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The casus irreducibilis in Cardano’s Ars Magna and De Regula Aliza

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Abstract

In Cardano’s classification in the Ars Magna (1545, 1570), the cubic equations were arranged in thirteen families. This paper examines the well-known solution methods for the families \(x^3 + a_1x = a_0\) and \(x^3 = a_1x + a_0\) and then considers thoroughly the systematic interconnections between these two families and the remaining ones and provides a diagram to visualize the results clearly. In the analysis of these solution methods, we pay particular attention to the appearance of the square roots of negative numbers even when all the solutions are real—the so-called casus irreducibilis. The structure that comes to light enables us to fully appreciate the impact that the difficulty entailed by the casus irreducibilis had on Cardano’s construction in the Ars Magna. Cardano tried to patch matters first in the Ars Magna itself and then in the De Regula Aliza (1570). We sketch the former briefly and analyze the latter in detail because Cardano considered it the ultimate solution. In particular, we examine one widespread technique that is based on what I have called splittings.

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Notes

  1. Namely, they are the Practica Arithmeticae (1539), the Ars Magna Arithmeticae (1663), the Ars Magna (1545, 1570), and the De Regula Aliza (1570).

    Concerning in particular the Ars Magna Arithmeticae, we remark that the only edition that we have available is the one posthumously printed in Cardano’s Opera Omnia. For a long time, it has been considered as a minor work and in any case as a late work (see Bortolotti 1926; Loria 1950, p. 298). Cardano himself did not mention this treatise in his autobiographies and never sent it to be printed. Moreover, the (supposedly chronological) position of the published version in the framework of the 1663 Opera Omnia contributed to the misunderstanding because the Ars Magna Arithmeticae was wrongly placed after the Ars Magna. Thanks to some recent reappraisals of Cardano’s mathematical writings (see Tamborini 2003, pp. 177–179; Gavagna 2012), it is nowadays commonly agreed that the Ars Magna Arithmeticae was conceived before—or at least at the same time as—the Ars Magna.

  2. I have deliberately chosen to expound Cardano’s mathematics using some terminology and symbols that he did not employ. Of course, this betrays to a certain extent the text and can lead to over-interpreting it. However, I believe that the advantages for the reader in terms of grasping the contents precisely are worth the risk. For the same reason and for the sake of brevity, I use certain notations, especially the discriminant \({\varDelta }_3\), that refer to Cardano’s equations. In this case, the discriminant is not to be understood in the modern sense, that is, as a symmetric function of the roots of the corresponding polynomial. Rather it is a shorthand for the number that is under the square root in Cardano’s formulae (even though this number has the same value up to a constant as the symmetric function evaluated at an appropriate point).

  3. Here and there in Cardano’s mathematical treatises, we also find equations of higher degrees. The prime degrees of the unknown greater than three that Cardano mentioned are the ‘first related (relatus primus)’ for degree five and the ‘second related (relatus secundus)’ for degree seven. The non-prime degrees are obtained by multiplication of prime factors, such as for instance the ‘square of a square (quadrati quadratus)’ for degree four, the ‘cube of a square (quadrati cubus)’ for degree six, the ‘square of a square of a square (quadrati quadrati quadratus)’ for degree eight, the ‘cube of a cube (cubi cubus)’ for degree nine.

  4. Interestingly, in Ars Magna, Chapter XXXVII Cardano gave a justification of the fact that one is also entitled to take into account negative solutions. Indeed, a negative solution of a certain family of equations is admissible as long as it corresponds to a positive solution of another family of equations where some of the coefficients are opposite in sign. For instance, this happens for \(x^2 = a_1x + a_0\) and \(y^2 + a_1y = a_0\), see Cardano (1545, Chapter XXXVII, rule I, p. 65v) but also for \(x^3 + a_0 = a_1x\) and \(y^3 = a_1y + a_0\), and for \(x^3 + a_2x^2 = a_0\) and \(y^3 + a_0 = a_2y^2\) (see Cardano 1545, Chapter I, Paragraphs 5–6 and 8, pp. 4r–4v and 5r–5v).

  5. Sometimes, Cardano wrote an equation such as \(x^3 = a_2x^2 + a_0\) in the form \(x^2 (x - a_2) = a_0\). Despite the fact that a minus sign appears, we observe that this does not mean that Cardano allowed negative numbers in an equation since \(a_0 > 0\) implies that \(x - a_2 > 0\) (but it would have been so if he had written \(x^3 - a_2x^2 = a_0\), which he never did).

  6. We remark that a classification in the same vein can already be found, for instance, in al-Khayyām’s algebra.

  7. The full title is Hieronymi Cardani, praestantissimi mathematici, philosophi, ac medici, artis magnae sive de regulis algebraicis, liber unus, qui et [sic] totius operis de arithmetica, quod opus perfectum inscripsit, est in ordine decimus, see Cardano (1545). This first edition was published in Nuremberg in 1545 by Iohannes Petreium. The second edition was published in Basel in 1570 by Oficina Henricpetrina and was a joint volume together with the De Proportionibus and the De Regula Aliza. There is, moreover, a third posthumous edition in the Opera Omnia. The second edition contains some additions, while the third edition is virtually the same as the second one. We also have the available manuscript Plimpton 510/1700: s.a., which is entitled L’algebra, at the Columbia University Library in New York.

    According to the title, the Ars Magna should have filled up the tenth volume of a certain Opus Arithmeticae Perfectum, which was never realized. This should have been an encyclopedic work on mathematics composed of fourteen books and was probably conceived between the 1530s and the 1560s. For an extensive explanation of this reference, see Gavagna (2010, p. 65) and Cardano (2004, pp. 64–66).

  8. Whether, and if so to what extent, these proofs contain geometrical arguments is up to discussion.

  9. These solution methods are indeed the most taken into account in the secondary literature: see, among others, Cantor (1892), Bortolotti (1926), Boyer (1991), Bashmakova and Smirnova (2000) and Maracchia (2005). For a detailed account of these two solution methods as well as all the others, see also Confalonieri (2013, pp. 103–151).

  10. In other words, I employ the term ‘geometrical’ only on a superficial linguistic level, and I do not associate to it a precise interpretation of what geometry was for Cardano. Note in particular that the fact that Cardano spoke about points and other such objects does not necessarily mean that he also employed the positional properties of these objects. Concerning the general relation between geometry and positional arguments, see Panza (2007).

  11. Historically speaking, it seems very likely that the idea came straight from Tartaglia’s poem (up to a change of variables). Indeed, its first two verses describe the substitutions and the systems:

    When the cube with the things next to it

    • is made equal to some other discrete number

    • find two others the difference of which [is] this.

    Hereafter, you will consider this customarily

    • that their product always will be exactly equal

    • to the third of the cube of the things.

    Its general remainder then

    • of their cube sides appropriately subtracted

    • will be the value of your principal unknown.

    In the second of these acts

    • when the cube remains alone,

    • you will observe these other arrangements,

    you will immediately make of the number two such parts

    • that the one times the other will produce straightforward

    • the third of the cube of the things

    of which then by common precept

    • you will take the cube sides joined together

    • and this sum will be your concept.

    (Quando che’l cubo con le cose appresso\se agguaglia à qualche numero discreto\trovan dui altri differenti in esso.\Da poi terrai questo per consueto\che ’l lor produtto sempre sia eguale\al terzo cubo delle cose neto.\El residuo poi suo generale\delli lor lati cubi ben sottratti\varra la tua cosa principale.\In el secondo de codesti atti\quandi che ’l cubo restasse lui solo\tu osservarai quest’altri contratti,\del numero farai due tal part’àl volo\che l’una in l’altra si produca schietto\el terzo cubo delle cose in stolo\delle qual poi, per comun precetto\torrai li lati cubi insieme gionti\e tal somma sara il tuo concetto), see Tartaglia (1959, Quesito XXXIII, p. 119). Nevertheless, this only displaces the difficulty to finding out how Tartaglia (or whoever else) got the key idea.

  12. We do not know exactly when this term came into use. The first occurrence of which I am aware is in Lagny (1697). I thank David Rabouin for this reference.

  13. This also happened in the case of quadratic equations. Considering \(x^2 + a_0 = a_1x\) in Chapter V, Cardano asked for a condition on the coefficients (namely, \(\left( \frac{a_1}{2}\right) ^2 - a_0 > 0\)) so that negative numbers never appear under square roots. More precisely, the condition is “[i]f that subtraction of the number from the square of the half of the number of the things cannot be done, the question itself is false and what has been proposed cannot be (“[q]uod si detractio ipsa numeri a quadrato dimidii numeri rerum fieri nequit, quæstio ipsa est falsa, necesse potest quod proponitu)”, see Cardano (1545, Chapter V, p. 11v).

    Nevertheless, there is an isolated passage in the Ars Magna where Cardano tried to justify the square roots of negative numbers. This is in Chapter XXXVII, while discussing “the rules for assuming a false (regula falsum ponendi)” and in particular in the proof of Rule II (see Cardano 1545, p. 66r). This is about solving a certain system equivalent to a quadratic equation with two complex, nonreal solutions. Cardano used the quadratic formula to find their (correct) value (namely, “5 p: ℞ m: 5 and 5 m: ℞ m: 5”, that is, \(5 \pm \sqrt{-15}\)), and then provided a proof, which is in short a geometrical interpretation of the quadratic formula. There, he had a try at linking the expression \(\sqrt{-15}\) to a geometrical object, namely to the difference between a surface and a segment (where the surface is called \(AD\) and measures \(25\) and the segment \(AB\) that measures \(10\) is taken four times: “imaginaberis ℞ m: 15, id est differentiae AD et quadrupli AB”). A few lines later, Cardano commented that this is “sophisticated” since one cannot carry it out with the same operations with a negative quantity (“quae vere est sophistica, quoniam per eam, non ut in puro m: nec in aliis, operationes exercere licet”). In my opinion, this amounts to one of the earliest investigations of the square roots of negative numbers. At the same time, it also shows how far Cardano was from being acquainted with these kinds of numbers. It is in fact significant that this chapter comes long after the ones that deliver the solution methods for quadratic and cubic equations and that it was a marginal comment.

  14. See Cardano (1545, Chapter XII, p. 31v). The condition means that Cardano took into account only the case where

    $$\begin{aligned} \left( \frac{a_0}{2}\right) ^2 > \left( \frac{a_1}{3}\right) ^3, \end{aligned}$$

    which is equivalent (up to a constant) to ask for \({\varDelta }_3 > 0\), see Footnote 2.

  15. From the modern viewpoint, these are not real changes of variable since the maps \(v \mapsto v^2\) and \(v \mapsto \sqrt{v}\) are not bijections. This is sometimes the case in Cardano’s substitutions. Anyway, if we consider that Cardano favored positive numbers, his substitutions are invertible. Indeed, in the examples that he provided, he used the substitutions in both directions, see for instance Footnote 19.

  16. Note that despite the fact that the equation in \(x\) has always \({\varDelta }_3 > 0\), this is not a priori the case for the transformed equation in \(y\). However, if we go and check the discriminant of the transformed equation, we get

    $$\begin{aligned} {\varDelta }_3 = \frac{a_0^2}{4} + \frac{1}{27}a_0a_2^3, \end{aligned}$$

    which is always positive since we took \(a_0, a_2\) positive.

  17. See [Cardano (1545, Chapter XIV, p. 33v). The second ‘dimidio’ in “residui radicem adde et minuo dimidio aggregati quod in se duxeras” is a typo (emended in none of the editions).

  18. See Franci (1985).

  19. See Cardano (1545, Chapter VII, Paragraphs 3–4 and 8–9, pp. 18r–18v and 19v–20r). More precisely, the substitutions are \(x = \frac{\root 3 \of {a_0}}{y}\) and \(x = y^2 + a_0\). As said, Cardano’s substitutions can be considered invertible as long as he only considers the positive roots while taking the \(2n\)-th roots. We remark that the substitution \(x = y^2 + a_0\) is also in Chapter XV, but is applied to a different family of equations. Concerning the first substitution, Cardano provided two examples: \(x^3 + 18x = 8\), which is transformed into \(x^3 = 36x^2 + 8\) (but Cardano wrongly wrote \(x^3 = 9x^2 + 8\)), and \(x^3 = 6x^2 + 16\), which is transformed into \(x^3 = \root 3 \of {3456}x + 16\).

  20. See Footnote 15.

  21. Nevertheless, see Footnote 16.

  22. See Tartaglia (1959, Libro IX, Quesito XL, pp. 125–126).

  23. See above Sect. 3.1.

  24. For a detailed account, see Confalonieri (2013, pp. 84–99). There I guess one possible origin of all these propositions. Let us limit to the first two propositions concerning the family of equations \(x^3 = a_1x + a_0\) since the line of reasoning for the other cases is analogous. If Cardano was aware of Vieta’s formulae

    $$\begin{aligned} {\left\{ \begin{array}{ll} 0 = - x_1 - x_2 - x_3\\ -a_1 = x_1x_2 + x_2x_3 + x_1x_3\\ -a_0 = -x_1x_2x_3, \end{array}\right. } \end{aligned}$$

    he could have rewritten them as

    $$\begin{aligned} {\left\{ \begin{array}{ll} a_1 = F + G^2\\ -a_0 = FG \end{array}\right. } \end{aligned}$$

    with \(G = x_1\) and \(F = x_1x_2 + x_2^2\). Solving the above system is equivalent to solve

    $$\begin{aligned} {\left\{ \begin{array}{ll} F = a_1 - G^2\\ G^3 - a_1G -a_0 = 0, \end{array}\right. } \end{aligned}$$

    which in turn is equivalent to solve the equation \(x^3 = a_1x + a_0\). The solutions are \(x_1 = G\), together with the solutions of \(x_2^2 + Gx_2 - F = 0\). In the end, we get

    $$\begin{aligned} x_1 = G, \quad x_2 = -\frac{G}{2} + \sqrt{\frac{G^2}{4} + F}, \quad \text {and}\quad x_3 = -\frac{G}{2} - \sqrt{\frac{G^2}{4} + F}. \end{aligned}$$

    We remark that if we take \(F = f\) and \(G = -\sqrt{g}\) (of course, from a modern viewpoint, these are not invertible substitutions, see Footnote 15), we obtain the system and the solution in the first proposition (where only \(x_2\) is considered since the other two solutions are always negative), and if we take \(F = -f\) and \(G = g\), we obtain the system and the solution in the second proposition (where only \(x_1\), which is always positive, is considered).

  25. See Confalonieri (2013, pp. 215–220).

  26. We have two editions of the De Regula Aliza available. The first one, printed in 1570 in Basel, is entitled Hieronymi Cardani mediolanensis, civisque bononiensis, medici ac mathematici praeclarissimi, de aliza regula, libellus, hoc est operi perfecti sui sive algebraicae logisticae, numeros recondita numerandi subtilitate, secundum geometricas quantitate inquirendis, necessaria coronis, nunc demum in lucem editæ. As said, it is coupled with the De Proportionibus and the second edition of the Ars Magna. The second edition of the Aliza, the title of which is simply De regula aliza libellus, is in the fourth volume of the 1663 edition of Cardano’s Opera Omnia, edited by Charles Spon almost a century after Cardano’s death. A few miscalculations and typos are emended, and the punctuation is here and there improved, but as a whole, this contains no major changes compared to the first edition. As far as I know [see the website www.cardano.unimi.it (last checked February 14, 2015] associated to the project of the edition of Cardano’s works supported by Consiglio Nazionale delle Ricerche in Italy and by the university of Milan), no manuscript of the Aliza is available.

  27. Over the centuries, the Aliza had a handful of readers. Among the generations of scholars contemporary to Cardano, Commandino (1572, Book X, Proposition 34, Theorem III, p. 149), Stevin (1585, Book II, p. 309), and Thomas Harriot [see Schemmel and Stedall (last checked February 14, 2015], Add. Ms. 6783, folio 121) studied few of its pages. It is also possible (though unlikely) that Rafael Bombelli got in touch with this treatise. During the nineteenth century and until the beginning of the twentieth century, Hutton (1812, pp. 219–224), Cantor (1892, pp. 532–537), and Loria (1950, pp. 298–299) reported on the Aliza from an historical viewpoint. More recently, Tanner (1980), Maracchia (2005, pp. 227, 331–335), and Stedall (2011, p. 10) discussed (parts of) this book.

    As far as I know, the one scholar who studied the Aliza in detail is the Italian mathematician and historian of mathematics Pietro Cossali (Verona 1748, Padua 1815). He was priest in Milan, professor at the university of Padua, member of the Società Italiana delle Scienze, and pensionnaire of the Reale Istituto Italiano di Scienze, Lettere e Arti. Mostly the Aliza, as well as the Ars Magna Arithmeticae, were handled on the same footing as the Ars Magna in the second volume of his history of algebra (see Cossali 1799a) and in the Storia del Caso Irriducibile (see Cossali 1966, which is the commented transcription by Romano Gatto of the unpublished manuscript). Nevertheless, it must be said that for the most of the time, Cossali’s accuracy as historian is not up to modern standards: He was doing mathematics from a historical starting point rather than history of mathematics. Indeed, Cossali was a mathematician, and this could partially explain his attitude.

    In 1781, the Academy of Padua announced a competition to prove either that the cubic formula could be freed from imaginary numbers or the contrary, but in the end, the prize was never assigned. However, this renewed mathematicians’ interest in the casus irreducibilis. Cossali was absorbed by this topic [his very first work in 1799 is devoted to it, and in 1813, he came back to the same issue, see Cossali (1799b) and Cossali (1813)]. He also wanted to take part in the 1781 competition, but he could not finish his contribution on time; he subsequently published it in 1782 (see Cossali 1782).

  28. In chronological order, we have already seen the first reference in Ars Magna, Chapter XII in Sect. 3.1. Unfortunately, the other occurrences step outside the context of cubic equations. The second reference is in the 1554 De subtilitate (and following editions), where Cardano dealt with the “reflexive ratio (proportio reflexa)”, see Cardano (1554, Book XVI, p. 428). The last reference is in the Sermo de plus et minus, which contains a specific mention of the Aliza, Chapter XXII, see Cardano (1663a, p. 435). Thereafter the term is mentioned nowhere else.

  29. In 1799 Pietro Cossali hinted that the term ‘aliza’ means ‘unsolvable’: “De Regula Aliza, cioè De regula Irresolubili”, see Cossali (1799a, volume II, p. 441) or Cossali (1966, Chapter I, Paragraph 2, p. 26). In 1892 Moritz Cantor related that Armin Wittstein suggests that this term comes from a wrong transcription of the Arabic word ‘a’izzâ’ and means ‘difficult to do’, ‘laborious’, ‘arduous’: “Titel De regula Aliza, der durch unrichtige Transkription aus dem arabischen Worte a’izzâ (schwierig anzustellen, mühselig, beschwerlich) entstanden sein kann [Diese Vermutung rührt von H. Armin Wittstein her.], und alsdann Regel der schwierigen Fälle bedeuten würde”, see Cantor (1892, p. 532). Finally in 1929, Gino Loria advanced as a common opinion that the term comes from a certain Arabic word that means ‘difficult’: “il titolo, sinora inesplicato De Regula Aliza (secondo alcuni aliza deriverebbe da una parola araba significante difficile)”, see Loria (1950, p. 298).

  30. By personal communication.

  31. It is very likely that the edition that Cardano had at hand was the one by Jacques Lefèvre d’Étaples printed in Paris in 1516. Indeed, it was explicitly mentioned by Cardano in the short preface to the manuscript Commentaria in Euclidis Elementa (see Gavagna 2003, p. 134). An accurate study to determine whether Cardano followed Campano or rather Zamberti’s interpretation is still lacking.

  32. “[N]ullus liber minus quam ter scriptum est”, see Cardano (1557, p. 78).

  33. See also Footnote 1.

  34. See Cardano (1663b, p. 40).

  35. See Cardano (1544, p. 426). This quotation raises the incidental question of how to know what that “Ars magna” was of which Cardano is speaking because all the editions that we have with that title have only forty chapters. Tamborini (2003, pp. 178–179), Ian Maclean in Cardano (2004, p. 65), and Gavagna (2012) discuss this incongruity, and I refer to them for an accurate discussion.

  36. See Cardano (1544, p. 426).

  37. See Cardano (1998, p. 9v).

  38. See Cardano (1998, p. 9v).

  39. Indeed, “[d]ecimus de regulis magnis, atque ideo ars magna vocatur: atque hic solus ex omnibus editus est” and “[t]ertiusdecimus quaestiones arithmeticas, ut quartusdecimus geometricas”, see Cardano (1557, pp. 37–38) and Cardano (1562, p. 16).

  40. “De proportionibus, et aliza regula addidi anno MDLXVIII ad librum artis magnae et edidi”, see Cardano (1663b, p. 41).

    This agrees with what Ercole Bottrigari (1531–1612) wrote in his La mascara overo della fabbrica de teatri e dello apparato delle scene tragisatirocomiche: “two or three years” before 1570, Bottrigari questioned Cardano about the Aliza and Cardano, avoiding answering, dropped that the Aliza was going to be printed in Germany. The reference can be found in Betti (2009, p. 163), and I thank Veronica Gavagna for it.

  41. For more details, see Confalonieri (2013, p. 235).

  42. This pattern is dealt with directly in Chapters I, II, VII, X, XVIII, XLV, XLVI, XLVIII, LIII, LX, and indirectly in many others.

  43. The first sentence in the description identifies three possible splittings, namely

    $$\begin{aligned} {\left\{ \begin{array}{ll} a_0 = 2y^2z + 2yz^2\\ a_1x = y^3 + y^2z + yz^2 + z^3 \end{array}\right. } \quad \text {or} \quad {\left\{ \begin{array}{ll} a_0 = y^2z + 3yz^2\\ a_1x = y^3 + 2y^2z + z^3 \end{array}\right. } \quad \text {or} \quad {\left\{ \begin{array}{ll} a_0 = 3y^2z + yz^2\\ a_1x = y^3 + 2yz^2 + z^3 \end{array}\right. } \end{aligned}$$

    The second sentence definitely points at the first splitting since only the following proportion holds

    $$\begin{aligned} (y^3 + y^2z + yz^2 + z^3) : (2y^2z + 2yz^2) = y^2 + z^2 : 2yz. \end{aligned}$$

  44. The numerical example that follows this description enables to clarify what a “connected” parallelepiped (to a cube) is. More generally (using the same formalism as above), in the Aliza, a parallelepiped \(y^2z\) is “connected (coherens, proximum)” to the cube \(y^3\), a parallelepiped \(yz^2\) is “opposite (adversum, alternum, altrinsecum, remotum)” to the cube \(y^3\), and the two parallelepipeds \(y^2z\) and \(yz^2\) are “mutual (mutua)”.

  45. We remark that as long as one do not put any further condition on \(y, z\), they are a priori interchangeable in Cardano’s phrasing. For example, the expression ‘one cube’ addresses both \(y^3\) and \(z^3\), or ‘one cube and one opposite parallelepiped’ addresses both \(y^3 + yz^2\) and \(z^3 + y^2z\). A few splittings, namely the ones where the assignation for \(a_0\) (and for \(a_1x\)) are symmetrical in \(y, z\), are defined in a univocal way. This is the case for instance in A I 1—A I 4. For the sake of clarity, I choose in the other cases to explicitly write all the combinations of \(y, z\).

  46. In the margin, Cardano made reference to “the 9th [Proposition] of the second [Book] of Euclid” and to “some dialectic rules ([p]er 9 secundi Elementorum et regula Dialec.)”. Note that both 1570 and 1663 editions have ‘regula’. Since the preposition ‘per’ takes the accusative case, we would have expected to find either ‘regulam’ or ‘regulas’, which is not the case. Therefore, I have chosen to translate this term in the plural to leave the interpretation as loose as possible. The mention of the ‘dialectic rules’ seems to be a blurry reference to some classical logical rules. Cardano also wrote a Dialectica, printed in 1566 in Basel [in the second volume of Cardano (1566)], which is a logic treatise where the rules are sometimes expounded through mathematical examples. Since Elements II.9 (both in Campano’s and Zamberti’s versions) implies that \(y^2 + z^2 \ge \frac{1}{2}x^2\), we are searching for one or more rules that enable to pass from the inequality with the squares to the inequality with the cubes, but I could find none in the Dialectica. It is nevertheless not hard to derive it by hand.

    Note, moreover, that in order to verify the inequality \(a_0 \ge \frac{1}{4}x^3\), we must assume both \(y\) and \(z\) are positive. Indeed, let us suppose for a contradiction that \(y \ge 0\) and \(z \le 0\), for example, \(y = 1\) and \(z = -2\). Then, \(-7 < -\frac{1}{4}\), which contradicts \(y^3 + z^3 \ge \frac{1}{4}x^3\).

  47. The condition yields to the following system

    $$\begin{aligned} {\left\{ \begin{array}{ll} a_0 \ge \frac{1}{4}x^3\\ a_1 \le \frac{3}{4}x^2, \end{array}\right. } \end{aligned}$$

    which in turn entails that

    $$\begin{aligned} {\left\{ \begin{array}{ll} \frac{1}{4} a_0^2 \ge \frac{1}{4} \left( \frac{1}{4}x^3\right) ^2\\ a_1^3 \le \left( \frac{3}{4}x^2\right) ^3 \end{array}\right. }, \end{aligned}$$

    that is \(\frac{1}{27}a_1^3 \le \frac{1}{4^3}x^6 \le \frac{1}{4}a_0^2\). Then, \(\frac{1}{4}a_0^2 - \frac{1}{27}a_1^3 \ge 0\), or \({\varDelta }_3 \ge 0\).

  48. “Giusta il metodo di Tartaglia questa equazione \((y^3 + 3y^2z + 3y\,z^2 + z^3 - p(y + z) - q = 0)\) si spezza nelle due \(y^3 + z^3 = q\), \(3y^2z + 3y\,z^2 = p(y + z)\). È egli di natura sua generale questo spezzamento? È egli l’unico? E qui stendendo Cadano lo sguardo su le varie combinazioni de’ termini, moltissimi gli si presentarono alla mente i supposti possibili a farsi”, see Cossali (1966, Chapter I, Paragraph 3, pp. 27–28).

  49. The 1663 edition has “constituunt”.

  50. Obviously it is not true that—strictly speaking—no condition on \(a_0\) is entailed. Indeed, there is at least the trivial condition \(a_0 \le x^3\) given by the equation \(x^3 = a_1x + a_0\) itself. Therefore, we better account for Cardano’s “no condition” as “no non-trivial condition”.

  51. See Confalonieri (2013, pp. 254–255).

  52. Its correlative ‘aut’ has been omitted and was—in my understanding—originally referred to “inutiles” in the next sentence. Moreover, it was common at that time to mix up ‘vel’ and ‘aut’ especially in listing, which should have happened in the two lists referred to the anomalous and useless splittings.

  53. By the way, according to Cardano’s description, these are all misshaped splittings.

References

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    Sara Confalonieri

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Communicated by: Jeremy Gray.

I would like to thank Veronica Gavagna, Marco Panza, Sabine Rommevaux, and Klaus Volkert for their help.

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Confalonieri, S. The casus irreducibilis in Cardano’s Ars Magna and De Regula Aliza . Arch. Hist. Exact Sci. 69, 257–289 (2015). https://doi.org/10.1007/s00407-015-0149-9

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