Horologium Oscillatorium
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English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
: ''The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks'') is a book published by Dutch physicist
Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
in 1673 and his major work on
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
s and
horology Horology (; related to Latin '; ; , interfix ''-o-'', and suffix ''-logy''), . is the study of the measurement of time. Clocks, watches, clockwork, sundials, hourglasses, clepsydras, timers, time recorders, marine chronometers, and atomic cl ...
. It is regarded as one of the three most important works on
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
in the 17th century, the other two being
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
’s '' Discourses and Mathematical Demonstrations Relating to Two New Sciences'' (1638) and
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
’s (1687). Much more than a mere description of clocks, Huygens's 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 Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
.Bruce, I. (2007).
Christian Huygens: Horologium Oscillatorium
'. Translated and annotated by Ian Bruce.
The book is also known for its strangely worded dedication to
Louis XIV , house = Bourbon , father = Louis XIII , mother = Anne of Austria , birth_date = , birth_place = Château de Saint-Germain-en-Laye, Saint-Germain-en-Laye, France , death_date = , death_place = Palace of Vers ...
.. The appearance of the book in 1673 was a political issue, since at that time the
Dutch Republic The United Provinces of the Netherlands, also known as the (Seven) United Provinces, officially as the Republic of the Seven United Netherlands (Dutch: ''Republiek der Zeven Verenigde Nederlanden''), and commonly referred to in historiography ...
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 , house = Bourbon , father = Louis XIII , mother = Anne of Austria , birth_date = , birth_place = Château de Saint-Germain-en-Laye, Saint-Germain-en-Laye, France , death_date = , death_place = Palace of Vers ...
..


Overview

The motivation behind ''Horologium Oscillatorium'' (1673) goes back to the idea of using pendulums to keep time, which had already been proposed by people engaged in astronomical observations such as
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
. Mechanical clocks at the time were instead regulated by balances that were often very unreliable.Bos, H. J. M. (1973)
Huygens, Christiaan
''Complete Dictionary of Scientific Biography'', pp. 597-613.
Moreover, without reliable clocks, there was no good way to measure
longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...
at sea, which was particularly problematic for a country dependent on sea trade like the
Dutch Republic The United Provinces of the Netherlands, also known as the (Seven) United Provinces, officially as the Republic of the Seven United Netherlands (Dutch: ''Republiek der Zeven Verenigde Nederlanden''), and commonly referred to in historiography ...
. Huygens interest in using a freely suspended
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
to regulate clocks began in earnest in December 1656. He had a working model by the next year which he patented and then communicated to others such as Frans van Schooten and
Claude Mylon Claude Mylon (1618–1660) was a French mathematician and member of the Académie Parisienne and the Académie des Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis ...
.van den Ende, H., Hordijk, B., Kersing, V., & Memel, R. (2018).
The invention of the pendulum clock: A collaboration on the real story
'.
Although Huygens’s design, published in a short tract entitled ''Horologium'' (1658), was a combination of existing ideas, it nonetheless became widely popular and many pendulum clocks by Salomon Coster and his associates were built on it. Existing
clock tower Clock towers are a specific type of structure which house a turret clock and have one or more clock faces on the upper exterior walls. Many clock towers are freestanding structures but they can also adjoin or be located on top of another buildi ...
s, such as those at
Scheveningen Scheveningen is one of the eight districts of The Hague, Netherlands, as well as a subdistrict (''wijk'') of that city. Scheveningen is a modern seaside resort with a long, sandy beach, an esplanade, a pier, and a lighthouse. The beach is po ...
and
Utrecht Utrecht ( , , ) is the List of cities in the Netherlands by province, fourth-largest city and a List of municipalities of the Netherlands, municipality of the Netherlands, capital and most populous city of the Provinces of the Netherlands, pro ...
, were also retrofitted following Huygens's design. Huygens continued his mathematical studies on
free fall In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on i ...
shortly after, in 1659, obtaining a series of remarkable results. 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 The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
. 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 In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
. 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 In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
, on a scale determined by the length of the pendulum.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, , p. 55 - 65.
These and other results led Huygens to develop his theory of evolutes and provided the incentive to write a much larger work, which became the ''Horologium Oscillatorium''. After 1673, during his stay in the ''
Academie des Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at th ...
'', Huygens studied
harmonic oscillation In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive consta ...
more generally and continued his attempt at determining longitude at sea using his pendulum clocks, but his experiments carried on ships were not always successful.


Contents

In the Preface, Huygens states: The book is divided into five interconnected parts. Parts I and V of the book contain descriptions of clock designs. The rest of the book is made of three, highly abstract, mathematical and mechanical parts dealing with pendular 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.


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 synodic day (or synodic rotation period or solar day) is the period for a celestial object to rotate once in relation to the star it is orbiting, and is the basis of solar time. The synodic day is distinguished from the sidereal day, which is ...
, a description on how to draw a
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
, and a discussion of the application of pendulum clocks for the determination of longitude at sea.


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 Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
and the law of composition of
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...
. He uses these three rules to re-derive geometrically Galileo's original study of
falling bodies Lection 0 A set of equations describing the trajectories of objects subject to a constant gravitational force under normal Earth-bound conditions. Assuming constant acceleration ''g'' due to Earth’s gravity, Newton's law of universal gravita ...
, including linear fall along inclined planes and fall along a curved path. He then studies constrained fall, culminating with a proof that a body falling along an inverted
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
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 In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
curve. In modern notation: (\pi/2)\surd(2D/g) The following propositions are covered in Part II:


Part III: Size and evolution of the curve

In the third part of the book, Huygens introduces the concept of an
evolute In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curv ...
as the curve that is "unrolled" (Latin: ''evolutus'') to create a second curve known as the
involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or ...
. He then uses evolutes to justify the cycloidal shape of the thin plates in Part I. Huygens originally discovered the isochronism of the
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
using infinitesimal techniques but in his final publication he resorted to proportions and
reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
, in the manner of
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, to rectify curves such as the cycloid, the
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
, and other higher order curves. The following propositions are covered in Part III:


Part IV: Center of oscillation or movement

The fourth and longest part of the book contains the first successful theory of the
center of oscillation The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is st ...
, together with special methods for applying the theory, and the calculations of the centers of oscillation of several plane and solid figures. Huygens introduces physical parameters into his analysis while addressing the problem of the
compound pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
. It starts with a number of definitions and proceeds to derive propositions using Torricelli's Principle: that the center of gravity of two or more objects joined together cannot lift itself, which Huygens used as a virtual work principle. 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 The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
and the constant of gravitational acceleration. It makes use, implicitly, of the formula for
free fall In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on i ...
. In modern notation: d = 1/2 gt^2 The following propositions are covered in Part IV:


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 In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
for uniform circular motion. These propositions were studied closely at the time, although their proofs were only published posthumously in the ''De Vi Centrifuga'' (1703).


Summary

Many of the propositions found in the ''Horologium Oscillatorium'' had little to do with clocks but rather point to the evolution of Huygens’s ideas. When an attempt to measure the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
using a pendulum failed to give consistent results, Huygens abandoned the experiment and instead idealized the problem into a mathematical study comparing free fall and fall along a circle. Initially, he followed Galileo’s approach to the study of fall, only to leave it shortly after when it was clear the results could not be extended to curvilinear fall. Huygens then tackled the problem directly by using his own approach to infinitesimal analysis, a combination of
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
,
classical geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, and contemporary infinitesimal techniques. Huygens chose not to publish the majority of his results using these techniques but instead adhered as much as possible to a strictly classical presentation, in the manner of
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
.


Reception

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. 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 ''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journa ...
'' (1673) likewise praised the author for his invention but mentions other contributors to the clock design, such as
William Neile William Neile (7 December 1637 – 24 August 1670) was an English mathematician and founder member of the Royal Society. His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and w ...
, that in time would lead to a priority dispute. 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 Johan de Witt (; 24 September 1625 – 20 August 1672), ''lord of Zuid- en Noord-Linschoten, Snelrewaard, Hekendorp en IJsselvere'', was a Dutch statesman and a major political figure in the Dutch Republic in the mid-17th century, the Fi ...
, and mathematicians such as
Gilles de Roberval Gilles Personne de Roberval (August 10, 1602 – October 27, 1675), French mathematician, was born at Roberval near Beauvais, France. His name was originally Gilles Personne or Gilles Personier, with Roberval the place of his birth. Biography ...
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. 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 , house = Bourbon , father = Louis XIII , mother = Anne of Austria , birth_date = , birth_place = Château de Saint-Germain-en-Laye, Saint-Germain-en-Laye, France , death_date = , death_place = Palace of Vers ...
and the Grand Duke Ferdinand II.


Mathematical style

Huygens's mathematics in the ''Horologium Oscillatorium'' and elsewhere is best characterized as geometrical analysis of curves and of motions. It closely resembled classical Greek geometry in style, as Huygens preferred the works of classical authors, above all
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
. He was also proficient in the
analytical geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineer ...
of Descartes and
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...
, and made use of it particularly in Parts III and IV of his book. With these tools, Huygens was quite capable of finding solutions to hard problems that today are solved using analytical methods, such as proving a uniqueness theorem for a class of
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, or extending approximation and inequalities techniques to the case of second order differentials.Bos, H. J. M. (1980). Huygens and mathematics. In H.J.M. Bos, M.J.S. Rudwick, H.A.M. Snelders, & R.P.W. Visser (Eds.), ''Studies on Christiaan Huygens'' (pp. 126-146). Swets & Zeitlinger B.V. Huygens's manner of presentation (i.e., clearly stated axioms, followed by propositions) also made an impression among contemporary mathematicians, including
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
, who studied the propositions on centrifugal force very closely and later acknowledged the influence of ''Horologium Oscillatorium'' on his own major work. Nonetheless, the Archimedean and geometrical style of Huygens's mathematics soon fell into disuse with the advent of the
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, making it more difficult for subsequent generations to appreciate his work.


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. Throughout this work Huygens showed not only his mastery of geometry and physics but also of
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
. His analysis of the
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
in Parts II and III would later lead to the studies of many other such curves, including the
caustic Caustic most commonly refers to: * Causticity, a property of various corrosive substances ** Sodium hydroxide, sometimes called ''caustic soda'' ** Potassium hydroxide, sometimes called ''caustic potash'' ** Calcium oxide, sometimes called ''caus ...
, the
brachistochrone In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...
, the sail curve, and the
catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superficia ...
. 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 Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
that would be carry on in the following centuries, albeit in the language of the calculus.


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 Leiden University (abbreviated as ''LEI''; nl, Universiteit Leiden) is a public research university in Leiden, Netherlands. The university was founded as a Protestant university in 1575 by William, Prince of Orange, as a reward to the city of Le ...
, now in the ''Codices Hugeniorum''. Much of the background material is in ''Oeuvres Complètes'', vols. 17-18. 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. 4+ 161 + page

* Later edition by W.J. ’s Gravesande. In ''Christiani Hugenii Zulichemii Opera varia'', 4 vols. Latin. Leiden: J. vander Aa, 1724, 15–192. epr. 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 Ostwalds Klassiker der exakten Wissenschaften (English: Ostwald's classics of the exact sciences) is a German book series that contains important original works from all areas of natural sciences. It was founded in 1889 by the physical chemist Wi ...
, no. 192). * Italian translation. ''L’orologio a pendolo'' (trans. C. Pighetti), Florence: Barbèra, 1963. lso_includes_an_Italian_translation_of_''Traité_de_la_Lumière''.html" ;"title="Traité_de_la_Lumière.html" ;"title="lso includes an Italian translation of ''Traité de la Lumière">lso includes an Italian translation of ''Traité de la Lumière''">Traité_de_la_Lumière.html" ;"title="lso includes an Italian translation of ''Traité de la Lumière">lso includes an Italian translation of ''Traité de la Lumière'' * 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. * Dutch translation. ''Christiaan Huygens: Het Slingeruurwerk, een studie'' (transl. J. Aarts), Utrecht: Epsilon Uitgaven, 2015.


References

{{Christiaan Huygens 1673 books Physics books Mathematics books 1670s in science Mathematics literature Historical physics publications 17th-century Dutch books Books by Christiaan Huygens