1. Archimedes' Clock

From the very beginning of Christiaan Huygens' career as a mathematician and natural philosopher his father referred to him as "my Archimedes", and friends and admirers soon followed suit. But not blindly: the Republic of Letters in the late seventeenth century chose its classical images carefully. The name of Archimedes conjured up not only the genius that established the mathematical principles of stability in the physical world but also the ingenuity that transformed those principles into machines that put nature to work. Archimedes was an engineer as well as a mathematician. It did not matter that, as Plutarch reported, the real Archimedes had disdained to commit his mechanical inventions to writing.² To have recognized the equal importance of works and words was the advantage the moderns enjoyed over the ancients. Yet, elevating mechanics to the rank of learning did not make any less rare those with the special talent (as Pierre Petit put it), "pas permis à tout le monde, de bien penser et de bien faire."³ Christiaan Huygens obviously had that talent, and contemporaries looked to it to yield fruits comparable to those found in Archimedes' writings and described in reports about him.

It would surely deepen our understanding of Huygens to know precisely what the cognomen meant to him. The repeated use of ευρηκα in his notes to signal the moment or occasion of discovery suggests that he took the comparison as more than a doting father's hyperbole. But to know how it influenced his choice of problems, his methods of investigation, and his style of presentation would require a study as yet undone. To do justice to the question, the study would have to explore also the differences that separate Huygens from Archimedes. These, I suspect, would turn out to have greater historical significance than have the similarities. In retrospect from today Huygens looks less like the reincarnation of Archimedes than like the prototype of Coulomb, Carnot, and the practitioners of the Ecole polytechnique's characteristic mélange de mathématique et physique. No facet of his career shows this more clearly than his research into the measurement of time and of longitude at sea. It established a model for a new sort of mechanics, a truly mathematical physics rooted firmly in the physical world.

From his first pendulum clock in 1657 to his last sketches for a "perfect marine balance", Huygens' efforts to capture the uniform flow of time in a reliable mechanism, and especially in a device immune to disturbance from its (possibly moving) surroundings, ranged from the most general theory to the most intricate practice. His explorations took him into uncharted regions of abstract mechanics, where he applied the newest mathematics of his day to the analysis of the constrained motion of points along curves and of points linked in rigid configurations; at one point, he had to create a new branch of mathematics, the theory of evolutes. But those explorations began with the actual physical mechanisms that were his successive clocks, and his discoveries took concrete form in practical improvements of the clocks. The theoretical research and the practical application of its results had one purpose: to increase the clocks's accuracy. Each step forward defined in precise quantitative terms the limits within which the next step would have to fall. Taking turns, theory and practice narrowed the tolerance between the abstract and the physical to fractions of seconds and inches. If predecessors such as Galileo, first emulating Archimedes and then going beyond him, had already shown how to reduce bodies in motion to geometrical configurations, none had yet insisted that the reduction satisfy such exacting standards of measured fit between the real and the ideal. In demanding that precision, Huygens did not follow the past; he led the future.

Huygens' work on clocks and on the method of longitudes has another facet that looks toward the future rather than reflecting the past. Although as mathematician Huygens aspired in the best classical tradition to fame, as mechanic he also jealously guarded the fortune that might result from his inventions.⁴ In the spirit of Archimedes he insisted that his magnum opus, the Horologium oscillatorium, wait for publication until he could present its theoretical content in the finished form of a mathematical treatise. In quite another spirit, he postponed publication when it seemed that his first marine clock would sustain sea trials and form the instrumental core of a practical (and profitable) method of longitudes. A written description of the clock might interfere with the exclusive license Huygens had awarded to a Hague clockmaker, and the successful outcome of even more rigorous trials might heighten the clock's value. For two years in the mid-1660s, while colleagues eagerly awaited the long-promised treatise, Huygens worked instead to secure patents and privilèges from the French, English, and Dutch governments, not to mention the reward offered by the Spanish throne for a reliable method of longitudes.⁵ Although by no means unprovoked, Huygens' disputes with the clockmakers Douw and Thuret over patent rights display an acrimony sharper than any of his contests over issues of mere theoretical consequence.⁶ The new mélange de mathématique et physique that brought the mechanic into the academy also lured the mathematician into the marketplace.

Huygens' life-long engagement with clocks passed through several distinguishable stages. The first begins in 1657 with his invention of a means of combining the pendulum and the mechanical clock into a single long-running and reliable device and extends through 1661, by which time he had discovered the tautochrony of the cycloidal pendulum, the cycloidal cheeks that make a pendulum cycloidal, the conical pendulum, the center of oscillation of a compound pendulum, and the sliding weight to vary the pendulum's period. The second stage covers the decade from 1662 to 1672 during which Huygens worked out, alone and in collaboration with others, means of suspending the clock from a shipboard mounting and the details of the measurement of longitude by means of clocks. The period ends with the final preparation of the Horologium oscillatorium, which Huygens felt forced to bring out after sea trials had proved only partly successful. The third stage stretches from his invention of a spring balance in 1674 to his final sketches of a marine clock in the years just preceding his death. Huygens' research during this period consisted largely of studies of physical mechanisms embodying his principle of incitation parfaite décroissante, laid down in 1675 to explain the tautochrony of the spring. But Huygens' investigations also reflect a troubling awareness that his earlier successes were illusory and that subsequent modifications were proving ineffectual. Sea trials on Dutch vessels in 1686-87 and 1690-92 did not live up to expectations. The fame Huygens so avidly sought as the inventor of the first reliable clock for determining longitude at sea eluded him.⁷ It would continue to elude horologers for almost fifty years more, to be captured ultimately by John Harrison in 1762. But if Harrison found the answer in metallurgy, he had Huygens to guide him through the mechanics.

2. The Horologium of 1658: Harnessing Driving Force and Regulating Motion

Commenting on the many rival claims to have invented the pendulum clock, Huygens wrote to Pierre Carcavi in 1660:

"C'est une chose étrange, que personne devant moi n'ait parlé de ces horloges, et qu'à cette heure il s'en découvre tant d'autres auteurs."⁸

Huygens may have belittled his achievement because neither of its main components was itself new. Mechanical clocks, driven by the force of a falling weight or of an uncoiling spring, had existed for centuries; in both the large format of elaborate orreries and the small format of intricate watches, they have reached a high level of sophistication.¹⁰ The pendulum was of more recent origin, having been brought to wide attention by Galileo. Astronomers had quickly adopted it for their observations; as Huygens described their procedure, "they impelled by hand a weight hung from a light chain and, counting its single vibrations, they measured as many equal moments of time."¹¹ The device had already produced improved measurements of eclipses and of celestial dimensions.

But each instrument had inherent flaws that counteracted its basic function. The clock lacked a regulator that by its intrinsic period of reciprocating motion could both advance the escapement in uniform steps and overcome the perturbing effect of variations in the force transmitted from the drive owing to imprecisely cut gears and uneven friction. The pendulum ran down unless repeatedly sustained by impulses from the operator's hand. Moreover, its use by astornomers meant tedious hours of counting its swings, with the attendant risk of error.¹² The pendulum required a motor and a counter, the clock a regulator. Huygens devised a means of combining them so that each supplied the other's need without otherwise interfering with it.

From Huygens' first pendulum clock, through all its subsequent versions, "the principle of its motion, and indeed of the whole invention," remained the same.¹³ The pendulum hung from a flexible cord clamped in a vise mounted on the clock's frame. The pendulum's motion was transmitted to the escapement via a small crank, or crutch, through one end of which the pendulum's rod passed freely yet snugly; the other end was joined to the verge either directly or through gearing. The crutch in turn transmitted from the escapement to the pendulum the small force necessary to sustain the latter's swing.

The independent suspension and the crutch linked the pendulum and the mechanical clock while still keeping them essentially separate and hence separately adjustable. For example, increasing the weight of the bob to resist small variations in the force delivered by the crutch, or to decrease the retarding effect of the air's resistance, did not add to the friction on the verge and hence to the resistance offered by the palettes to the escapement wheel.¹⁴ Similarly, changes in the driving force could be accommodated in the escapement and the crutch before reaching the pendulum. On the one hand, then, the small perturbations to which the pendulum in itself was subject were not overridden by grosser variations transmitted to it from the clock's works. On the other hand, the crutch transmitted those perturbations to the clock, which was thereby made sensitive to them.¹⁵ Without that arrangment of one-way sensitivity, precise analysis of small changes in the pendulum's motion served little practical purpose. With it, such theoretical analysis promised practical means of calibration. Thus Huygens' innovation made the clock an interface for theory and practice in mathematical physics.

3. Toward the Horologium oscillatorium

3.1 The Tautochronic Pendulum

In the late fall of 1659 two problems, one practical and one theoretical, met on that interface. The first stemmed from the clock: how can one control the wide excursions of the pendulum so as to make all its swings take the same time? Huygens knew from Mersenne and others that, despite Galileo's claims, the period of a pendulum varies with its amplitude. Only for swings in the immediate neighborhood of the centerline is the variation negligible and the pendulum effectively tautochronic.¹⁶ The second problem arose out of current research into the nature of accelerated free fall: what is the distance through which a body falls from rest in one second?¹⁷ The new clock seemed to offer a contribution to the latter questions by providing an independent means of calibrating a one-second pendulum, but the practical need for wide swings during experiments with falling objects overrode that help. What was needed was a solution to the problem Huygens set himself sometime in November: he sought "the ratio of the time of a very small oscillation ... to the time of perpendicular fall from the height of the pendulum."¹⁸ That resort to theoretical analysis yielded unexpected practical results.

Figure 1: Huygens' original diagram; HOC.XVI.392

Because of the dependence of period on amplitude, Huygens expected to have to make an approximation in his analysis of pendular motion. It turned out to lie deep in the mathematics of the problem. He began (Figure 1) with a bob swinging from rest at K over a small arc KZ and sought an expression for the time of motion over any infinitesimal segment E of that arc. Assuming that the bob moves uniformly over the segment at the speed acquired by fall along the curve from K to E --which is the same as the speed acquired in free fall through the vertical distance AB-- and taking as a base of comparison the bob's uniform motion over an infinitesimal segment B of the vertical AZ at the speed acquired in free fall through the whole interval AZ, he used Galileo's law of uniform motion to determine that By the tangent properties of the circle, which by constructing BG = TE one may express as Then, on the figure of the pendulum Huygens superimposed a parabola representing the relation of the successive distances AB to the speeds acquired by a body falling freely over those distances from rest at A; he chose the units of measure so as to set ZΣ --i.e. the speed of the body at Z-- equal to AK. That permitted him to express the ratio of the speeds over B and E as where The resulting expression, for the ratio of the times reduced the physical situation to a mathematical configuration to which Huygens then applied the analytical tools at his disposal, among them recent results by Fermat and others on the transformation and quadrature of curves. For, by constructing a curve of ordinates BX such that he had

where "all BX over AZ" is the sum of all ordinates BX standing on the base AZ, i.e. the area under the curve of ordinates BX.¹⁹ Determining the period of the pendulum meant finding that area.

It was that mathematical problem that required an approximation, thus reflecting the physical situation. Huygens could not find the area under the curve as given. Its ordinate is inversely proportional to the product of the ordinate of two base curves, a circle and a parabola, and it appeared to have no properties that would permit its reduction to simpler terms. But he could handle a curve which in the immediate neighborhood of Z --i.e. for small AK-- lay quite close to the original one. That new curve replaced the circular arc ZEK by a parabolic arc congruent to arc ADΣ.²⁰ For E lying on the parabola rather than the circle, the product BD.BE reduced to a constant multiple of the ordinate BI of a semicircle AIZ on AZ as diameter, and the quadrature of the adjusted curve of ordinates BX reduced to the rectification of the semi-perimeter AIZ; that is, ²¹ To relate the time over KZ to that over TZ required a turn back to the physical situation: by Galileo's laws of fall, and the final ratio reduces to .

The approximation of the period of minimal oscillation assumed, then, for the purposes of mathematical analysis that the relation between the normal and the ordinate of the bob's circular path could be represented by the relation between that normal and the ordinate to a parabola. As Huygens then turned to finding the physical situation for which the result just derived would be an exact solution, he looked for a path of descent for which the parabola would exactly represent the relation between the normal and the ordinate. He had to retain the parabola for both mathematical and physical reasons: by mirroring the parabola ADΣ it made the quadrature possible, and the parabola it mirrored was fixed by the kinematics of free fall. "From this," he recorded in the same worknote of 1 December 1659,

I saw that, if we want a curve on which the times of descent are equal over any arcs terminated at Z, it must be of such a nature that, if one sets the ratio of the curve's normal ET to the ordinate EB equal to that of a given line GB to another EB, the [second] point E falls on a parabola with vertex Z. This I found to agree with the cycloid, by a know method of drawing the tangent.

Figure 2

Suppose, that is, that ZKQ (see Figure 2) is the cycloid generated by circle MLZ, E any point on the cycloid, BE the ordinate to that point. The "known method of drawing the tangent" consists of drawing a line through E parallel to chord ZL of the circle. Hence, the normal ET is parallel to chord ML and By the properties of chords in circles, , whence . Now, through any point H on MQ draw HN parallel to MZ and intersecting BE at G, and construct point F on BG such that . Then , which holds if and only if F and H lie on a parabola with vertex at Z. If, then, in the derivation of the pendulum period one replaces circle ZEK by cycloid ZEKQ and constructs parabola ZFJH congruent to the parabola of free fall ADΣ, it follows mutatis mutandis that the time over arc KZ of the cycloid is the same for any starting point K; it stands to the time of free fall over diameter MZ as the semiperimeter of a circle to its diameter, i.e. as . Hence the time over any arc is to the time of fall through 2MZ as , and the approximately constant period of a narrowly oscillating simple pendulum of a given length becomes exactly constant when the bob is constrained to follow an inverted cycloid, the diameter of whose generating circle is half the given length.

Huygens did not include among the lemmas in his worknote the above derivation linking the parabola and cycloid. The nonchalance with which he adduced the relation was sincere. His recognition of the cycloid behind the conditions of the problem sprang from familiarity, no genius. Quite independently of his work on the clock, he had been following closely the controversy over the cycloid unleashed by Pascal in 1658.²² The results exchanged had honed his knowledge of the curve's properties to the extent that he could work backward from them. Indeed, his intimate knowledge of the curve enabled him soon to abandon the hybrid configuration of spatial path and kinematic diagram and to construct a "demonstration rendered better than heretofore" that used Galileo's results to embed the kinematics in ratios among chords of the cycloid's generating circle.²³ That better demonstration became the skeleton of the formal exposition published in the Horologium oscillatorium.

Huygens' knowledge of the cycloid perhaps also explains his quick recognition of another property of the curve pertinent to his needs. Beginning with the first clock, he had tried to control unequal excursions of the pendulum by enclosing the suspension cord between two strips of metal, or cheeks, thus pulling the bob back from the circle of its unrestricted swing. The shape of the cheeks had been a matter of cut-and-try, and Huygens reported having achieved some success.²⁴ But he could not precisely define the experimentally determined shape and in the Horologium of 1658 replaced the cheeks by gearing to keep the oscillations nattow, even at the expense of increasing the clock's sensitivity to disturbance.²⁵ Nonetheless, the cheeks remained on his mind, especially as he pondered the implications of motion along a cycloid.

25. Cf. Horologium (1658), HOC.XVII.69.

The transition from circle to cycloid in the derivation of the period of a pendulum had replaced the fixed length and center of the former by a normal that varied in both length and intersection with the cycloid's vertical axis (which was also the centerline of the pendulum). A cycloidal pendulum seemed at first to have no fixed length to serve as its parameter and to satisfy Galileo's law relating the period of a pendulum to the square root of its length.²⁶ Yet, precisely because Huygens constructed his cycloid to fit the parabola of free-fall velocities to which the circular arc had been an approximation, the new pendulum and the old shared within the close neighborhood of the centerline a common period measured by the fall of a body through a height equal to twice the diameter of the cycloid's generating circle. Hence, about the centerline, the cycloidal pendulum acted as if it were a simple pendulum of given length. For wider amplitudes, its arc lay within that of the simple pendulum, and it swung over a shortened radius about a center other than the point of suspension.

The cheeks themselves suggested to Huygens how to analyze that shortening and displacement. By winding along the surface of a cheek, the flexible cord followed the arc of the cheek's curve and was thus shortened by the difference between that arc and its subtending chord. As the cord wound around the cheek, the centrifugal force of the bob kept the free portion of the pendulum taut and thus tangent to the leaf at the cord's last point of contact, which momentarily acted as the center about which that free portion was swinging. For that moment, then, the bob followed a small circular arc coincident with the pendulum's trajectory and hence was normal to the latter. If the cheek extended all the way to intersect with the trajectory, and one began with the pendulum wound about it, then one could view the trajectory as the curve traced by the endpoint of the cord as it is unwound from the cheek while being kept taut and tangent to it. This physically inspired construction permitted exact mathematical definition in terms of tangents, normals, and arc lengths, and the cheeks of the clock thereby became a particular physical instance of a quite general mathematical concept. But the mathematical theory as first set out by Huygens retained a vestige of its inspiration. He called the curve corresponding to the leaf the evoluta or développée, the "unwound [curve]."²⁷

The pendulum clock gave rise, then, to the theory of evolutes which, especially in its application to the rectification of curves, yielded results far in excess of what Huygens had needed for the clock itself. Indeed, he admitted to the reader of the Horologium oscillatorium that most of Part III, "On the Evolution of Curves", was included for its "elegance and novelty" rather than for its application to present needs. Certainly, the determination of the shape of the leaves governing a cycloidal pendulum required very few of the mathematical techniques he eventually devised and published.²⁸ The initial conditions imposed ont he leaves all but dictated their form. First, each had to be tangent to the centerline at the point of suspension. Second, when extended to meet the cycloid at its widest point, each leaf had to be tangent to the baseline of the cycloid and perpendicular to the curve itself. Third, since at that point the leaf would coincide with the full length of the pendulum, it had to have an arc-length equal to twice the diameter of the cycloid's generating circle. But, as Christopher Wren had just shown, that was precisely the length of one half of the cycloid measured from base to vertex. In sum, then, the leaf had to follow a curve having the same base, height, and length as the cycloid to be traced. That it might be the cycloid itself was quickly confirmed in a rough sketch of a demonstration that (see Figure 3):

3.2 The Problem of Longitude

In the original Horologium Huygens had pointed to the development of a portable version of his clock as the key to transforming it from an astronomical instrument to a navigational tool.³² As an astronomical instrument, the stationary, long-pendulum clock making very narrow oscillations provided a uniform measure of time independent of the motions of the stars and planets and hence suitable for determining irregularities in those motions. In particular, since it was accurate to within seconds over a day, the pendulum clock could measure small variations in the length of the solar day as marked by the sun's successive passages through the meridian. When compared with theoretical calculations, the measurements offered astronomers empirical means of resolving longstanding questions about the parameters governing the inequality of the solar day and hence of settling on values of the equation of time to be used with the clock in recording solar events.³³ Huygens emphasized this new contribution to astronomical theory at the same time that he exploited its practical benefits. An accurate table of the equation of time made it possible to use the sun and its shadow, instead of the stars, to regulate the clock. That made regulation easier and more accessible to those using the clock.³⁴ More importantly, however, it completed the theoretical basis of a method of determining longitude by clocks and left open only the practical problem of portability.³⁵

Huygens first began investigating the equation of time as soon as he had built his first clock. He found some modern astronomers like Ismael Boulliau at odds with their predecessors over the structure of the phenomenon, and he used his clock first to decide between them.³⁶ He concluded that Ptolemy and Copernicus had been right: the inequality of the solar day stemmed solely from the eccentricity and obliquity of the ecliptic.³⁷ If the sun's circle coincided with the equator, then its annual motion, combined with the daily motion of the heavens, would produce a constant daily advance eastward of 59'8"20"' in right ascension.³⁸ But the angle of the ecliptic and its eccentricity join to produce a daily advance in longitude that does not correspond to that difference in right ascension. As the dialy differences accumulate, a clock regulated to the mean solar day (I.e. to the constant sidereal day plus 3m 56.5⁺s) and set to the sun at noon on 10 February will fall behind it almost 20m by 14 May. The clock will again catch up to within 9m by 25 July, only to fall behind again by 31⁺m on 1 November; by 10 February it will have caught up completely.³⁹ The choice of starting point is arbitrary unless one wants to compare astronomical observations with tables and parameters set to a standard epoch. For 10 February as base the cumulative inequality remains positive throughout the year, and Huygens chose it for that reason.

Huygens spent two years calculating and confirming his table of cumulative inequality, which by subtraction provided the inequality over any number of days, and sen out the first public copies in February 1662.⁴⁰ To use it in calibrating the clock, one set the clock to the sun on any given day. After the clock had run continuously for several days, one then noted the time of the sun's passage through the meridian. To that time one added the difference of the cumulative equation for the day of the second reading less that for the day of the setting; if the former was less than the latter, one subtracted the difference. The result was supposed to be 12:00; any discrepancy, divided by the number of days the clock had been running, gave the daily advance or retardation. When applied to finding longitude, the table first aided in calibrating the clock and then in correcting its readings for the elapsed inequality before they were compared with local sun time at sea. Hence the table formed as much a part of the navigator's kit as did the clock and the quadrant.

3.3 Adjusting the Pendulum

At about the same time that Huygens made public his table of equation of time, he also announced an addition to his clock that made its regulation by the table even easier and more accurate: the sliding weight. It was the practical payoff of another theoretical breakthrough, the determination of the center of oscillation of a compound pendulum. Neither the simple nor the cycloidal pendulum clock in itself demanded that breakthrough. Each already had provision for moving the bob up and down and thus for adjusting the clock to agree with the stars, with the corrected sun, or with another clock. But in musing on the wider implications of his invention Huygens had imagined that the length of a simple pendulum that beat isochronically with the 1-second pendulum of a precisely calibrated clock could serve as a universal standard of measure: one third of that length would be a pes horarius, a 'clock-foot'. Huygens' colleagues in the Royal Society proved especially receptive to the notion and in 1661 began experiments on both simple and cycloidal pendulums.⁴¹ They soon reported significant divergences between Huygens' theory and their observations. In particular, pendulums of different materials seemed to behave differently.

It is not clear from the evidence when Huygens hit upon the notion of the compound pendulum and the associated concept of the center of oscillation.⁴² By 1661, however, he had gone far enough to explain the English experiments, at least in part. His working notes show that he began with the case of two dimensionless weights swinging together on an inflexible weightless rods, and his solution established the paradigm for analyzing all more complex systems.⁴³ It relied on a principle Huygens had already used with success in deriving the laws of impact: when the center of gravity of a system falls from rest and then rises again to rest, it begins and ends at the same height, irrespective of the changes of relative position within the system itself.⁴⁴

From the situation of bodies falling through certain heights, communicating their motion to other bodies, and thereby driving the latter to certain heights, it follows, Huygens asserts,

that the composite center of gravity of the spheres G and F, after they have taken motion from E and D and have converted it upward as far as they can, that this center (I say) rises to the same height as that of the center of gravity of the spheres B and C.

if given length AD and weight D, and it is required to attach the given weight E to a certain point of the pendulum such that the compound pendulum is isochronous with a pendulum of given length HK, it follows from the preceding equation

In later versions Huygens eliminated the medium of impact and simply imagined the bodies constituting the compound pendulum to be simultaneously released from constraint and to be individually directed upward at the speeds acquired at the point of release. But the basic principle remained the same. At the start of the swing the pendulum's center of gravity, which is fixed by the weights of its constituent parts and by their distribution over it, lies at a certain height. Whether or not the parts remain connected over the length of the swing, they come to rest only when their common center of gravity has again reached that initial height. The difficulties of applying the principle lie in determining the initial and final positions of the center of gravity under the assumption of dissolution during the swing. In the case of a uniform rod of length a swinging about one end, Huygens expressed these two positions by a triangular and a parabolic area, respectively, equating an initial condition of and a final condition of to determine that .⁴⁸

As a corollary to this last derivation, Huygens noted that "whatever weight is added to the rod at that place, it will make the swings no faster or slower." The center of oscillation, he clearly implied, is the point at which the whole weight of the body concentrates to form a simple pendulum, the period of which is independent of its weight. From here he could move toward his goal of regulating pendular motion. Clearly, weight added to the bar below the center of oscillation would slow its swings, and added above the center would speed them up. Consider first a weight Q added to the very bottom (see Figure 5). If a is again taken as both the length and the weight of the rod, and n is to AD as the weight Q is to the weight of the rod, then n simply represents the weight of Q. To the determination of the initial center of gravity, Q adds the term nd; to that of the final center of gravity, the term Hence,, whence a value for x directly follows.

Adding another weight H = h at distance AC = c from A introduces two more terms to the equation: to the initial conditions on the left, to the final conditions on the right. They now yield the equation: . Of greater interest here than the solution of x is that for c:

, where e expresses the weight of the rod directly, rather than by means of a. c is the distance at which a given weight h must be placed in order to render the compound pendulum of weight H, weight Q, and suspension rod AB isochronous with a simple pendulum of known length x.⁴⁹

But if one knows the length of a simple one-second pendulum, one also knows the length of a simple pendulum that deviates from it by any fixed interval, say, 1' over 24 hours. Moreover, from the results obtained above one can determine the distance along a rod at which to place a fixed weight so that the compound pendulum has a period of 1 second (i.e. so that the distance from the point of suspension to the center of oscillation is the length of a simple 1-second pendulum). A small weight added to that pendulum at the center of oscillation will not affect its period. If placed elsewhere, the weight will change the period, and one can calculate where to place it to cause a precise change. One can, that is, mark off on either side of the center of oscillation the lengths c at which the small weight effects any desired deviation.

That is the principle behind the poids curseur, or sliding weight, that Huygens first announced to Moray on 30 December 1661:

The phrase "depuis quelque temps" cannot point very far back, since by all evidence the revision of the Horologium carried out in 1660 did not include any consideration of centers of oscillation. Hence, although the notes and calculations that appear to date from the fall of 1659 indicate that Huygens had made a start on the problem, he did not gain full control of it until sometime in 1661. It was probably then that he set down the derivations outlined above and calculated a table, found in his notebooks, which lists the settings of a sliding weight for gains over 24 hours of up to 2 minutes by 5-second intervals.⁵¹

Huygens' first set of results made some headway in accounting for the discrepances uncovered by English experiments, but it was not until 1664that he could explain why the size of the bob made a difference. By then he had developed the mathematical techniques to apply his basic principle to various solids, most important among them the sphere. He could not locate the precise center of oscillation of a pendulum consisting of a fine wire and a round bob. It lay a significant distance from the bob's geometrical center, and for a fixed length of wire that distance depended on the bob's radius.⁵²

By 1664, then, Huygens had all the elements of a theoretical treatise on the pendulum clock. But by then he also had reason to hope that the treatise could offer stunning evidence of the theory's practical value. In 1663 and 1664 the Royal Society had sent a marine version of the clock to sea to test its reliability and its accuracy as a means of determining longitude. The first results had exceed expectations.

4. Clocks at Sea

The first marine pendulum emerged from the collaboration of Huygens and the Scot Alexander Bruce during November and December 1662.⁵³ Huygens had already toyed with various ways of suspending the clock from a pivot so as to render it independent of the ship's motion, but Bruce's idea of a steel ball encased in a brass cylinder proved most effective in tests at The Hague. Two of the clocks went with Bruce to London early in 1663 and, after minor setbacks prompting the intervention of the Royal Society, accompanied Captain Robert Holmes on voyages first to Lisbon and then to Guinea and out into the Atlantic.⁵⁴ The first voyage yielded a log with encouraging data, including clocked longitudes that agreed well with accepted values in well known waters. The second voyage in 1664 added a dramatic touch.

Having set sail due west from the island of St. Thomas off the Guinea coast, Holmes went some 800 leagues before taking a northeast course back toward the African coast. After several days, supplies of fresh water began to run low, and Holmes's fellow masters urged that he head for the Barbados. They reckoned that the squadron was still some 100 leagues from the Cape Verde Islands. But Holmes's clock placed him only 30 leagues distant, a day's run. He continued his course and hit land the next afternoon. Huygens learned later that the ships had been becalmed for a time after heading northeast and had drifted with the current some 80 leagues eastward; traditional piloting could not detect that motion.⁵⁵

Huygens acted on this success by publishing Holmes's account in the Journal des Sçavans of 23 February 1665 and by placing the marine clock on public sale along with a Kort Onderwijs, or Brief Instruction, on its regulation and use in determining longitude.⁵⁶ Having secured an octroi from the States of Holland, he sought the same rights and recognition from France and England.⁵⁷ In the meantime, he withheld his new Horologium. He did not want to releasethe details of the clock, and he expected soon to have systematic data to back up the drama of the clock's seaworthiness.

Again in 1668-69 the clock met with initial success in measuring the longitudinal difference between Toulon, Crete, and several intermediate points.⁵⁸ But Huygens placed his greatest hope in long-range tests ordered by the Académie des Scioences as part of an expedition to the West Indies in 1670. The experiment would benefit from the lessons of the earlier voyages and, in particular, would silence the critical murmurs coming across the Channel in Oldenburg's letters. "Personnes intelligences" in the Royal Society doubted that Huygens' clocks could be trusted to remain vertical in a rolling ship and had reason to think that the pendulum was significantly sensitive to changes in the air around it.⁵⁹

According to Huygens' account, which no one at the time denied, the voyage was a disaster. Shortly after sailing, one of the clocks stopped in a storm. Contrary to instructions, Jean Richer, the elève detailed to carry out the experiment, did not immediately restart it. After a delay of some 10-12 hours, the second clock also stopped. Again ignoring instructions, Richer simply abandoned the tests altogether and allowed the clocks to deteriorate in their mountings and eventually to crash to the deck. Even more distressing that Richer's negligence was the apparent lack of concern about it among colleagues at the Académie. Writing from The Hague in February 1671, Huygens complained to J.B. Duhamel, "qu'on s'est fort peu applique à bien faire réussir cette experience," but his complaints seem to have gone unanswered.⁶⁰ A year later Huygens reported to Henry Oldenburg that, even if the newest version of his marine clock were ready for a new French expedition then departing for the West Indies, he would not send it. The ship was not properly fitted, nor wqas anyone suited to carry out the texts. "Je crois," he noted in resignation,

que j'irais quelque jour moi-même en quelque petit voyage, pour voir le succès de cette invention, car je vois qu'il dépend beaucoup de la diligence de ceux à qui on en commet et desquels je ne suis pas fort satisfait jusqu'à présent.⁶¹

It was, therefore, in a mood more of resignation than of triumph that Huygens decided in 1671 to proceed with the publication of the Horologium oscillatorium, which appeared in the spring of 1673. In it he presented the 1658 clock with the addition of cycloidal cheeks, a revised escapement, and an adjustable pendulum. He also described with illustrations the newest, as yet untested marine model with a triangular suspension of the pendulum and a Cardan mounting for the clock. He condensed his Kort Onderwijs to a few paragraphs and recounted the results of the 1663, 1664, and 1668 trials. He then took up in separate sections his analyses of the fall of heavy bodies and their motion along a cycloid, the evolution of curves, and the center of oscillation. The last concluded with the theory of the sliding weight, the design of a 1-second pendulum as a universal standard of measure, and the measure of the vertical distance through which a body falls freely in a given time. To this central theme he added a brief account of the conical pendulum with escapement and cheek to keep it on its tautochronic paraboloidal surface. As an appendix he offered his theorems --but not their demonstration-- on the measure of centrifugal force.

5. Strings, Springs, and Other Oscillators.

 In the years immediately following the appearance of the Horologium oscillatorium Huygens left the marine clock and the method of longitude in abeyance and turned instead to further exploration of the phenomenon of tautochronic oscillation. The new line of inquiry began with a theretofore unnoticed corollary of his analysis of motion on a cycloid: the effective motive force acting on a body at any point of the cycloid is proportional to the arc length from the vertex to that point.⁶² He first used the result to analyze the empirically know tautochrony of a vibrating string, but its connection with another area of investigation soon became evident.⁶³ Taking up again in 1674 the study of the nature of accelerative forces sketched out in the opening of the 1659 treatise on centrifugal force, he defined as incitation:

la force qui ait sur un corps pour le mouvoir quand il est en repos ou pour augmenter ou diminuer sa vitesse quand il est en movement.

se measure par la force qu'il faudrait employer pour empêcher le corps de commencer à se mouvoir, à l'endroit où il se trouve, et dans la direction qu'il a.⁶⁴

Within a year Huygens had transformed that theoretical insight into a practical mechanism. The eureka is dated 20 January 1675 for the first sketch of a watch balance regulated by a coiled spring.⁶⁶ A series of refinements during the next week led finally to an announcement sent in anagram to the Royal Society on 30 January: "the axis of a moving circle attached to the center of a metal coil."⁶⁷ Readers of the Journal des Sçavans learned of the invention in February.⁶⁸

Various preoccupations over the next few years, not the least of these being bitter priority disputes with Isaac Thuret, Robert Hooke, and Jean de Hautefeuille over the invention of the spring balance, distracted Huygens from extending it beyond its initial application to pocket watches.⁶⁹ But the idea of a larger format had occurred to him from the outset, and by 1679 he was speaking openly about a spring-regulated marine clock. Work on such a seq-going version reached by 1682 a state sufficiently promising to attract formal encouragement from the Directors of the (Dutch) East India Company as a possible means of determing longitude at sea.⁷⁰

Although the evidence is quite thin and circumstantial, it appears that Huygens' experiments with springs themselves and with spring-regulated clocks had by 1683 borne out Oldenburg's warning of 1675: springs are very sensitive to changes in temperature and humidity.⁷¹ Toward the end of 1683, having found that cold accelerated the oscillations of a clock spring, Huygens began looking for some other form of tautochronic oscillator, one that would have "the effect of the spring without the spring." On 4 December he found it in the pendulum cylindricum trichordon, consisting of three equal wires hanging parallel to one another from a fixed mounting and attached at equidistant points on the inside perimeter of a heavy ring free to rotate and in doing so to rise and fall along its perpendicular axis (see Figure 6).⁷² His analysis of the mechanism began on 7 December with the condition of tautochrony. Considering each cord as a pendulum with one third of the ring's weight as bob, he invoked the displacement principle to show that the motion would be tautochronic if the pendulum followed a certain parabola, which he then demonstrated to be very close to the actual space-curve generated by the endpoints of the cords. Noting that the period (but not the approximation to tautochrony) depended on the square root of the length of the cords and on the size (but not the weight for that size) of the ring, he foresaw calibrating the pendulum by small weights movable radially on the ring; one of his drawings even shows small cheeks about the suspension of one of the cords, apparently to guide the tricorn pendulum into the proper parabolic paths.

Figure 6. (HOC.XVIII.527)

A highly imaginative device posing significant difficulties for mathematical analysis, the tricorn pendulum nonetheless proved practicable within a short time. On 17 December drawings and a rough model went to the clockmaker Johannes van Ceulen, who soon produced two clocks with the new regulator. After some adjustments in design during the spring, Huygens had a clock capable of convincing the East India Company in July 1684 to pay van Ceulen and to place a boat at Huygens' disposal for trials on the Zuyder Zee.⁷³

At about the same time Huygens began to sketch the first versions of what emerged in early 1693 as his balancier marin parfait.⁷⁴ This new tautochronic regulator basically took the form of a vertical balance wheel over which a cord passed linking two systems of small weights, each of which by its rise and fall in following the swinging balance increased the force exerted alternately on the cord by the weights. Two of the systems had originally been designed in 1659 to counterbalance the centrifugal force of a conical pendulum and to pull it back into a paraboloidal envelope: the first consisted of two piles of chain resting on a common support; the second, of two weights partially immersed in mercury.⁷⁵ As one side rose and thus pulled more heavily on the cord, the other side rested more heavily on the support or was immersed more deeply in the mercury. In each case, the torque acting on the wheel varied as its displacement from equilibrium and hence produced tautochronic oscillation.

6. Final Test and New Problems

As noted above, the spring balance and its successors prompted Huygens as early as 1679 to propose new sea trials of his method of determining longitude by means of clocks. The time the Dutch were the first to respond, starting in 1682.⁷⁶ Formal encouragement became financial support in 1684, and when Huygens' personal experiments on the Zuyder Zee proved successful, the East India Company made provision for two clocks and two attendants to accompany a fleet to the Cape of Good Hope in 1686.⁷⁷

The finely detail instructions prepared by Huygens for one of the attendants, Thomas Helder, on 23 April 1686 incorporated not only all the procedures of the Kort Onderwijs but, with the memory of Richer's behavior perhaps still fresh set out explicit rules for the mounting, regulatin, and maintenance of the clocks.⁷⁸ Huygens' instructions make clear by words and drawings that those clocks were spring-driven remontoirs with triangularly suspended cycloidal pendulums.⁷⁹ That is, for reasons not stated, he abandoned the experimental designs that had stimulated the new trials and placed his reliance on the marine clock as it stood in 1671.

The ship Alkmaar carrying the clocks returned to Texel on 15 August 1687. Helder did not return with it; shortly after leaving the Cape he had died at sea. Moreover, on the outgoing boyage he had had trouble with the clocks, which had responded badly to heavy seas. The other attendant, Johannes de Graaf, picked up the experiment and brought home enough measurements for Huygens to plot a course that could be compared with that of the fleet's pilots.⁸⁰ According to the clocks, the Alkmaar had sailed right through Ireland and Scotland rather than around them.

Having examined the logs kept by both assistants, Huygens knew he could not fault their work. Instead, he concluded that he had to take seriously a perturbing effect of which he had first heard in about 1685 but which he had then considered unproved. On the 1672-73 expedition to Cayenne -- on which Huygens had refused to send his new clocks -- Richer had carried out instructions to determine the length of a 1-second pendulum. In his Observations astornomique et physiques faites en l'Isle de Caienne published in Paris in 1679,⁸¹ he reported that careful and repeated measurements agreed on a length shorter by 1-3/4 lignes than that found for Paris.⁸² Richer drew no horological consequences of that result, but on reading it Huygens did: a clock regulated to mean solar time in Paris and then carried to Cayenne would fall behind by just less than 2m 10s as a result of the apparent lengthening of its pendulum. The deviation was of the same order as the equation of time and hence had to affect the method of longitude.

Richer was not alone in reporting variations in the length of the 1-second pendulum. A Gabriel Mouton had published a value for Lyon that significantly differed from that for Paris, and most recently a group sent by the Académie to the Cape Verde Islands and the West Indies had announced differing lengths for various locations. Yet, astronomers and geographers of the stature of Jean Picard doubted these findings and insisted that the length was constant.⁸³ Hugyens felt as late as May 1687 that the matter was undecided; he could give theoretical reasons both for a variation in length and for a universaly constant length.⁸⁴ The trials of 1686-87 apparently decided him. To account for the seeming failure of his clocks on the Alkmaar, he reasoned that the earth's rotation produced a centrifugal force that diminished the weights of bodies by a factor dependent on their latitude. A body at either pole suffered no diminution; at the equator it underwent a maximum decreased of roughly of its weight.⁸⁵ Translatged into horological terms, the diminution of weight by centrifugal force meant that a clock regulated to mean time at the pole would fall behind at the equator by a bit more than 2-1/2 minutes a day. In navigational terms, a clock carried along the same meridian from higher latitudes toward the equator would appear to indicate a longitudinal shift to the east.

Huygens' theoretical calculations of the centrifugal effect agreed on the one hand with his treatise on the cause of weight, which he had composed in 1669 but which only now found reason to publish, and on the other with most of the reported lengths of 1-second pendulums. More importantly they brought de Graaf's raw data into general agreement with the course plotted by the fleet's pilots. The agreement was not complete; Huygens knew it could not be so. The pilots had based their calculations on an incorrect longitude for the Cape, which had meanwhile been determined quite accurately by means of the satellites of Jupiter.⁸⁶ Moreover, the charts used were demonstrably incorrect at several important points of reference.

Reviewed and generally approved by Burchard de Volder, Huygens' results encouraged the East India Company to undertake another trial, again under the supervision of de Graaf, in 1690-92. The outcome disappointed everyone. Although Huygens salvaged the measurements for the longitudinal distance from St. Iago (inthe Cape Verde Islands) to the Cape of Good Hope, and although he could cite errors in de Graaf's handling of the clocks, he had to admit that his marine pendulums did not perform well at sea.⁸⁷ Clearly, the method of longitude still hinged on a reliable clock. "I have found the business much more difficult than I thought at the outset," he had written to Bernad Fullenius in December 1683, thinking perhaps of the new tricord pendulum, "and it is still not accomplished; yet, there has been reason for no small hope." He still had reason to hope in 1693: what the pendulum lacked the "perfect marine balance" might still provide. He continued to look confidently toward yet further trials.⁸⁸ Death came first.

To the end, Huygens' work on time and longitude at sea continued the interplay between theory and practice that had characterized his earliest efforts. Each sea trial pitted the calculable world of theoretical mechanics against the arbitrary reality of wind and water. Each trial yielded new events and data to be incorporated as perturbations of the mathematical model. Yet, even while that combination of theory and practice led to ever more refined mechanics and timepieces, it also encountered the limits of Huygens' theoretical world and of the realm in which he was willing to practice. Faced in 1687 with the fact of geographical variations in the length of a one-second pendulum, Huygens turned to his mechanical theory of weight and to the measure of the perturbing centrifugal force. He did not turn back to the spring balance, which is immune to that perturbation.⁸⁹ The reason seems clear, and it points to the core of Huygens' science. The factors that perturbed the spring resided in its chemistry, in the matter within it and in the effects of temperature and humidity on that matter. Between these factors and the science of motion stood the tedious, dirty, empirical search of men like Mariotte, Boyle, and Hooke, a search in which Huygens had never taken part. As John Harrison showed, metallurgy, not mechanics, held the answer to longitude, and that answer lay behind Huygens' reach.