Table Of Content^CAPITALISM
AND
ARITHMETIC
The New Math of the 15 th
Century Including the full text of
the Treviso Arithmetic of 1478
Translated by David Eugene Smith
♦ FRANK J. SWETZ
•i*Open Court
La Salle, Illinois
Dedicated to my parents,
Frank Leo Swetz
and
Helen Curtis Swetz,
who taught me how to
persevere.
OPEN COURT and the above logo are registered in the U.S.
Patent and Trademark Office.
© 1987 by Open Court Publishing Company.
First printing 1987.
All rights reserved. No part of this pubhcation may be
reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission
of the publisher. Open Court Publishing Company, La Salle,
Illinois 61301.
Printed and bound in the United States of America.
Library of Congress Cataloging-in-Publicarion Data
Swetz, Frank.
Capitalism and arithmetic.
Bibhography: p.
Includes index.
1. Mathematics— History— 15th century. 2. Arithmetic—
History— 15th century. I. Smith, David Eugene, 1860-
1944. II. Title.
V QA23.S94 1986 510'.9'024 85-21694
HSBN 0-87548-438-7
ISBN 0-8126-9014-1 (pbk.)
This book includes the first English translation master: Wherefore in all great works are Clerks so
of the Treviso Arithmetic of 1478, the earliest much desired? Wherefore are Auditors so
well fed? What causeth Geometricians so
known example of a printed book on
highly to be enhaunsed? Why are
arithmetic. The Treviso Arithmetic bore no title
Astronomers so greatly advanced? Because
and no author’s name; it appears in many
that by number such things they finde,
library catalogues listed as Arte del Abbaco. The
which else would farre excell mans minde.
present Open Court edition was designed by
John Grandits, and is set in Bembo, an English scholar: Verily, sir, if it bee so, that these men by
Monotype design based on a face cut by numbering, their cunning do attain, at
Francesco Griffo for the printer Aldus whose great works most men do wonder,
Manutius of Venice in the early 1490s. The then 1 see well I was much deceived, and
small Maltese Cross motif was a feature of the numbering is a more cunning thing than 1
original Treviso volume.
took it to be.
Robert Recorde, The
Declaration of the Profit of
Arithmeticke (1540)
Contents
1 * Perspectives I
The Climate of European Social
and Intellectual Change I
The Venetian Republic and
Treviso 6
Welsche Praktik 10
The Italian Reckoning School i8
Printing and Early Arithmetic
Books 24
Some Features of the Treviso
Arithmetic 34
2 * A Translation of the Treviso
Arithmetic 37
Prologue 37
The Text 40
3 * An Introduction to the New
Knowledge 177
“Incommica . . 177
IX «
•i" David Eugene Smith: A XI «
«x Numeration i8i
CONTENTS Addition 185 Tribute 299 CONTENTS
Subtraction 191
Some Preliminary Impressions 194 «i< Bibliography of Historical
References 305
4 * Multiplication 197
Multiplication by the Column 201 General Bibliography 309
Cross Multiplication 203
Chessboard Multiplication 205
Notes 319
Division 2II
•i* Index 341
“Dura cosa e la partita”— division is
a difficult thing 213
Some Other Methods of Division 218
6 * Applications; “What tools to
use . . . ” 223
The Rule of Three 224
Tare and Tret 232
Partnership 234
Barter 238
Alligation 240
The Rule of Two 243
The Golden Number and the Full
Moon 248
7 * A Glimpse of Fifteenth-
Century Life and Mathematics
257
Trade and Industry 257
Society 264
Money and Finance 269
Weights and Measures 277
A Mathematical Testament 281 ,
The Merchant as an Intellectual
and Scientific Innovator 291
Some Final Thoughts 295
Preface
Several years ago, while searching through
miscellaneous papers and documents of the
David Eugene Smith Collection of the Butler
Library at Columbia University, I came across
a time-yellowed manuscript. The dusty bundle
of pages bore the simple title The Treviso
Arithmetic 1478, and consisted of over two
hundred pages of typed and handwritten notes.
In the main, the work was a translation Smith
had made of an Italian arithmetic book with
the intent of publishing it. In the nineteen
twenties, David Eugene Smith was an
extremely active scholar in the fields of
mathematics education and the history of
mathematics,* Chairman of the mathematics
department at Teachers College, Columbia
University, and a prolific author of textbooks,
Smith’s varied demands distracted him from
*See the brief historical sketch of the life and work of David
Eugene Smith in the epilogue following the text.
XIII#
#XIV his Treviso Arithmetic project, and the work was mathematical activity of this time: Die Practica XV#
PREFACE never completely prepared for publication. des Algorismus Ratisbonensis: Ein Rechenbuch des PREFACE
In glancing through the manuscript, I found Benediktiner klosters St. Emmeran aus der Mitte des
that it dealt with the concepts and processes of 15 Jahrhunderts (1954), and Ein italienisches
early Renaissance arithmetic, involving the use Rechenbuch aus dem 14 Jahrhundert (1977).
of Hindu-Arabic numerals, their computational However, the works of both these men are
algorithms, and the use of those algorithms in limited to readers of their respective native
solving problems of commercial arithmetic. Its languages, Italian and German, and are thus
problem situations focused on concerns of not readily available to a wide transatlantic
Italian merchants of the late fifteenth century, audience. Dr. Van Egmond’s scholarly study.
such as payment for goods received, currency The Commercial Revolution and the Beginnings of
exchange, and the determination of shares of Western Mathematics in Renaissance Florence,
profit derived through partnership ijoo-1300 (1976), focuses on the broad impact
arrangements. While the contents were of “abaci” manuscripts on mathematical
interesting, my main concern at the time thinking, and attempts to view their contents
focused on other matters, and I continued my and development within an economic and
search, dismissing the Treviso manuscript for sociological framework. Unfortunately, its
the moment. format and availability as a doctoral dissertation
In the ensuing months after my initial limits its accessibility and appeal to a general
encounter with Smith’s papers, my thoughts reading audience. Thus, I began to realize that
repeatedly returned to the Treviso translation. Smith’s work was the first and only English
Gradually, its importance and significance language translation of a fifteenth-century
dawned upon me. The fifteenth century was a European arithmetic book that I had
time of commercial, intellectual, and encountered. Its contents, if analyzed, could
mathematical ferment, yet there is a dearth of provide valuable information. A little further
specific information on the mathematical research revealed that the Treviso manuscript
climate of this period. Some work has been was, indeed, an intellectual and historical
done in this area, notably by Gino Arrighi of treasure for several reasons:
Italy, Kurt Vogel of Germany, and, most
recently, by Warren Van Egmond of the 1. It is the earliest known dated, printed
United States. Arrighi, perhaps the most active arithmetic book.
contemporary researcher on the history of 2. It is one of the first mathematics
mathematics in medieval and early Renaissance books written for popular consumption
Italy, has produced a rather extensive collection and, as such, marks a turning point in
of research publications that reflect on this the history of human knowledge. The
subject. Vogel, of the Deutsches Museum, book’s message, transcribed in the
Munich, has published two major works that common Venetian dialect of the period,
contribute to an understanding of the was intended for all who wished to learn
Seven focuses on the social, economic and XVII#
*XVI the art of computation, not just for a
commercial aspects of the book’s contents, and PREFACE
PREFACE privileged few, as had been the case
summarizes the findings of this excursion into
previously.
3. The contents of the work provide an early Renaissance mathematics.
Ostensibly, this study focuses on a book
early example of efforts to promote the
Hindu-Arabic numeral system and its and its contents but, perhaps more
computational algorithms. Use of the importantly, it also concerns a time (the early
abacus and Roman numerals was still Renaissance), a place (the Venetian Republic),
popular in much of Europe, although to and circumstances (the rise of mercantile
a lesser degree in Italy, where capitahsm and the economic beginnings of
commercial interests demanded more industrialization), and how these three aspects
efficient computational procedures. molded and affected the directions of human
Thus, the translation supplies insights involvement with mathematics. Ultimately, the
into the mathematical climate and object of this work is to contribute to a better
controversy of the late fifteenth century. understanding of the historical interaction of
the development of mathematical ideas and
More fully aware of the importance of the
techniques and their societal milieu.
manuscript I had uncovered, I undertook to
The undertaking and completion of a study
complete the task that David Eugene Smith
of this magnitude and scope, relied on the co
had begun nearly eighty years before— to
operation and assistance of many individuals
publish a study of early Renaissance arithmetic
and institutions. I would like to acknowledge
based on the contents of the 1478 Treviso
and thank some of the principal participants
tome. Smith’s concern was mainly pedagogical;
who assisted me in my research tasks. The
he was primarily interested in how early
libraries at Columbia University kindly
arithmetic was taught. The following study,
provided me with access to their copy of the
while considering the heuristics of early
Treviso Arithmetic housed in the George A.
European arithmetic instruction, will also
Plimpton Collection, Rare Book and
investigate the mathematical and sociological
Manuscript Library, and to the notes and
significance of the Treviso Arithmetic s contents.
translation of D. E. Smith contained in the
The first chapter of this study provides a
David Eugene Smith Collection. My own
general background in which the Treviso
library staff at the Heindel Library, Capitol
Arithmetic and its contents must be understood.
Campus, the Pennsylvania State University,
Chapter Two presents Smith’s free translation
particularly Carolyn Miller, reference librarian,
of the Arithmetic. The succeeding four chapters
and Ruth Runion, interlibrary loans clerk,
serve as a commentary on its contents and
correspond to the sequential ordering of topics demonstrated resourcefulness and
in the Arithmetic: numeration, addition and determination in securing elusive references for
subtraction, multiplication, and division, and my use. An uncle, Arthur Torelli, assisted in
the techniques of problem solving. Chapter the translation of Italian language materials.
# XVIII Dr. Alan M. Stahl, associate curator of
PREFACE medieval coins, the American Numismatic
Society, supplied information about early
European currency. Stillman Drake, of the
Institute for the History and Philosophy of
Science and Technology, University of
Toronto, clarified some issues on Renaissance
timekeeping. Finally, I wish to thank Dr. Peter
Hinton for his valuable editorial suggestions,
and my wife, Joan, for her careful proofreading
of the manuscript.
Frank J. Swetz
Harrisburg, PA
1986
^CAPITALISM
AND
ARITHMETIC
^ I
4
Perspectives
The Climate of European Social and
Intellectual Change
Fifteenth-century Europe witnessed a period of
great change— social, economic, and
intellectual— and was marked by both the
physical and transcendent movements of
peoples and ideas. The historical rise and
coming together of the forces and conditions
that bred these changes is a complex
phenomenon beyond the scope of this study;
however, the changes or movements
themselves are worth noting (see Table i.i).
Feudalism was on the wane. The beginnings of
a spirit of national or regional consciousness
were emerging among peoples who began to
feel a sense of societal cohesiveness and
purpose. Growth of towns and cities meant a
new distribution of political power, as well as
new types of power. Occupational
opportunities broadened from the realms of
I*
