If Cardan had left nothing but writings on astrology, mathematics, medicine, or morals, he would have passed among the rest as an eccentric genius, with a full share of all the folly and mysticism which pervaded the philosophy of his day. It is to his own account of him self that we are indebted for the quantity of description and speculation relative to his personal character which is found in all biographies. There may be in this production a touch of the insanity which delights In aconsing itself of crimes, or in exaggerating its foibles : as it is, and taking the character of Carden as he has given it himself, we see a man —of unequalled self-conceit, as when he says his book of logio (written in seven days, but hardly to be understood by any one else in a year) has not a letter either of omission or superfluity; and that he is born to deliver the world from a multitude of errors : of little benevolence, as when he avows that his greatest delight in conversation is to say things which be knows will be disagreeable to his hearers : of no veracity, as witness his assertion that be acquired a perfect knowledge of Greek, Latin, French, and Spanish in twenty-fonr hours from an edition of Apuleius : of violent temper, instanced by his striking one in the face with a dagger whom he discovered to be cheating him at play : and of little honesty, as evidenced by his justification of his refusal to return a pledge, namely, that it was deposited in presence of no witness. He was also a superstitions free-thinker; attached to his religion, but disposed to treat it in his own way, to an extent which made a worthy divine who claimed, we suppose, to be the adjutant-general of heretics, call him the " chief of the hidden Atheists of the second class." His refusal to accept an advantageous settlement in Denmark, on condition of apostatising, ought to establish his right to some principle. llis four gifts—I. the power of throwing his soul ont of his body (for his words can mean nothing less)-2. his faculty of seeing whatever he pleased with his eyes, °culls, non vi mantis '-3. his dreams, which uniformly and on.every occasion foretold what was to happen to him; and 4. his finger nails, which did the same thing; to say nothing of his astrology, his good demon, &c., ke.—establish his claim to be the chief of the visionaries "of the first class." Bayle has drawn the distinction between him and other men of equal talent with some point : he says that "nullum magnum ingenium sine mixtura dementim " is not a maxim which includes Cardan ; for that with him the folly is improved by talent, not the talent adulterated by folly.
It would hardly have been worth while to have entered into the preceding detail, if Cardan had been a common man. As a physician, his reputation extended through Europe, both as a practitioner and a writer. In 1552 he went to Scotland to the assistance of Hamilton, archbishop of St. Andrews, whom he cured : in the memoirs of Melvill the fact is stated, and Carden is mentioned by name, with the addition that he was an Italian magician. His medical writings have procured him no lasting reputation : those who follow such pursuits seem to have tacitly consented that Cardan shall be left to the mathematicians; and it is to his discoveries in algebra that he must be considered as entitled to a prominent place in biography. Before proceeding to consider him in this:, character, we shall only state that De Thou, who knew him personally, and records that he always dressed in a different manner from the rest of the world, says that it was commonly believed his end arose from starvation, voluntarily undergone, that he might not outlive the time which ha had predicted for his own death. This story has been frequently copied, as if the fact had beau positively asserted by the historian, whereas he only speaks of a rumour.
The 'Ara Magna,' published in 1545, contains the extensions which Cardan made in the solution of equations. Algebra was then an art contained not in formulae but in rules, and extended no further than the methods of solving numerical equations of the second degree. We shall not here enter into the celebrated dispute between Carden and Tartaglia, further than to specify the part taken by the former.
When he was informed of the solution of cubic equations which Tartaglia had discovered, he applied to the latter, March 25, 1539, and requested he would communicate his method, which Tartaglia declined, intending to reserve the same for the work which he published after.
wards in 1551. Carden then swore "upon the Holy Gospels, and the faith of a gentleman," that he would not only not divulge the secret, but would engage to write it in such a cipher as no one should be able to read in case of his death. Tartaglia, upon this assurance, commu nicated his method. This detail rests upon the authority of Tartaglia himself (' Qnesiti et Inventioni: folio, 120), but is amply confirmed by Cardan's subsequent letters, and was never denied by him. Notwith standing his word thus pledged Cardan published these methods in his 'Ars Magna' (1545), giving the credit of them indeed to Tartaglia, but concealing the promise he had made.
The communication made to Carden amounted to the solution, without demonstration, of e+ ax + b = 0, in the cases where a and b are, one or both, negative. Carden himself supplied the demonstrations, showed how to reduce all equations of the third degree to the preceding form, and how to extract the cube root of the binomial surd quantities which the well-known solution involves. He may be said to have arrived, in detached and isolated theorems, at as much, relative to equations of the third degree, as could afterwards be established, in the time of Des Cartes, for equations of all degrees. He was the first who considered negative roots, and comprehended the nature of the connection between them and the positive roots of other equations; and he even gave the first idea of a method of approximation.
The algebra of Cardan, owing to the want of general symbols, is difficult to read ; and Montucla, biassed perhaps in favour of his countryman Vista, has somewhat underrated his merits. On the other band we have Cossali (' Origine, &c., dell' Algebra,' Parma, 1797), whose object it seems to be to discover something like modern and symbolic analysis in the obscure and verbal rules of the Italians of the 16th century. If this learned and estimable writer be considered as holding a brief for Tartaglia, Carden, and Bombelli, his work may be highly usefuL For instance, when he shows, by collecting the various cases propounded by Carden, that the latter had all the elements which if put together would have been the celebrated rule of signs of Descartes, and thence affirms that Carden was in possession of that rule so far as equations of the third degree were concerned, he forgets that Carden neither did nor could put those elements together. And when he attributes a symbolic (or, as it was technically called, a specious) notation to Carden, because the latter sometimes uses a letter to stand for a number in his general enunciations, he does not remember that Euclid has a prior claim, if in that circumstance merely consists the leading feature of the method of Viet& There is in the algebra of Carden considerable power of developing the details of his subject, and of explaining the modifications presented by solutions, but not much inventive sagacity. He states himself that he was originally prevented from attempting the solution of cubic equations by the simple assertion of Lucas di Berge, in his work on algebra, that the solution was impossible ; though Cossali has shown that, had ho even read that author with attention, he would have seen that the assertion was not meant to apply to more than algebra as it then existed. In the case of biquadratie equations he attempted nothing himself, but requested his pupil, Ludovico Ferrari, to uuder take the investigation, who accordingly produced tho reduction now know by his name, and which was published by Carden. But if wo take the whole extent of the 'Ars Magna,' it is sufficiently obvious that Ouvian week! bare been an analyst of considerable power if be bad Wed alter Vieth.
There le in the second rolnme of Dr. Button's ' Tracts' an account of tie' Are Magna.' the most complete of which know in English. CA R D I. LU V I CO. (Ct Gott] the name of two very able Florentine painters, brother*, who vet tied and chiefly resided in Spain, whore, agreeably to Spaisieh orthography, they wrote their name Carducho.