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No evil disposition attributed to Cardan by himself or by his enemies is left unnoticed, and a lengthy catalogue of his offences is set down, but this list does not contain the particular sin of broken faith in the matter of Tartaglia's rules.
The letter goes on to contradict all Tartaglia's assertions by arguments which do not seem entirely convincing, and the case is not made better by the abusive passages interpolated here and there, and by the demonstration of certain errors in Tartaglia's book on Artillery.
If this discovery included and even went beyond Tartaglia's, so much the worse for Tartaglia. The case might be said to run on all fours with that where a man confides a secret to a friend under a promise of silence, which promise the friend keeps religiously, until one day he finds that the secret, and even more than the secret, is common talk of the market-place.
Of Buteon, who criticized the treatise on Arithmetic, he says: "Est plane stultus et elleboro indiget." Tartaglia's name is there, and he, according to Cardan, was forced to eat his words; "but he was ashamed to do what he promised, and unwilling to blot out what he had written.
If he understood the questions which he now sent for solution, he could not want to be taught this rule. Then Juan Antonio moderated his demand still farther, and said he would be satisfied with a copy of the questions which Fiore had put to Tartaglia, adding that the favour would be much greater if Tartaglia's own questions were also given.
The more clearly Tartaglia's jealous, suspicious nature displays itself, the greater seems the wonder that a man of such a disposition should ever have disclosed such a secret. The age was one in which the binding force of an oath was not regarded as an obligation of any particular sanctity if circumstances should arise which made the violation of the oath more convenient than its observance.
Tartaglia's record of his conversation with Messer Juan Antonio, the emissary employed by Cardan, and of all the subsequent details of the controversy, is preserved in his principal work, Quesiti et Inventioni Diverse de Nicolo Tartalea Brisciano, a record which furnishes abundant and striking instance of his jealous and suspicious temper.
The Book of the Great Art was not published till six years after Cardan had become the sharer of Tartaglia's secret, which had thus had ample time to germinate and bear fruit in the fertile brain upon which it was cast. It is almost certain that the treatise as a whole leaving out of account the special question of the solution of cubic equations must have gained enormously in completeness and lucidity from the fresh knowledge revealed to the writer thereof by Tartaglia's reluctant disclosure, and, over and beyond this, it must be borne in mind that Cardan had been working for several years at Giovanni Colla's questions in conjunction with Ferrari, an algebraist as famous as Tartaglia or himself. The opening chapters of the book show that Cardan was well acquainted with the chief properties of the roots of equations of all sorts. He lays it down that all square numbers have two different kinds of root, one positive and one negative, vera and ficta: thus the root of 9 is either 3. or
After pointing out to Cardan the blunders aforesaid, he concludes: "The whole of this work of yours is ridiculous and inaccurate, a performance which makes me tremble for your good name." Every succeeding page of Tartaglia's notes shows more and more clearly that he was smarting under a sense of his own folly in having divulged his secret.
On July 10 a letter came to him from one Maphio of Bergamo, a former pupil, telling how Cardan was about to publish certain new mathematical rules in a book on Algebra, and hinting that in all probability these rules would prove to be Tartaglia's, whereupon he at once jumped to the conclusion that Maphio's gossip was the truth, and that this book would make public the secret which Cardan had sworn to keep.
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