Jerome Cardan by William George Waters (important books to read .txt) 📕

them, ending with a warning hint to Cardan that, if he should publish the rules, either in the work he had in hand, or in any future one, either under the name of Tartaglia or of Cardan, he, the author, would put into print certain things which Messer Hieronimo would not find very pleasant reading.

After all Tartaglia was destined to quit Milan without paying his respects to D'Avalos. There is not a word in his notes which gives the reason of this eccentric action on his part. He simply says that he is no longer inclined to go to Vigevano, but has made up his mind to return to Venice forthwith; and Cardan, probably, was not displeased at this exhibition of petulant impatience on the part of his guest, but was rather somewhat relieved to see Messer Niccolo ride away, now that he had extracted from him the coveted information. From the beginning to the end of this affair Cardan has been credited with an amount of subtle cunning which he assuredly did not manifest at other times when his wits were pitted for contest with those of other men. It has been advanced to his disparagement that he walked in deceitful ways from the very beginning; that he dangled before Tartaglia's eyes the prospect of gain and preferment simply for the purpose of enticing him to Milan, where he deemed he might use more efficaciously his arguments for the accomplishment of the purpose which was really in his mind; that he had no intention of advancing Tartaglia's fortunes when he suggested the introduction to D'Avalos, but that the Governor of Milan was brought into the business merely that he might be used as a potent ally in the attack upon Tartaglia's obstinate silence. Whether this may have been his line of action or not, the issue shows that he was fully able to fight his battle alone, and that his powers of persuasion and hard swearing were adequate when occasion arose for their exercise. It is quite possible that Tartaglia, when he began to reflect over what he had done by writing out and handing over to Cardan his mnemonic rhymes, fell into an access of suspicious anger--at Cardan for his wheedling persistency, and at himself for yielding thereto--and packed himself off in a rage with the determination to have done with Messer Hieronimo and all his works. Certainly his carriage towards Cardan in the weeks ensuing, as exhibited in his correspondence, does not picture him in an amiable temper. On April 9 Jerome wrote to him in a very friendly strain, expressing regret that his guest should have left Milan without seeing D'Avalos, and fear lest he might have prejudiced his fortunes by taking such a step. He then goes on to describe to Tartaglia the progress he is making in his work with the Practice of Arithmetic, and to ask him for help in solving one of the cases in Algebra, the rule for which was indeed contained in Tartaglia's verses, but expressed somewhat obscurely, for which reason Cardan had missed its meaning.[98] In his reply, Tartaglia ignores Jerome's courtesies altogether, and tells him that what he especially desires at the present moment is a sight of that volume on the Practice of Arithmetic, "for," says he, "if I do not see it soon, I shall begin to suspect that this work of yours will probably make manifest some breach of faith; in other words, that it will contain as interpolations certain of the rules I taught you." Niccolo then goes on to explain the difficulty which had puzzled Cardan, using terms which showed plainly that he had as poor an opinion of his correspondent's wit as of his veracity.

Cardan was an irascible man, and it is a high tribute to his powers of restraint that he managed to keep his temper under the uncouth insults of such a letter as the foregoing. 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. He did not believe Cardan when he promised that he would not publish the rules in question without his (the discoverer's) consent--why then did he believe him when he swore by the Gospel? 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. However, the time was not yet come for Jerome to begin to quibble with his conscience. On May 12, 1539, he wrote another letter to Tartaglia, also in a very friendly tone, reproaching him gently for his suspicions, and sending a copy of the _Practice of Arithmetic_ to show him that they were groundless. He protested that Tartaglia might search from beginning to end without finding any trace of his jealously-guarded rules, inasmuch as, beyond correcting a few errors, the writer had only carried Algebra to the point where Fra Luca had left it. Tartaglia searched, and though he could not put his finger on any spot which showed that Messer Hieronimo had broken his oath, he found what must have been to him as a precious jewel, to wit a mistake in reckoning, which he reported to Cardan in these words:

"In this process your excellency has made such a gross mistake that I am amazed thereat, forasmuch as any man with half an eye must have seen it--indeed, if you had not gone on to repeat it in divers examples, I should have set it down to a mistake of the printer." 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."[99]

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. Night and day he brooded over his excess of confidence, and as time went by he let his suspicions of Cardan grow into savage resentment. His ears were open to every rumour which might pass from one class-room to another. 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. He left many of Cardan's letters unanswered; but at last he seems to have found too strong the temptation to say something disagreeable; so, in answer to a letter from Cardan containing a request for help in solving an equation which had baffled his skill, Tartaglia wrote telling Cardan that he had bungled in his application of the rule, and that he himself was now very sorry he had ever confided the rule aforesaid to such a man. He ends with further abuse of Cardan's _Practice of Arithmetic_, which he declares to be merely a confused farrago of other men's knowledge,[100] and with a remark which he probably intended to be a crowning insult. "I well remember when I was at your house in Milan, that you told me you had never tried to discover the rule of the _cosa_ and the _cubus_ equal to the _numerus_ which was found out by me, because Fra Luca had declared it to be impossible;[101] as if to say that, if you had set yourself to the task you could have accomplished it, a thing which sets me off laughing when I call to mind the fact that it is now two months since I informed you of the blunders you made in the extraction of the cube root, which process is one of the first to be taught to students who are beginning Algebra. Wherefore, if after the lapse of all this time you have not been able to find a remedy to set right this your mistake (which would have been an easy matter enough), just consider whether in any case your powers could have been equal to the discovery of the rule aforesaid."[102]

In this quarrel Messer Giovanni Colla had appeared as the herald of the storm, when he carried to Milan in 1536 tidings of the discovery of the new rule which had put Cardan on the alert, and now, as the crisis approached, he again came upon the scene, figuring as unconscious and indirect cause of the final catastrophe. On January 5, 1540, Cardan wrote to Tartaglia, telling him that Colla had once more appeared in Milan, and was boasting that he had found out certain new rules in Algebra. He went on to suggest to his correspondent that they should unite their forces in an attempt to fathom this asserted discovery of Colla's, but to this letter Tartaglia vouchsafed no reply. In his diary it stands with a superadded note, in which he remarks that he thinks as badly of Cardan as of Colla, and that, as far as he is concerned, they may both of them go whithersoever they will.[103]

Colla propounded divers questions to the Algebraists of Milan, and amongst them was one involving the equation _x^4 + 6x^2 + 36 = 60x_, one which he probably found in some Arabian treatise. Cardan tried all his ingenuity over this combination without success, but his brilliant pupil, Ludovico Ferrari, worked to better purpose, and succeeded at last in solving it by adding to each side of the equation, arranged in a certain fashion, some quadratic and simple quantities of which the square root could be extracted.[104] Cardan seems to have been baffled by the fact that the equation aforesaid could not be solved by the recently-discovered rules, because it produced a bi-quadratic. This difficulty Ferrari overcame, and, pursuing the subject, he discovered a general rule for the solution of all bi-quadratics by means of a cubic equation. Cardan's subsequent demonstration of this process is one of the masterpieces of the _Book of the Great Art_. It is an example of the use of assuming a new indeterminate quantity to introduce into an equation, thus anticipating by a considerable space of time Descartes, who subsequently made use of a like assumption in a like case.

How far this discovery of Ferrari's covered the rules given by Tartaglia to Cardan, and how far it relieved Cardan of the obligation of secresy, is a problem fitted for the consideration of the mathematician and the casuist severally.[105] An apologist of Cardan might affirm that he cannot be held to have acted in bad faith in publishing the result of Ferrari's discovery. If this discovery included and even went beyond Tartaglia's, so much the worse for Tartaglia. The lesser discovery (Tartaglia's) Cardan never divulged before Ferrari unravelled Giovanni Colla's puzzle; but it was inevitable that it must be made known to the world as a part of the greater discovery (Ferrari's) which Cardan was in no way bound to keep a secret. 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. Is the obligation of silence, with which he was bound originally, still to lie upon the friend, even when he may have sworn to observe it by the Holy Evangel and the honour of a gentleman; and is the fact that great renown and profit would come to him by publishing the secret to be held as