A Critical Edition of the Algebra of Maestro Dardi (1344)

A Critical Edition of the Algebra of Maestro Dardi (1344)

Dr. Warren van Egmond

The algebra of Maestro Dardi is the most important work of mathematics to have been written in Europe in the 350 years between the Liber abbaci of Leonardo Pisano in 1202 and the Ars magna of Girolamo Cardano in 1545. No other work of the period displays a comparable level of competence, organization, andscope, yet the full contents of this masterpiece have never been made available in an edition that meets the standards of modern scholarship.

According to its own testimony, Dardi’s work was composed in the year 1344, about two hundred years after the first works of algebra reached medieval Europe. While some of the basic techniques of this science can be found in some of the oldest surviving books of mathematics written in ancient Babylon and Egypt and were fully developed in the Arithmetica of the Greek mathematician Diophantus, the direct source of European algebra was the Kitab fi hisab al-jabr w’al-muqabala written by the Arab mathematician Mohammed ibn Musa al-Khwarizmi in Baghdad in the ninth-century A.D.[1] His basic treatment of the six simple and quadratic equations was copied by Leonardo Pisano and several other European authors in the twelfth and thirteenth centuries,[2] and had become a standard part of European mathematics texts by the fourteenth century.

Yet the algebra of al-Khwarizmi and Leonardo Pisano was comparatively primitive by modern standards. Because algebraists of the time did not recognize the possibility of including negative and zero terms in an equation, the practice of algebra consisted of listing all possible permutations of the positive terms and giving detailed rules or algorithms for solving each one. In the works of al-Khwarizmi and Leonardo only linear and quadratic values of the unknown were considered, producing only six possible cases:

1. ax = N

2. ax2 = N

3. ax2 = bx

4. ax2 + bx = N

5. ax2 + N = bx

6. ax2 = bx + N

(Note that even the possibility of reduction is ignored, so that ax2 = bx is treated separately from ax = N.) This standard list and sequence became the core of every European algebra book until the seventeenth century.

But already by the beginning of the fourteenth century, Italian algebraists had begun to expand the number of cases by introducing terms of the third, fourth, and even higher degrees, taking the root of a number or unknown as a term, and increasing the number of terms in an equation. Thus by the middle of the fourteenth century, one often finds cases like ax = √bx, ax4 + bx3 = cx2, and ax3 + bx2 + cx = N in many books of algebra.[3]

Yet overall the treatment remained naïve and primitive. New cases were added haphazardly, often without any logical order or evident understanding of the fundamental mathematics involved. For example, an early work of Paolo Gerardi included three “irreducible” cases of the cubic equation, ax3 = bx + N, ax3 = bx2 + N, and ax3 = bx2 + cx + N, which cannot be solved by any simple rule, yet Gerardi gave a solution by simply using the rule for a quadratic equation of similar form.[4] He then used these rules in the examples which accompanied each case, and accordingly produced an answer for each, which we can easily show is wrong simply by substituting the calculated answer into the original problem. Yet Gerardi never took this elementary step and accordingly accepted the wrong answers without question. Moreover, Gerardi’s false rules and incorrect solutions were copied and recopied in dozens of successive algebra texts for the next century and a half, without these errors ever being noticed.[5]

It is against this rather disappointing background that the algebra of Maestro Dardi stands out in so many respects. To begin with, it is a much larger and more systematic treatment of the subject than any of the earlier works.[6] Whereas all of the previous authors, following the model of Leonardo Pisano, had treated algebra only as a small part of a larger mathematics book, Dardi devoted his entire book to this single subject, writing more than 220 pages in the surviving manuscripts. He begins with a large and detailed chapter on the mathematics of radicals, which is essential to treating equations of higher orders, and follows this with a systematic and logically organized presentation of 198 different equation types, more than three times as many as those found in any other algebra of the period. He begins with the six quadratics known to the Arabs, then uses various combinations of powers and radicals to produce equations up to the equivalent of the 12th degree, and solves every one with complete accuracy. Unlike Paolo Gerardi and his copyists, there are no mistakes in Dardi’s algebra.

But the most remarkable feature of Dardi’s text is the fact that he includes four ‘special cases’ of third- and fourth-degree equations that cannot be reduced to lower degrees. These are the kinds of equations on which Gerardi had stumbled, and continued to bedevil European algebraists until the middle of the sixteenth century, yet Dardi succeeds in solving them correctly. Although the solutions are not general, in that they cannot be used to solve the equations for every possible value of the coefficients, they are correct for the cases that he gives. Moreover, Dardi is fully aware of these limitations and specifically notes this fact in his presentation of these four cases. This is the first time higher-order equations were correctly solved in Europe, and antedates by some 200 years the complete solution of third- and fourth-degree equations given by Girolamo Cardano and his student Ludovico Ferrari in the Ars magna of 1545.[7] Dardi’s algebra is thus a mathematical work of substantial mathematical and historical significance.

Other features of Dardi’s work provide additional evidence of his extraordinary mathematical abilities. For example, he frequently remarks on the possibility of reducing one equation to another, something his contemporaries consistently ignored. He uses zero as an explicit number several times, where all other mathematicians of the time regarded it as a fiction. He regularly uses proofs to check his results and remarks on the need to do this on a regular basis. He also uses an abbreviated symbolism for the unknowns, and a unique superscript system that writes the coefficients above them.

All in all, it is clear that Dardi was an algebraist of remarkable competence and originality who stood far above the level of his contemporary mathematicians, yet his work has remained largely unknown to historians of mathematics up to the present day. Its existence was not even noted until 1838, when the French historian of mathematics Guillaume Libri published a notice of it in volume II of his Histoire des sciences mathématiques en Italie,[8] followed by the complete text of the four ‘special cases’ in volume III,[9] yet even then it did not begin to receive any general attention until I published a complete summary of all the equations in Historia mathematica in 1983, and pointed out its correct solutions of the third- and fourth-degree equations.[10] However, to this date there is still no authoritative edition of the text. The only printed version of the full text is a transcription made from a single manuscript that was published in a limited edition in Siena in 2001.[11] This version, however, is based on one copy, makes no comparison with the other copies of the text, and provides a minimal description and mathematical summary with no detailed analysis. Thus Dardi’s important work has to this day never received the full scholarly edition that it so rightly deserves.

The purpose of this project is to produce a definitive, critical edition of Dardi’s algebra that will meet the highest standards of modern scholarship. It will be based on a complete collation of all available manuscripts, and print the most authoritative text with an apparatus noting all significant variants. English translations will be provided for the most significant and representative passages but, given the repetitive and tedious nature of mathematical writing, a complete translation of the entire text would not be fruitful. Instead, all rules, problems, and equations will be summarized in modern mathematical symbolism, which will be far more understandable for today’s readers.

The edited text will be accompanied by a historical and textual introduction, a mathematical summary and analysis of all the rules and problems, an index of equations and problems, a glossary of mathematical terms and concepts, and a lexicon of vocabulary and outline of grammar. Since no other Italian mathematical text of this length has been printed before, it also provides a representative sample of the mathematical idiom used in the vernacular Italian of the fourteenth and fifteenth centuries that can be used as a source for studies of Renaissance Italian. It is thus planned that this edition will meet the full standards of modern scholarship in both textual studies and the history of mathematics, and be of use in other areas of scholarship besides the history of mathematics.

[1] Modern edition and translation by Roshdi Rashed, Le commencement de l’algèbre (Paris: Blanchard, 2007) [in French]; Al-Khwarizmi: The Beginnings of Algebra (London: SAQI, 2009) [in English].

[2] Text edited by Baldassarre Boncompagni, Scritti di Leonardo Pisano, vol. I (Roma, 1857); English translation by L. E. Sigler, Fibonacci’s Liber Abaci (Springer, 2002).

[3] Warren Van Egmond, “The Study of Higher-Order Equations in Italy before Pacioli,” in: Mathematics Celestial and Terrestrial: Festschrift für Menso Folkerts zum 65. Geburtstag (Joseph W. Dauben, ed.) (Halle (Saale): Deutsche Akademie der Naturforscher Leopoldina, 2008), pp. 303-320.

[4] Warren Van Egmond, “The Earliest Vernacular Treatment of Algebra: The Libro di ragioni of Paolo Gerardi (1328),” Physis, 20 (1978), pp. 155-189.

[5] Van Egmond, “ The Study of Higher-Order Equations,” op. cit. n. 3

[6] Warren Van Egmond, “The Algebra of Master Dardi of Pisa,” Historia Mathematica, 10 (1983), pp. 399-421.

[7] Translated by T. Richard Witmer, The Great Art; or, The Rules of Algebra (M.I.T. Press, 1968).

[8] See the note printed on the bottom of pp. 519-20.

[9] At the end of Note XXXI in volume III (1840), pp. 349-356. Because the manuscript that Libri used (now Ashb. 1199) was anonymous, Dardi was not identified as the author at this time.

[10]Op. cit. n. 5.

[11] Maestro Dardi, Aliabraa argibra dal Mansocritto I.VIII.17 della Biblioteca Comunale di Siena (ed. R. Franci) [Quaderni del Centro Studi della Matematica Medioevale, 15] (Siena: Servizio Editoriale dell’ Università di Siena, 2001).