# Christian Huygens : Horologium Oscillatorium.

## translated and annotated byIan Bruce

This is an annotated translation of Huygens' Horologium Oscillatorium, taken from the 'Gallica' website of the French National Library from the Oeuvres of Huygens, this also includes a French translation of the latin text; here the latin text is inculded last, from my own transcription. I am indebted both to the editors of Huygens' works, and to the Bibliotheque numerique.... for making available Huygens' Works in an accessible form. This work inspired a generation of physicists and mathematicians, including Newton, Johanne Bernoulli, and Euler, for it reaches far beyond the working of pendulum clocks, and it still deserves a place in the historical presentation of the subject as presented today in elementary physics text books. For example, most of the intuitive ideas about centripetal forces are presented here in Part V. I am indebted on several points to the 1986 translation of Richard J. Blackwell, which I discovered only after most of my own translation; this admirable translation does not however include mathematical notes, apart from an introduction by H.J.M.Bos. It must be stressed that this present work is my own independent translation, in which I have gone through most of the relevant mathematics, and the Huygens scholar will of course be aware of Blackwell's work as well as French and German translation. This work also has the advantage of accessibility, and comes to you free of charge.

Part 1 describes the work on clocks carried out by Huygens, and his attempts at the determination of longitude at sea. The tautochronous behaviour of the cycloid curve is used as a means of correcting the erratic behaviour of pendulums on ships. Most of the first part is concerned with the making and testing of such clocks.

Part IIA relates to a careful geometrical examination of the speed and distance travelled by a body falling from rest in successive time intervals both vertically and down inclined planes; before embarking on motion down the cycloid curve, which is set out in part IIB. Huygens makes use of traditional Greek geometry and leans on Galileo in establishing the tautochronic behaviour of the cycloid, which is rather heavy going with such limited mathematical tools. If nothing else, it demonstrates the beauty of the analytical methods that were soon to appear. One presumes that Huygens made use of such tools in his derivation, which remain hidden.

Part III is a rather wonderful essay on the properties of the Evolutes of Curves and their lengths, starting with the cycloid, the parabola, and others derived from conic sections and higher order curves. Some results are only quoted, unfortunately, but this work gives a real insight into the state of mathematics in the middle of the 17th century. One can understand why Huygens' book made such an impression on his peers, including Newton.

Part IVA is a recipe for finding the centre of oscillation for a suspended shape. It is the first time the quantity that Euler later called the moment of inertia was investigated. You will find many familiar propositions here in an unfamiliar guise. Part IVB evaluates the centre of oscillation for many familiar shapes in two and three dimesions as canditates for the pendulum bob. The later theorems return to the clock, which is now standardised, and a use in determining a standard of length, as well as the acceleration of gravity, are investigated. Truly, a wonderful piece of work : Huygens is seen as the outstanding physicist of the pre-Newton age who pointed the way forwards. This work deserves to be read by all physicists and mathematicians with an interest in physics.

Part V shows a different kind of clock, where the motion of the pendulum is circular, and the string unwinds from the evolute of a parabola. This clock was presumably used to conduct experments on circular motion, and on the misnamed centrifugal force in particular, for which a number of theorems are set out without proof. A translation of Huygens 'On Centrifugal Force' presented in the Oeuvres, vol. XVI, pp 255-301, by M.S.Mahoney can be found on the web, which provides the analysis for the final theorems.

CONTENTS

Huygens' Pendulum Clock.

Part IVA : Concerning the centre of oscillation.

Part IVB : Concerning the centre of oscillation.

Part V : The construction of another kind of clock, and the enunciation of some theorems on centrifugal force.