History of complex numbers: the story continues

   However, four years later (in 1543) Cardano along with his disciple Luigi Ferrari (1522-1565) visited Bologna, where he was able to peruse the papers of the late Ferro concerning the solution to cubic equations. Convinced that Ferro had relevant results earlier than Tartaglia, Cardano, obviously, considered himself relieved of his oath. Furthermore, Ferrari determined a method for solving equations of 4th order.

   All of this gave Cardano cause to include these results into his scientific treatise "The Great Art, or, the Rules of Algebra," published in 1545.  The German mathematician Felix Klein (1849 - 1925) said about this outstanding work: "This extremely valuable piece contained the germs of the modern algebra (help with algebra assignment) beyond the ancient mathematics."

   In the preface to his "Great Art" Cardano, to do him justice, described in detail the background of the issue. Upon studying the contents of "Great Art", Tartaglia was shocked. He accused Cardano of oath breaking. It was Ferrari who replied to these accusations. First, he pointed out some shortcomings in Tartaglia’s book "Diverse Problems and Inventions" (1546), the last part of which contained his correspondence with Cardano and transcripts of conversations with him, and then challenged Tartaglia to a public dispute-battle "on geometry, arithmetic and related disciplines such as astrology, music, cosmography, perspective, architecture and so on." Tartaglia accepted this challenge with a heavy heart. On the battle itself we know only that it was held on August 10, 1548 in Milan and that Tartaglia was defeated. Thus ended this great drama of scientific ideas and human fate.

   The discovery of the Cardano formula was an outstanding scientific event. Unfortunately, this formula yielded "reliable" results only for equations of type (1). For it is then that p> 0 and therefore the expression:

under the square radical sign is always positive. Besides, the root extracted from x was real, that is positive. In the other two cases (2) and (3) an inadmissible root that was still negative could be extracted, while with the general formula was inapplicable. But the most paradoxical was the fact that it was in this latter case that the cubic equation had no less than three real roots, while equation (1) - only one such root.

   Probably it was Tartaglia’s attempt to "tame" this case, later called "irreducible" (casus irreducibilis), that did not allow him to publish his results earlier than Cardano. Cardano in his "Great Art" passed over the "irreducible" case.
But in his research? - Rather not. One fragment of his "Great Art" gives evidence that Cardano did sense, albeit intuitively, a possible way to solve this problem. But this way was too unusual, and therefore even a genius like Cardano did not dare to step on it.

   Cardano’s younger compatriot, Rafael Bombelli (c. 1526-1573), an engineer from Bologna, didn’t consider numbers like   unnecessary. Moreover, he successfully used them to study the "irreducible" case while solving cubic equations. Bombelli published these results in his "Algebra”. Bombelli used to write down the new numbers in the following form: , calling them imaginary, as well as "truly sophistic". Upon extending to these numbers the basic rules of operations with conventional numbers and algebraic binomials, Bombelli, in particular, was able to somehow "fight" the "irreducible" case.

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