VI.8 Rafael Bombelli

b. Bologna, Italy, 1526; d. Probably Rome, after 1572

Engineer–architect for the Roman nobleman Alessandro Rufini, later Bishop of Melfi

Bombelli was prompted to write his Algebra (1572) by a desire to make CARDANO’S [VI.7] Ars Magna (1545) accessible to the less sophisticated reader. The Algebra, which contains a systematic treatment of quadratics, cubics, and quartics, is noted for its advances in mathematical notation—it was the first printed text to include a notation for exponents—and for its role in disseminating awareness of the work of Diophantus. Above all, the Algebra was renowned for solving certain special cases of the so-called casus irreducibilis of the cubic, those in which Cardano’s rule appears to give rise to a complex or “impossible” solution. Cardano was aware that what we today call complex numbers (numbers of the form a + b ) could arise in the solution of quadratic equations. Bombelli made the important discovery that what at first sight appears to be a complex root of a cubic equation may in fact be a real root, because the imaginary parts cancel each other out. The Algebra included the first extensive discussion of complex numbers and Bombelli formulated the four basic operations of arithmetic for them.