The application of algebra was also at this period very limited; it was confined almost entirely to the resolution of certain questions of no great interest about numbers. No idea was then entertained of that extensive application which it has received in modern times.
The knowledge which the early algebraists had of their science was also circumscribed: it extended only to the resolution of equations of the first and second degrees; and they divided the last into cases, each of which was resolved by its own particular rule. The important analytical fact, that the resolution of all the cases of a problem may be comprehended in a single formula, which may be obtained from the solution of one of its cases, merely by a change of the signs, was not then known: indeed, it was long before this principle was fully comprehended. Dr Halley expresses surprise, that a formula in optics which he had found, should by a mere change of the signs give the focus of both converging and diverging rays, whether reflected or refracted by convex or concave specula or lenses; and Molyneux speaks of the universality of Halley's formula as something that resembled magic.
The rules of algebra may be investigated by its own principles, without any aid from geometry; and although in some cases the two sciences may serve to illustrate each other, there is not . now the least necessity in the more elementary parts to call in the aid of the latter in expounding the former. It was otherwise in former times. Lucas de Burgo found it to be convenient, after the example of Leonardo, to employ geometrical constructions to prove the truth of his rules for resolving quadratic equations, the nature of which he did not completely comprehend; and he was induced by the imperfect nature of his notation to express his rules in Latin verses, which will not now be read with the kind of satisfaction we receive from the perusal of the well-known poem, "the Loves of the Triangles."
As Italy was the first European country where algebra became known, it was there that it received its earliest improvements. The science had been nearly stationary from the days of Leonardo to the time of Paciolus, a period of three centuries; but the invention of printing soon excited a spirit of improvement in all the mathematical sciences. Hitherto an imperfect theory of quadratic equations was the limit to which it had been earned. At last this boundary was passed, and about the year 1505 a particular case of eqtiations of the third degree was resolved by Scipio reus. Ferreus.Ferreus, a professor of mathematics in Bononia. This was an important step, because it showed that the difficulty of resolving equations of the higher orders, at least in the case of the third degree, was not insurmountable, and a new field was opened for discovery. It was then the practice among the cultivators of algebra, when they advanced a step, to conceal it carefully from their contemporaries, and to challenge them to resolve arithmetical questions, so framed as to require for their solution a knowledge of their own new-found rules. In this spirit did Ferreus make a secret of his discovery: he communicated it, however, to a favourite scholar, a Venetian named Florido. About the year 1535 this person, having taken up his residence at taitu. Venice, challenged Tartalea of Brescia, a man of great ingenuity, to a trial of skill in the resolution of problems by algebra. Florido framed his questions so as to require for their solution a knowledge of the rule which he had learned from his preceptor Ferreu-s; but Tartalea had, five years before this time, advanced further than Ferreus, and was more than a match for Florido. He therefore accepted the challenge, and a day was appointed when each was to propose to the other thirty questions. Before the time came, Tartalea had resumed the study of cubic equations, and had discovered the solution of two cases in addition to two which he knew before. Floridb's questions were such as could be resolved by the single rule of Ferreus; while, on the contrary, those of Tartalea could only be resolved by one or other of three rules, which he himself had found, but which could not be resolved by the remaining rule, which was also that known to Florido. The issue of the contest is easily anticipated; Tartalea resolved all his adversary's questions in two hours, without receiving one answer from him in return.
Cardan.The celebrated Cardan was a contemporary of Tartalea. This remarkable person was a professor of mathematics at Milan, and a physician. He had studied algebra with great assiduity, and had nearly finished the printing of a book on arithmetic, algebra, and geometry; but being desirous of enriching his work with the discoveries of Tartalea, which at that period must have been the object of considerable attention among literary men in Italy, he endeavoured to draw from him a disclosure of his rules. Tartalea resisted for a time Cardan's entreaties. At last, overcome by his importunity, and his offer to swear on the holy Evangelists, and by the honour of a gentleman, never to publish them, and on his promising on the faith of a Christian to commit them to cypher, so that even after his death they would not be intelligible to any one, he ventured with much hesitation to reveal to him his practical rules, which were expressed by some very bad Italian verses, themselves in no small degree enigmatical. He reserved, however, the demonstrations. Cardan was not long in discovering the reason of the rules, and he even greatly improved them, so as to make them in a manner his own. From the imperfect essays of Tartalea he deduced an ingenious and systematic method of resolving all cubic equations whatsoever; but with a remarkable disregard for the principles of honour, and the oath he had taken, he published in 1545 Tartalea's discoveries, combined with his own, as a supplement to a treatise on arithmetic and algebra, which he had published six years before. This work is remarkable for being the second printed book on algebra known to have existed.
In the following year Tartalea also published a work on algebra, which he dedicated to Henry VIII., king of England.
It is to be regretted that in many instances the authors of important discoveries have been overlooked, while the honours due to them have been transferred to others having only secondary pretensions. The formulae for the resolution of cubic equations are now called Cardan's rules, notwithstanding the prior claim of Tartalea. It must be confessed, however, that he evinced considerable selfishness in concealing his discovery; and although Cardan cannot be absolved from the charge of bad faith, yet it must be recollected that by his improvements in what Tartalea communicated to him, he made the discovery in some measure his own; and he had moreover the high merit of being the first to publish this important improvement in algebra to the world.
The next step in the progress of algebra was the discovery of a method of resolving equations of the fourth order. An Italian algebraist had proposed a question which could not be resolved by the newly invented rules, because it produced a biquadratic equation. Some supposed that it could not be resolved at all; but Cardan was of a different opinion. Ferrari.He had a pupil named Lewis Ferrari, a young man Ferrari. of great genius, and an ardent student in the algebraic analysis: to him Cardan committed the solution of this difficult question, and he was not disappointed. Ferrari not only resolved the question, but he also found a general method of resolving equations of the fourth degree, by making them depend on the solution of equations of the third degree.
