(B) HISTORY OF ALGEBRA
(vii) Lewis Ferrari (Lodovico Ferrari). Rafael Bombelli (d. 1572). Stifellius. Scheubelius.
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. He had a pupil named Lewis Ferrari, a young man 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.
This was another great improvement; and although the precise nature of an equation was not then fully understood, nor was it indeed until half a century later, yet, in the general resolution of equations, a point of progress was then reached which the utmost efforts of modern analysis have never been able to pass.
There was another Italian mathematician of that period who did something for the improvement of algebra. This was Bombelli. He published a valuable work on the subject in 1572, in which he brought into one view what had been done by his predecessors. He explained the nature of the irreducible case of cubic equations, which had greatly perplexed Cardan, who could not resolve it by his rule; he showed that the rule would apply sometimes to particular example, and that all equations of this case admitted of a real solution; and he made the important remark, that the algebraic problem to be resolved in t his case corresponds to the ancient problem of the trisection of an angle.
There were two German mathematicians contemporary with Cardan and Tartalea, viz., Stifelius and Scheubelius. Their writings appeared about the middle of the 16th century, before they knew what had been done by the Italians. Their improvements were chiefly in the notation. Stifelius, in particular, introduced for the first time the characters which indicate addition and subtraction, and the symbol for the square root.
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