Keywords

1 Introduction

This paper is based on research for my doctoral dissertation, chapter two (Krüger 2014).

Around 1620 professor Frans van Schooten Sr. (1581–1645) composed his extensive lecture notes for the Engineering Course for surveyors and military engineers in Leiden. The course was offered by the University of Leiden, however the teaching language was Dutch, instead of Latin. At a time when Latin was the obligatory teaching language at all universities in Europe, a course in the vernacular was sufficiently exceptional to warrant the commonly used name ‘Duytsche Mathematique’ (Dutch Mathematics). Very soon the name ‘Duytsche Mathematique’ was used both for the course and its content, so the name had slightly different meanings, depending on the context in which it was used. The course had a well-defined formal aim, formulated by Simon Stevin as part of his teaching plan and written down in the Resolutions of the University governors.

Frans van Schooten taught the course for a longer time than either his predecessors or his successor did, from 1611 to 1645. Nevertheless, his name was not mentioned much after his passing away; he published very little, a trigonometric table for surveyors, which had several reprints and a translation of Euclid which was quickly forgotten. Both his predecessor and his successor remained well-known after their death. His predecessor was Ludolf van Ceulen , a well-known and highly respected reckonmaster and mathematician, who had published some mathematical works, some of which were still in use in the mid-eighteenth century. The successor of Frans van Schooten Sr . was Frans van Schooten Jr., his eldest son, at that time a mathematician of some fame, the center of a circle of Dutch students and mathematicians, amongst whom Christiaan Huygens. Frans Jr. published several mathematical works, in Latin and in Dutch, for mathematicians and practitioners; he was active in making Descartes’ work known, through translating Géométrie [Geometry] in Latin and through his private students. He was considered an expert in practical mathematics by mathematics users well into the eighteenth century. When the name Frans van Schooten is mentioned in literature, it is nearly always the name of his son. So was Frans van Schooten Sr. just an insignificant teacher, who dutifully implemented the teaching plan by Stevin, and continued the lectures in the same way as the respected professor van Ceulen, more of the same, until his brilliant son took over?

The question which provides a framework for this paper is

  • What significance had Frans van Schooten Sr. for the reputation of Duytsche Mathematique, the first mathematics course for surveyors and engineers in the Netherlands?

The answer to this question will help us to understand better an important period in the development of Dutch and European mathematics education. Fortunately, the lecture notes, written by Frans van Schooten around 1620, are preserved as manuscript BPL 1013Footnote 1 in the library of Leiden University. They are very extensive, beautifully illustrated and provide us with a unique insight in the planning of lessons for Duytsche Mathematique in the early 17th century. They represent van Schooten’s interpretation of the teaching plan by Stevin and so they tell us something about the content and also the style of teaching as planned by Frans van Schooten Sr. As these lecture notes were used by his successor as well, the discussion of the significance of Frans van Schooten Sr. as teacher of the Duytsche Mathematique will to a large extent be based on these lecture notes, however other sources will be taken into account as well. For practical reasons, only the parts on arithmetic and geometry will be used, the rather large part on fortification will be discussed at another occasion.

2 The Dutch Republic Around 1600

Since 1568 the Dutch were fighting an independence war with the king of Spain , a war which was mainly fought around the many small and larger towns in the Netherlands, both the Northern and the Southern Netherlands (at present Belgium). The regions which considered themselves independent of the Spanish king formed a federation of provinces, the Dutch Republic, with the States General, situated in The Hague as a government. Leiden University was established in 1575 by Prince Willem van Oranje [William of Orange], the leader of the rebellion against Spain. Prince Willem was assassinated in 1584; the following year his son, Prince Maurits (1567–1626), became stadtholder of Holland and Zeeland, at a rather young age. From 1590 Maurits was commander of the army of the Republic; he and his cousin Willem Lodewijk (re)captured a significant number of fortified towns. They also managed to hold on to most of these towns, through modernization of the fortifications. For this style of warfare well-trained surveyors and army engineers, who knew modern mathematical methods, using geometry and trigonometry, were indispensable.

Since the last decades of the 16th century, the conquests by the Spanish army of many towns caused a stream of migrants moving from the southern Netherlands to the north, often well-educated people and relatively wealthy. In the northern Netherlands in the 1590s the economic situation improved considerably: trade and navigation increased, and the population of towns, especially in the western part of the country, expanded considerably during the early years of the seventeenth century. This also led to a high demand for qualified surveyors and engineers, as land had to be measured and divided and the expanded towns must be fortified.

Italian architects and engineers were very influential in the Netherlands : from the mid-sixteenth century, Italian advisors were responsible for improving the defensive walls of towns. Their ideas also became widely known through publications of Italian mathematicians and of others, like Daniel Speckle and Simon Stevin . In the Stercktenbouwing [Fortification] (1594) Stevin used the Italian system, in which the preferred ground plan of a fortification was a regular polygon and the bastions were pentagonal.Footnote 2 Any surveyor and engineer who worked with the army had to have knowledge of geometry and also trigonometry, as a lot of the work was performed on inaccessible lands.Footnote 3 Thus there was a demand for surveyors and engineers, who were able to design and build modern fortifications, but who also could map the land and had other surveying skills.Footnote 4 The use of reliable trigonometric tables could improve the results of surveying techniques considerably; by the end of the 16th century surveyors had started to use plane trigonometry, for which they needed instruments to measure angles. The extensive calculations that were necessary could be simplified using decimal fractions, but these were not yet used by mathematical practitioners (Struik 1995). Thus in order to meet the demands of the developing nation there was a great need for good mathematical education.

In the Dutch Republic there was no centralized educational system. Schools were the responsibility of town councils; often there was a primary school, in which children could learn reading, writing and some arithmetic , depending on the skills of the teacher. However parents had to pay the schoolmaster a small sum for the teaching of each subject, arithmetic was the most expansive of these three. In addition there were private schools and private teachers of mathematics, many of whom had arrived from other countries or regions. There was no standard curriculum, nor any form of quality control (Krüger 2010). In 1600 there were two universities in the northern Netherlands, at Leiden and Franeker.

At Leiden University mathematics was taught from 1581 by Rudolphus Snellius (1546–1613) , who valued both pure and mixed mathematics. This distinction was common in the 17th century: ‘mathematica pura’ consisted of geometry and arithmetic, the abstract mathematics, without a relation to matter; ‘mathematica mixta’ consisted of those topics which had also a qualitative aspect, subsisting in matter. Some examples of mixed mathematics are navigation, astronomy, surveying, optics, mechanics, perspectives and geography Footnote 5(Mulder 1990; Alberts et al. 1999). Both pure and mixed mathematics could be studied for its own sake or could be used for practical purposes, by mathematical practitioners. The University of Franeker (Friesland) was established in 1585; from 1598 Adriaan Metius (1571–1635), a student of Snellius, taught pure and mixed mathematics.Footnote 6

Maurits had studied at Leiden University; in 1583 both he and Simon Stevin (1548–1620) were enrolled as students.Footnote 7 Stevin and Maurits probably both went to the mathematics courses by Snellius. Maurits was interested in mathematics and its applications and was well aware of its importance for modern warfare. Around 1593 he engaged the services of Simon Stevin as his personal tutor and as designer of the army’s quarters. In 1600 the course for surveyors and military engineers, the Duytsche Mathematique, was established at Leiden University on instigation of prince Maurits and Simon Stevin. The aim was to train engineers for the Dutch army thoroughly and quickly. That meant a solid mathematics background with the focus on practical mathematics.

3 Publications on Dutch Mathematics and Mathematics Education in the 17th Century

Mainly the recent and most important publications on the situation in the northern Netherlands are mentioned here. For more detailed information, older publications and articles see Krüger (2014).

Van Maanen (1987) provides an inventory and a systematic description of 17th century mathematical manuscripts in the library of Leiden University. His evidence made it clear that Frans van Schooten Sr . was indeed the author of BPL 1013, which until then was sometimes disputed. Boekholt and de Booy (1987) published on the history of the school in the Netherlands (including the teaching of arithmetic), from the Middle Ages until the 20th century. De Booy (1980) wrote in more detail on primary education in the towns of Utrecht from 1580 until the 19th century. Kool (1999) published research on Dutch arithmetic books from the 15th and 16th century. Those books could of course also be used for self-instruction.

Several authors have described the Duytsche Mathematique and the teaching plan by Stevin (Bierens de Haan 1878; Molhuysen 1913; Muller and Zandvliet 1987; Winter 1988). However, not much attention has been paid by these authors to the implementation of the teaching plan, the teachers or the characteristics of the teaching content.

Dopper (2014) published research on Frans van Schooten Jr. who, as the successor of his father, taught Duytsche Mathematique from 1645 until 1660. This publication also contains a systematic description, based on the method used by van Maanen (see van Maanen 1987), of the manuscripts in the Van Schooten collection, in the library of Groningen University. Moreover Dopper gives an overview of the teaching topics and their treatment by Frans van Schooten Jr.

Two more publications on mathematics teaching at the Dutch universities are relevant for this paper. De Wreede (2007) published research on Willebrord Snellius (1580–1626) , professor of mathematics at Leiden University from 1613 to 1626. Dijkstra (2012) describes the position of mathematics at the University of Franeker during the 17th century.

Van Gulik-Gulikers (2005) published research on mathematical practitioners in which she describes surveying instruments and practice of surveying in the 17th century. Sitters (2007) published his research on Sybrandt Hansz. Cardinaels, a teacher of mathematics and arithmetic , who published a popular collection of geometrical problems, Hondert geometrische question met hare solutien (Hundred geometrical questions with their solutions), around 1614.

4 The Duytsche Mathematique

In the Netherlands until well into the 19th century Latin was officially the language of the university courses. Thus, even if Snellius in Leiden and Metius in Franeker might occasionally offer lessons in Dutch, Latin was their official teaching language and the language of examinations. Consequently the Duytsche Mathematique could not be recognized as an academic course, the students who only came for Duytsche Mathematique were called ‘auditores’. However, the University appointed the professor, paid his salary, provided teaching locations and teaching materials, in short the University took responsibility for this mathematics course in vernacular. The choice for Leiden University as the place for this course was obvious. Not only did prince Maurits and Stevin both study at Leiden University; one of the curators, Jan Cornets de Groot, was a friend of Stevin. Another curator, Jan van Banchem , came from a family of surveyors and must have seen the value of a high level course for surveyors and engineers.

The Duytsche Mathematique was exceptional at a time when learning mathematics for professional purposes was mainly a matter of private instruction. The course was also unique because there was a formal plan for teaching, written by the mathematician and engineer Simon Stevin and usually named the Instructie [Instruction] (Krüger 2010, 2015). It was added as an addendum to the minutes of the meeting of the curators with prince Maurits and Stevin on 9 January 1600 (Molhuysen 1913). Stevin, who himself was a good mathematician, engineer and teacher, gave the aim of the course, an outline of the content, the order of the topics, what teaching activities he deemed necessary, i.e. practical work following theory, the teaching language etc. The Instruction formed the framework for the course; moreover it is the first example of such a formal plan for mathematics teaching in Dutch language.

4.1 The “Instruction”

The first paragraph of the Instruction,Footnote 8 which follows on a short introduction, states the intended aim and the global content of the course.

His Excellency has deemed it useful for the sake of the country and for those who will enroll to become an engineer, that there will be a plan for the teaching thereof which will take place at the Academy in Leiden, as follows.Footnote 9

  1. 1.

    The general opinion is that the auditors should be trained as quickly as possible to serve the country as engineers. To this goal they shall learn arithmetic or the counting and surveying, but of each of these only so much as is necessary for a practising ordinary engineer. Those who have come thus far are allowed to study more in depth if they wish to do so.

So the aim of the Engineering Course was to train the ‘auditors’ as fast as possible, in order that they could serve the country as engineers. In the Regional Archives in Leiden there is a manuscript, a copy of the Instruction, provided by the curators for the printer Jan Paedts Jacobsz. (Paedts 1600, RAL LB 36949). The first paragraph differs slightly from the same paragraph in the book of Molhuysen (1913). The most important difference is the expansion of the professions mentioned. Instead of “for those who will enroll to become an engineer,…” it has “for those […] who will become an engineer and other Mathematical Arts.”Footnote 10

The addition of “other Mathematical Arts” suggests that the curators preferred a broader aim than the training of engineers only. Wine gauger, schoolmaster and building master were examples of other “Mathematical Arts”.

Stevin was an advocate of the teaching of theory prior to the practical work, of the use of Dutch language instead of Latin, both in teaching and in mathematics and of the use of decimal notation by mathematical practitioners, as he wrote in his often translated De Thiende [The Tenth] (Stevin 1585) . All these characteristics are found in the Instruction, which consists of 13 short paragraphs, on content and other aspects of the course.

The second and third paragraphs of the Instruction are on arithmetic.

In the counting will be taught the four operations on whole numbers, on broken numbers and on decimal numbers, also the rule of three in each type of number.

When they have become skilled in this way of counting, they know sufficient arithmetic for ordinary engineering.

One could not be sure that the students would be sufficiently skilled in basic arithmetic, so Stevin included this in his teaching plan. He emphasized the teaching of decimal fractions (decimal numbers). At the same time he limited the extent of arithmetical techniques which should be taught. The use of decimal fractions, in the notation used by Stevin, was not at all common at the time. In manuscripts and publications, on surveying and fortification , one finds this notation rarely or not at all, as will be discussed in Sect. 7.

On geometry the Instruction contains the following passages (paragraph 3, 4, 7 and 11).

They will then start with surveying on paper in the way of surveyors, that is not to construct any lines through known lines, but only to find the area of plane figures through measured lines, drawn to scale, using arithmetic in decimals. […] they shall only learn with rectilinear figures, thereafter with curved figures in the way of surveyors, how to measure a plane by means of various subdivisions, such as triangles or other figures to check the different calculations. […] Once they have sufficient skills in measuring on paper and understand on a small scale what has to be done in reality, they shall start with the actual surveying in the field, they will be shown that instead of using ruler, compass and square on paper, one uses other tools in the field, with the same purpose. […] After that they will learn to draw on paper the perimeter of the land they have measured, and again the reverse, from a drawing on paper, in the open field lay out the plan using pickets. […] As they have learnt so much, that they can thus usefully serve the country, those who wish to do so are allowed, as has been mentioned before, to continue to study at Leiden in winter, in order to become more accomplished engineers.

On teaching activities and teaching language Stevin remarks (paragraph 12 and 13).

The lessons in counting and geometry will take half an hour in general; the other half hour will be filled with answering questions from individual students and explanations about what they haven’t understood from the lectures. As those who work in engineering, rarely talk in Latin with each other, but instead in each country the vernacular is used, so the lessons will not be given in Latin, French or similar languages, but only in Dutch.

There was no mention of an examination procedure in the Instruction.

4.2 Teachers and Students

To ensure that the new course would attract students an innovative teaching plan was not sufficient. Much depended on the quality of the teacher. What was needed was someone who was an expert in the required mathematics and the practice of surveying techniques, who was willing to teach a group of uneducated students in Dutch and who could teach well. Maurits and Stevin chose Ludolf van Ceulen (1540–1610), a well-known and respected ‘reckonmaster’, a private teacher in mathematics, who had published some mathematical treatises and who also taught fencing (Krüger 2014; Wepster 2010). Van Ceulen had not studied at a university, he did not speak or write Latin, but he had a very good knowledge of mathematics and he had a good reputation as a teacher. He participated in government committees, for example to find a method to determine the longitude at sea or to compose a table for taxes of inhabitants of Leiden. A second professor was appointed together with him, a surveyor and cartographer, also living in Leiden , Symon Fransz. van Merwen (1548–1610). As van Ceulen was already 60 years old when he was appointed, it is likely that van Merwen was supposed to take care of the lessons in the practice of surveying techniques. Soon after the start of the course van Ceulen and van Merwen informed the University governors that the students had repeatedly asked to receive a testimony on their surveying capabilities. In August 1600 the University governors decided that there would be an optional final exam, which would serve as a testimony in the admittance procedure for surveyor by the provincial authorities. A final exam resulting in a testimony for military engineer was not realized.

The students had very different educational backgrounds and they were of very different ages, roughly from 16 to 45 years old (Krüger 2014). Some were regular students, enrolled at the University; most were already craftsmen, such as carpenter, stonemason, surveyor, draughtsman or teacher. The reputation of van Ceulen as a reckonmaster and teacher was sufficiently good to attract students during the first ten years of the existence of Duytsche Mathematique.

Not much is known about the teaching by van Ceulen and van Merwen. In 1615 a book by van Ceulen was published posthumously, De arithmetische en geometrische fondamenten [Fundaments of arithmetic and geometry]. This book has six chapters, on arithmetic, geometric constructions, geometric transformations, geometrical applications, advanced applications and regular polygons, with inscribed and circumscribed circles, a specialty of van Ceulen. It seems likely that these topics were to some extent part of his lectures in Duytsche Mathematique . However, there is much more on species of numbers and on circles than was necessary for Duytsche Mathematique; there is not much on surveying and there is no use of decimal numbers as mentioned by Stevin. The Regional Archives in Leiden possess a handwritten booklet by Symon van Merwen , the colleague of van Ceulen . The title is De vijff spetie inde tiende getalen [The five species in decimal numbers]. There is no date but van Merwen refers to himself as professor of mathematics in Leiden; he also mentions professor van Ceulen, so it must be written between 1600 and 1610. It is a small booklet, containing 148 written pages. The first 12 of those are on the use of decimal numbers, with a notation which is reminiscent of the one used by Stevin. However, the other 136 pages are about different types of numbers, very much like the first chapter in the book by van Ceulen, with many examples concerning currencies. Decimal numbers remained for van Merwen a separate topic, for which he had little use. This booklet possibly shows mainly topics which were taught in private lessons.

More is known of the work of van Ceulen’s and van Merwen’s successor, Frans van Schooten Sr.

5 Frans van Schooten

Frans van Schooten Sr. was the son of migrants from the southern Netherlands. His father, a baker, was citizen of Leiden since 1584. Frans, the oldest of four brothers, studied Duytsche Mathematique with van Ceulen and van Merwen and took more lessons in mathematics with van Ceulen; he was officially admitted as a surveyor in 1608. For some years he was an assistant in van Ceulens small private school for mathematics. During 1610, after van Merwen died, van Schooten assisted professor van Ceulen with the lessons in Duytsche Mathematique . After Ludolf van Ceulen also passed away in December 1610, van Schooten continued the lessons on request of the students, in a temporary position and without an official appointment by the University. He taught both the theory and the practice of surveying, the practice probably on the field which had been purchased by the University for that purpose. From 1611 until 1614 at least three formal requests to the curators of the University were made by Frans van Schooten and his students, to have van Schooten appointed as successor to professor van Ceulen. However, there was a serious competitor, a candidate who got the support of prince Maurits. Samuel Marolois (1572–before 1627) was born in the Netherlands shortly after his parents emigrated from France. He was a military engineer and technical advisor; he taught mathematics and produced books on geometry, perspective, architecture and fortification. These books were popular, as they contained many engravings of good quality. However, at the start of 1615 Frans van Schooten was appointed Professor Duytsche Mathematique. He continued his teaching of theory and practice in this official capacity until December 1645.

Frans van Schooten Sr . left some teaching texts on mathematics. The most impressive of those is BPL 1013, the extensive lecture notes for the Duytsche Mathematique . Van Schooten probably wrote the manuscript around 1620; it was also used by his successors, Frans van Schooten Jr . and Petrus van Schooten , so there was a long-lasting influence of this teaching text. Next to teaching Duytsche Mathematique Frans van Schooten worked as a surveyor and cartographer for the army during the summer months, he was an advisor on fortification and examiner for the surveyor admission committee in Holland (Dopper 2014; Krüger 2014).

In 1627 he published a trigonometric table, Tabulae Sinuum Tangentium Secantium ad Radium 10 000000. Met ‘t gebruyck der selve in Rechtlinische Triangulen [Table of sine, tangent, secant based on radius 10000000. With the use of the same in plane triangles]. The book had several reprints, possibly also because of the clear explanation and the systematic treatment of examples, with many drawings, in rectangular, acute and obtuse triangles. In the book van Schooten explained that the accuracy of the tables is dependent on the accuracy of the instruments for measuring angles in the field.

As working on the land it is with the instruments available not possible to measure angles more accurately than in minutes, I have calculated the tables in as many rectangular triangles as the number of minutes which can be represented on a quarter of a circle (Tabulae, p. 2).

So, the tables have an accuracy of minutes.

As far as we know his students appreciated him, Duytsche Mathematique had a good reputation, also in neighboring countries; the University governors evidently thought well of him, even if he did not have a university education.

6 BPL 1013

BPL 1013 is a fairly large manuscript, 425 × 280 mm. The writing on the pages is in brown ink, on each page is some text and pen drawings, of geometrical figures (Fig. 1), landscapes with geometrical figures (Figs. 2, 3 and 4) or geometrical figures enhanced with a landscape (Fig. 7). Occasionally calculations are added, written with a different pen, sometimes one sees a correction of an error in a calculation. The text is structured following the teaching plan by Stevin, the style is what one would expect of a teacher who prepares carefully his lessons, with some explanations and short notes as scaffolding for his teaching. The carefully executed illustrations may well have served as visualization for his students, to support explanations and exercises. A lot of the content consists of mathematical exercises, mostly with solutions. The last part of the manuscript, roughly one third, is on fortification and contains some watercolor drawings, pen drawings and calculations. The manuscript was in very bad condition due to ink-corrosion. Recently it has been restored and preserved; it is now digitized.Footnote 11

Fig. 1
figure 1

Exercise to practice calculations in decimal numbers, f.4v

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Fig. 2
figure 2

Measuring angles, use of tangent, f.58r

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Fig. 3
figure 3

Measuring angles, use of law of sines, f.57v

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Fig. 4
figure 4

Use of similar triangles, f.58v

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On f.113r, just before the treatment of fortification, is written

Begonnen den 25 November Anno 1622 door Frans van Schooten professor der Fortificatien en Dependerenden Scientien in de Universiteit tot Leyden” [Started 25 November 1622 by Frans van Schooten , professor of fortifications and related sciences at the University of Leiden]

Van Schooten must have written the preceding content in the years before 1622.

As already indicated, the topics in the lecture notes and their order are according to Stevin’s Instruction, however, Frans van Schooten left out the start of arithmetic, i.e. the four operations with whole numbers and rational numbers and the rule of three (Table 1). These operations were probably known to the students; van Schooten may have taught them separately to those who lacked knowledge in arithmetic . He included some topics which had nothing to do with surveying, such as the construction of a spiral or a geometric rose and wine gauging. In the part on surveying he added notes on a variety of geometrical methods, which were not commonly used.

Table 1 Index of the content of BPL 1013
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The content of the lecture notes may be described as in Table 1.

Some differentiation is noticeable in the content. The numbers 1–3 seem to cover a basic course, 4, 5 and 6 are more advanced geometry and cover everything the best surveyors had to know; 5 has a more mathematical theoretical character and 7 may be considered a further specialization, that of military engineer, which at that time included building fortifications. The differentiation noticeable in the content suggests that van Schooten catered for different demands of students.

6.1 Interesting Features of BPL 1013

The order of topics is roughly the same as in the Instruction; the treatment of topics is very well structured. Within topics there is a built-up from simple to more complex situations, with occasionally a reminder of what was previously taught.

Remarkable is the consistent use of decimal notation, starting on f.4r and always linked to units of measurement, with the rod as unit of length. This is a very early example of the use of this notation throughout a manuscript. I know of no other publications or manuscripts from the early 17th century, in which decimal notation is used throughout the text, with such apparent ease; at most this notation is used occasionally or demonstrated as a special type of number.

Another interesting feature is the systematic use of instruments to measure, combined with the use of trigonometric tables and the demonstration of how to create a triangulation network, starting from a line of known length (f.65v).

The manuscript is strong on visualization; van Schooten used illustrations throughout the manuscript, with every topic and exercise, very neatly executed; some are of a high artistic quality (f.66r–f.89r).

He paid much attention to the minimalizing of errors in the results. In his publication on trigonometric tables (Schooten van 1627) he pointed out the importance of choosing a suitable radius for measurements, depending on the magnitude of the length to be measured and citing Stevin on this topic. As mentioned above, he also drew attention to the limitations of the measuring instruments available.

Van Schooten systematically showed more than one method to solve a problem. This served as a way to check outcomes of measurements and calculations, as Stevin had prescribed. But it also gave the future practitioner a choice of methods, depending on personal preference and, in surveying, the tools available.

6.2 Arithmetic (See Table 1, Row 1)

This part differs slightly from Stevin’s plan as there is no treatment of the four main operations; presumably the students already mastered basic arithmetic. The manuscript starts with calculations of square and cube roots from whole numbers and fractions, followed by the introduction of decimal notation. For example to introduce calculation of a cube root (f.2v), van Schooten uses a drawing of a cube with a volume of 110 592 cubic feet. Calculation of the cube root gives the length of the edge, 48 feet. This is followed by a table of roots 1–9 with their cubes, and calculation of the edge of a cube with volume 30 517 578 125 cubic rods, which turns out to be 3 125 rod. On the last page of this topic one finds another example of taking the cube of a very large number and two small tables of cubes of some fractions, based on calculation of the third power of the fraction (f.3v). All examples are illustrated with a drawing of a cube, with the numbers written in it. Looking at the handwriting and some other characteristics the drawings probably were added some time after the calculations were written. One reason may be that the calculations were written by van Schooten while he was studying with or assisting Ludolf van Ceulen and that he added the drawings as a didactic tool when he was teaching independently as van Ceulens successor.

Introduction of the unknown decimal notation as promoted by Stevin takes place on f.4r, linked to the surveyor’s rod, divided in ten equal parts. At the time, division in twelve equal parts was still more common, though division in ten parts occasionally was used (van Gulik-Gulikers 2005). The notation is the same as used by Stevin in The Tenth (De Thiende 1585), the indication of value above the digits (Fig. 1).Footnote 12 However, van Schooten preferred the more convenient indication of the number of decimals at the end (Figs. 1, 2, 3 and 4). The first example is a length of 346 rods, 8 feet, 7 inches, 5 grain, which in modern notation would become 346,875.Footnote 13 Most examples are on measuring in plane geometry, such as in Fig. 1. From here on, van Schooten used decimal notation only in all calculations.

The geometry (see Table 1, row 2) starts with 32 definitions, each one illustrated. The part on constructions and transformations seems to be rather similar to the posthumous publication by van Ceulen (1615). The topics were of importance for the practice of surveying and fortification, with some additions such as the construction of a spiral and of a geometric rose. There are only mathematical contexts.

6.3 Examples from (Advanced) Practical Geometry (See Table 1, Row 3–5)

A closer look at the geometrical methods used, shows that in the part on basic surveying (3 and part of 4) only a few mathematical techniques were taught, applied in different situations and depending also on the surveying tools. For example, in measuring distances in inaccessible lands van Schooten repeatedly combined measuring angles with the law of sines or use of tangents or he used proportionality in similar triangles. This enabled students to practice the mathematical methods they could use in many different situations and with different surveying tools, such as a Dutch circle or an astrolabe for the measuring of angles or a quadrant. See for examples Figs. 2, 3 and 4, in which the distance to a tower or between two towers is to be determined, the context in each case is measurement from the opposite bank of a river or an island in the river. So, some or all of the towers are on inaccessible land for the surveyor.

In part of 4 and in 5 (Table 1) many more techniques are present, many of which most surveyors might never need. In Advanced surveying techniques (see Table 1, row 4) there are many examples of measuring heights, not just the habitual church tower or statue, as found in the Netherlands. Indeed, there are 16 examples of measuring heights and depths of buildings on top of or half way a mountain, a situation in which an engineer for the army would find himself when working in neighboring countries. Some of these drawings are of a very high quality; they look the work of a professional engraver or painter (van Maanen 1987). The mathematical methods are of a higher complexity than in previous parts of the manuscripts, in the sense that one needs combinations of previous used techniques.

Some mathematical procedures, known at the time, do rarely or not at all occur in the manuscript, e.g. the law of cosines. Examples of less common mathematical methods in Advanced geometry (Table 1, row 5) are on ff.88v–89v (Figs. 5, 6 and 7); in each case the length of the sides of the triangle has to be determined. It is important to remember that these are notes to use during the lessons, so the method used is more often implied than elaborately written down. The description below is an interpretation based on drawings and calculations written on the folia discussed. As the Duytsche Mathematique was a course for mathematical practitioners the methods used would have been less formal than in lectures on ‘mathematica pura’.

Fig. 5
figure 5

f.88v

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Fig. 6
figure 6

f.89r

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Fig. 7
figure 7

f.89v

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In the problem presented on f.88v (Fig. 5) two angles and the sum of the sides are known. The area of the triangle is also requested. The lengths of the sides are determined through a variation of the law of sines. In slightly more modern notation: (sin A + sin B + sin C): (BC + AC + AB) = sin A: BC = sin B: AC = sin C: AB. The perpendicular BD is also determined by using proportionality. The area is calculated by multiplying ½ AC by BD.

In the triangle depicted on f.89r (Fig. 6) the area and two angles are known.

The solution, which is suggested in and partially written under the drawing, starts with the construction of a circle with center B and radius BD. The sides AD and DC represent the value of the tangent of respectively ∠ABD and ∠CBD, the sum of AD and DC represents the ‘parts’ of AC. See Fig. 8. Van Schooten assumes here the existence of a virtual triangle, which is proportional to the triangle depicted, but with the value of the tangent as the lengths of the sides. The trigonometric table used to find these values is based on a radius of 100,000, so BD in the virtual triangle would consist of 100,000 units or ‘parten [parts]’ as van Schooten writes. Take half of BD as 50,000 (‘parts’) and calculate the area in ‘parts’ as ½ BD × AC. Proportionality in the form of the rule of three gives the real value of BD. Repeated use of the rule of three combined with the cosecant of ∠A and ∠B provides the real length of the sides.

Fig. 8
figure 8

The calculations of the solution to the problem on f.89r

Full size image

In the triangle with the beautiful illustration on f.89v (Fig. 7) the area is known, as well as the proportion of the three sides (13:14:15). In this case the angles have to be determined as well. Van Maanen (1997) provides a good example of treatment of this interesting problem in a modern secondary school.

7 Comparison with Other Texts

Around 1620 there were some books on surveying and fortification available in Dutch language; they were of course relatively expensive, but the University library possessed copies. How does BPL 1013 compare to these books, which were available at the time when Frans van Schooten Sr. started lecturing? The most important for comparison are publicationsFootnote 14 by Stevin (1585–1608) , Sems and Dou (1600), Marolois (1614) and van Ceulen (1615).

Publications by Stevin which were relevant to Duytsche Mathematique are De Thiende,‘t Weereltschrift and De Meetdaet.

In De Thiende [The Tenth], published in 1585, Stevin introduced decimal notation and promoted its use by mathematical practitioners, such as surveyors. The notation used in De Thiende had didactical value, it made very clear in which way surveyor’s measurements could very simply be translated into decimal notation. In his calculations van Schooten used a simpler notation, also used by Stevin in his De Meetdaet [The Act of Measuring]. In ‘t Weereltschrift [Describing the world] Stevin gives a theoretical treatment of trigonometry, starting with many definitions, many examples in triangles and no applications in surveying (Stevin 1605a). The terms he uses are ‘houckmaet, raeklyn, snylyn’; van Schooten has an introduction of one page and uses drawings of a half-circle, divided into degrees and minutes and with the names and definitions of sine, secant and tangent added. All exercises after this introduction are in the context of surveying (Figs. 2, 3 and 4) until part 5 on Advanced geometric techniques (Table 1; Figs. 5, 6, 7 and 8). So the approach and terminology in BPL 1013 is practical, starting with the theory which is necessary for surveying techniques and later on more in-depth theoretical examples. De Meetdaet (Stevin 1605b) is on geometrical constructions, transformations and surveying techniques. The topics in part 2–4 of BPL 1013 (Table 1) are very similar to what one finds in De Meetdaet, but the applications in BPL 1013 are more systematic in the treatment of different instruments, increase in complexity and in the continuous use of surveying contexts through illustrations. Though some topics and their treatment are comparable, it is clear that van Schooten was inspired by, but did not imitate Stevin.

Johan Sems and Jan Pietersz. Dou were surveyors. They published Practijck des lantmetens [Practice of surveying] together with Van het ghebruyck der geometrische instrumenten [The use of geometrical instruments] as one volume in 1600, when the lectures Duytsche Mathematique started. The topics are roughly the same as in BPL 1013, but they occur in a different order and the treatment is less well structured. The authors use a division in rods, feet and inches in twelve and in ten parts. Other differences with BPL 1013 are: no use of decimal notation for fractions, with as a consequence extensive calculations with fractions and the different measurements, not much of Euclid, mentioning of sine and related terms in the definitions and no mentioning of secant or tangent. There is nothing on fortification. This volume could be used by students, if they had some money to spend, for the surveying part of the course. The lack of decimal notation made the calculations more complicated, the trigonometry was less up-to-date than in BPL 1013. The text by van Schooten is also more concise; that may be due to the difference between teaching notes as preparation for a course by a mathematical expert and a publication meant for self-instruction.

Opera Mathematica [Mathematical Works] by Marolois contained chapters on geometry, perspective, architecture and fortification. His books were illustrated with many engravings of high quality; there were several reprints during the 17th century, usually with improvements. Albert Girard added the use of trigonometric tables; Frans van Schooten Sr. made improvements for a French edition of Fortificatie [Fortification], in 1628. Marolois paid much attention to perspective, not a topic in Duytsche Mathematique . Marolois did not use decimal fractions systematically and he preferred the more common division of measuring units in 12 parts. Frans van Schooten Sr. referred to Marolois in his lectures on fortification, with regards to calculation of angles of a regular polygon as basis (BPL 1013 f.140v), but the treatment of fortification in BPL 1013 is different from Marolois.

De arithmetische en geometrische fondamenten by van Schooten’s former teacher, Ludolf van Ceulen , has been discussed in 4.2. The chapters on geometrical constructions and transformations are very similar to BPL 1013, ff.8v–44v, again bearing in mind the difference in style between a book and preparatory notes for teaching. However, the rest of the book is very different from van Schooten’s manuscript, though van Ceulen is mentioned a few times as an authority.

The analyzed manuscripts from the 17th century on similar subjects, in the libraries of Leiden and Utrecht, show a less complete treatment of the topics, none uses decimal notation consistently, sometimes not at all. A few examples from LeidenFootnote 15 are BPL 2084 (first half of 17th century) and BPL 1970 (1658). Both originate from Zeeland, the southwestern part of the Netherlands and treat similar topics as in BPL 1013, but much more limited in scope. They are instructional texts, BPL 1970 is a copy from another text, the scribe mentions several experts amongst whom are Ludolf van Ceulen, Frans van Schooten and the surveyors Sems and Dou . Another example is BPL 1351 (1655), notes from a student of Kechelius , a private mathematics teacher in Leiden . In this manuscript one finds more or less the same subjects as in BPL 1013, without use of decimal notation and sometimes written in Latin. In Utrecht UBU 1363 is a manuscript on geometry, in two parts, written around 1700. The scribes are Anemaet (1690) and Hub. Anemaet (1726–1727). The manuscript deals with geometry and surveying, with limited treatment of the topics. It probably represents student’s notes.

8 Discussion and Conclusion

The Duytsche Mathematique had to attract sufficient number of students to make it worthwhile for the University to continue paying the salary of a professor. As the students, or ‘auditores’, who came solely for Duytsche Mathematique, usually were not registered as such, there is no direct way to find information about the numbers. Those who were registered as university students sometimes also followed the lectures Duytsche Mathematique , as becomes clear from various sources; some examples are Martinus Wimmers (1607), Tielmannus Wyntgens and Steffen van Blitterswijck (1609), Jean Gillot (1630), Levyn Baers (1631), Willem van Schoonhoven (1633) , Caspar Poorten (1639) and Johannes Steen (1644) (Krüger 2014; Witkam 1969). An important source of information is formed by the archives on admission of surveyors in Holland until 1646; there are also some letters of contemporaries in which Duytsche Mathematique is mentioned. Quite a few of the candidates for admission as surveyor mentioned that they had studied Duytsche Mathematique; this served as a testimony.

During the first ten years the reputation of professor Ludolf van Ceulen as a mathematician and as a teacher must have been part of the attraction. Between 1611 and 1614 a group of students lobbied for the appointment of Frans van Schooten as successor of van Ceulen and van Merwen . Such a request was sometimes signed by more than 30 people. In the request of 1612 they wrote that

… the lessons have been continued by Frans van Schooten during one year and ten months, both in the lecture hall and in the fields… (UBL AC1 Archief van Curatoren, 1574–1815, inv. 42)

So, Frans van Schooten taught both the theory and the practice of surveying. Amongst the signatures were those of at least three well-known surveyors from Leiden who evidently thought well of his lessons. The governors of the University paid for his services at the end of each year; in 1612 he also did some surveying for the University; at the end of that year the governors asked him to continue teaching in 1613. At the end of 1613 he was remunerated for the costs of four wooden instruments which he had ordered for the teaching. After van Ceulen’s passing away, the number of students clearly remained sufficiently high to appoint a new professor, who actually did the work of the previous two professors. Frans van Schooten got appointed, with support from the students. During the period 1615–1645 one finds in the various archives many names of surveyors, who indicate that they studied Duytsche Mathematique , with professor van Schooten. There is less information about engineers, however there are engineers who studied Duytsche Mathematique, for example Axel Arup , Geraert van Belcum , Joris Gerstecoorn and Jean Gillot . Some of the engineers were employed by the king of Sweden. Comments from contemporaries also was in general favourable. In 1646, after his father passed away, Frans Jr. stated in a letter to Christiaan Huygens that the Duytsche Mathematique had produced engineers, surveyors, wine gaugers, reckonmasters (teachers of mathematics) and other professionals; students also came from abroad.

We may conclude that Frans van Schooten attracted students sufficiently to maintain the course.

What made his lectures so valuable? We only have his lecture notes, the manuscript BPL 1013, there are no reports on his teaching known. I discuss the content and some characteristics which are important in teaching.

If the content of BPL 1013 is compared with the publications on surveying and fortification , which were available around 1620, it becomes clear that none of the books treated all the topics which occur in BPL 1013. Also, van Schooten was the only one, apart from Stevin , who used decimal notation consequently and in a seemingly natural way. For those who mastered decimal notation, calculations were less cumbersome than with ordinary fractions. The manuscripts from that time which have been analysed also show a less comprehensive treatment and far less consistent use of decimal notation, or not at all. Van Schooten also showed and let the students practice with several instruments and up-to-date mathematical techniques, with attention to limitations of measurements accuracy and the question of accuracy of calculations.

The characteristics of the manuscript which are important in teaching:

  • careful planning, resulting in a well-structured, coherent and consistent curriculum, in which relevant mathematical techniques for surveyors and engineers were taught and practised;

  • extensive use of visualization; throughout the manuscript, illustrations are used with every exercise and with every definition, axiom or proposition;

  • a gradual increase in complexity within topics;

  • many exercises, a great many examples from surveying, but also from other practices, such as building;

  • distinction between the core of the curriculum and more advanced topics;

  • differentiation in breadth (more topics) and in depth (advanced geometry).

In BPL 1013 neither logarithms nor algebra are mentioned. Logarithmic tables were new and not yet available in a Dutch edition.Footnote 16 Algebra was not mentioned in the teaching plan by Stevin; at the time when BPL 1013 was composed it was still coss algebra, for which neither surveyors nor engineers had much use. There is a manuscript on coss algebra by Frans van Schooten Sr . (Dopper 2014), he probably taught it privately, but it was not part of Duytsche Mathematique . Topics such as taking square roots and wine gauging, both in BPL 1013, were not mentioned by Stevin in his teaching plan. However, calculating roots was a necessary arithmetic skill for practitioners. Wine gauging was a very useful skill as most towns employed a professional wine gauger. This is an example of aims, which were more diverse than the training of military engineers, of Duytsche Mathematique.

BPL 1013 shows a very extensive collection of teaching notes by someone who evidently was an accomplished mathematician, surveyor, and an excellent teacher. As Frans van Schooten Sr. taught the Duytsche Mathematique for a long time, 35 years and must have had many students, he was influential in reaching the aims of Duytsche Mathematique during those years.

However, he did more. Through his well-designed and carefully written and illustrated lecture notes he left a legacy for his son who was his successor. Frans van Schooten Jr. (1615–1660) made use of these notes of his father during many years and so the influence of Frans van Schooten Sr. lasted for years after he passed away.

We may conclude that in a quiet way Frans van Schooten Sr . during and after his lifetime was very important for the reputation of the Duytsche Mathematique and for the achievement of the aims of this course.