Before his time very little advance had been made in the solution of any equations high er than the second degree ; except that, as we are told, about the year 1508, Scipio Ferrei, professor of mathematics at Bologna, had found out a rule for resolving one of the cases of cubic equations, which, however, he concealed, or communicated only to a few of his scholars. One of these, Florido, on the strength of the secret he possessed, agreeably to a practice then common among mathematicians, challenged Tartalea of Brescia, to con tend with him in the solution of algebraic problems. Florid() had at first the advantage ; but Tamales, being a man of ingenuity, soon discovered his rule, and also another much more general, in consequence of which, he came off at last victorious. By the report of this victory, the curiosity of Cardan was strongly excited ; for, though he was himself much versed in the mathematics, he had not been able to discover a method of resolving equa tions higher than the second degree. By the most earnest and importunate solicitation, he wrung from Tartalea the secret of his rules, but not till he had bound himself, by pro mises and oaths, never to divulge them. Tartalea did not communicate the demonstra tions, which, however, Cardan soon found out, and extended, in a very ingenious and systematic manner; to all cubic equations whatsoever. Thus possessed of an important discovery, which was at least in a great part his own, he soon forgot his promises to Tar talea, and published the whole in 154.5, not concealing, however, what he owed to the latter. Though a proceeding, so directly contrary to an express stipulation, cannot be defended, one does not much regret the disappointment of any man who would make a mystery of knowledge, or keep his discoveries a secret, for purposes merely selfish.
Thus was first published the rule which still bears the name of Cardan, and which, at this day, marks a point in the progress of algebraic investigation, which all the efforts of succeeding analysts have hardly been able to go beyond. As to the general doctrine of equations, it appears that Cardan was acquainted both.witt the negative and positive roots, the former of which he called by the name of false roots. He also knew that the num ber of positive, or, as he called them, true roots, is equal to the number of the changes of the signs of the terms ; and that the coefficient of the second term is the difference be tween the sum of the true and the false roots. He also had perceived the difficulty of that case of cubic equations, which cannot be reduced to his own rule. He was not able to overcome the difficulty, but showed how, in all cases, an approximation to the roots might be obtained.
There is the more merit in these discoveries, that the language of Algebra still remain ed very imperfect, and consisted merely of abbreviations of words. Mathematicians were then in the practice of putting their rules into verse. Cardan has given his a poetical dress, in which, as may be supposed, they are very awkward and obscure ; for whatever assistance in this way is given to the memory, must be entirely at the expellee of the un derstanding. It is, at the same time, a proof that the language of Algebra was very im
perfect. Nobody now thinks of translating an algebraic formula into verse ; because, if one has acquired any familiarity with the language of the science, the formula will be more easily remembered than any thing that can be substituted in its room.
Italy was not the only country into which the algebraic analysis had by this time found its way; in Germany it had also made considerable progress, and Stiphelius, in a book of Algebra, published at Nuremberg in 1544, employed the same numeral exponents of powers, both positive and negative, which we now use, as far as integer numbers are con cerned ; but he did not carry the solution of equations farther than the second degree. He introduced the same characters for plus and minus which are at present employed.
Robert Recorde, an English mathematician, published about this time, or a few years later, the first English treatise on Algebra, and he there introduced the same sign of equa lity which is now in use.
The properties of algebraic equations were discovered, however, very slowly. Pelitarius, a French mathematician, in a treatise which bears the date of 1658, is the first who ob served that the root of an equation is a divisor of the last term ; and he remarked also this curious property of numbers, that the sum of the cubes of the natural numbers is the square of the sum of the numbers themselves.
The knowledge of the solution of cubic equations was still confined to Italy. Bombelli, a mathematician of that country, gave a regular treatise on Algebra, and considered, with very particular attention, the irreducible case of Cardan's rule. He was the first who made the remark, that the problems belonging to That case can always be resolved by the trisection of an arch.' Vieta was a very learned man, and an excellent mathematician, remarkable both for in dustry and invention. He was the first who employed letters to denote the known as well as the unknown quantities, so that it was with him that the language of algebra first be came capable of expressing general truths, and attained to that extension which has since rendered it such a powerful instrument of investigation. He has also given new demonstrations of the rule for resolving cubic, and even biquadratic equations. He also discovered the relation between the roots of an equation of any degree, and the coefficients of its terms, though only in the case where none of the terms are wanting, and where all the roots are real or positive. It is, indeed, extremely carious to remark, how gradually the truths of this sort came in sight. This proposition belonged to a general truth, the greater part of which remained yet to be discovered. Vieta's treatises were originally pub lished about the year 1600, and were afterwards collected into one volume by Schooten, in 1646.