Descartes: A Biography

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 Descartes: A Biography

It was inspired by a well-established tradition in geometry that accepted

only lines and circles as legitimate elements for constructing all geomet-

rical figures.Another way of expressing the same restriction is that one

must be able to construct any geometrical figure by using only a com-

pass and ruler. There were well-known problems that could not be solved

given these limitations, such as constructing a square with the same area

as a given circle or finding two lines that were mean proportionals to two

given lines. The extent to which geometry, as a discipline, could express

problems that it could not solve was made even more acute by the pub-

lication, in,ofaLatin translation of Pappus’ problem.Descartes’

ambition was to release geometry from such limitations or to introduce

new methods that could address previously insoluble problems. The focus

of his contribution was to develop algebraic methods for describing geo-

metrical figures, especially curves, and to propose novel approaches for

solving problems of construction.

The ambition of his previous engagements with these problems, dating

back toand the mid-s,reappears in the first sentence of the

Geometry.‘All the problems of geometry can be reduced easily to such

terms that we then need to know only the length of certain straight lines

in order to construct them’ (vi.). What catches the eye immediately is

the ‘all’ and the ‘easily’. It is true that, in theGeometry, Descartes displays

the power of his new curves to solve the Pappus problem for five lines, and

that he endorses the use of algebraic equations as the criterion of exactness

forthe construction of curves. However, there is an element of wishful

thinking in the claim that he has provided a method of solving all problems

in geometry, and he was much too good a mathematician not to have

seen some of the fundamental problems that remained in the discipline.

Rather than take more time to tackle them, however, he returned to the

fundamental philosophical issues that featured in the unpublishedWorld;

and while he defended his significant contribution to mathematics and

scorned the claims of many of his contemporaries, he effectively stopped

doing new work in mathematics once theGeometrywas published.

TheDiscourse on Method()

TheDiscoursewas a rather hastily written Preface designed to introduce

the three scientific essays of. Its rambling, repetitive, and uneven

character is easily understood from the context of its composition. As