Table Of ContentHIERONYMI CAR
PRJESTANTISSIMI MATHE
DANI~
• A TIC I~ P]I I x.. 0 , 0 P H r. A C II B Die I~
AR TIS MAGNiE,
SIVE DE REGVLIS ALGEBRAICIS.
Lib. unus. QUi & tonus operis de Aridunctica. quod
OPVS PERFECTVM
infaiplit.dtin ordinc DccimUi.
H
Am in hocJibro,Gudiofc Lcdor,ReguYas AlgEbnkas (Itati, dela cor
fa lIocant) nouis adinumtiorubus .ac dcmonftrationibus ab Authorc ita
locuplctatas,ut pro pauculis anrca uulgo tritis.iam kptuaginta cuafttint.NOI
" folum , ubilmus numerus altcri,aut duo uni,ucrum etiam,ubiduo duobus.
aut trcs uni {qualcs nlcrinr,nodum txplicant. HuncatJt hbnnnidco feors.
ft m edcre placuit,ut hoc abftruftlSimo, & plant! incxhaufto tonus Arithmcri
cz thefauro in luccm cruro, & quafi in thcatro quodam omnibus ad fpedan
dum txpofito. Lcdorcs incirarirur,ut rcliquos Operit P~rfedibbros, qui per
Tomos cdentur,tanto auidiusamplcdantur,ac minorc faftidio pcrdifcant.
ARS MAGNA
or
The Rules of Algebra
GIROLAMO CARDANO
Translated and Edited by
T. RICHARD WITMER
With a Foreword by Oystein Ore
DOVER PUBLICATIONS, INC.
New York
Copyright
Copyright © 1968 by The Massachusetts Institute of Technology.
All rights reserved under Pan American and International Copyright
Conventions.
Published in Canada by General Publishing Company, Ltd., 30
Lesmill Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd., 3
The Lanchesters, 162-164 Fulham Palace Road, London W6 9ER.
Bibliographical Note
This Dover edition, first published in 1993, is an unabridged repub
lication of the edition published by The MIT Press in 1968 under the
title The Great Art or The Rules if Algebra. It was translated and edited
from the 1545 edition of Artis maguac, sive de regillis algebraic is. Lib.
Qui & tot ius operis de aritlunetica, quod opus Peifectlllll inscripsit,
U/lUS.
est in ordine decimus with additions from the 1570 and 1663 editions.
This Dover edition is published by special arrangement with The MIT
Press, 55 Hayward Street, Cambridge, Massachusetts 02142.
Library of Congress Cataloging-in-Publication Data
Cardano, Girolamo, 1501-1576.
lArs magna. Liber l. English)
Ars magna, or, The rules of algebra / Girolamo Cardano
translated and edited by T. Richard Witmer ; with a foreword by
Oystein Ore.
p. cm.
Originally published: The great art, or The rules of algebra.
Cambridge, Mass. : M.I.T. Press, 1968, which was translated and
edited from the 1545 ed. of Ars magna, liber unus, with additions
from the 1570 and 1663 eds.
Includes bibliographical references and index.
ISBN 0-486-67811-3 (pbk.)
l. Algebra-Early works to 1800. I. Witmer, T. Richard, 1909-
II. Title. III. Title: Ars magna. IV. Title: Rules of algebra.
QAl54.8.C3713 1993
512.9'4-dc20 93-34446
CIP
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
of
Table Contents
FOREWORD by Oystein Ore Vll
PREFACE xv
The Great Art, or The Rules of Algebra
by Girolamo Cardano
ApPENDIX
Portions of Euclid's Elements Cited by Cardano 262
INDEX 265
Foreword
It has often been pointed out that three of the greatest masterpieces
of science created during the Rinascimento appeared in print almost
simultaneously: Copernicus, De Revolutionibus Orbium Coelestium (1543) ;
Vesalius, De Fabrica Humani Corporis (1543), and finally Girolamo
Cardano, Artis Magnae Sive de Regulis Algebraicis (1545). But while the
two first works have been readily available in magnificent editions and
excellent translations, Cardano's Ars Magna has remained relatively
obscure, its material confined to early and now rare Latin editions.
The Ars Magna has always been highly praised as a milestone in the
history of mathematics, yet it is true that the number of modern scholars
who can claim to have examined it in detail is extremely small. Thus
the present translation, making it available to a much wider circle of
readers, is a most significant addition to the literature of the history of
science.
The creation of a modern version of an ancient scientific text poses
a number of problems-the suitable choice of new or old technical
terms, the rendering and interpretation of obscure or confused passages,
and also, peculiar to mathematics, the decision to which extent one
should introduce present-day symbolism. These matters of judgment
seem to have been happily resolved in this translation. Cardano's
work, cumbersome as it is in its original form, complicated by the
rudimentary mathematical language of the period, now emerges as a
book which can be studied by any college undergraduate. The systema
tic use of ordinary school algebra in Cardano's reasonings contributes
greatly to this simplification.
Nowadays, the Ars Magna would be characterized as a text on
algebraic equations. To Cardano's contemporaries it was a break
through in the field of mathematics, exhibiting publicly for the first
time the principles for solving both cubic and biquadratic equations,
giving the roots by expressions formed by radicals, in a manner similar
viii Foreword
to the method which had been known for equations of the second
degree since the Greeks, or even the Babylonians. Cardano actually
does not claim either of the two innovations entirely as his own; he
rather considers the special third degree equation first solved by
Scipione del Ferro, and the fourth degree equation solved by Lodovico
Ferrari, Cardano's secretary, as toeholds which enable him to create
his own general theory embracing all possible cases.
These many cases, produced largely by the necessity for separating
the arguments for positive and negative numbers and by the lack of an
efficient algebraic notation, lead to the elaborate lists of equation types.
Cardano studies what he considered to be properties of general
equations, for instance, relations between roots and coefficients, rules
for the signs or the locations of the roots. He barely touches upon the
numerical solution of equations, but here he brings little new. One
notable aspect of Cardano's discussion is the clear realization of the
existence of imaginary or complex solutions. They appear as necessary
consequences of the formulas, and he does not avoid them or brush
them aside as unimportant, as often done by earlier writers. On the
contrary, Cardano constructs examples for the express purpose of
dealing with problems with imaginary rools (Chapter 37).
In dealing with problems as complicated as the solution of higher
degree equations, it is evident that Cardano is straining to the utmost
the capabilities of the algebraic system available to him. Shortly
afterwards his successors set to work to create a more general and more
efficient algebraic language. This is noticeable already in Bombelli's
Algebra (1572); in 1591 appeared Viela's work, 111 Artem AlIalyticam
Isagoge, which brought mathematical terminology to a stage approach
ing the modern one.
Cardano complicates his notational difficulties further by basing
most of his proofs upon geometric arguments, thus emulating the
reasoning of Euclid as he repeatedly emphasises. To us there is no
necessity for the use of such methods; on the contrary, they appear
strongly incongruent in this connection. But this was a period of the
highest veneration of the methods of Greek mathematics, and Euclid's
Elemmts represented the pinnacle of logical stringency. Euclid's
geometric solution of second degree equations consisted in constructing
squares, and so the proper approach to cubic equations would be
through the construction of cubes. But from then on the guidance of the
geometric intuition disappeared, and {t is curious to note that Cardano
seems to feel that there is a connection between the fact that space as
Foreword lX
he knew it was three dimensional, and the difficulties in solving
higher degree equations: "Nature does not permit it" (Chapter I).
The Ars Magna, in spite of its novelty, is influenced to some extent by
the centuries of Italian algebra from Leonardo Pisano's Liber Abaci
(1202) to the Summa (1494) of Fra Luca Paccinolo. As in the lectures
of the Abacista, the problems are often clothed in practical garb to
make them more attractive to the readers. For equations of higher
degree this is no easy task, so in most cases Cardano falls back upon
straight numerical examples. But a number of the examples are of the
types reminiscent of mercantile Italian arithmetics of late medieval
times. Among them let us mention the distribution of monies among
soldiers, profit on repeated business trips, and the ever present questions
concerning partnerships (regola della compagnia, Chapter 5). All these
problems have a highly artificial aspect. The only ones in which the
cubic equations appear in nearly natural fashion are the extension of
the Delian problem concerning the doubling of the cube (Chapter 17),
geometric problems concerning triangles (Chapters 32, 38) and ques
tions concerning compound interest (Chapters 18, 20). Chapter 37 is
one of the few where Cardano deals freely with negative numbers. One
of the examples is quite illustrative, and this situation was probably
not rare. The dowry of the bride is compared to the estate of the hus
band, and it turns out in the end that he is so heavily in debt that the
dowry barely bails him out.
The Ars Magna, besides being an outstanding mathematical achieve
ment, is also famous as the direct cause of one of the most violent feuds
in the history of science. Cardano acknowledges abundantly, in fact
three times (in Chapters 1,6, and II) that he had originally received
the solution to the special cubic equation
X3 + ax = b
from his friend Niccolo Tartaglia of Brescia, at the time abacista in
Venice. Cardano's account in Chapter II runs as follows: "Scipio
Ferro of Bologna well-nigh thirty years ago discovered this rule and
handed it on to Antonio Maria Fior of Venice, whose contest with
Niccolo Tartaglia of Brescia gave Niccolo occasion to discover it. He
[Tartaglia] gave it to me in response to my entreaties, though with
holding the demonstration." This places the original discovery of the
solution by Scipione del Ferro, professor of mathematics in Bologna,
at approximately the year ISIS. Scipione died in 1526, and at that
x Foreword
time his manuscripts passed to his son-in-law and successor Annibale
della Nave. These facts are fully confirmed by the rediscovery by
E. Bartolotti of del Ferro's original papers in the library of the university
of Bologna.
At his death the solution was known by certain of del Ferro's pupils,
certainly by Annibale della Nave and Antonio Maria Fiore. The
latter returned to his native town of Venice, probably to make a living
as a teacher of mathematics. To make his name known, he in 1535
challenged Tartaglia to a public problem-solving contest. Fiore's
problems all concerned del Ferro's solution of the equation: The cosa
and the cube equal to a number. Tartaglia describes his agony before
the contest, but in the nick of time, the night before the contest, he
succeeded in finding the method, and Fiore suffered a humiliating
defeat.
Word of the contest reached Cardano in Milan who at the time was
preparing the manuscript for his Practica Arithmeticae Generalis, which
appeared in 1539. Cardano had obviously hoped that Tartaglia would
divulge the secret and give him permission to include the method
for the cosa and the cube in his book. This Tartaglia flatly refused with
the statement that in due time he would write a book on the subject.
So far the facts seem clear, but for the subsequent events we are
dependent almost exclusively upon Tartaglia's printed accounts, which
by no stretch of imagination can be regarded as objective. It appears
that not long afterwards Tartaglia had at least a partial change of heart.
He accepted an invitation from Cardano to visit him in Milan,
possibly in the hope that through Cardano's influence he could secure
a position as advisor to Alfonso d'Avalos, the Spanish viceroy and
commander in chief in Milan. Tartaglia had written a book on artillery
and constructed instruments for measuring distances in the field. At a
meeting in Cardano's house the secret was divulged and, according to
Tartaglia, Cardano swore a most solemn oath, by the Sacred Gospels
and his word as a gentleman, never to publish the method, and he
pledged by his Christian faith to put it down in cipher, so that it would
be unintelligible to anyone after his death. At this point the word of
one man stands against that of another. Ferrari, then a youth of
eighteen years and a servant in the Cardano household, related later
that he was present at the meeting, and there had been no question of
secrecy involved.
After the publication of the Ars Magna, Tartaglia's rage knew no
bounds. Already the following year he published another book Quesiti
