The onset of the mathematical revolution was marked by a surge of interest
in improving the efficiency of problem solving all across the board. We see the
same impulse in many areas. The development of abbreviated notation in
commercial arithmetic was one version; the expansion of trigonometric tools
for astronomy was another; the search for general algebraic methods of solu-
tion was yet another. Only the third of these explicitly led to an arena of pure
mathematics, but the competition soon pulled the other branches into ongoing
contact.
These were the leading areas of activity in mathematics, but not the only
kind. Traditional Greek mathematics, which consisted largely of geometry, was
revived, extended, and combined with the newer branches. Among Humanist
scholars there was an increased interest in Greek mathematical texts (Rose,
1975). The fact that most of these had already been available in medieval
Christendom suggests that the concern for new translations was the result of
the upsurge of mathematical creativity rather than vice versa. In the late 1500s
and into the 1600s, interest peaked in a Humanistic brand of mathematical
puzzle: restoring the lost portions of texts such as Apollonius (Boyer, 1985:
330, 351, 380). Since the mathematical revolution was already launched, it
appears that the most advanced Greek work was sought out and extended
because the Europeans under their own impulse now were working at this
level. Transmission of old texts, in other words, was not the cause but to a
considerable extent the effect of the crescendo of new mathematics.
Greek geometry was recruited to the new mathematics but was certainly
not its leading edge. Geometry was the most conservative area of mathematics
from the point of view of problem-solving technology, attached as it was to
concrete representations without general notation or higher-order rules for
solution. Descartes, who put the final touches on the mathematical revolution,
introduced his reform of geometry by explaining that his aim was to free
geometry from the figures that fatigue the mind; like Fermat, who simultane-
ously developed a version of coordinate geometry, Descartes was concerned to
break out of the clumsy geometric methods of the Greeks (Discourse on
Method 2.17, in Descartes, 1985: 119; see also Mahoney, 1980). Descartes
was a militant advocate of the modern algebraic approach, freeing the last
remaining area of mathematics from the Humanistic revival of the classics and
turning it into a rapid problem-solving technique.
Even in classical geometry, new methods were formulated as the mathe-
matical revolution got under way. The initial concern came from painters
interested in the theory of perspective; the first new curve since ancient times
was constructed by Durer around 1525, about the same time that Copernicus
produced a new curve by compounding two circular motions (Boyer, 1985:
320, 326). The work of Kepler in the early 1600s on planetary orbits and of
Galileo in the 1630s on projectiles fitted these motions to conic sections known
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