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Horologium Oscillatorium

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Horologium Oscillatorium
AuthorChristiaan Huygens
LanguageLatin
GenrePhysics, Horology
Published1673

Horologium Oscillatorium: Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks) is a book published by Christiaan Huygens in 1673 and his major work on pendulums and horology.[1][2] It is regarded as one of the three most important works on mechanics in the 17th century, the other two being Galileo’s Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) and Newton’s Philosophiæ Naturalis Principia Mathematica (1687).[3]

Much more than a mere description of clocks, Huygens' Horologium Oscillatorium is the first modern treatise in which a physical problem (the accelerated motion of a falling body) is idealized by a set of parameters then analyzed mathematically and constitutes one of the seminal works of applied mathematics.[4][5][6] The book is also known for its strangely worded dedication to Louis XIV.[7] The appearance of the book in 1673 was a political issue, since at that time the Dutch Republic was at war with France; Huygens was anxious to show his allegiance to his patron, which can be seen in the obsequious dedication to Louis XIV.[8]

Background

The use of pendulums to keep time was not new but had already been proposed by people engaged in astronomical observations such as Galileo.[4] Mechanical clocks, on the other hand, were regulated by balances that were often very unreliable.[9][10] Moreover, without reliable clocks, there was no good way to measure longitude at sea, which was particularly problematic for a country dependent on sea trade like the Dutch Republic.[11]

Huygens interest in using a freely suspended pendulum to regulate clocks began in earnest in December of 1656 and he had a working model by the next year, which he patented and then communicated to other scholars such as Frans van Schooten and Claude Mylon.[8][12] Although Huygens’ design, published under the title Horologium (1658), was a combination of existing ideas, it nonetheless became widely popular and led to many pendulum clocks being built and even retrofitted to existing clock towers such as those of Scheveningen and Utrecht.[9][13]

Huygens began to study the problem of free fall mathematically shortly after in 1659, obtaining a series of remarkable results.[13][14] At the same time, he was aware that the periods of simple pendulums are not tautochronous, that is, they do not keep exact time but depend on their amplitude.[4][9] Huygens was interested in finding a way to make the bob of a pendulum move reliably and independently of its amplitude. The breakthrough came later that same year when he discovered that the ability to keep perfect time can be achieved if the path of the pendulum bob is a cycloid.[10][15] However, it was unclear what form to give the metal cheeks regulating the pendulum to lead the bob in a cycloidal path. His famous and surprising solution was that the cheeks must also have the form of a cycloid, on a scale determined by the length of the pendulum.[9][16][17] These and other results led Huygens to develop his theory of evolutes and provided the motivation to write a much larger work, which became the Horologium Oscillatorum (1673).[8][13]

After 1673, during his stay in the Academie des Sciences, Huygens studied harmonic oscillation more generally and continued his attempt at determining longitudes at sea using his pendulum clocks, but his experiments carried on ships were not very successful.[9][11][18]

Contents

Illustration of Huygens' 1673 experimental pendulum clock from Horologium Oscillatorium.

The book is divided into five interconnected parts. The first and last parts of the book contain descriptions of clock designs. The rest of the book is devoted to the analysis of pendulum motion and a theory of curves. Except for part IV, written in 1664, the entirety of the book was composed in a three-month period starting in October 1659.[4][5]

Part I: Description of the oscillating clock

Huygens spends the first part of the book describing in detail his design for an oscillating pendulum clock. It includes descriptions of the endless chain, a lens-shaped bob to reduce air resistance, a small weight to adjust the pendulum swing, an escapement mechanism for connecting the pendulum to the gears, and two thin metal plates in the shape of cycloids mounted on either side to limit pendular motion. It ends with a table to adjust for the inequality of the solar day, a description on how to draw a cycloid, and a discussion of the application of pendulum clocks for the determination of longitude at sea.[5][8]

Part II: Fall of weights and motion along a cycloid

In the second part of the book, Huygens states three hypotheses on the motion of bodies. They are essentially the law of inertia and the law of composition of motion. He uses these three rules to re-derive geometrically Galileo's original study of falling bodies, including linear fall along inclined planes and fall along a curved path.[4][19] He then studies constrained fall, culminating with a proof that a body falling along an inverted cycloid reaches the bottom in a fixed amount of time, regardless of the point on the path at which it begins to fall. This in effect shows the solution to the tautochrone problem as given by a cycloid curve.[8][20] In modern notation:

Part III: Size and evolution of the curve

An illustration of a rolling circle forming a cycloid.

In the third part of the book, Huygens introduces the concept of an evolute as the curve that is "unrolled" (Latin: evolutus) to create a second curve known as the involute. He then uses evolutes to justify the cycloidal shape of the thin plates in Part I.[8] Huygens originally discovered the isochronism of the cycloid using infinitesimal techniques but in his final publication he resorted to proportions and reductio ad absurdum, in the manner of Archimedes, to the rectification of curves such as the cycloid, the parabola, and other higher order curves.[5][16]

Part IV: Center of oscillation or movement

The fourth and longest part of the book is concerned with the study of the center of oscillation. Huygens introduces physical parameters into his analysis while addressing the problem of the compound pendulum. It starts with a number of definitions and proceeds to derive propositions using Torricelli's Principle: that the center of gravity of heavy objects cannot lift itself, which Huygens used as a virtual work principle.[4] In the process, Huygens obtained solutions to dynamical problems such as the period of an oscillating pendulum as well as a compound pendulum, the center of oscillation and its interchangeability with the pivot point, and the concept of moment of inertia and the constant of gravitational acceleration.[5][8][21] In modern notation:

Part V: Alternative design and centrifugal force

The last part of the book returns to the design of a clock where the motion of the pendulum is circular, and the string unwinds from the evolute of a parabola. It ends with thirteen propositions regarding bodies in uniform circular motion, without proof, and states the laws of centrifugal force for uniform circular motion.[22] The proofs of these propositions were published posthumously in the De Vi Centrifuga (1703).[4]

Reception

Initial reviews of Huygens' Horologium Oscillatorium in all three major research journals at the time were generally positive. An anonymous review in Journal de Sçavans (1674) praised the author of the book for his invention "which brings the greatest honor to our century because it is of utmost importance... for astronomy and for navigation" while also noting the elegant, but difficult, mathematics needed to fully understand the book.[23] Another review in the Giornale de Letterati (1674) repeated many of the same points than the first one, with further elaboration on Huygens' trials at sea. The review in the Philosophical Transactions (1673) likewise praised the author for his invention but mentions other contributors to the clock design, such as William Neile, that in time would lead to a priority dispute.[11][23]

Huygens' manner of presentation also made an impression among mathematicians, including Newton, who later acknowledged the influence of Horologium Oscillatorium on his own major work.[17] Nonetheless, the Archimedean and geometrical style of Huygens' mathematics soon fell into disuse with the advent of the calculus, making it more difficult to appreciate his work.[9]

Huygens’ most lasting contribution in the Horologium Oscillatorum is his through application of mathematics to explain pendulum clocks, which were the first reliable timekeepers fit for scientific use.[4] With his exacting mathematical dissections of physical problems into a minimum of parameters, Huygens opened the way for others (such as the Bernoullis) to go on and continue work in applied mathematics.[8]

References

  1. ^ Huygens, Christiaan; Blackwell, Richard J., trans. (1986). Horologium Oscillatorium (The Pendulum Clock, or Geometrical demonstrations concerning the motion of pendula as applied to clocks). Ames, Iowa: Iowa State University Press. ISBN 0813809339.{{cite book}}: CS1 maint: multiple names: authors list (link)
  2. ^ Herivel, John. "Christiaan Huygens". Encyclopædia Britannica. Retrieved 14 November 2013.
  3. ^ Bell, A. E. (30 Aug 1941). "The Horologium Oscillatorium of Christian Huygens". Nature. 148 (3748): 245–248. doi:10.1038/148245a0. S2CID 4112797. Retrieved 14 November 2013.
  4. ^ a b c d e f g h Yoder, Joella G. (1988). Unrolling Time: Christiaan Huygens and the Mathematization of Nature. Cambridge: Cambridge University Press. ISBN 978-0-521-34140-0.
  5. ^ a b c d e Bruce, I. (2007). Christian Huygens: Horologium Oscillatorium. Translated and annotated by Ian Bruce.
  6. ^ "Christiaan Huygens, book on the pendulum clock (1673)". Landmark Writings in Western Mathematics 1640-1940: 33–45. 2005-01-01. doi:10.1016/B978-044450871-3/50084-X.
  7. ^ Levy, David H.; Wallach-Levy, Wendee (2001), Cosmic Discoveries: The Wonders of Astronomy, Prometheus Books, ISBN 9781615925667.
  8. ^ a b c d e f g h Yoder, Joella G. (2005), "Christiaan Huygens book on the pendulum clock 1673", Landmark Writings in Western Mathematics 1640-1940, Elsevier, ISBN 9780080457444.
  9. ^ a b c d e f Bos, H. J. M. (1973). Huygens, Christiaan. Complete Dictionary of Scientific Biography, pp. 597-613.
  10. ^ a b Lau, K. I.; Plofker, K. (2007), Shell-Gellasch, A. (ed.), "The Cycloid Pendulum Clock of Christiaan Huygens", Hands on History: A Resource for Teaching Mathematics, Mathematical Association of America, pp. 145–152, ISBN 978-0-88385-182-1
  11. ^ a b c Howard, Nicole (2008). "Marketing Longitude: Clocks, Kings, Courtiers, and Christiaan Huygens". Book History. 11: 59–88. ISSN 1098-7371.
  12. ^ van den Ende, H., Hordijk, B., Kersing, V., & Memel, R. (2018). The invention of the pendulum clock: A collaboration on the real story.
  13. ^ a b c Dijksterhuis, Fokko J. (2008). "Stevin, Huygens and the Dutch Republic". Nieuw archief voor wiskunde (in Dutch). S 5, dl 9 (2): 100–107. ISSN 0028-9825.
  14. ^ Ducheyne, Steffen (2008). "Galileo and Huygens on free fall: Mathematical and methodological differences". Dynamis. 28: 243–274. ISSN 0211-9536.
  15. ^ Lodder, J. (2018). The Radius of Curvature According to Christiaan Huygens, pp. 1-14.
  16. ^ a b Mahoney, M. S. (2000), Grosholz, E.; Breger, H. (eds.), "Huygens and the Pendulum: From Device to Mathematical Relation", The Growth of Mathematical Knowledge, Synthese Library, Springer Netherlands, pp. 17–39, doi:10.1007/978-94-015-9558-2_2
  17. ^ a b Chareix, F. (2004). Huygens and mechanics. Proceedings of the International Conference "Titan - from discovery to encounter" (April 13-17, 2004). Noordwijk, Netherlands: ESA Publications Division, ISBN 92-9092-997-9, p. 55 - 65.
  18. ^ Erlichson, Herman (1996-05-01). "Christiaan Huygens' discovery of the center of oscillation formula". American Journal of Physics. 64 (5): 571–574. doi:10.1119/1.18156. ISSN 0002-9505.
  19. ^ Ducheyne, Steffen (2008). "Galileo and Huygens on free fall: Mathematical and methodological differences". Dynamis : Acta Hispanica Ad Medicinae Scientiarumque. Historiam Illustrandam. 28: 243–274. doi:10.4321/S0211-95362008000100011. ISSN 0211-9536. Retrieved 2013-12-27.
  20. ^ Mahoney, Michael S. (March 19, 2007). "Christian Huygens: The Measurement of Time and of Longitude at Sea". Princeton University. Archived from the original on 2007-12-04. Retrieved 2013-12-27.
  21. ^ Bevilaqua, Fabio; Lidia Falomo; Lucio Fregonese; Enrico Gianetto; Franco Giudise; Paolo Mascheretti (2005). "The pendulum: From constrained fall to the concept of potential". The Pendulum: Scientific, Historical, Philosophical, and Educational Perspectives. Springer. pp. 195–200. ISBN 1-4020-3525-X. Retrieved 2008-02-26. gives a detailed description of Huygens' methods
  22. ^ Huygens, Christian (August 2013). "Horologium Oscillatorium (An English translation by Ian Bruce)". Retrieved 14 November 2013.
  23. ^ a b Howard, N. C. (2003). "Christiaan Huygens: The construction of texts and audiences - ProQuest". pp. 162–177.{{cite web}}: CS1 maint: url-status (link)


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