1 1 100 NAW 5/9 nr. 2 June 2008 The golden age of mathematics. Stevin, Huygens and the Dutch republic 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 power 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 work 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 forces 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 Christiaan Huygens (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, optics, 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 1 1 2 2 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 Dutch Golden age 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 interests of his new country, of machines 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 couple the elements of weighing was appended with years in Holland 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 time, 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 interest 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 Archimedes 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 hydrostatics 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 Leiden university 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 motion 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.