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‘Following in the footsteps of geometry’: the mathematical world of Christiaan Huygens
 
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Four years after Galileo's death, a seventeenyear old in the Netherlands, unaware of Galileo's results, discovered the law of falling bodies mathematically, using the technique conjectured by Drake to explain Galileo's discovery. The young man sent his results to his brother, who was travelling with their father; the brother showed the letter to their father; the father forwarded the results to his friend in Paris, Marin Mersenne.5. Thus began the international career of Christiaan Huygens. Appropriately for our sense of order, Huygens was in retirement when the Principia burst upon the scene. In one stroke it both confirmed and challenged Huygens' view of nature. In the intervening years, however, no one represented the union of mathematics and physics better than Huygens. So, who better to study in order to see what was involved in the mathematization of physics during the late seventeenth century? What can be gleaned from Huygens' career? What does it take to mathematize nature?
First, it helps to have a natural aptitude for mathematics. Recognizing just such a talent in his precocious son, Constantijn had sent Christiaan to study in Leiden with Frans van Schooten Jr., who at the time was preparing a Latin translation and extensive commentary on René Descartes' Géométrie. The second edition, in 1659, was even more extensive, including contributions from his former pupil now turned collaborator.6. This textbook, much more than the original treatise itself, established a new discipline, now called analytic geometry. It joined the heuristic power of algebraic symbolism to traditional geometry. And Christiaan Huygens had been one of its first students. But Huygens was also well versed in the classical geometry of Euclid and, most tellingly, of Archimedes. His first two published works were extensions of the Archimedean corpus. De circuli magnitudine inventa (1654) presented an advanced algorithm for computing pi that improved upon Archimedes' approximations by means of inscribed and circumscribed polygons. In Theoremata de quadratura hyperboles, ellipsis et circuli, ex dato portionum gravitatis centro (1651), one of many seventeenth century books that attempted to reconstruct Archimedes' lost proofs, Huygens supplied theorems for computing the areas of the hyperbola and ellipse in relation to their centers of gravity that paralleled Archimedes' famous quadrature of the parabola. And later in the Horologium Oscillatorium (1673) Huygens presented similar results for the surface areas of the solid conics made by revolving the twodimensional ones about their axes. Perhaps even more important to his development was an early unpublished treatise on floating bodies, in which he emulated Archimedes' mathematical treatment of problems in statics. No wonder that, not long after they began corresponding, Mersenne dubbed Christiaan a new Archimedes, a very apt title that the poetical Constantijn frequently reiterated.7. Mechanical aptitude also seems a good prerequisite for mathematizing nature. Early on, Christiaan and his older brother began grinding their own lenses for telescopes, and with one of their telescopes Christiaan discovered Saturn's largest moon. He also correctly explained Saturn's strange shape by arguing that it was surrounded by a ring. Indeed, he claimed that his results were proof of their superior lenses, which promptly led to disputes both with one of the foremost lens makers of the period and with one of its greatest astronomers.8. Similar disputes  
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arose because of Huygens' ability to improve a mechanical device significantly by revising its fundamental design. Hence, he modified the air pump, created the first accurate pendulum clock, invented the spiral spring watch, built a magic lantern and devised what is now know as the Huygens' eyepiece for telescopes. Although the mathematical and the mechanical traditions are often placed in opposition to each other (the philosopher versus the artisan), the two do seem to be functioning together in Huygens' case.9. They are joined together by a naive realism. Echoing Galileo's credo, Huygens argued in his Cosmotheoros (1695) that the inhabitants of other planets would still develop Euclidean geometry because the same mathematical principles abide throughout the universe.10. In other words, mathematics is not an abstract construct of our earthly minds but exists independently in nature. Such a viewpoint permits, although it does not necessarily demand, an easy translation of the physical into the geometrical, substituting point for body or particle and line for magnitude or trajectory. It is the Archimedean method writ large. Huygens himself cited Archimedes' quadrature of the parabola by means of weighted segments when, on the draft of ‘De vi centrifuga’, he attempted to formulate his mathematical view of the physical world: ‘Whatever you will have supposed not impossible either concerning gravity or motion or any other matter, if then you prove something concerning the magnitude of a line, surface, or body, it will be true; as for instance, Archimedes on the quadrature of the parabola, where the tendency of heavy objects has been assumed to act through parallel lines.’11. In other words, translate the real world into mathematics, manipulate the equations, and what comes out says something valid about nature. Huygens never seems to have questioned why this should be true. His success at expanding mathematics beyond Archimedean statics to the Galilean science of motion only strengthened his bias. Also strongly reinforcing this attitude was his general acceptance of the Cartesian mechanical philosophy, with its explanation of natural causes by means of fundamental particles in motion. Indeed, his first major achievement in treating motion mathematically was his repudiation and correction of Descartes' rules regarding the impact of colliding bodies. One key to his successful solution was his use of relative motion. Paralleling Galileo's famous ship analogy, he imagined an observer on shore and one in a passing boat. They hold hanging balls and compare results when the balls collide.12. The success of this technique and other early achievements must certainly have convinced him that he was approaching nature properly. His discovery of the isochronism of the cycloid in 1659 is a perfect example of his mature mathematical method in action. He wanted to compare the speed of a freely falling body with the motion of a pendulum's bob. He began by overlapping the circular path of the pendulum with a parabola representing free fall. As he proceeded with his proof, every physical parameter was either absorbed into the geometry or eliminated by the use of proportions. Then, a series of geometric and algebraic equations led to a constant. The physical interpretation of the final equation is that the time of fall of an object along a cycloid is independent of its starting point. Continuing on with his mathematical derivation, Huygens discovered that if the pendulum's chord banks along a similar cycloidal arch, the bob will be constrained to fall along the requisite path. Theoretically, a cycloidal clock based on this design keeps perfect uniform time, because it is independent of the pendulum's swing.13.  
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One of the most persistent topics to which Huygens applied his mathematical skills was optics. In his youth he began a treatise on refraction in lenses, and throughout his life he added to this ‘Dioptrica’, although he withheld it from publication once Newton had published his analysis of spherical aberration. The introductory chapter of this massive book did appear under the title Traité de la lumière (1690).14. In it, Huygens explained the double refraction of Iceland spar by means of his wave theory of light. The mathematical derivation of the refracted wave front involves a type of curve that he discovered during his work on the cycloidal clock. In the course of his argument Huygens treated the wave front as the curve tangential to all the small waves being generated at the previous moment; this concept still goes by the name ‘Huygens' principle.’ Huygens also applied mathematics to a study of corona and perihelia (that is, halos and false suns). These phenomena result from a very complex confluence of light rays, and modern analyses depend heavily on computer simulation.15. Huygens' solution was to theorize that the phenomena are caused by cylinders of ice in the sky that are falling with their main axes parallel to the earth's surface. Naturally, in his treatise on the subject he presented a full mathematical argument, with appropriate derivations of the angles at which the light is refracted and reflected in order to produce the given events. Although, in fact, Huygens achieved only a partial solution, his basic model is correct; there really are cylinders of ice in the sky, albeit they are hexagonal shaped.16. But mathematics is not merely a tool, passively waiting to be applied to nature. There was something else besides the mathematization of physics taking place in the seventeenth century and in Huygens' work in particular. For the sake of symmetry, call it the physicalization of mathematics. The joining of the real and the ideal was bidirectional. The ongoing contemporary debate over theories of space is the most familiar example of the questions that needed to be addressed if one was to build a firm metaphysics upon mathematics. However, true to his image as a nonphilosopher, Huygens did not participate much in the discussion over absolute space, except as it impinged upon his early and strong commitment to relative motion. His contribution was the development of rational mechanics; that is, the study of physical curves and related subjects. Ancient geometers had already introduced into mathematics curves that could not be defined as a simple locus but had to be constructed from the compound motion of other curves. For example, it was with one of these ‘mechanical curves’ that the angle was ‘trisected.’ In the seventeenth century new curves arose that were generated by physical processes. Galileo investigated the first two: the catenary, which is the curve traced by a hanging chain; and the cycloid, which is the curve traced out by a stone embedded in a turning wheel. The young Huygens, true to his Archimedean roots, treated the catenary as a discrete set of weights in order to show Mersenne that Galileo had been wrong when he claimed that the catenary was a parabola. But it was Huygens' successful analysis of the cycloid that prompted a whole series of studies regarding curves that satisfy a specified physical property, curves that included the caustic (refracted wave front), the brachistochrone (curve of fastest descent), and the sail curve. This expansion of its traditional domain reinvigorated mathematics. However, the introduction of these curves was not unproblematical. Conven  
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tion, codified by Descartes in his Géométrie no less, had relegated mechanical curves to a secondary status. Cartesian geometry only recognized algebraic curves, not because they could be represented by a polynomial but because they could be constructed by acceptable means, including, but not restricted to, ruler and compass. Although Descartes himself had represented curves by algebraic expressions (indeed, this is what comes to mind when we think of his geometry), the proper definition of a curve was still by its construction. And Descartes had essentially rigged the requirements for constructibility so that only algebraic curves qualified. The mechanical curves, also called transcendental curves, needed to be validated as real curves, and the legitimation had to come from an acceptable construction. The tractrix is another physical curve, made by a heavy object when it is dragged over a horizontal surface by a chord of fixed length as its endpoint moves along a straight line. Huygens argued that the tractrix should be considered a legitimate curve because it could be drawn as exactly as a circle is drawn by an ordinary compass. To substantiate his claim, he had to invent an instrument that would match a compass in its simplicity and feasibility. His unpublished notes show a mechanism by which he would produce the tractrix by dragging a stylus over a smooth surface of syrup. Why syrup? Because he needed to make the plane as truly horizontal as the side of a ruler is straight. Since most mechanical curves are generated by idealized motion, in some sense all of them could qualify as legitimately geometrical under this sticky construction method. Leibniz, ever eager to generalize, even leapt to the conclusion that any curve imagined to be generated by a continuous motion is legitimate. However, most mathematicians still required proof that the construction was feasible, although apparently the constructing device did not actually have to be built, as Huygens seems to have done with his. In the published paper that resulted from his studies, Huygens explained why his construction of the tractrix was important: ‘Si cette description, qui par les loix de la Mechanique doit être exacte, pouvoit passer pour Geometrique, de même que celles des sections de Cone qui se font par les instrumens l'on auroit par elle, avec la quadrature de l'Hyperbole, la construction parfaite des Problemes qui se reduisent à cette quadrature; comme sont entre autres la determination des points de la Catenaria, ou Chainette, & les Logarithmes.’17. Legitimizing the transcendental curves increased the set of allowable solutions to many important mathematical problems. Johann Bernoulli dealt the Cartesian division its final blow when he discovered that, if the problem of the tractrix is generalized by allowing the chord's length or the path of the endpoint to vary, the curves that result can be either algebraic or transcendental depending upon the initial parameters.18. Another example of Huygens' physicalization of geometry was his contribution to the problem of rectification. As the pendulum of his cycloidal clock unrolled off one of the plates that restricted its swing it was literally straightening, or rectifying, the plate. It is the common method by which we use a tape measure to find the girth of an irregularlyshaped object. Huygens generalized his results in the Horologium Oscillatorium, calling the curve that was rectified the ‘evolute,’ Latin for ‘unrolled.’ Jakob Bernoulli expanded Huygens' work into the concept of the radius of curvature and transformed it for the new calculus.19.  
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Clearly, the demarcation between the geometrical and the physical has become thoroughly blurred. Is it the cycloid that is isochronous or the clock whose bob is made to follow that curve's path? The terminology can be loose because the pairing of mathematical entity and physical object has been made so precise. The cycloid is a mechanical curve in the ancient sense of having its locus defined by a mechanical motion, but it is also mechanical in a new sense, because it is the path of uniform timekeeping. In the Horologium Oscillatorium Huygens referred to ‘the power of this line to measure time, which we found not by expecting this but only by following in the footsteps of geometry.’20. But, of course, Huygens himself chose to ‘follow in the footsteps of geometry,’ by concentrating on problems that straddled the boundary between physics and mathematics. In a letter to the Marquis de l'Hospital regarding Leibniz's work on the calculus, he concluded by saying, ‘mais il est vray aussi que je n'avois pas beaucoup meditè alors ces matieres, m'estant tousjours plus d'avantage à chercher l'utilitè de la Geometrie dans les choses de physique et de mechanique.’21. With Leibniz he was even more specific regarding the need to apply mathematics to physics: ‘Vous croiez, à ce qu'il semble, qu'il ne seroit pas extremement difficile d'achever de tout point la Science des Lignes et des Nombres. En quoy je ne suis pas jusqu'icy de vostre avis, ni mesme qu'il seroit à souhaiter qu'il ne restast plus rien à chercher en matiere de Geometrie. Mais cette etude ne doit pas nous empescher de travailler à la physique, pour la quelle je crois que nous scavons assez, et plus de geometrie qu'il n'est besoin; mais il faudroit raisonner avec methode sur les experiences, et en amasser de nouvelles, à peu pres suivant le projet de Verulamius.’22. His concluding reference to Baconian experiments highlights another aspect of early modern science. What role did experiments play in Huygens' work? If ‘experiences’ means the acute observation of phenomena under controlled circumstances, Huygens definitely experimented and ‘reasoned systematically’ from what he observed. But his manuscripts contain very little evidence of repeated observations, especially tables of values such as those that are associated with modern experimental science and that are abundant in Galileo's papers, once someone has decided to look for them. Now, of course, such tables contain inherent problems, because the numerous results must be reduced to one unequivocal value. Giambattista Riccioli, reporting his attempt to determine the constant of gravitational acceleration, published tables with irreconcilable values. Although quantification and the concept of precision measure is usually associated with experimental science, in fact the measurement of that first physical constant was as much a product of mathematics as of experimentation. Huygens was the first to measure accurately the ‘constant’ of gravitational acceleration (‘little g’), but he achieved it indirectly by way of the mathematical theory underpinning his greatest technical contribution. For Huygens gave science its fundamental measuring tool  time. His cycloidal pendulum clock was theoretically a perfect timekeeper, and his own model was accurate to within 7 seconds a day. Huygens was able to use his clock to measure the constant of gravitational acceleration, because in the course of developing the cycloidal clock he had discovered the mathematical relationship between the constant and the length of the pendulum.23.  
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In addition, his efforts to measure longitude using marine variants of that clock led to his development of standards by which universal gravity (‘big G’) could be verified.24. The attempt by the Royal Society of London to test the accuracy of Huygens' clock demonstrates vividly how underdeveloped the science of measurement was at the time. Because of incompatible measures, the fellows had to ask him to mail them an example of something that measured the length of his pendulum.25. Although weights and measures were indispensable in the marketplace, the units of measure were usually locally determined. This lack of universal standards impeded international scientific discourse. For one brief moment the Royal Society became enamored of using Huygens' clock as a universal measure. As usual, their attention soon wandered elsewhere, but Huygens persisted with the idea, defining 3 horological feet to be the length of a pendulum that makes a single oscillation in one second.26. Unfortunately, his new unit of length was undermined by the discovery that it varied with temperature and location. One of Huygens' more lasting contributions to the quantification of nature was a forerunner of the micrometer, which he made by inserting little metal wedges into the eyepiece of his telescope so that he could measure the diameters of the planets. Like quantifying, axiomatizing is another way to the mathematize the world. Mathematics has always held a privileged place among the sciences because of its role as a model for correct reasoning. The axiomatic method of propositions based on first principles, codified by Euclidean geometry, guarantees truth  or at least give the semblance of such assurance. This, I would argue, was the use to which Galileo put mathematics. His results followed from his experiments. He might present them in mathematical language, but he did not discover them through mathematics. Always the polemicist, he drew upon the image of rigor that mathematics invokes to justify what he already knew or believed. In contrast, Huygens seems to have needed to convince himself through mathematics. His manuscripts abound with small quasitreatises that are narrowly focused but complete analyses of specific problems. ‘De vi centrifuga’ is the most famous one. They are not finished drafts for projected publication, although his editors have treated them as such. Usually they consist of a set of theorems introduced by a paragraph that specifies certain physical presumptions, such as Torricelli's principle. The final stage of Huygens' process of discovery was this axiomatization of the results. It is as if the study was not complete until rigorous analysis had assured a high degree of probability, by identifying the physical postulates and reducing the remainder to mathematics. But Huygens went beyond mathematics as proof structure to mathematics as a means of discovery. Indeed, his commitment to a mathematical methodology led him to the unambiguous statement of the hypotheticodeductive method.27. What does he himself say about the proper way to approach nature? He would certainly agree with our choice of Galileo as the founder of the new physics, or as he called it, ‘la vraye Philosophie dans laquelle on conçoit la cause de tous les effets naturels par des raisons de mechanique. Ce qu'il faut faire à mon avis, ou bien renoncer à toute esperance de jamais rien comprendre dans la Physique.’ Huygens made this statement in the Traité de la lumière in order to justify his claim that ‘L'on ne sçauroit douter que la lumiere ne consiste dans le mouvement  
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de certaine matiere.’28. But the assertion reflects his general commitment to a mechanical philosophy, or more precisely to a metaphysics for which mechanics/kinematics is the fundamental science. Now, if we presume that the study of matter in motion is basic to our understanding of natural phenomena, then Galileo does indeed become the founder of the true mechanical philosophy, for, as Huygens declared, ‘Galilee premier qui ait bien examinè le mouvement.’29. Further, when Huygens came to describe his own work on falling bodies and pendula for the Horologium Oscillatorium, he characterized it as ‘le fruit principal que l'on pouvoit esperer de la science de motu accelerato, que Galilée a l'honneur d'avoir traictée le premier: et je m'assure que les Geometres estimeront infiniment plus cette addition que tout le reste de cet automate.’30. And how do we study matter in motion? By abstracting from nature and applying mathematics, in accordance with Galileo's dictum. While still a teenager, Huygens criticized another researcher with the terms: ‘However he has not considered motion abstractly enough, since he considers the motion of a stone or metal sphere falling through the air from on high; truly, far greater does the thought of Galileo agree with experience, excepting that the resistance of air impedes it. Therefore we will consider accelerated motion in a better way.’31. Later he would comment that Bacon had very good methods, as shown in Bacon's analysis of heat as motion of particles, but Bacon was deficient mathematically. ‘Mais au reste il n'entendoit point les Mathematiques et manquoit de penetration pour les choses de physique.... Galilee avoit du costè de l'esprit, et de la connoissance des Mathematiques tout ce qu'il faut pour faire des progres dans la Physique, et il faut avouer qu'il a estè le premier à faire de belles decouvertes touchant la nature du mouvement, quoy qu'il en ait laissè de tres considerable à faire.’32. Obviously, Huygens is interpreting Bacon and Galileo from his own perspective. Whatever we might think of the accuracy of his descriptions, however, he thought that he was identifying his precursors. And most definitely he was indicating his values: mechanics is the fundamental science of nature and mathematics is the fundamental method of mechanics. Yet, the mathematical approach to nature does have its limits. At the end of that long deductive chain of reasoning we must return to the real world and ask: Does the conclusion seem reasonable, or is it contradicted by observation? If the latter, then we have erred  not in our deductive reasoning, for that belongs to the perfect realm of mathematics, but in the postulates that we have used to translate the physical into the geometrical. Huygens stated this attitude very clearly in his introduction to the Traité: ‘On y verra de ces sortes de demonstrations, qui ne produisent pas une certitude aussi grande que celles de Geometrie, & qui mesme en different beaucoup, puisque au lieu que les Geometres prouvent leurs Propositions par des Principes certains & incontestables, icy les Principes se verifient par les conclusions qu'on en tire; la nature de ces choses ne souffrant pas que cela se fasse autrement. Il est possible toutefois d'y arriver à un degré de vraisemblance, qui bien souvent ne cede guere à une evidence entiere. Sçavoir lors que les choses, qu'on a demontrées par ces Principes supposez, se raportent parfitement aux phenomenes que l'experience a fait remarquer; sur tout quand il y en a grand nombre, & encore principalement quand on se forme & prevoit  
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des phenomenes nouveaux, qui doivent suivre des hypotheses qu'on employe, & qu'on trouve qu'en cela l'effet repond à nostre attente.’33. Elsewhere he seems to suggest that the degree of probability that a given postulate, or Principle, is true depends directly upon the number and variety of the experimental phenomena that confirm it.34. An example of a postulate whose degree of probability he believed to be ‘scarcely less than complete proof’ is Torricelli's principle that the center of gravity of a system cannot raise itself. Using it, he derived many beautiful results  although, significantly, not all of them could be experimentally verified. Indeed, in practice, he was never that tied to the experimental confirmation of his results, for he was always too aware that observation is biased and imprecise to grant it priority. For instance, when the Royal Society had trouble verifying his results regarding the length of the pendulum that beats in one second, he responded that any anomalies were merely the effect of air resistance (because he felt that they used bobs of insufficient weight).35. He was distressed that the Fellows could not reproduce his value, but he was not deterred in his approach nor in his belief that his derivations were correct. In fact, he readily admitted that experiment could not confirm his value except in the negative sense.36. Of course, many researchers, including his hero Galileo, quite often erroneously waved aside objections to conclusions that couldn't be substantiated with allusions to unaccounted physical variables like air resistance. Huygens' strength was his consistent appeal to mathematics when faced with anomalies. Thus, when objections persisted, he analyzed the mass of the pendulum's bob and the mass of its cord and tried to subject air resistance to the same mathematical approach. Every problem ultimately had a mathematical solution. And its verification came not so much from some crucial experiment as from the internal logical rigor of mathematics. In fact, there is a strong sense of determinism about Huygens' mathematical approach to nature. In Huygens' world, an isochronous clock must have a regulator, such as a pendulum, that moves according to the mathematics of perfect oscillating motion. In 1657 he created a clock whose period was kept more nearly uniform because its pendulum banked along curved plates, the shape of which were empirically derived. He soon abandoned the design because of mechanical problems, but reinstated it in 1659 when he discovered mathematically what shape the plates should have in order for the bob to trace the isochronous cycloidal path. Yet, the mechanical problems that had plagued him in 1657 were still present in 1659. Nonetheless, although the curved plates were dispensable when they were the result of experimentation, they became essential once they were the product of mathematical analysis. Whatever objections might be advanced regarding the actual value of curved plates, such as the interference of friction, at that point become irrelevant. Physical aberration should be met by additional derivations, not by the abandonment of the design proven by mathematics. Time and again in Huygens' work we find that a complete mathematical description suffices as the solution, or at least delimiter, of a physical problem. The wave theory of light must be correct because the mathematics of waves explains the double refraction of Iceland spar. Certainly Huygens did not always succeed in his  
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mathematical approach. Thus, it was Edme Mariotte who truly dealt with the corona, building upon Huygens' work nonetheless. And sometimes Huygens did not apply mathematics in situations that would seem obvious choices. For example, he never tried to analyze Saturn's ring. But, in the seventeenth century it was truly rare to find such persistent adherence to a mathematical methodology. The supreme example of his fidelity to mathematics is his reaction to the Principia. The vortices of Descartes had to be rejected once Newton had shown them to be mathematically unsound. The Principia had destroyed one of the fundamental concepts of the natural philosophy to which he was committed, yet he would not deny its conclusions, because he would have to repudiate the foundation of his truth system. ‘Mais voiant maintenant par les demonstrations de Mr. Newton, qu'en supposant une telle pesanteur vers le Soleil, & qui diminue suivant la dite proportion, elle contrebalance si bien les forces centrifuges des Planetes, & produit justement l'effet du mouvement Elliptique, que Kepler avoit deviné, & verifié par les observations, je ne puis guere douter que ces Hypotheses touchant la pesanteur ne soient vrayes, ni que le Systeme de Mr. Newton, autant qu'il est fondé la dessus, ne le soit de mesme.’37. Would action at a distance ever have been seriously debated if it had not been associated with an overwhelming mathematical analysis? And would that mathematical analysis have been considered sufficient as an explanation without the groundwork of Christiaan Huygens?
Of course there have always been people who have applied mathematics to nature. Even in the classical era, when the science of number (arithmetic) was distinguished from the science of quantity (geometry), the mathematical sciences included music and astronomy as the real world analogues of the other two fields. Archimedes expanded the mathematical sciences to include statics, which Galileo further refined with the first of his two new sciences. The Arabs developed optics beyond Euclid, and the European medievalists contributed sophisticated studies of ratios of quantities.38. There have also always been people who have applied mathematics to human activities  the ‘practioners’ rather than the ‘philosophers.’ From the days of the very first traders, number has been essential to daily commerce, including the problem of standard weights and measures. The mechanical arts, such as architecture, ship navigation, and the construction of military machines have all required mathematical skills. In the Renaissance these two traditions of theory and practice had already begun to merge. For example, new treatises on perspective blended practical information for the artist with theoretical explanations drawn from optics. Likewise, the new science of mechanics joined the mechanical art of machine construction to a pseudoAristotelian natural philosophy.39. Galileo and Huygens thereafter were following a welltrod path. What was different in the seventeenth century, and what was Huygens' contribution? Foremost, we see the complete integration of physics and mathematics, so much so that the line between them becomes difficult to demarcate. The modern tendency has been to use Newtonian dynamics as the dividing knife, and to call the work of those outside that tradition, including Huygens, mathematics. Of  
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course, this approach requires the further broadening of the definition of mathematics to incorporate rational mechanics. Eliminating all divisions, Isaac Barrow defined physics to be nothing other than the mathematical study of quantity, including both theoretical and practical mechanics.40. Huygens himself summarized: ‘L'on a joint ensemble les Mathematiques et la Physique non seulement a fin qu'on s'accoustumait en physique s'approcher le plus qu'on pourroit du la solidite de raisonnement des geometres, mais aussi parce qu'on a assez recevoir que la pluspart des choses naturelles ne se doivitur expliquer par des raisons de geometrie et de mechanique.’41. Besides the application of mathematics to physics and physics to mathematics, other aspects of the mathematization of nature have emerged in this survey of Huygens' work. Mathematization involves quantification; it also comprises axiomatization. But if pressed to identify what was new, I would focus on mathematics as methodology. It is the predictive aspect: ‘following in the footsteps of geometry,’ to use Huygens phrase, leads to new knowledge about the world. Which brings up a very awkward question: just who else was ‘following in the footsteps of geometry’ earlier in the seventeenth century? I am hard pressed to find anyone before Huygens who so systematically applied mathematics to physical problems, except perhaps Kepler. Naturally, I am referring to the use of mathematics as a method of discovery or analysis, not merely as rhetoric or synthesis. Yet, the approach was so obvious, so natural, to Huygens, that he felt no need to justify it. Can it be that he, like Kepler before and Newton after, simply saw the world through the eyes of a born mathematician and that the cumulative effect of the highly successful examples that they independently produced in the seventeenth century created a norm for the eighteenth century  or, at least for Johann Bernoulli? True contextualists all, you as well as I should beware of episodic history and its overdependency upon heros. Yet, I cannot help closing with a line from l'Hospital, who replied to Huygens' complaint, regarding the penchant of the Bernoullis and Leibniz to generalize, with the florid praise: ‘Je suis de vostre avis que la geometrie n'est qu'un ieu d'esprit si on ne l'applique à a phisique et aux inventions de mecaniques, mais il est rare, qu'on y reussisse et il faut des siecles entiers pour produire un Hugens.’42. Presented to the Congress 1695 Christiaan Huygens 1995 of the Werkgroep de XVIIe Eeuw on 7 July 1995. An earlier version of this paper was presented to the New York Academy of Sciences on 22 March 1995. 
