Opkomst van de propaedeutische functie van de wiskunde in Nederland, 1750-1850

Proefschrift

ter verkrijging van de graad van doctor
aan de Katholieke Universiteit Nijmegen,
openbaar te verdedigen op 3 juli 2003
des namiddags om 3.30 uur precies
door

Danny Beckers

The decades around 1800 were the time of reformulating and rigorizing geometry, algebra, and —last but certainly not least— analysis. No Dutch contributions from those days are still relevant today. However, without the hindsight of today, the Netherlands were actively involved in developing mathematics. Dutch mathematicians were productive in fundamental subjects: finding suitable definitions and axioms to prove mathematical theorems. Furthermore, Dutch mathematicians were actively involved in getting mathematics on the curricula of primary, secondary and tertiary education. It turns out that 1750–1850 was quite a crucial time for the development of mathematics in the Netherlands. It was the time that a propaedeutic function for mathematics was succesfully claimed: mathematics would be a suitable way to train the mind, and would even provide a way to become a good citizen. Some of the main characters of that time (De Gelder, Lobatto, Van Swinden) were still held in esteem at the beginning of the twentieth century. This suggests that the work of these people ought to be taken seriously. In the present study I propose to consider the work of the Dutch mathematical community as a mathematical culture in its own right. To investigate this community properly it seems a good idea to start looking at the activities of people calling themselves mathematicians (and generally regarded as such), and describe what they thought mathematics was about. Subsequently, the (changing) role of this mathematics in education, (popular) culture and society can be researched.

In chapter two I describe what mathematics was according to eighteenth century practitioners. It turns out there were two mainstream views of mathematics, both holding their own ideas about definitions, proofs and what mathematics was about. One of them I call “academic mathematics”, the other “middle-class mathematics”. The distinction can be made on the basis of text analysis.

Middle class mathematicians were not very strict in their definitions and axioms. For example, a popular ‘definition’ of an ellipse was by means of a smooth connection of circle segments, according to a widely-used technique in building works. Furthermore a definition was not something that was referred to in the proof. Definitions served as a kind of specialist vocabulary. Mathematical reasoning could, for example, contain empirical or metaphorical elements. Surveying, bookkeeping and such professions were, without question, considered mathematical, but there was a vague notion of a common (arithmetical / geometrical) background that was held in esteem. In accordance with this esteem, the group of middle class mathematicians showed an interest, although immature, in foundational subjects.

The academic mathematicians taught, or at least knew, the works of Euclid and Apollonius. They regarded The Elements as a perfect example of sound reasoning. However, they did not attempt to cast algebra or calculus in a similar conceptual framework of axioms and definitions. On the contrary, there was hardly any interest in foundational matters. Rather, the academic mathematicians distinguished between two kinds of mathematics: the one they called Mathesis Pura and the one they called Mathesis Applicata. The Mathesis Pura was a tool for use in the latter kind of math, and was, although considered with due respect, not really a subject for research. Subjects in the field of Mathesis Applicata that were studied intensively were fortress- and dike-building. These subjects were considered as mathematics, mixed with knowledge of an empirical nature. Unlike the point of view taken by the middle class mathematicians, these empirical elements were considered as the ‘weak spots’ in the (essentially) mathematical theory.

The academic and middle class mathematicians started evolving towards one another by the end of the Eighteenth Century. The two groups experienced a fruitful symbiosis within the Dutch Mathematical Society. The result of this symbiosis was what one might call a mathematical community. Although there was no community of people doing research in the Mathesis Pura as a profession and the professional interests of the members of the Mathematical Society were rather diverse, a small group of academically oriented mathematicians managed to direct the Society towards a common goal. These common goals included the promotion of a clear understanding of elementary subjects in mathematics by all civilized (Dutch) people. Mathematics, in the early Nineteenth Century, meant foremost what of old had been the field of the Mathesis Pura, with the notable extension of the definition–axiom–theorem–proof approach to all mathematical subjects (algebra, analysis, arithmetic). Bookkeeping, fortress building and hydraulics came to be regarded as subjects to which mathematics could be applied, but they were no longer mathematical in essence.

A clear understanding of mathematics meant that new textbooks were necessary, since the eighteenth century textbooks on algebra and calculus were aimed at learning techniques, rather than understanding. To attain the newly set goal mathematicians promoted rigorizing mathematics. For inspiration they looked abroad, to people like Lagrange, Lacroix, Legendre and Gauss. Dutch mathematicians in the early Nineteenth Century realised that this new way of viewing mathematics fitted their purposes better. Contrasting the views of Gauss and Lacroix, they were of the opinion that some form of metaphor could be allowed in proofs, since mathematics remained a sensible abstraction of (some aspect of) reality, of the world surrounding us. Axioms denying that, or definitions not in that vein were not at all fashionable. Of course spherical and projective geometry existed, but these were treated as extensions of plane geometry. A spherical triangle, for example, was not considered to be a manifestation of a more general concept of a triangle, from which one could start building one’s geometry, but rather as an abnormality that vaguely resembled the original thing, and therefore was named after its Euclidean predecessor —the plane triangle preceding in both the mathematical and the paedogigical sense.

The new mathematical élite since 1815 established within the newly founded faculties of mathematics and physics (counting four or five full professors, including one for biology, one for experimental physics, and one for applied mathematics) at the Dutch universities, and within the Koninklijk Instituut (Royal Institute). The latter institute was founded by Louis Napoleon in 1808 as a Dutch Académie des Sciences. Under the reign of William I this institute continued to publish a journal and a series of books on various subjects. Among other things, the Institute helped setting a generally accepted level of mathematics. The mathematics it promoted was the new kind of mathematics also promoted by the managing élite of the Mathematical Society. This illustrates not only that the same people were involved in both institutes, but also that there was a commonly felt desire to view mathematics differently from what it was from the eighteenth century perspective.

Although university professors were hardly unanimous about the way mathematics should be taught, there were similarities. Roughly two schools may be distinguished, exemplified by the Leyden professor Jacob de Gelder, and the Utrecht professor Johan Schröder. De Gelder had begun his career as a mathematics teacher and engineer, and was one of the driving forces of the Dutch Mathematical Society. Schröder was a pupil of the Heidelberg professor Jacob Fries. Like his teacher, Schröder was a neo-Kantian philosopher which implied that he regarded mathematics as a rather strict way of doing philosophy. The aim of his mathematics teaching was (apart from creating a toolkit for physical research) getting a clear philosophical foundation of his mathematical intuition. This opinion made him clash with De Gelder, who regarded mathematics as a perfect way of sound reasoning, and the only way of obtaining certain knowledge. According to De Gelder, Schröder allowed himself too much freedom in choosing his axioms. Schröder, attempting to rigorize elementary geometry, started by scrutinizing the essential objects of geometry, and concluded that in essence it all came down to points, distances and directions. He claimed that geometry could only be rigorized if the entire geometrical opus was recast in these concepts. De Gelder, in contrast, very much convinced of the eternal truth that mathematics provided, was of the opinion that one needed a very good reason to diverge so drastically from Euclid. He had done so too, most notably he had actually put a definition of an angle into his geometry, but not without showing that it was essentially just a refinement of what Euclid had intended —more productive in proofs.

In chapter three the developments in mathematics education are discussed. In the eighteenth century Dutch republic, schools fell under local jurisdiction. It is noted that mathematics played hardly a role in the curricula of the Latin schools (preparing for university) and the schools for the lower classes (where one learned to read). Mathematics was taught at the so-called French schools: schools for the upper middle classes. Like all other subjects mathematics was taught individually. Neither the school utilities (mostly the teacher was the only one possessing a textbook) nor the practice of school attendance (children helped getting the crops in) allowed for another way of teaching.

The middle class tradition in mathematics was responsible for a number of textbook publications. As is to be expected these were textbooks destined for vocational training. Most of them are reprints of popular arithmetics, but also bookkeeping and surveying (and the geometrical, algebraical or arithmetical backgrounds) were popular subjects. These books were based on a rote learning didactics. Notably the rule of threes was presented in various guises, to prevent that the jeweller’s son would be confronted with exercises that only needed to be brought to the attention of a carpenter’s or merchant’s son.

University teachers either used printed Latin or Greek edition of The Elements, or taught from their own manuscripts. All students read Euclid I-VI, XI and XII. Most professors also taught some elementary algebra. Other courses that were quite common at Dutch universities were calculus and conics. Sometimes even subjects from the Mathesis Applicata were taught, but only few students attended these courses.

When the political circumstances in the Republic forced the stadholder to leave the country, things started changing drastically. The institutional framework of higher and lower education, educating for the higher resp. lower social classes, remained in existence (also afterwards, during the reign of William I), but the local control over schools and curricula was discontinued during the time of the Batavian Republic. A system of (central) state control was established to guarantee the quality of teaching in all primary and secondary schools. All schools were visited by inspectors, and except for the Latin schools, fell under direct supervision of these departmental schoolopzieners. Teaching ideally took place in a class, no longer individually —thus also stressing the pupils’ equality. Assessment of both teachers and curriculum became customary. Teachers had to earn degrees and were stimulated to keep educating themselves —most notably in mathematics. They were stimulated to do so in local teacher societies that were (if not arising spontaneously) called to life by the schoolopzieners. Primary schools were obliged to teach several new subjects, such as writing, geography and national history. Among these new subjects there was also reckoning, and some of these schools even went as far as elementary geometry.

In primary education it was the Dutch Society for the Common Good (’t Nut) that triggered changes. Founded in 1786, this Society favoured good primary education for the lower classes in order to prevent poverty. Better educated people would provide better for themselves, and even more important, would actually make better citizens. Education was believed to bring also moral advancement, for example, because the educated would be able to see the justice behind their humble state, and would seek their diversions in science or reading, rather than gambling or drinking. Furthermore, there would perhaps be a few enlightened spirits, who would be able to learn more, thus actually improving their state, and contribute to the wellfare of the nation. The Society was soon in close contact with government officials, and helped in realizing the new primary schools. Meanwhile, its ideals also helped reshaping other educational institutions, such as the French schools.

William I continued the educational policy which had been started during the years of the revolution. He saw how it could help him build a nation from the bundle of provinces in economic debris he had inherited. William I even extended the educational policy of his predecessors to the level of higher education. In 1815 Latin schools were obliged to teach mathematics, and university students were obliged to take exams in mathematics, to a level much beyond what had been customary in the Eighteenth Century.

New ways of teaching mathematics came into being. In accordance with changing views on mathematics, teachers began to think that certainly the well-educated should have a solid knowledge of mathematics, since it would stimulate their (thinking) capabilities. In the most exalted view a good citizen was not complete as long as he had not completed his (pure) mathematical training. In the more down-to-earth version mathematics was considered to be a good way of testing a pupil’s ability to reason. In both cases, mathematics became a prerequisite to almost every profession —mathematics in its propaedeutic function. The fact that both D. J. van Ewijck, Minister of Education (1824–1831) and H. Wijnbeek, national school inspector, were very much in favour of a propaedeutic function for mathematics did help to make the legislation (aiming for mathematics to become fruitful as a propaedeutic discipline) effective.

Three didactical approaches became customary during the early Nineteenth Century. The mathematics textbooks by the aforementioned De Gelder (and similar ones) were popular at the Latin schools and universities. Some of the French schools also used them. De Gelder attempted to teach his pupils all ins and outs of pure mathematics. He built on axioms and definitions, providing examples and counter-examples to give the pupil the opportunity to get into the subject. What ensued was a coherent and well thought-through book of theorems and problems with proofs. De Gelder always told his pupil which things were necessarily so, and which were the result of deliberate choices. For example, in his arithmetic textbook he told the pupil, when teaching him the decimal system of writing numbers, that choosing a fixed base for one’s number system was a choice that made reading and reckoning easier. The choice for base 10 was circumstantial, etcetera. He also showed several of the alternatives, partly in the appendices. In the exercises the pupil was shown that all the theory he had learned was applicable in all kinds of situations, varying from merchant situations to engineering practice, from physics experiments to the everyday practice of the carpenter or jeweler.

Closest to the eighteenth century didactics of rote learning were the books by teachers like H.G. Witlage. They regarded mathematics as an important subject as well, but more from the way it was used in the profession. These textbook authors wanted their pupils first of all to become good bookkeepers or merchants. Of course mathematics also had the formative aspect that was promoted by De Gelder, but the effect of this was restricted to the way textbooks were organized. Regularly, exercises were presented in a random order, which confronted the pupil with the additional problem of choosing which rule fitted the question best. But the rules were not explained, and there were no proofs.

A third didactical approach was presented by teachers at French schools, such as P.J. Baudet. These teachers (and textbook authors) strove for a practical purpose, but also wanted their pupils to catch the idea of implication and proof. Instead of striving for rigorous mathematics, they relied on the common sense of the pupil. Also, they were more practically oriented. This is illustrated, for example, by the fact that they did not mention alternative ways of representing numbers, but simply showed how to write them —without hinting at possible alternatives.

William I confirmed the abolishment of the guilds (and thereby the manufactures training) in 1818. Unlike the situation in other forms of education, the government did virtually nothing to stimulate the propaedeutic function of mathematics in the branch of technical education (the exception to this rule being, of course, military education, where the government stimulated a curriculum resembling that of the French écoles militaires). Nevertheless, several textbooks that were published in the early Nineteenth Century, show that at least mathematicians bothered about the propaedeutic function of mathematics for engineers.

In 1842, when the government was convinced that the laisser faire policy was not very fruitful, and several attempts to found technical education had failed, it was decided that a polytechnical institute was to be founded in Delft. It did not receive generous financial support, but mathematics played a major role in the education of the Delft engineers. Professors like R. Lobatto, a very capable mathematician who had been a government advisor for years, guaranteed that (the ‘Dutch’ view on) pure mathematics was the prerequisite for all the Delft engineers. Although financial troubles marked the first decades of the new academy, the Dutch government did recognize the quality of its engineers. Within a few decades all government positions in engineering required a diploma from Delft.

Chapter four continues the story by looking at the role of mathematics in Dutch popular culture. First of all it is noted that around 1800 the Netherlands were in a rather poor state. Both economically and militarily the Netherlands were no longer the leading nation, as they had been during their Golden Age. Dutch popular culture attributed this decline to the loss of certain ‘Dutch’ values, which, once revived and restored, would lead to prosperity. The size of the country, with respect to its much bigger and more modernly governed neighbours, was regarded as an advantage, rather than a disadvantage.

Attempts to perfect the Dutch society were taken from within the various (learned) Societies that were founded during the Eighteenth Century. Also the popular press was active in printing ideas about ways to revive the country. This accounts for the two particular fields which are examined in this chapter: the role of mathematics in popular journals and the role of mathematics within Society life. Two of the most important topics in attempts to revive the country were religion and nation. The protestant religion was considered the most important Dutch asset. Without religion Society could not be perfected. The greatest common denominator of the values advocated by the various protestant churches was emphasized. Nationalism was rather new in the Eighteenth Century. It was certainly promoted by William I, who needed to unify his new state. For this purpose the Glorious Seventeenth Century was a very welcome starting point.

The Eighteenth Century witnessed the emergence of a new type of journal, adressing the well-educated and broadly-interested public, publishing book reviews and news of a scientific or polemic nature, ranging from papers on theology, law, history, astronomy and mathematics, to discussion on the current state of affairs in the Netherlands or abroad. The two most succesful (in duration and circulation) were the Vaderlandsche Letteroefeningen (since 1761) and the Algemeene Konst- en Letterbode (since 1794). Several of these journals paid attention to mathematics. A peculiar change occurred here around 1800. Wereas journals at first more or less tried to keep in touch with mathematical developments, this changed in the early nineteenth century. New developments in mathematics were no longer published. The interested reader was expected to buy more specialized mathematical journals. After 1800 mathematics remained present in the quires of the journals, but now in another guise. Textbooks were reviewed (sometimes in considerable detail), and popular papers on mathematics, as well as recreational mathematics were to be found. Also the discussions on the role of mathematics in education took place within the pages of these journals. In this way, journals like the Algemeene Konst- en Letterbode did play their part in the dissemination of the propaedeutic function of mathematics.

During the Eighteenth Century mathematics was a more or less obvious topic of interest within the learned societies. Usually the focus was on the Mathesis Apllicata, but some subjects from the Mathesis Pura occasionally found their way to the journals of the learned societies. Curiously, after 1800, the learned societies showed hardly any interest in pure mathematics, although the university professors of mathematics were present within these organisations. I don’t try to give an explanation for this phenomenon, but notice that the scientific climate within the learned societies regarding the new, purified form of mathematics was kind of old-fashioned. As a tentative conclusion one could say that mathematicians attended because they wanted to be a part of the established academical scene.

The Society for the Common Good played an important role in developments in education. For the sake of good arithmetic education they published a textbook. There were more attempts to publish mathematics textbooks. Since the Society was depending on the readiness of the public to hand in drafts for textbooks in a prize competition, these projects never bore fruit, with the notable exception of a geometry book for the common man, published in 1824. Many textbooks on arithmetic, algebra and geometry were published that would have fitted the ideals of the Society —containing, for example, exercises that demonstrated the foolishness of drinking, spending money mindlessly, etcetera. It probably was more lucrative to publish a textbook privately, than to submit it to the Society, running the risk to be turned down.

Most initiatives for the founding of mathematical institutes to educate the children of the local well-to-do, were taken from within the Society for the Common Good. Also, more often than not, local departments of the Society supported these institutes, either financially or by giving weight to its activities, for example by letting the board attend the annual festivities of such an institute.

The final Society that I consider in detail is the Koninklijk Instituut (Royal Institute). Founded by Louis Napoleon in 1808, it had never offered real job opportunities, but it did grant a certain esteem to its members. Among the members of the first department of the Institute (the department of mathematics and physics) were several renowned mathematicians. Although both Louis Napoleon and William I regarded the first department of the Institute as think-tank in technical matters, its journal published quite a number of papers on pure mathematical subjects. Also a series of treatises was published, in which, for example, a book on the foundation of negative numbers by the aforementioned Jacob de Gelder. Even though the Royal Institute did not stimulate professionalisation of mathematics, it did induce a norm for pure mathematics in the Netherlands.

Chapter five is concerned with the role of mathematics in society. For this purpose there are three focal points. First, the role of statistics in society. Second, the role of mathematics within the technical branches of society. Third, the introduction of the metric system of measures and weights.

1. Until the end of the Nineteenth Century the word ‘statistics’ had two meanings. First, it was the quantitative description of a State. In this sense the statistics was never complete unless accompanied by a qualitative judgment of the author. The second meaning was that of error estimating or error calculations. In this field Laplace in 1812 wrote the paradigm text that would determine the view on the subject for the next century. Laplace regarded a series of measurements as the result of a probability distribution centered around the actual value (the mean). Thus the theory of probability could be applied to measurement errors. Dutch mathematicians knew of these developments, so application of statistics was to be expected. In the fields where mathematicians had found work of old (e.g. surveying) the applications easily found their way. Not so straightforward was the role of statistics in two newish areas of society, and these two are examined here (other fields would have been possible candidates as well, but have been left out of this study). First in the way it was used within the insurance business and second in the way it was used in the rising field of economy.

Insurance had a long history in the Netherlands. Ships and windmills were insured for damage, either by spreading the risk, or by contracts that would guarantee financial backup. Annuities had been sold by the Dutch municipalities since the Sixteenth Century, in cases where quick money was needed. The guilds and village communities, ‘insured’ each member in their midst of their way of life. Insurance calculations were rare, although the mathematical theory had been developed. Partly, this may be explained by the informality of the insurance system and the absence of life tables, but mostly by the fact that people viewed the events that were insured (life, loss of goods etcetera) as the course of fate. Buying an insurance was therefore more like buying a lottery ticket than buying security.

This way of thinking changed in the early Nineteenth Century. Well-educated and enlightened directors of the Hollandsche Sociëteit attracted a mathematician as an expert to carry out the annuity calculations. They viewed insurance as a way to back up the striving of the Society for the Common Good to make the lower classes morally better. These views were not shared by everybody, and many insurance companies simply offered a competitive premium. It was the mathematician R. Lobatto who advocated more government surveillance on insurance companies. According to Lobatto good (life) insurances would help prevent people from falling into poverty, and the regular payments would be benificiary to the moral advancement of the nation (i.e. it would keep people from the booze). William I in 1833 issued a law —becoming effective in 1840— which obliged insurance companies to have their calculations approved. Although some of the companies were not eager to cooperate in making their data known, in the end they agreed. Thus mathematical expertise became a common thing in insurance matters.

In the field of economics (Staathuishoudkunde) the application of statistics took quite another direction. Although the initiative for starting the publication of government statistics was taken in 1826 by a mathematician (R. Lobatto), after the democratic revolution of 1848 economists (members of the liberal party which had been ignored by king William) took over his job, and established an exclusive ‘description of state’ kind of government statistics.

2. During the Eighteenth Century Dutch trade and industry were in a crisis. Initiatives from within learned Societies were taken to counter this situation. Most notably was the founding of the Oeconomische Tak in 1777. After its London predecessor, the Society for the Encouragement of Arts, Manufactures and Commerce, it used prizes to encourage new developments. No mathematical considerations seem to have played a role in this Society, but it purely restricted itself to economically sound initiatives. It even sponsored the —in mathematical sense— rather old-fashioned navigation school in Amsterdam.

During the reign of William I every well-educated civilian was confronted with mathematics. This is to be noted in textbooks on arts and commerce as well, since from the early Nineteenth Century onwards they contain mathematical notations and formulae (that are expected to be understandable). Some textbooks for artisans show that the authors expected even more mathematical skills from their readers. There was no explicit reason for promoting the propaedeutic function of mathematics. It was simply stated that more mathematical skills would be benificial to the number of inventions and their application. These promotors of a propaedeutic function for mathematics in the arts and manufacture sector did not get much influence in the training colleges. Even the Tijdschrift ter bevordering van Nijverheid, a Journal for Arts and Manufacture, sponsored heavily by the Dutch government, with a mathematician in its editorial committee, did not assume much mathematical skills among its readers.

This changed when the Delft Polytechnic was founded. The Delft engineers soon mingled with their military counterparts from Breda and together founded the Royal Institute for Engineers, issuing a journal from whose quires it is immediately clear that mathematics was an absolute prerequisite to an engineering position. The readers at least had to be able to understand mathematical argumentation or reconstruct some of the reckoning. Some of the mathematicians and engineers publishing in this journal even had the exalted vision of mathematics morally advancing society. One of the more illuminating exceptions was commercial training. Delft started off with a training college that never would florish, since less theoretical alternatives existed. Thus there was a considerable part of society that could (and would) do without the blessing of mathematics.

The Netherlands always had been engaged in water works. The mathematical expertise of surveyors, dike builders and other engineers always had been accepted naturally. When mathematics started changing its face, and turned into pure mathematics, this acceptation was no longer natural. In fact, mathematicians involved in the technical sector were, from the early Nineteenth Century onwards, often accused of being too theoretical-minded to be taken seriously —mostly when their expertise denied the opinion of the technician or banker.

Mathematical expertise, in the new meaning of the word, was not guaranteed in education —with the exception of military engineers. Still, textbooks for technical education show a steady increase in the readers’ ability to read mathematical formulae. In the Breda Military Academy (as in the Delft Polytechnic Institute from 1843 onwards) pure mathematics was the key to all success. All textbooks published for the Military Academy, such as the hydraulic engineering books, assumed that the reader was familiar with mathematical notation and (algebraic and analytical) reckoning techniques. That in itself did not justify an introduction to pure mathematics, but the other way around this introduction would make it easier for the technicians to learn these basic mathematical skills. It is striking, that the propaedeutic function of mathematics was best received within those institutions that trained for government positions. The technical literature for railway engineers and the commercial sector, for example, hardly show any affinity for a mathematical language. Judging from textbooks, reading formulae seems to have been the most advanced mathematical skill these students encountered.

3. The metric system was introduced in the Netherlands during the time of the Batavian republic. Jan Hendrik van Swinden and Hendrik Aeneae were very much in favour of the new system of measures and weights. They had attended the French conference where the metric system had been developed. The system met with considerable support within the governing classes, not least since history could be read in such a way that several of the Dutch ‘scientists’ from the Golden Age (Stevin and Huygens, most notably), had been involved in developing the system. Their involvement, as well as the presence of Van Swinden and Aeneae was regularly used in propaganda for the metric system.

Mathematicians were involved in the introduction in three ways. First, they helped develop the system, by doing calculations and giving advice. Both the Koninklijke Akademie and several individuals were approached for their mathematical expertise. A whole new army of gaugers (inspectors of weights and measures) was called to life, and from the late 1810s onwards, put their mathematical and technical expertise in service of the Dutch nation. Secondly, mathematicians helped to teach the new subject. Numerous new textbooks were written or revised, and many mathematics teachers started teaching the new metric system. Thirdly, mathematicians helped with the propaganda that accompanied the introduction of the metric system. In this last role they were certainly not alone, but as textbook authors and teachers their role was invaluable. By taking upon themselves these three roles, mathematicians showed their usefulness, and the usefulness of their profession, to their contemporaries.

Finally, chapter six attempts to give an overview of the results. It is noted that no evidence for a Dutch national kind of mathematics is as yet found. The idea of ‘Dutch mathematics’ would certainly have been kindled by the mathematicians at the time, but there are a few reasons to believe that it would be more appropriate to see nationalistic tendencies at work. The ‘great’ mathematicians have been misunderstood, and their view on mathematics certainly was not universal in France, Germany or Britain. Quite a lot of contacts between mathematicians in the several countries existed, and there was also rather a lot of esteem for each others work. Furthermore, the ideas of linkage between the creation of God and mathematical reality was not exclusively Dutch (it existed in Britain as well), and was quite a logical way out of the tension between rigorising mathematics and romantic pietism.

History of mathematics education in the Netherlands shows a incredible rise of the propaedeutic function of mathematics. This seems part of a European tendency, but mathematics assumed its propaedeutic function easier and more quickly in those countries where the governments exercised a strong influence on the educational system. The rise of the propaedeutic function of mathematics, therefore, might be seen as the successful mix of belief in progress, educational ideas and the rise of modern nations.

With respect to the role of mathematics in Western culture, it is noted that, at least in the Netherlands, the fear for Spinozism accompagnied the rising importance of mathematics. This book carries as a title the exclamation of an anonymous author in 1842 who was annoyed that university students in his days had to take exams in mathematics. He, and several others, claimed that mathematical reasoning constituted a moral threat, and was better not given too much attention —unless as a necessary tool in technical studies. The sound of their voices faded in time, and mathematics claimed and achieved an important place in Dutch culture.