The alien realm of the minus: Deviatory mathematics in Cardano's writings

References

• 1545 . Ars magna, quam vulgo cossam vocant, sive regulas algebraicas, …; et est liber decimus sui arithmeticae Nuremburg
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• See Ore Oystein Cardano, the gambling scholar Princeton 1953 53 107 ‘The battle of the scholars’.
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• A succinct account is given by Hofmann J.E. Geschichte der Mathematik I, Von den Anfängen bis zum Auftreten von Descartes , 2nd ed. Berlin 1963 132 133
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• The whole volume is known from the shortened title of its first part as Cardano's Opus novum de proportionibus, or simply De proportionibus Basle 1570 Each of the three works is separately paginated, and this reference is at Ars magna, 62. Cardano's recognition of the difficulty is stated in the first edition (footnote 1), 31v.
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• Hieronymi Cardani Mediolensis, civisque Bononensis, philosophi, medici, et mathematici clarissimi, opus novum, de proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum, non solum geometrico more stabilitum, sed etiam variis experimentis et observationibus rerum in natura, solerti demonstratione illustratum, ad multiples usus accomodatum, et in v libros digestum. Praeterea, Artis magnae, sive de regulis algebrae liber unus, abstrusissimus et inexhaustus plane totius arithmeticae thesaurus, ab authore recens multis locis recognitus et auctus. Item, de Aliza regula liber, hoc est, algebraicae logisticae suae, numeros recondita numerandi subtilitate, secundum geometricas quantitates inquirentis, necessaria coronis. Opus physicis et mathematicis imprimis utile et necessarium Basle 1570 The word ‘coronis’ was taken from the Greek to signify a flourish at the end of a manuscript in the Middle Ages.
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• Cardano . 1545 . Ars magna, quam vulgo cossam vocant, sive regulas algebraicas, …, et est liber decimus sui arithmeticae fol. 3v – fol. 3v . Nuremburg page 6 in the 1570 edition
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• The suggestion came from Armin Wittstein: see Cantor M. Vorlesungen über die Geschichte der Mathematik , 2nd ed. Leipzig 1913 2 532 532 reprint The Arabic word is a' izzâ.
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• Cantor . 1913 . Vorlesungen über die Geschichte der Mathematik , 2nd ed. Vol. 2 , 500 – 500 . Leipzig reprint 503, where the reference is to G. Cardano, Opera omnia (ed. Caroli Sponii: 10 vols., Lyon, 1663: repr. New York, 1967), vol. 1, 103
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• The running title in the Ars Magna of 1545 is Hieronymi Cardani De Arithmetica Lib.X , 1570 edition unchanged in the
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• Hutton , C. 1812 . Tracts on mathematical and philosophical subjects Vol. 2 , 219 – 224 . London 1737–1823 3 vols Tract 33, ‘History of algebra’ ‘Of CARDAN's Libellus de Aliza Regula’.
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• Cossali , P. 1787–1799 . Origine, trasporto in Italia, primi progressi in essa dell'algebra Parma 2 vols
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• For instance Cantor Vorlesungen über die Geschichte der Mathematik , 2nd ed. Leipzig 1913 2 536 536 reprint
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• Cossali . 1787–1799 . Origine, trasporto in Italia, primi progressi in essa dell'algebra 369 – 369 . Parma 2 vols Cardano's chapter is discussed on pp. 367–369.
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• Cardano . 1539 . Practica arithmeticae generalis Milan Compare Cantor (footnote 7), 507–508.
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• Hofmann , Compare . 1963 . Geschichte der Mathematik I, Von den Anfängen bis zum Auftreten von Descartes , 2nd ed. 137 – 138 . Berlin
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• Heath , Compare T.L. 1908 . The thirteen books of Euclid's Elements , 2nd ed. Cambridge 1925; Dover reprint New York, 1956), Introduction, 104–105.
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• Commandino , F. 1572 . Euclidis elementorum libr. xv. Una cum scholijs antiquis Pisa fol. 149 recto
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• B.L.Add.Ms.6783 f. 121 – f. 121 . See section 5 of my preceding paper ‘The ordered regiment of the minus sign: off-beat mathematics in Harriot's manuscripts’, Ann. sci., 37 (1980), 127–158.
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• See Cajori F. A history of elementary mathematics New York 1917 244 244
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• See my The ordered regiment of the minus sign: off-beat mathematics in Harriot's manuscripts Ann. sci. 1980 37 127 158 sections 3 and 6
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• Cossali . 1787–1799 . Origine, trasporto in Italia, primi progressi in essa dell'algebra Vol. 2 , 369 – 369 . Parma
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• ‘Ex quo intelliges veram rationem ducendi m: & dividendi per m: & accipiendi ℞ tam quadratam quam cubicam (nam decuba dubium non est, quod est m:) non antea cognita’. The parenthesis is given the marginal reference: ‘Cor.I supra Ca.VI’; it refers to chapter 6 of De Aliza, Corollary 1. This chapter is entitiled: ‘De operibus p: et m: secundum communem usum’ (‘Of the operations plus and minus according to common usage’) De Aliza f. 15 f. 15 Hutton (footnote 12), 219 gives details of this chapter.
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• For instance in Colenso J.W. Elements of algebra adapted for the use of national and adult schools , 3rd ed. London 1849 London, 1868, 108.
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• See my preceding paper The ordered regiment of the minus sign: off-beat mathematics in Harriot's manuscripts Ann. sci. 1980 37 127 158 section 4
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• Loria , G. 1929 . Storia delle matematiche , 1st ed. 40 – 41 . Turin repr. Turin, 1931 (2nd ed. Milan, 1950), 298–299. Compare Cantor (footnote 7), vol. 2, 537
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• Hutton . 1812 . Tracts on mathematical and philosophical subjects Vol. 2 , 222 – 222 . London 3 vols
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• Loria . 1929 . Storia delle matematiche , 1st ed. 299 – 299 . Turin repr. Turin, 1931 It was from Niccolo Tartaglia (1500?–1557) that Cardano gleaned the famous formulae; but Tartaglia had not ‘imagined’ them on his own. See Ore (footnote 2), chapter 3, particularly pp. 62–107: Hofmann (footnote 3), 133; Cantor (footnote 7), 482–497; and their source: E. Giordani. I sei cartelli di matematica disfida (Milan, 1876).
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• Loria . 1929 . Storia delle matematiche , 1st ed. 299 – 299 . Turin repr. Turin, 1931
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• Bombelli , Rafael . 1572 . L'algebra, parte maggiore dell'arithmetica, divisa in tre libri Bologna two further books on geometry are included in L'algebra, prima edizione integrale (prefazione di Ettore Bortolotti e di Umberto Forti: Bologna 1929; repr. Milan, 1966). The dates of birth and death (1526–1572) of Bombelli are established in S. A. Jayawardene, ‘Documenti inediti negli archivi di Bologna intorno a Raffaele Bombelli e la sua famiglia’, Rend. Atti Acc. Sc. Ist. Bologna, cl. sc. fis., (11) 10 (1963), 235–247
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• See Ore Cardano, the gambling scholar Princeton 1953 21 23 and compare Cantor (footnote 7), 495
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• Cardano . 1663 . Opera omnia Edited by: Sponii , Caroli . Vol. 4 , 435 – 439 . Lyon Sermo de plus et minus
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• In the Euclidean tradition which still persisted, equations and equalities were widely expressed as proportions, which achieved the remarkable feat of conveying a new and little-known technique under a semblance of accustomed phrases and a set magisterial mode. Stevin Simon Arithmétique Leyde 1585 (1548–1620) habitually gave this form as alternative and final formulation in his which may serve us for illustration in an equation that Harriot transcribed in B.L.Add.6782., f. 311. The modern form is Stevin exhibits it, with the root x = 10 resolving it, as (p. 331): presented in Harriot's notation as The convention that the ratio represented in a proportion is here to be 1 appears to have been accepted like other liberties taken with the logic of symbolic and verbal presentation.
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• Cardano's second point concerns the geometrical solution of the same cubic equation in the three cases of non-positive coefficients; it makes use of two carpenters' squares, as shown in Bombelli's L'algebra, parte maggiore dell'arithmetica, divisa in tre libri Bologna 1572 298 299 See E. Bortolotti, ‘La trisectione dell'angolo ed il caso irreducibile della equazione cubica nell’ “Algebra” di Rafael Bombelli’, Rend. R. Acc. Sc. Inst. Bologna, 27 (1922–23), 125–138.
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• Wieleitner , H. 1927 . Zur Frühgeschichte des Imaginären . Jber. Dtsch. Math.-Ver. , 36 : 74 – 88 . (p. 87). This is the only mention noted of Cardano's misunderstanding
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• Crossley , John N. 1977–78 . “ The introduction of complex numbers ” . In History of mathematics preprint no. 4 , 40 – 40 . Monash University . quotes this comment, which is by Bortolotti in his edition of Bombelli's Algebra (footnote 42)
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• Biblioteca Comunale dell'Archiginnasio in Bologna Ms. (Bombelli), Codex B.1569; see Jayawardene S.A. The influence of practical arithmetics on the Algebra of Rafael Bombelli Isis 1973 64 510 523
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• The date is placed after 1557 and probably before 1560 in Jayawardene S.A. Rafael Bombelli. engineer-architect: some unpublished documents of the Apostolic Camera Isis 1965 56 298 306 particularly pp. 299, 304
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• Bombelli . 1572 . L'algebra, parte maggiore dell'arithmetica, divisa in tre libri Bologna lib. 1 (1572), 90; (1929), 77. In normal speech ‘via’ is equivalent to the Latin ‘per’ (or our ‘by’), also used for products
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• The ‘demonstration’ is reproduced in full by Crossley The introduction of complex numbers History of mathematics preprint no. 4. Monash University 1977–78 36 37 with translation p. 80. note 114
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• Bombelli . 1973 . The influence of practical arithmetics on the Algebra of Rafael Bombelli . Isis , 64 f.51v. Compare Algebra (footnote 42), (1929), 62, reproduced by Crossley (footnote 56), 34, without the two colloquial and Dantesque truncations of ‘meno’ to ‘men’
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• Loria . 1929 . Storia delle matematiche , 1st ed. 320 – 320 . Turin
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• A facsimile contrasting the three presentations of one passage of text, in manuscript, 1572 printing and 1966 edition, may be seen in Hofmann J.E. Bombellis Algebra—eine geniale Einzelleistung und ihre Einwirkung auf Leibniz Studia Leibnitiana 1972 4 196 252 Tafel 2, facing p. 209. Page numbers are not shown, and from which example it is taken, is not immediately obvious. However, the notation is consistently the same throughout the manuscript.
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• ‘Ancora si puo precedere nella equazione di questo capitulo [this word abbreviated] in un altro modo, come se si havesse ad agguagliare 1a 15p. 4’; this page (f.72v) of the Bombelli manuscript is shown in facsimile in Bortolotti E. L'algebra, opera di Rafael Bombelli … Archiv Gesch. Math. 1929 11 407 424 (p. 414)
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• See E. Bortolotti Origine e primo inizio del calcolo degli imaginari Scientia 1923 33 385 394 (p. 387)
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• For the cube root, the printed version has ‘lato’ (as Latin ‘latus’ in the Euclidean tradition), where the original had ‘creator’, a word used only in the manuscript as the originator of a power; for the unknown, the original uses ‘cose’, Italian equivalent of Latin ‘res’ (‘thing’), which was changed to ‘tanti’ Bombelli L'algebra, parte maggiore dell'arithmetica, divisa in tre libri Bologna 1572 225 225 Lib.2
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• Bombelli . 1572 . L'algebra, parte maggiore dell'arithmetica, divisa in tre libri 169 – 169 . Bologna (1929), 133
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• Bortolotti cites the whole passage in his preface Bombelli L'algebra, parte maggiore dell'arithmetica, divisa in tre libri Bologna 1572 xlix xlix as does Crossley (footnote 56), 38, 39
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• The rules are set forth in Crossley The introduction of complex numbers History of mathematics preprint no. 4 Monash University 1977–78 41 41
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• Cardano . 1663 . Opera omnia Edited by: Sponii , Caroli . Vol. 4 , 435 – 435 . Lyon
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• Bombelli . 1572 . L'algebra, parte maggiore dell'arithmetica, divisa in tre libri 293 – 293 . Bologna ‘resta m. 121 (il qual si chiamera puì di meno) che di questo pigliata la ℞.c. sarà p.di m. 11’.
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• Cardano . 1663 . Opera omnia Edited by: Sponii , Caroli . Vol. 4 , 435 – 435 . Lyon ‘(est hoc 121.m. vocatur p.di m. cuius ut notum ℞ est 11.)’.
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• Cardano . 1663 . Opera omnia Edited by: Sponii , Caroli . Vol. 4 , 435 – 439 . Lyon continued: et ita una pars erit 2.p.11 alia 2.m.p. 11. quarum ℞ cu.1 efficiunt rem quam constat esse 4’
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• Cardano . 1663 . Opera omnia Edited by: Sponii , Caroli . Vol. 4 , 435 – 439 . Lyon ‘ut dictum est, dicemus quod quadratum 2.m.3. est 4.m.21. & quadratum quod plus est 3.m.2. est 9.m.16. vere’.
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• Cardano . 1663 . Opera omnia Edited by: Sponii , Caroli . Vol. 4 , 435 – 439 . Lyon eontinued: ‘Tertia dubitatio est quoniam nesciamus quae quantitates verae sint quae tot miracula faciunt, nec ipse ausus est explicare nec reddere rationem huius rei’
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• Loria . 1929 . Storia delle matematiche , 1st ed. 299 – 299 . Turin repr. Turin, 1931 R. Bombelli—whose work crossed the attention of Cardano, now decrepit, as is attested by the short memoir … Sermo de plus et minus, which can be read in his Complete Works; memoir of a philosophical character …’.
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• Loria . 1929 . Storia delle matematiche , 1st ed. 40 – 41 . Turin repr. Turin, 1931
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• See Hutton Tracts on mathematical and philosophical subjects London 1812 2 219 224 3vols and text
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• See Cossali Origine, trasporto in Italis, primi progressi in essa dell'algebra Parma 1787–1799 2 vols
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• See Loria Storia delle matematiche , 1st ed. Turin 1929 40 41 repr. Turin, 1931 Cantor (footnote 7), 540–541 says of the Sermo only that it was written under the influence of Bombelli.
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• The list is in the autobiographical De vita propria liber, often separately reprinted after its appearance in the Opera (footnote 8), and translated by Jean Stoner as The book of my life New York 1930 repr. New York, 1962), where the list of manuscript mathematical works is pp. 222–224 of chapter 45, ‘Books written by me …’ (pp. 220–237). The heading of chapter 4 gives the end of October 1575 as the date
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