THE ART OF ALGEBRA FROM AL-KHWARIZMI TO VIÈTE: A STUDY IN THE NATURAL SELECTION OF IDEAS


Karen Hunger Parshall
University of Virginia (formerly at the University of Illinois at Urbana-Champaign)

This document previously appeared as an article in the June 1988 issue of History of Science, Vol. 26, No. 72, pp.129-164. The HTML-encoded version which you are reading appears here with the kind permission of the Editors of Science History Publications, Ltd., London, England, and includes ancillary material not present in the original article, © Karen Hunger Parshall, 1995, 1999. Additional biographical and historical material provided by Karen H. Parshall and Patti Wilger Hunter. HTML-encoding was performed by Fred O'Bryant, Science and Engineering Library, University of Virginia. (E-Mail: JFO@VIRGINIA.EDU). Note that, at the time of original HTML-encoding (May 1995), common WWW browsers were unable to properly render various mathematical and typographical symbols used in the original published text. Compensations and compromises have been made in this WWW document that take these limitations into account, hopefully without loss of clarity or meaning in the text. Improvements in WWW browser technology have allowed some of these limitations to be overcome. However, persons still using older browsers may encounter typographical anomalies. You are encouraged to upgrade to the latest browser versions to minimize this possibility.

If you have questions or comments regarding this document, you may address them to Karen H. Parshall, Professor of Mathematics and History, University of Virginia, (E-Mail KHP3K@VIRGINIA.EDU). Please visit Ms. Parshall's web page for more information about the author and her work.


CONTENTS

  1. Introduction
  2. Algebra in the Arab World: al-Khwarizmi, Abu- Kamil, and al-Karaji
  3. Algebra in the West: Leonardo Fibonacci and the Maestri d'Abaco
  4. The Ars Magna of Girolamo Cardano
  5. The Rediscovery of Diophantus's Arithmetica and the Work of Raphael Bombelli and François Viète
  6. References
  7. Index of Supplemental Biographies and Illustrations

I. INTRODUCTION

Traditionally, historians of mathematics have most often adopted a presentistic approach to their subject. From the vantage point of the state of the discipline in their own times, they have tended to portray the development of mathematics as fundamentally linear in nature. In other words, looking back into mathematical history, they have picked and chosen from among the various contributions and constructed a logical, straight line progression from the past to the present. This kind of history serves to anchor contemporary mathematics in the past by providing it with a clear sense of direction, but at the same time it profoundly distorts the view of the mathematical climate at any given time in history. In the search for predecessors of a particular type of equation, theorem, or idea, other concepts which may have been of prime importance to the authors under scrutiny tend to be ignored or trivialized. Furthermore, competing approaches and underlying philosophies often fall into total obscurity. Thus, relative to the goal of reconstructing the dynamics of mathematics at a particular point in time, the traditional historical method falls short. This end may be more nearly met, however, by regarding mathematics as the result of the natural selection of ideas.

Suppose that at any given time and place, we define the mathematical environment as the known body of mathematical facts, techniques, theories, and ideas together with the mathematicians who deal with them. Within this context, every idea which presents itself, whether new or newly rediscovered, effects a change in the environment. Thus, the individual mathematician, by generating new ideas, by remaining ignorant of an idea, or by failing to absorb an idea, shapes the particular niche within which his or her own theories develop.

A particular mathematician's theory relative to a given mathematical question inherits largely the characteristics of the theories of that mathematician's immediate predecessors. Yet a given theory may present individual variations which occur as the result of the introduction of the mathematician's new ideas. Those individual variations which in any way favour the theory, whether by clarifying some fact, however small, or by correcting some point, however minor, will make the theory more fit and so will be naturally selected. The successive accumulation of individual differences through the natural selection of ideas yields a variety which differs more and more from its parent-theory. Then, as different varieties of theories interbreed through the combination, reorganization, and introduction of ideas, and as natural selection acts upon the favourable variations which result, the varieties may gradually develop into clear and distinct species of mathematical theories. If at any time during the evolutionary process, however, a variation, that is, an idea or an approach occurs which is neither useful nor injurious, it would persist essentially unaltered until changes in the mathematical environment rendered it either advantageous or disadvantageous. Viewed with respect to this kind of an evolutionary framework, what the modern mathematician and some historians might regard as the false starts, ill-conceived techniques, and imperfectly formed theories of the past, actually appear as intermediate steps in the evolutionary process of descent with modification.

The development of algebra from al-Khwarizmi to Viète provides a good test case for this model of the natural selection of ideas. In the sixteenth century, algebra became the stage for the confrontation of the more or less continuous and adapting Arabic line of al-Khwarizmi (c. 800-c. 847) and the previously latent but newly rediscovered approach of Diophantus of Alexandria (fl. A.D. 250). Writing at midcentury, Girolamo Cardano (1501-76) opened his Ars magna by declaring algebra's indebtedness to the Arab world. He asserted that "this art originated with Mahomet the son of Moses the Arab [i.e., al-Khwarizmi]"1 and proceeded to expound the findings of al-Khwarizmi and his successors in the Arabic line of descent. By the end of the century, though, this Arabic approach to algebra no longer held sway. Long neglected manuscripts of Diophantus's Arithmetica had come to light, and mathematicians like Raphael Bombelli (1526- 72) and François Viète (1540-1603) not only absorbed the ideas presented there but also recognized the Arithmetica as a mathematical work significantly different from the usual Arabic-inspired text. In his In artem analyticem isagoge of 1591, Viète clearly expressed his humanistic desire to purge algebra of its Arabic corruptions and to return it to a more pristine state inspired by the classical Greeks. He bade his readers:

Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbarians, that I considered it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudotechnical terms lest it should retain its filth and continue to stink in the old way, but since till now ears have been little accustomed to them, it will be hardly avoidable that many will be offended and frightened away at the very threshold. And yet underneath the Algebra or Almucabala which they lauded and called "the great art", all Mathematicians recognized that incomparable gold lay hidden, though they used to find little ... our art [i.e., the analytical art of algebra] is the surest finder of all things mathematical.... 2
In this paper, we interpret the development of the Arabic, algebraic line from the time of al-Khwarizmi to the sixteenth century in light of an evolutionary framework and examine the way in which natural selection may be thought of as having acted on and modified this approach in the presence of the reintroduced, Diophantine concepts.
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2. ALGEBRA IN THE ARAB WORLD: AL-KHWARIZMI, ABU-KAMIL, AND AL-KARAJI

Sometime around A.D. 830, Muhammad ibn Musa al-Khwarizmi composed the earliest known Arabic treatment of algebra and started an algebraic line in the Arabic world that persisted for several centuries. As a scholar at the House of Wisdom, which began in Baghdad as the library of Harun al-Rashid (reigned 786-809) and flourished under Caliph al-Ma'mun (reigned 813-33), al-Khwarizmi, together with his noted colleagues the Banu Musa, directed and engaged in intellectual interests ranging from algebra and geometry to astronomy and the translation of Greek scientific manuscripts.3 It was in this atmosphere of protection and patronage that al-Khwarizmi wrote his treatise entitled al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala or The compendious book on calculation by completion and balancing.4 Conceived as an elementary textbook of practical mathematics, the Al-jabr wa'l-muqabala began with a discussion of the algebra of first and second degree equations and moved on in its final two parts to the business of practical applications to questions of mensuration and legacies.

Judging from evidence internal to al-Khwarizmi's Al-jabr wa'l- muqabala, the mathematical environment in which his ideas developed included facts, theories, and approaches from several recognizable sources. Al-Khwarizmi's use of geometrical justifications of algebraic manipulations together with the fact that the Elements (Sample Page - 322K) existed in two distinct translations from Greek into Arabic by his contemporary at the House of Wisdom, al-Hajjaj ibn Yusuf ibn Matar,5 suggest a line of descent from Euclid. On the other hand, because his treatment of practical geometry so closely followed that of the Hebrew text, Mishnat ha Middot, which dated from around A.D. 150, the evidence of Semitic ancestry exists.6 Al-Khwarizmi's concern with practical algebra and his treatment of equations through the second degree betray a vestige of the Babylonian line,7 while his totally rhetorical style points to a remote Hindu ancestor and a lack of contact with later Greek texts, particularly the Arithmetica of Diophantus. In fact, since the first known Arabic translation of the Arithmetica was not completed by Qusta ibn Luqa until the middle of the ninth century or later,8 we can be fairly certain that the more theoretical ideas of Diophantus had not yet entered the environment of, and so had not come into competition with, Arabic mathematics. Given this complex mathematical environment with its well-defined varieties of algebraic theories, we must now examine how the theory al-Khwarizmi presented in his Al-jabr wa'l-muqabala could have arisen through a natural selection of ideas.

In the opening algebraic part of the Al-jabr wa'l- muqabala, al-Khwarizmi distinguished and solved six types of algebraic equations up to and including the quadratic, namely, squares equal to roots, squares equal to numbers, roots equal to numbers, squares and roots equal to numbers, squares and numbers equal to roots, and roots and numbers equal to a square. ln modern notation these become ax² = bx, ax² = c, bx = c, ax² + bx = c, ax² + c = bx, and bx + c = ax², respectively, with the presence of six separate cases following from the fact that mathematicians up to and well beyond this time acknowledged neither zero coefficients nor negative numbers. Al-Khwarizmi systematically presented the algebraic solutions, known since Babylonian times, of particular cases of these equations and then provided geometric justification for his algebraic rules. Consider his discussion of squares and roots equal to numbers:

... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.9
In modern notation, the problem was to solve the equation x² + 10x = 39 for x². 10 The method, given step by step, translates as 1/2 X 10 = 5, 5² = 25, 39 + 25 = 64, [SQUARE ROOT OF] 64 = 8, 8 - 1/2 X 10 = 3 so x = 3 and x² = 9.

However, al-Khwarizmi went beyond merely providing the sort of algebraic recipe found in Babylonian texts. He insisted upon superadding a Euclidean style of geometrical proof for algebraic fact. Thus, after explicitly stating that "... it is necessary that we should demonstrate geometrically the truth of the same problems which we have explained in numbers",11 he proceeded to justify the above example with two different geometrical constructions, both of which yielded a completion of the square. In the second construction he required that

... to the square as representing the square of the unknown we add ten roots and then take half of these roots giving 5. From this we construct two areas added to the sides of the square figure ab.

Figure 1

These again are called ag and bd. The breadth of each is equal to the breadth of one side of the square ab and each length is equal to 5. We now have to complete the square by the product of 5 and 5, which, representing the half of the roots, we add to the two sides of the first square figure, which represents the second power of the unknown. Whence it now appears that the two areas which we joined to the two sides, representing ten roots, together with the first square, representing x², equals 39. Furthermore it is evident that the larger or whole square is formed by the addition of the product of 5 by 5. This square is completed and for its completion 25 is added to 39. The sum total is 64. Now we take the square root of this, representing one side of the larger square and then we subtract from it the equal of that which we added, namely 5. Three remains, which proves to be one side of the square ab, that is, one root of the proposed x². Therefore three is the root of this x², and x² is 9.12

Although not as formal in style, this argument paralleled that given by Euclid in the Elements for II.6: "If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line."13

Figure 2

For a straight line AB bisected at C and a straight line BD added to it, Euclid proved II.6 by showing that rectangle ADMK equals the sum of rectangles CDML and HMFG. Thus, by adding the square on CB, that is, by adding square LHGE to CDML and HMFG, we have square CDFE, as desired.14 With respect to al-Khwarizmi's proof above, square BDMH played the role of ab, the equal rectangles CBHL and HMFG were the geometric results of dividing the ten roots into two groups of five roots each and corresponded to ag and bd, and finally square LHGE for Euclid and al-Khwarizmi's square bh completed the larger square. The two arguments hinged on exactly the same sequence of steps.15

By incorporating a certain measure of Euclidean geometrical rigour into a practical textbook on algebraic manipulation, al-Khwarizmi effected a variation upon which the natural selection of ideas could act. His idea amounted to uniting aspects of two previously distinct varieties of algebraic thought, namely, the calculationally oriented Babylonian approach to algebra and Euclid's formal geometrical interpretation of algebra. In the present context, this may be viewed as the interbreeding of two varieties yielding an offspring, a new variety, which, through the preservation of the favourable characteristics of both parents by natural selection, was distinct from both. The Babylonians wanted accurate techniques for solving practical problems involving both linear and quadratic equations, while as Sir Thomas Heath explained, Euclid wished "... to show the power of the method of geometrical algebra as much as to arrive at results".16 Al-Khwarizmi's new variety of algebra presented the favourable variation of practical computation justified by mathematical proof.17 Thus, in our model, the natural selection of ideas should have preserved this favourable variation. That it was indeed preserved may be seen in the fact that al-Khwarizmi's Al-jabr wa'l-muqabala served as the point of departure for many succeeding Arabic treatments of algebra.

In the generation just after al-Khwarizmi, Abu-Kamil (c. 850-c. 930) based his own Kitab fi al-jabr wa'l-muqabala or Book on completion and balancing on al-Khwarizmi's work. In his text, Abu-Kamil not only quoted directly from al-Khwarizmi, but he also incorporated almost half of al-Khwarizmi's forty examples into his work with little more than numerical changes.18 The mathematical environment in which Abu-Kamil's thought developed involved more than the work of al-Khwarizmi, however. Whereas the evidence of Euclidean ancestry in the mathematical thought of al-Khwarizmi, though strong, was purely morphological, Abu-Kamil actually cited Euclid in his geometrical proofs.19 Thus, after only one generation, Euclid's text and ideas appear to have become more widespread within the environment of algebraic ideas. Since any variation which better adapts a theory to changed conditions should be preserved under the action of natural selection, Abu-Kamil's idea of establishing al-Khwarizmi's geometrical arguments with complete Euclidean formality and rigour should have proved favourable.

Compare, for example, Abu-Kamil's proof of the solution of the equation x² + 10x = 39 with the proofs of al-Khwarizmi and Euclid discussed above:

... the obvious solution is the root when one lays out a surface of a square quadrilateral on it--ABGD:

Figure 3

One adds the roots to it which were originally associated with the square--it is 10--they are ABWH. One knows that line BH is 10 because the side AB of the surface ABGD multiplied by unity is a root of the surface ABGD. It is one multiplied by 10; it is 10 roots of the surface ABGD. Thus it is line BH or 10. The entire surface WHGD is 39 because it was set up as the square and 10 of its roots; it is the product of line HG by line GD. But line GD is equal to line GB. Also, the product of line HG by line GB is 39. Line HB equals 10. Divide it in half by the point H. Add line GB to its length. And so, the surface is the result of the product of [HG by itself just as the surface is the result of the product of] HG by line BG added to the square quadrilateral, the product of HB by itself just as Euclid stated in the second part of his book. But the product of line HG by line GB is set at 39. The product of line HB by itself is 25; the total is 64. Thus, the product of line HG by itself is 64; the root of 64 is 8. Then, the line HG is 8. One knows that line HB is 5 and line BG remains as 3. It is the root of the square; the square is 9.20

Unlike al-Khwarizmi, Abu-Kamil systematically constructed and labelled the geometrical pieces of his proof and carefully linked them to the numerical components of the algebraic procedure. This rendered his treatment more rigorous (in the sense of Euclid's proof of II.6) than al-Khwarizmi's, but his use of numbers from the particular example at hand anchored his work in practical algebra as distinct from the theoretical geometrical algebra of Euclid. As Martin Levey explained, Abu-Kamil "... utilized the theoretical Greek mathematics without destroying the concrete base of al-Khwarizmi's algebra and evolved an algebra based on practical realities derived from Babylonian roots and strengthened by Greek theory"21 By infusing al-Khwarizmi's algebra with proofs inspired by Euclid's Elements, Abu-Kamil introduced a variation that made al-Khwarizmi's ideas more competitive in a more strongly Euclidean environment. Thus, in the natural selection of ideas the favourable variation presented by Abu-Kamil's variety of algebra should also have been preserved.

In fact, this line of descent from al-Khwarizmi to Abu-Kamil persisted relatively unchanged through the tenth century. At the House of Wisdom in Baghdad, Abu'l-Wafa' al-Buzjani (940-97/98) commented on al-Khwarizmi's Al-jabr wa'l-muqabala as well as on the works of Euclid and composed a practical arithmetic for the use of scribes and businessmen.22 Of particular interest, however, is Abu'l-Wafa's commentary on Diophantus's Arithmetica, which although now lost, significantly altered the mathematical environment of the Arab world by providing access to the Diophantine brand of determinate and indeterminate analysis of equations through the eighth degree. One mathematician whose work adapted itself to this change in environment was al-Karaji.

Also working in Baghdad sometime toward the end of the tenth century and through the beginning of the eleventh, al-Karaji united aspects of the geometrical algebra of al-Khwarizmi and Abu-Kamil with the indeterminate algebra of Diophantus in his treatise entitled al-Fakhri. There, for the first time in the Arabic literature, he presented a formal algebraic calculus in which he exhibited rhetorically relations such as these modern equivalents: 1/x² X 1/x = 1/x³, 1/x X x³ = x², and l/x : l/x² = x²/x.23 He also incorporated a large number of problems and solutions from the Arithmetica into his extensive collection of examples. Among these we find indeterminate equations of degrees two and three in up to three unknowns,24 problems that transcended the theories of al-Khwarizmi and Abu-Kamil. According to Roshdi Rashed, "the more-or-less explicit aim of this exposition was to find the means of realizing the autonomy and specificity of algebra, so as to be in a position to reject, in particular, the geometric representation of algebraic operations".25 Within the present context, however, al-Karaji's new variation of algebra merely marked an adaptation to a changed environment. In the presence of Diophantus's ideas, al-Karaji, who was thoroughly imbued with the geometrical algebra of al-Khwarizmi, accepted algebraic techniques from both of these sources. At the same time, he rejected aspects of both approaches, specifically, al-Khwarizmi's complete reliance on geometry and Diophantus's syncopated notation. For reasons which we shall examine in the next section, al-Karaji's algebraic variation competed with only limited success in an environment which continued to be dominated by the ideas of al-Khwarizmi.


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3. ALGEBRA IN THE WEST: LEONARDO FIBONACCI AND THE MAESTRI D'ABACO

In 711, one hundred years before al-Khwarizmi wrote his Al-jabr wa'l-muqabala, Muslim armies invaded Spain from North Africa and within several years they had conquered virtually the entire peninsula. In the three centuries that followed, this new Islamic state flourished until conflicts over succession ended the ruling Umayyad Dynasty in 1031. During the century prior to its collapse, however, the Umayyad court, set up at Cordova under Abd al-Rahman III (ruled 912-61), became a focal point of scholarship and learning. Under Abd al-Rahman's successor, al-Hakam II (ruled 961-76), scholars filled the court and the Cordovan library grew to some 400,000 volumes secured from all over the Islamic world. With the fall of the dynasty and the concurrent destruction of the library, the books that managed to survive the desolation found their ways into the court libraries of the new states which succeeded the old regime. The presence of these Arabic texts, which both presented Arabic scholarship and preserved ancient Greek knowledge, made Spain one of the three primary centres for the translation from Arabic into Latin during the next two centuries.26

Furthermore, by opting to deal with a particular text. the translators unwittingly shaped the mathematical environment of medieval Europe. Since, by and large, only those Arabic texts which were translated into Latin were accessible to the Western reader, only the mathematics presented therein had a chance for survival. Thus, when al-Khwarizmi's Al-jabr wa'l-muqabala came to light in two twelfth-century Latin translations by Robert of Chester (fl. 1141-50) and Gerard of Cremona (c. 1114-87), his algebraic ideas escaped extinction in the West.27 Whether Leonardo Fibonacci of Pisa (c. 1170- c. 1240) studied one or both of these Latin translations of al-Khwarizmi's algebra28 or whether he learned of algebra more directly during the business trips that took him to Egypt, Syria, Greece, Sicily, and Provence,29 the fact remains that al-Khwarizmi's ideas figured prominently in his mathematical environment.

The son of a secretary of the Republic of Pisa, Fibonacci encountered early on the sort of practical, commercial mathematics involved in his father's duties as overseer of the Pisan trading colony of Bugia (now Bugie, Algeria)30 This contact piqued his mathematical interest in general and his algebraic interest in particular. Throughout his adult life, then, as business took him to various Mediterranean ports, Fibonacci fashioned his mathematical environment by seeking out texts from which, and people from whom, he could learn more of the intricacies of arithmetic and algebra. One result of these studies, his most influential book, entitled Liber abbaci (1202, revised 1228), attested to his mastery not only of the Hindu-Arabic techniques of practical calculation but also of the theory of quadratic equations as found in the works of al-Khwarizmi, Abu-Kamil, and al-Karaji.31

In his work, Fibonacci put forth not so much an original exposition (although he showed a certain amount of innovation in some of his solutions) as a compilation of the techniques of Arabic arithmetic and algebra.32 For instance, in the fifteenth and final chapter of his book, he turned to an investigation of "algebrae et almichabile". There, he presented the usual Arabic classification of equations, namely, the three simple cases of squares equal to roots, squares equal to number, and root equal to number, and the three composite cases of squares and roots equal to number, roots and number equal to squares, and squares and number equal to roots.33 Then, following his above named Arabic ancestors, he gave specific examples written out rhetorically, solved algebraically, and justified geometrically.34 Leonardo's mathematical environment encompassed more than this Arabic theory of algebra however. Within his sphere of commercial activities, there was also a need for comprehensive catalogues of techniques for solving day-to-day problems. Through the action of the natural selection of ideas, then, any algebraic treatment which presented the favourable variation of combining the practical and the theoretical should have been preserved.

The next three centuries evidenced the dominance of Leonardo's presentation not only of the theory of algebra but also of the techniques of practical problem-solving. Particularly during the fourteenth and fifteenth centuries, the mathematical environment changed with the rise of the merchant class in Italy and the establishment of so-called "abacus schools". This ever greater commercial emphasis created a need for honed-down, practical mathematics textbooks written in the vernacular, as opposed to long and sometimes theoretical treatises in Latin like the Liber abbaci or the Latin translations of al-Khwarizmi's Al-jabr wa'l-muqabala.35 In this environment, since Leonardo's theoretical, geometrical demonstrations of algebraic facts were largely unnecessary, they proved neither useful nor injurious to the work of the writers of purely practical texts. Thus, this aspect of his work was not affected by the natural selection of ideas. It merely persisted in a dormant state until conditions changed so as to bring it back into competition. The practical part of Leonardo's treatise, however, did come into direct competition with the ideas of these "maestri d'abaco". A careful examination of their "trattati" or "libri d'abaco" reveals the authority of Fibonacci's approach as well as evidence of the continuing process of the natural selection of favourable variations.

One of the earliest known vernacular treatments of algebra,36 Paolo Gerardi's Libro di Ragioni of 1328 represented a variety of algebraic treatise which, although clearly descendant from Leonardo and through him al-Khwarizmi, Abu-Kamil, and al-Karaji, presented variations favourable or uninjurious within the fourteenth century commercial environment. First, it consisted of 193 rhetorically presented examples of which all but the last fifteen were commercial in nature. Second, these final fifteen problems gave solutions with no proofs, geometrical or otherwise, for fifteen different algebraic equations. Third, while six of these last fifteen equations were the standard six equations we have seen in al-Khwarizmi, Abu-Kamil, al-Karaji, and Leonardo, nine of them were cubic and of these five were irreducible.37 For the first time in Western mathematical literature, Gerardi gave general, albeit incorrect, solutions for the irreducible cubics: ax³ = bx + N, ax³ = bx² + N, and ax³ = bx² + cx + N.38 His solutions were merely naïve applications of the quadratic formula to cubic equations. Thus, for ax³ = bx + N, he claimed

that is, the solution of the quadratic ax² = bx + N. Since he did not check his answers by reapplying them to the original problem, he did not recognize that his solution techniques yielded erroneous results. Nevertheless, Gerardi's treatment of irreducible cubics categorically proved that the quest for solutions to such equations did not begin in the sixteenth century with the celebrated controversy involving Cardano and Niccolò Tartaglia (c. 1499-1557). In fact, "... Gerardi's rules, his problems, and even his erroneous formulations are repeated in similar abacus manuscripts dating from about 1340 to the time of Paciolo .... Thus Gerardi's treatise was only the beginning of a long tradition in the study of higher order equations that did not bear fruit until the sixteenth century.39 Interpreted in the light of the present point of view, however, Gerardi's text presented favourable variations which endured through the action of natural selection.

By the middle of the fourteenth century, two more libri d'abaco introduced yet another important variation, the irreducible quartic equation, into the mathematical environment and so into the struggle for existence. The anonymous Trattato dell'alcibra amuchabile (c. 1340) and the Aliabraa argibra (midcentury, possibly 1344) of Master Dardi of Pisa belied the notion that the search for solutions to fourth degree equations began with the successful general solution of Ludovico Ferrari (1522-65) in the sixteenth century.40 Furthermore, in his 1463 Trattato di praticha d'arismetrica, Maestro Benedetto of Florence selected many of the findings of the maestri d'abaco for inclusion in his discussion of the work of Fibonacci and al-Khwarizmi. Of importance for the present development, however, Benedetto questioned the pretended general solutions of the cubic equations and thereby introduced the variation represented by this new research problem into the mathematical environment. He also mentioned the abbreviations in use for the various powers of the unknown in his treatise, namely. [GREEK LETTER RHO] = "cosa" = x, c = "census" = x², b = "cubo" = x³, cc = "censo di censo" = x4, br = "cubo relato cosa" = x5, and bb = "cubo di cubo cosa" = x6. Although he basically used only the symbol for "cosa" in his text, this underscored the shift that was taking place during the fifteenth century away from the purely rhetorical writing style of al-Khwarizmi, Leonardo, and the fourteenth century authors and toward an algebraic notation. Finally, influenced by two centuries of these practical tracts, Benedetto's work reflected the gradual abandonment of strict geometrical demonstrations and the progressive rise of more abstract algebraic justification.41

Over the more than two centuries between the appearance of Leonardo's Liber abbaci and the work of Maestro Benedetto, the natural selection of favourable variations within a heavily commercial environment had resulted in a well-marked variety of algebraic treatment which had diverged from its parent-species in the range of problems considered, in the type of justification presented, and in the language and form of presentation. As the need for problem-solving texts gradually diminished over the course of the fifteenth century, however, the practical tracts of the maestri d'abaco became less competitive and were supplanted by texts of a more theoretical nature. The first of these treatments, Fra Luca Pacioli's (c. 1445-c. 1517) Summa de arithmetica, geometria, proportioni e proportionalità (1494, second edition 1523) dealt with the ideas and findings of the maestri d'abaco while also drawing from the theoretical portion of works such as Euclid's Elements and Fibonacci's Liber abbaci. This theory with its emphasis on the geometrical proof of algebraic fact had lain dormant in the environment of practical problem-solving characteristic of the intervening three centuries. The reintroduction of such notions at the turn of the sixteenth century represented a new and favourable variation upon which natural selection acted.

Owing largely to the fact that Pacioli's Summa was the first work on algebra to appear in print as opposed to manuscript, it reached a relatively wide audience and established Pacioli, rightly or wrongly, as an important mathematical contributor.42 In essence, little of the mathematics presented in the Summa was due to Pacioli. His contribution lay rather in bringing virtually all realms of mathematical knowledge together in one work. Written in a curious blend of Italian, regional dialect, and Latin, the Summa was subdivided into parts on arithmetic, algebra, commercial mathematics, and geometry. With its arithmetic and algebraic parts drawn primarily from Fibonacci's Liber abbaci, its presentation of Archimedean geometry from his Practica geometriae, and its number-theoretic sections from his Liber quadratorum,43 the Summa effected a change in the mathematical environment which brought the advances, techniques, and geometrical standards of proof evident in these works back to the fore.44 As they remarked in their respective works, Cardano, Tartaglia, and Bombelli had read and absorbed the work presented in Pacioli's mathematical encyclopedia. They were each in a position to accept or reject ideas they found there in light of the continuing, but no longer dominant, practical line of algebra which they each appreciated.

While in content Pacioli's Summa contained little that had not already appeared, the presentation of these known facts differed significantly from the originals. As we have seen, the works of the thirteenth and fourteenth centuries were purely rhetorical in style with everything except the numerals written out in words. Benedetto's work of 1463 evidenced a slight movement away from this with the introduction of a symbol, [GREEK LETTER RHO], for the unknown. In Pacioli's Summa of 1494, however, algebraic computations took on an even more abbreviated form. Consider the following sequence from the Summa:45

In modern notation this becomes:

Thus, Pacioli's Summa reflected the fifteenth century trend toward greater abbreviation of the old rhetorical style which gave algebraic manipulations a more compact look and set them out in the text.46 Still, it is important to acknowledge that this did not represent a true notation. In the Summa, co. was merely a shortened form of "cosa", ce. abbreviated "census", R derived from "radix" or "root", and p[tilde] and m[tilde] came from the first letters of "piu [plus]" and "meno [minus]", respectively. Furthermore, these abbreviations occurred in a largely rhetorical setting. Abbreviation had not fully taken over, and the beginning of a true algebraic symbolism would not begin to appear for about a century.

Aside from the changes in mathematical exposition which Pacioli's Summa embodied, the work directly or indirectly spurred the search for general solutions of the cubic equations. As we have seen, incorrect solutions to various forms of cubics and quartics had already appeared in the libri d'abaco of Paolo Gerardi and Master Dardi of Pisa. Maestro Benedetto had recognized the errors which had been passed down by successive generations of maestri d'abaco and had noted that general solutions continued to elude mathematicians. In the Summa, Pacioli perpetuated the variation which Benedetto had introduced by asserting the impossibility of such general solutions within the context of the algebra of the day. Yet he left open the possibility that solutions might someday be found.47 Through the visibility of Pacioli and his work, Benedetto's observation dominated mathematical research in Italy for almost fifty years.

In the years from 1501 to 1502, Pacioli's name appeared on the roster of professors at the University of Bologna. There, he undoubtedly gave lectures which reflected the contents of his Summa. Although it is not known whether Scipione dal Ferro (1465-1526), a professor of mathematics at the University of Bologna, heard these lectures or whether Pacioli dealt therein with the problem of higher degree equations, it is known that dal Ferro succeeded in solving the cubic ax³ + bx = c sometime between 1500 and 1515, and possibly in 1504.48 In keeping with the customs of the time, dal Ferro kept his discovery a closely guarded secret, revealing it only to a very privileged few. Among the privileged were his son-in-law, the mathematician Annibale della Nave (c. 1500-58) and his student, Antonio Maria Fiore. The solution was not published; it was by no means disseminated; it was private and precious property.

At the turn of the sixteenth century in Italy, the teacher of mathematics lived in a highly competitive world. At this time, students paid their professors directly for each course they took. Thus, if they became dissatisfied with the level or quality of instruction, payment could be summarily suspended, and the instructor could be forced to leave the school and even the town. To uphold their reputations and to insure their livelihoods, professors engaged in public contests with the winner gaining prestige and, presumably, greater numbers of students. These contests were generally initiated by an underdog who proposed a series of problems to an established figure. The better-known mathematician then prepared a comparable set of examples for the challenger. After a predetermined length of time, the participants came together in public to present their solutions, the one with the greater number of correct answers taking the contest.49 In such an atmosphere, the guardian of a new solution or technique gained a distinct advantage over potential opponents and enjoyed job security by virtue of his secret. Given the system, it was simply not in one's best interests to publicize major discoveries. Hence, although dal Ferro introduced an important variation into the body of mathematical knowledge, namely, his solution to the cubic ax³ + bx = c, it did not immediately come into competition with other algebraic ideas. In fact, it lay dormant, outside the scope of the natural selection of ideas, until Cardano published it in his Ars magna in 1545.

Dal Ferro's death in 1526 released his confidants from their pledge of secrecy, and in 1530 Fiore challenged Zuannin de Tonini da Coi, a mathematician from Brescia, to a contest which involved the irreducible cubic. Unable to resolve the challenge problems, Tonini da Coi in turn put them to a local rival, Niccolò Tartaglia. In 1530, Tartaglia responded that such problems were impossible. In 1535, when Fiore challenged him directly with thirty examples requiring the same secret formula, Tartaglia independently discovered the solution and won the contest.50

Cardano heard of Tartaglia's feat and petitioned him to share his findings so that it could be included, with all due credit, in the book Cardano was busy preparing. Wishing to see his discovery first published in one of his own forthcoming works, however, Tartaglia declined to divulge the secret. At Cardano's subsequent entreaties, though, he capitulated sometime in 1539. By publishing the result in 1545, Cardano sparked one of the most spectacular priority controversies in the history of mathematics, but that need not concern us here.51 Of importance to our study is the fact that Cardano's publication of solutions of the cubic equations brought these variations into competition within the mathematical environment. In particular, the struggle for existence between these new facts and the algebraic theory as previously held took place within the context of Cardano's own statement on algebra, his text entitled Ars magna.

Earlier writers on the history of mathematics often saw in Cardano a beacon of the renaissance of mathematics and a modern who completely cast off the mathematical bonds of the past. For instance, Morris Kline assessed Cardano in these words in his Mathematics in western culture: "In his lewdness and rejection of authoritarian doctrines, as well as in his searching mathematical, physical, and medical studies, Cardano symbolized the revolt from a thousand years of intellectual serfdom...."52 In his biography of Cardano, Oystein Ore expressed a similar point of view: "A revolution took place in mathematics during the first half of the sixteenth century. The classical works of the Greek mathematicians had been for nearly two thousand years the unsurpassable pinnacles of mathematical attainment. And then, within a few years the shackles were broken and new fields with golden opportunities lay open. The theories of higher equations and algebra were created and some of the more visionary mathematicians, especially Cardano, began to see the general principles which were to occupy mathematicians in the centuries to come."53 As we have seen, neither the theory of higher degree equations in particular nor algebra in general sprang fully matured from the minds of the sixteenth century. Cardano's systematic exposition of the theory of algebra marked not a revolution in mathematics but a step in the continuing process of the natural selection of ideas. He did not revolt against the ideas of his predecessors. He selected from among them and, in combination with his own ideas, produced a new variety of algebra.


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4. THE ARS MAGNA OF GIROLAMO CARDANO

Cardano opened his Ars magna with a brief historical passage in which he first traced his intellectual lineage from al-Khwarizmi through Fibonacci to Pacioli and then outlined the sixteenth century developments on the solution of the cubic equation. In this passage, he not only indicated the mathematical environment in which his thoughts had developed but also, like so many of his contemporaries, expressed a firm conviction in the unlimited capabilities of the art of algebra thereby justifying his researches. As he put it: "Since this art surpasses all human subtlety and the perspicuity of mortal talent and is a truly celestial gift and a very clear test of the capacity of men's minds, whoever applies himself to it will believe that there is nothing he cannot understand."54 As a true Renaissance natural philosopher, Cardano sought the keys to a deeper understanding of nature. He believed that his Ars magna perfectly revealed at least one such key, the key to the art of algebra.

Cardano began this work by specifying the limits which his environment forced on his mathematics. He explained that only those problems which described some aspect of three-dimensional space were real and true. In his words: "For as positio [the first power of the unknown] refers to a line, quadratum [the square of the unknown] to a surface, and cubum [the unknown cubed] to a solid body it would be very foolish for us to go beyond this point. Nature does not permit it."55 Thus, Cardano preserved the standards of his acknowledged ancestors. He held that only equations of or reducible to degrees one, two, or three made sense because only equations of those degrees described nature. Furthermore, since he also selected the standard of geometrical proof of algebraic fact evident in the work of the Arabic mathematicians, Fibonacci, and Pacioli, geometry restricted him to a consideration of third degree equations at most. From this viewpoint, consider then his demonstration of the infamous cube and first power equal to the number:

Figure 4

For example, let GH³ plus six times its side GH equal 20, and let AE and CL be two cubes the difference between which is 20 and such that the product of AC, the side [of one], and CK the side [of the other], is 2, namely one-third the coefficient of x. Marking off BC equal to CK, I say that, if this is done, the remaining line AB is equal to GH and is, therefore, the value of x, for GH has already been given as [equal to x].

In accordance with the first proposition of the sixth chapter of this book [on the formula, in modern notation, (a + b)³ = a³ + 3a²b + 3ab² + b³], I complete the bodies DA, DC, DE, and DF; and as DC represents BC³, so DF represents AB³, DA represents 3(BC x AB²) and DE represents 3(AB x BC²).56

Here, Cardano asked his readers to complete the cubes formed on AB + BC, in just the same way the Arabs or Leonardo completed the square.

Figure 5

This resulted in the body shown in Figure 5, where, for instance, DE is the shaded region. In Cardano's set-up, AC³ corresponded to the cube AE and CK³ represented the cube CL and was equivalent to BC³, so

AC³ - BC³ = AC³ - CK³ = 20,
by hypothesis, and
AC³ - BC³ = AC³ - CK³ = DA + DE + DF
(1)
= 3(BC x AB²) + 3(AB x BC²) + AB³,
by construction. Furthermore, "since ... AC x CK equals 2, AC x 3CK will equal 6, the coefficient of x; therefore AB x 3(AC x CK) makes 6x or 6AB, wherefore three times the product of AB, BC, and AC is 6AB."57

This gave Cardano an essentially algebraic (as opposed to geometric) equivalent of 6AB, namely,

(2)
6AB = 3(AB x BC x AC).
Invoking the second proposition of the sixth chapter, that is,
AC³ + 3(AC x CB²) = (AC - CB)³ + CB³ + 3(CB x AC²),58
Cardano used the equivalent formulation
AB³ = (AC - CB

(3)
= AC³ + 3(AC x CB²) + (-CB³) + 3(-CB x AC²).
Substituting (3) into (1) and verifying that
3(BC x AB²) + 3(AB x BC²)
equalled (2), Cardano found that
(4)
AC³ - BC³ = AC³ + 3(AC x CB²) + 3(-CB x AC²) + (-BC³) + 6AB = 20
Adding 6AB to both sides of (3) and using (4), he concluded that AB³ + 6AB = 20, and so the GH desired was the AB constructed. Cardano completed the chapter with a rhetorical description of the formula for the cube and first power equal to the number followed by three specific examples.59

In the natural selection of ideas, the standard of rigorous, geometrical demonstration applied to algebraic fact which Cardano adopted obviously represented a favourable variation in the theory of algebra. As we have seen, it had endured from the time of Euclid, through the medieval period, and into the sixteenth century but always in applications to first and second degree equations. In the Ars magna, Cardano extended its realm of applicability to cubic equations and thereby introduced a new variation. For quadratic equations, Cardano, like his ancestors, built squares; but for third degree equations, he constructed cubes.

In spite of his selection of the geometrical standard of proof for cubics, however, Cardano rejected it for higher degree equations and generated another variation. He followed the statement in which he acknowledged the geometrical foolishness of trying to go beyond the cubic with this: "Thus, it will be seen, all those matters up to and including the cubic are fully demonstrated, but the others which we will add, either by necessity or out of curiosity, we do not go beyond barely setting out."60 These "others" were the solutions of irreducible equations of the fourth degree discovered by his student, Ludovico Ferrari. Hence, for equations of or reducible to degree three, Cardano adhered to one standard of proof, the geometrical construction and he extended it to the completion of the cube for degree three. For irreducible equations of degree four, however, geometry failed him. He was unable to conceive of, much less complete, a four-dimensional figure. Thus, in order to deal with the quartic equations which sparked his interest, he had to reject geometry and rely instead on algebraic algorithm alone.

In fact, Cardano did not give a general proof of the solution to the biquadratic. In the thirty-ninth chapter of the Ars magna, he explained that: "there is another rule, more noble than the preceeding. It is Ludovico Ferrari's, who gave it to me on my request. Through it we have all the solutions for equations of the fourth power, square, first power and number, or of the fourth power cube, square, and number...."61 Following a partial list of the biquadratic types, however, Cardano gave only a demonstration for completing the square on x4 + bx² = c, where he interpreted x4 as the square of side x². This did not constitute a proof of the solution rule for a fourth degree equation, it merely justified one step in the algebraic process toward such a solution. He followed this demonstration with a rhetorical summary of the steps involved in effecting the rule and proceeded to work out eight different types of quartic equations.62 Cardano's relaxation of the justification criteria in the case of fourth degree equations represented an important variation. In the presence of such higher degree equations, the old standard, which had competed successfully in the struggle for existence since the time of Euclid, no longer proved useful and so was rejected. Of necessity, equations of degree four or greater could not he held accountable to geometry. They required some new technique, some new variation. By treating them algebraically instead of geometrically, Cardano generated just such a variation.

Geometric considerations also created a problem with respect to the meaning of negative numbers. What did it mean for a line to have negative length, a square to have negative area, or a cube to have negative volume? What did it mean to subtract a larger quantity from a smaller one? Euclid, the Arabs, Fibonacci, the maestri d'abaco, Pacioli, and Cardano all handled this matter in the same way.

As we have seen, they never allowed negative coefficients and were thereby forced to consider many cases of equations of each degree. In his discussion of quadratic equations, for example, Cardano dealt with

1.quad.aeq.10.pos.p.144 or x² = 10x + 144,
144.aeq.10.pos.p.1.quad. or 144 = 10x + x², and
l.quad.p.16.aeq.10.pos. or x² + 16 = 10x.63
Since each of these equations represented a different way of partitioning up a square into rectangles of smaller areas, the sides of the smaller rectangles could not have negative length. Furthermore. the geometrical interpretation of the solutions of such equations as the side of a square precluded negative solutions. Again, though, Cardano produced an important variation by rejecting the strict, geometrical interpretation of negative numbers and by allowing them an independent algebraic existence. As he admitted, negative numbers did, after all, satisfy certain equations. For instance, "if the square of a square is equal to a number and a square, there is always one true solution and another and fictitious solution equal to it. Thus, in x4 = 2x² + 8, x equals 2 or -2."64 For Cardano, the "true solution" made geometrical sense whereas the "fictitious" one did not. So although he generally accepted the sovereignty of the geometrical standard, here Cardano accepted something different, something algebraic with no obvious geometric meaning.65

Negative numbers created an even greater conundrum when they occurred under a square root. In his Chapter 5, Cardano dealt with solutions of quadratic equations and rhetorically stated the same rules as we have seen in texts from the Al-jabr wa'l-muqabala on. He explained that "if the first power is equal to the square and the number, multiply as before one-half the coefficient of the first power by itself and, having subtracted the number from the product, subtract the root of the remainder from one-half the coefficient of the first power or add the two of them, and the value of x will be both the sum and the difference."66 In modern notation, if ax = x² + b, then

a solution involving a square root. After applying this rule to two examples, Cardano warned his readers that "if the number [i.e., b in (5)] cannot be subtracted from the square of one-half the coefficient of the first power [i.e. (1/2a)² in (5)], the problem itself is a false one and that which has been proposed cannot be. It must always be observed throughout this treatise that, when those things which have been directed cannot be carried out, that which is proposed is not and cannot be."67 The geometrical criteria necessarily rendered problems which resulted in a negative under the radical invalid. Such problems simply lay outside the bounds of the art.

In Chapter 37, however, Cardano introduced another key variation when he rejected geometry once more in order to deal with another purely algebraic construct. He wrote: "If it should be said, Divide 10 into two parts the product of which is 30 or 40, it is clear that this case is impossible. Nevertheless, we will work thus: We divide 10 into two equal parts, making each 5. These we square, making 25. Subtract 40, if you will, from the 25 thus produced, ... leaving a remainder of -15, the square root of which added to or subtracted from 5 gives parts the product of which is 40. These will be 5 + [SQUARE ROOT SYMBOL] -15 and 5 - [SQUARE ROOT SYMBOL] -15." 68 Since this problem translated into a quadratic of type x² + b = ax, Cardano tried to justify its algorithm geometrically. (See Equation (5) above.) When he computed (1/2a)² - b, -15 in this case, he explained to his readers that "since such a remainder is negative, you will have to imagine [SQUARE ROOT SYMBOL] -15", 69 and he concluded his discussion admitting that "this truly is sophisticated, since with it one cannot carry out the operations one can in the case of a pure negative and other [numbers]".70 Thus, the rejection of the geometrical constraints produced a new algebraic entity which behaved very differently from anything previously known, an entity which had no physical interpretation. and "so progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless"71

In Cardano's Ars magna, we witness a definite struggle for existence between the old, venerated tradition of Euclid and the Arabs and the new, untried ideas of the sixteenth century. At the same time he tried to maintain the Euclidean standard, Cardano acknowledged the solution of the fourth degree equations, a discovery which he could not completely justify within his chosen framework. While desirous of purely geometrical interpretations of algebraic facts, Cardano admitted that negative numbers satisfied certain equations in spite of the fact that negative lengths, areas, and volumes made no sense. Even in light of the thorny geometrical problem presented by unadorned negative numbers, Cardano conceded that their square roots yielded to algebraic manipulation and provided solutions to equations. Far from casting off his mathematical heritage as Kline, Ore, and others have suggested, Cardano worked to uphold it in the face of new mathematical facts and constructs. In Cardano's work, we see evidence not of the complete rejection of the past but rather of the struggle for existence of the old and venerated geometrical standard for algebraic proof within an environment changed by fourth degree equations and imaginary numbers. In an environment so altered, geometrical justification was obviously no longer the "fittest" way of proving all algebraic facts. In Raphael Bombelli's Algebra of 1512 and François Viète's In artem analyticem isagoge of 1591, we see evidence of yet another struggle for existence. By the last quarter of the century, Cardano's variety in the geometrical line came into stiff competition with the ideas found in a newly rediscovered Greek text, the Arithmetica of Diophantus.


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5. THE REDISCOVERY OF DIOPHANTUS'S ARITHMETICA AND THE WORK OF RAPHAEL BOMBELLI AND FRANÇOIS VIÈTE

Around 1463 the noted humanist, Johannes Müller (1436-76) or Regiomontanus as he was most often called, reported that he had found a Greek text of "...the fine thirteen Books of Diophantus, in which the very flower of the whole of Arithmetic lies hid, the ars rei et census, which today they call by the Arabic name of Algebra".72 Actually, the manuscript Regiomontanus discovered in Venice contained only six books of the Arithmetica73 and dealt not so much with the determinate type of algebra of the day but with indeterminate algebra largely unknown to the fifteenth century.74 Had Regiomontanus succeeded in fulfilling his desire to translate Diophantus's text into Latin, the Arithmetica might have entered into the Western mathematical environment a century earlier than it finally did.

As we have seen, the text consistently failed to compete successfully within the mathematical environment. Such was the case when Diophantus first put forth his ideas and methods around A.D. 250 and then again when al-Karaji reintroduced them in his al-Fakhri almost eight hundred years later.75 Given the dominance of the geometrical approach to algebra from at least the time of Euclid through that of al-Khwarizmi, Abu-Kamil, and al-Karaji himself, this should come as no surprise. In his text, rather than giving geometrical justifications for solution algorithms of problems with a practical bent, Diophantus presented a series of problems, mostly of an indeterminate nature and provided algebraic techniques for solving them.

Consider, for example, the following problem from Book II: "To find three numbers such that the square of any one of them minus the next following gives a square."76 In modern notation, Diophantus took x + 1, 2x + 1, and 4x + 1 as his three numbers and noted that they satisfied two of the conditions. In other words, by careful choices of his indeterminate numbers he clearly had

(x + 1)² - (2x + 1) = x², a square, and
(2x + 1)² - (4x + 1) = 4x², a square.
However, (4x + 1)² - (x + 1) = 16x² + 7x = 25x², a square, to get x = 7/9. Substituting this value in the expressions for the three numbers, he found the solution 16/9, 23/9, 37/9.77 The arbitrariness of this problem left Diophantus open to postulate any convenient indeterminate form for the desired numbers and allowed him to give one particular answer rather than any sort of general solution. Furthermore, his approach to the problem was algebraic; he provided no geometrical demonstration of the validity of the result.

Diophantus also developed an abbreviated way of writing algebraic expressions not unlike that which had evolved by the sixteenth century from the rhetorical texts of the Arabs and Fibonacci. For Diophantus, the equation 630x² + 73x = 6 appeared as

Here [GREEK=DELTAY] abbreviated [GREEK=DYNAMIS] meaning "power" and was equivalent to the modern x²; [GREEK=SIGMA] stood for [GREEK=ARITHMOS] or "number" and corresponded to x; [GREEK=IS] abbreviated [GREEK=ISOS] or "equals"; [GREEK=MU with a small circle over it] denoted [GREEK=MONADES] or "units"; the specific numbers were barred versions of the usual Greek alphabetic number system; and addition was indicated by simple juxtaposition.78 Thus, Diophantus's treatment of the determinate problem I.1 looked like this79:

Diophantus's use of abbreviations such as [GREEK=SIGMA], [GREEK=DELTAY], and [GREEK=KAPPAY] for algebraic unknowns within his rhetorically expressed solution algorithm appeared, as we shall see in a moment, remarkably congenial to sixteenth century mathematicians accustomed to their corresponding abbreviations co., ce., and cu.

By virtue of its rejection of the dominant, Euclidean, geometrical algebra, the variety of algebra which Diophantus developed in his Arithmetica simply did not compete within the mathematical environment. Its variation, namely a purely algorithmic presentation applied in particular to problems of an indeterminate nature, endowed it with the capability of solving a type of problem outside the realm of geometrical algebra. Within that realm, then, the variation proved neither useful nor injurious and was unaffected in the natural selection of ideas. Whenever indeterminate problems did arouse interest, as they apparently did with Hypatia in A.D. 400 and with al-Karaji at the start of the eleventh century, however, Diophantus's ideas did enter into competition. Thus, as a result of the dominance of geometrical algebra with its emphasis on determinate problems, the theory Diophantus presented in the Arithmetica lay in a virtual state of dormancy from its advent in A.D. 250 to its rediscovery in the 1560s.

By that time, significant changes in the mathematical environment had taken place. The solution of the fourth degree equation had pushed determinate algebra to a point where geometry had proven insufficient to its needs. The realization that negative and imaginary numbers algebraically satisfied certain equations further undermined the authority of geometry in algebra. In such a changed environment, the variations introduced by Diophantus might finally prove favourable and allow for the successful completion of his theory. That they did may be seen in the Algebra of Raphael Bombelli.

An engineer-architect by profession, Bombelli wrote the first draft of his treatise L'Algebra parte maggiore dell'Arithmetica divisa in tre libri most probably between 1557 and 1560 when his work on the reclamation of the marshes of the Val di Chiana in Tuscany had temporarily come to a halt.80 He consciously undertook this task in order to provide the mathematical community with a clear and in-depth treatment of this most vital of areas. Although he felt that Cardano's Ars magna represented such a study, Bombelli and others found Cardano's work difficult to read. As he explained in his introduction, "no one has explored the secrets of algebra except Cardano of Milan, who in his Ars magna dealt with the subject at length, but was not clear in his exposition".81 Bombelli wished to combine the comprehensiveness of the Ars magna with an unquestionable clarity of presentation.

As he later admitted, the first draft of the Algebra clearly reflected the environment in which he worked, an environment dominated by the work of al-Khwarizmi, Fibonacci, and Pacioli.82 In its original form, the text consisted of five books each of which contained material of a particular nature. Logically, the first book contained definitions of all of the concepts Bombelli intended to use in the remainder of the work. There, he explained notions like powers and roots and gave the first systematic exposition of negative numbers under the radical. He also discussed the various techniques by which all of these algebraic entities could be manipulated.83 This done, he turned in Book II to his prime concern, an elucidation of the solutions of equations up to and including the biquadratics. He modelled this part of the treatise on the Ars magna and so approached the task case-by-case. Following the lead of Fibonacci, the maestri d'abaco, and Pacioli, he then devoted the third book to multifarious examples and practical problems.84 In the fourth and fifth books which remained incomplete, he planned to present further applications of geometry to algebra as well as applications of algebra to geometry. This draft of the Algebra with the planned fourth and fifth books never made its way into print.85

Sometime during the second half of the 1560s, Antonio Maria Pazzi a reader in mathematics at the University of Rome, discovered a manuscript copy of Diophantus's Arithmetica in the Vatican Library and showed it to Bombelli.86 Convinced of its merits, the two men set about to translate the work and managed to complete five of the books before other duties called them away. The discovery and translation of this text represented a significant change in the mathematical environment. At a time when geometry's competitive edge in handling algebraic questions had just been undermined by the discoveries of the solution of the quartic and of imaginary and negative numbers as solutions to equations, Diophantus's non-geometric approach to algebra entered successfully into competition.

In fact, by 1572 when his Algebra finally appeared in print, Bombelli had accepted the Diophantine point of view and had altered his own work accordingly. Thus, while the Algebra's first version was phrased in terms of the Arabic-inspired "cosa" and "census" for the unknown and its square, its 1572 rewriting employed the translations "tanto" and "potenza" of Diophantus's "number [GREEK=ARITHMOS]" and "power [GREEK=DYNAMIS]".87 Furthermore, Bombelli consciously excised most of the practical problems taken from the maestri d'abaco and replaced them by 143 indeterminate problems taken from Diophantus. In his introduction to Book III, he announced that he had broken with the usual custom of casting problems "... in the guise of human actions (buying, selling, barter, exchange, interest, defalcation, coinage, alloys, weights, partnership, profit and loss, games and other numerous transactions and operations relating to daily living)."88 He wished to teach "the higher arithmetic (or algebra) in the manner of the ancients".89

Responding to the broader intellectual environment which was dominated by the humanistic revival of Greek texts, the mathematician Bombelli sought to purify an algebra tainted by the practical, untheoretically motivated problems of the maestri d'abaco. The new variety that he produced presented the characteristics of both of its parent-varieties: for problems of a determinate nature, the Algebra employed Cardano's geometrical algebra, but for the new, indeterminate problems, it used Diophantus's new, ungeometrical, indeterminate analysis. The variation which Bombelli's algebra presented was the focusing on a type of algebraic problem which was not solved using geometrical devices and, by implication, the acknowledgement that all algebraic problems did not require geometrically justified solutions. Given geometry's inability to deal successfully with the solution of the quartics and with negative and imaginary solutions in general, Bombelli's work presented a favourable variation which the action of the natural selection of ideas should have preserved. The work of François Viète evidenced the preservation of just this sort of variation.

Viète prepared his In artem analyticem isagoge or Introduction to the analytic art of 1591 in a humanistic environment in which the dominance of geometrical algebra was being challenged by newly rediscovered ideas. Thus, although firmly grounded in algebra as presented by Cardano in the Ars magna Viète also drew from such works as Diophantus`s Arithmetica, Federico Commandino's (1509-75) Latin translation of Pappus of Alexandria's (fl. A.D. 320) Mathematical collection, and the humanist texts of Petrus Ramus (1505- 72).90 In Ramus's writings, for example, Viète read of the algebraic content of Book II of Euclid's Elements, of Diophantus's kind of indeterminate analysis, and of the equation of algebra and analysis as opposed to its equation to geometrical synthesis. Furthermore, as indicated by the quote from the In artem analyticem isagoge which we cited earlier,91 Viète's humanistic leanings predisposed him to reject the geometric variety of algebra with its Arabic line of descent. An algebra more of recipes, albeit geometrically justified ones, than of general problem-solving techniques, from Viète's stance, this variety failed to uncover the "incomparable gold [that] lay hidden".92 By selecting certain characteristics of the works of Pappus and Diophantus and by combining these with his own ideas, Viète developed just such a general method of problem-solving. At its heart lay his new notion of a "species".

As we have already mentioned, Pappus's Mathematical collection (in the Commandino translation which appeared in Pesaro in 1588) formed a part of Viète's mathematical environment. In its seventh book, Viète found a complete exposition of Greek analysis and synthesis as applied primarily to geometry. According to Pappus: "Analysis, then, is the way from what is sought, taken as admitted by means of a previous synthesis ... but in synthesis going in reverse, we supposed as admitted what was the last result of the analysis, and, arranging in their natural order as consequences what were formerly the antecedents, and connecting them with one another, we arrive at the completion of the construction of what was sought, and this we call synthesis."93 He went on to break analysis down into two basic kinds where "the one is searching for the truth [i.e., zetetic from ... 'to search'], which is called theoretical and the other is for supplying what is required [i.e., poristic from ... 'to supply'], which is called problematical".94 Thus, since Pappus understood analysis and synthesis as converse procedures, each of the two forms of analysis had its corresponding synthesis. Direct proof was the synthesis of the zetetic art; proof generated by geometric construction was that of the poristic art. Diophantus tacitly recognized these same correspondences, but whereas Pappus used them only in solving geometrical problems, he applied them only to algebraic ones.95 By means of his new concept of a species, Viète formally extended Pappus's notions to encompass both geometric and algebraic questions and thereby created a unified and universal theory of problem-solving. He introduced his new variation on these old ideas in the opening chapter of The analytic art.

After defining analysis and synthesis in virtually the same way as had Pappus, Viète continued by noting that

although the ancients set forth a twofold analysis, the zetetic and the poristic, to which Theon's definition primarily refers, it is nevertheless fitting that there be established also a third kind, which may be called rhetic or exegetic, so that there is a zetetic art by which is found the equation or proportion between the magnitude that is being sought and those that are given, a poristic art by which from the equation or proportion the truth of the theorem set up is investigated, and an exegetic art by which from the equation set up or the proportion there is produced the magnitude itself which is being sought. And thus, the whole threefold analytical art, claiming for itself this office, may be defined as the science of right finding in mathematics.96
In other words, Viète recognized the need for a set of clearly expounded rules for actually solving equations and proportions. Reacting against the kind of unmotivated solution of one example after another which characterized Diophantus's Arithmetica, Viète underscored the importance of a lucid explanation of general procedures to motivate the solutions given.

Viète's addition of the rhetic art to Pappus's twofold analysis represented an important variation in that "... it encompasse[d] nothing less than the theory of equations addressed to their solutions".97 Correlated to this variation, however, was another of equal or greater import. A little further on in The analytic art he wrote: "In the zetetic art, however, the form of proceeding is peculiar to the art itself; inasmuch as the zetetic art does not employ its logic on numbers--which was the tediousness of the ancient analysts--but uses its logic through a logistic which in a new way had to do with species. This logistic is much more successful and powerful than the numerical one ...."98 His new logistic, the logistice speciosa, in contradistinction to Diophantus's reckoning with indeterminate numbers, or logistice numerosa, operated "... with species or forms of things, as, for example, with the letters of the alphabet"99

For Viète, a species was a placeholder for an undetermined unknown or a given magnitude. He called for "... the given magnitudes [to] be distinguished from the undetermined unknowns by a constant and very clear symbol, as, for instance, by designating the unknown magnitude by means of a letter A or some other vowel E, I, O, U, or Y, and the given magnitudes by means of letters B, G, and D or other consonants".100 Then, unlike Diophantus but in keeping with the views of his predecessors in the line of descent of the geometrical algebra, Viète attached dimension to the species in any given equation and insisted that only expressions of equal dimension were commensurate. He stated this formally as follows:

The supreme and everlasting law of equations or proportions, which is called the law of homogeneity because it is conceived with respect to homogeneous magnitudes is this:
1. Only homogeneous magnitudes are to be compared [comparari] with one another.
For ... it is impossible to know how heterogeneous magnitudes may be conjoined.
And so, if a magnitude is added to a magnitude, it is homogeneous with it.
If a magnitude is multiplied by a magnitude, the product is heterogeneous in relation to both.
If a magnitude Is divided by a magnitude, it is heterogeneous in relation to it.
Not to have considered these things was the cause of the darkness and blindness of the ancient analysts.101
Thus, for Viète, the notions of dimensionality and its homogeneity bound his new notational system together. Without them, a symbolism such as his lacked an internally dictated and coherent set of rules for operation and so lacked meaning. To Viète's way of thinking, then, Diophantus's system suffered precisely from the absence of this philosophical glue.

In Viète's new system, (A cube) + (B solid) denoted the addition of a three-dimensional unknown and a three-dimensional magnitude, (A square) + (B plane) stood for the addition of a two-dimensional unknown and a two-dimensional magnitude, and A + B (with the dimensionality terms suppressed) represented the analogous situation in one-dimension. Subtraction, multiplication, and division behaved similarly.102 With this system in his employ, Viète could write expressions such as

... let it be required to add Z to A plane / B. The sum will be103

an expression which translates in modern terms as

A glance at these two versions of the same mathematical statement reveals that although Viète introduced and operated on symbols for unknown as well as known quantities, his work exhibited an obvious vestige of his geometrical algebraic ancestry, namely, dimensionality. In the natural selection of ideas, the variations represented by Pappus's twofold analysis, Diophantus's indeterminate analysis, and the geometrical algebraic concept of dimensionality all survived in Viète's work and proved favourable in his conception of a general method of problem-solving. In a mathematical environment changing through the reintroduction of Greek texts and their contents, characteristics like the geometrical justification of algebraic fact and the use of specific numerical constants in an ostensibly general context were no longer advantageous and were rejected. As the subsequent development of algebra would attest, the analytic art which resulted from the fusion of ideas adopted, ideas rejected and Ideas newly generated, provided favourable as well as ultimately injurious variations on which the natural selection of ideas acted.104

Viète explicitly demonstrated the immediate efficacy of his new art in the Zeteticorum libri quinque, a work published in 1593 but probably written in 1591.105 Since he aimed to contrast his logistice speciosa directly with Diophantus's logistice numerosa in this text, Viète juxtaposed dozens of Diophantus's solutions with his own in order to demonstrate the superiority of his methods. Consider his version of Diophantus's I.1 which we examined above:

Given the difference of two "sides" and their sum, to find the "sides".
Let the differences B of the two "sides" be given, and also let their sum D be given.
It is required to find the "sides".
Let the less "side" be A; then the greater will be A + B. Therefore, the sum of the "sides" will be A2 + B. But the same sum is given as D. Wherefore, A2 + B is equal to D. And, by antithesis, A2 will be equal to D - B, and if they are all halved, A will be equal to D1/2 + B1/2.
Or, let the greater "side" be E. Then the less will be E - B. Therefore, the sum of the "sides" will be E2 - B. But the same sum is given as D. Therefore, E2 - B will be equal to D, and by antithesis, E2 will be equal to D + B, and if they are all halved, E will he equal to D1/2 + B1/2.
Therefore, with the difference of two "sides" given and their sum, the "sides" are found.
For, indeed, half the sum of the "sides" minus half their difference is equal to the less "side", and half their sum plus half their difference is equal to the greater.
Which very thing the zetesis shows.
Let B be 40 and D 100. Then A becomes 30 and E becomes 70.106
By recasting Diophantus's problem into more general terms, Viète highlighted the algebraic forms involved in the problem and the algebraic manipulations necessary to effect a general solution in terms of B and D. Then he simply substituted in the given values for B and D, namely, 40 and 100, to generate the particular solution.

In his work, On the origin of species, Darwin explained that since "... there will be a constant tendency in the improved descendants of any one species to supplant and exterminate in each stage of their descent their predecessors and their original parent, ... all the intermediate forms between the earlier and the later states that is between the less and the more improved state of a species, as well as the original parent-species itself will generally tend to become extinct"107 Viète's analytic art, with its more general notation and emphasis on general problem-solving, represents such a highly improved and competitive descendant, a descendant which totally supplanted both its Diophantine and its geometrical algebraic predecessors through the continued and continuing action of the natural selection of ideas.


[TOP] [PREVIOUS] [CONTENTS] [REFERENCES]

ACKNOWLEDGEMENTS

This research was partially supported by National Science Foundation Scholars Award # SES-8509795 held while I was in the Department of Mathematical Sciences, Sweet Briar College, Sweet Briar, VA 24595. I would like to thank the referees for their helpful suggestions, but I am especially grateful to Professor Allen G. Debus for his role in seeing this study to fruition.

If you have questions or comments regarding this document, you may address them to Karen H. Parshall, Professor of Mathematics and History, University of Virginia, (E-Mail KHP3K@VIRGINIA.EDU). Please visit Ms. Parshall's web page for more information about the author and her work.


REFERENCES

1. Girolamo Cardano, The great art or the rules of algebra, trans. by T. Richard Witmer (Cambridge, 1968), 7.[RETURN]
2. François Viète, Introduction to the analytic art, trans. by Rev. J. Winfree Smith, in Jacob Klein, Greek mathematical thought and the origin of algebra (Cambridge, 1968), 318-19. Smith's translation of Viète's work forms an appendix to Klein's book and may be found on pp. 315-53. In what follows, I shall cite "Viète, Introduction" when drawing quotations directly from this translation of Viète's work. More recently, nine of Viète's works, including the Introduction to the analytic art, have been translated by T. Richard Witmer in François Viète, The analytic art: Nine studies in algebra, geometry and trigonometry from the Opus restitutae mathematicae analyseos, seu algebra nova, trans. by T. Richard Witmer (Kent, Ohio, 1983). For a reprint of the original 1646 edition of Viète's collected works, see François Viète, Opera mathematica, recognita francisci à Schooten, ed. by Joseph E. Hofmann (Leiden, 1646; reprint edn, Hildesheim--New York, 1970).[RETURN]
3. Gerald Toomer, "Al-Khwarizmi, Abu Ja'far Muhammad ibn Musa", Dictionary of scientific biography, ed. by C. C. Gillispie (New York,1970-80; hereafter DSB), vii, 358-65, p. 358; J. Al-Dabbagh, "Banu Musa, three brothers- -Muhammad, Ahmad, and al-Hasan", DSB, i, 443-6, p. 444; and David C. Lindberg, "Transmission of Greek and Arabic learning to the West", in David C. Lindberg (ed.), Science in the Middle Ages (Chicago, 1978), 52-90, pp. 55-58.[RETURN]
4. Here "al-jabr" translates as "completion" and signifies the elimination of negative quantities from an equation. For example, "al-jabr" transforms x = 10-3x into 4x = 10. The term "al-muqabala" translates as "balancing" and refers to reducing positive quantities of the same power on both sides of an equation. For example, "al-muqabala" turns 10x + 64 = 5x + 36 + x² into 5x + 28 = x². See Toomer, op. cit. (ref. 3), 359.[RETURN]
5. Sir Thomas L. Heath, The thirteen books of Euclid's Elements, i (Cambridge, 1926), 73-84.[RETURN]
6. See Solomon Gandz, "The sources of al-Khwarizmi's algebra", Osiris, i (1936), 263-77.[RETURN]
7. See Otto Neugebauer, The exact sciences in Antiquity (New York, 1969), 147.[RETURN]
8. Jacques Sesiano, Books IV to VII of Diophantus' Arithmetica in the Arabic translation attributed to Qusta ibn Luqa (New York, 1982), 8. In this work, Sesiano presents an English translation of and commentary on four of the thirteen books of the Arithmetica which had been considered lost for well over six hundred years. For a new translation (in French) of the ten known books of the Arithmetica, see Diophantus, Les Arithmétiques, trans. by Roshdi Rashed (Paris, 1984). In his translation, Rashed claims priority for the discovery of the four newly-discovered books of the Arithmetica and strongly criticizes Sesiano's version. See, for example, Les Arithmétiques, iii, ref. 63, pp. lix-lxii. On Qusta ibn Luqa's dates, see ibid., iii, pp. xvi-xxii.[RETURN]
9. Al-Khwarizmi. "Six types of rhetorical algebraic equations", in Edward Grant (ed.), A source book in medieval science (Cambridge, 1974), 106-11, p. 108.[RETURN]
10. Notice that al-Khwarizmi solves not for the unknown x, as we would today, but for its square. His aim is to complete or solve the square, literally speaking.[RETURN]
11. Al-Khwarizmi, op. cit., 110.[RETURN]
12. Ibid., 111.[RETURN]
13. Heath, Elements (ref. 5), i, 385.[RETURN]
14. Ibid., 385-6.[RETURN]
15. Scholars disagree on the role of Euclid in al-Khwarizmi's mathematics. In his article on al-Khwarizmi's sources, Gandz argues against a Euclidean influence, but in his DSB article, Toomer argues for such an influence. See Gandz, op. cit. (ref. 6), 264-7, and Toomer, op. cit. (ref. 3), 360. For yet another view, see "L'Idée de l'algèbra selon al-Khwarizmi", in Roshdi Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984), 17- 29. (This book by Rashed brings together many of his influential articles on Arabic mathematics originally published in Archive for history of exact sciences.)[RETURN]
16. Heath, Elements (ref. 5), i, 377. On differing views concerning the significance and meaning of "geometrical algebra", see B. L. van der Waerden, Geometry and algebra in ancient civilizations (New York, 1983), 75-96; B. L. van der Waerden Defence of a shocking point of view", Archive for history of exact sciences, xv (1976), 199-210; and Sabetai Unguru "On the need to rewrite the history of Greek mathematics", Archive for history of exact sciences, xv (1975), 67-114.[RETURN]
17. Of course, it is impossible to say with total certainty that this approach to algebra actually originated with al-Khwarizmi. His Al-jabr wa'l-muqabala, nevertheless, is the earliest fossil remain of this line.[RETURN]
18. Martin Levey, The Algebra of Abu Kamil: Kitab fi al-jabr wa'l-muqabala (Milwaukee, 1966), 13-18.[RETURN]
19. While the exact translation which Abu-Kamil used is unclear, by his period of activity, the Elements existed not only in the translations made by al-Hajjaj but also in versions by Abu-Kamil's near contemporaries, Ishaq ibn Hunain and Thabit ibn Qurra. See Heath, Elements (ref. 5), i, 75- 77, 84, 87-88, and Lindberg, op. cit. (ref. 3), 56-57.[RETURN]
20. Levey, op. cit., 34. Note Abu-Kamil's direct citation of Book II of Euclid's Elements.[RETURN]
21. Ibid., 4.[RETURN]
22. A. P. Youschkevitch, "Abu 'l-Wafa' Al-Buzjani, Muhammad ibn Muhammad ibn Yahya ibn Isma'il ibn Al-'Abbas", 39-43, p. 43. See also Heath's Elements (ref. 5), i, 85-86. Somewhat before Abu'l-Wafa wrote his commentary on the Arithmetica, Abu Ja'far Al-Hazin had directly referred to Diophantus in dealing with several questions in indeterminate analysis. See Sesiano, op. cit. (ref. 8), 10.[RETURN]
23. Roshdi Rashed, "Al-Karaji (or al-Karkhi), Abu Bekr ibn Muhammad ibn al Husayn (or al-Hasan)" DSB, vii, 240-6, p. 241.[RETURN]
24. Ibid, 244.[RETURN]
25. Ibid, 241. For more on al-Karaji's mathematics, see Roshdi Rashed, "L'induction mathématique: al-Karaji, as-Samaw'al", Archive for history of exact sciences, ix (1972), 1-21; Rashed, Entre arithmétique et algèbre (ref. 15), 71-91.[RETURN]
26. Lindberg, op. cit. (ref. 3), 58. The other two centres for translation were Sicily and the Latin kingdoms in the Near East.[RETURN]
27. See ibid., 63-66, and Michael S. Mahoney, "Mathematics", in David C. Lindberg (ed.), Science in the Middle Ages (ref. 3), 145-78, pp. 157-8 for more on these translators. On a third translation due to William de Lunis, and so dating from the thirteenth century, see Barnabas Hughes, "The medieval Latin translations of al-Khwarizmi's Al'Jabr'", Manuscripta, xxvi (1982), 31-37.[RETURN]
28. It is known that Leonardo had some direct contact with the translations of Gerard of Cremona. He used Gerard's Latin translation of the work of the Banu Musa, entitled Verba filiorum, in preparing his Practica geometria (1220). See Marshall Clagett Archimedes in the Middle Ages, i (Madison, 1964), 7-32, 224, and ibid., iii (Philadelphia, 1976-80), 215.[RETURN]
29. Leonardo mentions his travels in the biographical introduction to his Liber abbaci. For the Latin text, see Guillaume Libri, Histoire des sciences mathématiques en Italie depuis la Renaissance des lettres, ii (Paris, 1838-41), 287-90. Libri uses the 1228 revised version of the 1202 Liber abbaci.[RETURN]
30. Ibid., 287. Also, in the biographical opening remarks of the Liber abbaci, Leonardo says that his father saw to it that he was instructed in the "Indian" art of calculating.[RETURN]
31. On the calculating tradition which predated Fibonacci, see Mahoney, "Mathematics" (ref. 27), 146-52. It is important to note that Fibonacci incorporated problems of a Diophantine nature in the Liber abbaci. His contact with Diophantus's work, however, seems not to have been direct but through the texts of al-Karaji. In fact, Ettore Picutti, in his "Il Libro dei Quadrati di Leonardo Pisano e i Problemi di Analisi indeterminata nel Codice Palatino 557 della Biblioteca Nationale di Firenze", Physis, xxi (1979), 195-339, argues for Leonardo's independence of Diophantus's ideas and techniques. Although some Diophantine notions did filter down through time, Diophantus was not really introduced into Western mathematics until the sixteenth century.[RETURN]
32. Kurt Vogel, "Fibonacci, Leonardo or Leonardo of Pisa", DSB, iv, 604-13. Vogel gives a summary of the contents of the whole work.[RETURN]
33. Libri, op. cit. (ref. 29), 356-8.[RETURN]
34. Leonardo's exposition is purely rhetorical although he does employ Hindu-Arabic numerals. It is also important to note that he frequently gives more than one geometrical demonstration of a given algebraic fact.[RETURN]
35. Warren Van Egmond, "The earliest vernacular treatment of algebra: The Libro di Ragioni of Paolo Gerardi (1328)", Physis, xx (1978), 155-89, pp. 155-7.[RETURN]
36. The earliest known vernacular treatment of algebra actually dates from the end of the thirteenth century. See Van Egmond, "The earliest vernacular treatment of algebra", 157.[RETURN]
37. A polynomial is said to be irreducible if it cannot be simplified to a polynomial of lower degree.[RETURN]
38. Van Egmond, "The earliest vernacular treatment of algebra", 187-8. These equations are given in modern notation. Gerardi wrote mathematics rhetorically. These were not the first cubic equations to appear in the Western mathematical literature. Borrowing either directly or indirectly from al-Khayyami (Omar Khayyam), Fibonacci included the cubic equation x³ + 2x² + 10x = 20 in his text entitled, Flos (c. 1225). See Mahoney, "Mathematics" (ref 27), 160.[RETURN]
39. Van Egmond, "The earliest vernacular treatment of algebra", 163.[RETURN]
40. Warren Van Egmond, "The algebra of Master Dardi of Pisa", Historia mathematica, x (1983), 399-421. For a more detailed study of the algebraic work being done in Italy from the thirteenth through the end of the fifteenth century, see Raffaella Franci and Laura Toti Rigatelli, "Towards a history of algebra from Leonardo of Pisa to Luca Pacioli", Janus, lxxii (1985), 17-82.[RETURN]
41. Raffaella Franci and Laura Toti Rigatelli, "Maestro Benedetto de Firenze e la Storia dell'Algebra", Historia mathematica, x (1983), 297-317, p. 314.[RETURN]
42. Franci and Toti Rigatelli, "Towards a history of algebra" (ref. 40), 61. See pp. 61-66 on the Summa and what the authors term Pacioli's "unmerited fame".[RETURN]
43. Completed around 1225, the Liber quadratorum represented Leonardo's main foray into indeterminate analysis in the style of Diophantus. See ref. 31 above on Leonardo's sources.[RETURN]
44. P. Speziali, "Luca Pacioli et son œuvre", in Sciences de la Renaissance: VIII Congrès international de Tour (Paris, 1973), 93-106, p. 96. Pacioli drew the commercial sections of the Summa principally from the maestri d'abaco.[RETURN]
45. Florian Cajori, A history of mathematical notations, i (Chicago, 1928), 90.[RETURN]
46. See ibid. for a historical development of the cossist school and the notation it developed.[RETURN]
47. Speziali, "Luca Pacioli et son œuvre" (ref. 44), 98. Al-Khayyam also left open the possibility for general solutions of higher degree equations. See A. P. Youschkevitch and B. A. Rosenfeld, "Al-Khayyami (or Khayyam), Gheyath al-Din Abu'L-Fath 'Umar ibn Ibrahim al-Nisaburi (or al-Naysaburi), also known as Omar Khayyam", DSB, vii, 323- 34, p. 328. The work of al-Khayyami was unknown in the West at this time, however.[RETURN]
48. In his Ars magna written in 1545, Cardano said that the solution was discovered "well-nigh thirty years ago", which would put the date at around 1515. See Cardano, The great art (ref. 1), 96. However, P. Speziali in "L'École algébriste italienne du XVI siècle et la résolution des equations des 3e et 4e degrés", in Sciences de la Renaissance (ref. 44), 107-20, gives the date "1504 or maybe even the year before", without any further evidence. See ibid., 110.[RETURN]
49. See Speziali, "L'École algébriste italienne du XVIe siècle", 108, and Oystein Ore, Cardano: The gambling scholar (Princeton, 1953), 62-63.[RETURN]
50. Speziali, "L'École algébriste italienne du XVIe siècle", 111, and Ore, op. cit., 63-65.[RETURN]
51. See Ore, op. cit., 77-107. For biographical information on Cardano, see Ore, op. cit., 3- 52; Girolamo Cardano, The book of my life, trans. by Jean Stoner (Toronto, 1931); Mario Gliozzi, "Cardano, Girolamo", DSB, iii, 64-67; and for a Jungian slant on his life, see Markus Fierz, Girolamo Cardano (1501-1576): Philosopher, natural philosopher, mathematician, astrologer, and interpreter of dreams, trans. by Helga Niman (Boston, 1983).[RETURN]
52. Morris Kline, Mathematics in Western culture, (New York, 1953), 100.[RETURN]
53. Ore, op. cit., 47-48. Here when Ore refers to "the Greek mathematicians", he means mathematicians like Euclid whose work enjoyed a more or less continuous tradition.[RETURN]
54. Cardano, The great art (ref 1), 8.[RETURN]
55. Ibid., 9.[RETURN]
56. Ibid., 96-97.[RETURN]
57. Ibid., 97. I would like to thank my colleague, George Francis, for his rendering of Fig. 5.[RETURN]
58. In modern notation, this says a³ + 3ab² = (a - b)³ + b³ + 3a²b, an equivalent version of (a - b)³ = a³ - 3a²b + 3ab² - b³.[RETURN]
59. Cardano, The great art (ref. 1), 96- 101.[RETURN]
60. Ibid., 9. My emphasis.[RETURN]
61. Ibid., 237.[RETURN]
62. Ibid., 237-53.[RETURN]
63. Ibid., 38-39.[RETURN]
64. Ibid., 11. My emphasis. In the remainder of the text, Cardano inconsistently mentions negative roots. For example, he gives only 18 as a solution to x² = 10x + 144. See Ibid., 36.[RETURN]
65. In his subsequent treatment of negatives, particularly in Chap 37, he does give the traditional interpretation of negatives as debits or defects.[RETURN]
66. Cardano, The great art (ref. 1), 39.[RETURN]
67. Ibid., 39.[RETURN]
68. ibid., 219 (my emphasis).[RETURN]
69. ibid., 219.[RETURN]
70. ibid., 220.[RETURN]
71. Ibid. The problem of negatives continued to haunt Cardano. As R H C. Tanner has pointed out, Cardano returned to the issue in an appendix to the 1570 edition of the Ars magna. Furthermore, in a work entitled, "Sermo de plus et minus", which only appeared in 1663 in Cardano's collected works, he responded to the way in which Bombelli treated negatives in his Algebra of 1572. See R. H. C. Tanner, "The alien realm of the minus: Deviatory mathematics in Cardano's writings", Annals of science, xxxvii (1980), 159-78, pp. 168-77.[RETURN]
72. Sir Thomas L. Heath, Diophantus of Alexandria: A study in the history of Greek algebra (New York, 1964), 20. See also Sir Thomas L. Heath, A history of Greek mathematics, ii (Oxford, 1921; reprint edn, New York, 1981), 448.[RETURN]
73. In the early 1970s a manuscript containing four more of the thirteen books was found to exist in the Mashhad Shrine Library. Both recent translators of these four newly-discovered books, Roshdi Rashed and Jacques Sesiano, have determined that the four new books should be interposed between what have been considered Books III and IV up until now. See ref. 8 above.[RETURN]
74. As we have noted, in the Liber abbaci Fibonacci did give some indeterminate problems which may have been inspired indirectly by Diophantus through al-Karaji. See ref. 31 above.[RETURN]
75. When Diophantus wrote the Arithmetica around A.D. 250, the mathematical environment, although dominated by Euclid's geometrical algebra, still included the Babylonian, algorithmic approach to algebra. (See Heath, A history of Greek mathematics (ref 72), ii, 448.) From the present point of view, Diophantus, in selecting a more Babylonian mode of thought and presentation, rendered his new ideas largely uncompetitive. That the Arithmetica did generate some interest relatively early on, however, is evidenced by the fact that Hypatia wrote a commentary on the text around A.D. 400. Diophantus's work is also mentioned in scattered, later texts through the time of Fibonacci. (See Sesiano, op. cit. (ref. 8), 8-20.[RETURN]
76. Heath, Diophantus (ref. 72), 155.[RETURN]
77. ibid., 155.[RETURN]
78. For a complete discussion of Diophantus's notation, see ibid., 32-53, or Kurt Vogel "Diophantus of Alexandria", DSB, iv, 110-19, p. 112.[RETURN]
79. Viète, Introduction (ref. 2), 330-1. See ref. 22 above.[RETURN]
80. For detailed archival studies which clarify the particulars of Bombelli's life, see S. A. Jayawardene, "Unpublished documents relating to Raphael Bombelli in the archives of Bologna", Isis, liv (1963), 391-5; "Raphael Bombelli, engineer-architect: Some unpublished documents of the Apostolic camera", Isis, lvi (1965), 298-306; and S. A. Jayawardene, "Bombelli, Raphael", DSB, ii, 279-81.[RETURN]
81. S. A. Jayawardene, "The influence of practical arithmetics on the Algebra of Raphael Bombelli", Isis, lxiv (1973), 510-32, p. 513. See ref. 15 above. See also Rafael Bombelli L'Algebra, with an Introduction by U. Forti and a Preface by E. Bortolotti, 1st integral edn (Milan, 1966), 9.[RETURN]
82. Paul Lawrence Rose, The Italian renaissance of mathematics: Studies of humanists and mathematicians from Petrarch to Galileo (Geneva, 1975), 146.[RETURN]
83. See Bombelli, op. cit., 11-154. For a discussion of Bombelli's treatment of the so-called imaginaries and of Cardano's reaction to them, see Tanner, op. cit. (ref. 71), 168-77.[RETURN]
84. See Bombelli, op. cit., 155-314, 315-476. For a treatment of these problems and their sources, see Jayawardene, "The influence of practical arithmetics" (ref. 81), 513-21.[RETURN]
85. For texts of the fourth and fifth books, see Bombelli, op. cit., 477-618, 619-69.[RETURN]
86. Jayawardene, "Unpublished documents relating to Raphael Bombelli" (ref. 80), 392.[RETURN]
87. On Bombelli's nomenclature, see Bombelli, op. cit., 155-6. On his notation in general, see Ettore Bortolotti, "Sulla rappresentazione simbolica della incognita e delle potenze di essa introdotta dal Bombelli", Archivo di storia della scienza, viii (1927), 49-63.[RETURN]
88. Jayawardene, "The influence of practical arithmetics" (ref. 81), 511, see ref. 7 for the translation. For the original Italian, see Bombelli, op. cit., 317.[RETURN]
89. ibid, 511.[RETURN]
90. On Viète's sources, see Karin Reich, "Diophant, Cardano, Bombelli, Viète ein Vergleich ihrer Aufgaben", in Rechenpfennige: Aufsätze zur Wissenschaftsgeschichte Kurt Vogel zum 80. Geburtstag (Munich, 1968), 131-50; Michael S. Mahoney, "Die Anfänge der Algebraische Denkweise in 17. Jahrhundert", Rete, i (1971), 15-31; and Mahoney, The mathematical career of Pierre de Fermat (1601-1665) (Princeton, 1973), 26-48. It is not clear whether Viète had studied Bombelli's Algebra by 1591, but his reading of Diophantus had led him to many of the same conclusions regarding the role of geometry in algebra.[RETURN]
91. See Section I above.[RETURN]
92. Viète, Introduction (ref. 2), 319.[RETURN]
93. Ibid., ref. 218, 260, and Pappus of Alexandria, La Collection mathématique, trans. by Paul Ver Eecke, ii (Paris, 1933), 477.[RETURN]
94. Klein, op. cit. (ref. 2), 155, and Pappus, op. cit., ii, 478.[RETURN]
95. Klein, op. cit. (ref. 2), 155-7.[RETURN]
96. Viète, Introduction (ref. 2), 320-1. In the In artem analyticem isagoge, Viète attributed the definitions he gave of analysis to Theon. Klein explained this as a result of the "general humanistic tendency to derogate the authority of those writers who were recognized as authorities in the schools, on the grounds of a 'better' knowledge of the ancients". See Klein, op. cit. (ref. 2), ref. 217, 260-1.[RETURN]
97. Mahoney, The mathematical career of Pierre de Fermat (1601-1665) (ref. 90), 34.[RETURN]
98. Viète, Introduction (ref. 2), 321-2.[RETURN]
99. Ibid., 328.[RETURN]
100. Ibid., 340. In the natural selection of ideas, this variation proved favourable and persists today with René Descartes's (1596-1650) modification of it calling for letters at the beginning of the alphabet, such as a, b, c, to denote the indeterminate magnitudes and letters at the end, such as x, y, z, to indicate the unknowns.[RETURN]
101. Ibid., 324-5.[RETURN]
102. See ibid., 325-38. It is important to note that Viète did not acknowledge negative numbers. For him subtraction signified taking the smaller from the larger. In this he did not differ from Cardano, Fibonacci, or any of his predecessors. Like Cardano, the limitations of dimensionality did not stop Viète from dealing with fourth and higher degree equations. For these, he resorted to so-called mechanical methods.[RETURN]
103. Ibid., 338.[RETURN]
104. On the reactions of Descartes and Fermat to Viète's ideas on dimension, for example, see Mahoney, The mathematical career of Pierre de Fermat (1601-1665) (ref. 90), 42-44.[RETURN]
105. For a translation of this work into English, see Viète, The analytic art: Nine studies (ref. 2), 83-153.[RETURN]
106. Viète, Introduction (ref. 2), ref. 23, 331, or Viète, The analytic art: Nine studies (ref. 2), 83-84. Note that Viète employs no notation for equality. I. G. Bachmakova and E. I. Slavutin have examined Viète's treatment of Diophantus's indeterminate problems in "'Genesis triangulorum' de François Viète et ses recherches dans l'analyse indeterminée", Archive for history of exact sciences, xvi (1977), 289-306.[RETURN]
107. Charles Darwin, On the origin of species, with an Introduction by Ernst Mayr, facs. of 1st edn (Cambridge. 1964), 121.[RETURN]

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This document previously appeared as an article in the June 1988 issue of History of Science, Vol. 26, No. 72, pp.129-164. The HTML-encoded version which you are reading appears here with the kind permission of the Editors of Science History Publications, Ltd., London, England, and includes ancillary material not present in the original article. © Karen Hunger Parshall, 1995, 1999. Additional biographical and historical material provided by Karen H. Parshall and Patti Wilger Hunter. HTML-encoding performed by Fred O'Bryant, Science and Engineering Library, University of Virginia.