# Horologium Oscillatorium

Author Christiaan Huygens Latin Physics, Horology 1673

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's 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

Invention of the pendulum clock by Christiaan Huygens by Georg Sturm (c. 1885)

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 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’s 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 perfectly tautochronous, that is, they do not keep exact time but depend to some extent 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 Oscillatorium (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 longitude 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.

In the Preface, Huygens states:[1][5]

For it is not in the nature of a simple pendulum to provide equal and reliable measurements of time… But by a geometrical method we have found a different and previously unknown way to suspend the pendulum… [so that] the time of the swing can be chosen equal to some calculated value

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. This part 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:

${\displaystyle (\pi /2)\surd (2D/g)}$

The following propositions are covered in Part II:[8]

Propositions Description
1-8 Bodies falling freely and through inclined planes.
9-11 Fall and ascent in general.
12-15 Tangent of cycloid, history of the problem, and generalization to similar curves.
16-26 Fall through a cycloid.

### 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 rectify curves such as the cycloid, the parabola, and other higher order curves.[5][16]

The following propositions are covered in Part III:[8]

Propositions Description
1-4 Definitions of evolute, involute, and their relationship.
5-6, 8 Evolute of cycloid and parabola.
7, 9a Rectification of cycloid, semicubical parabola, and history of the problem.
9b-e Circle areas equal to surfaces of conoids; rectification of the parabola equal to

quadrature of hyperbola; approximation by logarithms.

10-11 Evolutes of ellipses, hyperbolas, and of any given curve; rectification of those

examples.

### 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] It makes use, implicitly, of the formula for free fall. In modern notation:

${\displaystyle d=1/2gt^{2}}$

The following propositions are covered in Part IV:[8]

Propositions Description
1-6 Simple pendulum equivalent to a compound pendulum with weights equal to its

length.

7-20 Center of oscillation of a plane figure and its relationship to center of gravity.
21-22 Centers of oscillation of common plane and solid figures.
23-24 Adjustment of pendulum clock to small weight; application to a

cyclodial pendulum.

25-26 Universal measure of length based on second pendulum; constant of

gravitational acceleration.

### 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 proofs, 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

A page from Horologium Oscillatorium (1673) showing Huygens's mathematical style.

Initial reviews of Huygens's Horologium Oscillatorium in 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 of the pendulum clock "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's 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.[12][23]

In addition to submitting his work for review, Huygens sent copies of his book to individuals throughout Europe, including statesmen such as Johan De Witt, and mathematicians such as Gilles de Roberval and Gregory of St. Vincent. Their appreciation of the text was due not exclusively on their ability to comprehend it fully, but rather as a recognition of Huygens’s intellectual standing, or of his gratitude or fraternity that such gift implied.[11] Thus, sending copies of the Horologium Oscillatorium worked in a manner similar to a gift of an actual clock, which Huygens had also sent to several people, including Louis XIV and the Grand Duke Ferdinand II.[23]

### Mathematical style

Huygens's mathematics in the Horologium Oscillatorium and elsewhere is best characterized as geometrical analysis of curves and of motions. In style it closely resembled classical Greek geometry, and Huygens was well versed in the works of both Apollonius and Archimedes.[1][13] He was also proficient in the analytical geometry of Descartes and Fermat, and made use of it particularly in Parts III and IV of his book. Using these tools, Huygens was quite capable of finding solutions to hard problems that today are solved using analytical methods.[4]

Huygens's manner of presentation (i.e., clearly stated axioms, followed by propositions) also made an impression among contemporary 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's mathematics soon fell into disuse with the advent of the calculus, making it more difficult for subsequent generations to appreciate his work.[9]

### Legacy

Huygens’s most lasting contribution in the Horologium Oscillatorium is his thorough application of mathematics to explain pendulum clocks, which were the first reliable timekeepers fit for scientific use.[4] His analysis of the cycloid in Parts II and III would later lead to the studies of many other such curves, including the caustic, the brachistochrone, the sail curve, and the catenary.[9] Additionally, Huygens's exacting mathematical dissection of physical problems into a minimum of parameters provided an example for others (such as the Bernoullis) on work in applied mathematics that would be carry on in the following century.[8]

## Editions

Huygens’s own manuscript of the book is missing, but he bequeathed his notebooks and correspondence to the Library of the University of Leiden, now in the Codices Hugeniorum. Much of the background material is in Oeuvres Complètes, vols. 17-18.[8]

Since its publication in France in 1673, Huygens’s work has been available in Latin and in the following modern languages:

• First publication. Horologium Oscillatorium, Sive De Motu Pendulorum Ad Horologia Aptato Demonstrationes Geometricae. Latin. Paris: F. Muguet, 1673. [14] + 161 + [1] pages.[1]
• Later edition by W.J. ’s Gravesande. In Christiani Hugenii Zulichemii Opera varia, 4 vols. Latin. Leiden: J. vander Aa, 1724, 15–192. [Repr. as Christiani Hugenii Zulichemii opera mechanica, geometrica, astronomica et miscellenea, 4 vols., Leiden: G. Potvliet et alia, 1751].
• Standard edition. In Oeuvres Complètes, vol. 18. French and Latin. The Hague: Martinus Nijhoff, 1934, 68–368.
• German translation. Die Pendeluhr (trans. A. Heckscher and A. von Oettingen), Leipzig: Engelmann, 1913 (Ostwalds Klassiker der exakten Wissenschaften, no. 192).
• Italian translation. L’orologio a pendolo (trans. C. Pighetti), Florence: Barbèra, 1963. [Also includes an Italian translation of Traite de la Lumiere]
• French translation. L’Horloge oscillante (trans. J. Peyroux), Bordeaux: Bergeret, 1980. [Photorepr. Paris: Blanchard, 1980.]
• English translation. Christiaan Huygens’ The Pendulum Clock, or Geometrical Demonstrations Concerning the Motion Of Pendula As Applied To Clocks (trans. R.J. Blackwell), Ames: Iowa State University Press, 1986.

## References

1. ^ a b c 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.
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. Yoder, Joella G. (1988). Unrolling Time: Christiaan Huygens and the Mathematization of Nature. Cambridge: Cambridge University Press. ISBN 978-0-521-34140-0.
5. 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. Yoder, Joella G. (2005), "Christiaan Huygens book on the pendulum clock 1673", Landmark Writings in Western Mathematics 1640-1940, Elsevier, ISBN 9780080457444.
9. 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. ^ a b 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 d 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. 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 c Howard, N. C. (2003). "Christiaan Huygens: The construction of texts and audiences - ProQuest". pp. 162–177.{{cite web}}: CS1 maint: url-status (link)