The Literal Calculus of Viete and Descartes

1. The contribution of Fransois Viete (1540-1603). Viete tried to create a new science (he called it ars analytica, or analytic art) that would combine the rigor of the geometry of the ancients with the operativeness of algebra. This analytic art was to be powerful enough to leave no problem unsolved: nullum non problema solvere. Viete set down the foundations of this new science in his An introduction to the art of analysis (In artem analyticem isagoge) of 1591. In this treatise he created a literal calculus. In other words, he introduced the language of formulas into mathematics. Before him, literal notations were restricted to the unknown and its powers. Such notations were first introduced by Diophantus and were somewhat improved by mathematicians of the 15th and 16th centuries. The first fundamentally new step after Diophantus was taken by Viete, who used literal notations for parameters as well as for the unknown. This enabled him to write equations and identities in general form. It is difficult to overestimate the importance of this step. Mathematical formulas are not just a compact language for recording theorems. After all, theorems can also be stated by means of words; for example, the formula