Cardan and Tartaglia. 3. Jerome Cardan is, as it were, the founder of the higher algebra; for, whatever he may have borrowed from others, we derive the science from his Ars Magna, published in 1545. It contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan’s name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. |Cubic equations.| The original inventor appears to have been Scipio Ferreo, who, about 1505, by some unknown process, discovered the solution of a single case; that of x3 + p x = q. Ferreo imparted the secret to one Fiore, or Floridus, who challenged Tartaglia to a public trial of skill, not unusual in that age. Before he heard of this, Tartaglia, as he assures us himself, had found out the solution of two other forms of cubic equation; x3 + p x2 = q; and x3 - p x2 = q. When the day of trial arrived, Tartaglia was able not only to solve the problems offered by Fiore, but to baffle him entirely by others which resulted in the forms of equation, the solution of which had been discovered by himself. This was in 1535; and four years afterwards Cardan obtained the secret from Tartaglia under an oath of secrecy. In his Ars Magna, he did not hesitate to violate this engagement; and though he gave Tartaglia the credit of the discovery, revealed the process to the world.[835] He has said himself, that by the help of Ferrari, a very good mathematician, he extended his rule to some cases not comprehended in that of Tartaglia; but the best historian of early algebra seems not to allow this claim.[836]
[835] Playfair, in his second dissertation in the Encyclopædia Britannica, though he cannot but condemn Cardan, seems to think Tartaglia rightly treated for having concealed his discovery; and others have echoed this strain. Tartaglia himself says in a passage I have read in Cossali, that he meant to have divulged it ultimately; but in that age money as well as credit was to be got by keeping the secret; and those who censure him wholly forget, that the solution of cubic equations was, in the actual state of algebra, perfectly devoid of any utility to the world.
[836] Cossali, Storia Critica d’Algebra (1797), ii. 96, &c. Hutton’s Mathematical Dictionary-Montucla, i. 591. Kästner, i. 152.
Beauty of the discovery. 4. This writer, Cossali, has ingeniously attempted to trace the process by which Tartaglia arrived at this discovery;[837] one which, when compared with the other leading rules of algebra, where the invention, however useful, has generally lain much nearer the surface, seems an astonishing effort of sagacity. Even Harriott’s beautiful generalisation of the composition of equations was prepared by what Cardan and Vieta had done before, or might have been suggested by observation in the less complex cases.[838]
[837] Ibid. p. 145. Tartaglia boasts of having discovered that the cube of p + q = p3 + p2 q + p q2 + q3. Such was the ignorance of literal algebra; yet in this state of the science he solved cubic equations.
[838] Cardan strongly expresses his sense of this recondite discovery. And as the passage in which he retraces the early progress of algebra is short, and is quoted from Cardan’s works, which are scarce in England, by Kästner, who is himself not very commonly known here, I shall transcribe the whole passage, as a curiosity for our philomaths. Hæc ars olim a Mahomete Mosis Arabis filio initium sumpsit. Etenim hujus rei locuples testis Leonardus Pisanus. Reliquit autem capitula quatuor, cum suis demonstrationibus quas nos locis suis ascribemus. Post multa vero temporum intervalla tria capitula derivativa addita illis sunt, incerto autore, quæ tamen cum principalibus a Luca Paciolo posita sunt. Demum etiam ex primis, alia tria derivativa, a quodam ignoto viro inventa legi, hæc tamen minimè in lucem prodierant, cum essent aliis longe utiliora, nam cubi et numeri et cubi quadrati æstimationem docebant. Verum temporibus nostris Scipio Ferreus Bononiensis, capitulum cubi et rerum numero æqualium [x3 + p x = q] invenit, rem sane pulchram et admirabilem: cum omnem humanam subtilitatem, omnis ingenii mortalis claritatem ars hæc superet, donum profecto cœleste, experimentum autem virtutis animorum, atque adeo illustre, ut qui hæc attigerit nihil non intelligere posse se credat. Hujus æmulatione Nicolaus Tartalea Brixellensis, amicus noster, cum in certamen cum illius discipulo Antonio Maria Florido venisset, capitulum idem ne vinceretur invenit, qui mihi ipsum multis precibus exoratus tradidit. Deceptus enim ego verbis Lucæ Pacioli, qui ultra sua capitula generale ullum aliud esse posse negat (quanquam tot jam antea rebus a me inventis sub manibus esset, desperabam) tamen [et?] invenire q. quærere non audebam. [sic, sed perperam nonnihil scribi liquet]. Inde autem illo habito demonstrationem venatus, intellexi complura alia posse haberi. Ac eo studio, auctaque jam confidentia, per me partim, ac etiam aliqua per Ludovicum Ferrarium, olim alumnum nostrum, inveni. Porro quæ ab his inventa sunt, illorum nominibus decorabuntur, cætera quæ nomine carent nostra sunt. At etiam demonstrationes, præter tres Mahometis, et duas Ludovici, omnes nostræ sunt, singulæque capitibus suis præponentur, inde regula addita, subjicietur experimentum. Kästner, p. 152. The passage in Italics is also quoted by Cossali, p. 159.
Cardan’s other discoveries. 5. Cardan, though not entitled to the honour of this discovery, nor even equal, perhaps, in mathematical genius to Tartaglia, made a great epoch in the science of algebra; and, according to Cossali and Hutton, has a claim to much that Montucla has unfairly or carelessly attributed to his favourite Vieta. “It appears,” says Dr. Hutton, “from this short chapter (lib. x. cap. 1. of the Ars Magna), that he had discovered most of the principal properties of the roots of equations, and could point out the number and nature of the roots, partly from the signs of the terms, and partly from the magnitudes and relations of the coefficients.” Cossali has given the larger part of a quarto volume to the algebra of Cardan; his object being to establish the priority of the Italian’s claim to most of the discoveries ascribed by Montucla to others, and especially to Vieta. Cardan knew how to transform a complete cubic equation into one wanting the second term; one of the flowers which Montucla has placed on the head of Vieta; and this he explains so fully, that Cossali charges the French historian of mathematics with having never read the Ars Magna.[839] Leonard of Pisa had been aware that quadratic equations might have two positive roots; but Cardan first perceived, or at least first noticed, the negative roots, which he calls “fictæ radices.”[840] In this perhaps there is nothing extraordinary; the algebraic language must early have been perceived by such acute men as exercised themselves in problems to give a double solution of every quadratic equation; but, in fact, the conditions of these problems, being always numerical, were such as to render a negative result practically false, and impertinent to the solution. It is therefore, perhaps, without much cause that Cossali triumphs in the ignorance shown of negative values by Vieta, Bachet, and even Harriott, though Cardan had pointed them out;[841] since we may better say, that they did not trouble themselves with what, in the actual application of algebra, could be of no utility. Cardan also discovered that every cubic equation has one or three real roots; and that there are as many positive or true roots as changes of signs in the equation; that the coefficient of the second term is equal to the sum of the roots, so that where it is wanting, the positive and negative values must compensate each other;[842] and that the known term is the product of all the roots. Nor was he ignorant of a method of extracting roots by approximation; but in this again the definiteness of solution, which numerical problems admit and require, would prevent any great progress from being made.[843] The rules are not perhaps all laid down by him very clearly; and it is to be observed that he confined himself chiefly to equations not above the third power; though he first published the method of solving biquadratics, invented by his coadjutor Ferrari. Cossali has also shown that the application of algebra to geometry, and even to the geometrical construction of problems, was known in some cases by Tartaglia and Cardan; thus plucking another feather from the wing of Vieta, or of Descartes. It is a little amusing to see that, after Montucla had laboured with so much success to despoil Harriott of the glory which Wallis had, perhaps with too national a feeling, bestowed upon him for a long list of discoveries contained in the writings of Vieta, a claimant by an older title started up in Jerome Cardan, who, by help of his very accomplished advocate, seems to have established his right at the expense of both.
[839] P. 164.
[840] Montucla gives Cardan the credit due for this; at least in his second edition (1799), p. 595.
[841] i. 23.