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Jerome Cardan - A Biographical Study
by William George Waters
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[47] Compare De Vita Propria, chaps. iv. and xxxi. pp. 13 and 92.

[48] De Vita Propria, ch. xxxi. p. 92. In taking the other view he writes: "Vitam ducebam in Saccensi oppido, ut mihi videbar, infelicissime."—Opera, tom. i. p. 97.

[49] De Utilitate, p. 235.

[50] He gives a long and interesting sketch of his father-in-law in De Utilitate, p. 370.

[51] De Vita Propria, ch. xxvi. p. 68; Opera, tom. i. p. 97.

[52] De Vita Propria, ch. xli. p. 149.

[53] De Utilitate, p. 350.

[54] De Utilitate, p. 357: "Nam in urbe nec collegium recipere volebat nec cum aliquo ex illis artem exercere licebat et sine illis difficillimum erat." He writes thus while describing this particular visit to Milan.

[55] Ill fortune seems to have pursued the whole family in their relations with learned societies. "Nam et pater meus ut ab eo accepi, diu in ingressu Collegii Jurisconsultorum laboravit, et ego, ut alias testatus sum, bis a medicorum Patavino, toties filius meus natu major, a Ticinensi, uterque a Mediolanensi rejecti sumus."—Opera, tom. i. p. 94.

[56] De Utilitate, p. 358.

[57] He became a priest, and died Archbishop of Milan in 1552. Cardan dedicated to him his first published book, De Malo Medendi.

[58] De Vita Propria, ch. xxxvii. p. 119.

[59] De Vita Propria, ch. xxv. p. 67.

[60] The Xenodochium, which was originally a stranger's lodging-house. By this time places of this sort had become little else than succursales of some religious house. The Governors of the Milanese Xenodochium were the patrons of the Plat endowment which Cardan afterwards enjoyed.

[61] "Hoc unum sat scio, ab ineunte aetate me inextinguibili nominis immortalis cupiditate flagrasse."—Opera, tom. i. p. 61.

[62] "Minimo tamen honorario, et illud etiam minimum suasu cujusdam amici egregii praefecti Xenodochii imminuerunt; ita cum hujus recordor in mentem venit fabellae illius Apuleii de annonae Praefecto."—Opera, tom. i. p. 64.

[63] De Utilitate, p. 351.

[64] The following gives a hint as to the treatment followed: "Referant leprosos balneo ejus aquae in qua cadaver ablutum sit, sanari."—De Varietate, p. 334.

[65] De Vita Propria, ch. xxxvii. p. 121. This dream is also told in De Libris Propriis, Opera, tom. i. p. 64.

[66] De Vita Propria, ch. xxxvii. p. 121.



CHAPTER IV

JEROME CARDAN is now standing on the brink of authorship. The very title of his first book, De Malo Recentiorum Medicorum Medendi Usu, gives plain indication of the humour which possessed him, when he formulated his subject and put it in writing. With his temper vexed by the persistent neglect and insult cast upon him by the Milanese doctors he would naturally sit down con amore to compile a list of the errors perpetrated by the ignorance and bungling of the men who affected to despise him, and if his object was to sting the hides of these pundits and arouse them to hostility yet more vehement, he succeeded marvellously well. He was enabled to launch his book rather by the strength of private friendship than by the hope of any commercial success. Whilst at Pavia he had become intimate with Ottaviano Scoto, a fellow-student who came from Venice, and in after times he found Ottaviano's purse very useful to his needs. Since their college days Ottaviano's father had died and had left his son to carry on his calling of printing. In 1536 Jerome bethought him of his friend, and sent him the MS. of the treatise which was to let the world learn with what little wisdom it was being doctored.[67]

Ottaviano seems to have expected no profit from this venture, which was manifestly undertaken out of a genuine desire to help his friend, and he generously bore all the costs. Cardan deemed that, whatever the result of the issue of the book might be, it would surely be to his benefit; he hazarded nothing, and the very publication of his work would give him at least notoriety. It would moreover give him the intense pleasure of knowing that he was repaying in some measure the debt of vengeance owing to his professional foes. The outcome was exactly the opposite of what printer and author had feared and hoped. The success of the book was rapid and great.

Ottaviano must soon have recouped all the cost of publication; and, while he was counting his money, the doctors everywhere were reading Jerome's brochure, and preparing a ruthless attack upon the daring censor, who, with the impetuosity of youth, had laid himself open to attack by the careless fashion in which he had compiled his work. He took fifteen days to write it, and he confesses in his preface to the revised edition that he found therein over three hundred mistakes of one sort or another. The attack was naturally led by the Milanese doctors. They demanded to be told why this man, who was not good enough to practise by their sanction, was good enough to lay down the laws for the residue of the medical world. They heaped blunder upon blunder, and held him up to ridicule with all the wealth of invective characteristic of the learned controversy of the age. Cardan was deeply humbled and annoyed. "For my opponents, seizing the opportunity, took occasion to assail me through the reasoning of this book, and cried out: 'Who can doubt that this man is mad? and that he would teach a method and a practice of medicine differing from our own, since he has so many hard things to say of our procedure.' And, as Galen said, I must in truth have appeared crazy in my efforts to contradict this multitude raging against me. For, as it was absolutely certain that either I or they must be in the wrong, how could I hope to win? Who would take my word against the word of this band of doctors of approved standing, wealthy, for the most part full of years, well instructed, richly clad and cultivated in their bearing, well versed in speaking, supported by crowds of friends and kinsfolk, raised by popular approval to high position, and, what was more powerful than all else, skilled in every art of cunning and deceit?"

Cardan had indeed prepared a bitter pill for his foes, but the draught they compelled him to swallow was hardly more palatable. The publication of the book naturally increased the difficulties of his position, and in this respect tended to make his final triumph all the more noteworthy.

It was in 1536 that Cardan made his first essay as an author.[68] The next three years of his life at Milan were remarkable as years of preparation and accumulation, rather than as years of achievement. He had struck his first blow as a reformer, and, as is often the lot of reformers, his sword had broken in his hand, and there now rested upon him the sense of failure as a superadded torment. Yet now and again a gleam of consolation would disperse the gloom, and advise him that the world was beginning to recognize his existence, and in a way his merits. In this same year he received an offer from Pavia of the Professorship of Medicine, but this he refused because he did not see any prospect of being paid for his services. His friend Filippo Archinto was loyal still, and zealous in working for his success, and as he had been recently promoted to high office in the Imperial service, his good word might be very valuable indeed. He summoned his protege to join him at Piacenza, whither he had gone to meet Paul III., hoping to advance Cardan's interests with the Pope; but though Marshal Brissac, the French king's representative,[69] joined Archinto in advocating his cause, nothing was done, and Jerome returned disappointed to Milan.

In these months Cardan, disgusted by the failure of his late attack upon the fortress of medical authority, turned his back, for a time, upon the study of medicine, and gave his attention almost entirely to mathematics, in which his reputation was high enough to attract pupils, and he always had one or more of them in his house, the most noteworthy of whom was Ludovico Ferrari of Bologna, who became afterwards a mathematician of repute, and a teacher both at Milan and Bologna. While he was working at the De Malo Medendi, he began a treatise upon Arithmetic, which he dedicated to his friend Prior Gaddi; but this work was not published till 1539. In 1536 he first heard a report of a fresh and important discovery in algebra, made by one Scipio Ferreo of Bologna; the prologue to one of the most dramatic incidents in his career, an incident which it will be necessary to treat at some length later on.

Cardan was well aware that his excursions into astrology worked to his prejudice in public esteem, but in spite of this he could not refrain therefrom. It was during the plentiful leisure of this period that he cast the horoscope of Jesus Christ, a feat which subsequently brought upon him grave misfortune; a few patients came to him, moved no doubt by the spirit which still prompts people suffering from obscure diseases to consult professors of healing who are either in revolt or unqualified in preference to going to the orthodox physician. In connection with this irregular practice of his he gives a curious story about a certain Count Borromeo. "In 1536, while I was attending professionally in the house of the Borromei, it chanced that just about dawn I had a dream in which I beheld a serpent of enormous bulk, and I was seized with fear lest I should meet my death therefrom. Shortly afterwards there came a messenger to summon me to see the son of Count Carlo Borromeo. I went to the boy, who was about seven years old, and found him suffering from a slight distemper, but on feeling his pulse I perceived that it failed at every fourth beat. His mother, the Countess Corona, asked me how he fared, and I answered that there was not much fever about him; but that, because his pulse failed at every fourth beat, I was in fear of something, but what it might be I knew not rightly (but I had not then by me Galen's books on the indications of the pulse). Therefore, as the patient's state changed not, I determined on the third day to give him in small doses the drug called Diarob: cum Turbit: I had already written my prescription, and the messenger was just starting with it to the pharmacy, when I remembered my dream. 'How do I know,' said I to myself, 'that this boy may not be about to die as prefigured by the portent above written? and in that case these other physicians who hate me so bitterly, will maintain he died through taking this drug.' I called to the messenger, and said there was wanting in the prescription something which I desired to add. Then I privately tore up what I had written, and wrote out another made of pearls, of the horn of unicorn,[70] and certain gems. The powder was given, and was followed by vomiting. The bystanders perceived that the boy was indeed sick, whereupon they called in three of the chief physicians, one of whom was in a way friendly to me. They saw the description of the medicine, and demanded what I would do now. Now although two of these men hated me, it was not God's will that I should be farther attacked, and they not only praised the medicine, but ordered that it should be repeated. This was the saving of me. When I went again in the evening I understood the case completely. The following morning I was summoned at daybreak, and found the boy battling with death, and his father lying in tears. 'Behold him,' he cried, 'the boy whom you declared to ail nothing' (as if indeed I could have said such a thing); 'at least you will remain with him as long as he lives.' I promised that I would, and a little later the boy tried to rise, crying out the while. They held him down, and cast all the blame upon me. What more is there to say? If there had been found any trace of that drug Diarob: cum Turbit: (which in sooth was not safe) it would have been all over with me, since Borromeo all his life would either have launched against me complaints grave enough to make all men shun me, or another Canidia, more fatal than African serpents, would have breathed poison upon me."[71]

In this same year, 1536, Lucia brought forth another child, a daughter, and it was about this time that Cardan first attracted the attention of Alfonso d'Avalos, the Governor of Milan, and an intimacy began which, albeit fruitless at first, was destined to be of no slight service to Jerome at the crisis of his fortunes.[72] In the following year, in 1537, he made a beginning of two of his books, which were subsequently found worthy of being finished, and which may still be read with a certain interest: the treatises De Sapientia and De Consolatione. Of the last-named, he remarks that it pleased no one, forasmuch as it appealed not to those who were happy, and the wretched rejected it as entirely inadequate to give them solace in their evil case. In this year he made another attempt to gain admission to the College at Milan, and was again rejected; the issue of the De Malo Medendi was too recent, and it needed other and more potent influences than those exercised by mere merit, to appease the fury of his rivals and to procure him due status. But it would appear that, in 1536 or 1537, he negotiated with the College to obtain a quasi-recognition on conditions which he afterwards describes as disgraceful to himself, and that this was granted to him.[73]

Whatever his qualifications may have been, Cardan had no scruples in treating the few patients who came to him. The first case he notes is that of Donato Lanza,[74] a druggist, who had suffered for many years with blood-spitting, which ailment he treated successfully. Success of this sort was naturally helpful, but far more important than Lanza's cure was the introduction given by the grateful patient to the physician, commending him to Francesco Sfondrato, a noble Milanese, a senator, and a member of the Emperor's privy council. The eldest son of this gentleman had suffered many months from convulsions, and Cardan worked a cure in his case without difficulty. Shortly afterwards another child, only ten months old, was attacked by the same complaint, and was treated by Luca della Croce, the procurator of the College of Physicians, of which Sfondrato was a patron. As the attack threatened to be a serious one, Della Croce recommended that another physician, Ambrogio Cavenago, should be called in, but the father, remembering Cardan's cure of Lanza, wished for him as well. The description of the meeting of the doctors round the sick child's bed, of their quotations from Hippocrates, of the uncertainty and helplessness of the orthodox practitioners, and of the ready resource of the free-lance—who happens also to be the teller of the story—is a richly typical one.[75] "We, the physicians and the father of the child, met about seven in the morning, and Della Croce made a few general observations on death, for he knew that Sfondrato was a sensible man, and he himself was both honoured and learned. Cavenago kept silence at this stage, because the last word had been granted to him. Then I said, 'Do you not see that the child is suffering from Opisthotonos?' whereupon the first physician stood as one dazed, as if I were trying to trouble his wits by my hard words. But Della Croce at once swept aside all uncertainty by saying, 'He means the backward contraction of the muscles.' I confirmed his words, and added, 'I will show you what I mean.' Whereupon I raised the boy's head, which the doctors and all the rest believed was hanging down through weakness, and by its own weight, and bade them put it into its former position. Then Sfondrato turned to me, and said, 'As you have discovered what the disease is, tell us likewise what is the remedy therefor.' Since no one else spoke, I turned towards him and—careful lest I should do hurt to the credit I had gained already,—I said, 'You know what Hippocrates lays down in a case like this—febrem convulsioni'—and I recited the aphorism. Then I ordered a fomentation, and an application of lint moistened with linseed-oil and oil of lilies, and gave directions that the child should be gently handled until such time as the neck should be restored; that the nurse should eat no meat, and that the child should be nourished entirely by the milk of her breast, and not too much of that; that it should be kept in its cradle in a warm place, and rocked gently till it should fall asleep. After the other physicians had gone, I remember that the father of the child said to me, 'I give you this child for your own,' and that I answered, 'You are doing him an ill turn, in that you are supplanting his rich father by a poor one.' He answered, 'I am sure that you would care for him as if he were your own, fearing naught that you might thereby give offence to these others' (meaning the physicians). I said, 'It would please me well to work with them in everything, and to win their support.' I thus blended my words, so that he might understand I neither despaired of the child's cure, nor was quite confident thereanent. The cure came to a favourable end; for, after the fourteenth day of the fever—the weather being very warm—the child got well in four days' time. Now as I review the circumstances, I am of opinion that it was not because I perceived what the disease really was, for I might have done so much by reason of my special practice; nor because I healed the child, for that might have been attributed to chance; but because the child got well in four days, whereas his brother lay ill for six months, and was then left half dead, that his father was so much amazed at my skill, and afterwards preferred me to all others. That he thought well of me is certain, because Della Croce himself, during the time of his procuratorship, was full of spite and jealousy against me, and declared in the presence of Cavenago and of Sfondrato, that he would not, under compulsion, say a word in favour of a man like me, one whom the College regarded with disfavour. Whereupon Sfondrato saw that the envy and jealousy of the other physicians was what kept me out of the College, and not the circumstances of my birth. He told the whole story to the Senate, and brought such influence to bear upon the Governor of the Province and other men of worship, that at last the entrance to the College was opened to me."

Up to the time of his admission to the College, Jerome had never felt that he could depend entirely upon medicine for his livelihood. He now determined to publish his Practica Arithmeticae, the book which he had prepared pari passu with the ill-starred De Malo Medendi. It seems to have been thoroughly revised and corrected, and was finally published in 1539, in Milan; Cardan only received ten crowns for his work, but the sudden fame he achieved as a mathematician ought to have set him on firm ground. His friends were still working to secure for him benefits yet more substantial. Alfonso d'Avalos, Francesco della Croce, the jurisconsult whose name has already been mentioned, and the senator Sfondrato, were doing their best to bring the physicians of the city into a more reasonable temper, and they finally succeeded in 1539; when, after having been denied admission for twelve years, Jerome Cardan became a member of the College, and a sharer in all the privileges appertaining thereto.

Though Cardan was now a fully qualified physician, he spent his time for the next year or two rather with letters than with medicine. He worked hard at Greek, and as the result of his studies published somewhat prematurely a treatise, De Immortalitate Animorum, a collection of extracts from Greek writers which Julius Caesar Scaliger with justice calls a confused farrago of other men's learning.[76] He published also about this period the treatise on Judicial Astrology, and the Essay De Consolatione, the only one of his books which has been found worthy of an English translation.[77] In 1541 he became Rector of the College of Physicians, but there is no record of any increase in the number of his patients by reason of this superadded dignity. A passage in the De Vita Propria, written with even more than his usual brutal candour, gives a graphic view of his manner of life at this period. "It was in the summer of the year 1543, a time when it was my custom to go every day to the house of Antonio Vicomercato, a gentleman of the city, and to play chess with him from morning till night. As we were wont to play for one real, or even three or four, on each game, I, seeing that I was generally the winner, would as a rule carry away with me a gold piece after each day's play, sometimes more and sometimes less. In the case of Vicomercato it was a pleasure and nothing else to spend money in this wise; but in my own there was an element of conflict as well; and in this manner I lost my self-respect so completely that, for two years and more, I took no thought of practising my art, nor considered that I was wasting all my substance—save what I made by play—that my good name and my studies as well would suffer shipwreck. But on a certain day towards the end of August, a new humour seized Vicomercato (either advisedly on account of the constant loss he suffered, or perhaps because he thought his decision would be for my benefit), a determination from which he was to be moved neither by arguments, nor adjurations, nor abuse. He forced me to swear that I would never again visit his house for the sake of gaming, and I, on my part, swore by all the gods as he wished. That day's play was our last, and thenceforth I gave myself up entirely to my studies."[78]

But these studies unfortunately were not of a nature to keep the wolf from the door; and Jerome, albeit now a duly qualified physician, and known to fame as a writer on Mathematics far beyond the bounds of Italy, was well-nigh as poor as ever. His mother had died several years before, in 1537; but what little money she may have left would soon have been wasted in gratifying his extravagant taste for costly things,[79] and at the gaming-table. He found funds, however, for a journey to Florence, whither he went to see d'Avalos, who was a generous, open-handed man, and always ready to put his purse at the service of one whom he regarded as an honour to his city and country. There can be little doubt that he helped Cardan liberally at this juncture. The need for a loan was assuredly urgent enough. The recent resumption of hostilities between the French and the Imperialists had led to intolerable taxation throughout the Milanese provinces, and in consequence of dearth of funds in 1543, the Academy at Pavia was forced to close its class-rooms, and leave its teachers unpaid. The greater part of the professors migrated to Pisa; and the Faculty of Medicine, then vacant, was, pro forma, transferred to Milan. This chair was now offered to Cardan. He was in desperate straits—a third child had been born this year—and, though there must have been even less chance of getting his salary paid than when he had refused it before, he accepted the post, explaining that he took this step because there was now no need for him to leave Milan, or danger that he would be rated as an itinerant teacher. It is not improbable that he may have been led to accept the office on account of the additional dignity it would give to him as a practising physician. When, a little later on, the authorities began to talk of returning to Pavia, he was in no mind to follow them, giving as a reason that, were he to leave Milan, he would lose his stipend for the Plat lectureship, and be put to great trouble in the transport of his household, and perhaps suffer in reputation as well. The Senate was evidently anxious to retain his services. They bade him consider the matter, promising to send on a certain date to learn his decision; and, as fate would have it, the question was conveniently decided for him by a portent.

"On the night before the day upon which my answer was to be sent to the Senate to say what course I was going to take, the whole of the house fell down into a heap of ruins, and no single thing was left unwrecked, save the bed in which I and my wife and my children were sleeping. Thus the step, which I should never have taken of my own free will or without some sign, I was compelled to take by the course of events. This thing caused great wonder to all those who heard of it."[80]

This was in 1544. Jerome hesitated no longer, and went forthwith to Pavia as Professor of Medicine at a salary of two hundred and forty gold crowns per annum; but, for the first year at least, this salary was not paid; and the new professor lectured for a time to empty benches; but, as he was at this time engaged in the final stage of his great work on Algebra, the leisure granted to him by the neglect of the students must have been most acceptable. He published at this time a treatise called Contradicentium Medicorum, and in 1545 his Algebra or Liber Artis Magnae was issued from the press by Petreius of Nuremberg. The issue of this book, by which alone the name of Cardan holds a place in contemporary learning, is connected with an episode of his life important enough to demand special and detailed consideration in a separate place.

His practice in medicine was now a fairly lucrative one, but his extravagant tastes and the many vices with which he charges himself would have made short work of the largest income he could possibly have earned, consequently poverty was never far removed from the household. Hitherto his reputation as a man of letters and a mathematician had exceeded his fame as a doctor; for, even after he had taken up his residence as Professor of Medicine at Padua, many applications were made to him for his services in other branches of learning. It was fortunate indeed that he had let his reading take a somewhat eclectic course, for medicine at this time seemed fated to play him false. At the end of 1544 no salary was forthcoming at Pavia, so he abandoned his class-room, and returned to Milan.

During his residence there, in the summer of 1546, Cardinal Moroni, acting on behalf of Pope Paul III., made an offer for his services as a teacher of mathematics, accompanied by terms which, as he himself admits, were not to be despised; but, as was his wont, he found some reason for demur, and ultimately refused the offer. In his Harpocratic vein he argued, "This pope is an old man, a tottering wall, as it were. Why should I abandon a certainty for an uncertainty?"[81] The certainty he here alludes to must have been the salary for the Plat lectureship; and, as this emolument was a very small one, it would appear that he did not rate at a high figure any profits which might come to him in the future from his acceptance of the Pope's offer; but, as he admits subsequently, he did not then fully realize the benevolence of the Cardinal who approached him on the subject, or the magnificent patronage of the Farnesi.[82] It is quite possible that this refusal of his may have been caused by a reluctance to quit Milan, the city which had treated him in such cruel and inhospitable fashion, just at the time when he had become a man of mark. In the arrogance of success it was doubtless a keen pleasure to let his fellow-townsmen see that the man upon whom they had heaped insult after insult for so many years was one who could afford to let Popes and Cardinals pray for his services in vain. But whatever may have been his humour, he resolved to remain in Milan; and, as he had no other public duty to perform except the delivery of the Plat lectures, he had abundant leisure to spend upon the many and important works he had on hand at this season.

Cardan had now achieved European fame, and was apparently on the high road to fortune, but on the very threshold of his triumph a great sorrow and misfortune befell him, the full effect of which he did not experience all at once. In the closing days of 1546 he lost his wife. There is very scant record of her life and character in any of her husband's writings,[83] although he wrote at great length concerning her father; and the few words that are to be found here and there favour the view that she was a good wife and mother. That Jerome could have been an easy husband to live with under any circumstances it is hard to believe. Lucia's life, had it been prolonged, might have been more free of trouble as the wife of a famous and wealthy physician; but it was her ill fortune to be the companion of her husband only in those dreary, terrible days at Sacco and Gallarate, and in the years of uncertainty which followed the final return to Milan. In the last-named period there was at least the Plat lectureship standing between them and starvation; but children increased the while in the nursery, and manuscripts in the desk of the physician without patients, and Lucia's short life was all consumed in this weary time of waiting for fame and fortune which, albeit hovering near, seemed destined to mock and delude the seeker to the end. Cardan was before all else a man of books and of the study, and it is not rare to find that one of this sort makes a harsh unsympathetic husband. The qualities which he attributes to himself in his autobiography suggest that to live with a man cursed with such a nature would have been difficult even in prosperity, and intolerable in trouble and privation. But fretful and irascible as Cardan shows himself to have been, there was a warm-hearted, affectionate side to his nature. He was capable of steadfast devotion to all those to whom his love had ever been given. His reverence for the memory of his tyrannical and irascible father had been noted already, and a still more remarkable instance of his fidelity and love will have to be considered when the time comes to deal with the crowning tragedy of his life. If Cardan had this tender side to his nature, if he could speak tolerant and even laudatory words concerning such a father as Fazio Cardano, and show evidences of a love strong as death in the fight he made for the life of his ill-starred and unworthy son, it may be hoped—in spite of his almost unnatural silence concerning her—that he gave Lucia some of that tenderness and sympathy which her life of hard toil and heavy sacrifice so richly deserved; and that even in the days when he sold her trinkets to pay his gambling losses, she was not destined to weep the bitter tears of a neglected wife. If her early married life had been full of care and travail, if she died when a better day seemed to be dawning, she was at least spared the supreme sorrow and disgrace which was destined to fall so soon upon the household. Judging by what subsequently happened, it will perhaps be held that fate, in cutting her thread of life, was kinder to her than to her husband, when it gave him a longer term of years under the sun.

FOOTNOTES:

[67] De Libris Propriis, Opera, tom. i. p. 102.

[68] Besides the De Malo Medendi Usu, he published in 1536 a tract upon judicial astrology. This, in an enlarged form, was reprinted by Petreius at Nuremburg in 1542.

[69] Cardan writes of Brissac: "Erat enim Brissacus Prorex singularis in studiosis amoris et humanitatis."—De Vita Propria, ch. iv. p. 14.

[70] "Mirumque in modum venenis cornu ejus adversari creditur."—De Subtilitate, p. 315. Sir Thomas Browne (Vulgar Errors, Bk. iii. 23) deals at length with the pretended virtues of the horn, and in the Bestiary of Philip de Thaun (Popular Treatises on Science during the Middle Ages) is given an account of the many wonderful qualities of the beast.

[71] De Vita Propria, ch. xxxiii. p. 105. He also alludes to this case in De Libris Propriis (Opera, tom. i. p. 65), affirming that the other doctors concerned in the case raised a great prejudice against him on account of his reputation as an astrologer. "Ita tot modis et insanus paupertate, et Astrologus profitendo edendoque libros, et imperitus casu illustris pueri, et modum alium medendi observans ex titulo libri nuper edito, jam prope ab omnibus habebar. Atque haec omnia in Urbe omnium nugacissima, et quae calumniis maxime patet."

[72] The founder of this family was Indico d'Avalos, a Spanish gentleman, who was chosen by Alfonso of Naples as a husband for Antonella, the daughter and heiress of the great Marchese Pescara of Aquino. This d'Avalos Marchese dal Guasto was the grandson of Indico. He commanded the advanced guard at the battle of Pavia, and took part in almost every battle between the French and Imperialists, and went with the Emperor to Tunis in 1535. Though he was a brave soldier and a skilful tactician, he was utterly defeated by d'Enghien at Cerisoles in 1544. He has been taxed with treachery in the case of the attack upon the messengers Rincon and Fregoso, who were carrying letters from Francis I. to the Sultan during a truce, but he did little more than imitate the tactics used by the French against himself; moreover, neither of the murdered men was a French subject, or had the status of an ambassador. D'Avalos was a liberal patron of letters and arts, and was very popular as Governor of Milan. He was a noted gallant and a great dandy. Brantome writes of him—"qu'il etait si dameret qu'il parfumait jusqu'aux selles de ses chevaux."—He died in 1546.

[73] "Violentia quorundam Medicorum adactus sum anno MDXXXVI, seu XXXVII, turpi conditione pacisci cum Collegio, sed ut dixi, postmodum dissoluta est, anno MDXXXIX et restitutus sum integre."—De Vita Propria, ch. xxxiii. p. 105.

[74] De Vita Propria, ch. xl. p. 133.—He gives a long list of cases of his successful treatment in Opera, tom. i. p. 82.

[75] There is a full account of this episode in De Libris Propriis, p. 128, and in De Vita Propria, ch. xl. p. 133.

[76] Exotericarum exercitationum, p. 987.

[77] Cardanus Comforte, translated into Englishe, 1573. It was the work of Thomas Bedingfield, a gentleman pensioner of Queen Elizabeth.

[78] De Vita Propria, ch. xxxvii. p. 116.

[79] "Delectant me gladii parvi, seu styli scriptorii, in quos plus viginti coronatis aureis impendi: multas etiam pecunias in varia pennarum genera, audeo dicere apparatum ad scribendum ducentis coronatis non potuisse emi."—De Vita Propria, ch. xviii. p. 57.

[80] De Vita Propria, ch. iv. p. 15.

[81] "At ego qui, ut dixi, Harpocraticus sum dicebam:—Summus Pont: decrepitus est: murus ruinosus, certa pro incertis derelinquam?"—De Vita Propria, ch. iv. p. 15. It is quite possible that Paul III. may have desired to have Cardan about him on account of his reputation as an astrologer, the Pope being a firm believer in the influence of the stars.—Vide Ranke, History of the Popes i. 166.

[82] "Neque ego tum Moroni probitatem, nec Pharnesiorum splendorem intelligebam."—De Vita Propria, ch. iv. p. 15.

[83] In writing of his own horoscope (Geniturarum Exempla, p. 461) he records that she miscarried thrice, brought forth three living children, and lived with him fifteen years. He dismisses his marriage as follows: "Duxi uxorem inexpectato, a quo tempore multa adversa concomitata sunt."—De Vita Propria, ch. xli. p. 149. But in De Rerum Subtilitate, p. 375, he records his grief at her death:—"Itaque cum a luctu dolor et vigilia invadere soleant, ut mihi anno vertente in morte uxoris Luciae Bandarenae quanquam institutis philosophiae munitus essem, repugnante tamen natura, memorque vinculi cojugalis, suspiriis ac lachrymis et inedia quinque dierum, a periculo me vindicavi."



CHAPTER V

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 Geometriae, which he followed up shortly after by the publication of a work, Quindecim Libri Novae Geometriae. 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 Magnae 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 (x^3 + 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 x^3 + px = q, and the new forms of cubic equation which Tartaglia elaborated were as follows: x^3 + px^2 = q: and x^3 - px^2 = 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:

"Quando chel cubo con le cose apresso Se agualia a qualche numero discreto Trouan dui altri differenti in esso Dapoi terrai questo per consueto Ch'el lor' produtto sempre sia eguale Al terzo cubo delle cose neto El residuo poi suo generale Delli lor lati cubi ben sottratti Varra la tua cosa principale. In el secondo de cotesti atti Quando chel cubo restasse lui solo Tu osseruarai quest' altri contratti Del numer farai due tal part 'a uolo Che luna in l'altra si produca schietto El terzo cubo delle cose in stolo Delle qual poi, per commun precetto Torrai li lati cubi insieme gionti Et cotal summa sara il tuo concetto Et terzo poi de questi nostri conti Se solve col recordo se ben guardi Che per natura son quasi congionti Questi trouai, et non con passi tardi Nel mille cinquecent' e quatro e trenta Con fondamenti ben sald' e gagliardi Nella citta del mar' intorno centa."

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 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 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 Naude.[106] In the course of his essay Naude 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, Naude 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 Naude,[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 x^3 + 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. x^3 + bx = c. x^3 = bx + c. and x^3 + 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, Ptolemaeus, 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]

FOOTNOTES:

[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, misere vitam ibi, ne dicam crudeliter, finivit: nam per quindecim dies in profundissima gorgyne fuit, ut vivus sepeliretur."

[85] There is a reference to Osiander in De Subtilitate, p. 523. Cardan gives a full account of his relations with Osiander and Petreius in Opera, tom. i. p. 67.

[86] November 1536.

[87] Ferrari was one of Cardan's most distinguished pupils. "Ludovicus Ferrarius Bononiensis qui Mathematicas et Mediolani et in patria sua professus est, et singularis in illis eruditionis."—De Vita Propria, ch. xxxv. p. 111. There is a short memoir of Ferrari in Opera, tom. ix.

[88] Opera, tom. i. p. 66.

[89] Fra Luca's book, Summa de Arithmetica Geometria Proportioni e Proportionalita, extends as far as the solution of quadratic equations, of which only the positive roots were used. At this time letters were rarely used to express known quantities.

[90] The early writers on Algebra used numerus for the absolute or known term, res or cosa for the first power, quadratum for the second, and cubus for the third. The signs + and - first appear in the work of Stifelius, a German writer, who published a book of Arithmetic in 1544. Robert Recorde in his Whetstone of Wit seems first to have used the sign of equality =. Vieta in France first applied letters as general symbols of quantity, though the earlier algebraists used them occasionally, chiefly as abbreviations. Aristotle also used them in the Physics.—Libri. Hist. des Sciences Mathematiques. i. 104.

[91] Opera, tom. iv. p. 222.

[92] In the conclusion of the Treatise on Arithmetic, Cardan points out certain errors in the work of Fra Luca. Fra Luca was a pupil of Piero della Francesca, who was highly skilled in Geometry, and who, according to Vasari, first applied perspective to the drawing of the human form.

[93] Tartaglia, i.e. the stutterer.

[94] Papadopoli, Hist. Gymn. Pata. (Ven. 1724).

[95] "Balbisonem post relatam jurisprudentiae lauream redeuntem Brixiam Nicolaus secutus est, caepitque ex Mathematicis gloriam sibi ac divitias parare, aeque paupertatis impatiens, ac fortunae melioris cupidus, quam dum Brixiae tuetur, homo morosae, et inurbanae rusticitatis prope omnium civium odia sibi conciliavit. Quamobrem alibi vivere coactus, varias Italiae urbes incoluit, ac Ferrariae, Parmae, Mediolani, Romae, Genuae, arithmeticam, geometricam, ceteraque quae ad Mathesim pertinent, docuit; depugnavitque scriptis accerrimis cum Cardano ac sibi ex illis quaesivit nomen et gloriam. Tandem domicilium posuit Venetiis, ubi non a Senatoribus modo, ut mos Venetus habet eruditorum hominum studiosissimus, maximi habitus est, at etiam a variis Magnatum ac Principum legatis praemiis ac muneribus auctus sortem, quam tamdiu expetierat visus sibi est conciliasse. Ergo ratus se majorem, quam ut a civibus suis contemneretur, Brixiam rediit, ubi spe privati stipendii Euclidis elementa explanare coepit; sed quae illum olim a civitate sua austeritas, rustica, acerba, morosa, depulerat, eadem illum in eum apud omnes contemptum, et odium iterum dejicit, ut exinde horrendus ac detestabilis omnibus fugere, atque iterum Venetias confugere compulsus fuerit. Ibi persenex decessit."—Papadopoli, Hist. Gymn. Pata., ii. p. 210.

[96] This work is the chief authority for the facts which follow. The edition referred to is that of Venice, 1546. There is also a full account of the same in Cossali, Origine dell' Algebra (Parma, 1799). vol. ii. p. 96.

[97] Quesiti et Inventioni, p. 115.

[98] Cardan writes: "Vi supplico per l'amor che mi portati, et per l'amicitia ch'e tra noi, che spero durara fin che viveremo, che mi mandati sciolta questa questione. 1 cubo piu 3. cose egual a 10." Cardan had mistaken (1/3 b)^3 for 1/3 b^3, or the cube of 1/3 of the co-efficient for 1/3 of the cube of the co-efficient.—Quesiti et Inventioni p. 124.

[99] Quesiti et Inventioni, p. 125.

[100] "Non ha datta fora tal opera come cose composto da sua testa ma come cose ellette raccolte e copiate de diverse libri a penna."—Quesiti et Inventioni, p. 127.

[101] Cardan repeats the remark in the first chapter of the Liber Artis Magnae (Opera, tom. iv. p. 222). "Deceptus enim ego verbis Lucae Paccioli, qui ultra sua capitula, generale ullum aliud esse posse negat (quanquam tot jam antea rebus a me inventis, sub manibus esset) desperabam tamen invenire, quod quaerere non audebam." Perhaps he wrote them down as an apology or a defence against the storm which he anticipated as soon as Tartaglia should have seen the new Algebra.

[102] Subsequently Tartaglia wrote very bitterly against Cardan, as the latter mentions in De Libris Propriis. "Nam etsi Nicolaus Tartalea libris materna lingua editis nos calumniatur, impudentiae tamen ac stultitiae suae non aliud testimonium quaeras, quam ipsos illius libros, in quibus nominatim splendidiorem unumquemque e civibus suis proscindit: adeo ut nemo dubitet insanisse hominem aliquo infortunio."—Opera, tom. i. p. 80.

[103] Quesiti et Inventioni, p. 129.

[104] Montucla, Histoire de Math. i. 596, gives a full account of Ferrari's process.

[105] In the De Vita Propria, Cardan dismisses the matter briefly: "Ex hoc ad artem magnam, quam collegi, dum Jo. Colla certaret nobiscum, et Tartalea, a quo primum acceperam capitulum, qui maluit aemulum habere, et superiorem, quam amicum et beneficio devinctum, cum alterius fuisset inventum."—ch. xlv. p. 175.

[106] Prefixed to the De Vita Propria.

[107] In a question of broken faith, Cardan laid himself open especially to attack by reason of his constant self-glorification in the matter of veracity.

[108] Leonardo knew that quadratic equations might have two positive roots, and Cardan pursued this farther by the discovery that they might also have negative roots.

[109] "Caput xxviii. De capitulo generali cubi et rerum aequalium numero, Magistri Nicolai Tartagliae, Brixiensis—Hoc capitulum habui a prefato viro ante considerationem demonstrationum secundi libri super Euclidem, et aequatio haec cadit in [Symbol: Rx]. cu v binomii ex genere binomii secundi et qunti m. [Symbol: Rx]. cuba universali recisi ejusdem binomii."—Opera, tom. iv. p. 341.

[110] Montucla, who as a historian of Mathematics has a strong bias against Cardan, gives him credit for the discovery of the fictae radices, but on the other hand he attributes to Vieta Cardan's discovery of the method of changing a complete cubic equation into one wanting the second term.—Ed. 1729, p. 595.

[111] Opera, tom. i. p. 66.



CHAPTER VI

IT has been noted that Cardan quitted Pavia at the end of 1544 on account of the bankruptcy of the University, and that in 1546 a generous offer was made to him on condition of his entering the service of Pope Paul III.; an offer which after some hesitation he determined to refuse. In the autumn of this same year he resumed his teaching at Pavia, a fact which sanctions the assumption that this luckless seat of learning must have been once more in funds. In the year following, in 1547, there came to him another offer of employment accompanied by terms still more munificent than the Pope's, conveyed through Vesalius[112] and the ambassador of the King of Denmark. "The emolument was to be a salary of three hundred gold crowns per annum of the Hungarian currency, and in addition to these six hundred more to be paid out of the tax on skins of price. This last-named money differed in value by about an eighth from the royal coinage, and would be somewhat slower in coming in. Also the security for its payment was not so solid, and would in a measure be subject to risk. To this was farther added maintenance for myself and five servants and three horses. This offer I did not accept because the country was very cold and damp, and the people well-nigh barbarians; moreover the rites and doctrines of religion were quite foreign to those of the Roman Church."[113]

Cardan was now forty-six years of age, a mathematician of European fame, and the holder of an honourable post at an ancient university, which he might have exchanged for other employment quite as dignified and far more lucrative. In dealing with a character as bizarre as his, it would be as a rule unprofitable to search deeply for motives of action, but in this instance it is no difficult matter to detect upon the surface several causes which may have swayed him in this decision to remain at Pavia. However firmly he may have set himself to win fame as a physician, he was in no way disposed to put aside those mathematical studies in which he had already made so distinguished a name, nor to abandon his astrology and chiromancy and discursive reading of all kinds. At Pavia he would find leisure for all these, and would in addition be able to make good any arrears of medical and magical knowledge into which he might have fallen during the years so largely devoted to the production of the Book of the Great Art. Moreover, the time in question was one of the prime epochs in the history of the healing art. A new light had just arisen in Vesalius, who had recently published his book, Corporis Humani Fabrica, and was lecturing in divers universities on the new method of Anatomy, the actual dissection of the human body. He went to Pavia in the course of his travels and left traces of his visit in the form of a revived and re-organized school of Anatomy. This fact alone would have been a powerful attraction to Cardan, ever greedy as he was of new knowledge, but there was another reason which probably swayed him more strongly still, to wit, the care of his eldest son's education and training. Gian Battista Cardano was now in his fourteenth year, and, according to the usages of the time, old enough to make a beginning of his training in Medicine, the profession he was destined to follow. It is not recorded whether or not he chose this calling for himself; but, taking into account the deep and tender affection Jerome always manifested towards his eldest son, it is not likely that undue compulsion was used in the matter. The youth, according to his father's description, strongly favoured in person his grandfather Fazio.[114] He had come into the world at a time when his parents' fortunes were at their lowest ebb, during those terrible months spent at Gallarate,[115] and in his adolescence he bore divers physical evidences of the ill nurture—it would be unjust to call it neglect—which he had received. At one time he was indeed put in charge of a good nurse, but he had to be withdrawn from her care almost immediately through her husband's jealousy, and he was next sent to a slattern, who fed him with old milk, and not enough of that; or more often with chewed bread. His body was swollen and unhealthy, he suffered greatly from an attack of fever, which ultimately left him deaf in one ear. He gave early evidence of a fine taste in music, an inheritance from his father, and was, according to Cardan's showing, upright and honest in his carriage, gifted with talents which must, under happier circumstances, have placed him in the first rank of men of learning, and in every respect a youth of the fairest promise. The father records that he himself, though well furnished by experience in the art of medicine, was now and again worsted by his son in disputation, and alludes in words of pathetic regret to divers problems, too deep for his own powers of solution, which Gian Battista would assuredly have mastered in the course of time. He does not forget to notice certain of the young man's failings; for he remarks that he was temperate of speech, except when he was angered, and then he would pour forth such a torrent of words that he scarce seemed in his right mind. Cardan professes to have discerned a cause for these failings, and the calamities flowing therefrom, in the fact that Gian Battista had the third and fourth toes of his right foot united by a membrane; he declares that, if he had known of this in time, he would have counteracted the evil by dividing the toes.[116] Gian Battista eventually gained the baccalaureat in his twenty-second year, and two years after became a member of the College.

The life which Cardan planned to lead at Pavia was unquestionably a full one. He had several young men under his care as pupils besides his son, amongst them being a kinsman of his, Gasparo Cardano, a youth of sterling virtue and a useful coadjutor in times to come. He was at this time engaged on his most important works in Medicine and Physical Science. He worked hard at his profession, practising occasionally and reading voraciously all books bearing on his studies. He wrote and published several small works during the four years—from 1547 to 1551—of his Professorship at Pavia; the most noteworthy of which were the Book of Precepts for the guidance of his children, and some Treatises on the Preservation of Health. He also wrote a book on Physiognomy, or as he called it Metoposcopy, an abstract of which appears as a chapter in De Utilitate (lib. iii. c. 10), but the major part of his time must have been consumed in collecting and reducing to form the huge mass of facts out of which his two great works, De Subtilitate and De Varietate Rerum, were built up.

A mere abstract of the contents of these wonderful books would fill many pages, and prove as uninteresting and unsuggestive as abstracts must always be; and a commentary upon the same, honestly executed, would make a heavy draft on the working life-time of an industrious student. In reference to each book the author has left a statement of the reasons which impelled him to undertake his task, the most cogent of which were certain dreams.[117] Soon after he had begun to write the De Astrorum Judiciis he dreamt one night that his soul, freed from his body, was ranging the vault of heaven near to the moon, and the soul of his father was there likewise. But he could not see this spirit, which spake to him saying, "Behold, I am given to you as a comrade." The spirit of the father then went on to tell the son how, after various stages of probation, he would attain the highest heaven, and in the terms of this discourse Cardan professed to discern the scheme of his more important works.

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