|
2.2
Geometry-Algebra interplay
In
his book La Geometrie (an appendix to Discours de la Méthode)
Descartes begins by looking at "How multiplication, division and
extraction of square roots are performed geometrically". This is
essentially a description of some Greek constructions. Only the algebraic
notation is new. We say "only", but of course the application
of algebra to geometry was Descartes's great achievement. Nevertheless
this application of algebra is not yet what we understand as Analytical
Geometry. The central idea of Analytical Geometry is the study of
curves using algebraic equations. Let us look at what all this means
by exploring the situations he elaborates at the beginning of La Geometrie.
Click
on the pages below for a larger view of the first two pages of La
Geometrie. We will analyse the two problems in the figures on the
second page.
In
the first problem Descartes shows how to multiply the length
BD by BC. He starts with the segment AB that he takes
to be unity. This length is not necessarily equal to
one, nevertheless it is treated arithmetically as if
it is equal to one. He then joins A to C and B to C.
Finally he draws DE parallel to AC and then claims that
BE = BD.BC
Convince yourself that Descartes is correct
by dragging points D and C and looking at the lengths and
the calculations …
Drag point A. How come AB remains constant at 1?
|
|
Now prove
that BE = BD.BC.
Click
here for a discussion:  |
|
|
Now
let’s look at how Descartes found square roots geometrically.
Prove
that GI is the square root of GH.
Click
here for the solution: 
|
|
|