 |
 |
 |
|
 |
|
 |
|
 |
|
 |
|
 |
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
 |
|
|
|
Contents
Analytic Geometry
|
Analytic
Geometry
marriage of algebra & geometry 
Analytic
geometry was introduced in the 1630s. It was an important mathematical
development, for it laid the foundations for modern mathematics as well as
aided the development of calculus. The brainchild of Pierre de Fermat and Rene
Descartes. The cartesian plane, the integral part of analytic geometry, is
named after Descartes.
Pierre
de Fermat (1601-1665)
Fermat was
well-versed with Francois Viete's ideas for symbolization in algebra, and also
tried to reconstruct Apollonius' theorems. His replacement of Apollonius'
geometric analysis with algebraic ones led him to analytic geometry. He
considered the relation between geometric loci and algebraic equations in
two or more variables, as well as the framework for this, a system of axes
where lengths can be measured against.
In
his work, he came up with the idea of origin. He also used horizontal and
vertical co-ordinates. Although he developed the basic ideas of modern
analytic geometry, there are differences. The most notable are that he used
only one axis, and only accepted positive solutions as "proper".
He constructed a parabola, but did not consider the negative portion of it.
Rene
Descartes (1596-1650)
Descartes
made a greater impact with his analytic geometry because he published his
work, which Fermat never did. His mathematical inclinations were discovered
in a dream which he had whilst serving, under harsh conditions, in the
Bavarian army. Some researchers believe he thought of the principles of
analytic geometry in this dream.
Descartes'
most significant writing is the Discours de la Methode pour bien conduire
sa Raison et cher la Verite dans les Science (Discourse on the Method of
Rightly Conducting the Reason in the Search for Truth in the Sciences).
Analytic geometry appears in his section of La Geometrie. It is
believed that he formulated the idea whilst watching a fly crawl along the
ceiling of his room near a corner--he began expressing the path of the fly
in terms of distance from the walls.
The
algebraic notation that we use today was introduced by Descartes, where
unknown quantities were represented by the last few letters of the alphabet
and the constants by the first few. He also broke away from traditional
trains of thought; instead of interpreting exponents in their geometrical
sense such as areas or volumes, he regarded them as lines.
With
the introduction of two axes of co-ordinates (however, he never formally
introduced the second, or y, axis) and the idea of an origin, the
co-ordinate system was born (Descartes never used the word co-ordinate,
it was coined by Leibniz). He believed that curves are more complex than
lines. He also said that geometric curves can be represented by an algebraic
equation in two variables defining all their points. This is the fundamental
idea of analytic geometry.
|
