c CUNYB/Clarke December, :
Magic, Mathematics, and Mechanics intellectual contacts and a clearinghouse for exchanges with many other
correspondents, including even members of the Descartes family who
lived in Brittany.
During this period Descartes began to work on a number of projects,
all of which were left unfinished. He reflected on his apparent lack of
resolution when he wrote to Mersenne in: ‘If you find it odd that I
began a number of treatises when I was in Paris and did not finish them, I
can explain why. As I was working on them, I gained a little more knowledge
about them than I had when I began, and in trying to accommodate that I
was forced to begin a new project....’ (i.–). The unfinished projects
included: (a) an outline of a general method, usually called the Rules for
Guiding One’s Intelligence in Searching for the Truth; (b) an early draft of
the mathematical papers that were eventually published as theGeometry
in; and (c) research on the theory and practice of grinding lenses.
This last project was much assisted by Claude Mydorge, who introduced
Descartes to a skilled lens grinder, Jean Ferrier. These draft essays and
the discussion of solutions to various problems in mathematics and optics
were shared with others, to such an extent that there are contemporary
references inquiring about the progress being made by this otherwise
unknown scholar.
This apparently fallow period included a significant transition in
Descartes’ mathematical outlook. One of the standard challenges to math-
ematicians at the time was to devise a method for constructing mean pro-
portionals. For example, if two line segments are given with lengthsa
andb,one is asked to find two other lengthsxandy, such thata:x=
x:y=y:b. Despite its apparent simplicity, this could not be constructed
using only the traditionally accepted means of ruler and compass, and
those who proposed solutions had to rely on sliding rules and comparative
measurements that were generally regarded as too inexact to be acceptable
in geometry. Descartes devised a solution to this problem using a circle and
a parabola. He shared the results with Mydorge, who produced a proof of
the result. When Descartes told Beeckman about the solution, in, the
Dutch physicist noted that ‘Mr. Descartes thinks so much of this discov-
ery that he claims to have never discovered anything more significant and
that, in fact, nothing more significant has ever been discovered by anyone’
(x.). Descartes’ characteristic lack of modesty is evident here. In fact,
this great ‘discovery’ represented a somewhat late recognition on his part
of the significance of algebraic methods for solving geometrical problems,