since the Greeks. The upheaval in mathematical method incorporated these
classical curves into the central achievements of the new mathematical science.
Kepler generalized Archimedean methods to various solids of revolution, and
showed that the familiar conic sections were transformations of one another;
Galileo reinterpreted conic sections as products of the combination of two
motions (Boyer, 1985: 356–358). These discoveries were results of the new
algebraic vision applied to geometry.
Sustained discovery in mathematics was well along by the time of Viète in
the 1580s. With Viète the combination of different mathematical fields became
a discovery-making technique: the new higher algebra was combined with
geometric methods of solution, the new trigonometry turned into algebraic
functions. The new algebraic geometry developed by Viète, Descartes, and
Fermat went beyond traditional plane and solid figures into a more abstract
space, in which lines, squares, cubes, and higher powers were all treated as
quantities in the same equation. Descartes’s Geometry closes with an explicit
overview of the theory of equations. By the mid-1600s, whole regions of higher
mathematics were emerging through recombining subspecialties. New methods
were developed for the algebraic solution of problems relating to curves,
especially where the new curves raised issues of motion and of infinitesimal
changes and their summation. Out of these problems arose a crude calculus in
the hands of Galileo, Roberval, Cavalieri, and Torricelli, perfected in the next
generation by Newton and Leibniz.
Between the time of Viète and Descartes, mathematics was transformed
into a machinery for manipulating equations. In part this marked a change
from verbal arguments to abbreviations to the invention of symbols for givens,
unknowns, and operations. The decisive step was to set up systems of equations
with explicit rules for how to substitute and recombine them; by following
these rules of manipulation, one could gain the same advantages that resulted
from introducing a forefront of research equipment into science. Tinkering
with the mathematical machinery could open up new areas of application and
generate new results, turning mathematics into a moving forefront of research.
And perfecting the procedures for manipulating equations—the equivalent of
tinkering with the machinery until it is reliable—yields absolute certainty of
results, because results are absolutely repeatable.
This procedure is not exclusively modern; the history of mathematics
consists in building up just such a technology for manipulating classes of
expressions so that results are highly repeatable. One might say that this is
what defines mathematics: it is the cumulated practices of tinkering with the
operations of counting and measuring, proceeding on to higher-order gener-
alizations about classes of such operations. These skills have always been
embodied in a technology, though usually tacit and not sufficiently portable to
542 • (^) Intellectual Communities: Western Paths