
• Rossum, G. D-V.: 1996, History of the Hour: Clocks and Modern Temporal Orders, Chicago University Press, Chicago.
• Scheffler, I.: 1973, Reason and Teaching, Bobbs- Merrill, Indianapolis.
• Schneider, C.G. &Shoenberg, R.: 1998, 'Contemporary Understandings of Liberal Education', Liberal Education, Spring, 32-37.
• Schweitzer, A.: 1910, The Quest for the Historical Jesus, Adam and Charles Black, London.
• Shayer, M. &Adey, P.: 1981, Towards a Science of Science Teaching: Cognitive Development and Curriculum Demand, Heinemann, London.
• Siegler, R.S., Liebert, D.E. &Liebert, R.M.: 1973, 'Inhelder and Piaget's Pendulum Problem: Teaching Preadolescents to Act as Scientists', Developmental Psychology 9(1), 97-101.
• Sobel, D.: 1995, Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time, Walker &Co., New York.
• Sommerville, S.: 1974, 'The Pendulum Problem: Patterns of Performance Defining Development Stages', British Journal of Educational Psychology 44, 266-281.
• Suchting, W.A.: 1995, 'The Nature of Scientific Thought', Science &Education 4(1), 1-22.
• Suchting, W.A.: 1995, 'The Nature of Scientific Thought', Science &Education 4(1), 1-22.
• Tavel, M.: 2002, Contemporary Physics and the Limits of Knowledge, Rutgers University Press, New Brunswick.
• Tristram, H. (ed.): 1952, The Idea of a Liberal Education:A Selection from the Works of Newman, Harrap, London.
• Westfall, R.S.: 1990, 'Making a World of Precision: Newton and the Construction of a Quantitative Physics'. In F. Durham &R.D Purrington (eds.), Some Truer Method. Reflections on the Heritage of Newton, Columbia University Press, New York, pp. 59-87.
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Footnotes

1. The Project is coordinated by Michael Matthews at the University of New South Wales, and has some support from the Australian Research Council. The book Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion can Contribute to Science Literacy (Matthews 2000) provides an overview of some of the scholarly and pedagogical matters with which the Project is concerned. ... back
2. Many books deal with the physics of the pendulum. Specifically: Tavel (2002, pp.219-231) deals with the progressive elaboration of the pendulum from simple to chaotic; Barger & Olsson (1973, pp.63-75) work through the mathematics of Lagrangian formulations of pendulum motion; Rogers (1960), a text written for the PSSC Physics Course, has an excellent chapter on the pendulum. ... back
3. Dava Sobel has given the Longitude Problem enormous exposure (Sobel 1995); other more detailed and wide-ranging treatments are in Andrewes (1998), Howse (1980), and Gould (1923). ... back
4. Many books deal with the social and cultural history of timekeeping, among them are: Landes (1983), Rossum (1996), Macey (1980) and Cipolla (1967). ... back
5. Macey 1980, Pt.II is a nice introduction to the utilisation of the clock in eighteenth century philosophy and theology. ... back
6. Accounts of the development of the standard metre can be found in Alder (1995), Kline (1988, chap. 9), Kula (1986, chaps. 21-23), Heilbron (1989), and Berriman (1953, chap. XI). Some of the methodological and political story is told in Matthews (2000, pp.141-150). ... back
7. Some insightful discussions of Galileo's methodological revolution are McMullin (1978, 1990), Mittelstrass (1972) and Machamer (1998). ... back
8. For an exemplary discussion of this paradigm case alternative to essentialism and nominalism, see Suchting (1995) ... back
9. Ernst Mayr, in the opening pages of his The Growth of Biological Thought, commends historical study to scientists in these terms: I feel that the study of the history of a field is the best way of acquiring an understanding of its concepts. Only by going over the hard way by which these concepts were worked out - by learning all the earlier wrong assumptions that had to be refuted one by one, in other words by learning all past mistakes - can one hope to acquire a really thorough and sound understanding. In science one learns not only by one's own mistakes but by the history of the mistakes of others. (Mayr 1982, p. 20) ... back
10. Some contributions are: Shayer &Adey (1981), Sommerville (1974), Bond &Bunting (1995), Kuhn &Brannock (1977), and Siegler, Liebert, &Liebert, (1973). ... back
11. Some of the more prominent advocates of liberal education have been: John Henry Newman (Tristram 1952), Mortimer Adler (Adler 1939/1988), Richard Peters (Peters 1966), Richard McKeon (McKeon 1994), Paul Hirst (Hirst 1974), Israel Scheffler (Scheffler 1973) and G.H. Bantock (Bantock 1981). See also Schneider &Shoenberg (1998) and contributions to Orrill (1995). Elliot Eisner, in his review of curriculum ideologies, calls this educational tradition 'rational humanism' (Eisner 1992). ... back