The Transition to Incipient Modern Algebra

The changes that were traced in the previous chapters might have remained unimportant, if a new mathematical culture had not emerged. Traditional Humanism was no protection against modern gunnery and irrelevant for overseas expansion; in consequence, courts took up interest in mathematics, and some Humanists (being Humanists) prepared editions and translations of Greek mathematical classics. In a generally agonistic intellectual environment, dilettante mathematicians (not least French) challenged each other with geometric problems of Greek inspiration; first Viète, then Descartes choose to apply algebra to such problems. Viète keeps secret where he had learned about algebra (Diophantos would not have helped him), but Mennher is likely to be among his sources; Descartes knew algebra from Clavius, but never read the pages where several unknowns are treated. He seems to have learned about that technique from Mennher’s book in 1629. Both give names to all those segments in a diagram that appear to be involved, known as well as unknown. Those which are known become (arbitrary) coefficients. Neither thus invented abstract coefficient B they received them as a gift, as a consequence of applying algebra with several unknowns to geometry. Descartes introduces braces to delimit composite coefficients, and also the long root sign B both special-purpose parentheses. Newton uses both, but only for the same purposes. When multiplying two polynomials, Newton makes use of a vinculum (a line above), the first general parenthesis. Others borrow it but use it rarely, having little need. In infinitesimal calculus, it was needed; once the round brackets were established there, they spread backwards to ordinary algebra.