The Art of Algebra from Al-Khwārizmī to Viète: A Study in the Natural Selection of Ideas

Cardano Girolamo , The great art or the rules of algebra, trans. by Witmer T. Richard (Cambridge, 1968), 7.

Viète François , Introduction to the analytic art , trans. by Rev. Smith J. Winfree , in Klein Jacob , 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 Viète François , The analytic art: Nine studies in algebra, geometry and trigonometry from the Opus restitutae mathematicae analyseos, seu algebra nova , trans. by Witmer T. Richard (Kent, Ohio, 1983). For a reprint of the original 1646 edition of Viète's collected works, see Viète François , Opera mathematica, recognita francisci à Schooten , ed. by Hofmann Joseph E. (Leiden, 1646; reprint edn, Hildesheim–New York, 1970).

Toomer Gerald , “Al-Khwārizmī, Abū Ja'far Muhammad ibn Mūsā”, Dictionary of scientific biography , ed. by Gillispie C. C. (New York, 1970–80; hereafter DSB), vii, 358–65, p. 358; Al-Dabbagh J. , “Banū Mūsā, three brothers—Muhammad, Ahmad, and al-Hasan”, DSB , i, 443–6, p. 444; and Lindberg David C. , “Transmission of Greek and Arabic learning to the West”, in Lindberg David C. (ed.), Science in the Middle Ages (Chicago, 1978), 52–90, pp. 55–58.

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-muqābala” translates as “balancing” and refers to reducing positive quantities of the same power on both sides of an equation. For example, “al-muqābala” turns 10x + 64 = 5x + 36 + x² into 5x + 28 = x². See Toomer , op. cit. (ref. 3), 359.

Heath Thomas L. Sir , The thirteen books of Euclid's Elements, i (Cambridge, 1926), 73–84.

See Gandz Solomon , “The sources of al-Khwarizmi's algebra”, Osiris, i (1936), 263–77.

See Neugebauer Otto , The exact sciences in Antiquity (New York, 1969), 147.

Sesiano Jacques , 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 Rashed Roshdi (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 Qustā ibn Lūqā's dates, see ibid., iii, pp. xvi–xxii.

Al-Khwārizmī , “Six types of rhetorical algebraic equations”, in Grant Edward (ed.), A source book in medieval science (Cambridge, 1974), 106–11, p. 108.

Notice that al-Khwārizmī 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.

Al-Khwārizmī , op. cit., 110.

ibid., 111.

Heath , Elements (ref. 5), i, 385.

ibid., 385–6.

Scholars disagree on the role of Euclid in al-Khwārizmī's mathematics. In his article on al-Khwārizmī'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'ldée de l'algèbra selon al-Khwārizmī”, in Rashed Roshdi , Entre arithmétique el 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.).

Heath , Elements (ref. 5), i, 377. On differing views concerning the significance and meaning of “geometrical algebra”, see van der Waerden B. L. , Geometry and algebra in ancient civilizations (New York, 1983), 75–96; van der Waerden B. L. , “Defence of a shocking point of view”, Archive for history of exact sciences, xv (1976), 199–210; and Unguru Sabetai , “On the need to rewrite the history of Greek mathematics”, Archive for history of exact sciences, xv (1975), 67–114.

Of course, it is impossible to say with total certainty that this approach to algebra actually originated with al-Khwārizmī. His Al-jabr wa'l-muqābala, nevertheless, is the earliest fossil remain of this line.

Levey Martin , The Algebra of Abū Kāmil: Kītab fi al-jabr wa'l-muqābala (Milwaukee, 1966), 13–18.

While the exact translation which Abū-Kāmil used is unclear, by his period of activity, the Elements existed not only in the translations made by al-Hajjāj but also in versions by Abū-Kāmil's near contemporaries, Ishāq ibn Hunain and Thābit ibn Qurra. See Heath , Elements (ref. 5), i, 75–77, 84, 87–88, and Lindberg , op. cit. (ref. 3), 56–57.

Levey , op. cit., 34. Note Abū-Kāmil's direct citation of Book II of Euclid's Elements.

ibid., 4.

Youschkevitch A. P. , “Abū‘l-Wafū’ Al-Būzjānī, Muhammad ibn Muhammad ibn Yahyā ibn Ismā'īl ibn Al-'Abbās”, 39–43, p. 43. See also Heath , Elements (ref. 5), i, 85–86. Somewhat before Abū‘1-Wafā’ wrote his commentary on the Arithmetica, Abū Ja'far Al-Hazīn had directly referred to Diophantus in dealing with several questions in indeterminate analysis. See Sesiano , op. cit. (ref. 8), 10.

Rashed Roshdi , “Al-Karajī (or al-Karkhī), Abū Bekr ibn Muhammad ibn al Husayn (or al-Hasan)”, DSB, vii, 240–6, p. 241.

ibid., 244.

Ibid ., 241. For more on al-Karajī's mathematics, see Rashed Roshdi , “L'induction mathématique: Al-Karajī, as-Samaw'al”, Archive for history of exact sciences, ix (1972), 1–21; Rashed , Entre arithmétique et algèbre (ref. 15), 71–91.

Lindberg , op. cit. (ref. 3), 58. The other two centres for translation were Sicily and the Latin kingdoms in the Near East.

See ibid ., 63–66, and Mahoney Michael S. , “Mathematics”, in Lindberg David C. (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 Hughes Barnabas , “The medieval Latin translations of al-Khwārizmī's Al'Jabr”', Manuscripta, xxvi (1982), 31–37.

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 Banū Mūsā, entitled Verba filiorum, in preparing his Practica geometria (1220). See Clagett Marshall , Archimedes in the Middle Ages , i (Madison, 1964), 7–32, 224, and ibid ., iii (Philadelphia, 1976–80), 215.

Leonardo mentions his travels in the biographical introduction to his Liber abbaci. For the Latin text, see Libri Guillaume , Histoire des sciences mathématiques en Italie depuis la Renaissance des letlres , ii (Paris, 1838–41), 287–90. Libri uses the 1228 revised version of the 1202 Liber abbaci.

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.

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-Karajī. In fact, Picutti Ettore , in his “Il Libro dei Quadrati di Leonardo Pisano e i Problemi di Analisi indeterminate 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.

Vogel Kurt , “Fibonacci, Leonardo or Leonardo of Pisa”, DSB, iv, 604–13. Vogel gives a summary of the contents of the whole work.

Libri , op. cit. (ref. 29), 356–8.

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.

Van Egmond Warren , “The earliest vernacular treatment of algebra: The Libro di Ragioni of Paolo Gerardi (1328)”, Physis, xx (1978), 155–89, pp. 155–7.

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.

A polynomial is said to be irreducible if it cannot be simplified to a polynomial of lower degree.

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-Khayyāmī (Omar Khayyām), Fibonacci included the cubic equation x³ + 2x² + 10x = 20 in his text entitled, Flos (c. 1225). See Mahoney , “Mathematics” (ref. 27), 160.

Van Egmond , “The earliest vernacular treatment of algebra”, 163.

Van Egmond Warren , “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 Franci Raffaella and Rigatelli Laura Toti , “Towards a history of algebra from Leonardo of Pisa to Luca Pacioli”, Janus, lxxii (1985), 17–82.

Franci Raffaella and Rigatelli Laura Toti , “Maestro Benedetto de Firenze e la Storia dell'Algebra”, Historia mathematica, x (1983), 297–317, p. 314.

Franci and Rigatelli Toti , “Towards a history of algebra” (ref. 40), 61. See pp. 61–66 on the Summa and what the authors term Pacioli's “unmerited fame”.

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.

Speziali P. , “Luca Pacioli et son œuvre”, in Sciences de la Renaissance: VIII^e Congrès international de Tour (Paris, 1973), 93–106, p. 96. Pacioli drew the commercial sections of the Summa principally from the maestri d'abaco.

Cajori Florian , A history of mathematical notations, i (Chicago, 1928), 90.

See ibid. for a historical development of the cossist school and the notation it developed.

Speziali , “Luca Pacioli et son œuvre” (ref. 44), 98. Al-Khayyāmī also left open the possibility for general solutions of higher degree equations. See Youschkevitch A. P. and Rosenfeld B. A. , “Al-Khayyāmī (or Khayyām), Gheyāth al-Dīn Abū'L-Fath 'Umar ibn Ibrāhīm al-Nīsābūrī (or al-Naysābūrī), also known as Omar Khayyam”, DSB, vii, 323–34, p. 328. The work of al-Khayyāmī was unknown in the West at this time, however.

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, Speziali P. in “L'École algébriste italienne du XVI^e siècle et la résolution des equations des 3^e et 4^e 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.

See Speziali , “L'École algébriste italienne du XVI^e siècle”, 1081, and Ore Oystein , Cardano: The gambling scholar (Princeton, 1953), 62–63.

Speziali , “L'École algébriste italienne du XVI^e siècle”, 111, and Ore , op. cit., 63–65.

See Ore , op. cit., 77–107. For biographical information on Cardano, see Ore , op. cit., 3–52; Cardano Girolamo , The book of my life , trans. by Stoner Jean (Toronto, 1931); Gliozzi Mario , “Cardano, Girolamo”, DSB , iii, 64–67; and for a Jungian slant on his life, see Fierz Markus , Girolamo Cardano (1501–1576): Philosopher, natural philosopher, mathematician, astrologer, and interpreter of dreams, trans. by Niman Helga (Boston, 1983).

Kline Morris , Mathematics in Western culture (New York, 1953), 100.

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.

Cardano , The great art (ref. 1), 8.

ibid., 9.

ibid., 96–97.

Ibid ., 97. I would like to thank my colleague, George Francis, for his rendering of Fig. 5.

In modern notation, this says a³ + 3ab² = (a – b)³ + b³ + 3a²b, an equivalent version of (a – b)³ = a³ – 3a²b + 3ab² – b³.

Cardano , The great art (ref. 1), 96–101.

ibid., 9. My emphasis.

ibid., 237.

ibid., 237–53.

ibid., 38–39.

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.

In his subsequent treatment of negatives, particularly in Chap. 37, he does give the traditional interpretation of negatives as debits or defects.

Cardano , The great art (ref. 1), 39.

ibid., 39.

ibid., 219 (my emphasis).

ibid., 219.

ibid., 220.

Ibid . The problem of negatives continued to haunt Cardano. As Tanner R. H. C. 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 Tanner R. H. C. , “The alien realm of the minus: Deviatory mathematics in Cardano's writings”, Annals of science, xxxvii (1980), 159–78, pp. 168–77.

Heath Thomas L. Sir , Diophantus of Alexandria: A study in the history of Greek algebra (New York, 1964), 20. See also Heath Thomas L. Sir , A history of Greek mathematics , ii (Oxford, 1921; reprint edn, New York, 1981), 448.

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.

As we have noted, in the Liber abbaci Fibonacci did give some indeterminate problems which may have been inspired indirectly by Diophantus through al-Karajī. See ref. 31 above.

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.

Heath , Diophantus (ref. 72), 155.

ibid., 155.

For a complete discussion of Diophantus's notation, see ibid ., 32–53, or Vogel Kurt , “Diophantus of Alexandria”, DSB, iv, 110–19, p. 112.

Viète , Introduction (ref. 2), 330–1. See ref. 22 above.

For detailed archival studies which clarify the particulars of Bombelli's life, see Jayawardene S. A. , “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 Jayawardene S. A. , “Bombelli, Raphael”, DSB, ii, 279–81.

Jayawardene S. A. , “The influence of practical arithmetics on the Algebra of Raphael Bombelli”, Isis, lxiv (1973), 510–32, p. 513. See ref. 15 above. See also Bombelli Rafael , L'Algebra, with an Introduction by Forti U. and a Preface by Bortolotti E. , 1st integral edn (Milan, 1966), 9.

Rose Paul Lawrence , The Italian renaissance of mathematics: Studies of humanists and mathematicians from Petrarch to Galileo (Geneva, 1975), 146.

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.

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.

For texts of the fourth and fifth books, see Bombelli , op. cit., 477–618, 619–69.

Jayawardene , “Unpublished documents relating to Raphael Bombelli” (ref. 80), 392.

On Bombelli's nomenclature, see Bombelli , op. cit., 155–6. On his notation in general, see Bortolotti Ettore , “Sulla rappresentazione simbolica della incognita e delle potenze di essa introdotta dal Bombelli”, Archive di storia della scienza, viii (1927), 49–63.

Jayawardene , “The influence of practical arithmetics” (ref. 81), 511, see ref. 7 for the translation. For the original Italian, see Bombelli , op. cit., 317.

Ibid, 511.

On Viète's sources, see Reich Karin , “Diophant, Cardano, Bombelli, Viète ein Vergleich ihrer Aufgaben”, in Rechenpfennige: Aufsätze zur Wissenschaftsgeschichte Kurt Vogel zum 80. Geburtstag (Munich, 1968), 131–50; Mahoney Michael S. , “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.

See Section 1 above.

Viète , Introduction (ref. 2), 319.

Ibid ., ref. 218, 260, and Pappus of Alexandria, La Collection mathématique, trans. by Ver Eecke Paul , ii (Paris, 1933), 477.

Klein , op. cit. (ref. 2), 155, and Pappus , op. cit., ii, 478.

Klein , op. cit. (ref. 2), 155–7.

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.

Mahoney , The mathematical career of Pierre de Fermat (1601–1665) (ref. 90), 34.

Viète , Introduction (ref. 2), 321–2.

ibid., 328.

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.

ibid., 324–5.

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.

ibid., 338.

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.

For a translation of this work into English, see Viète , The analytic art: Nine studies (ref. 2), 83–153.

Viète , Introduction (ref. 2), ref. 23, 331, or Viète , The analytic art: Nine studies (ref. 2), 83–84. Note that Viète employes no notation for equality. Bachmakova I. G. and Slavutin E. I. 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.

Darwin Charles , On the origin of species, with an Introduction by Ernst Mayr, facs. of 1st edn (Cambridge, 1964), 121.