HUYGENS, Christiaan.

Horologium oscillatorium, sive de motu pendulorum ad horologia aptato demonstrationes geometricae.

Paris: François Muguet, 1673.

First edition, a very fine copy in contemporary calf, of one of the greatest books in the history of science — the birth of modern dynamics. The Horologium oscillatorium, sive de motu pendulorum ad horologia aptato demonstrationes geometricae is, in Joella Yoder’s phrase, a superb tapestry woven from the three strands of Huygens’s science: mathematics, mechanics, and technology. Printing and the Mind of Man judges it the most original work of this kind since Galileo’s Discorsi. Charles Singer describes it more strongly still, as a work of the highest genius which has influenced every science through its mastery of the principles of dynamics, and as second in scientific importance perhaps only to the Principia, which is in some respects based on it. The book is Huygens’s definitive treatise on pendulum motion and is simultaneously the first rigorous mathematical analysis of pendulum dynamics, the foundational study of evolutes, the earliest accurate determination of the constant of gravitational acceleration, and the source of the thirteen propositions on circular motion from which Newton would draw the centrifugal-force arguments of the Principia. In the history of mathematical science, Printing and the Mind of Man observes, Huygens stands next to Newton, drawing — as Newton did too — upon Galileo and Descartes.

Galileo, unknown to Huygens, had proposed the construction of a pendulum clock and believed that the pendulum was isochronous — that its time of swing is the same no matter how large the arc. Huygens knew that this is not exactly so: a pendulum swinging through an arc of more than a degree or two takes slightly longer than one swinging a shorter arc. He discovered that if curved strips of metal were placed at the top of the suspension, the pendulum would ride up against these cheeks on wider swings, effectively shortening itself and slightly reducing the time of its swing; and he showed that if the cheeks were shaped like a cycloid — the curve traced by a point on a circle as the circle rolls along a flat plane — the resulting motion was truly isochronous. He was the first to have such a clock built, and he thereby revolutionised the practice of timekeeping. The pendulum clock is described in the first of the five parts of the Horologium oscillatorium.

Important as Huygens’s clock was — both as a practical instrument and as a tool of astronomy — the book is a general work on dynamics, and especially a mathematical analysis of pendulum motion. In it Huygens treated many problems of the dynamics of bodies in motion: he determined the tautochronous character of the cycloid — a curve that a generation later would also prove to be the brachistochrone, the path of quickest descent — and applied it to invent an isochronous pendulum clock; he added to the theory of curves the theory of evolutes; he extended the analysis of fall to motion along curves; he determined the first value of the force of gravity by using a compound pendulum, a pendulum whose weight is distributed along its length rather than concentrated at the end; and he closed the book with thirteen theorems on the theory of centrifugal force in circular motion — theorems that aided Newton in the determination of universal gravitation (Dibner). Newton in the Scholium to Proposition IV, Book I, Section II of the Principia acknowledged that Huygens’s propositions on centrifugal force were the first step towards resolving the question of the force of gravity.

The exchange of copies between the two men is among the most consequential in the history of science. Huygens sent copies of the Horologium to London through Henry Oldenburg in the summer of 1673, one of them destined for Newton, and Newton acknowledged the gift in a letter to Oldenburg of 23 June 1673, Old Style — 3 July by the Continental calendar — finding the book “full of very subtile and usefull speculations very worthy of ye Author”. Newton reciprocated in 1687 by sending Huygens a copy of the Principia. Huygens, while acknowledging the importance of Newton’s work, could not accept the theory of universal gravitation, which was irreconcilable with his own adherence to a strictly mechanistic philosophy of the laws of nature. The copy Huygens sent to Newton was recorded by Printing and the Mind of Man in the Barchas collection at Tucson, Arizona; with the rest of that collection it passed in 1982 to Stanford University Libraries.

The construction of a reliable timekeeping device was a matter of acute practical urgency in seventeenth-century Europe, not least because of the problem of longitude. Galileo had believed the pendulum to be isochronous and therefore a perfect regulator for a clock, but it was Huygens who discovered how to couple a pendulum to a clockwork mechanism. He dated the invention to 25 December 1656 — meaning that he had a workable clock on that date — and the patent, granted on 16 June 1657, named the Hague clockmaker Salomon Coster, who built the clocks to Huygens’s design. In 1658 Huygens published the Horologium, a brief fifteen-page treatise illustrated with a woodcut of the mechanism: the swing of the pendulum rocks a small L-shaped wire back and forth, which in turn rotates a pair of tabs on the verge, which allows the crown-shaped wheel to advance in steps; the same L-shaped wire also powers the pendulum, giving it a small boost on every swing, driven by a slowly falling weight. At this stage Huygens had reduced the swing of the pendulum by inserting additional gearing — a crown wheel and pinion — that allowed the pendulum to swing with so small an arc that any variation had a negligible effect on the time. But Huygens had already recognised what Galileo had not: that the isochronism of the pendulum is only approximately true, and only when the width of the swing is small.

In 1659 Huygens solved the problem exactly. The discovery — “the most fortunate finding which ever befell me”, as the Dictionary of Scientific Biography preserves his own words — rested on a detailed mathematical analysis of circular motion and of dynamics more widely, and it gave him the cycloidal cheeks that would appear in the Horologium oscillatorium. In 1663 he was elected a Fellow of the Royal Society, and in 1666 his achievements in mechanics and mathematics earned him an invitation to join the newly founded Académie Royale des Sciences in Paris. He welcomed the move; his many Parisian correspondents, among them Jean Chapelain, urged him continually to publish a second edition of the 1658 Horologium. In August 1660 Chapelain reported that the savants of Paris were awaiting the new book, to which Huygens replied that his treatise on the clock had been finished for a while, but that there had not been an opportunity to have it printed before his trip. Chapelain reiterated the plea in December 1661, congratulating Huygens in advance on the improvement that the invention would have received from his fortunate attention. The publication was delayed for years, not least because Huygens waited for better results from sea trials of his marine clock. The printing finally began in the autumn of 1672: Huygens had the first sheet in hand around the permis d’imprimer of 30 September, the royal privilege was dated 4 November, and Chapelain approved the text of the dedication on 4 February 1673. The work was completed at the press of François Muguet — printer to the king and to the archbishop of Paris, at the sign of the Three Kings in the rue de la Harpe — in the early spring of 1673.

The book was dedicated to Louis XIV, an unavoidable but politically fraught choice. In 1672 Louis had invaded the Low Countries, and the fate of the Dutch state — whose States of Holland and West Friesland had received the dedication of the 1658 Horologium — was seriously threatened by the French offensive. That Huygens could dedicate his work to the Great King put him in a precarious situation, all the more because his family had close ties to the Stadtholder William III. But he was tethered to the patronage of the French academy, and his longstanding connections to the intellectual and social communities of Paris prevailed over any sense of loyalty to his homeland; they required him to present the book to the French king (Howard).

The Horologium oscillatorium is organised in five closely related parts. The first describes the new clock and how to build it. Many features of the earlier design are carried over — the small weight by which the timing of the swing is adjusted, the verge-and-crown-wheel escapement by which the pendulum is coupled to the gears, the endless cord or chain by which the clock is wound without stopping it. The radical innovation is that the pendulum no longer swings freely but is limited in its motion by thin metal plates mounted on either side of the pivot point. These plates are bent to a cycloidal shape in the vertical plane; Huygens describes how to construct a cycloid by rolling a circular object along a straight edge. At the end of Part 1 he describes a variant of the clock for use at sea, in which the regulating apparatus is stretched out in the horizontal, non-cycloidal plane to compensate for the fore-and-aft tilting of the ship; the pendulum here is a triangular contrivance, a cord fastened at either end to a pair of cycloidal plates, with the bob hung at its midpoint. Part 1 also contains a table for the Equation of Time, by which the clock, which beats at a constant rate and measures a mean solar day, is adjusted for the inequality of the true solar day — an adjustment essential if the clocks are to be used for astronomical observation, on land or at sea.

The primary purpose of Parts 2 and 3 is to prove that the cycloidal pendulum is isochronous: that the bob completes its swing at a uniform rate independent of the magnitude of the swing, so that the clock keeps exact time irrespective of anomalies in the pendulum’s oscillation. Galileo’s claim that a freely swinging pendulum is isochronous is, despite the enduring myth, wrong — though if the arcs are very small, as in a grandfather clock or in Huygens’s 1658 mechanism, the variation is negligible. Part 2 begins with the hypothesis of rectilinear inertial motion and introduces the compound motion owing to gravity. The object in motion is an idealised point mass; gravity is assumed and not explained. In this section Huygens takes up Galileo’s analysis of free fall and of fall along inclined planes and extends it to fall along curves, where the curve is approximated at each point by its tangent plane. These propositions are completely general, but they are background for the main theorem: given an inverted cycloid erected in the vertical plane with its axis perpendicular and its vertex at the bottom, no matter where along the curve a body is released it will reach the lowest point in a fixed amount of time. Fall along an inverted cycloid is isochronous.

Part 3 goes further into the mathematics embodied in the clock. Huygens introduces the concept of an evolute, a curve that is unrolled (evolutus in Latin) to create a second curve, which he calls ‘that described by the unrolling’ and which later mathematicians would label the evolvent or involute. He never discusses the technological application behind this theory in Part 3, but the pendulum of his clock plainly served as his model: if a cord is unrolled from one of the curved plates, the bob traces out the involute. Given the results of Part 2, he wanted the curve swept out by the bob to be a cycloid. When he first discovered the isochronism of the cycloid in 1659 he had determined its evolute by infinitesimal techniques — taking two points very close together on the cycloid, finding the intersection of the perpendiculars through them, and deducing the curve defined by all such points. That evolute would be the shape into which the curved plates should be bent. The extraordinary outcome was that the evolute of a cycloid is another cycloid — a self-dual relationship of great elegance. For the published derivation Huygens banished infinitesimals and proved the result by a sequence of propositions that defined the reciprocal relationship between the tangent of the evolute and the perpendicular of the involute, in the classical manner.

In Part 4 Huygens introduces physical parameters into his analysis and addresses the compound pendulum. He defines a pendulum as any figure that can continue reciprocal motion about a point or axis by means of its own weight; a simple pendulum as one with a weightless cord and a point-mass bob; and a compound pendulum as any suspended object with weight distributed throughout. Two pendulums are defined as isochronous when they swing through equal arcs in equal times, and the centre of oscillation of a suspended object is defined as the point along its axis at which to situate the bob of a simple pendulum isochronous with the compound one. The fundamental hypothesis of this section is the principle, usually called after Torricelli, that a system of weights cannot rise of its own accord above its centre of gravity. Applying this hypothesis, Huygens derives a simple pendulum isochronous to a given compound pendulum — first for a set of weights distributed along a pendulum’s cord, then for a general plane figure, and finally, as a tour de force, for a whole set of plane and solid figures. Commentators have observed that he did not break free of gravity to treat the generalised system of masses, and in particular the rotating body; but his achievement was nonetheless epoch-making. At the close of Part 4 he draws together everything that has come before and applies it to the clock of Part 1: Parts 2 and 3 had justified the cycloidal plates by demonstrating that the bob of a pendulum banking against them traces a cycloid, so the clock described in Part 1 is theoretically isochronous; Part 4 provides the means of accounting for the mass of the cord and bob and of adjusting the rate of swing of the pendulum by means of a small weight that can be moved up and down the cord, so the clock can be made as physically exact as possible.

In the final proposition of Part 4 Huygens returns to the problem that had instigated the entire investigation: to find the distance traversed in a given time by a body falling perpendicularly under gravity. If the time is one second, the distance fallen from rest is numerically one-half the constant of gravitational acceleration. By the last proposition of Part 2, the time of free fall can be related to the length of a cycloidal pendulum that swings one arc in a second; inverting the concluding proportion of that proposition yields the first accurate value of the constant of gravity — Riccioli’s rougher measurement of falling bodies in 1651 being its only predecessor — expressed, in Huygens’s geometric manner, as a proportion between the time of fall through half the pendulum length, half a second, and the ratio of the diameter of a circle to its semi-circumference.

Part 5 introduces a second clock, one based on a three-dimensional mathematical model that parallels the design elements of the cycloidal clock. The earlier clock had of course existed in three dimensions physically, but Huygens had treated it as a two-dimensional system: the pendulum was assumed to swing in a plane, and the curved plates along which it banked were shaped only for that plane. The progenitor of the second clock had a freely swinging pendulum whose bob rotated in a circle, so that the cord traced out a cone. As with the cycloidal clock, Huygens modified this design so that a curved plate constrained it to move isochronously. In one of his 1659 discoveries he had shown that a ball circulating inside a paraboloidal chalice — the surface generated by rotating a parabola about its axis — completed any horizontal circle in the same time, regardless of how high or low the ball lay in the chalice; the paraboloid was isochronous. Using the theory of evolutes, he could transfer this result to the rotating pendulum and deduce the proper shape for a curved plate that would hold the bob onto the imaginary surface of a paraboloid. By Part 3, Proposition 8, the evolute of a parabola is a semi-cubical parabola, so a pendulum mounted to a plate shaped like a semi-cubical parabola and rotated about an axis would sweep out a paraboloid isochronously. Like its cycloidal brother, the paraboloidal clock would theoretically keep perfect time.

With the description of the paraboloidal pendulum clock, the Horologium oscillatorium ends. Huygens did not continue with a detailed mathematical study to accompany the second clock that would parallel his analysis of the first. Instead he appended a list of thirteen theorems on motion in a circle — the theorems that had guided his creation of the cycloidal clock in 1659 — but withheld their proofs. His stated intent was to save the results for a larger work that would present his definitive explanation of circular motion and centrifugal force; but that work never materialised, and the original proofs appeared only in the 1703 posthumous Opuscula postuma under the title De vi centrifuga. The thirteen theorems, even unproven, provided Newton with the starting point for his own treatment of circular motion in the Principia, and it is the priority that Newton acknowledged at Proposition IV of Book I.

Parts 2 and 3 summarise many of the achievements of seventeenth-century mathematics, and particularly Huygens’s own contributions to them. They are written in the classical style of Archimedean geometry, with every step strictly substantiated by an appropriate proposition and with nothing displayed in the modern form of equations — everything is expressed through the classical language of proportions. In a quintessential tribute to Archimedes, Huygens pauses in the middle of Part 3 to reduce the areas of surfaces of conoids to the areas of circles; the results parallel the propositions of his early Theoremata de quadratura of 1651, in which he had related the areas of conics to their centres of gravity. Despite this classicism, Huygens was at pains to place his discoveries in the context of the contemporary priority debates that accompanied the seventeenth-century advances in quadrature and rectification — the determination of the area of a curved figure and of the length of a curve — all material that set the stage for the growth of the calculus. In Part 2 he presented his method for finding the tangent to the cycloid, with a review of earlier methods; in Part 3 he asserted his place at the forefront of discoveries regarding the rectification of important curves, including the cycloid. He developed, in fact, a general method of rectification based on evolutes: as the cord unwound from the evolute it literally straightened, or rectified, the curve. The method had a major flaw in that it rectified the evolute of the given curve and not the curve itself, but it led him to derive the evolutes of the ellipse, hyperbola, and higher-order conics and to rectify those evolutes. More significantly, he reduced the rectification of the parabola to the quadrature of the hyperbola, which he solved by numerical approximation using logarithms. In Part 4, ever seeking his due, he sketched an abbreviated history of the problem of the centre of oscillation, particularly referring to his youthful correspondence with Marin Mersenne.

Fine copies with distinguished histories have long been prized. Huygens gave copies away carefully: besides the Newton copy now at Stanford, a presentation copy inscribed “Donné par l’autheur” survives with a seventeenth-century provenance, and the mathematician Jacques Ozanam’s copy, given by Huygens with Ozanam’s note “Ex dono clarissimi Authoris” on the title, passed through the Norman Library. Other early owners ranged from Esprit Fléchier, bishop of Nîmes, whose arms were stamped on his copy, to the Capuchins of Meudon, who received theirs from the Capuchin Fiacre de Paris; a copy figured in the Honeyman dispersal. Institutional copies are widespread, but copies in unrestored contemporary bindings appear only seldom on the market, and public collections have competed for them — National Museums Liverpool acquired its copy as recently as 1988 with charitable-grant support. The present copy, complete as issued in its text-block and in a contemporary calf binding that has needed only the most minor attention, stands close to the state in which it left a seventeenth-century Paris bindery.

Huygens met Newton twice, during his third visit to England in 1689, and exchanged important letters with him. A magnificent edition of his letters, writings, and notebooks appeared in twenty-two volumes at The Hague between 1888 and 1950. The Horologium oscillatorium remains the greatest of his published works and perhaps the most consequential single mathematical book of the period between Descartes and Newton. Its arguments entered the bloodstream of the mathematical sciences; its evolutes founded a new branch of geometry; and its thirteen unproven theorems supplied the conceptual seed from which the theory of universal gravitation would, in other hands, be grown.

References: Dibner, Heralds of Science, 145 — Grolier/Horblit 53 — Norman 1137 — PMM 154 — Evans 1934, 31 — J. G. Yoder, ‘Christiaan Huygens, book on the pendulum clock (1673)’, ch. 3 in Grattan-Guinness (ed.), Landmark Writings in Western Mathematics 1640–1940 (Elsevier, 2005), pp. 33–45 — J. G. Yoder, Unrolling Time: Christiaan Huygens and the Mathematization of Nature (Cambridge, 1988) — J. G. Yoder, A Catalogue of the Manuscripts of Christiaan Huygens (Brill, 2013) — R. J. Blackwell (trans.), Christiaan Huygens’ The Pendulum Clock (Iowa State University Press, 1986), with introduction by H. J. M. Bos — H. J. M. Bos, ‘Huygens, Christiaan’, Dictionary of Scientific Biography VI, pp. 597–613 — N. Howard, ‘Marketing Longitude: Clocks, Kings, Courtiers, and Christiaan Huygens’, Book History 11 (2008), pp. 59–88 — Huygens, Oeuvres complètes, 22 vols (The Hague, 1888–1950), vol. XVIII — The Correspondence of Isaac Newton, vol. I, pp. 292–294 — Baillie, Watchmakers and Clockmakers of the World, 1966 — Singer, A Short History of Science to the Nineteenth Century (Oxford, 1941), p. 357.

Folio (292 × 201 mm), pp. [xiv], 161, [1, errata], complete as issued (the second preliminary quire of three leaves, the fourth cancelled in all normal copies); ornamental royal armorial woodcut device of Louis XIV on title, full-page woodcut of the pendulum clock on p. 4, approximately 100 woodcut diagrams in text, woodcut head- and tail-pieces and initials. Contemporary mottled calf, spine with raised bands richly gilt in compartments with palmette centre-tools, red morocco lettering-piece, board-edges with gilt roll, edges sprinkled red, marbled endpapers (very minor restoration, corners lightly worn). Internally clean and crisp with only light occasional spotting at the margins.

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Item #6498

Price: $95,000.00