Simon Stevin and the Art of War
by
Guido Vanden Berghe
Universiteit Gent, Belgium
and
Jozef T. Devreese
Universiteit Antwerpen / Technische Universiteit Eindhoven

General Introduction
To a general educated public Simon Stevin is mainly associated with the introduction of the decimal fractions, which led later on to the foundation of the decimal system of weights and measures. He realised numerous technical inventions out of which the construction of a 28passenger sailing carriage used along the seashore was best known to his contemporaries.
Figure 1: Token with the portrait of Simon Stevin, Penningkabinet Koninklijke Bibliotheek Brussel, afb. 3 in “Simon Stevin, De geboorte van de nieuwe wetenschap”.
Caption: Onesided silver token, casted in 1607, 65 mm. This is the only portrait made of Stevin during his life. The artist is unknown.
For those less familiar with Stevin we recall^{1} that he was born in Brugge (now Belgium) in 1548 as the illegitimate child of Anthuenis (Anton) Stevin and Catelyne (Catherine) van der Poort^{2}. Recent studies^{3} prove that his father was a cadet son of a burgomaster (mayor) of Veurne; his mother was the daughter of a burgher family of Ieper (Ypres), who married later a merchant in carpetweaving and silktrade, who belonged to a family with sympathy for the Calvinism religion^{4}. Practically nothing more is known of Stevin's youth and education. He was at first employed in Antwerp as a bookkeeper and cashier in one of the tradingfirms; there he became acquainted with business techniques. In 1577 he accepted a post at the financial administration of the jurisdiction of Brugge (Brugse Vrije). A few years later he is registered at Leiden (now The Netherlands). The reasons for this emigration, possible aversion for the Spanish oppression of the southern part of the Low Countries, or protestant sympathies, are not known. In 1583 we find him inscribed on the roll of the just founded University of Leiden, where he made the acquaintance of the young prince Maurits (Maurice) of Orange. A lifelong friendship developed between the two men: Prince Maurice found in Stevin an excellent tutor and later a capable and loyal counsellor; on the other hand Stevin could always reckon on support and protection of his princely friend. Prince Maurice, elected in 1584 as stadtholder of Holland and Zeeland, appointed Stevin in his personal service. Some sources quote Stevin as QuarterMaster General of the States' army (Staatse Leger). Recently one has discovered in the Public Record Office (Rijksarchief) of The Hague a journal and corresponding ledger, identified as an application in the year 1604  by Stevin  of his ideas on "princely" bookkeeping, containing as an entry Stevin's annual salary of 600 Dutch guilders^{5}. This high amount confirms the high status of Stevin at the court of the Prince. In 1600 Stevin, on initiative of the Prince, founded within the University of Leiden a school of engineering, where the courses were taught in the Dutch (Nederduyts) language. Several authors mention that Stevin has travelled extensively through Europe. Unfortunately there exists only one known record of such travel. It concerns a visit to Dantzig ^{6} (now Poland), where Stevin was invited to give his expert advice on harbourworks. Around 1614 Stevin married young Catherine Cray; they had four children: Frederic, Hendrik, Susanna and Levina. It was his second son, Hendrik who published some posthumous work of his father. Stevin died at The Hague between February 20 and the beginning of April 1620, probably in his house at the Raamstraat, which he had bought in 1612 for 3800 Dutch guilders^{7}.
Figure 2: Oilpainting of Stevin, Leiden, Universiteitsbibliotheek, office of C. Van den Heuvel, icones 40.
Caption: Portrait of Simon Stevin, oil colours on panel, 82 x 68 cm. On the upper left one can read the place and date of birth, Brugge 1548, at the upper right the place and date of death, Den Haag 1620. The text under the portrait: Simon Stevin, excellent mathematician, advisor of the Prince Maurice.
Figure 3: The house of Stevin; figure 2.10 in “Wonder is gheen wonder, De geniale wereld van Simon Stevin”, foto Guido Vanden Berghe.
Caption: The house of Simon Stevin in Den Haag, Raamstraat 42, where Stevin lives with his wife Catharina Craiy and their four children.
There still exists an important interest for those contributions and endeavours of Stevin which have been recognised as pioneering or influential since a long time. One of the first books Stevin published was "Tafelen van Interest" ^{8} (Tables of interest, 1582), in which  for the first time in Western Europe interesttables were made public. Before that time only manuscripts were available, of which copies were sold at very high prices to tradesmen, merchants and bankers. It is without doubt that Stevin introduced for the first time a complete and systematic description of decimal fractions and the operations which can be carried out with them in a booklet, "De Thiende" ^{9} (the Tenth, 1585). It also treats the practical application of decimal fractions to the surveying of land, to the measurements of weights and to the partition of money. The English translation by Robert Norton (London 1608), "Disme, The Arts of Tenths or Decimal Arithmetike", has inspired Thomas Jefferson when he proposed a decimal monetary unit for the newly created United States of America. The tenth of the present dollar is still called a dime today. The Scottish Laird and mathematician, John Napier was inspired by Stevin's work for the invention of logarithms. In his works on physics Stevin introduced new original ideas. The study of statics of rigid bodies is exposed in "De Beghinselen der Weegconst" ^{10}(Principles of Statics, 1585). It contains the fabulous "clootcrans" theorem, an ingenious Thought Experiment, by which the condition of equilibrium of weights on an inclined plane is obtained. As a consequence of this theorem Stevin succeeded in obtaining the sum of two concurrent forces by the parallelogram rule. In "De Beghinselen des Waterwichts" ^{11} (Principles of Hydrostatics, 1586) Stevin gives an improved demonstration of Archimedes' law, stating that a body immersed in a fluid undergoes an upwards force and determining it. He also succeeds in calculating the force a fluid exerts on the bottom of the vessel in which it is contained, which leads him to the formulation of the socalled hydrostatic paradox, many years before this was done by Blaise Pascal, to whom the paradox is usually attributed. Stevin published in 1586 his experiment in which two lead spheres, one 10 times as heavy as the other, fell from a height of 30 feet probably in Delft (Holland) in the same time. His report preceded by three years Galilei's first treatise concerning gravity and by 18 years Galileo's theoretical work on falling bodies.
Stevin was also a great mathematician. As all scientists of the Renaissance he was active, as already shown, in many disciplines. His innovating mathematical work dates from 1582 up to 1585. In that short period he calculates his interest tables; the theory behind these tables is very clearly developed and explained. In his "The Tenth" he introduced all operations  addition, subtraction, multiplication, division and the calculation of square roots  on decimal fractions, a real novelty for that time. In his work "L' Aritmetique" ^{12}  the only work written in French  he gives a review of the algebra knowledge of his time. He introduces new ideas and new theories in algebra and describes a numerical method for finding a zero real root of an function. In his "Problematum Geometricorum" ^{13}, his only work written in Latin, the geometry, known at the end of the 16^{e} century is discussed. In his “Wisconstige Gedachtenissen”^{14} (Mathematical Memoirs, 16051608) he gives a pedagogically very well documented review of mathematics in general.
Figure 4: Portrait of Prince Maurice, painting of Michile Jansz. Van Mierevelt, Rijksmuseum Amsterdam (zie figure 2.8 in “Wonder is gheen wonder, De geniale wereld van Simon Stevin”.
Caption: Maurice, Prince of Orange (15671625) was an important friend of Simon Stevin. Stevin instructed him in many scientific problems and advised him in many military issues,
From 1590 on Stevin is mainly working in the service of Prince Maurice. The exhibition in the "Rijksmuseum", Amsterdam entitled "Maurice, prins van Oranje (2000)" ^{15} contributes to a better insight in the StevinMaurice interaction. The catalogue accompanying this exhibition describes many details of this relationship. Most of Stevin's publications from 1590 on are devoted to subjects of interest to his friend or to the nation. In 1590 he published a booklet "Vita Politica. Het Burgherlick leven" ^{16} (Civil life) in which he exposes how a citizen as a good subject should comply with the rules of the authorities. At that time the Republic of the Provinces of the Netherlands was being organised and with this publication Stevin aims at order and regularity. In the same spirit Stevin published two books of practical use for the defence of the country and for the extension of the fleet. In "De Sterctenbouwing" ^{17} (The construction of fortifications, 1594) the new way of fortification following the Italian system was accommodated to the geographical conditions and the available means of the Low Countries; in this way Stevin's name remains associated with the socalled old Dutch method of fortification. This book was probably used as course material in the engineering school. In the second book, "De Havenvinding"^{18}, (literally: “harbour finding”, in fact dealing with position finding, 1599) Stevin describes the determination of a place on earth by the knowledge of the geographical latitude and the magnetic variation of the needle of the compass. This technique was of great importance for the ships of the "Verenigde OostIndische Compagnie" (VOC, United EastIndian Company), which has built up a monopoly position in the trade between the (far) east and WesternEurope. In de period 16051608 the lessons in the numerous sciences (algebra, geography, astronomy, bookkeeping, physics, etc.) given by Stevin to prince Maurice were collected and published as "Wisconstige Gedachtenissen" (Mathematical Memoirs). Stevin's last publication, dated from 1617, is a double book: in "Castrametatio, dat is legermeting" ^{19} (Castrametatio, That is Campmeasurement) he describes the establishment and the furnishing of a wellorganized military camp; in "Nieuwe Maniere van Stercktebou door Spilsluysen", (New methods of fortification by spindlesluices) he describes the use of this special sluices in the defensive works, principally to keep the moats of a desirable depth.
Besides the wellknown aspects of Stevin's work, nowadays the less publicised contributions receive an increasing appreciation. In "De Spiegheling der Singconst" ^{20} (Theory of the art of singing (music)), a manuscript, recovered in 1884 by Bierens de Haan, Stevin was the first to give a correct theory of the division of the octave into twelve equal intervals. In the field of perspective, treated in "Van de beghinselen der Spiegelschaeuwen" (On the principles of mirror images), and "Van de Deursichtighe" (On the transparency) both parts of "Wisconstige Gedachtenissen" Stevin was the first to build on the pioneering work of Guidobaldo del Monte; he contributed fundamental and new theorems regarding projections, which are found under Stevin's name in mathematical works of prominent mathematicians. The influence of Stevin’s concepts in the field of architecture, home building and townplanning receive considerable attention. We find Stevin's ideas in "de Huysoirdening" (Planning of the house) and "Stedenoirdening" (townplanning) included in the "Materiae Politicae, Burgherlicke Stoffen" ^{21} (Dissertations on political and civil affairs, 1649), published by Stevin's son Hendrik. This material is only a small part of a project for a more elaborated book, planned by Stevin, "Huysbou" (Building of houses), which was never published. Parts of the handwritten manuscript of this "huysbou" were partially reproduced in the journal of Isaac Beeckman ^{22} and were recently discovered in the Rijksarchief (State Archive) of the province Zeeland (the Netherlands); they show the importance of it for the history of the technique of civil architecture in Stevin's time.
The majority of Stevin's works were published in his mother tongue, the Dutch language. Stevin's goal was to bring science and technology also to nonacademic people having no knowledge of Latin, but possessing a certain understanding of science. In the introduction of "De Thiende" he gives the categories of people for which he wrote that pamphlet: stargazers, surveyors, carpetmakers, winegaugers, mintmasters and all kind of merchants. He extensively exposed his ideas about the Dutch language in his "Uytspraeck van de Weerdicheyt der Duytsche Tael" (Enunciation about the dignity of the Dutch language), written as an introduction to "De Weeghconst". For Stevin the "Nederduyts" was a preeminently language to express ideas, especially scientific thoughts, because of its short words and the possibility of word combinations. Stevin's great merit has been that he has enriched the Dutch language by the introduction of new words or combinations of known words as translations of their Latin counterparts. A typical example is the Dutch word for mathematics, "wiskunde" derived from "wisconst".

Stevin’s writings on the art of war.
Stevin’s military works were not all published in his lifetime. The three treatises on military science, published by himself in bookform are the already mentioned “De Sterctenbouw, Castrametatio, Dat is Legermeting” and “Nieuwe Maniere van Sterctebou door Spilsluysen”. At the moment that Stevin passed away, in the beginning of 1620, he left behind for his widow a pile of unfinished manuscripts. The scholar Isaac Beeckman, who has visited the remarried widow in 1624 several times, made an inventory of Stevin’s work and copied part of it. He noted down in his famous journal a list of 28 titles and made several extracts of items which were devoted to different military subjects, with the following titles ^{23}: Van de Spabijlhouw (About the Spabijlhouw = Spadeaxepick), Van de geduerighe verlegginghe des crychsvolckx, (About the continual displacement of troops), Van de weerdicheden der cryghsampten (About the worthiness of the ArmyDuties), Oorden der verkiesinge (Order of selection), Calis int groot afgebeelt (Calais depicted in detail), Chryghssaken (Miltary matters), Teghen verdruckingen (Against reprisals), Veltslachoordeningh (Battlearray), Pyckschansinghe (Pike redoubts), Chryghskonst, seer veel daarvan (The art of war, very much of it), Verscheyden Chryghstochten, dadelick van hem gesien ende geordineert (Various campaigns, observed on the spot and controlled as to their order). It seems that those texts were not yet suitable in 1608 to be published in “Wisconstige Gedachtenissen”, where some military tracts were treated. However these manuscripts seemed to be more than some idle notes. According to Stevin’s own statement ^{24} this manuscript had not been entirely finished in the year 1608, but that it was to have developed into an extensive work is shown by its large number of subjects and the space they occupy. All these subjects have been included by Beeckman in his journal under the title or collective name: Vijfde deel der ghemeynghde stoffen. Van de Crijchconst. Hendrick Stevin published a number of his father’s papers concerning military arts in his “Materiae Politicae, Burgherlicke Stoffen”. What he presents is not completely in accordance with the survey of the contents found in Beeckman’s journal and the contents of manuscripts discussed by de Waard ^{25}. Schukking ^{26} even expressed some criticism about the way Hendrick has published his father’s valuable military treatises, by agreeing with the disapproval mentioned by de Waard. He formulated his comments as follows ^{27}:

Hendrik Stevin wanted to render his own book more readable by merely inserting from the Conduct of War matters that could be understood without the practice of mathematics; yet he went too far in this, omitting not only all the drawings, but even very often the calculations as well!

He has been deliberately incomplete in his description of the arrays of battles, as in this instance he referred ^{28} to his father’s Crychconst, at the time still to be published, which however never appeared in print; thus we find with him a great lack of detail in this very important part of army tactics, which consequently has been more amply recorded by Beeckman in his annotations.

Hendrick has very often inserted in his father’s texts personal remarks (it is true under the headings H.S.); (…)

In the very arbitrary classification of chapter VIII: About the theory of war of his book, into 17 chapters, he has unnecessarily deviated from the original framework and has caused much trouble to an expert compiler such as Brialmont, who erroneously thought to have discovered Stevin’s complete Théorie de l’art de la guerre.
To finalize this discussion about the manuscripts on war matters we can refer to the contribution of Van den Heuvel ^{29}, where the still existing manuscripts are described in detail. In volume IV of the Principal Works Works of Simon Stevin, Schukking discusses under the title Vant belegheren der Steden en Sterckten (Of besieging towns and fortresses) the contents of many of these manuscripts with many details.
Figure 5: The “spabijlhou”, Brussel, Koninklijke Bibliotheek van België, Kostbare Werken, VH 28529A. See also figure 4.9 in “Wonder en is gheen wonder”.
Caption: During the siege of towns tunnels are dug under the walls. Stevin suggested for this labour a handy equipment, the socalled “spabijlhou” (spadeaxepick).

The “nederduytsche Mathematicque”: in summer time in the field, in winter time inside.
The foundation of the University of Leiden in 1575 must be seen in the war of the Protestant northern part of the Low Countries against the Spanish troops in the South. In the request for the foundation made by William the Silent (Willem de Zwijger) the University is called the bastion for the protection of the whole country. Some professors, such as Justus Lipsius and Scaliger wrote essays about military arts. The education was rather theoretical, with emphasis on the “artes militares” of the antiquity, and not at all practically oriented. Moreover the courses were given in Latin. The war with Spain needed practically trained engineers, who were able to fortify towns very rapidly during the many campaigns. To foresee in this demand Prince Maurice requested Stevin to draw up a program by which engineers could be educated in the Dutch language, the socalled Duytsche Mathematicque. Although this education did not have an academic character, it could be seen as part of the university activities. One can read in the resolutions of the curators of the university of January 10ty 1600 ^{30}:
Sijne Excellentie heeft tot dienst van den lande ende bevorderinge der geenre, die hun tot oeffeninge van het ingenieurscap sullen begeven, orbaer verstaen seeckere ordre gevolcht te worden in de leeringe, die men daer aff in de Academie tot Leyden doen sal.
It was the intention of the founders of the school that the students became as quickly as possible full engineers:
Hyer toe sal men leeren die arithmeticque oft het tellen ende het landtmeten maer alleenlyck van elck soe veel, als tottet dadelyck gemeene ingenieurscap nodich is. Die soe verre gecomen sijn, hebben se alsdan lust die diepsinnige dingen grondelycker t'ondersoucken dat sullen sy mogen doen.
One expects from the students following the courses that they have some mathematical background and the way the different aspects of mathematics are taught is of importance. In first instance one has to learn the different mathematical operations (addition, subtraction, multiplication, division, extraction of roots) on integers, on fractions and decimal fractions and the rule of three or the invention of the fourth proportional of quantities, by which one solves quadratic equations. There were enough textbooks available on these matters at that time ^{31}. This knowledge of arithmetic was enough for this kind of engineers. Next topic was geometry and in particular the land surveying. The main topics in that field were:

"het inhoudt vant plat" (oppervlakte) te vinden door verdeling in driehoeken,^{32}

"het meten opt papyer van dijcken, wallen ende eerdewercken, te weten hoe veel schachten of voeten den voergestelde wal ofte hoop eerden in heeft" ^{33}.
Once this theoretical knowledge was instructed the students has to exercise on the field:
deur 't cleyn verstaende, watter int groot moet gedaen sijn, soe sal men comen totte datelycke landtmetinge int velt, hun wijsende hoe men in plaets van regel, passer ende winckelhaeck op papier, ander gereedtscap opt velt gebruyct
The purpose of this field work is to become familiar with other instruments than the compass, the ruler and the try square, which they have used during their activities on paper. After this fieldwork the students have to learn to draw on paper what they have measured on the field and conversely they must be taught to measure out by means of beacons on the field what is given as a sketch on a figure ^{34}.
After all these preliminary instructions the students were guided to the main activities of an engineer:
sullen bequaem sijn om totte fortificatie of sterckbouwinge te comen, waertoe bereyt sullen worden houtten of eerden botsen van schantsen ende bolwercken
i.e. the construction of fortifications. The students have to construct models in wood or clay of entrenchments and bastions and they must learn the typical terminology connected to the building of fortresses. Once arrived at this point they can be used in the practical construction of fortifications. In winter time they can be go on with further studies as one can read in the text of the resolution:
Dus verre gecomen sijnde sullen mogen in de somer trecken nae tleger of ter plaetsen daer stercten gebouwt werden, twelck den geenen best gelegen sal sijn, die als soldaten in dienst wesende, dan te velde moeten comen
[...] sullen henlyen, die willen, des winters tot Leyden mogen oeffenen als voeren geseit is, in diepsinniger stoffen
The text of the Prince, inspired by Stevin, concerning the start of these studies is finalized with a sentence in which all the participants are requested to swear that no information will be transferred to the enemy:
Men verstaet oeck dat alle die geene, die tottet leeren deser const van ingenieurscap toegelaten worden, eerst sullen beloven ende zweeren aen den viandt deser landen daermede geenen dienst te doen.

Stevin’s De Sterctenbouwing (The Art of Fortification).
Figure 6: Front page of “De Sterctenbouwing”.
Caption: In “De Sterctenbouwing” Stevin describes how fortifications can be constructed in the Netherlands.
Stevin’s Art of Fortification of 1594 is to be regarded as a textbook. He teaches not only the theory but also the practice of building fortresses, just as many of his Italian and French contemporaries. He, however, surpasses them in clarity. Stevin understood that he wrote for simple people and not for learned man. As explained in the Dedication he used the vernacular ^{35}:
De sterckten deses tijts, welckemen sterckten noemen mach, te weten die door wisconstich beleyt, met sichtstralen {Mathematicam operationem radiis visualibus.} opt meeste voordeel gheteyckent worden, sijn inde Italiaensche ende Françoische spraken soo overvloedelick beschreven, dat ymant dencken mocht daer af ghenouch ghedaen te wesen: Doch twee merckelicke redenen hebben my beweeght derghelijcke oock in onse tael te doen. Ten eersten, om daer mede te gherieven veel Nederduytschen in die spraken onervaren, als Kriegsluyden, Boumeesters {Architectis.}, ende ander tot wetenschap van dien belusticht, waer uijt volghen can, niet alleen vernoughinghe van soodanighe besonder persoonen, maer oock daden {Effecta.} streckende tot dienst des ghemeene Landts. Ten tweeden, om dat onse tale het selve (ghelijc oock alle stof {Materiam.} daer swaricheyt in gheleghen is) veel beter uijtbeelden, ende grontlicker verclaren can als eenighe ander ^{36}.
