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Ars magna, or, The rules of algebra Artis magnae, sive de regulis algebraicis. Lib. unus. Qui & totius operis de arithmetica, quod Opus Perfectum inscripsit, est in ordine decimus PDF

291 Pages·1993·12.27 MB·English
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Preview Ars magna, or, The rules of algebra Artis magnae, sive de regulis algebraicis. Lib. unus. Qui & totius operis de arithmetica, quod Opus Perfectum inscripsit, est in ordine decimus

HIERONYMI 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

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