a complete square he added to both members the expression 2 ( x 2 + 6 ) y + y 2 , containing a new unknown quantity y . This gave him ( x 2 + 6 + y ) 2 = ( 6 + 2 y ) x 2 + 60 x + ( 12 y + y 2 ) . The condition that the right member be a complete square is expressed by the
cubic equation . Extracting the square root of the biquadratic, he got . Solving the cubic for~ and substituting, it remained only to determine from the resulting quadratic. Ferrari pursued a similar method with other numerical biquadratic equations.7 Cardan had the pleasure of publishing this discovery in his Ars Magna in 1545. Ferrari's solution is sometimes ascribed to Bombelli, but he is no more the discoverer
of it than Cardan is of the solution called by his name.
To Cardan algebra is much indebted. In his Ars Magna he takes notice of negative roots of an equation, calling them
fictitious, while the positive roots are called real. Imaginary
roots he does not consider; cases where they appear he calls impossible. Cardan also observed the difficulty in the irreducible case in the cubics, which, like the quadrature of the circle, has since "so much tormented the perverse ingenuity of mathematicians." But he did not understand its nature. It remained for Raphael Bombelli of Bologna, who published in 1572 an algebra of great merit, to point out the reality of the apparently imaginary expression which the root assumes, and thus to lay the foundation of a more intimate knowledge of imaginary quantities.
After this brilliant success in solving equations of the third and fourth degrees, there was probably no one who doubted, that with aid of irrationals of higher degrees, the solution of equations of any degree whatever could be found. But all attempts at the algebraic solution of the quintic were fruitless, and, finally, Abel demonstrated that all hopes of finding algebraic
solutions to equations of higher than the fourth degree were purely Utopian.
Since no solution by radicals of equations of higher degrees
could be found, there remained nothing else to be done than the devising of rules by which at least the numerical values of the roots could be ascertained. Cardan applied the Hindoo rule of "false position" (called by him regula aurea) to the cubic, but this mode of approximating was exceedingly rough. An incomparably better method was invented by Franciscus Vieta, a French mathematician, whose transcendent genius
enriched mathematics with several important innovations. Taking the equation , wherein is a polynomial containing different powers of , with numerical coefficients, and is a given number, Vieta first substitutes in a known approximate value of the root, and then shows that another figure of the root can be obtained by division. A repetition of the same process gives the next figure of the root, and so on. Thus, in , taking for the approximate root, and placing , we get
Since 174 b is much greater than b 2 , we place 174 b = 409 , and obtain thereby b = 2 . Hence the second approximation is 82 . Put x = 82 + c , then ( 82 + c ) 2 + 14 ( 82 + c ) = 7929 , or 178 c + c 2 = 57