II THE PRINCIPAL WORKS OF SIMON STEVIN E D IT E D BY ERNST CRONE, E. J. DIJKSTERHUIS, R. J. FORBES M. G. J. MINNAERT, A. PANNEKOEK A M ST E R D A M C. V. SW ETS & Z E IT L IN G E R J m THE PRINCIPAL WORKS OF SIMON STEVIN VOLUME II MATHEMATICS E D IT E D BY D. J. STRUIK PROFESSOR AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE (MASS.) A M S T E R D A M C. V. SW ETS & Z E IT L IN G E R 1958 The edition of this volume II of the principal works of SIMON STEVIN devoted to his mathematical publications, has been rendered possible through the financial aid of the Koninklijke. Nederlandse Akademie van Wetenschappen (Royal Netherlands Academy of Science) Printed by Jan de Lange, Deventer, Holland The following edition of the Principal Works of SIMON STEVIN has been brought about at the initiative of the Physics Section of the Koninklijke Nederlandse Akademie van Weten­ schappen (Royal Netherlands Academy of Sciences) by a committee consisting of the following members: ERNST CRONE, Chairman of the Netherlands Maritime Museum, Amsterdam E. J. DIJKSTERHUIS, Professor of the History of Science at the Universities of Leiden and Utrecht R. J. FORBES, Professor of the History of Science at the Municipal University of Amsterdam M. G. J. M INNAERT, Professor of Astronomy at the University of Utrecht A. PANNEKOEK, Former Professor of Astronomy at the Municipal University of Amsterdam The Dutch texts of STEVIN as well as the introductions and notes have been translated into English or revised by Miss C. Dikshoorn. THE MATHEMATICAL WORKS OF SIMON STEVIN 1. GENERAL INTRODUCTION When, about 1581, Stevin settled in the Northern Netherlands, he found a country ready to appreciate his talents. The young Republic, at war with Spain and entering a period of great maritime expansion, needed instructors for its navigators, merchants, surveyors, and military engineers. Teachers of mathe­ matics, surveying, navigation and cartography, instrument-makers and engineers found encouragement; their number increased and soon no commercial town was without some of them i). Before the sixteenth century came to an end text­ books in arithmetic, algebra, geometry, and the applied mathematical sciences were available, many written in the vernacular. The teachers and those who patronized them included a great many immigrants from neighbouring countries, expecially from the Southern Netherlands, long known for its learning — the country in which Stevin himself was born. The Stadtholder, Prince Maurice of Orange (1567-1625), was also greatly interested in the mathematical sciences, and so was the new University of Leyden, founded in 1575. Several of these early Dutch mathematicians and teachers of mathematics are still remembered. Apart from Stevin, we find among them Adriaen Anthonisz (c. 1543-1620), several times burgomaster of Alkmaar and a military engineer, who made the value ~ for tt known in Europe2); Ludolph Van Ceulen ( 1540-16 1 0 ), fencing master at Delft, who computed ir first in 20, then in 33 and finally in 35 decimals by the ancient Archimedean method of inscribed and circumscribed polygons; and Claes Pietersz or Nicolaus Petri, after 1567 school­ master at Amsterdam, who wrote a series of Dutch textbooks, which show con­ siderable knowledge of contemporary science. Rudolf Snel, or Snellius (1546- 1612), taught at Leyden University and edited the mathematical works of Petrus Ramus, the Parisian educator. A popular school for navigators at Amsterdam was conducted by the Reverend Petrus Plancius ( 1552-1 6 2 2), cartographer and instrument-maker. Among the scientific amateurs we find Jan Cornets De Groot (1554-1640), patrician of Delft, whose attainments have been eclipsed by the fame of his son, known as Hugo Grotius. With several of these men Stevin entered into correspondence or personal contact, in particular with De Groot and Van Ceulen at Delft. *) This was a development typical of the period. E. G. R. Taylor, in The Mathematical Practitioners of Tudor and Stuart England (London, 1954), lists 582 such practitioners active between 1485 and 1715. 2) This value is sometimes called that of Metius through a confusion between Anthonisz and his son, who adopted the name of Metius. The intellectual climate of Holland seems to have agreed with Stevin. During the years 1582-86 several of his books appeared, first his Tables of Interest, then his Problemata Geometrica, then his Tenth, his L’Arithmétique, a Pratique d’Arithmétique, and the three books on mechanics, which also contain creative mathematical thoughts. These are the books that have established Stevin’s po­ sition in the history of mathematics. It is of some interest to sketch, in some­ what greater detail than in Vol. I, pp. 16-19, the nature of his contributions. ' 2. In Stevin’s formative years the decimal position system, based upon the Hindu- Arabic numerals in their present form 0 , 1, ..., 9, was already widely accepted in Europe and commonly used by those who professed the mathematical sciences. Elementary arithmetic, using this system, could be learned from many textbooks, available in Latin, French, German, and Flemish. Stevin specially mentions the French Arithmétique of Jean Trenchant, first .published in 1558. From books such as these he could also learn the application of arithmetic to commercial transactions, as well as the computation of single and compound interest. They also often contained operations with radicals such as V2, V3, etc. Some features of these books must have been irksome to him. One of them was their reluctance to recognize 1 as a number and their tendency to designate other numbers as “irrational” or “surd”, as if they belonged to a lower class. Other objections were of a more practical nature, such as the reluctance of the authors to illustrate their rules of interest by tables, which still were held as a secret by banking houses, or the clumsy fractional calculus, which used either the numerator-de- nominator notation or the sexagesimal system, but only rarely the more con­ venient decimal .notation. This decimal notation was almost exclusively confined to trigonometric tables, available in several forms, including those published by Rhaeticus (1551), later expanded into the Opus Palatinum (1596). Stevin, in his first published works, tried to remedy some of these shortcomings, and also to improve on the exposition. Thus, in the Tables of Interest, he not only gave a lucid presentation of the rules of single and compound interest, but also published a series of tables, together with a rule for computing them. Some years later, in his Tenth (1585), he showed the use of the decimal system in the calculus of fractions. He took this opportunity to suggest the introduction of the decimal system also into the classification of weights and measures, a proposal which had to wait for partial acceptance until the time of the French Revolution. His theoretical ideas he laid down in his book 11Arithmétique and in a geometrical manuscript, of which only a part was published. Since UArithmétique also contained Stevin’s algebra, while his books on mechanics included several applications of the calculus of infini­ tesimals, Stevin’s. work of these years 1582-1586 can be considered as a fair and often original exposition of most features of the mathematics of his day. In his arithmetical and geometrical studies Stevin pointed out that the analogy between numbers and line-segments was closer than was generally recognized. He showed that the principal arithmetical opérations, as well as the theory of pro­ portions and the rule of three, had their counterparts in geometry. Incommen­ surability existed between line-segments as well as numbers, and since the nature 5 of line-segments was independent of the number that indicated their length, all ¡numbers, including unity, also were of the same nature. All numbers were, squares, all numbers were square roots. Not only was \/2 incommensurable with 2 and :V3, but so was 2 with \J2 and V3; incommensurability was a relative property, and there was no sense in calling numbers “irrational”, “irregular” or any other name which connoted inferiority. He went so far as to say, in his Traiclé des incommensurables grandeurs, that the geometrical theory of incommensurables in Euclid’s Tenth Book had originally been discovered in terms of numbers, and translated the content of this book into the language of numbers. He compared .the still incompletely understood arithmetical continuum to the geometrical continuum, already explained by the Greeks, and thus prepared the way for that correspondence of numbers and points on the line that made its entry with Des­ cartes’ coordinate geometry. Stevin recognized several kinds of quantities: arithmetical numbers, which are abstract numbers, and geometrical numbers, connected with lines, squares, cubes, and rectangular blocks (figures in more than three dimensions were be- yond the compass of the age), which we now denote by a, a2, a?, ..., a 2, a 5 , etc. From this he passed on to linear combinations of geometrical numbers, which he called algebraic numbers. Thus he came to algebra—the theory of equations—, which to him, in his attempt to construe analogies between geometry and arith­ metic, hence between geometrical and arithmetical numbers, consisted in the application of the rule of three to algebraic quantities. His algebra thus forms part of his general ’’arithmetic”. The theory of equations had made considerable progress in the course of the sixteenth century. Cubic equations had been solved, though the ’’casus irreducibilis” still presented difficulties. The new results were laid down by Jerome Cardan in his Ars magna (1545), which became the sixteenth-century standard text on the theory of equations, eclipsing even the Arithmetica integra (1544) of Michael Stifel.