Vestigia mathematica: Studies in medieval and early modern mathematics in honour of H.L.L. Busard

238 Pages·1993·12.66 MB·English, German, French

by  M. Folkerts and J.P. Hogehdijk (eds)

VESTIGIA MATHEMATICA
Studies in medieval and early modern mathematics
in honour of H.L.L. Busard
edited by
M. Folkerts and J.P. Hogehdijk
Amsterdam - Atlanta, GA 1993
Contents
Introduction.........................................................................................................7
H. L. L. Busard; Publications .........................................................................11
P. Bockstaele A challenge to the mathematicians of the University
of Leuven as a New Year’s gift for 1639 .................. 15
H. J. M. Bos Johann Molther’s ‘Problema Deliacum’, 1619.........29
S. Brentjes Varianten einer Haggäg-Version von Buch II der
Elemente ........................................................................47
C. Burnett Ocreatus .........................................................................69
A. Djebbar Deux mathématiciens peu connus de l’Espagne du
XI' siècle: al-Mu’taman et Ibn Sayyid ......................79
Y. Dold-Samplonius The volume of domes in Arabie mathematics ..........93
M. Folkerts Die Rithmachm des Werinher von Tegernsee ...........107
J. P. Hogendijk The Arabie version of Euclid’s On Division ............143
B. Hughes Robert Recorde and the first published equation . 163
W. Kaunzner Über die beiden nachgelassenen mathematischen
Handschriften von Adam Ries ..................................173
P. Kunitzsch ‘The Peacock’s Tail”: On the names of some
theorems of Euclid’s Elements .................................205
R. Lorch Abu Kamil on the pentagon and decagon ...............215
J. van Maanen The ’double-meaning’ method for dating
mathematical texts ....................................................253
G. Molland Roger Bacon’s Geometria Speculativa ......................265
B. A. Rosenfeld ‘‘Geometric trigonometry” in treatises of
al-Khwarizrru, al-Mähänl and Ibn al-Haytham ... .305
C. J. Scriba Zur Aufgabe S6 des Byzantischen Rechenbuchs
Cod. Vindob. Phil. Gr. 65 ....................................... 309
J. Sesiano La version latine médiévale de TAlgèbre
d’.Abü Kâmil ...............................................................315
.A. Simi, Some 14th and 15th century texts on practical
L. Toti Rigatelli geometry .......................................................................453
Addresses of the authors ...............................................................................471
ISBN; 90-5183-536-1 (CIP)
®Editions Rodopi B.V., Amsterdam - Atlanta, GA 1993
Printed in The Netherlands
JiMTKODUCTION
Introduction
Hubertus Lainbertus Ludovicus Busard Wcis born on 21 August 1923 in
Sittard (The Netherlands, Province of Limburg). He had three sisters and
no brothers. After finishing grammar school in Sittard with certificates a
(1942) and /? (1943), he studied mathematics, physics and astronomy at the
University of Utrecht. In 1949 he graduated in mathematics. In 1951 he was
appointed as a lecturer in mathematics and physics at the Technical College
in Venlo, a city in the South of the Netherlands, very close to the German
border. He taught there until his retirement in 1984. Since 1951 he has been
happily married to Yvonne (née Bollen). They have two children and five
grandchildren.
It was E. J. Dijksterhuis (1892-1965), professor of history of science in
Utrecht (since 1953) and Leiden (since 1955), who first interested Busard in
the history of science, and he became Dijksterhuis’ last student. Busard was
awarded his doctorate in science (Dr.rer.nat.) in Utrecht in 1961 for a thesis
on Nicole Oresme’s Quaestiones super geometriam Euclidis, which he edited
for the first time (1). This dissertation was the beginning of Busard’s involve­
ment in Western medieval mathematics. The only articles he has published
outside the field of medieval mathematics are on Viète (3, 1964) and Mydorge
(6, 1965); they were related to studies by J. E. Hofmann, who invited him
to his meetings on history of science at the Mathematical Research Institute
Oberwolfach (Black Forest). Connected with his thesis are two articles on
infinite series in the Middle Ages (2, 4) and another (18) on the sources of
Oresme.
Almost all Busard’s publications are editions of Latin mathematical texts.
They cover all branches of medieval western mathematics. In geometry he
edited the influential treatise of Dominicus de Clavéïsio (5, 1965) and three
texts on surveying which were translated by Gerard of Cremona in the 12th
century (Abû Bekr, Liber Saydi Abuothmi, Liber Aderameti: 8, 11, 1968-
69), Ahmed ibn Jusuf's book on similar arcs, also translated from the Arabic
by Gerard (19, 1973), the treatise on isoperimetry translated directly from
the Greek (27, 1980), and John of Gmunden's work on trigonometry (23,
1971). Arithmetical texts edited by Busard are the well-known treatise on
the computation with fractions by Johannes de Lineriis (9, 1968), the Arith-
vnetica speculativa by Johannes de Mûris, which is closely connected with
Boethius’ work on arithmetic (14, 1971), and the highly interesting and in­
fluential treatise on theoretical arithmetic by Jordanus de Nemore, who was
one of the most important mathematicians in the Latin West during the
Middle Ages (32, 1991; the main results in 34, 1992). Yet another work by
Jordanus was edited by Busard, namely the treatise on proportions, which
appeared together with a similar work attributed to Campanus (17, 1971). In
the same year he published an edition of Albert of Saxony’s treatise on pro­
portions (12). Some results of Busard’s early research became part of a small
8 INTRODUCTION INTRODUCTION 9
booklet on mathematics in the Middle Ages which is based on a paper given Busard’s work on the history of mathematics was honoured when he was
in Paris (10, 1969). In 1974 Busard published the text of an algebraic chapter elected as corresponding member of the Académie Internationale d’Histoire
in Johannes de Mûris’ De arte mensurandi (23). Later he decided to edit the des Sciences in 1971 and as effective member in 1978.
whole text of this compilation. He has now finished this work, and it will be Until his retirement in 1984, Busard did all his scientific work in his spare
published soon (37). time. The example of Busard shows that a high scientific output need not
Although the editions mentioned above are very substantial contribu­ conflict with family commitments. In fact, his harmonious family life has
tions to the history of medieval Western mathematics. Busard has done even been the inexhaustible source of energy for his work. Busard has always been
more: perhaps the most important part of his œuvre consists of his numer­ a very cooperative and helpful person, ais most of the contributors to this
ous articles and books on the Latin Euclid. Before Busard’s time only the volume know from personal experience.
reworking by Campanus (ca. 1255) had been printed - in editions of the The plan to honour Busard on his 70th birthday with a volume of studies
15th and 16th centuries -, and almost all information on the Latin Euclid in dates from 1990, and it was secretly discussed at a conference on medieval
the Middle Ages depended on an article by Marshall Clagett in 1953 (The mathematics in Wolfenbüttel (this was somewhat difficult because Busard
Medieval Latin Translations from the Arabic of the Elements of Euclid, with was of course present). It was very easy to find enthusiastic authors. Many of
Special Emphasis on the Versions of Adelard of Bath, in: /sis, 44 (1953), the articles in this volume of studies are in the style of Busard’s own work,
16-42). It was Busard who put our knowledge of the history of this most and some of these articles are the fruit of hundreds or even thousands of
important Greek mathematical text in the Latin West on a solid base: in the hours of labour. Together they reflect the high esteem of their authors for
last 25 years he h2is produced critical editions of six complete versions of the Dr. Busard.
Elements which originated in the 12th and 13th centuries, and in addition We thank the Department of Mathematics of the University of Utrecht
he has edited some scholia and commentaries connected with Euclid’s main for generously providing access to its computer facilities, and we are grateful
work (20, 21, 22) - to say nothing of his articles on the Euclid tradition to Gerhard Brey (München), Benno van Dalen, Wilberd van der Kallen and
in the West (15, 25, 26, 30, 35). From Busard’s research we know that André de Meijer (all of Utrecht) for their help in solving computer prob­
Euclid’s Elements were translated in the 12th century from the Arabic three lems, to Prof. Klaus Lagally (Stuttgart) for providing his program ArabT^X
times: by Adelard of Bath (the so-called “Adelard I” version), Hermann of for typesetting .Arabic, and to the Managing Director of Rodopi Publish­
Carinthia and Gerard of Cremona. Based upon Adelard’s and Hermann’s ers (Amsterdam-Atlanta), F. van der Zee, for his willingness to publish this
translations is the version which Clagett called “Adelard II”, but which, as volume of studies.
Busard showed, was probably written by Robert of Chester. This reworking We would like to conclude this introduction by wishing Dr. Busard and
was the most influential of these texts: it became the basis of the Campanus his family many more happy and fruitful years.
edition (“the medieval Euclid”) and of another version, unknown to Clagett,
upon which the version sometimes attributed to Albertus Magnus depends. The editors:
Except for the Campanus text. Busard has edited all of them using all known
manuscripts (“Adelard I”: 29, Hermann: 7, Gerard: 28, Version II: 33, the M. Folkerts J. P. Hogendijk
anonymous redaction based upon it: 36). Besides the translations from the
Arabic and the later reworkings, there is also a translation of the Elements
made directly from the Greek in the middle of the 12th century, and this
translation, too, has been edited by Busard (31). Therefore, we now have
editions of almost all Latin texts, apart from the so-called “Adelard III” ver­
sion and some special redactions of which only one manuscript is known. It is
almost incredible that this huge task has been performed by only one man:
Dr. Busard. He has done for the Latin Euclid what Marshall Clagett has
done for Archimedes in the Middle Ages, and no other historians of mathe­
matics in this century have edited such a large amount of important material
from the Middle Ages as Clagett and Busard have done. It is to their credit
that our knowledge of the history of mathematics in the M'ddle Ages has
advanced so considerably and that almost all important Latin texts are now
available in modern editions.
10 INTRODUCTION
H. L. L. Busard: Publications
(1) Nicole Oresme, Quaestiones super geometriam Euclidis^ Leiden
(E.J. Brill), 1961 (XIV + 179 pp.).
(2) Über unendliche Reihen im Mittelalter, in; L’Enseignement
Mathématique, tome 8, fase. 3-4 (1962), 281-290.
(3) Über einige Papiere aus Vlètes Nachlaß in der Pariser Bibliothèque
Nationale (mit Wiedergabe des bisher ungedruckten Textes aus nouv. acqu.
lat. 1643), in: Centaurus, 10 (1964), 65-126.
(4) Unendliche Reihen in A est unum calidum, in: Archive for History of
Exact Sciences, 2 (1965), 387-397.
(5) The Practica geometriae of Dominicus de Clavasio, in: Archive for
History of Exact Sciences, 2 (1965), 520-575.
(6) Über die Verwandlung eines Quadrats in ein regelmäßiges Vieleck
und die Konstruktion dieser Vielecke über einer gegebenen Linie bei Claude
Mydorge, in: Janus, 52 (1965), 1-39.
(7) The Translation of the Elements of Euclid from the Arabic into Latin
by Hermann of Carinthia (?): books I-VI, Janus, 54 (1967), 1-140, and
published separately, Leiden (E. J. Brill), 1968 (142 pp.); books VII-IX,
Janus, 59 (1972), 125-187; books VII-XII, Amsterdam (Mathematisch
Centrum), 1977 (198 pp.).
(8) L'algèbre au moyen âge; Le “Liber mensurationum” d’Abû Bekr, in:
Journal des Savants, 1968, 65-124.
(9) Het rekenen met breuken in de middeleeuwen, in het bijzonder bij
Johannes de Lineriis, in; Mededelingen van de Koninklijke Vlaamse Academie
Dr. H. L. L. Busard at work on an edition of Euclid’s Elements voor Wetenschappen, Letteren en Schone Künsten van België. Klasse der
Wetens chap pen, Jaargang 30, nr. 7, Brussels 1968 (36 pp).
(10) Quelques sujets de l’histoire des mathématiques au moyen-âge, Paris
1969 (Université de Paris, Palais de la Découverte, D 125) (32 pp.).
(11) Die Vermessungstraktate Liber Saydi Abuothmi und Liber
Aderameti, in; Janus, 56 (1969), 161-174.
(12) Der Tractatus proportionum von .Albert von Sachsen, in: Öster­
reichische Akademie der Wissenschaflen, math.-nat. Klasse, Denkschriften,
116. Band, 2. Abhandlung, Wien 1971 (pp. 43-72).
(13) Der Traktat De sinibus, chordis et arcubus von Johannes von
Gmunden, in: Österreichische Akademie der Wissenschaften, math.-nat.
Klasse, Denkschriften, 116. Band, 3. .Abhandlung, Wien 1971 (pp. 73-113).
(14) Die “.Arithmetica speculativa” des Johannes de Mûris, in;
Scientiarum historm, 13 (1971), 103-132.
12 BUSARD
PUBLICATIONS 13
(15) [Commentary to:] J. E. Murdoch, The Medieval Euclid: Salient
(29) The First Latin Translation of Euclid’s Elements Commonly
aspects of the translations of the Elements by Adelard of Bath and
Ascribed to .Adelard of Bath, Toronto (Pontifical Institute of Mediaeval
Campanus of Novara, in: XID Congrès International d’Hlstoire des Sciences,
Studies), 1983 (VI -h 425 pp.).
Paris 1968, Actes, tome I B, Paris (Albert Blanchard) 1971, 88-90.
(30) Some Early Adaptations of Euclid’s Elements and the Use of its
(16) Der Codex orientalis 162 der Leidener Universitätsbibliothek, in:
Latin Translations, in: Mathemata. Festschrift fir Helmrith Gericke, ed.
XID Congrès International d’Histoire des Sciences, Paris 1968, Actes, tome
M. Folkerts and U. Lindgren, Stuttgart (Franz Steiner), 1985, 129-164.
III A, Paris (Albert Blanchard) 1971, 25-31.
(31) The Mediaeval Latin Translation of Erie lid’s Elements Made Directly
(17) Die Traktate De Proportionibus von Jordanus Nemorarius und
from the Greek, Stuttgart (Franz Steiner), 1987 (411 pp.).
Campanus, in: Centaims, 15 (1971), 193-227.
(32) Jordanus de Nemore, De elementis arithmetice artis. A Medieval
(18) Die Quellen von Nicole Oresme, in: XIII. Internationaler Kongreß
Treatise on .Number Theory. Part I: Text and Paraphrase, Part II: Conspectus
für Geschichte der Wissenschaft, UdSSR, Moskau, 18-24 August 1971,
Siglorum and Critical .Apparatus. Stuttgart (Franz Steiner), 1991 (372 pp.,
Colloquium: Wissenschaft im Mittelalter; Wechselbeziehungen zwischen dem
188 pp.).
Orient und Okzident, Moskau (Nauka) 1971 (55 pp.). Separately published
(33) Robert of Chester’s (?) Redaction of Euclid’s Elements: the .so-called
in: Janus 58 (1971), 161-193.
Adelard II Version, 2 Vols., Basel. Boston, Berlin (Birkhäuser), 1992 (950
(19) Der Liber de arcubus similibus des Ahmed ibn Jusuf, in: Annals of pp.) (with M. Folkerts).
Science, 30 (1973), 381-406 (with P. S. vtin Koningsveld).
(34) The Arithmetica of Jordanus Nemorarius, in: Amphora. Festschrift
(20) Uber einige Euklid-Kommentare und Scholien, die im Mittelalter für Hans Wussing, ed. S. Demidov, M. Folkerts, D. Rowe, C.J. Scriba, Basel,
bekannt waren, in: Janus, 60 (1973), 53-58. Boston, Berlin (Birkliäuser), 1992, 121-132.
(21) Uber einige Euklid-Scholien, die den Elementen von Euklid, über­
setzt von Gerard von Cremona, angehängt worden sind, in: Centaurus, 18
In press:
(1974), 97-128.
(22) Ein mittelalterlicher Euklid-Kommentar, der Roger Bacon zuge­
(35) Lateinische Euklidiibersetzungen und -bearbeitungen aus dem 12. und
schrieben werden kann, in: Archives Internationales d’Histoire des Sciences,
13. Jahrhundert. To be published in the Wolfenbuttel colloquium on history
24 (1974), 199-218.
of medieval mathematics.
(23) The second part of chapter 5 of the De arte mensurandi by Johannes
(36) A Thirteenth-Century .Adaptation of Robert of Chester’s Version of
de Mûris, in: For Dirk Struik, ed. R. S. Cohen et al., Dordrecht (D. Reidel)
Euclid’s Elements. To be published in: Algorismus, München (Institut für
1974, 147-167.
Geschichte der Naturwissenschaften).
(24) Zum Gedenken an Prof. Dr. J. E. Hofmann, in: RETE, 2 (1974),
(37) Johannes de .Maris. De arte mensurandi. A Geometrical Hand­
298-302.
book of the Fourteenth Century. To be published with Franz Steiner Verlag
(25) Über die Übermittlung der Elemente Euklids über die Länder des (Stuttgart).
Nahen Ostens nach West-Europa, in: XIVth International Congress of the
History of Science, Tokyo & Kyoto, Japan, 19-27 August, 1974- Proceedings
Dr. Busard contributed the following articles to C.G. Gillispie (ed.). Dictio­
No. 2, Tokyo 1975, 31-34.
nary of Scientific Biography, New York, Charles Scribner’s sons, 1970-1976:
(26) Über die Überlieferung der Elemente Euklids über die Länder des Bouvelles, Charles (voi. II, 360-361); Buot, Jacques (voi. II, 592-593);
Nahen Ostens nach West-Europa, in: Historia Mathematica, 3 (1976), 279- Carcavi, Pierre de (vol. III. 63-64); Clavius, Cristoph (voi. Ill, 311-312);
290. Deparcieux, Antoine (voi. IV. 38-39); Despagnet, Jean (voi. IV, 74-75);
(27) Der Traktat De isoperimetns, der unmittelbar aus dem Griechischen Frenicle de Bessy, Bernard (voi. V. 158-160); Guldin. Paul (voi. V, 588-
ins Lateinische übersetzt worden ist, in: Mediaeval Studies. 42 (1980), 61-88. 589); Hardy, Claude (voi. VI. 112-113); Henry of Hesse (voi. VI, 275-276);
La Faille. Charles de [vol. VII, 557-558); Lansberge, Philip van (voi. Vili,
(28) The Latin translation of the Arabic version of Euclid’s Elements
27-28); Le Paige. Constantin (vol. Vili. 250); Pitiscus, Bartholomeo (vol. XI,
commonly ascribed to Gerard of Cremona, Leiden (New Rhine Publishers),
3-4); Roomen. .-kdriaan van (voi. XI. 532-534); Ver Eecke, Paul (voi. XIII,
1983 (XXVIII pp. -t- 503 coll.).
615-616); Viète. François (vol. XIV'. 18-25).
14 BUSAKÜ
Dr. Busard contributed the following articles to the Lexicon des Mittel­ A challenge to the mathematicians of the
alters, München/Zürich (Artemis), 1980 - ... :
University of Leuven as a New Year’s gift
Bartholomaeus von Parma: Bd. 1, Sp. 1496; Bonatti, Guido v. Porli: Bd. 2,
Sp. 402; Bonfils (Immanuel ben Jakob): Bd. 2, Sp. 411; Bradwardine, for 1639
Thomas: Bd. 2, Sp. 538-539; Domninos von Larissa: Bd. 3, Sp. 1226;
Dorotheus von Sidon: Bd. 3, Sp. 1321-1322; Geometrie/Erdmessung: Bd. 4,
Sp. 1271-1273; Gerhard von Brüssel: Bd. 4, Sp. 1317; Hermann von Carinthia: Paul Bockstaele
Bd. 4, Sp. 2166; Johannes (Danck) de Seixonia (J. Danekow de Magdeborth):
Bd. 5, Sp. 568; Johannes de Mûris (Jehan de Murs): Bd. 5, Sp. 591; Jordanus
In the Albert I Royal Library in Brussels, there is, bound together with math­
Nemorarius (J. de Nemore): Bd. 5, Sp. 628.
ematical and astronomical works,^ a rare document that throws some light on
the practice of mathematics at the University of Leuven ip the first half of the
seventeenth century. It is a broadside in which an anonymous author offers
three mathematical problems to the mathematicians of the Leuven Univer­
sity as a New Year’s Gift for 1639 (Figure 1). The title is: Strenae Mathe-
maticae, Ex Scientia occultorum Numerorum desumptae, omnibusque Mathe-
maticis huius almae Universitatis Lovaniensis anno 1639, Kalendis lanuarij
propositae. It was printed in Leuven by Cornelius Coenesteyn, probably in
1638. What follows is an analysis of this document and an investigation of
the identity of the author. First we give a translation of the paper. The
mathematical symbols used in it have been left unchanged.
Mathematical new year’s gifts
Selected from the Science of the Hidden Numbers^ and offered to all the
Mathematicians of the salutary University of Leuven on the first day of the
year 1639.
First
In the introduction to his Methodus Polygonorum, the most noble and illus­
trious Adrianus Romanus presents to all the mathematicians of the entire
world a problem derived from this science. For three examples he gives the
solution. It is asked that these solutions be proven, and if errors occur in the
formulation or in the solution, to indicate and correct them.
Second
Given four straight lines .4, B, C, D whereof, if A is 1(1), B is equal to
5(1) — 5(3) -(- 1(5). If B is 1(1), then C is 3(1) — 1(3) and D is 9(1) —
30(3) -I- 27(5) — 9(7) + 1(9). C and D form together with the hypotenuse 2
a right triangle. It is asked that the aforesaid straight lines be expressed in
radical signs. However, if one takes a fifth straight line E to be 1(1), then
A is 5(1) — 5(3) -I- 1(5). It is asked how great is E, so close that it neither
falls short of, nor exceeds [the value] by 1/10000000000000000 in absolute
numbers.
‘Inc. B 625.
^By the “scientia occultorum numerorum’’ is meant algebra.
16 BOCKSTAELE NEW YEAR’S GIFT FOR 1639 17
S T R E N Æ M A T H E M A T I C Æ , Third
A general analytic method for the solution of algebraic equations, which is
Ex Sdcntia occultoruin Numerorumdcfumpcse, so much desired by many learned men, is asked for.
omnibufq*, Machcmaticis hiiius almæ Vniucrficacis But
L o v a n ie n s is anno cio. idc. xxxix« Kalcndis Who will read this? Maybe two, maybe nobody,
A shame and a pity.
lanuarij propoficae.
Printed bv COENESTENIUS
t R I M A. With three algebraic problems, the unknown author challenges the Leuven
mathematicians to demonstrate their learning and knowledge. However, he
SVmtnuÉ Vir àc Ciarifs. D. D. ADRIANVS ROMANVSiii
did not expect much of a result. The first problem was taken from a work
Præfatione Methodi Polygonorum, omnibus tociusorbis Mathemacicis
of Adriaan van Roomen,^ who taught medicine and mathematics in Leuven
Problema,es bac fcientia dedaâum, propofuic, caias tria exempla ipfe fol*
from around 1586 to 1592. At the beginning of 1593, he left for Wurzburg
nit. HxigiturSolaiiones demouftraodæ propooaaior j Sc, li alibi iu propo-
to teach medicine at the just established University. As a mathematician, he
neodo ant folueado error fublic,hic decegendos arque emendandos peticor.
was interested in the compilation of goniometric tables and everything related
to it: the calculation of the sides of regular polygons, the n-section problem
S E C V N D A.
of an angle, squaring the circle, and so on. And these are precisely the prob­
lems raised in the work cited in the Strenae. It appeared in 1593 under the
SVot quatuor RedtaS, A. B* C- D. qnarunt prima A iifiieric I fi), B
title of Ideae Mathematicae Pars Prima, sive Methodus Polygonorum qua la-
erit s O;— 5 ® Quodû B fueric 1 fi;.erit C} Ct J
terum, perimetrorum et artarum cujuscunqut polygoni investigandorum ratio
(3) Dvcrôjri^-^jo (3) -H27(5) > {7) < (?)• CverôacDcom
exactissima et certissima; una cum circuli quadratura continentur.* In the
hypotheQaià 2 triaugulum couftitaubt reâaogoiam* Quxmntnr Reâte introduction, van Roomen told how he had devoted himself for years to the
priifdiâx A. B. C. D. ia oumeris Radicaiibus adxqnatis. Si rerò aflù- finding of a general rule for the calculation of the sides of regular polygons.
matur quinta E fucricqj I (ijtum A fît y (i) 5 (i J He had discovered three methods, one of which made use of algebraic equa­
t tions. For ail regular polygons from the triangle to the 80-sided polygon,
tur, quanta fît E? Tarn propè ne t'ôoooooooooaaoooa ant deficiatant
he had already formulated the matching equation. He hoped to publish the
excédât in numerts abfoiutis.
results of his research in some ten books. The first four constitute his Ideae
Mathematicae Pars Prima. The rest of this planned work never appeared,
T £ R T I A.
apart from a few fragments published under other titles. In these first four
books, van Roomen gives the sides of the regular convex 3-, 4-, 5-, and 15-
OMnium Æqnationnin Algebraicaram, à plurimis profundioris doàri'*
sided polygon inscribed in a circle with radius 1, and of polygons that result
nae Viris haâenns tantopere defîderata, Vniuerfàlis Methodus Aiuly*
by repeated doubling of the number of sides, accurate to 32 decimal places.
cica petitur.
He continued the calculations to the polygon with 15 x 2®° sides and derived
the number ir to 16 decimcil places.
S E D At the end of the Praefatio appears the problem to which the pamphlet
refers (Figure 2). Under the title “Mathematical problem presented for so­
leget hdtcf *vel Duo *vel Memo^
lution to all the mathematicians of the world”, van Roomen asks that the
^urpe GJ* miferdbitc. following equation of the 45th degree be solved:
Typis CoMtniatt.
®On Adriaan van Roomen or Adrianus Romanus (1561-1615) see P. Bockst;iele in Na-
Uonaat Biographisch Woordenboek, vol. 2 (Brussels 1966), s. v. Roomen, col. 751-765.
Figure 1 ^Printed in Leuven by Joannes Masius. A part of the edition, however, was taken over
by the Antwerp printer and publisher Joannes van Keerbergen. In some copies, therefore,
Copyright Koninklijke Bibliotheek Albert I, Brussels
Masius’s name and address are replaced by Aniwerptae, Aped loannem Keerbergtum.
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' fit,nt i Q . 4<f45 © - J795 (T) + 9,5654
for
m -^115, 850^(7^ + 781,1375 (f.) ~ 545I) i°75 (ii) +
1» 0«o, 6075 til) — ^ ^ 7, 6180 (1^) + 5, 8494,237J 6 = V 1--W ■
4 16 8 V 64
8384,1800 (^)_-- 3,78^5,
(2£) -1,1767,9100 (^) + 4<$Pj', This is, at first sight - even for an exceptionally gifted calculator like van
Roomen - an impossible task. It is obvious that the meaning of the ques­
^ , *i?S 504o QD ± }?6,4565 (ii) - 74f
tion cannot be the solution of an arbitrarily chosen nth-degree equation. The
lliujjo (5) —1^1300 (3^) + 945 - 45 (4i) + way to the solution must lie somewhere else, namely, in the recognition of the
* i^^dcCU^^^uc terminuspoftcrior,invenirepriorem. geometric problem that is offered under the guise of this algebraic equation.
Perhaps to give his readers a hint, van Roomen gives three examples:
SxtmfkmpriwMmiAtum*
O Utenaiaatpd&enotr^bim., >f r bin. j> + rbin i 4>rz. qu^tnr rerminus Example 1: If, in the given equation,
l3prior.SoLyrio.Dico cermiaupriorein c'^er^ 2 -r ^ .i «f rbin.z^
s ■- Exempimliewukm Utaml b=y2+]/2 + t/2 + V2,
STl^nrlimi^fKlflerforrbm. rbin. z-r 'm.i ~ rbin. i - rbin. z> r
qBaerifrirtemiritu prior. S oi VTib. Terminus prior eft rbin* z - rbin. i + then the solution is
fbm.xJigrVmrzitrbitn.1 ^ rj.
Exempiam tertiam datum*
2 - p + ^ 2 + ^ 2 + ^ .
Sit tenninospofterior rbim. .z <i> r i, qturritar terminus prior.
3>o&Tcwninu5pri6tgftr^i».x- r^aadtin,z.4» r i ■{» rbim. < - r <
Example 2\ If
■'h‘ . . 16 16 S tf+
Sitemimarisabiblarisibltnoaijsisiprbponerelibaeric; Sitpofteriortmni-
807». 5696,71 »7,587S-
> J000tf,'000e,0000,0000,0000,0000,0000,0000,0000,0000,0000.0000, b = 2 - 1/2 V^2- %/2,
‘‘^QmrritimTierniinasprior. Solvtio. Tcrminuspriorerii
»7.-*o9}.d*90,9fZ2,}Z4.i,joi},SZ3i,ziiz,6t3t,iJto.
then the solution is
^ ioooo,o«oOiOoeo,90Qo,ooao,ooao,0000,oSoo,0000,0000,
£ x B14?I.VM Q© ^ S I T V m.
5 IrpofteriortermmusrtrhtQmiai _L — r Bin,i-L — r
2-F 1/2-F V2+ 1/24- v/3.
x = v-’ -N
«facrimrterniiniuprior. HDcexemplum ooinibos Matbenarictsadcon-
ibraendumfitpropoiinun, I ^ dobtoquin LttdalfwmCeimcyu^ohi»
Example 3\ If one takes b — \j2 A \/2, then the solution is
cttaan.fiirea in xwmem fi^nomijs fit iavemurar.
. M E-
o , 3 /1_5 £ _ /-I
16 \1 8 ~ V 64 ■
Figur/2
In the last example, van Roomen gives the value of b and of the solution also
in decimals, b accurate to 1/10^®, the solution to 1/10“*°.
20 BOCKSTAELE NEW YEAR’S GIFT FOR 1639 21
Among the problems that captured the attention of mathematicians in van to Henry IV of France, published in 1595 in Paris his Ad problema quod om­
Roomen’s time were the squaring of the circle and the compilation of go­ nibus Mathematicis totius orbis construendum proposuit Adrianus Romanus
niometrie tables. In this, the calculation of the sides of regular polygons as Responsum. Already in the 1570s, Viète had worked on the problem of the
a function of the radius of the circumscribed circle occupied an important angle division and had obtained some remarkable results. We are particu­
place. For those working actively on it, the complicated radical expressions larly interested here in the formulas in which he expressed the chord of an
in van Roomen’s problem were easily recognizable as sides or diagonals of uneven multiple of an arc a in terms of the chord of a. If the radius of the
regular polygons inscribed in a circle with radius 1. Once one has grasped circle is 1, and if chd a = x, then one has:
this, one has the first indication about the direction in which the solution
has to be sought. chd 3a = 3x —
Let us analyse the three examples van Roomen gives. As abbreviations, chd 5a = 5x — 5x^ -|- x^
we will use the following symbols: chd 7a = 7x — 14x^ -t- 7x® — i'
Sn for the side of the regular n-gon inscribed in a circle with reidius 1; chd 9a 9x — 30x^ -I- 27x^ — 9x' -h x^
s„_p for the diagonal of the regular n-gon that subtends an arc of px360°/n; etc.
chd a for the chord of arc a;
arc a for the arc subtended by chord a. Viète also tells how the coefficients can be Ccilculated for an arbitrary n. For
In Example 1, the second member 6 of the equation appears to be the n=45 the formula becomes:
chord of the supplement of arc S32 which is the diagonal S3245:
chd 45a = 45x - 3795x^ + 95634x° - ... - 45x" + a:"*®,
b = chd (180°- arc S32) = 532,15 = chd 168°45'.
which is nothing other than van Roomen's famous equation. Viète immedi­
As a solution, van Roomen gives sgg = chd 3°45'. And it turns out that
ately recognized it. In his answer, he gives not only the solution chd 32' =
45 X 3°45'= 168°45'. A solution of the equation is thus the chord of the
2 sin 16', but also the 22 other positive roots:
forty-fifth of arc b.
X = 2sin (16'-hA:-2§|!), À: = 1,2,3,..., 22.
In the second example, van Roomen gives as the solution the side 3392 of
the 192-sided polygon, which is the chord of 1°52'30". As the value for b, one After this long excursus let us read again the first problem of the Leuven
now expects the chord of 45 x 1°52'30"= 84°22'30", which is pamphlet. It is asked to prove the three examples van Roomen gives, and
to indicate and correct any oversights or errors in the formulation or in the
solution. From the way the question is formulated, it is clear that the author
chd (90°- arc sg-t) = 564,i5 2- \ 2- 2 + v^2 + v^. of the paper knew Viète’s Responsum. Indeed, in Viète we find criticisms of
the way in which van Roomen formulated the problem as well as corrections
Van Roomen apparently made a mistake here, for the value given for b is of the examples and supplements to the solutions. Viète begins by noting
5128,43 = chd 120°56'15". “that van Roomen’s problem is ridiculous, if it is not corrected.”® For van
In the third example, b = chd (180°- arc sg) = sg,3 = chd 135°. As a Roomen formulated it as follows: when a : b = x : (45x — 3795x® + ... + 1^®),
solution, van Roomen gives S120 = chd 3°. Because 45 x 3°= 135°, one finds then calculate a when b is given. Correctly, Viète pointed out that this
here, too, that the given solution is the chord of the 45th part of arc b. problem is totally undetermined, because, with b given, one can choose x
The problem hidden behind van Roomen’s equation thus comes down to arbitrarily and from it then determine a.®
this: Given the chord of an arc, find the chord of the forty-fifth of that arc. We have already pointed out that the value given for 6 is wrong in van
Once one grasps this, it is no longer so difficult to solve the problem. What Roomen’s second problem. Viète also observed this. As a correction he
he gives as the value for b is Sis = chd 24°. The solution will thus be proposed
X — S675 = chd (^- 24°) = chd 32'= 2 sin 16'.
b= ^2-\j2 + \j2A\j2 + \A = 5i28,3i = chd 87°11'I5",
In contreust with the chords from the examples, this can no longer be ex­
pressed by radicals. An approximate value is given by the sine table.
®“reXoìov autem est Adriani Problema, nisi emendetur.” Responsum, f. 2r.
Van Roomen’s chcdlenge to all the mathematicians of his time did not ®This curious interpretation of equations as proportions van Rjoomen took from Stevin’s
remain without response. François Viète, maitre des requêtes and counselor AriihmeUqut. p. 264-266.
nxi/' »/y