Table Of ContentMathematics from
Manuscript to Print
1300-1600
Edited by
CYNTHIA HAY
Open Uf>iversi(y
CLARENDON PRESS • OXFORD
1988
Oxford University Press, Walton Street, Oxford 0X2 6DP
Oxford Sew York Toronto
Delhi Bombay Calcutta Madras Karachi
Petaling Java Singapore Hong Kong Tokyo
Nairobi Dar es Salaam Cape Town
Melbourne Auckland PREFACE
and associated companies in
Berlin Ibadan
Oxford is a trade mark of Oxford University Press This volume records the proceedings of a joint conference of the British
Society for the History of Mathematics and the Open University on Renais
Published in the United States
by Oxford University Press, New York sance mathematics, held at Keble College, Oxford, September 27-30, 1984.
It should be said here that the idea of holding an Anglo-French conference
I Cynthia Hay, I98S
on Renaissance mathematics in 1984, to celebrate the quincentenary of the
All righis reserved. No part of this publication may be reproduced, stored in a retrieval completion of Nicolas Chuquet’s Triparty, originated with Graham Fiegg,
system, or transmitted, in any form or bv any means, electronic, mechanical, photocopying, who would have jointly edited the proceedings with me, but for an unfortu
recording, or otherwise, without the prior permission of Oxford University Press
nate bout of ill health. The conference proceedings, with a wide range of
Briitsh I.ibrarv Caialoguing in Piiblicalion Data mathematical notation and references, were efficiently and ingeniously
Mathematics from manuscript to print, 1300-1600. word-processed by Barbara Moss; her expertise in both the history of math
I. .Mathematics—History
ematics and in French has been invaluable in preparing these proceedings for
I. Hav, Cynthia
51 O'.9 023 OA23 publication. Several of the papers were originally written in French. M
ISBN 0-19-S53909-6 L’Huillier’s paper was translated by Barbara Moss; the papers by Professor
Beaujouan, M Benoit and Dr Cassinet were translated by Cynthia Hay.
Library of Congress Cataloging in Publication Data
Financial support for the conference came from the Royal Society, the
.Mathematics from maniiscripi to print. 1300-1600
London Mathematical Society, the Open University, the Societe frangaise
edited by Cynthia Hay.
Includes index. d’histoire des sciences et des techniques, and the European Educational
I. Mathematics—History. 1. Hay. Cynthia. Research Trust.
QA23..M2S 1987 510 .9—dcI9 87-16002 CIP
ISBN 0-19-853909-6 Milton Keynes C.H.
April 1987
Set by Colset Private Ltd.
Printed in Great Britain by
Si Edrnundsbury Press,
Bury St Edmunds, Suffolk
CONTENTS
Contributors IX
Introduction
C. HAY
PART I: Italian and Provengal mathematics 9
Fourteenth-century Italian algebra 11
R. FRANCl AND L. TOTl RIGATELLI
On an algorithm for the approximation of surds from a Provencal
treatise 30
J. SESIANO
PART II: Nicolas Chuquet and French mathematics 57
Nicolas Chuquet—an introduction 59
G. FLEGG
The place of Nicolas Chuquet in a typology of fifteenth-century
French arithmetics 73
G. BEAUJOUAN
Concerning the method employed by Nicolas Chuquet for the ex
traction of cube roots 89
H. L’HUILLIER
The commercial arithmetic of Nicolas Chuquet 96
P. BENOIT
Chuquet’s mathematical executor: could Estienne de la Roche have
changed the history of algebra? 117
B. MOSS
How algebra came to France 127
W. VAN EGMOND
PART III: Mathematics in the sixteenth century 145
What could we learn from Master Christianas van Varenbraken? 147
M. KOOL
A note on Rudolf Snellius and the early history of mathematics in
Leiden 156
K. VAN BERKEL
Contents
\
The first arithmetic book of Francisco Maurolico, written in 1557
and printed in 1575: a step towards a theory of numbers 162
J. CASSINET
PART IV: Mathematics and its ramifications 181 CONTRIBUTORS
Is translation betrayal in the history of mathematics? 183
C. HAY
G. BEAUJOUAN, Ecole Pratique des Hautes Etudes, IV' section, Sciences
Renaissance mathematics (and astronomy) in Baldassare
historiques et philologiques, 45-47 rue des Ecoles, 75005 Paris, France.
Boncompagni’s Bullettino di bibliografia e di storia delle scienze
P. BENOIT, Centre National de la Recherche Scientifique, Universite de
matematiche e fisiche (1868-87) 190
Paris 1, 9 rue Malher, 75004 Paris, France.
S. A. JA YAWARDENE
K. VAN BERKEL, Rijksuniversiteit Utrecht, Instituut voor geschiedenis der
Some early sources in recreational mathematics 195
natuurwetenschappen, Janskerhof 30, 3512 BN Utrecht, Netherlands.
D. SINGM ASTER J. CASSINET, Universite Paul Sabatier, Toulouse, France.
Cornelius Agrippa’s mathematical magic 209 vv. VAN EGMOND, Institut fur Geschichte der Wissenschaften der
A. G. MOLLAND Universitat Miinchen, Deutsches Museum, D-8000 Miinchen 26, Federal
The market place and games of chance in the fifteenth and sixteenth Republic of Germany.
centuries 220 J. V. FIELD, Science .Museum, South Kensington, London SW7 2DD, UK.
G. FLEGG, Faculty of Mathematics, The Open University, Walton Hall,
I. SCHNEI DER
Milton Keynes MK7 6AA, UK.
Perspective and the mathematicians: Alberti to Desargues 236
R. FRANC 1, Universita di Siena, Centro Studi della Matematica Medioevale,
J. V. FIELD
Via del Capitano 15, Siena 53100, Italy.
Why did mathematics begin to take off in the sixteenth century? 264 C. HAY, Faculty of Mathematics, The Open University, Walton Hall,
G. J. WHITROW Milton Keynes MK7 6AA, UK.
H. L HUiLLiER, Compagnie Fran^aise des Petroles, TOTAL, 5 rue Michel-
Index 271 Ange, 75781 Paris, France.
S. A. JAYAWARDENE, Science Museum, South Kensington, London SW7
2DD, UK.
M. KOOL, Jhr. Ranweg 28b, 3998 JR. Schalkwijk, Netherlands.
G. MOLLAND, Department of History and Philosophy of Science, King’s
College, University of Aberdeen, Aberdeen AB9 2UB, UK.
B. MOSS, 36 Haroldstone Road, London E17 7AW, UK.
I. SCHNEIDER, Institut fur Geschichte der Wissenschaften der Universitat
Munchen, Deutsches Museum, D-8000 Miinchen 26, Federal Republic of
Germany.
J. SESiANO, Departement des Mathematiques, Ecole Polytechnique
Federale de Lausanne, CH-1015, Lausanne, Switzerland.
D. siNGMASTER, Department of Mathematics, Polytechnic of the South
Bank, Borough Road, London SEl OAA, UK.
L. TOTi RIGATELLI, Universita di Siena, Centro Studi della Matematica
Medioevale, Via del Capitano 15, Siena 53100, Italy.
G. J. WHITROW, Department of Mathematics, Imperial College of Science
and Technology, Huxley Building, Queen’s Gate, London SW7 2BZ, UK.
Introduction
‘Renaissance mathematics? Was there any?’ This was how one mathemati
cian responded to my remark that the proceedings of an international
conference of Renaissance mathematics, held at Keble College, Oxford in
September 1984, were to be published. This introduction will attempt briefly
to answer this question. Some historians might dismiss my colleague’s query
as a typical mathematician’s ignorance of his subject’s history. There is,
however, another way of construing his question: not as an indication of
ignorance but as a reflection of deeper doubts. Was the period, say, from
1450 to 1600 significant for the development of mathematics; were there
innovations sufficient to justify the sustained attention of historians or even
of mathematicians? And a question of a different order: does it make any
sense at all to include mathematical activities in the fifteenth and sixteenth
centuries under the Ooppy and shapeless umbrella of the Renaissance? These
questions are considered in reverse order: the Renaissance and mathematics,
and the mathematics of the Renaissance.
To start with objections: several generations of historians have stripped
the term ‘Renaissance’ of whatever content it once had; it is now no more
than a way of avoiding writing dates. An appeal to return to the original
concept of the Renaissance does not help. The classical concept of the Re
naissance centred on developments in art and culture that were ascribed to
humanism and the revival of antiquity; as Thorndike and Sarton argued
some fifty years ago, this concept of the Renaissance had little to do with
contemporary developments in science and mathematics.' If the invention of
printing and its impact are considered to be part of the definition of the
Renaissance—and Elizabeth Eisenstein sees the relation between printing
and the Renaissance as more complex-—then it should be noted that the use
I Lynn Thorndike. A History of magic and experimental science, 8 vols, Macmillan and Co..
New York, 1923-1958. There are surprisingly few references in all these volumes to the Renais
sance, but those that there are are uniformly hostile to the Renaissance which in Thorndike’s
view was a ‘bad thing’ for science. (W.C. Sellar and R.J. Yeatman’s historiographical
categories in 1066 and all that, Methuen, London, 1930 are still useful.) Thorndike expresses
similar views in Science and thought in the fifteenth century, Columbia University Press,
New York, 1929, pp. 2-3 and 9-10, George Sarton’s views at the same time were similar if less
vehement, in ‘Science in the Renaissance’, in The Civilization of the Renaissance, ed. James
Westfall Thompson, University of Chicago Press. Chicago, 1929, pp. 75-6, 85-6, 92-3. How
ever, by 1957 he saw the Renaissance as a productive period in science, in Six wings: men of
science in the Renaissance, Bodley Head, London, 1957, pp. 2-4.
- Elizabeth Eisenstein discusses the complexities of the relationship between printing and the
Renaissance in The printing press as an agent of change, Cambridge University Press,
Cambridge, 1979, vol. 1, pp. 163-302. She argues that the Renaissance preceded printing and
2 Introduction
Introduction 3
of printing developed more slowly in mathematics than in the natural sci
have recently been described as the first modern histories of science,^ were
ences and that, moreover, the initial impact of printing on mathematics has
consciously modelled on Vasari’s Lives of the artists. Mathematicians were
been exaggerated.^
part of the general culture of their times, for which the description ‘Renais
The term Renaissance has been loosened to refer to a period in time rather
sance’ is helpful.
than to an identifiable set of attitudes. However, it can be argued that this
It is accepted among historians of mathematics that a central achievement
chronological use of the term Renaissance antedates the modern
of Renaissance mathematics was in the development of notation, particu
historiography, stemming from Jacob Burckhardt. One of the first histo
larly for algebra. The importance of notation is clear; the lack of notation or
rians to use the term Renaissance to refer to a period was an historian of
limited notation impeded both the general formulation of problems and
mathematics—of sorts; this was Count Libri in his Histoire des sciences
manipulation towards their solutions. The main lines of the story of the
mathematiques en Italic depiiis la Renaissance jusqu’a la fin du XVIIieme
development of algebraic notation are familiar. In mathematics in Western
siecle* * •* This observation may be more in the nature of a historiographical
Europe from the high .Middle Ages, the development of notation was inter
curiosity than a ‘knock-down’ argument. However, both historical evidence
mittent and halting. A partial symbolism developed by one mathematician
and the work of recent historians indicate that there is more to the con
was often ignored for centuries. Hindsight may provoke impatience with
nection between mathematics and the Renaissance than a set of dates, which
these earlier mathematicians—why could they not see how much help some
are in any case disputed.
notation would be in their work, and set about developing it accordingly?
The sceptical case against speaking of Renaissance mathematics is less
But the halting development of mathematical notation is historically signifi
plausible than it appears; it conflates outdated criticism and neglects more
cant; what may now seem to be intrinsic to mathematics was an immensely
recent historiography. The iconoclastic view of the Renaissance is hard to
slow acquisition, over centuries. Its intermittent development reflects the
sustain. Contemporaries spoke of their own times in terms of a renaissance,
difficulties of developing notation and, as well, the relative isolation of many
and mathematicians did so as well. For example, the publishing programme
mathematicians in the earlier part of this period; communication between
in mathematics that Regiomontanus proposed in the late fifteenth century,
mathematicians is also something that developed during the Renaissance, as
that was unfortunately cut short by his death, exemplifies a characteristic
many of the sources testify, in a variety of intriguing and entertaining ways.
Renaissance endeavour of recovering the works of Greek antiquity. Com-
Some aspects of Nicolas Chuquet’s achievements exemplify this pattern of
mandino and Maurolicus, among others, were to make good this pro
the uneven development of mathematical notation. The occasion for the
gramme about a century later, by translating a number of major Greek
conference on Renaissance mathematics was the quincentenary of the com
mathematical works. Paul Lawrence Rose has argued that the characteristic
pletion of the mathematical manuscript of Nicolas Chuquet. Little if any
attitudes of the Renaissance shared the outlook of most Italian mathemati
thing is known of the life of Nicolas Chuquet, beyond what can be gleaned
cians at the time.* And—an interesting link of the history of mathematics
from his manuscript and the tax registers of Lyon, where he lived in the latter
with that of art—Bernardino Baldi’s biographies of mathematicians, which
part of his life. His mathematical manuscript appears to have been written in
part as a product of his teaching some of the youth of Lyon the mathematics
they needed for commercial and practical purposes. But the algebra that he
was itself modified by printing. Her analysis of the relationship might be compared with Max
Weber’s account of the affinity between Protestantism and capitalism, in The Protestant ethic developed in the Triparty goes well beyond what was required for practical
and the spirit of capitalism, George Allen and Unwin Ltd., London, 1930. purposes, to innovate in matters of notation. He had a workable notation for
* Cf. Robert S. Westmann, ‘On communication and cultural change’, Isis 71 (1980)
powers of an unknown which enabled him to deal effectively with problems
pp. 474-7 and Anthony T. Grafton, ‘The importance of being printed’, Journal of interdiscip
linary history, 1980, pp. 265-289. that we would now call algebraic. His ability to deal with ideas that were both
•* Johan Huizinga, ‘The problem of the Renaissance’ in his Men and ideas. Free Press, New mathematical and unrealistic, such as fractions of a child in an inheritance
York, 1959, p. 253;
problem,' are probably not unconnected with his achievements in notation:
According to Walter Goetz, the first person who assumed that the term Renaissance without any both are evidence of a movement towards greater abstraction which might be
further restriction was familiar as a specific term for a circumscribed cultural period was the
Florentine Count Libri of dubious memory who, after having fled to France in 1838, published a
s Paul Lawrence Rose, letter entitled ‘The history of science’. Times literary supplement,
work entitled Histoire des sciences mathimatiques en Italie depuis la Renaissance jusqu 'a la fin
February 15, 1985, p. 171. Rose argues that Baldi, in his biographies of mathematicians, written
du XVIIiime slide. This is not correct, however. Libri was merely following a custom that
largely in the 1580s, ‘clearly envisages the history of science as an authentic field of enquiry and
had won ground in French literary circles.
[writes) with clearly defined historigraphical categories in mind.’ Rose is making a priority claim
* Paul Lawrence Rose, The Italian Renaissance of mathematics, Droz, Geneva, 1975. for Baldi as antedating Kepler by fifty years in being the first modern historian of science.
t Cf. Graham Flegg’s paper.
4 Introduction Introduction 5
taken as an identifying mark of a mathematician. As is told in detail in and more distinguished of their predecessors. It is reminiscent of the stuff of
Graham Flegg’s paper, his achievements in notation were neglected, and his many a novel, in which an upwardly mobile individual ‘forgets’ or ignores
work—in manuscript form—was lost from view for several centuries. Other the lowly associates who helped along the way. The social and intellectual
aspects of Chuquet’s work, in comparison with his predecessors and succes rise of algebra may have contributed to these manuscripts having been
sors, are discussed in a number of papers in this volume. In the context of ‘hidden from history’ until quite recently. It may be time for a figure as
Renaissance mathematics, part of his significance is in connection with the famous in the history of mathematics as Pacioli to be reassessed in the light
piecemeal development of notation. But the fitful development of algebraic of these manuscripts.
notation is only one—important—part of the contributions to mathematics One theme that recurs in a number of the papers on manuscripts is that of
made in the late medieval and Renaissance periods, and discussed in this tracing ‘priority claims’—that is, investigating whether a particular math
volume. ematical technique has been correctly attributed to its originator, if indeed its
Chuquet’s mathematical manuscript was effectively lost from view for originator can be established at all. Different tactics are used in these studies.
nearly 400 years. To lose one manuscript is unfortunate; for some two Sometimes the results are agnostic, in the sense that they demolish rather
hundred and fifty manuscripts to be lost from view for nearly five hundred than establish such claims, and matters are left open for further investi
years might have occasioned scathing remarks from Oscar Wilde’s Lady gation. M L’Huillier explores techniques for extracting cube roots, and
Bracknell.'* For historians, the recovery of this body of fourteenth- and argues that the technique attributed to Cardano is not original with him;
fifteenth-century Italian manuscripts provides material for rellection. The moreover, credit for this cannot be given to Chuquet, despite his having used
prehistory of the development of algebra has traditionally been told in terms a similar technique. The ultimate originator is as yet unknown. It is a matter
of a few great men; for late medieval mathematics, this meant a lengthy gap of disproving the accepted view, withdrawing credit from one of the ‘hero-
between, in the early thirteenth century, Leonardo of Pisa and Jordanus de figures’ of Renaissance mathematics, and locating the origins earlier in time
Nemore, and late fifteenth-century mathematicians such as Pacioli and'more than had previously been thought.
recently Chuquet. Italian mathematical manuscripts, from the late thir Dr Jacques Sesiano explores the extraction of surds in a Provencal manu
teenth century to the sixteenth century, are evidence of a more mundane and script of c. 1430, with references and comparisons that range from Aristotle
practical tradition. to Cantor. Both he and Professor Beaujouan have independently shown, by
Professor Franci and Professor Toti Rigatelli discuss the algebraic content reference to this manuscript, that the one passage in his manuscript where
of eighteen Italian manuscripts of the fourteenth century. In these manu Chuquet appears to claim originality must be read with some care for
scripts, algebra is embedded in the practical context of teaching everyday Chuquet’s meaning, which is precisely expressed. Chuquet wrote, with refer
mathematical skills for commercial activities. It has been argued elsewhere ence to the rule of intermediate terms, a technique for approximating roots,
by Dr Warren Van Egmond that the algebraic tradition of these manuscripts that ‘formerly [he] had been [its] inventor’." This pasage is on the face of it
was both a seed-bed and a precondition for the sixteenth- and seventeenth- puzzling. There are similar approximation techniques to be found earlier, so
century development of algebra.^ On the principle that the skill of walking that Chuquet cannot be said to have invented this technique in its entirety;
precedes that of running, the level of mathematical competence diffused by furthermore, why should it matter that Chuquet thought he had invented
these manuscripts preceded and was presupposed by the later development this technique formerly? Professor Beaujouan reconstructs the intellectual
of algebra from a practical skill to a more abstract and general kind of genealogies of a number of fifteenth-century French manuscripts, and re
mathematics. solves this puzzle. He gives thereby a fascinating picture of mathematical
The ascent of algebra appears to have been associated with a discounting activity in fifteenth-century France, as recorded in extant manuscripts. The
of more humble antecedents. Insofar as Renaissance algebraists acknowl technique in question, the rule of intermediate terms, is to be found in the
edged their sources, they tended to follow something like an early version of Pamiers manuscript analysed by Dr Sesiano. Chuquet’s contribution, by
Robert Merton’s Matthew effect:'® that is, to acknowledge the better known comparison with this manuscript, is to have refined this technique, rather
than to have devised it. Professor Beaujouan argues that Chuquet worked on
* Oscar Wilde, The importance of being earnest, act I, ‘To lose one parent, Mr. Worthing,
the rule of intermediate terms before encountering the Pamiers manuscript;
may be regarded as a misfortune; to lose both looks like carelessness.’
’ Warren van Egmond, The commercial revolution and the beginnings of Western math on reading this manuscript, he realized that he had been less original than he
ematics in Renaissance Florence, 1300-1500, Ph.D. thesis, Indiana University, 1976.
'® Robert Merton, ‘The Matthew effect in science’, pp. 439-459, in his The sociology of " In A. Marre’s edition, cited in the papers referred to: 'll y a aussi la rjgle des nombres
science. University of Chicago Press, Chicago, 1973. moyens de laquelle Jadizjefuz Inuenteur. . .’
6 Introduction Introduction 7
had thought; he had thought formerly that he had invented it, but now were put on what he could teach, perhaps because of the practical slant of his
recognized that he had only modified and improved an existing rule. That mathematical interests. Only after practical mathematics was successfully
Chuquet did not mention the Pamiers manuscript specifically is in keeping institutionalized, in a Dutch mathematical school established in 1600, was
with his own practice, and a more general intellectual style of the time, of not Snellius made a full professor. A new university, Leiden was not immedi
citing sources and references. As Professor Beaujouan remarks at the end of ately receptive to new developments in mathematics, but followed suit quite
his paper, there are still discoveries to be made in the field of manuscripts. quickly when these new developments were institutionalized elsewhere, in
Barbara Moss’s study of the relationship of de la Roche’s Larismethique close proximity.
nouellement composee to Chuquet’s Triparty can be considered in the con Dr Jean Cassinet discusses Francisco Maurolico’s Arithmetic in con
text of priority claims. It has often been said that, at the very least, de la nection with the development of number theory, and in particular of math
Roche took advantage of Chuquet, in that Chuquet’s mathematical manu ematical induction. Although the recovery of Diophantus was considerably
scripts passed into the hands of de la Roche and remained unpublished. De la more significant, Maurolico’s work on figurate numbers came to the atten
Roche’s Arismethique incorporated much of Chuquet’s work, while omit tion of both Fermat and Pascal.
ting most of what subsequent historians have found interesting and sophisti The papers in the final section are concerned with the ramifications of
cated in the way of mathematics. What might now be called plagiarism can mathematics and its history. Dr Cynthia Hay rellects, somewhat ruefully, on
perhaps be seen in a different light at a time when ideas of intellectual whether the enterprise of providing a representative selection of Chuquet’s
property were not quite the same as ours. De la Roche was by no means the mathematical manuscripts, translated into English, can only be a pale
only mathematician to incorporate the work of others into his own publica shadow of the original.
tions; Vasari, in his Lives of the artists, accused Pacioli of having passed off There are two surveys of work in progress, aided and abetted by comput
parts of Piero della Francesca’s mathematical work as his own.'- Barbara ers. ."Vlr Jayawardene provides a foretaste of his promised systematic guide to
Moss re-examines the relationship between the work of Chuquet and that of exploiting the resources of the Bullettino di bibliografia e di storia delle
de la Roche, and gives an apologia for the latter. scienze matematiche e fisiche for the history of mathematics. Most histo
Two papers in this section on Chuquet and French mathematics consider rians of medieval or Renaissance mathematics will have, at some time,
mathematics in broader historical settings. M Paul Benoit provides a de consulted one or two volumes of this nineteenth-century periodical with a
tailed account of the contents of Chuquet’s Commercial Arithmetic and specific topic in mind; to browse through one or two volumes is tantalizing.
considers the links between this work and commercial practices at the time; Mr Jayawardene’s guide will make possible a systematic exploitation of its
he examines as well the complex and difficult question of whether commer riches. Dr David Singmaster surveys recreational mathematics, with particu
cial activities can be said to have influenced the development of mathemati lar reference to the period of the Renaissance, in connection with an exten
cal work in fields other than commercial arithmetic. Dr Warren van Egmond sive bibliography that he is preparing in this field, and a series in recreational
provides a survey and an analysis of the development of French mathemat mathematics that he is editing for Oxford University Press.
ics, with the much more rapid and fruitful development of Italian mathemat Dr George Molland looks at a little-known and perhaps ‘disreputable’
ics as a point of reference. He argues that the reasons for the relative back aspect of the literature on magic in this period. Dr Molland examines an
wardness of French mathematics lie in the lack of an active and continuing aspect of the Renaissance interest in magic which had considerable accep
social tradition in which an intellectual tradition of mathematics could take tance, and which has subsequently been neglected; mathematical magic, as
root. Algebra, in the form of a coherent algebraic tradition, did not come illustrated in the sixteenth-century work De occulta philosophia by Henry
into being in France prior to Viete. Cornelius Agrippa. Among the topics discussed are Agrippa’s numerology,
In the section on mathematics in the sixteenth century, Marjoiein Kool his views on magic squares, and on the abstract virtues. Molland assesses,
summarizes the contents of a sixteenth-century Dutch manuscript which she sceptically, claims that aspects of Agrippa’s mathematical magic can help
has studied in great depth. Dr Klaas van Berkel examines how mathematics fully be viewed in structuralist terms or as an anticipation of modern group
was incorporated into the university curriculum In Leiden. The first profes theory. He suggests, perhaps not too seriously, that ultimately Agrippa may
sor of mathematics in Leiden, Rudolf Snellius, was originally only reluc not have had the courage of his convictions in that he did not develop his
tantly admitted as a teacher at the University of Leiden; some restrictions ideas on mathematical magic to their limits.
Professor Ivo Schneider makes a forceful case that the solution of the
'2 A. Giorgio Vasari, L/ves o///iearUsw, Penguin, Harmondsworth, 1965, p, 191. central problem in probability theory, the problem of points, was not simply
8 Introduction
a matter of pure mathematical invention or discovery. An analysis of one of
the Italian manuscripts discussed above indicates that a correct algebraic
solution had been found as early as the fourteenth century. Discussions of P A R T I
games of chance in the fifteenth and sixteenth centuries did not reach the
standards of mathematical rigour displayed in this early solution, nor did
they show any knowledge of its approach. This relapse in mathematical
thinking about the solution of the problem of points is, Professor Schneider
Italian and Provenfal
argues, closely linked with religious ideology and the development of the
economy.
Dr J. V. Field examines the question of the relationship between perspec mathematics
tive in the works of Renaissance artists and the mathematical understanding
of perspective. She argues that, with the exception of Piero della Francesca,
artists only had recourse to the mathematical rules of perspective at a ‘rudi
mentary’ level. The importance of the mathematical theory of perspective is
to be found, rather, in its contribution to later developments in mathematics,
which Dr Field traces.
The volume ends, appropriately, with Professor Whitrow’s paper which
raises a major question about the development of mathematics. The
phrasing of the question, ‘Why did mathematics begin to take off in the
sixteenth century?’, suggests analogies with W. W. Rostow’s classic analysis
of the Industrial Revolution, in The stages of economic growth,'^ which
endeavoured to identify the prerequisites for the English economy to ‘take
off’ in the late eighteenth century, into sustained economic growth. The
analogy with economic development is a useful heuristic device which directs
attention to the preconditions for sustained mathematical development.
Professor Whitrow reviews a number of the factors which influenced the
development of mathematics, and concludes with his question clarified but
still unsolved.
The papers in this volume contribute to our understanding of Renaissance
mathematics, both in terms of its technical contents and achievements, and
also in terms of its ramifications for culture and society more generally. They
show how much has been discovered, recovered and analysed in detailing the
transition of mathematics from manuscript to print.
Cambridge University Press, Cambridge, 1960.
Fourteenth-century Italian algebra
R. FRANCI and L. TOTI RIGATELLI
The significance of sixteenth century Italian algebra is due, not only to the
quality of the results themselves, but also to the fact that they appeared
suddenly and almost without justification: that is, they seem to be manifesta
tions of genius independent of the study of previous results.
The only precedents usually recalled are the Liber abaci (1202) of
Leonardo Pisano and the Siimma de arithmeiica (1494) of Luca Pacioli, and
they seem negligible by comparison with sixteenth-century algebra texts.
Only recently has publication of some fourteenth- and fifteenth-century
manuscripts shown a continuity of development with the earlier authors
mentioned above.
The conclusions which we have drawn as to the high level reached by
fourteenth-century algebra are based on a systematic study of eighteen
manuscripts.
Among the more characteristic elements which we have found are the
broad treatment of algebraic calculations and the presentation of the solu
tions of a large number of types of equations. In particular, concerning this
latter subject, attempts to solve the cubic equation must be emphasized, even
if the general solution was not found. Nevertheless, many noteworthy partial
results were obtained, and some of them were used by Scipione dal Ferro and
Girolamo Cardano. Fourteenth-century algebraists also turned their atten
tion to the solution of some types of fourth degree equations and to
equations of the form ax^" + bx" + c = 0. Finally, concerning the application
of algebra to the solution of problems, we can observe that it is always used
more to solve business problems, even though the fourteenth-century
algebraists did not neglect the elaboration of new and difficult theoretical
problems such as those which Leonardo Pisano presented in the fifteenth
chapter of Liber abaci.
1. The sources
The primary sources for reconstructing the history of fourteenth century
Italian algebra are about fifteen manuscripts which contain treatises on
algebra.
Further, we must consider, as secondary sources, three very important
manuscripts of the following century. These texts, which are compendia of
the mathematical knowledge taught in abbacus schools, contain large
11
