Algebras

European Developments To The Seventeenth Century

Common algebra came into an awakening Europe during the thirteenth century. Among the various sources involved, Latin translations of some Arab authors were important. A significant homegrown source was the Italian Leonardo Fibonacci, who rendered the theory into Latin, with res, census, and cubus denoting the unknown and its powers. He and some translators of Arabic texts also adopted the Indian system of numerals. Communities developed, initially of Italian abbacists and later of German Rechenmeister, practicing arithmetic and common algebra with applications—some for a living.

The title of al-Khwarizmi's book included the word al-jabr, which named the operation of adding terms to each side of an equation when necessary so that all of them were positive. Maybe following his successor Thabit ibn Qurra (836–901), in the sixteenth century Europeans took this word to refer to the entire subject. Its theoretical side principally tackled properties of polynomial equations, especially finding their roots. An early authority was Girolano Cardano (1501–1576), with his Ars magna (1545); successors include François Viète (1540–1601) with In artem analyticem isagoge (1591), who applied algebra to both geometry and trigonometry. The Europeans gradually replaced the words for unknowns, their powers, means of combination (including taking roots), and relationships by symbols, either letters of the alphabet or special signs. Apart from the finally chosen symbols for the arithmetic numerals, no system became definitive.

In his book Algebra (1572), Rafael Bombelli (1526–1572) gave an extensive treatment on the theory of equations as then known, and puzzled over the mystery that the formula for the (positive) roots of a cubic equation with real coefficients could use complex numbers to determine them even if they were real; for example (one of his), given

The formula involved had been found early in the century by Scipione del Ferro (1465–1526), and controversially published later by Cardano. It could be adapted to solve the quartic equation, but no formula was found for the quintic.