m THE PRINCIPAL WORKS , jH E PRINCIPAL WORKS OF OF SIMON STEVIN SIMON STEVIN EDITED BY VOLUME II ERNST CRONE, E. J. DIJKSTERHUIS, R. J. FORBES MATHEMATICS M. G. J. MINNAERT, A. PANNEKOEK EDITED BY D. J. STRUIK PROFESSOR AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE (MASS.) AMSTERDAM AMSTERDAM C. V. SWETS & ZEITLINGER C V. SWETS & ZEITLINGER 1958 ^ f.fi.1 .t. ..A IV The edition of this volume ll of the principal works of The following edition of the Principal Works of SIMON SIMON STEVIN STWIN has been brought about at the initiative of the Physics Section of the Koninklijke Nederlandse Akademie van Weten­ devoted to his mathematical publications, has been rendered possible through schappen (Royal Netherlands Academy of Sciences) by a the financial aid of the committee consisting of the following members: Koninklijke Nederlandse Akademie van Wetenschappen ERNST CRONE, Chairman of the Netherlands Maritime Museum, Amsterdam (Royal Netherlands Academy of Science) 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. MINNAERT, Professor of Astronomy at the University of Utrecht A. PANNEKOEK, Former Professor of Astronomy at the Muniiipal University of Amsterdam The Dutch texts of STEVIN as well as the introductions and notes have been translated into En^sh or revised by Miss C. Dikshoorn. Printed by J«n de Lange, Deventer, Holland THE MATHEMATICAL WORKS ,4w.-v OF SIMON STEVIN 1. GENERAL INTRODUCTION . When about 1581, Stevin settled in the Northern Netherlands, he found a eeuntry ready to appreciate his talents. The young Republic, at war with Spain atid 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 fdund encouragement; their number increased and soon no commercial town was i^ithout some of themi). Before the sixteenth century came to an end text- ;4&oks in arithmetic, algebra, geometry, and the applied mathematical sciences were pllrailable, many written in the vernacular. The teachers and those who patronized jjj'llem included a great many immigrants from neighbouring countries, expecially ilom the Southern Netherlands, long known for its learning — the country in ■Z' ^ich Stevin himself was born. The Stadtholder, Prince Maurice of Orange ipi'567-1625), was also greatly interested in the mathematical sciences, and so 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 1543-1620), several times burgomaster of Alkmaar and a military engineer, who made the value — for tt known in Europe 2); Ludolph Van Ceulen fl540-l6l0), fencing master at Delft, who computed tt first in 20, then in 33 and finally in 35 decimals by the ancient Archimtfdean method of inscribed and drcumscribed 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 (1552-1622), 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 ot 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 1713. *) 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 of line-segments was independent of the number that indicated their length, ail the years 1582-86 several of his books appeared, first his Tables of Interest, then numbers, including unity, also were of the same nature. All numbers were squares, his Problemata Geometrica, then his Tenth, his VArithmétique, a Pratique all numbers were square roots. Not only was V2 incommensurable with 2 and d*Arithmétique, and the three books on mechanics, which also contain creative V3, but so was 2 with V2 and V3; incommensurability was a relative property, mathematical thoughts. These are the books that have established Stevin’s po­ ^nd there was no sense in calling numbers “irrational”, “irregular” or any other sition in the history of mathematics. It is of some interest to sketch, in some­ name which connoted inferiority. He went so far as to say, in his Traictê des what greater detail than in Vol. I, pp. 16-19, the nature of his contributions. 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 In Stevin’s formative years the decimal position system, based upon the Hindu- correspondence of numbers and points on the line that made its entry with Des­ Arabic numerals in their present form 0, 1, ..., 9, was already widely accepted in cartes’ coordinate geometry. Europe and commonly used by those who professed the mathematical sciences. Stevin recognized several kinds of quantities: arithmetical numbers, which Elementary arithmetic, using this system, could be learned from many textbooks, are abstract numbers, and geometrical numbers, connected with lines, squares, available in Latin, French, German, and Flemish. Stevin specially mentions the cubes, and rectangular blocks (figures in more than three dimensions were be- French Arithmétique of Jean Trenchant, first published in 1558. From books 1 ^ such as these he could also learn the application of arithmetic to commercial yond the compass of the age), which we now denote by a, a^, a^....... ,a^ , etc. transactions, as well as the computation of single and compound interest. They From this he passed on to linear combinations of geometrical numbers, which he also often contained operations with radicals such as V2, V3, etc. Some features called algebraic numbers. Thus he came to algebra—the theory of equations—, of these books must have been irksome to him. One of them was their reluctance which to him, in his attempt to construe analogies between geometry and arith­ to recognize 1 as a number and their tendency to designate other numbers as metic, hence between geometrical and arithmetical numbers, consisted in the “irrational” or “surd”, as if they belonged to a lower class. Other objections application of the rule of three to algebraic quantities. His algebra thus forms were of a more practical nature, such as the reluctance of the authors to illustrate part of his general ’’arithmetic”. their rules of interest by tables, which still were held as a secret by banking The theory of equations had made considerable progress in the course of the houses, or the clumsy fractional calculus, which used either the numerator-de­ sixteenth century. Cubic equations had been solved, though the ’’casus irreducibilis” nominator notation or the sexagesimal system, but only rarely the more con­ still presented difficulties. The new results were laid down by Jerome Cardan venient decimal notation. This decimal notation was almost exclusively confined in his Ars magna (1545), which became the sixteenth-century standard text on to trigonometric tables, available in several forms, including those published by the theory of equations, eclipsing even the Arithmetica integra (1544) of Michael Rhaeticus (1551), later expanded into the O^us Palatinum (1596). Stevin, in Stifel. Cardian’s book also contained Ferrari’s reduction of the fourth-degree e- his first published works, tried to remedy some of these shortcomings, and also quation to one of the third degree, Stevin knew these books intimately, and also to improve on the exposition. studied Bombelli’s VAlgebra (1572), which treated the “casus irreducibilis” Thus, in the Tables of Interest, he not only gave a lucid presentation of the with complex numbers and introduced an improved notation. Stevin did not have rules of single and compound interest, but also published a series of tables, much use for these complex numbers, because he did not see a possibility of together with a rule for computing them. Some years later, in his Tenth (1585), finding a numerical approximation for a number like V 4 + 5/, in contrast to he showed the use of the decimal system in the calculus of fractions. He tO(^ such a number as y6, where a numerical approximation can be obtained. How­ this opportunity to suggest the introduction of the decimal system also into the ever, he liked Bombelli’s notation, and availed himself of it in his own book. classification of weights and measures, a proposal which had to wait for partial Against negative numbers, with which Cardan had played, he had no objection, acceptance until the time of the French Revolution. His theoretical ideas he laid even if he did not use them as freely as we do now. In the light of our present down in his book VArithmétique and in a geometrical manuscript, of which only knowledge we are inclined to wonder why in his speculations on the analogies a part was published. Since VArithmétique also contained Stevin’s algebra, while between the arithmetical and the geometrical continuum he did not assign a his books on mechanics included several applications of the calculus of infini­ geometrical meaning to negative numbers, but even Descartes and his immediate tesimals, Stevin’s work of these years 1582-1586 can be considered as a fair and successors did not use negative coordinates. The study of directed quantities be­ often original exposition of most features of the mathematics of his day. longs to a much later stage of mathematical development. In his arithmetical and geometrical studies Stevin pointed out that the analogy The main merit of Stevin’s I!Arithmétique is the systematic way in which he between numbers and line-segments was closer than was generally recognized. He discusses operations with rational, irrational, and algebraic numbers, and the showed that the principal arithmetical operations, as well as the theory of pro­ theory of equations of the first, second, third, and fourth degrees. To our feeling portions and the rule of three, had their counterparts in geometry. Incommen­ he went too far in stressing the analogy between arithmetical and algebraic en­ surability existed between line-segments as well as numbers, and since the nature tities, even the theory of equations becoming an application of the rule of three.' However, this latter point of view met with little success, even among his con­ temporaries and the algebrists who followed him. His particular notation for and straightedge alone. But Archimedes’ influence is also visible in Stevin’s books equations was also soon abandoned i). on mechanics, where Stevin, modifying Archimedes’ later so-called exhaustion Geometry, during the sixteenth century, still followed closely the track of method, appears as one of the first Renaissance pioneers in the field of mathe­ Euclid, whose Elements, from 1482 on, were available in several printed editions matics afterwards known as the theory of limits and the calculus. and translations. Stevin was especially familiar with the Latin editions prepared Archimedes’ handling of what we now call limit and integration processess by Zamberti (1546) and by Clavius (1574). Christopher Clavius (1537-1612), was still on the extreme confines of knowledge. Only a few mathematicians as Stevin’s contemporary, who was the Vatican’s astronomer, excelled as a writer of yet were able to emulate Archimedes, among them Commandino, who had ap­ plied his methods in the determination of centres of gravity. Stevin’s friend Van textbooks, which embraced well-nigh the whole of the mathematical and astrono­ Geulen was engaged in improving on Archimedes’ computation of n. One diffi­ mical sciences of his day. There is reason to believe that Stevin was quite culty in Archimedes was his cumbersome method of demonstration in dealing familiar with these books, and that Clavius equally remained in contact with Stevin’s with limit processes (which had already appeared in Euclid and was typical of work. To his study of Euclid we owe Stevin’s Traicté des incommensurables Antiquity). When Archimedes wanted to demonstrate that a certain quantity Q, grandeurs, already mentioned, and his Problemata geometrica, the former pro­ e.g. the area of a parabolic segment, was equal to A, he showed that the two bably, the latter certainly forming part of that longer geometrical manuscript hypotheses g < /4 and (g > /1 both led to an absurdity, so that Q = A was which was to do for geometry what UArithmétique had done for arithmetic. the only possibility. Stevin replaced this indirect proof by a direct one. Demon­ Euclid’s influence in the Problemata is particularly evident in the sections dealing strating that the centre of gravity of a triangle lies on the median, he argues that with proportional division of figures and with regular bodies, enriched with a if the difference between two quantities B and A can be made smaller than any description of the semi-regular bodies, which had a touch of originality. Stevin assignable quantity e, and \B — A \ <C s, then B = A (see Vol. I, p. 43). Here knew several of them through Albrecht Durer, who had described them in his Stevin entered upon a course which was to lead to the modern theory of limits. IJnderweysung of 1525, but he added some others, while rejecting one of them. We can discern a certain impatience with the method of the Ancients in Stevin He does not seem to have known that all thirteen semi-regular bodies had been and his successors; an impatience quite conspicuous in Kepler. These men applied described in Antiquity by Pappus, who had mentioned Archimedes as the dis­ short cuts in what wc call the integration process, because they wanted resultç coverer, information not readily available in the 1580’s, since Pappus’ text was rather than exact proofs. They used methods of far more dubious rigour than only published in 1589- We do not know whether Stevin was aware of other Stevin’s, even though they knew that the only rigorous proof was the Archi­ books which appeared in the sixteenth century, with descriptions of semi-regular medean one. Stevin must have experimented with such short cuts, as we can see bodies, sometimes beautifully illustrated: the only source he quotes is Dürer. in his paper on Van de Molens (On the Mills', Work XVI; Vol. V). If we like, we The Problemata also show Stevin as a student of Archimedes. The editio prin­ can see a topic related to the calculus in Stevin’s determination of the equation of ceps of Archimedes appeared in 1544, when Venatorius published the Greek text the loxodrome on a sphere, in his book on Cosmography, by means of the series of all the works, a Latin translation, and the commentaries of Eutocius. More­ over, a selection of the works in Latin appeared in 1558 through the care of tan K (sec 10' -f sec 20' + ....... -f sec n. 10'). 10', Commandino. The theories of Archimedes, the most advanced mathematician of where K is the angle between the loxodrome and the meridian. The expression Antiquity, were not easily understood, and creative work based on them was even more difficult. Stevin was among the first Renaissance men to study Ar­ is an approximation of tan X ƒ J sec epdq), expressed in degrees. chimedes with a certain amount of independence. In the Problemata he took some problems he had found in Archimedes’ An the Sphere and Cylinder and gener­ During the latter part of Stevin’s life the mathematical sciences continued to alized them somewhat; this gave him an opportounity to apply the methods given flourish in Holland. This was the period in which he wrote, or rewrote, the by Eutocius for the construction of the tw'o mean proportionals between two lines; different books which he assembled in 1605-1608 in the Wisconstighe Ghe- a : X — X \ y — y : b, ^ problem which cannot be solved by means of compass dachtenissen. The short Appendice algébraique, which contains a method for ap­ proximating a real root of an algebraic equation of any degree, dates from 1590. *) The criticism of K. Menger on the promiscuous use of the symbol x in modern This was also the period in which Stevin acted as a teacher and adviser to Prince mathematics, and in particular of its use as a dummy index in expressions like Maurice of Orange. He remained in personal contact and correspondence v/ith ƒ f{x)dx, which he writes Sf, or as „indeterminates” in expressions like = many of his colleagues, including representatives of the younger generation, out­ standing among whom was Rudolf Snel’s son Willebrord (1580-1626), a gradu­ X - which he writes = * -f i, lends a touch of modernity to Stevin’s ate of Leyden University. This younger Snellius, who translated the Wiscon­ notation. The latter expression, in the symbolism of L'Arithmétique, is written in stighe Ghedachtenissen into Latin, later succeeded his father in the chair at Ley­ den, and is remembered as the discoverer of “Snellius’ law” in the theory of the form -------- = (J) q. very much in Professor Monger’s spirit. See K. optics and the first man on record to perform an extensive triangulation. An­ Menger, Calculus. A modern approach. Boston 1955, or Math. Gazette 40 (1956), pp. other Leyden mathematician was Frans Van Schooten (1581/82-1645), who 246-255. after Van Ceulen’s death in 1610 taught at the engineering school founded by Prince Maurice. His son and namesake (1615-1660), who became professor öf The other mathematical sciences represented in the Ghedachtenissen are plane mathematics at Leyden University and was the teacher of Christiaan HuygenS, and spherical trigonometry, with tables of sines, tangents, and secants. They showed in his works Stevin’s influence. Older than these men was Philippiis contain little that was new at the time, though the spherical trigonometry was Van Lansbergen (1561-1632), a minister in Zeeland and an able mathematiciairt, somewhat simplified as compared with previous expositions. His understanding who shared Stevin’s preference for the Copernican system. We also know that of the geometry of the sphere also led Stevin, in his books on navigation, to a Stevin was in personal contact with Samuel Marolois (c. 1572-before 1627)', a careful discrimination between sailing along great circles and along rhumb lines military engineer who wrote on perspective, and we may safely assume that (orthodromes and loxodromes, as Willebrord Snellius called them in his trans­ Stevin was in touch with the surveyors Jan Pietersz. Dou (1572-1635), the first lation of the Ghedachtenissen'). This was still an enigma to most sailors and to publish a Dutch edition of some of Euclid’s books, and Ezechiel De Decker, teachers of navigation, although the difference had already in 1546 been clearly whose work shows considerable influence of Stevin. This was also a period in stated by Pedro Nunes, mathematician in the University of Coimbra; Mercator, the which appeared many elementary mathematical textbooks, of which those of Duisburg cartographer, had represented the loxodromes by straight lines on his Willem Bartjes were used for more than two centuries and made his name well-known world map of 1569 (they already appear on his terrestrial globe proverbial in Dutch. Dutch cartographers, among them Plancius, Willem Ba- of 1541). The mathematics of the loxodrome was still poorly understood; as rendtz (of Nova Zembla fame), Jodocus Hondius (son-in-law to Mercator), anid a matter of fact, this understanding only matured when the calculus began to take William Jansz. Blaeu, were building up an international reputation. It would be shape, in the latter part of the seventeenth century. Stevin was able to compute interesting to know something about the relationship between Stevin and Isaac tables which for a variable point of each loxodrome, belonging to seven given Beeckman (1588-1637), the Dordrecht physician and teacher, who through his bearings 11°15', 22°30' ....... 78°45' with the meridian, gives the latitude as contact with Descartes forms one of the links connecting the Stevin period of a function of the longitudinal difference with the point where the loxodrome in­ Dutch mathematics with that of Descartes. We do know that after Stevin’s death, tersects the equator. Stevin also caused copper curves to be made, which had the in 1620, he visited his widow and studied some of her late husband’s manuscripts. form of rhumb lines, for the seven principal bearings and by means of which The most original of the mathematical books published in the Ghedachtenis- on a globe of suitable size the loxodrome could be drawn for any given initial sen is the Perspective. Its subject was developed by the Italian artists of the point. Stevin can thus also be considered as a contributor to mathematical carto­ fifteenth century and during the sixteenth century several books on it had ap­ graphy. peared, some with beautiful pictures. These books were written by and for painters and engineers and contained a rather loose presentation of the mathe­ matical theory involved, which often was not more than a set of prescriptions for foreshortening. The first systematic exposition of the mathematical theory öf perspective appeared in 1600, when Guidobaldo Del Monte published his Pèf- spectivae libri sex. It is likely that by the time this book appeared Stevin’s mathe­ matical theory of perspective, the result of his reflections on architecture, military engineering, and the technique of drawing in general, was already far advanced. It is also probable that in the final draft of the manuscript Stevin was influenced by Del Monte. In the book Stevin develops the laws of perspective in his usual systematic and didactic way (the Prince may well have been no easy pupil!), derives the laws of the vanishing poinits, discusses the case that picture plane and ground plane are not at right angles, and also investigates what niây be called the inverse problem of perspective: to find the eye when a plane figure and its perspective are given. Despite a certain long-windedness the book can still serve as an introduction to perspective; it is among the writings of Stevin which are least antiquated. The Meetdaet, another book of the Ghedachtenissen, was based on the manu­ script on geometry to which Stevin referred at the time when he was writing UArithmétique and of which he published a section in the Problemata, It also shows the influence of Prince Maurice, which may have improved the expo­ sition and added a practical touch. The name became Meetdaet, French Pratique de Géométrie, a counterpart to the Pratique d'Arithmétique which Stevin had added to his VArithmétique in order to give some practical applications of his theoty. Most of the subject matter of the Problemata reappears in the Meetdaet, some­ times in a slightly modified form. TAFELEN VAN INTEREST TABLES OF INTEREST 13 INTRODUCTION The Tables of Interest, the first book published by Stevin, represented a kind of challenge to an ancient and established tradition. Money-lending leads to problems concerning the payment of interest, and with the expansion of mercan­ tile activity and of banking in the later Middle Ages such problems had a tendency to become complicated. Many banking houses engaged in large-scale dealings of varied aspects, involving questions of insurance, of annuities and other payments at set intervals, of discounting of sums due at a later date and related transactions. Against their power, objections based on canon law, pro­ hibiting or circumscribing the taking of interest, were of little or no avail. The Baldi and Medici of Florence, the Welsers and Fuggers of Augsburg at one time or another ruled financial empires, respected and feared by king, emperor, and pope. In a period where even multiplication and division of integers were considered difficult operations, only experts could answer with authority questions involving the computation of interest. The larger and more established houses had found it convenient to have such experts compute tables of interest and to keep them on file as confidential information. Such tables remained, as Stevin expressed it, “hidden as mighty secrets by those who have got them.” They could remain hidden as long as the number of skilled computers was small. This period came to an end with the spread of arithmetical instruction in the sixteenth century. One of these early manuscript tables, composed about 1340, has been preserved in a copy finished in 1472. It was prepared for the Florentine house of the Baldi by their commissary Francesco Balducci Pcgolotti as part of his Pratica della Mer­ catura. This book was published in 1766 (i), an English translation appeared in 1936 (2). The tables of interest appear as an insert between other topics (3); they record the increase, at compound rate of interest of 1, lè, 2, ..., 8 per cent, of 100 lires. Each of the 15 tables has 20 terms. Here follows, as an example, the table for 2 per cent: Le 100 lire a 2 per cents I’anno 1. lire 102.—.— 11. lire 124. 6. 8 2. lire 104.—.10 12. lire 126.16. 4 P) Della Decima e di varie altre grave^r^e imposte dal comune di Firenze, Della moneta e della mercatura de Florentini fine al secolo XVI, 4 vols., Lisbon and Lucca 1765-1766. The book was published anonymously, but the author became known as Gian-Francesco Pagnini della Ventura (1715-1789), Florentine Chancellor of the Tithe. See A. Evans, next ref., pp. IX-X. (*) A. Evans, Francesco Balducci Pegolotti La pratica della mercatura. The Mediaeval Academy of America, Cambridge, Mass., 1936, LIV -I-443 pp. See pp. XV-XXVI on the life of Pegolotti. (®) A. Evans, /.c. **) pp. 301-302; Pagnini, l.cF) pp. 302-304. 14 15 3. lire 106. 2. 13. lire 129. 7. 1 4. lire 108. 4. 14. lire 131.18.10 téé" (9). The date of publication is important. Lyons was famous as a money 5. lire 110. 8. 15. lire 134.11. 7 market, where kings and other nobles bargained for huge loans with the most 6. lire 112.12. 16. lire 137. 5. 3 important bankers of Europe. A first attempt was made in 1555 by King Henry II 7. lire 114.17. 17. lire 140.—. 2 and his financiers to consolidate the many haphazard royal loans of the past and 8. lire 117. 3. 18. lire 142.16. 2 to establish a regular system of amortization. This was the "Grand Parti”, famous 9. lire 119.10. 19. lire 145.13. 3 in its days, and so popular that wide strata of the population hastened to sub­ 10. lire 121.17.11 20. lire 148.11. 6 scribe (10), Trenchant’s book, with its extensive chapter on simple and compound interest, reflects the public desire for understanding the intricacies of the money [1 lira = 20 soldi, 1 soldo = 6 denari} market. The third part of his book contains four interest tables, of which two were specially compiled to illustrate the “Grand Parti”. This transaction, to which It is interesting to note that the Baldi computed the accumulation of capital later also Coignet (H) and Stevin return, is described in the following problem. not at simple, but at compound interest. This practice was already old in their "En l’an 1555, le Roi Henri pour ses affaires de guerre, prenait argent des days. At any rate, Leonardo of Pisa, whose Ltber Abaci dates from 1202, and banquiers, à raison de 4 pour 100 par foire (12); c’est meilleure condition pour whose problems reflect early thirteenth-century mercantile practice, also accepts eux, que 16 pour 100 par an. En ce même an avant la foire de la Toussaint il compound interest (4). Its legititmcy was a subject of juridical controversy for reçut aussi par les mains de certains banquiers la somme de 3954941 ecus et many centuries (5). plus, qu’ils appelaient le grand parti, à condition qu’il payerait à raison de 5 pour 100 par foire, jusqu’à la 4l-ième foire; à ce paiement il demeurerait quite de It is not unlikely that further search in European libraries will reveal other tout; à savoir laquelle de ces conditions est meilleure pour les banquiers? La pre­ treatises on interest, with or without tables. An example is a manuscript text on mière à 4 pour 100 par foire est évidente, c’est à dire on voit son profit évidemment. arithmetic by Rucellai, a Florentine citizen, bearing the date April 23, 1440, and Mais la dernière est difficile; de sorte que les inventeurs de cette condition-là found in the Bibliotheca Nazionale in Florence. It contains tables of ne l’ont trouvée qu’à tâtons et presque avec un labeur inestimable. Maintenant je interest computed, it says, by Antonio Mazinghi as part of an exposition on simple veux montrer à faire telles calculations légèrement (facilement) et précisément and compound interest (6). avec raison démonstrative facile à entendre.” Luca Pacioli, in his widely read Summa of 1494, also mentions tables of in­ The question raised is therefore the following. The king borrows 3,954,941 terest and sketches the way how to compute them (7). There are no tables in the écus. Everyq uarter year he has to pay interest and the total debt must be paid Summa, only a number of problems on interest, simple and compound. In order off after 41 payments. What is more advantageous to the bankers: payment of to find tables in print we still have to wait for half a century. Then we meet a few 4 per cent interest each quarter and return of the principal at the 41st payment, in the Arithmétique of Jean Trenchant (8). or payment of 5 per cent interest each quarter and no extra payment at the end? Nothing is known about Trenchant except that he was a teacher of mathe­ ’Trenchant, in solving this problem, introduces two tables. 'The first one is a matics at Lyons, who in 1558 published a book called L!Arithmétique départie es trois livres, which passed through many editions, occasionally 'Wevue et augmen- table which lists the increase in value of 107 (1.04)” , n — 0,1,..., 40: 1 0 0 0 0 0 q^o 1 0 4 0 0 0 1 0 (*) Liber Abaci, Scritti diLeonardo Pisano, ed. B. Boncompagni, vol. 2 (1862) p. 267. 1 0 8 1 6 0 0 0 (®) Leibniz, in his essay Meditatio iuridico-mathematica de interusurio simplice, Acta Erudi­ torum 1683, defended the use of compound interest according to the formula 4 6 1 6 3 6 5 9 = Cj (i -f iy. He was attacked by other jurists with the argument that the taking 4 8 0 1 0 2 0 6 of interest on non-paid interest is prohibited. See M. Cantor, Politische Arithmetik (Leipzig, 1898, X + 1^6 pp.), p. 35. (*) The fourth edition has the title : Varithmétique de Ian Trenchant départie en trois livres. (•) The manuscript is in the Biblioteca Nazionale, Florence, call number Palatino 573, Ensemble un petit discours des Changes avec Tart de calculer aux Getons. Revue et augmentée pour author Girolamo di Piero di Chardinale Rucellai (This informition is due to Dr. R. De la quatrième édition, de plusieurs règles et articles, par TAutheur. A Lyon, pat Michel love, Roover, Aurora, NY). 1578, 375 pp. Trenchant was therefore alive in 1578. The edition of 1363 is also „revue et (’) L. Pacioli, Summa de Arithmetica Geometria Proportioni et Proportionalité (Venice, augmentée”. 1494, second ed., Toscolano, 1523), first part, 9th distinctio, 5th tractatus. Pacioli writes ('®) R. Doucet, Le grand parti de Lyon au siècle. Revue historique /77 (1933), pp. "del modo a sapere componere le tavole del merito”. The term “m^rito”, French 473-313 ; '7* (1933)» PP- î Ehrenberg, Das Zeitalter der Fugger II (Jena, 1896, “mérite”, stands for what Stevin calls "profitable interest.” Compound interest is “a capo 18+367 pp), p. I o I ff. ; translated as Capital and Finance in the Age oft he Renaissance (New Y qrk, d’anno, o altro tempo, o termine”. See footnote'®). 1928, 590 pp.). Information oxile grand parti is duc to Mrs C. B. Davis, Ann Arbor. Mich. (•) On Jean Trenchant, see H. Bosmans, V Arithmétique de Jean Trenchant, Annales Soc. (") Livre d’arithmétique... composé par Valentin Mennher Allemand: revue, corrigée et aug- Sc. Bruxelles 33 (1908-09), le partie, pp. 184-192; G. Sarton,/w» Trenchant, French , mentée... par Michiel Coiÿtet. Anvers, 1373, 141 pp. Doucet and Ehrenberg /.r.'®) write Mathematician of the Second Half of the Sixteenth Century, Isis 21 (19*4), pp. 207-208 ;C.M. Coquet instead of Coignet. Waller Zeper, De oudste intresttafels in Itali'é, Frankrijk en Nederland met een herdruk van ('*) There were four fairs a year at Lyons; „par foire” therefore means : “every quarter Stevins „Tafelen van Interest”, Diss. Leiden, (Amsterdam, 1937, 93 -}- 92 pp.), esp. Ch. III. year”.