At this point it may not be inopportune to make a break in the record of Cardan's life and work, and to treat in retrospect of that portion of his time which he spent in the composition of his treatises on Arithmetic and Algebra. Ever since 1535 he had been working intermittently at one or other of these, but it would have been impossible to deal coherently and effectively with the growth and completion of these two books—really the most important of all he left behind him—while chronicling the goings and comings of a life so adventurous as that of the author.

The prime object of Cardan's ambition was eminence as a physician. But, during the long years of waiting, while the action of the Milanese doctors kept him outside the bounds of their College, and even after this had been opened to him without inducing ailing mortals to call for his services, he would now and again fall into a transport of rage against his persecutors, and of contempt for the public which refused to recognize him as a master of his art, and cast aside his medical books for months at a time, devoting himself diligently to Mathematics, the field of learning which, next to Medicine, attracted him most powerfully. His father Fazio was a geometrician of repute and a student of applied mathematics, and, though his first desire was to make his son a jurisconsult, he gave Jerome in early youth a fairly good grounding in arithmetic and geometry, deeming probably that such training would not prove a bad discipline for an intellect destined to attack those formidable tomes within which lurked the mysteries of the Canon and Civil Law. Mathematical learning has given to Cardan his surest title to immortality, and at the outset of his career he found in mathematics rather than in medicine the first support in the arduous battle he had to wage with fortune. His appointment to the Plat lectureship at Milan has already been noted. In the discharge of his new duties he was bound, according to the terms of the endowment of the Plat lecturer, to teach the sciences of geometry, arithmetic, and astronomy, and he began his course upon the lines laid down by the founder. Few listeners came, however, and at this juncture Cardan took a step which serves to show how real was his devotion to the cause of true learning, and how lightly he thought of an additional burden upon his own back, if this cause could be helped forward thereby. Keenly as he enjoyed his mathematical work, he laid a part of it aside when he perceived that the benches before him were empty, and, by way of making his lectures more attractive, he occasionally substituted geography for geometry, and architecture for arithmetic. The necessary research and the preparation of these lectures led naturally to the accumulation of a large mass of notes, and as these increased under his hand Jerome began to consider whether it might not be worth his while to use them in the composition of one or more volumes. In 1535 he delivered as Plat lecturer his address, the Encomium GeometriÆ, which he followed up shortly after by the publication of a work, Quindecim Libri NovÆ GeometriÆ. But the most profitable labour of these years was that which produced his first important book, The Practice of Arithmetic and Simple Mensuration, which was published in 1539, a venture which brought to the author a reward of ten crowns.[84] It was a well-planned and well-arranged manual, giving proof of the wide erudition and sense of proportion possessed by the author. Besides dealing with Arithmetic as understood by the modern school-boy, it discusses certain astronomical operations, multiplication by memory, the mysteries of the Roman and Ecclesiastical Calendars, and gives rules for the solution of any problem arising from the terms of the same. It treats of partnership in agriculture, the Mezzadria system still prevalent in Tuscany and in other parts of Italy, of the value of money, of the strange properties of certain numbers, and gives the first simple rules of Algebra to serve as stepping-stones to the higher mathematics. It ends with information as to house-rent, letters of credit and exchange, tables of interest, games of chance, mensuration, and weights and measures. In an appendix Cardan examines critically the work of Fra Luca Pacioli da Borgo, an earlier writer on the subject, and points out numerous errors in the same. The book from beginning to end shows signs of careful study and compilation, and the fame which it brought to its author was well deserved.

Cardan appended to the Arithmetic a printed notice which may be regarded as an early essay in advertising. He was fully convinced that his works were valuable and quite worth the sums of money he asked for them; the world was blind, perhaps wilfully, to their merits, therefore he now determined that it should no longer be able to quote ignorance of the author as an excuse for not buying the book. This appendix was a notification to the learned men of Europe that the writer of the Practice of Arithmetic had in his press at home thirty-four other works in MS. which they might read with profit, and that of these only two had been printed, to wit the De Malo Medendi Usu and a tract on Simples. This advertisement had something of the character of a legal document, for it invoked the authority of the Emperor to protect the copyright of Cardan's books within the Duchy of Milan for ten years, and to prevent the introduction of them from abroad.

The Arithmetic proved far superior to any other treatise extant, and everywhere won the approval of the learned. It was from Nuremberg that its appearance brought the most valuable fruits. Andreas Osiander,[85] a learned humanist and a convert to Lutheranism, and Johannes Petreius, an eminent printer, were evidently impressed by the terms of Cardan's advertisement, for they wrote to him and offered in combination to edit and print any of the books awaiting publication in his study at Milan. The result of this offer was the reprinting of De Malo Medendi, and subsequently of the tract on Judicial Astrology, and of the treatise De Consolatione; the Book of the Great Art, the treatises De Sapientia and De Immortalitate Animorum were published in the first instance by these same patrons from the Nuremberg press.

But Cardan, while he was hard at work on his Arithmetic, had not forgotten a certain report which had caused no slight stir in the world of Mathematics some three years before the issue of his book on Arithmetic, an episode which may be most fittingly told in his own words. "At this time[86] it happened that there came to Milan a certain Brescian named Giovanni Colla, a man of tall stature, and very thin, pale, swarthy, and hollow-eyed. He was of gentle manners, slow in gait, sparing of his words, full of talent, and skilled in mathematics. His business was to bring word to me that there had been recently discovered two new rules in Algebra for the solution of problems dealing with cubes and numbers. I asked him who had found them out, whereupon he told me the name of the discoverer was Scipio Ferreo of Bologna. 'And who else knows these rules?' I said. He answered, 'Niccolo Tartaglia and Antonio Maria Fiore.' And indeed some time later Tartaglia, when he came to Milan, explained them to me, though unwillingly; and afterwards I myself, when working with Ludovico Ferrari,[87] made a thorough study of the rules aforesaid. We devised certain others, heretofore unnoticed, after we had made trial of these new rules, and out of this material I put together my Book of the Great Art."[88]

Before dealing with the events which led to the composition of the famous work above-named, it may be permitted to take a rapid survey of the condition of Algebra at the time when Cardan sat down to write. Up to the beginning of the sixteenth century the knowledge of Algebra in Italy, originally derived from Greek and Arabic sources, had made very little progress, and the science had been developed no farther than to provide for the solution of equations of the first or second degree.[89] In the preface to the Liber Artis MagnÆ Cardan writes:—"This art takes its origin from a certain Mahomet, the son of Moses, an Arabian, a fact to which Leonard the Pisan bears ample testimony. He left behind him four rules, with his demonstrations of the same, which I duly ascribe to him in their proper place. After a long interval of time, some student, whose identity is uncertain, deduced from the original four rules three others, which Luca Paciolus put with the original ones into his book. Then three more were discovered from the original rules, also by some one unknown, but these attracted very little notice though they were far more useful than the others, seeing that they taught how to arrive at the value of the cubus and the numerus and of the cubus quadratus.[90] But in recent times Scipio Ferreo of Bologna discovered the rule of the cubus and the res equal to the numerus (x³ + px=q), truly a beautiful and admirable discovery. For this Algebraic art outdoes all other subtlety of man, and outshines the clearest exposition mortal wit can achieve: a heavenly gift indeed, and a test of the powers of a man's mind. So excellent is it in itself that whosoever shall get possession thereof, will be assured that no problem exists too difficult for him to disentangle. As a rival of Ferreo, Niccolo Tartaglia of Brescia, my friend, at that time when he engaged in a contest with Antonio Maria Fiore, the pupil of Ferreo, made out this same rule to help secure the victory, and this rule he imparted to me after I had diligently besought him thereanent. I, indeed, had been deceived by the words of Luca Paciolus, who denied that there could be any general rule besides these which he had published, so I was not moved to seek that which I despaired of finding; but, having made myself master of Tartaglia's method of demonstration, I understood how many other results might be attained; and, having taken fresh courage, I worked these out, partly by myself and partly by the aid of Ludovico Ferrari, a former pupil of mine. Now all the discoveries made by the men aforesaid are here marked with their names. Those unsigned were found out by me; and the demonstrations are all mine, except three discovered by Mahomet and two by Ludovico."[91]

This is Cardan's account of the scheme and origin of his book, and the succeeding pages will be mainly an amplification thereof. The earliest work on Algebra used in Italy was a translation of the MS. treatise of Mahommed ben Musa of Corasan, and next in order is a MS. written by a certain Leonardo da Pisa in 1202. Leonardo was a trader, who had learned the art during his voyages to Barbary, and his treatise and that of Mahommed were the sole literature on the subject up to the year 1494, when Fra Luca Pacioli da Borgo[92] brought out his volume treating of Arithmetic and Algebra as well. This was the first printed work on the subject.

After the invention of printing the interest in Algebra grew rapidly. From the time of Leonardo to that of Fra Luca it had remained stationary. The important fact that the resolution of all the cases of a problem may be comprehended in a simple formula, which may be obtained from the solution of one of its cases merely by a change of the signs, was not known, but in 1505 the Scipio Ferreo alluded to by Cardan, a Bolognese professor, discovered the rule for the solution of one case of a compound cubic equation. This was the discovery that Giovanni Colla announced when he went to Milan in 1536.

Cardan was then working hard at his Arithmetic—which dealt also with elementary Algebra—and he was naturally anxious to collect in its pages every item of fresh knowledge in the sphere of mathematics which might have been discovered since the publication of the last treatise. The fact that Algebra as a science had made such scant progress for so many years, gave to this new process, about which Giovanni Colla was talking, an extraordinary interest in the sight of all mathematical students; wherefore when Cardan heard the report that Antonio Maria Fiore, Ferreo's pupil, had been entrusted by his master with the secret of this new process, and was about to hold a public disputation at Venice with Niccolo Tartaglia, a mathematician of considerable repute, he fancied that possibly there would be game about well worth the hunting.

Fiore had already challenged divers opponents of less weight in the other towns of Italy, but now that he ventured to attack the well-known Brescian student, mathematicians began to anticipate an encounter of more than common interest. According to the custom of the time, a wager was laid on the result of the contest, and it was settled as a preliminary that each one of the competitors should ask of the other thirty questions. For several weeks before the time fixed for the contest Tartaglia studied hard; and such good use did he make of his time that, when the day of the encounter came, he not only fathomed the formula upon which Fiore's hopes were based, but, over and beyond this, elaborated two other cases of his own which neither Fiore nor his master Ferreo had ever dreamt of.

The case which Ferreo had solved by some unknown process was the equation x³ + px = q, and the new forms of cubic equation which Tartaglia elaborated were as follows: x³ + px² = q: and x³-px² = q. Before the date of the meeting, Tartaglia was assured that the victory would be his, and Fiore was probably just as confident. Fiore put his questions, all of which hinged upon the rule of Ferreo which Tartaglia had already mastered, and these questions his opponent answered without difficulty; but when the turn of the other side came, Tartaglia completely puzzled the unfortunate Fiore, who managed indeed to solve one of Tartaglia's questions, but not till after all his own had been answered. By this triumph the fame of Tartaglia spread far and wide, and Jerome Cardan, in consequence of the rumours of the Brescian's extraordinary skill, became more anxious than ever to become a sharer in the wonderful secret by means of which he had won his victory.

Cardan was still engaged in working up his lecture notes on Arithmetic into the Treatise when this contest took place; but it was not till four years later, in 1539, that he took any steps towards the prosecution of his design. If he knew anything of Tartaglia's character, and it is reasonable to suppose that he did, he would naturally hesitate to make any personal appeal to him, and trust to chance to give him an opportunity of gaining possession of the knowledge aforesaid, rather than seek it at the fountain-head. Tartaglia was of very humble birth, and according to report almost entirely self-educated. Through a physical injury which he met with in childhood his speech was affected; and, according to the common Italian usage, a nickname[93] which pointed to this infirmity was given to him. The blow on the head, dealt to him by some French soldier at the sack of Brescia in 1512, may have made him a stutterer, but it assuredly did not muddle his wits; nevertheless, as the result of this knock, or for some other cause, he grew up into a churlish, uncouth, and ill-mannered man, and, if the report given of him by Papadopoli[94] at the end of his history be worthy of credit, one not to be entirely trusted as an autobiographer in the account he himself gives of his early days in the preface to one of his works. Papadopoli's notice of him states that he was in no sense the self-taught scholar he represented himself to be, but that he was indebted for some portion at least of his training to the beneficence of a gentleman named Balbisono,[95] who took him to Padua to study. From the passage quoted below he seems to have failed to win the goodwill of the Brescians, and to have found Venice a city more to his taste. It is probable that the contest with Fiore took place after his final withdrawal from his birthplace to Venice.

In 1537 Tartaglia published a treatise on Artillery, but he gave no sign of making public to the world his discoveries in Algebra. Cardan waited on, but the morose Brescian would not speak, and at last he determined to make a request through a certain Messer Juan Antonio, a bookseller, that, in the interests of learning, he might be made a sharer of Tartaglia's secret. Tartaglia has given a version of this part of the transaction; and, according to what is there set down, Cardan's request, even when recorded in Tartaglia's own words, does not appear an unreasonable one, for up to this time Tartaglia had never announced that he had any intention of publishing his discoveries as part of a separate work on Mathematics. There was indeed a good reason why he should refrain from doing this in the fact that he could only speak and write Italian, and that in the Brescian dialect, being entirely ignorant of Latin, the only tongue which the writer of a mathematical work could use with any hope of success. 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,[96] a record which furnishes abundant and striking instance of his jealous and suspicious temper. Much of it is given in the form of dialogue, the terms of which are perhaps a little too precise to carry conviction of its entire sincerity and spontaneity. It was probably written just after the final cause of quarrel in 1545, and its main object seems to be to set the author right in the sight of the world, and to exhibit Cardan as a meddlesome fellow not to be trusted, and one ignorant of the very elements of the art he professed to teach.[97]

The inquiry begins with a courteously worded request from Messer Juan Antonio (speaking on behalf of Messer Hieronimo Cardano), that Messer Niccolo would make known to his principal the rule by means of which he had made such short work of Antonio Fiore's thirty questions. It had been told to Messer Hieronimo that Fiore's thirty questions had led up to a case of the cosa and the cubus equal to the numerus, and that Messer Niccolo had discovered a general rule for such case. Messer Hieronimo now especially desired to be taught this rule. If the inventor should be willing to let this rule be published, it should be published as his own discovery; but, if he were not disposed to let the same be made known to the world, it should be kept a profound secret. To this request Tartaglia replied that, if at any time he might publish his rule, he would give it to the world in a work of his own under his own name, whereupon Juan Antonio moderated his demand, and begged to be furnished merely with a copy of the thirty questions preferred by Fiore, and Tartaglia's solutions of the same; but Messer Niccolo was too wary a bird to be taken with such a lure as this. To grant so much, he replied, would be to tell everything, inasmuch as Cardan could easily find out the rule, if he should be furnished with a single question and its solution. Next Juan Antonio handed to Tartaglia eight algebraical questions which had been confided to him by Cardan, and asked for answers to them; but Tartaglia, having glanced at them, declared that they were not framed by Cardan at all, but by Giovanni Colla. Colla, he declared, had sent him one of these questions for solution some two years ago. Another, he (Tartaglia) had given to Colla, together with a solution thereof. Juan Antonio replied by way of contradiction—somewhat lamely—that the questions had been handed over to him by Cardan and no one else, wishing to maintain, apparently, that no one else could possibly have been concerned in them, whereupon Tartaglia replied that, supposing the questions had been given by Cardan to Juan Antonio his messenger, Cardan must have got the questions from Colla, and have sent them on to him (Tartaglia) for solution because he could not arrive at the meaning of them himself. He waved aside Juan Antonio's perfectly irrelevant and fatuous protests—that Cardan would not in any case have sent these questions if they had been framed by another person, or if he had been unable to solve them. Tartaglia, on the other hand, declared that Cardan certainly did not comprehend them. If he did not know the rule by which Fiore's questions had been answered (that of the cosa and the cubus equal to the numerus), how could he solve these questions which he now sent, seeing that certain of them involved operations much more complicated than that of the rule above 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. He probably felt that it would be mere waste of breath to beg again for Tartaglia's answers. The end of the matter was that Tartaglia handed over to the messenger the questions which Fiore had propounded in the Venetian contest, and authorized Juan Antonio to get a copy of his own from the notary who had drawn up the terms of the disputation with Fiore. The date of this communication is January 2, 1539, and on February 12 Cardan writes a long letter to Tartaglia, complaining in somewhat testy spirit of the reception given to his request. He is aggrieved that Tartaglia should have sent him nothing but the questions put to him by Fiore, thirty in number indeed, but only one in substance, and that he should have dared to hint that those which he (Cardan) had sent for solution were not his own, but the property of Giovanni Colla. Cardan had found Colla to be a conceited fool, and had dragged the conceit out of him—a process which he was now about to repeat for the benefit of Messer Niccolo Tartaglia. 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. In short a more injudicious letter could not have been written by any man hoping to get a favour done to him by the person addressed.

In the special matter of the problems which he sent to Tartaglia by the bookseller Juan Antonio, Cardan made a beginning of that tricky and crooked course which he followed too persistently all through this particular business. In his letter he maintains with a show of indignation that he had long known these questions, had known them in fact before Colla knew how to count ten, implying by these words that he knew how to solve them, while in reality all he knew about them was the fact that they existed. Tartaglia in his answer is not to be moved from his belief, and tells Cardan flatly that he is still convinced Giovanni Colla took the questions to Milan, where he found no one able to solve them, not even Messer Hieronimo Cardano, and that the mathematician last-named sent them on by the bookseller for solution, as has been already related.

This letter of Tartaglia's bears the date of February 13, 1539, and after reading it and digesting its contents, Cardan seems to have come to the conclusion that he was not working in the right way to get possession of this secret which he felt he must needs master, if he wanted his forthcoming book to mark a new epoch in this History of Mathematics, and that a change of tactics was necessary. Alfonso d'Avalos, Cardan's friend and patron, was at this time the Governor of Milan. D'Avalos was a man of science, as well as a soldier, and Cardan had already sent to him a copy of Tartaglia's treatise on Artillery, deeming that a work of this kind would not fail to interest him. In his first letter to Tartaglia he mentions this fact, while picking holes in the writer's theories concerning transmitted force and views on gravitation. This mention of the name of D'Avalos, the master of many legions and of many cannons as well, to a man who had written a Treatise on the management of Artillery, and devised certain engines and instruments for the management of the same, was indeed a clever cast, and the fly was tempting enough to attract even so shy a fish as Niccolo Tartaglia. In his reply to Jerome's scolding letter of February 12, 1539, Tartaglia concludes with a description of the instruments which he was perfecting: a square to regulate the discharge of cannon, and to level and determine every elevation; and another instrument for the investigation of distances upon a plane surface. He ends with a request that Cardan will accept four copies of the engines aforesaid, two for himself and two for the Marchese d'Avalos.

The tone of this letter shows that Cardan had at least begun to tame the bear, who now seemed disposed to dance ad libitum to the pleasant music of words suggesting introductions to the governor, and possible patronage of these engines for the working of artillery. Cardan's reply of March 19, 1539, is friendly—too friendly indeed—and the wonder is that Tartaglia's suspicions were not aroused by its almost sugary politeness. It begins with an attempt to soften down the asperities of their former correspondence, some abuse of Giovanni Colla, and an apology for the rough words of his last epistle. Cardan then shows how their misunderstanding arose chiefly from a blunder made by Juan Antonio in delivering the message, and invites Tartaglia to come and visit him in his own house in Milan, so that they might deliberate together on mathematical questions; but the true significance of the letter appears in the closing lines. "I told the Marchese of the instruments which you had sent him, and he showed himself greatly pleased with all you had done. And he commanded me to write to you forthwith in pressing terms, and to tell you that, on the receipt of my letter, you should come to Milan without fail, for he desires to speak with you. And I, too, exhort you to come at once without further deliberation, seeing that this said Marchese is wonted to reward all men of worth in such noble and magnanimous and liberal fashion that none of them ever goes away dissatisfied."

The receipt of this letter seems to have disquieted Tartaglia somewhat; for he has added a note to it, in which he says that Cardan has placed him in a position of embarrassment. He had evidently wished for an introduction to D'Avalos, but now it was offered to him it seemed a burden rather than a benefit. He disliked the notion of going to Milan; yet, if he did not go, the Marchese d'Avalos might take offence. But in the end he decided to undertake the journey; and, as D'Avalos happened then to be absent from Milan on a visit to his country villa at Vigevano, he stayed for three days in Cardan's house. As a recorder of conversations Tartaglia seems to have had something of Boswell's gift. He gives an abstract of an eventful dialogue with his host on March 25, 1539, which Cardan begins by a gentle reproach anent his guest's reticence in the matter of the rule of the cosa and the cubus equal to the numerus. Tartaglia's reply to this complaint seems reasonable enough (it must be borne in mind that he is his own reporter), and certainly helps to absolve him from the charge sometimes made against him that he was nothing more than a selfish curmudgeon who had resolved to let his knowledge die with him, rather than share it with other mathematicians of whom he was jealous. He told Cardan plainly that he kept his rules a secret because, for the present, it suited his purpose to do so. At this time he had not the leisure to elaborate farther the several rules in question, being engaged over a translation of Euclid into Italian; but, when this work should be completed, he proposed to publish a treatise on Algebra in which he would disclose to the world all the rules he already knew, as well as many others which he hoped to discover in the course of his present work. He concludes: "This is the cause of my seeming discourtesy towards your excellency. I have been all the ruder, perhaps, because you write to me that you are preparing a book similar to mine, and that you propose to publish my inventions, and to give me credit for the same. This I confess is not to my taste, forasmuch as I wish to set forth my discoveries in my own works, and not in those of others." In his reply to this, Cardan points out that he had promised, if Tartaglia so desired, that he would not publish the rules at all; but here Messer Niccolo's patience and good manners gave way, and he told Messer Hieronimo bluntly that he did not believe him. Then said Cardan: "I swear to you by the Sacred Evangel, and by myself as a gentleman, that I will not only abstain from publishing your discoveries—if you will make them known to me—but that I will promise and pledge my faith of a true Christian to set them down for my own use in cypher, so that after my death no one may be able to understand them. If you will believe this promise, believe it; if you will not, let us have done with the matter." "If I were not disposed to believe such oaths as these you now swear," said Tartaglia, "I might as well be set down as a man without any faith at all. I have determined to go forthwith to Vigevano to visit the Signor Marchese, as I have now been here for three days and am weary of the delay, but I promise when I return that I will show you all the rules." Cardan replied: "As you are bent on going to Vigevano, I will give you a letter of introduction to the Marchese, so that he may know who you are; but I would that, before you start, you show me the rule as you have promised." "I am willing to do this," said Tartaglia, "but I must tell you that, in order to be able to recall at any time my system of working, I have expressed it in rhyme; because, without this precaution, I must often have forgotten it. I care naught that my rhymes are clumsy, it has been enough for me that they have served to remind me of my rules. These I will write down with my own hand, so that you may be assured that my discovery is given to you correctly." Then follow Tartaglia's verses:

Having handed over to his host these rhymes, with the precious rules enshrined therein, Tartaglia told him that, with so clear an exposition, he could not fail to understand 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 x⁴ + 6x² + 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 an additional reason for keeping silence, or as a justification for speech? In forming a judgment after a lapse of three and a half centuries as to Cardan's action, while having regard both to the sanctity of an oath at the time in question, and to the altered state of the case between him and Tartaglia consequent on Ludovico Ferrari's discovery, an hypothesis not overstrained in the direction of charity may be advanced to the effect that Cardan might well have deemed he was justified in revealing to the world the rules which Tartaglia had taught him, considering that these isolated rules had been developed by his own study and Ferrari's into a principle by which it would be possible to work a complete revolution in the science of Algebra.

In any case, six years were allowed to elapse before Cardan, by publishing Tartaglia's rules in the Book of the Great Art, did the deed which, in the eyes of many, branded him as a liar and dishonest, and drove Tartaglia almost wild with rage. That his offence did not meet with universal reprobation is shown by negative testimony in the Judicium de Cardano, by Gabriel NaudÉ.[106] In the course of his essay NaudÉ lets it be seen how thoroughly he dislikes the character of the man about whom he writes. 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. On the contrary, after abusing and ridiculing a large portion of his work, NaudÉ breaks out into almost rhapsodical eulogy about Cardan's contributions to Mathematical science. "Quis negabit librum de Proportionibus dignum esse, qui cum pulcherrimis antiquorum inventis conferatur? Quis in Arithmetica non stupet, eum tot difficultates superasse, quibus explicandis Villafrancus, Lucas de Burgo, Stifelius, Tartalea, vix ac ne vix quidem pares esse potuissent?" It seems hard to believe, after reading elsewhere the bitter assaults of NaudÉ,[107] that he would have neglected so tempting an opportunity of darkening the shadows, if he himself had felt the slightest offence, or if public opinion in the learned world was in any perceptible degree scandalized by the disclosure made by the publication of the Book of the Great Art.

This book was published at Nuremberg in 1545, and in its preface and dedication Cardan fully acknowledges his obligations to Tartaglia and Ferrari, with respect to the rules lately discussed, and gives a catalogue of the former students of the Art, and attributes to each his particular contribution to the mass of knowledge which he here presents to the world. Leonardo da Pisa,[108] Fra Luca da Borgo, and Scipio Ferreo all receive due credit for their work, and then Cardan goes on to speak of "my friend Niccolo Tartaglia of Brescia, who, in his contest with Antonio Maria Fiore, the pupil of Ferreo, elaborated this rule to assure him of victory, a rule which he made known to me in answer to my many prayers." He goes on to acknowledge other obligations to Tartaglia:[109] how the Brescian had first taught him that algebraical discovery could be most effectively advanced by geometrical demonstration, and how he himself had followed this counsel, and had been careful to give the demonstration aforesaid for every rule he laid down.

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,[110] vera and ficta: thus the root of 9 is either 3. or-3. He shows that when a case has all its roots, or when none are impossible, the number of its positive roots is the same as the number of changes in the signs of the terms when they are all brought to one side. In the case of x³ + 3bx = 2c, he demonstrates his first resolution of a cubic equation, and gives his own version of his dealings with Tartaglia. His chief obligation to the Brescian was the information how to solve the three cases which follow, i.e. x³ + bx = c. x³ = bx + c. and x³+c=bx, and this he freely acknowledges, and furthermore admits the great service of the system of geometrical demonstration which Tartaglia had first suggested to him, and which he always employed hereafter. He claims originality for all processes in the book not ascribed to others, asserting that all the demonstrations of existing rules were his own except three which had been left by Mahommed ben Musa, and two invented by Ludovico Ferrari.

With this vantage ground beneath his feet Cardan raised the study of Algebra to a point it had never reached before, and climbed himself to a height of fame to which Medicine had not yet brought him. His name as a mathematician was known throughout Europe, and the success of his book was remarkable. In the De Libris Propriis there is a passage which indicates that he himself was not unconscious of the renown he had won, or disposed to underrate the value of his contribution to mathematical science. "And even if I were to claim this art (Algebra) as my own invention, I should perhaps be speaking only the truth, though Nicomachus, PtolemÆus, Paciolus, Boetius, have written much thereon. For men like these never came near to discover one-hundredth part of the things discovered by me. But with regard to this matter—as with divers others—I leave judgment to be given by those who shall come after me. Nevertheless I am constrained to call this work of mine a perfect one, seeing that it well-nigh transcends the bounds of human perception."[111]

[84] It was published at Milan by Bernardo Caluschio, with a dedication—dated 1537—to Francesco Gaddi, a descendant of the famous family of Florence. This man was Prior of the Augustinian Canons in Milan, and a great personage, but ill fortune seems to have overtaken him in his latter days. Cardan writes (Opera, tom. i. p. 107):—"qui cum mihi amicus esset dum floreret, Rexque cognomine ob potentiam appellaretur, conjectus in carcerem, miserÉ vitam ibi, ne dicam crudeliter, finivit: nam per quindecim dies in profundissima gorgyne fuit, ut vivus sepeliretur."