omy was a fertile ground for innovations in calculation, such as trigonometry
and logarithms, as well as in geometry. And astronomy attracted ambitious
and innovative intellectuals because it was a focus of concern for a dominant
field such as theology, just now in the midst of controversies because of the
crisis in church politics (cf. Westman, 1980).
Careers of other early mathematicians show the practical and commercial
math coming together with the status-centered activities of the intellectuals.
Pacioli, a pupil in the atelier of the painter Piero della Francesca, published in
1494 the first important mathematics textbook; Pacioli stressed the practice of
bookkeeping, but incorporated geometry from the works of Piero, and his
figures were drawn by Leonardo da Vinci.^28 None of this constituted a new
discovery in math, but the painters added public attention to the subject, and
soon the network gave rise to the first great breakthrough in modern algebra.
Cardano’s father was a friend of Leonardo; Cardano himself moved through
the same universities where Copernicus had studied 20 years before, picking
up a secret formula for solving particular types of equations of the third degree
from a pupil of Ferro, a probable acquaintance of Copernicus at Bologna.
Tartaglia, a teacher of commercial mathematics at Venice, engaged in public
problem-solving contests with several men in this network; by the 1530s and
1540s, this set of contestants had come up with general solutions for both
cubic and quartic equations.
Mathematics was becoming a matter of public prestige. Cardano and his
assistant defended a mathematical challenge from Tartaglia in the cathedral of
Milan in 1548 with the governor acting as judge. Cardano somewhat unscru-
pulously took secret formulae from his acquaintances and published them, but
this in itself shows that he was playing to a different arena than local mathe-
matics teachers advertising their skills by winning public contests by means of
secret techniques. Cardano was a medical professor at the major Italian uni-
versities who wrote widely on philosophy and theology as well as science and
mathematics. The great upsurge in the innovativeness of mathematics came
just at the time when it was shifting from a humble commercial activity to an
attention-getting contest among high-status intellectuals. These contests were
pushing activity into the realm of pure discovery making, far beyond issues of
practicality. By the turn of the century Galileo, taught by a pupil of Tartaglia,
was in the core intellectual networks only a few links from leading philosophers
such as Suarez.
With Descartes, the leading edge of creativity in mathematics and in phi-
losophy merged. The prestige of one became the basis of the elevated prestige
of the other. In mathematics Descartes was not the progenitor of the revolution,
but he was its first culmination. The symbolic notation had first been developed
in commercial arithmetic books; Descartes raised its level of generality and558 • (^) Intellectual Communities: Western Paths