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

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....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.^{2}

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 *ax*² = *bx*, *ax*² = *c*, *bx* =
*c*, *ax*² + *bx* = *c*, *ax*² + *c* = *bx*,
and *bx* + *c* = *ax*², 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.In modern notation, the problem was to solve the equation^{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 figureAlthough not as formal in style, this argument paralleled that given by Euclid in theab.

These again are called

agandbd. The breadth of each is equal to the breadth of one side of the squareaband 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, representingx², 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 squareab, that is, one root of the proposedx². Therefore three is the root of thisx², andx² is 9.^{12}

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
*x*² + 10*x* = 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--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"ABGD:

One adds the roots to it which were originally associated with the square--it is 10--they are

ABWH. One knows that lineBHis 10 because the sideABof the surfaceABGDmultiplied by unity is a root of the surfaceABGD. It is one multiplied by 10; it is 10 roots of the surfaceABGD. Thus it is lineBHor 10. The entire surfaceWHGDis 39 because it was set up as the square and 10 of its roots; it is the product of lineHGby lineGD. But lineGDis equal to lineGB. Also, the product of lineHGby lineGBis 39. LineHBequals 10. Divide it in half by the pointH. Add lineGBto its length. And so, the surface is the result of the product of [HGby itself just as the surface is the result of the product of]HGby lineBGadded to the square quadrilateral, the product ofHBby itself just as Euclid stated in the second part of his book. But the product of lineHGby lineGBis set at 39. The product of lineHBby itself is 25; the total is 64. Thus, the product of lineHGby itself is 64; the root of 64 is 8. Then, the lineHGis 8. One knows that lineHBis 5 and lineBGremains as 3. It is the root of the square; the square is 9.^{20}

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/*x*² X
1/*x* = 1/*x*³, 1/*x* X *x*³ = *x*², and l/*x* :
l/*x*² = *x*²/*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.

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 algebra^{28} 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: *ax*³
= *bx* + *N*, *ax*³ = *bx*² + *N*, and *ax*³ =
*bx*² + *cx* + *N*.^{38} His solutions were merely naïve applications of the quadratic
formula to cubic equations. Thus, for *ax*³ = *bx* + *N*, he
claimed

that is, the solution of the quadratic *ax*² = *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" = *x*², *b* = "cubo" =
*x*³, *cc* = "censo di censo" = *x*^{4}, *br* =
"cubo relato cosa" = x^{5}, and *bb* = "cubo di cubo cosa" =
*x*^{6}. 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 *ax*³ + *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 *ax*³ + *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.

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:

For example, letHere, Cardano asked his readers to complete the cubes formed onGH³ plus six times its sideGHequal 20, and letAEandCLbe two cubes the difference between which is 20 and such that the product ofAC, the side [of one], andCKthe side [of the other], is 2, namely one-third the coefficient ofx. Marking offBCequal toCK, I say that, if this is done, the remaining lineABis equal toGHand is, therefore, the value ofx, forGHhas already been given as [equal tox].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 bodiesDA,DC,DE, andDF; and asDCrepresentsBC³, soDFrepresentsAB³,DArepresents 3(BCxAB²) andDErepresents 3(ABxBC²).^{56}

This resulted in the body shown in Figure 5, where, for instance, *DE*
is the shaded region. In Cardano's set-up, *AC*³ corresponded to the cube
*AE* and *CK*³ represented the cube *CL* and was equivalent to
*BC*³, so

by hypothesis, andAC³ -BC³ =AC³ -CK³ = 20,

by construction. Furthermore, "since ...(1) AC³ -BC³ =AC³ -CK³ =DA+DE+DF= 3( BCxAB²) + 3(ABxBC²) +AB³,

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

(2)Invoking the second proposition of the sixth chapter, that is,6 AB= 3(ABxBCxAC).

Cardano used the equivalent formulationAC³ + 3(ACxCB²) = (AC-CB)³ +CB³ + 3(CBxAC²),^{58}

Substituting (3) into (1) and verifying thatAB³ = (AC-CB)³

(3)= AC³ + 3(ACxCB²) + (-CB³) + 3(-CBxAC²).

equalled (2), Cardano found that3( BCxAB²) + 3(ABxBC²)

(4)Adding 6AC³ -BC³ =AC³ + 3(ACxCB²) + 3(-CBxAC²) + (-BC³) + 6AB= 20

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
*x*^{4} + *bx*² = *c*, where he interpreted
*x*^{4} as the square of side *x*². 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 orSince 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 onex² = 10x+ 144,

144.aeq.10.pos.p.1.quad. or 144 = 10x+x², and

l.quad.p.16.aeq.10.pos. orx² + 16 = 10x.^{63}

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* = *x*² + *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.
(1/2*a*)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 *x*² +
*b* = *ax*, Cardano tried to justify its algorithm geometrically. (See
Equation (5) above.) When he computed (1/2*a*)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.

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, 2*x* + 1,
and 4*x* + 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

However, (4( x+ 1)² - (2x+ 1) =x², a square, and

(2x+ 1)² - (4x+ 1) = 4x², a square.

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 630*x*² + 73*x* = 6 appeared as

Here [GREEK=DELTA^{Y}] abbreviated [GREEK=DYNAMIS] meaning "power"
and was equivalent to the modern *x*²; [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 this^{79}:

Diophantus's use of abbreviations such as [GREEK=SIGMA],
[GREEK=DELTA^{Y}], and [GREEK=KAPPA^{Y}] 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.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^{96}

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: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.

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}

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 be^{103}

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".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

Let the differencesBof the two "sides" be given, and also let their sumDbe given.

It is required to find the "sides".

Let the less "side" beA; then the greater will beA+B. Therefore, the sum of the "sides" will beA2 +B. But the same sum is given asD. Wherefore,A2 +Bis equal toD. And, by antithesis,A2 will be equal toD-B, and if they are all halved,Awill be equal toD1/2 +B1/2.

Or, let the greater "side" beE. Then the less will beE-B. Therefore, the sum of the "sides" will beE2 -B. But the same sum is given asD. Therefore,E2 -Bwill be equal toD, and by antithesis,E2 will be equal toD+B, and if they are all halved,Ewill he equal toD1/2 +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.

LetBbe 40 andD100. ThenAbecomes 30 andEbecomes 70.^{106}

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.

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.

2. François Viète,

3. Gerald Toomer, "Al-Khwarizmi, Abu Ja'far Muhammad ibn Musa",

4. Here "al-jabr" translates as "completion" and signifies the elimination of negative quantities from an equation. For example, "al-jabr" transforms

5. Sir Thomas L. Heath,

6. See Solomon Gandz, "The sources of al-Khwarizmi's algebra",

7. See Otto Neugebauer,

8. Jacques Sesiano,

9. Al-Khwarizmi. "Six types of rhetorical algebraic equations", in Edward Grant (ed.),

10. Notice that al-Khwarizmi solves not for the unknown

11. Al-Khwarizmi,

12.

13. Heath,

14.

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

16. Heath,

17. Of course, it is impossible to say with total certainty that this approach to algebra actually originated with al-Khwarizmi. His

18. Martin Levey,

19. While the exact translation which Abu-Kamil used is unclear, by his period of activity, the

20. Levey,

21.

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

23. Roshdi Rashed, "Al-Karaji (or al-Karkhi), Abu Bekr ibn Muhammad ibn al Husayn (or al-Hasan)"

24.

25.

26. Lindberg,

27. See

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

29. Leonardo mentions his travels in the biographical introduction to his

30.

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

32. Kurt Vogel, "Fibonacci, Leonardo or Leonardo of Pisa",

33. Libri,

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)",

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

39. Van Egmond, "The earliest vernacular treatment of algebra", 163.[RETURN]

40. Warren Van Egmond, "The algebra of Master Dardi of Pisa",

41. Raffaella Franci and Laura Toti Rigatelli, "Maestro Benedetto de Firenze e la Storia dell'Algebra",

42. Franci and Toti Rigatelli, "Towards a history of algebra" (ref. 40), 61. See pp. 61-66 on the

43. Completed around 1225, the

44. P. Speziali, "Luca Pacioli et son œuvre", in

45. Florian Cajori,

46. See

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",

48. In his

49. See Speziali, "L'École algébriste italienne du XVI

50. Speziali, "L'École algébriste italienne du XVI

51. See Ore,

52. Morris Kline,

53. Ore,

54. Cardano,

55.

56.

57.

58. In modern notation, this says

59. Cardano,

60.

61.

62.

63.

64.

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,

67.

68.

69.

70.

71.

72. Sir Thomas L. Heath,

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

75. When Diophantus wrote the

76. Heath,

77.

78. For a complete discussion of Diophantus's notation, see

79. Viète,

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",

81. S. A. Jayawardene, "The influence of practical arithmetics on the Algebra of Raphael Bombelli",

82. Paul Lawrence Rose,

83. See Bombelli,

84. See Bombelli,

85. For texts of the fourth and fifth books, see Bombelli,

86. Jayawardene, "Unpublished documents relating to Raphael Bombelli" (ref. 80), 392.[RETURN]

87. On Bombelli's nomenclature, see Bombelli,

88. Jayawardene, "The influence of practical arithmetics" (ref. 81), 511, see ref. 7 for the translation. For the original Italian, see Bombelli,

89.

90. On Viète's sources, see Karin Reich, "Diophant, Cardano, Bombelli, Viète ein Vergleich ihrer Aufgaben", in

91. See Section I above.[RETURN]

92. Viète,

93.

94. Klein,

95. Klein,

96. Viète,

97. Mahoney,

98. Viète,

99.

100.

101.

102. See

103.

104. On the reactions of Descartes and Fermat to Viète's ideas on dimension, for example, see Mahoney,

105. For a translation of this work into English, see Viète,

106. Viète,

107. Charles Darwin,

- The Banu Musa (Muhammad, Ahmad, and al-Hasan)
- Rafael Bombelli
- Girolamo Cardano (includes Portrait)
- Charles Darwin (includes Portrait)
- Diophantus
- Sample page (Filesize = 322K) from Euclid's
*Elements* - Euclid (includes Portrait)
- Leonardo Fibonacci (includes Portrait)
- The House of Wisdom
- Hypatia
- Abu Kamil
- Muhammad ibn Musa al-Khwarizmi
- Pappus of Alexandria
- Niccolò Tartaglia (includes Portrait)
- François Viète

This document previously appeared as an article in the June 1988 issue of