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100 NAW 5/9 nr. 2 June 2008 The golden age of . Stevin, Huygens and the Dijksterhuis

Fokko Jan Dijksterhuis Science, Technology, Health and Policy Studies (STeHPS) Universiteit Twente, Faculteit MB Postbus 217 7500AE Enschede The Netherlands [email protected]

History The golden age of mathematics Stevin, Huygens and the Dutch republic

Mathematics played its role in the rise of the Dutch Republic. Simon Stevin and Christiaan its formation. Half a century later, the wealth, Huygens were key figures in this. Though from very different backgrounds, both gave shape to the and the splendour of the republic Dutch culture. Fokko Jan Dijksterhuis, who recently received an NWO-vidi grant for his research had been established (and had even begun to on the history of mathematics, discusses their influence on mathematics and its historical wane) with Huygens as its main mathematical significance. representative. The hands-on mathematics of Stevin differed greatly from Huygens’ aris- The 17th century was the golden age of the Recent developments in cultural history have tocratic geometry and yet, as will be argued Dutch republic. A small state lacking sub- opened new perspectives on the mathemat- here, they had essential features in common. stantial natural resources, from the 1580s on- ical pursuits of the Dutch republic.[1] Under- In this article, the of Stevin and Huygens wards it quickly became a wealthy trade cen- standing the brilliant ideas of Stevin and Huy- will be sketched as part of, and giving shape tre that played a leading part in international gens in their historical context promises a new to, the golden age. politics. It produced cultural tour de and deeper insight into their achievements. in the arts, sciences and technology. Mathe- Stevin and Huygens lived at different stages Mixed mathematics matics is represented by Simon Stevin (1548– of the golden age and their mathematics like- A history of mathematics in the golden age 1620) and (1629–1695), wise differed. Stevin witnessed the birth of begins with the realisation that early modern spanning the main part of the golden age. the Dutch republic and played an active role in mathematics was an endeavour quite differ- ent from its present form. Viewed with our modern conception of mathematics, our main heroes Stevin and Huygens instantly disap- pear from the history of mathematics. Their contributions to mathematics per se were mi- nor (Huygens) or virtually zero (Stevin). More importantly, by trying to single out their math- ematical achievements we lose sight of the structure of their work and consequently their ingenuity. The first thing we need to realise is that the idea of pure mathematics dates from the early nineteenth century. Prior to La- grange’s and Cauchy’s rationalist purification of mathematics, the whole range of mathe- matical sciences (geometry, arithmetic, as- tronomy, music, , etc.) were seen as parts of mathematics. The idea of an a pri- ori given pure mathematics applied to phys- ical domains simply did not exist. In early modern conceptions ‘mathematica pura’ was distinguished from ‘mathematica mixta’, in- Figure 1 Christiaan Huygens, pastel drawing Bernard Vail- Figure 2 Simon Stevin, Universiteitsbibliotheek Leiden lant, 1686. Collectie Huygensmuseum Hofwijck, Voorburg dicating the level of abstraction of the object

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Dijksterhuis The golden age of mathematics. Stevin, Huygens and the Dutch republic NAW 5/9 nr. 2 June 2008 101

of study.[2] Geometry and arithmetic stud- al, cashiers, and all merchants”. With works ied quantity as subsisting in itself; astron- like these Stevin contributed to the growing omy, music and optics studied quantity as amount of calculating in all kinds of profes- subsisting in matter. The mixed parts were sions. He did so by proposing specific meth- not separated, let alone subordinate to the ods and elucidating their character and appli- pure parts of geometry and arithmetic: math- cation — to the point of disclosing profession- ematics consisted of the study of reality in its al secrets. Twenty years later Stevin got the quantitative aspect. From this historical point opportunity to put his ideas about sophisti- of view, Stevin and Huygens were full-blown cated bookkeeping into practice when he be- mathematicians, and brilliant at that. came administrator of Maurits’ domains. He was given permission to introduce Italian dou- The beginning of the ble entry bookkeeping. In Vorstelicke bouck- When the northern Netherlands in 1579 revolt- houding (Royal bookkeeping, 1608) he de- ed against Spanish rule and liberated them- scribed and explained his ideas and meth- selves, a 50-year exodus of capital, intellect ods.

and skills from the southern Netherlands be- Stevin has achieved lasting fame with De Figure 3 The ‘clootcrans’ gan. Although his motives are unclear, Stevin Beghinselen der Weeghconst (Elements of moved too and he turns up in Leiden in 1581, the Art of Weighing, 1586), appended with ature and elaborating his account in geomet- where he enrolled at the university in 1583. De Weeghdaet (The Act of Weighing) and De ric fashion by formulating principles, draw- He had been born in Brugge in 1548, ille- Beghinselen des Waterwichts (The Elements ing up thought-experiments and deducing gitimate son from a well-to-do family, and of Waterweight). These publications reflect propositions. He did so, however, in an id- had worked as a bookkeeper in Antwerp and the of his new country, of iosyncratic manner. He explicitly combined at the Brugge tax office.[3] Although little is and of water management. Stevin employed his Archimedean analyses with practical pur- known of his scholarly and practical educa- his findings in new mill designs, for which he suits. The titles of his books reflect his views: tion, the documented activities in his early got patents and which he realised in a the elements of weighing was appended with years in show that he was both lit- of places — albeit not fully satisfactorily.[5] At the act of weighing, and the elements of wa- erate and skilled, and that he applied him- the same , however, these works reflect- terweighing contained a commencement of self to the whole spectrum of mathematics. ed an international — most notably Italian — the act of waterweighing. He called these During the 1580s, he published a range of in the mathematics of mechanisms two aspects ‘spiegheling en daet’, reflection books in Dutch, French and Latin, applied for and the Renaissance of Greek mathematics. and action.[6] Both were requisite: just like patents for the construction of mills and per- In particular the rediscovery of a foundation without a building is futile “so formed experiments. These activities on the had sparked a lively interest in this partic- the reflection on the principles of the arts is one hand reflected his previous experiences ularly ‘tactile’ mathematics and methods of lost labour where the end does not tend to and on the other hand opened up new areas analysis. the action”.[7] In all his work Stevin conse- of interest, areas that fitted particularly well Stevin contributed to this Archimedean quently combined practical work and literary with his new environment. During these early renaissance with an inventive inquiry into pursuits. In bookkeeping or mill building he Leiden years Stevin also developed relation- statics and that extended well did not content himself with getting the pa- ships with notable Dutchmen, for example beyond the original of Archimedes. The perwork right and putting machines to work. the future Delft burgomaster Johan Cornets de ‘clootcrans’ (wreath of spheres) is the most fa- He also committed his opinions and inquiries Groot (1554–1640) with whom he collaborat- mous example. He proved by reductio ad ab- to paper, learnedly elaborating them. ed on mill projects, and the future stadholder surdum that two bodies on an inclined plane ‘Spiegheling’ and ‘daet’ are usually inter- Maurits van Nassau (1567–1625) who stud- are in balance when their weights are propor- preted as ‘theory’ and ‘practice’ but we should ied at at the same time. In tional to the length of the planes. Imagine be careful interpreting these words with our short, we see a Flemish exile in his thirties a wreath of equal spheres at equal distances modern understanding. By ‘spiegheling’ who energetically and quite successfully pur- that can move freely. The number of spheres Stevin understood the intellectual analysis sued his interests in matters mathematical. is proportional to the length of the sides of the aimed at disclosing principles and causes triangle. If the spheres on the one side would that should precede the ‘daet’ of understand- Simon Stevin not keep those at the other side in balance, ing and employing phenomena in the real The publications that reflected Stevin’s pre- the wreath would begin to move. However, it world. The Weeghconst and the Weegh- vious occupations were Nieuwe Inventie van would soon reach the same state and thus daet both contain theories of weighing, with rekeninghe van compaignie (New invention a perpetual would occur. Because principles, deductions and so on. The for- of business calculation, 1581) and Tafelen a perpetuum mobile is impossible, the sup- mer considers weights per se whereas the van Interest (Tables of Interest, 1582).[4] De position is false, and thus the spheres must latter considers them in their physical ap- Thiende (The Dime, 1585) also built on his be in balance. Stevin was so proud of this pearance, Stevin said.[8] For us (early 21st- bookkeeping experience but had a much proof that he used it as the frontispiece of the century mathematicians) such a distinction is wider scope. It was an inspired plea for Weeghconst. not self-evident but Stevin’s explanation cor- the use of decimal fractions in all calculat- With the Weeghconst Stevin expressly pre- responds quite well with the distinction ‘pure’ ing professions: “stargazers, surveyors, mea- sented himself as a man of learning, dis- and ‘mixed’. This is remarkable, because surers of carpet, of wine, of bodies in gener- playing his knowledge of mathematical liter- ‘spiegheling en daet’ may best be translated

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102 NAW 5/9 nr. 2 June 2008 The golden age of mathematics. Stevin, Huygens and the Dutch republic Dijksterhuis

Figure 4 The ‘Duytsche Mathematique’ found its first home in the fencing school of its first professor, Ludolf van Ceulen, on the ground floor of the University Library. This picture nicely il- lustrates the close links that existed between mathematics and the military at that time. Copper engraving: Jan Cornelisz. van’t Woudt (1610)

as ‘reflection’ and ‘action’. That was a second The Weeghconst opened with a long digres- of ‘spiegheling en daet’ Stevin envisioned in distinction drawn in early modern mathemat- sion on the virtues of Dutch, being more clear his writings. The teaching was in the vernac- ics, signifying the goal of study rather than the and more efficient than any other language. ular, hence Duytsche mathematics. Although object.[9] Stevin’s oblique use of reflection He rejected the tendency of other writers in Stevin never taught at the engineering school, and action may be explained by realising that the vernacular to Dutchify Latin words and he drew up a detailed curriculum in which a he was primarily interested in the mathemat- proposed a whole list of truly Dutch words. careful selection of mathematical theory was ics of an artisan — bookkeeping, machines Irrespective of the virtues of Stevin’s linguis- combined with field practice. For example: and, later on, fortifications and navigation — tic theories, he did give Dutch its own word for “The measuring of circles with segments to which he applied his considerable specula- mathematics: wisconst, nowadays wiskunde, of that sort, further the area of spheres. The tive ingenuity in finding principles and caus- literally ‘art of knowing’. shapes named ellipsis, , es. The mathematics of natural phenomena and the like, that is not necessary here, be- considered outside a context of artefacts and The establishing of the Leiden engineering cause are very seldom made to per- ‘practices’ lay beyond his interest. school in 1600 form such measurements; but only they shall The use of the vernacular was important Stevin’s greatest success in realising his learn with straight planes, after that curvilin- for Stevin. Not only did he consider Dutch the ideas was the establishment in 1600 of the ear in surveyor’s manner, measuring thus a best way to reach his intended audience (men ‘Duytsche Mathematique’, the engineering plane by various division, like in triangles or of action), he even went so far as to regard school at Leiden University. It would train for- other planes to see how this matches with Dutch the language best suited for learning. tificationists with precisely the combination that.” [10]

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Dijksterhuis The golden age of mathematics. Stevin, Huygens and the Dutch republic NAW 5/9 nr. 2 June 2008 103

as a victory of Stevin over the practically in- ed the renaissance mathematics inspired by was. His case is different, however, from his clined engineers in the Dutch army. The idea Archimedes. The beautiful Dutch word ‘ver- fellow students. He managed to devote his that engineers should also learn from books nufteling’ is most appropriate for Stevin, indi- entire life to his main pleasure: mathematics. was not obvious, and would not be for at least cating cleverness of both hand and mind. Huygens was the second son of Constantijn a century. Huygens, secretary of the stadholder and a Stevin’s success can be explained by his French mathematics brought to Leiden leading man of learning and art in the Dutch close ties to Count Maurits, the Dutch stad- The historical link between Stevin and Huy- republic.[15] His predilection for mathematics holder and army commander.[11] They proba- gens is established by showed at an early age, when he and his older bly knew each other from their student year Jr. (1615–1660). Van Schooten’s career re- brother Constantijn were educated by private in Leiden in the 1580s, and in the 1590s flects the transition from the Dutch-spoken, tutors. He helped his brother with mathemati- they established a close relationship. In utilitarian mathematics of Stevin to the Latin- cal exercises and was soon bored with writing 1593 Stevin entered the service of Maurits written, academic mathematics of Huygens. poems. After the brothers had gone through as quartermaster of the army.[12] From this After his studies at Leiden university, he be- university, their father could set out to plot time his publications dealt with issues of gan replacing his father, Frans van Schooten a diplomatic career for his sons. However, state building and reinforcement, for exam- Sr., the then professor of ‘Duytsche Mathema- in 1650 the first stadholderless period began ple Sterctenbouwing (Fortification, 1594) and tique’ in the 1630s, before succeeding him in and with the Oranges out of power no posts De Havenvinding (The finding of harbours, 1645. In the meantime, he had established were available to the Huygenses. Huygens 1599). These had been preceded by a trea- himself as an advocate of the new mathemat- was free to pursue his interests (and happy tise on citizenship Vita Politica. Het Burger- ics coming from France. He gathered and pub- too, we may presume). He would do so for the lick Leven (Political Life. The Civil Life, 1590). lished papers by Fermat and Viète and collab- rest of his life, even when his brother made Stevin’s ideas and pursuits fitted particular- orated with René Descartes. He comment- a diplomatic career some years later. For a ly well with his employer’s interests. Mau- ed on the manuscript of La Géométrie and man of learning of Huygens’ rank few public rits was greatly interested in mathematics made the illustrations. His claim to fame is positions were available and in the republic and shared Stevin’s idea of combining practi- Geometria a Renato Des Cartes, a Latin trans- next to none. A university professorship was cal improvements with theoretical reflection. lation with commentaries and supplements far below his standing and the republic did The two had lengthy conversations on these disclosing Descartes’ new geometry to the in- not have a lustrous court. The only options matters, which Stevin eventually put down ternational mathematical community. A first were the great courts of Italy, France and Eng- in his Wisconstighe Ghedachtenissen (Math- edition appeared in 1649, followed by an ex- land. His bid for Florence was unsuccessful ematical Memoirs, 1605–1608). These reveal panded edition in 1659–1661. This edition but in 1666 Colbert brought him to Paris to several original contributions of Maurits.[13] contained contributions from elite students chair Louis XIV’s Académie des Sciences. He Maurits played a central role in the military re- Van Schooten had attracted to Leiden: patri- would stay there until the early 1680s, spend- forms that turned the Dutch army into a high- cian sons like Johan (future governor ing his last years at his family residences in ly skilled, professional and quite successful of the republic), (future bur- and Voorburg: a life in mathemat- . Part of these reforms was the ‘Duytsche gomaster of ) and Christiaan Huy- ics, free from public obligations — a savant’s Mathematique’ that Maurits installed high- gens. paradise. handedly at Leiden university. What did this pastime look like? In the The promotion of mathematics in warfare, Christiaan Huygens course of his life, Huygens worked on a wide state building and land development was not It remains one of the miracles of the gold- range of topics: the catenary and floating just a Dutch affair. The second stadholder of en age that patricians like De Witt and Hud- bodies; Cartesian ovals, quadrature and rec- the republic Willem Lodewijk of Friesland had de developed and cultivated an interest in tification; lenses, telescopes, heavenly ob- similar ideas about the intellectual and mate- mathematics, a field of study that apparently servations and light waves; impact, fall, cir- rial reinforcement of the new state. His schol- bore no relevance to their administrative am- cular and motion, clocks and lon- arly interests were mainly aimed at classical bitions. Their amateur interest in the mathe- gitude; ; tuning; and so on. His- military theories but he provided for the culti- matical sciences may be a major key to under- torians have lamented that Huygens’ oeuvre vation of mathematics by appointing Adriaan standing the unprecedented rise of the epis- lacks a unifying idea, that it is an eclectic col- Metius in the chair of mathematics at Franek- temic and cultural status of mathematics in lection of puzzles that came his way through er University. Like Stevin, Metius disclosed the 17th century. In the course of the scientif- coincidental encounters and the like. From new developments in the mathematical sci- ic revolution, mathematics rose from being an the viewpoint of early modern natural philos- ences to a broad audience by publishing a art pursued by ingenious professionals to the ophy and in comparison with men like Galileo, range of books in both Dutch and Latin.[14] In language of philosophy — and even nature — Descartes and Newton, Huygens’ oeuvre in- Franeker as well as Leiden, mathematics fig- cultivated by savants of all ranks. The growing deed looks fragmentary. Yet, Huygens had ured next to (Calvinist) theology as the means interest in natural inquiry among Dutch patri- no taste for philosophy, as his pupil Leibniz of intellectually reinforcing the new republic. cians has been explained by the ‘aristocrati- would aptly observe. He was a mathemati- Stevin’s religious inclinations are somewhat sation’ of this group. They were no nobles but cian. From the perspective of early modern diffuse but we have acquired a good under- they increasingly took up noble activities like mathematics, Huygens’ oeuvre does not look standing of his mathematical persona. His investing in landed property, building dynas- incoherent at all. All his activities were math- specific and explicit mixing of reflection and ties of power, and taking up arts and letters. ematical and he covered almost the entire action resulted in a series of textual and ma- Huygens too was an amateur in mathemat- range of the mathematical sciences: geom- terial works in which he ingeniously extend- ics; until the 19th century any true scientist etry, (hydro)statics, optics, astronomy, me-

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chanics, arithmetics and music. Furthermore, Lenses and telescopes rigorously elaborating the dioptrics of lens- and this is important to realise, he barely went In 1653 Huygens became ‘totally absorbed’ by es. Still, he selected mathematical topics beyond the domain of mathematics, except dioptrics, the mathematical analysis of lens- for their relevance to understanding the tele- for one important case that will be discussed es and their configurations.[17] Although the scope, leaving out sophisticated problems below. In other words Huygens was a full- sine law had been published by Descartes in like obliquely incident rays. blown, 17th-century mathematician. 1637, Huygens was the first to apply it to lens- After he finished his theory of the tele- Huygens’ first steps in the Republic of Let- es actually used in telescopes. Descartes had scope, Huygens turned to the practice of lens- ters earned him the epithet ‘little Archimedes’. only been interested in determining the shape es. He inquired about the grinding and pol- A discussion of projectile motion greatly im- of lenses with a perfect focus, finding them ishing of lenses and began manufacturing pressed Mersenne, who wrote to Huygens’ elliptic or hyperbolic. Those lenses, howev- telescopes with his brother Constantijn. In father saying that his boy would soon be- er, turned out to be very difficult to manufac- 1655 he discovered a satellite of Saturn with come a new Archimedes.[16] From then on, ture and telescopes were fitted with ordinary one of his telescopes, the first new body in the Constantijn would call Christiaan his little spherical lenses. In a study that would be solar system since Galileo’s famous discover- Archimedes. published only posthumously, Huygens em- ies in 1610. Soon after he was able to figure Huygens’ style was indeed Archimedean. ployed the sine law to derive the properties of out the cause of the strange shape of Saturn: He had a visual and tangible approach, in lenses in a rigorous fashion. Due to spherical the planet was surrounded by a flat, circular which he saw the ratios fundamental to clas- aberration the focus is not perfect and Huy- disc. These discoveries gained him fame, but sical geometry and favoured the kinematic gens defined it as the limit of intersections there is something odd about Huygens’ use of understanding of curves Van Schooten had of the axis. He then derived exact expres- the telescope. Apart from these and a couple taught him in his introduction to Descartes’ sions for the focal distance of any kind of lens: of other observations, Huygens never became new geometry. His theory of evolutes is based plano-convex, plano-concave, bi-convex, etc. a systematic astronomical observer. His inter- on the actual motion of a thread unwinding Only then did he show how these cumber- est in the telescope on the other hand was vir- from a curve. Archimedes was also promi- some expressions reduced to the simple lens tually unlimited. He endlessly manufactured nent in Huygens’ first publication in 1651, a formulas when the thickness of the lens is ne- lenses and telescopes, developing configu- treatise on quadrature in which he revealed glected. rations with excellent optical properties and mistakes in a recent treatise on the quadra- Huygens’ theory of the telescope reveals a inventing improvements like the diaphragm ture of the circle. He proposed an improved second feature of his mathematics: the rig- and a rudimentary micrometer. With instru- method and applied this to conic sections as orous elaboration of mathematical problem ments so central to Huygens’ mathematics it well. He showed how to limit the amount of focused on concrete topics. His oeuvre con- is remarkable how little interested he was in calculations for the method of exhaustion by sists of a couple of clusters of topics that are their actual employment, as though the tin- using the centre of of line segments. bound together by a marked interest in in- kering with them, both intellectually and ma- The visual and tangible style of Huygens’ ear- struments. This is not to say that material terially, sufficed. liest work would be a distinctive feature of his problems guided his mathematical studies. In the 1660s he continued his mathemat- mathematics. He valued the mathematics of things greatly, ical analysis of lenses by elaborating on a theory of spherical aberration, relating the amount of aberration to the properties of a lens. He then managed to determine a con- figuration of two lenses that mutually can- celled out their spherical aberration. ‘Eure- ka’, Huygens exclaimed on 1 February 1669 in Archimedean style — a perfect telescope made of ordinary, spherical lenses. Soon, the novice Newton would prove that his invention would never work in practice due to the na- ture of chromatic aberration. On 25 October 1672, Huygens crossed out his invention and wrote ‘useless’. This debacle is probably the reason he never published his dioptrics: a theory without an impressive invention was futile.[18]

Mechanics In mechanics things were different. Huygens gained fame with the invention in 1657 of the pendulum clock and this was a perfect stepping-stone for publishing his mathemat- ics. That winter he invented a mechanism to regulate a clock with a pendulum. Mechani- Left, Huygens’s design of the lens (1669); right: the word Eureka is crossed out. cal clocks were grossly inaccurate at that time.

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The motion of a pendulum was known to be very regular and used by astronomers and the like to measure time. The combination of these two elements yielded a clock that was accurate within a couple of seconds a day. In 1658 Huygens published Horologium, a de- scription of his clock, that contained an addi- tional invention. He hung the pendulum be- tween little ‘cheeks’, so that its length dimin- ished for larger displacements. A pendulum was, after all, not isochronous: the period in- creases slightly with the amplitude. Modify- ing the path of them both slightly would yield an isochronous pendulum and thus a per- fect clock. However, Huygens did not know the exact path required (or the shape of the cheeks) and for this he needed a thorough understanding of pendulum motion. Around the same time Huygens had be- gun to study free fall and in 1659, his annus mirabilis, he tackled a whole series of prob- lems relating to accelerated motion.[19] The problem was to determine g, the constant of gravitational , or in 17th-century phrasing: the distance an object in free fall covers in one second. Several experiments had been performed without a conclusive out- come. Huygens himself tried to measure the distance by letting a seconds pendulum hit the wall and having a hit the ground at the same time. It did not yield any con- vincing results so he turned to a mathemati- cal analysis of the situation. A pendulum is a body in and weight, he ar- gued, is identical to the tension exerted by the body on the cord. Comparing a horizontally moving body deflected by gravity (a parabol- ic path) and by a chord (a circular path) and noting the identical centres of curvature, Huy- gens derived a measure for centrifugal accel- eration equal to our modern mv2/r . He then devised a set-up in which gravitational and centrifugal acceleration are in balance: a pen- dulum rotating in a horizontal plane. Know- ing now how to calculate the centrifugal ac- celeration of this conical pendulum, Huygens could determine gravitational acceleration to be 9.79m/s2 (in modern terms) — quite accu- rate indeed! Huygens then attacked the problem of the isochronous pendulum and this shows all the features of his mathematical genius.[20] What is the ratio between an with a small amplitude EZ and free fall over the height of the pendulum TZ? Huygens com- pared the infinitesimals at E and B and con- structed the parabola ADΣ representing ve- locities along the path of the pendulum. He Figure 5 Top: the deduction of the isochronism of the cycloid (Codex Hugeniorum 26, f.72r.). Bottom: diagram of this fig- ure (construction taken from: Joella G. Yoder, Unrolling Time. Christiaan Huygens and the Mathematization of Nature (1989), then considered the to traverse the Cambridge, Cambridge University Press)

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infinitesimals and constructed curve LXN. chanics he explicitly rejected non-kinematic construct a wave refracted in Iceland crystal. Notice that the curves are constructed within concepts like force. And in optics too, he He had it propagate unequally in the crystal the diagram and directly represent the math- kept to the beaten track when he said that producing an elliptical instead of a circular ematics of the . The time to tra- the coloured fringes that disturbed his tele- wave and this did the trick. The tangent to verse arc KEZ is equal to the indented rectan- scopic images exceeded mathematical anal- these was parallel to the refracting gle between AZ and LXN, but this involved ysis.[21] Traditionally colours were not sub- surface but propagated somewhat askew and the uniform over AZ. To get around ject to mathematical analysis and Huygens thus the ray was refracted. ‘Eureka’, Huygens that, Huygens approximated arc KEZ with a was not someone to change that. Newton noted again and this time nobody would take parabola ZKℵ and determined the time anew. would turn the science of colours ‘mathemati- it away from him. He concluded that the time is independent of cal’ with his prism experiments. And he would Unwittingly, Huygens took a revolutionary the amplitude, and so he established isochro- turn mechanics into by introducing step when he tackled strange refraction. His nism. However, this applied only to small a new concept of force. Still, Huygens would principle of wave propagation made mathe- amplitudes. Huygens then showed his math- go on to make a revolutionary step beyond matical the properties of hypothetical enti- ematical ingenuity and his masterly eye in ge- the established borders of mathematical in- ties in the unobservable realm of the ether; ometry. He went through his diagram again quiry; however, he did not truly realise he he had transgressed the borders of mathe- and found the solution by recognizing the was doing so. In the , Huygens tack- matics, for mathematics did not treat the un- properties of the curve needed. The radius led an intriguing optical phenomenon: the observable motions of unobservable matter TE is perpendicular to the circle ZEK but not strange refraction of Iceland crystal.[22] This that explained natural phenomena. Huygens to the parabola ZKℵ he had used as an ap- crystal displays a double refraction and one thus transferred mathematical inquiry to the proximation. He needed a new curve that did of these does not follow the sine law of re- domain of natural philosophy and he was the preserve the conditions of his derivation. And fraction: a perpendicular ray is broken and first to do so. Yet to Huygens this went without then, out of the blue, Huygens said: “I saw an oblique ray passes unrefracted. Huygens saying: of course the motions of hypothetical this was the cycloid because of the familiar was by then convinced that light consists of waves in the ether ought to be discussed in method to draw its tangent”. Thus he found waves in a material ether but he could not the same exact manner as the motions of rays, out that an isochronous pendulum should fol- figure out how a wave traversing a refracting pendula and billiard balls. Bear in mind that low a cycloidal path. And it turns out that the surface in a straight direction was diverted to the principle came from mathematics rather involute of a cycloid is also a cycloid. So the produce a refracted ray. The flash of genius than physics. Huygens devised a way to ge- cheeks of his clocks should be cycloids. In came in August 1677 and the way it came cor- ometrically construct caustics and strangely 1673 Huygens published these results in his roborates the thorough mathematical charac- refracted rays; he did not consider the nature magnum opus , in ter of Huygens’ work. He was working on a so- of light waves at this point. He would do so which the mathematics of motion was pre- phisticated dioptrics topic that also seemed only afterwards when preparing a presenta- sented with the construction of his ingenious to defy his understanding of waves: caustics, tion of his account for the Académie, which invention. the bright curves produced when light passes he eventually published in 1690 as Traité de through lenses and other curved bodies (like la Lumière. It presents the principle of wave The Huygens principle a glass of water in the sun). Trying to con- propagation as a much better means of de- Huygens’ work on the of fall, circu- struct geometrically a wave propagating after riving the laws of optics and as a unique way lar and pendulum motion has all the charac- traversing the curved boundary, Huygens hit of enabling the penetration of strange refrac- teristics of his mathematics: exact, rigorous, upon an ingenious idea that we now know as tions. But it does not present it as we see it: a visual and concrete. It was a brilliant speci- his principle of wave propagation. At this time revolutionary transformation of mathematics men of the new Galilean science of motion. It it was only a tiny sketch jotted down to note (and natural philosophy). If Huygens was a was not revolutionary, though. Huygens did the method he used in constructing caustics revolutionary, he was sleepwalking. not break new grounds but stayed well with- by means of tangents to wavelets. And then Huygens’ wave theory was part of his in the established domains of 17th century to tackle strange refraction, without pausing dioptrics. Although he never published his mathematics. He consciously did so. In me- to elaborate on his idea, Huygens went on to theory of the telescope, he cut the umbili-

Figure 6 Strange refraction. On the left, Huygens’s sketch of this phenomenon (1672); middle: his sketch of the ‘Huygens principle’ (1677); right: the solution to the problem

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Dijksterhuis The golden age of mathematics. Stevin, Huygens and the Dutch republic NAW 5/9 nr. 2 June 2008 107

cal cord only at the very last. Until 1690, he as an amateur pastime. He could be called a times were changing and so were their math- planned a book on optics that included both, mathematical virtuoso but Huygens may have ematics. Early modern mathematics was a eventually deciding to publish his wave theo- been somewhat ill at ease with such a qual- much different affair to mathematics today. ry separately [23]. Thus this cluster of activ- ification. He saw himself as a physicomath- Considering the pursuits of Stevin and Huy- ities was pulled apart, obscuring the coher- ematician like Galileo; consider the evalua- gens from this historical perspective gives ence of telescopes and waves. Light waves tion of 17th century savants he wrote at the both an insight into their achievements and were part of Huygens’ mathematical consid- end of his life [24]. His father would surely reinforces their ingenuity. Such are the fruits eration of telescopes that had first produced have balked at the idea. He once received of recent developments in the historiogra- his dioptrical analysis of lenses. a letter that praised his ‘mathématicien’ son. phy of mathematics. Inquiry into the cul- In Huygens we see a similar combination Constantijn replied irritably that he did not ture of early modern mathematics raises excit- of ‘spiegheling en daet’ as in Stevin. Yet the know that he had craftsmen in his family. A ing questions, for example on the role math- nature and the context of his mathematics learned mathematician was a ‘géomètre’ af- ematicians played in the formation of the was entirely different. Huygens was a ‘ver- ter all. For him, Christiaan always remained Dutch republic and the way patrician ama- nufteling’ like Stevin, but in the learned world his little Archimedes. teurs paved the way for the mathematization of instruments, academies and letters, rather of natural philosophy. It promises to yield a than the artisan’s world of mills, bookkeeping Conclusion better understanding of the way mathemati- and fortifications. Huygens tinkered with in- Stevin and Huygens were children of their cians began shaping our modern world. k struments for the sake of tinkering with them, times and so were their mathematics. The

Referenties 1 Apart from my own work, some examples Stevin en de financiële wereld’, pp. 73–81 in Si- er of Holland and Zeeland in 1585, at his 18th are: Peter Dear, Discipline and Experience. The mon Stevin. birthday. Although he used the title ‘Prince of Mathematical Way in the Scientific Revolu- Orange’ from 1584, he officially became prince 5 Dijksterhuis, E. J., Simon Stevin. Science in The tion (Chicago, 1995); Andrew Warwick, Mas- only in 1618. Netherlands around 1600, Den Haag, 1970, pp. ters of Theory. Cambridge and the Rise of 93–98. 12 Berghe, Simon Stevin, 23. (Chicago, 2003); Er- ic Ash, Power, Knowledge, and Expertise in 6 In the Wisconstighe Ghedachtenissen Stevin 13 Heuvel, Charles van den, ‘Wisconstighe Ghedacht- Elizabethan (Baltimore, 2004); Wolf- later elaborated his ideas about the ‘mixing enissen. Maurits over de kunsten en weten- gang Lefèvre (ed.), Picturing Machines. 1400- of reflection and action’ (part of the introduc- schappen in het werk van Stevin.’, pp. 107– 1700 (Boston, 2004); Volker Remmert, Wid- tion of the Eertclootschrift in the Weereltschrift, 121 in Kees Zandvliet, Maurits Prins van Oranje mung, Welterklärung und Wissenschaftslegit- part 1 of the Memoirs). “Spiegheling is een ver- (Zwolle, 2000). imierung: Titelbilder und ihre Funktionen in dochten handel sonder natuerlicke stof, ghe- 14 See: Dijksterhuis, F. J., ‘Duytsche Mathematique der Wissenschaftlichen Revolution (Wiesbaden, lijck onder anderen sijn de Spieghelinghen and the building of a new society: pursuits of 2005); Matthew Jones, The Good Life in the Sci- des Spieghelaers Euclides, handelende deur mathematics in the seventeenth-century Dutch entific Revolution. Descartes, Pascal, Leibniz, stelling (per hypotesim) van grootheden en republic’ in L. Cormack (ed.), Mathematical and the Cultivation of Virtue (Chicago, 2006). ghetalen, maer elck ghescheyden van natuer- Practitioners and the Transformation of Natural licke stof. Daet is een handel die wesentlick met 2 “With regard to their Object, Mathematics are Knowledge in early modern Europe (forthcom- natuerlicke stof gheschiet, als lant en wallen divided into pure or abstract; and mix’d. Pure ing). meten, de menichte der roen of voeten tellen Mathematics consider Quantity, abstractedly; dieder in sijn, en dierghelijcke. T’ besluyt vande 15 A good, concise biography in Dutch is Vermij, R., and without any relation to Matter: Mix’d Math- voorstellen der Spieghelingh is volcommen, Christiaan Huygens. De Mathematisering van ematics consider Quantity as subsisting in ma- maer der daet onvolcommen: ...” de Werkelijkheid, Diemen, 2004. A highly read- terial Beings, and as continually interwove.” ible, although somewhat fictitious biography Chambers, Ephraim, Cyclopædia, or, ‘An uni- 7 Weeghdaet, 3. “Alsoo is de spiegheling (theo- in English is Andriesse, C. D., Huygens. The versal dictionary of arts and sciences : contain- ria) inde beghinselen der consten verloren ar- Man behind the Principle, Cambridge, 2005. ing the definitions of the terms, and accounts beydt, daer t’einde totte deat (effectum) niet en Vermij does not go into the mathematics while of the things signify’d thereby, in the sever- strect.” Andriesse does in enlightening and inspiring al arts, both liberal and mechanical, and the 8 Weeghconst, 1. ways. several sciences, human and divine : the fig- ures, kinds, properties, productions, prepara- 9 “Mathematics are distinguish’d with regard to 16 Huygens, Chr., Oeuvres Complètes de Christi- tions, and uses, of things natural and artificial : their End, into Speculative, which rest in the aan Huygens. 22 vols, The Hague, 1888-1950: the rise, progress, and state of things ecclesi- bare Contemplation of the Properties of Things; vol. 1, 47. astical, civil, military, and commercial : with the and Practical, which apply the Knowledge of 17 Dijksterhuis, F. J., Lenses and Waves. Christi- several systems, sects, opinions, &c : among those Properties to some Uses in Life.” Cham- aan Huygens and the Mathematical Science of philosophers, divines, mathematicians, physi- bers, Cyclopædia, 509. Optics in the Seventeenth Century, , cians, antiquaries, critics, &c : the whole intend- 10 Molhuysen, Bronnen, 390*. “Het meten des 2004, pp. 11–24. ed as a course of ancient and modern learning’. rondts mette gedeelten van dien aengaende, The Second Volume (1728), 509. 18 Dijksterhuis, Lenses and Waves, pp. 67–91. voerts het vlack des cloots, de formen genaemt 3 Berghe, Guido Vanden, ‘Simon Stevin. Een lev- ellipsis, parabola, hyperbole ende diergelijcke, 19 Yoder, J. G., Unrolling Time. Christiaan Huygens en in de schaduw van de macht’, pp. 19–25 in dat en is hyer nyet nodich, wantet den inge- and the Mathematization of Nature, Cambridge, H. Elkadem (red.), Simon Stevin 1548-1620. De nieurs seer selden te voeren compt, sulcke 1989, pp. 9–32. geboorte van de nieuwe wetenschap. metinge te moeten doen; maer alleenlyck sul- 20 Yoder, Unrolling Time, pp. 44–63. lense leeren met rechtlinige platten, daer na 4 Tafelen was published by Plantijn in Antwerp cromlinige landtmetersche wijse, metende al- 21 Huygens, Oeuvres Complètes, vol. 17, p. 359. (dedicated to the aldermen of Leiden), Nieuwe soe een plat deur versceyde verdeelinge, als in Inventie by Hendricsz in Delft (dedicated to 22 Dijksterhuis, Lenses and Waves, pp. 169–184. dryehoucken of ander platten om te syen hoe the Amsterdam burgomasters). Nieuwe Inven- t’een besluyt met het ander overcompt.” 23 Dijksterhuis, Lenses and Waves, pp. 219–223. tie has only been recently discovered and now turns out to be Stevin’s first publication. Heir- 11 After his father Willem van Oranje had been as- 24 Huygens, Oeuvres Complètes, vol. 10, pp. 399– wegh, Jean-Jacques, and Frédéric Métin, ‘Simon sassinated in 1584, Maurits became Stadhold- 406.

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