Abstract
The aim of this chapter is to clarify what is meant by the “invention of complex numbers“ by the Renaissance Italian algebraists Girolamo Cardano and Rafael Bombelli.
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Notes
- 1.
After the editio princeps printed in Nuremberg in 1545, Cardano published a new edition of the Ars magna in 1570, the same version (filled by many misprints) published in the fourth volume of the Opera omnia, edited in 1663 by Charles Spon. For the references, we have used, quite freely, the English translations by T. R. Witmer Cardanus (1968).
- 2.
On the Practica Arithmetica and the abacus tradition, see Gavagna (2010).
- 3.
In fact, he wrote in the Summa: “But of number, thing and cube together being composed … it was not possible to find general rules … except sometimes gropingly for some particular cases … the art has not yet shown them as there are no ways to square the circle” (“Ma de numero, cosa e cubo tra loro stando composti … non se possuto finora troppo bene trovar regole generali … se non ale volte a tastoni in qualche caso particulare … larte ancora a tal caso non a dato modo si commo ancora non e dato modo al quadrare del cerchio” (Pacioli 1994, c. 150r)).
- 4.
The history of the solution formula for cubic equations and the challenge between Cardano and Tartaglia before, and between Ludovico Ferrari and Tartaglia next, is one of the best known in the history of mathematics and for this reason we will not enter into details. However, for a slightly different reconstruction, based on a new reading of the extant documents, see Gavagna (2012).
- 5.
The manuscripts Fond.Princ.II.V.152 and Conv.Sop.G.7.1137 kept by the National Library of Florence, presumably written in Florence in the last decade of the fourteenth century, have preserved some examples in which cubic equations without a linear term are transformed by a linear replacement, to cubic equations without a quadratic term, which are then solved gropingly (“a tastoni”). Nothing is known about the distribution of such results in the abacus environments. On this question, see Franci (1985). For a transcription of the algebraic section of the ms. Fond.Princ.II.V.152 we refer to Franci and Pancanti (1988).
- 6.
The device of the rhyme is less bizarre than it could seem at first sight. Tartaglia was an abacus teacher and it was a common practice to expect from the students the memorization of the most important procedures via acronyms or rhyming verses. Cardano himself, in Chapter V of the Ars Magna, offers three carmina for solving quadratic equations: “Querna da bis, Nuquer admi, Requan minue dami” (Cardanus 1545, ff. 10v–11v). With regards to mathematics and poetry in the Renaissance, see Saiber (2014).
- 7.
“Quando chel cubo con le cose appresso/Se agguaglia a qualche numero discreto/Trovan due altri differenti in esso/Ch’el lor produtto sempre sia eguale/Al terzo cubo delle cose netto/El residuo poi suo generale/Delli lor lati cubi ben sottratti/Varra la tua cosa principale” (Tartaglia 1546, f. 123r.).
- 8.
“In el secondo de cotesti atti/Quando che’l cubo restasse lui solo/Tu osservarai quest’altri contratti/Del numero farai due tal part’à volo/Che l’una in l’altra si produca schietto/El terzo cubo delle cose in stolo/Delle qual poi, per comun precetto/Terrai li lati cubi insieme gionti/Et cotal somma sara il tuo concetto” (Tartaglia 1546, ff. 123r–123v).
- 9.
Tartaglia was in fact completely aware that the sum of the roots (with opposite sign) and their product are respectively equal to the coefficients of the linear term and the constant term.
- 10.
“El terzo poi de questi nostri conti/Se solve col secondo se ben guardi/Che per natura son quasi congionti/Questi trovai, et non con passi tardi/Nel mille cinquecenté quatro e trenta/Con fondamenti ben sald’é gagliardi/Nella citta dal mare intorno centa”. I would like to thank Arielle Saiber for providing me with this translation, that will appear in Saiber (2014).
- 11.
On August 4, 1539 Cardano wrote to Tartaglia: “I have asked you for the answer to several questions you have never answered, e.g. the one on the cube equal to things and number … when the cube of the third part of the things exceeds the square of the half of the number, then I cannot make them follow the equation as it appears” (“io ve ho mandato a domandare la resolutione de diversi quesiti alli quali non mi haveti risposto, et tra li altri quello di cubo equale a cose e numero … quando che il cubo della terza parte delle cose eccede il quadrato della mita del numero, allora non posso farli seguir la equatione come appare” (Tartaglia 1546, ff.125v–126r)); on August 7 Tartaglia replied “And therefore I reply, and say, that you have not used a good method for solving such a case; also I say that such proceeding of yours is entirely false” (“E pertanto ve rispondo, et dico che voi non haveti appresa la buona via per risolvere tal capitolo; anci dico che tal vostro procedere è in tutto falso” (Tartaglia 1546, ff.126r–127r)).
- 12.
In the Ars Magna, Cardano also displayed out the solution formula of fourth degree equations, giving the credit to his pupil Ludovico Ferrari . Ferrari’s procedure reduced the solution of the equation to that of a third degree resolvent and it is therefore evident that in this context the irreducible case had to be managed.
- 13.
It is one of the three canonical quadratic equations \(x^{2} = \mathit{ax} + b\), \(x^{2} + \mathit{ax} = b\) and \(x^{2} + b = \mathit{ax}\) where a, b > 0 whose solution formulas, even if in rhetorical form, only provided positive roots.
- 14.
The terms binomial (binomium) and residual (recisum) or apotome are taken from the Latin translations of Book X of the Elements, devoted to the classification of the quadratic irrationals. They respectively denote expressions of the form \(a + \sqrt{b}\) and \(a -\sqrt{b}\) or, more generally, a + x and a − x, where the terms involved have different natures (“quantitas quae additur vel detrahitur, non est eiusdem naturae cum prima”).
- 15.
The three most important editions are dated back to 1550, 1554 and 1560. For a critical edition of the first seven books, compare Nenci (2004).
- 16.
The chapter It is shown how any proposition of Euclid’s Elements can be proved without a change of opening of the compass (Quomodo quaecumque in Elementis Euclidis demonstrata sunt absque ulla propositi unus tantum circuli mutatione ostendi possint) from Book XV of the De subtilitate essentially represents the Latin translation of the answer to Tartaglia, published by Ludovico Ferrari in the Fifth of the Cartelli di matematica disfida Masotti (1974). In fact, Tartaglia challenged Cardano and Ferrari to prove some Euclidean propositions by using, in addition to the ruler, a fixed opening compass, by replacing the third Euclidean Postulate, which allows one to describe a circle with any centre and any distance, with the possibility of describing a circle with any centre, but fixed radius. In the Quinto Cartello, Ferrari claimed to be able to prove all the Euclidean propositions with a fixed aperture compass and not only those indicated by Tartaglia. In the De subtilitate, Cardano recounted that he had re-proved all the Euclidean Elementa with Ferrari in very few days (“paucis in diebus”), by using a ruler and a compass with fixed aperture, but he did not mention any of the querelle with Tartaglia.
- 17.
The first, in fact, proved that “any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone” (Mascheroni Theorem), while the latter came to demonstrate that “all Euclidean geometric constructions can be carried out with a straightedge alone, if given a single circle and its centre in addition” (this result is known as “Poncelet-Steiner Theorem” and was definitely proved in 1833).
- 18.
On the topic, see, in particular Malet (2006).
- 19.
“Subiectum Arithmeticae numerus est integer, per analogiam quatuor subiecta sunt: videlicet numerus integer ut 3, fractus ut \(\frac{3} {7}\), surdus ut Radix 7, denominatus ut census tres, quae omnia explicabo” (Cardanus (1539), Caput primum, De subiectis arithmetice, Italics mine).
- 20.
The only printed edition of the Ars magna arithmeticae is the one contained in the fourth volume of the Opera omnia published by Charles Spon in Lyon in 1633. On the role of the work in the development of Cardano’s mathematics, see Gavagna (2012). The work consists of 40 chapters and 40 problems; the 38th problem also deals with the equation \(x^{2} + 16 = 6x\) and, concerning the negative discriminant, he observes: “Note that \(\sqrt{9}\) is either +3 or −3, for a plus or a minus times a minus yields a plus. Therefore \(\sqrt{-9}\) is neither +3 nor −3 but is some recondite third sort of thing” (“Et nota quod R. p̃ 9 est 3 p̃ vel 3 m̃ nam p̃ & m̃ in m̃ faciunt p̃. Igitur R. m̃ 9 non est p̃ 3 nec m̃ sed quaedam tertia natura abscondita” (Cardanus 1663a, p. 373)).
- 21.
Aliza, or aluza, is a mispronunciation based on Byzantine pronunciation ɑʅɪʤʝɑ from the Greek ἀ-λυθει̃α, composed by privative α and the aorist passive singular feminine participle of λὐω, loose, solve. I wish to thank Paolo d’Alessandro, who kindly provided this information to me. Up until now, the most complete studies of the De regula aliza are due to Cossali (1996) and Confalonieri (2013). For an excursus of the various approaches to the irreducible case, also see Gatto (1992).
- 22.
For a detailed analysis of this topic, see Tanner (1980).
- 23.
“Et ideo patet communis error dicentium, quod m̃ in m̃ producit p̃ neque enim magis m̃ in m̃ producit p̃ neque enim magis m̃ in m̃ producit p̃ quam p̃ in p̃ producat m̃. Et quia nos ubique diximus contrarium, ideo docebo causam huius, quare in operatione m̃ in m̃ videatur producere p̃ et quomodo debeat intelligi” (Cardanus 1570a, p. 44).
- 24.
“If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments”.
- 25.
“If a straight line is cut at random, then the sum of the square on the whole and that on one of the segments equals twice the rectangle contained by the whole and the said segment plus the square on the remaining segment”.
- 26.
“Ideo in recisis necesse est operari per septimam propositionem secundi Euclidis loco quartae: & ita quia in illa includitur additio illa quadrati m̃ in multiplicatione unius in partis integrae, in partem dectractam bis supra gnomonem, ideo oportet addere ad p̃ quantum est quadratum partis illius quae est m̃. Ideo ut in binomiis operamur per quartam propositionem, & secundum substantiam quantitatis compositae, ita etiam in recisis quo ad substantiam & vere operamur cum eadem: sed ad nominum cognitionem operamur in virtute septima eiusdem” (Cardanus 1570a, p. 400).
- 27.
It is worth mentioning that the first printed edition of Diophantus’ Arithmetica was published in 1575 in Basel by Xylander.
- 28.
The most recent edition is Bortolotti and Forti (1966). The drafting of Books IV and V found by Bortolotti, however, shows a text still imperfect, certainly not ready for publication.
- 29.
In the first chapter Diffinitione del numero quadrato, Bombelli indirectly evokes Euclid, explaining that “even if the unit is not a number, in the operations it is useful like numbers” (“se bene l’unità non è numero, pur nelle operationi serve come li numeri”). On the concept of numbers according to Bombelli , also see Wagner (2010).
- 30.
For example, the square root is defined in this way: “The square root is the side of a non-square number; it is impossible to be denominated: however, it is denoted as surd Root or indiscreet, as it would be if one has to take the square of side 20, which does not mean anything else than finding a number that, multiplied by itself, would give 20; which is impossible to be found, 20 being a non-square number” (“La Radice quadrata è il lato di un numero non quadrato; il quale è impossibile poterlo nominare: però si chiama Radice sorda, overo indiscreta, come sarebbe se si havesse a pigliare il lato di 20, il che non vuol dire altro, che trovare un numero, il quale moltiplicato in se stesso faccia 20; il ch’è impossibile trovare, per essere il 20 numero non quadrato” (Bombelli 1572, pp. 3–4).). The definitions of n-th root (n = 3, 4, 5) that follow are quite similar.
- 31.
There are only few authors presenting the geometric construction of the cube root of a given segment, e.g. Fibonacci, Pacioli and Tartaglia. On this topic see Rivolo and Simi (1998).
- 32.
On this aspect see, for example, Giusti (1992).
- 33.
“If in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the straight line so drawn is a mean proportional between the segments of the base”.
- 34.
As noticed by Ettore Bortolotti: “This chapter should rather be entitled: Proof of how it is necessary to put \(-\cdot - = +\) so that the distributive property of the product remains valid” (Bortolotti and Forti 1966, p. 77, n.30).
- 35.
“… un’altra sorte di Radici cubiche legate, molto differenti dall’altre, la qual nasce dal capitolo di cubo eguale a tanti e numero, quando il cubato del terzo delli tanti è maggiore del quadrato della metà del numero” (Bombelli 1572, p. 133). For an analysis of this remark and the irreducible case in Bombelli, compare La Nave and Mazur (2002), Kenney (1989).
- 36.
The very common interpretation of this rule in terms of the imaginary unit i makes the comprehension easier for the modern reader (and this is the reason why I write it in the square brackets), but it is, as we will see, a forcing of Bombelli ’s way of thinking. It is also worth underlining that the symbol i was not introduced until the end of the eighteenth century. “La qual sorte di Radici quadrate ha nel suo Algorismo diversa operatione dall’altre e diverso nome; perché quando il cubato del terzo delli tanti è maggiore del quadrato della metà del numero, lo eccesso loro non si può chiamare né più né meno, però lo chiamarò più di meno quando egli si dovrà aggiongere, e quando si doverà cavare lo chiamerò men di meno, e questa operatione è necessarissijma più che l’altre Radici cubiche legate per rispetto delli capitoli di potenze di potenze, accompagnati con li cubi, o tanti, o con tutti due insieme, ché molto più sono li casi dell’agguagliare dove ne nasce questa sorte di Radici che quelli dove nasce l’altra, la quale parerà a molto più tosto sofistica che reale, e tale opinione ho tenuto anch’io, sin che ho trovato la sua dimostratione in linee […] e prima trattarò del moltiplicare, ponendo la regola del più e del meno: Più via più di meno, fa più di meno; Meno via più di meno, fa meno di meno; Più via meno di meno, fa meno di meno; Meno via meno di meno, fa più di meno; Più di meno via più di meno, fa meno; Più di meno via men di meno, fa più; Meno di meno via più di meno, fa più; Meno di meno via men di meno, fa meno” (Bombelli 1572, p. 169).
- 37.
See the paragraph Modo di trovare il lato cubico di simil qualità di radici (Bombelli 1572, from p. 180 on). The same problem appeared even when the unknown was expressed as the sum or difference of the usual linked cube roots of the form \(\root{3}\of{a \pm \sqrt{b}}\). Both in the Ars magna and in the Algebra there are procedures aimed at rationalizing these expressions in particular cases.
- 38.
Applying this method, Bombelli obtains, for instance, \(\root{3}\of{2 \pm \sqrt{-121}} = 2 \pm \sqrt{-1}\), \(\root{3}\of{52 \pm \sqrt{-2209}} = 4 \pm \sqrt{-1}\). The first case deals with the cube roots obtained by the Cardano formula for the equation \(x^{2} = 15x + 4\): by summing \(2 + \sqrt{-1}\) and \(2 -\sqrt{-1}\) the true solution 4 is obtained. Compare (Bortolotti and Forti 1966, pp. 180–5).
- 39.
In the Ars Magna, Cardano attributes the solution formula to Scipione Del Ferro and Tartaglia, but strongly lays claim to the geometrical proof of the formula, which rigorously legitimizes the validity of the arithmetic algorithm.
- 40.
“Et hoc nos docet facere Eutocius Ascalonita in secundum de Sphaera et Cylindro bifariam, sed sufficiat adduxisse primam illius demonstrationem” (Cardanus 1570a, p. 25).
- 41.
- 42.
… e perché si sa che a trovare le due medie proportionali fra due linee date non ci è via reale, ma si opera a tentoni (come si è mostrato nella estrattione delle Radici cubiche in linea) però non si deve tenere questa dimostratione di poco valore per havere ad alzare et abbassare lo squadro.g. tanto che la.bc. sia pari alla.hm. perché dove intervengono corpi non si può fare altrimente.” (Bombelli 1572, pp. 287–288).
- 43.
The examined case is \(x^{3} = 12x + 9\); add 27 to both sides so that they are both divisible by x + 3.
- 44.
“Piglisi il terzo delli Tanti, ch’é 5, cubisi fa 125 e questo si cavi del quadrato della metà del numero, ch’è 4, resta − 121. Il qual si chiamerà più di meno che di questo pigliata la Radice quadrata sarà + di − 11, che pigliatone il lato cubico ed aggionto col suo residuo fa 2 + di − 1 et 2 − di − 1, che gionti insieme fanno 4 e 4 è la valuta del Tanto. Et benché a molti parerà questa cosa stravagante, perché di questa opinione fui ancho già un tempo, parendomi più tosto fosse sofistica che vera, nondimeno tanto cercai che trovai la dimostratione, la quale sarà qui sotto notata, sì che questa ancora si può mostrare in linea, che pur nelle operationi serve senza difficultade alcuna, et assai volte si trova la valuta del Tanto per numero (come si è trovato in questo esempio). (Bombelli 1572, pp. 293–294).
- 45.
“this decomposition cannot be made in the mentioned way, but since it did not seem general to me, I investigated until I found a very general proof in a plane surface; since where bodies are considered mean proportional lines cannot be found if not by instruments; nobody should be surprised if this proof shows the same difficulty and if there was not such a difficulty, the invention of Plato and Archita from Taranto together with many other talented scientists about the duplication of the altar, that is a cube, would have been vain, (as widely treated by Barbaro in his Commentary on Vitruvius). Having the shield of so many talented men I will not strive to support the fact that the proof could not be carried out without involving the instrument [the sliding square]” (“tal agguagliatione non si potrà fare con detto taglio, però non parendo tale agguagliatione generale sono andato tanto investigando che ho trovato una dimostratione in superficie piana generalissima, ma perché dove intervengono li corpi le linee medie non si possono ritrovare se non per via d’instromento, però non paia ad alcuno strano se questa dimostratione haverà la medesima difficultà, che quando non l’havesse saria stata vana la inventione di Platone ed Archita Tarentino con tanti altri valent’huomini nel voler duplare l’altare, overo Cubo (come largamente ne ha parlato il Barbaro nel Comento del suo Vitruvio), però havendo lo scudo di tanti valent’huomini non mi affaticarò in volere sostentar tal dimostratione non di potere far altramente che con l’instromento”) (Bombelli 1572, p. 297).
- 46.
Given the triangle mgi, we deduce from the proportion ml: li = li: lg, with ml = 1 and li = x, that lg = x 2 and the area of the rectangle rlg is equal to x 3. Since the two rectangles abfl and fgh are equal, the rectangle rlg is also equal to 6x + 4.
- 47.
In particular, the example offered by Bombelli is \(\root{3}\of{4 + \sqrt{-11}} + \root{3}\of{4 -\sqrt{-11}}\) and the equation to which it belongs is \(x^{3} = 9x + 8\).
- 48.
“Propositio sexagesimasexta. Proportionem laterum eptagoni et subtensarum considerare et quae a reflexa proportione pendent” (Cardanus 1570b, pp. 55–56).
- 49.
The construction of the regular heptagon is one of the themes on which Ferrari and Tartaglia challenged each other in the Cartelli Masotti (1974): Tartaglia failed to propose a solution in the context of the challenge, but published the construction of the heptagon in (Tartaglia 1557, Part IV, Book I, c. 17). In Chap. XVI of the De subtilitate, especially since the edition of 1554, and in the De proportionibus, Cardano handles problems related to the construction of the regular heptagon. On this topic, see Field (1994) and Gavagna (2003).
- 50.
The contents of the Commentaria in Euclidis Elementa (Par. Lat. 7217, Bibliothque Nationale de France, Paris) and the meaning in Cardano’s mathematical work are discussed in Gavagna (2003a).
- 51.
“It is the circle abcdef, whose diameter be is \(\sqrt{192}\), inside which I would like to inscribe a regular nonagon; I was wondering about the length of one of his sides” (“Egli è il circulo abcdef, che ’l diametro be è \(\sqrt{192}\) dentro del quale vorrei fare un nove faccie di lati eguali; addimandasi quanto sarà uno de detti lati” (Bortolotti and Forti 1966, pp. 639–641)).
- 52.
“Questa dimanda sino ad hora la tengo impossibile, fino a tanto che non sia retrovato il modo generale di agguagliare il capitolo di cubo et numero eguale a cose et dato che detto Capitolo ancor si ritrovi, dificil cosa sarà che in detto agguagliamento non intravenga qualche Radice cuba, che darebbe inditio, che potendosi formare detto nove faccie, non si potrebbe fare se non per via instrumentale, benché da Horontio et Alberto Duro siano state date regole da fare detto nove faccie, le quali sono falsissime: et per essere cose chiare non mi affaticarò in volerle dimostrare.” Bombelli (Bortolotti and Forti 1966, pp. 639–641).
- 53.
The historical development of this problem is described in Stedall (2011).
- 54.
“Notice that this kind of root cannot be obtained if not together with its conjugate” (“Si deve avertire che tal sorte di Radici legate non possono intravenire se non accompagnato il Binomio col suo Residuo”).
- 55.
“Toutes les equations d’algebre reçoivent autant des solutions que la denomination de la plus haute quantité le demontre excepté les incomplettes” (Girard 1629, p. 45).
- 56.
The translation is not literal; the original passage is “Donc il se faut resouvenir d’observer tousjours cela: on pourroit dire à quoy sert ces solutions qui sont impossibles, je respond pour trois choses, pour la certitude de la reigle generale, & qu’il ny a point d’autre solutions, & pour son utilité” (Girard 1629, p. 47).
- 57.
“We should always choose with care the simplest curve that can be used in the solution of a problem, but it should be noted that the simplest means not merely the one most easily described, nor the one that leads to the easiest demonstration or contruction of the problem, but rather the one of the simplest class that can be used to determine the required quantity” (“Il faut avoir soin de choisir tousiours la plus simple, par laquelle il soit possible de le resoudre. Et mesme il est a remarquer, que par les plus simples on ne doit pas seulement entendre celles qui peuvent le plus aysement estre descrites, ny celles qui rendent la construction, ou la demonstration du Probleme proposé plus facile, mais principalement celles qui sont du plus simple genre, qui puisse servir a determiner la quantité qui est cherchée”) (Smith and Latham 1954, pp. 152–155). See also Bos (1990) and Bos (2001).
- 58.
“il faut que ie die quelque chose en general de la Nature des equations” (Smith and Latham 1954, pp. 156–157).
- 59.
“Scachés donc qu’en chasque Equation, autant que la quantité inconnue a de dimensions, autant peut il y avoir de diverses racines, c’est a dire de valeurs de cete quantité”. (Smith and Latham 1954, pp. 158–159).
- 60.
A similarly weak formulation was already contained in the Arithmetica Philosophica (1608) by Peter Roth, who instead excluded the imaginary roots from the sets of the acceptable ones and did not correctly compute the multiple ones. On the possible influence of the Arithmetica Philosophica on Descartes , compare Manders (2006).
- 61.
“Au reste tant le vrayes racines que les fausses ne sont pas toujours reelles, mais quelquefois imaginairea, c’est a dire qu’on peut bien tousiours en imaginer autant que iay dit en chasque Equation, mais qu’il n’y a quelquefois aucune quantité, qui corresponde a celles qu’on imagine, comme encore qu’on puisse imaginaire trois en celle cy \(x^{3} - 6\mathit{xx} + 13x - 10 = 0\), il n’y en a toutefois qu’une reelle, qui est 2, & pour les deux autres, quoy qu’on les augmente, ou diminue, ou multiplie en la façon que ie viens d’expliquer, on ne sçauroit les rendre autres qu’imaginaires.” (Smith and Latham 1954, pp. 174–175).
- 62.
On the concept of “mathematical object” and its birth, see (Giusti 1999, pp. 87–93), where a chapter is entirely devoted to complex numbers.
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Acknowledgements
I would like to express my gratitude to Jackie Stedall, for competent remarks and generous help in translation, and to Nicole Jones, Arielle Saiber and John Stillwell for the careful reading of this essay. I would like to thank also Francesca Gallori of the National Central Library of Florence for her precious help and kindness.
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Gavagna, V. (2014). Radices Sophisticae, Racines Imaginaires: The Origins of Complex Numbers in the Late Renaissance. In: Lupacchini, R., Angelini, A. (eds) The Art of Science. Springer, Cham. https://doi.org/10.1007/978-3-319-02111-9_7
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