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History 291 Fall 2002 History 291Lecture 20 Pendulums and Falling Bodies: Clocking Longitude

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History 291 Fall 2002 Galileo math’l mechanics weakly connected to non-mechanistic cosmology Descartes mechanistic cosmology weakly connected to math’l mecanics laws of accelerated motion esp. v 2 h laws of impact: cons. motion pendulum non-tautochronism (Mersenne et al.) determination of [g] as measured by incorrect as tested by

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History 291 Fall sec. pendulum

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History 291 Fall 2002 h1 h2 h’1 h’2

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History 291 Fall 2002 Christiaan Huygens ( )

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History 291 Fall 2002

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cycloid

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History 291 Fall 2002

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Springs and Other Regulators Why the cycloid is tautochronic Hooke’s law of springs The spring-balance regulator Other tautochronic mechanisms

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History 291 Fall 2002

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Huygens’ original sketch of his balance-spring regulator 20 January 1675

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History 291 Fall 2002

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Pendulums, Gravity, and the Shape of the Earth The great voyage of correcting for latitude Descartes’s vortices or Newton’s gravity?

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History 291 Fall 2002

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12/16/56 pendulum clock 1658 Horologium 1673 Horologium oscillatorium 1657 first efforts at using leaves to temper swing of pendulum 12/1/1659 tautochronism of cycloid 12/20/1659 cycloidal leaves 1/13/1660 tested cycloidal clock against sun 1661 first efforts with sliding weight to adjust C osc C osc, C G for various solid, esp. wedges by 1664 complete theory of C osc & sliding weight relation of vibrating string to cycloid spring as source of incitation parfaite 2/1675 art. in Journal des Sçavans on spring-balance watch 10/5/1659 sketch of conical pendulum clock 10/21/59 ms. on centrifugal force marine clocks calculation of period of conical pendulum 1671 triangular suspension 1675 anchor escapement 1663 Holmes to Lisbon 1664 Holmes to Guinea 1669 Duc de Beaufort, de la Voye Richer to Cayenne Helder and de Graaf de Graaf first sketch of balancier marin parfait 1683 pendulum cylindricum trichordon - abandoned application of triangular suspension to spring-driven marine clock 1-2/1693 balancier marin parfait 1694 studies on marine clock 6/1658 Pascal’s challenge problems re: cycloid sent to H. via Boulliau late 1659 challenge of priority by Italians (Leopoldo de’ Medici) Sea Trials

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Laws of fall Vortex theory cycloid astronomy Pendulum clock 1657 Tautochrone 1659 Cycloidal pendulum 1659 Center of oscillation Marine clock method of longitude equation of time 1662 Spring balance 1675 Tautochronic oscillators Constrained motion along arbitrary curve Isochrone, brachistochrone Newton Bernoulli Varignon calculus of variations Harmonic oscillation theory of springs Bernoulli Dynamics of rigid bodies moment of inertia “potential ascent. = actual descent” Daniel Bernoulli (Hydrodynamica, 1738) Theory of evolutes higher differentials Analytical dynamics on variously described orbits, e.g. polar coords. Varignon, 1700ff. Analytic kinematics Centrifugal force Conical pendulum pendulum Evolute of circle Evolute of cycloid impact center of percussion Torricelli’s Principle Evolute of parabola Period of pendulum Galileo Descartes Huygens and the Pendulum Clock, msm 98

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History 291 Fall 2002

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