the 1650s, gave his theorems rhetorically; nevertheless he was in the swing of
the new abstract procedures, and he gave the first clear explanation of the
method of mathematical induction (Boyer, 1985: 335, 397).
More significant for mathematical discovery was the drive to find devices
for improving scientific calculations. First came the expansion of trigonometric
tables as a tool for improving the precision and speed of astronomical calcu-
lations. In the 1460s, Regiomontanus compiled a text of trigonometry from
Greek and Arab sources, based primarily on sine functions. Two generations
later Copernicus produced new trigonometric tables, and his assistant Rheticus
worked out a trigonometry of all six functions with elaborate tables for each.
Such tables may be regarded as a rather low-level empirical extension of
existing mathematical methods into adjacent areas, but the effort spilled over
into shortcuts in calculation. Formulae for substituting the addition of cosines
for their multiplication were a great improvement in the clumsy arithmetic of
the time; these algorithms of trigonometric algebra were developed in the 1580s
by Viète and others. This labor-saving device was quickly picked up by as-
tronomers such as Tycho Brahe. From there a visitor brought it to Napier in
Scotland, who generalized the idea into logarithms in the early 1600s. The
concern for a technology of calculation led to mechanical devices as well as
conceptual ones. Galileo in 1597 constructed and marketed computational
compasses, a device analogous to a slide rule for quick computations. The
focus of concern at this time is shown by the fact that Galileo was competing
against a similar device invented by Burgi, a rival of Napier in the discovery
of logarithms. Still later, in 1642 we find Pascal at the very beginning of his
career building and selling a mechanical calculating machine (Boyer, 1985:
338–340, 351, 396).
The revolution in algebra followed the same path in a more abstract way.
Algebra initially consisted of shortcuts in arithmetic, principles which cover
whole classes of calculations. Algebra advanced to new terrain when it formu-
lated such methods in the form of meta-rules about how to solve abstract
equations. The very substance of algebra, and of abstract mathematics gener-
ally, comprises the methods for solving lower-order problems. Pure mathe-
matics becomes an independent activity when intellectuals concentrate on
developing algorithms apart from their application. The takeoff of abstract
mathematics in the early 1500s came as just such a research forefront was
emerging, with the discovery of general methods for solving the cubic and
quartic equations by the mathematicians around Tartaglia and Cardan. By the
time of Viète, techniques were being developed for problems of much higher
degree. In the process, other areas of mathematics such as trigonometry and
geometry were being brought in as tools, resulting in cross-fertilization among
these fields. The genealogies of mathematical techniques were beginning to
propagate.
540 • (^) Intellectual Communities: Western Paths