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.