A further analysis of Cardano’s main tool in the De Regula Aliza: on the origins of the splittings

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Abstract

In the framework of the De Regula Aliza (1570), Cardano paid much attention to the so-called splittings for the family of equations \(x^3 = a_1x + a_0\); my previous article (Confalonieri in Arch Hist Exact Sci 69:257–289, 2015a) deals at length with them and, especially, with their role in the Ars Magna in relation to the solution methods for cubic equations. Significantly, the method of the splittings in the De Regula Aliza helps to account for how Cardano dealt with equations, which cannot be inferred from his other algebraic treatises. In the present paper, this topic is further developed, the focus now being directed to the origins of the splittings. First, we investigate Cardano’s research in the Ars Magna Arithmeticae on the shapes for irrational solutions of cubic equations with rational coefficients and on the general shapes for the solutions of any cubic equation. It turns out that these inquiries pre-exist Cardano’s research on substitutions and cubic formulae, which will later be the privileged methods for dealing with cubic equations; at an earlier time, Cardano had hoped to gather information on the general case by exploiting analogies with the particular case of irrational solutions. Accordingly, the Ars Magna Arithmeticae is revealed to be truly a treatise on the shapes of solutions of cubic equations. Afterwards, we consider the temporary patch given by Cardano in the Ars Magna to overcome the problem entailed by the casus irreducibilis as it emerges once the complete picture of the solution methods for all families of cubic equations has been outlined. When Cardano had to face the difficulty that appears if one deals with cubic equations using the brand-new methods of substitutions and cubic formulae, he reverted back to the well-known inquiries on the shape of solutions. In this way, the relation between the splittings and the older inquiries on the shape of solutions comes to light; furthermore, this enables the splittings to be dated 1542 or later. The last section of the present paper then expounds the passages from the Aliza that allow us to trace back the origins of the substitution \(x = y + z\), which is fundamental not only to the method of the splittings but also to the discovery of the cubic formulae. In this way, an insight into Cardano’s way of dealing with equations using quadratic irrational numbers and other selected kinds of binomials and trinomials will be provided; moreover, this will display the role of the analysis of the shapes of solutions in the framework of Cardano’s algebraic works.

References

  1. Bortolotti, E. 1926. I contributi del Tartaglia, del Cardano, del Ferrari e della scuola matematica bolognese alla teoria algebrica delle equazioni cubiche. Studi e memorie per la storia dell’Universitá di Bologna 10: 55–108.Google Scholar
  2. Cardano, G. 1539. Heorinymi C. Cardani Medici mediolanensis, Practica arithmetice, et mensurandi singularis. In quaque preter alias continentur, versa pagina demonstrabit. Io. Antonins Castellioneus for Bernardini Calusci, Milan.Google Scholar
  3. Cardano, G. 1545. Hieronymi Cardani, præstantissimi mathematici, philosophi, ac medici, Artis magnae sive de regulis algebraicis, lib. unus. Qui et totius operis de arithmetica, quod Opus perfectum inscripsit, est in ordine decimus. Iohannes Petrius, Nuremberg.Google Scholar
  4. Cardano, G. 1570a. Hieronymi Cardani mediolanensis, civisque bononiensis, philosophi, medici et mathematici clarissimi, Opus novum de proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum, non solum geometrico more stabilitum, sed etiam varijs experimentis et observationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, et in V libros digestum. Prætera Artis magnæ, sive de regulis algebraicis, liber unus, abstrusissimus et inexhaustus plane totius arithmeticæ thesaurus, ab authore recens multis in locis recognitus et auctus. Item De aliza regula liber, hoc est, algebraicæ logisticæ suæ, numeros recondita numerandi subtilitate, secundum geometricas quantitates inquirentis, necessaria coronis, nunc demum in lucem edita, Oficina Henricpetrina, Basel, chap Artis magnae sive de regulis algebraicis, lib. unus. Qui et totius operis de arithmetica, quod Opus perfectum inscripsit, est in ordine decimus. Ars magna, quam volgo cossam vocant, sive regulas algebraicas, per D. Hieronymum Cardanum in quadraginta capitula redacta, et est liber decimus suæ Arithmeticæ.Google Scholar
  5. Cardano, G. 1570b. Hieronymi Cardani mediolanensis, civisque bononiensis, philosophi, medici et mathematici clarissimi, Opus novum de proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum, non solum geometrico more stabilitum, sed etiam varijs experimentis et observationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, et in V libros digestum. Prætera Artis magnæ, sive de regulis algebraicis, liber unus, abstrusissimus et inexhaustus plane totius arithmeticæ thesaurus, ab authore recens multis in locis recognitus et auctus. Item De aliza regula liber, hoc est, algebraicæ logisticæ suæ, numeros recondita numerandi subtilitate, secundum geometricas quantitates inquirentis, necessaria coronis, nunc demum in lucem edita, Oficina Henricpetrina, Basel, chap In Librum de Propotionibus Hieronymi Cardani Mediolanensis, civisque Bononiensis, Medici.Google Scholar
  6. Cardano, G. 1570c. Hieronymi Cardani mediolanensis, civisque bononiensis, philosophi, medici et mathematici clarissimi, Opus novum de proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum, non solum geometrico more stabilitum, sed etiam varijs experimentis et observationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, et in V libros digestum. Prætera Artis magnæ, sive de regulis algebraicis, liber unus, abstrusissimus et inexhaustus plane totius arithmeticæ thesaurus, ab authore recens multis in locis recognitus et auctus. Item De aliza regula liber, hoc est, algebraicæ logisticæ suæ, numeros recondita numerandi subtilitate, secundum geometricas quantitates inquirentis, necessaria coronis, nunc demum in lucem edita, Oficina Henricpetrina, Basel, chap De aliza regula libellus, hoc est Operis perfecti sui sive algebraicæ Logisticæ, numeros recondita numerandi subtilitate, secundum geometricas quantitates inquirenti, necessaria coronis, nunc demum in lucem editæ.Google Scholar
  7. Cardano, G. 1663. Hieronymi Cardani mediolanensis Opera omnia in decem tomos digesta, vol 4, Ioannis Antonii Huguetan and Marci Antonii Ravaudn, Lyon, chap Ars magna arithmeticæ, seu liber quadraginta capitulorum et quadraginta quæstionum. Edited by Charles Spon.Google Scholar
  8. Cardano, G. 1968. Ars magna or the rules of algebra. Dover, edited and translated by T. Richard Witmer.Google Scholar
  9. Confalonieri, S. 2015a. The importance of the casus irreducibilis in the Ars Magna and Cardano’s attempt to put a patch in the De Regula Aliza. Archive for History of Exact Sciences 69: 257–289.MathSciNetCrossRefMATHGoogle Scholar
  10. Confalonieri, S. 2015b. The unattainable attempt to avoid the casus irreducibilis for cubic equations. Gerolamo Cardano’s De Regula Aliza. New York: Springer.MATHGoogle Scholar
  11. Confalonieri, S. 2018. Cardano’s De Regula Aliza. Bollettino di Storia delle Scienze Matematiche Accepted.Google Scholar
  12. Cossali, P. 1799. Origine, trasporto in Italia, primi progressi in essa dell’Algebra. Reale Tipografia Parmense, two volumes.Google Scholar
  13. Cossali, P. 1966. Storia del caso irriducibile. Istituto veneto di scienze, lettere ed arti, edited by Romano Gatto.Google Scholar
  14. Gavagna, V. 2003. Cardano legge Euclide. I Commentaria in Euclidis Elementa. In: Cardano e la tradizione dei saperi. Atti del Convegno internazionale di studi, eds. M Baldi, G Canziani, 125–144. Milano (23–25 maggio 2002), Franco Angeli.Google Scholar
  15. Gavagna. V. 2012. Dalla Practica arithmeticæ all’Ars magna. Lo sviluppo dell’algebra nel pensiero di Cardano. In Pluralité de l’algèbre à la Renaissance, Champion, eds. Maria-Rosa Massa-Esteve, Sabine Rommevaux, and Maryvonne Spiesser, 237–268.Google Scholar
  16. Kichenassamy, S. 2015. Continued proportions and Tartaglia’s solution of cubic equations. Historia Mathematica 42: 407–435.MathSciNetCrossRefMATHGoogle Scholar
  17. Lefèvre d’Étaples, J. 1516. Euclidis megarensis geometricorum elementorum libri XV. Campani galli transalpini in eosdem commentariorum libri XV. Theonis alexandrini Bartholamaeo Zamberto veneto interprete, in tredecim priores, commentariorum libri XIII. Hypsiclis alexandrini in duos posteriores, eodem Bartholamaeo Zamberto veneto interprete, commentariorum libri II. Utcunque noster valuit labor conciliata sunt haec omnia, ad studiosorum non paruam (quam optamus) utilitatem: id Magnifico D. Francisco Briconneto postulante. Si haec beneuole suscipiantur, et fructum adferant quem cupidimus: alia eiusdem authoris opera prodibunt in lucem, successum praestante deo, et adiutoribus (ubiubi gentium sint) ad bonarum literarum institutionem probe affectis Gallis, Italis, Germanis, Hispanis, Anglis. Quibus omnibus prospera imprecamur: et puram pro dignitate veramque cognitionis lucem. Henrici Stephani, Paris.Google Scholar
  18. Loria, G. 1931, 1950. Storia delle matematiche. Dall’alba della civiltà al secolo XIX, Hoepli.Google Scholar
  19. Tamborini, M. 2003. Per una storia dell’Opus Arithmeticæ Perfectum. In Cardano e la tradizione dei saperi. Atti del Convegno internazionale di studi. Milano (23–25 maggio 2002), Franco Angeli, eds. Marialuisa Baldi and Guido Canziani, 157–190.Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Bergische Universität WuppertalWuppertalGermany

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