history of mathematics - National STEM Centre

The full title is Discours
de la methode pour bien
conduire sa raison et
chercher la verite dans
les sciences.
8 An overview of La Geometrie
to construct all problems, more and more complex, ad infinitum; for in the
case of a mathematical progression, whenever the first two or three terms
are given, it is easy to find the rest.
40 I hope that posterity will judge me kindly, not only as to the things
which I have explained, but also as to those which I have intentionally
omitted so as to leave to others the pleasure of discovery.
The method of normals
Descartes wrote La Geometrie as an appendix to his philosophical work Discours.
Another appendix is Dioptrique, a treatment on geometrical optics. To draw the
light path made by an incoming ray on a lens, it is essential to construct the normal
to a curve. In La Geometrie Descartes shows a method for constructing normals.
Figure 8.1 shows part of an ellipse, as it appeared in La Geometrie.
Here is Descartes's solution to finding the normal to a curve, when the curve is an
ellipse. The method is based on the following idea.
41 ... observe that if the point P fulfils the required conditions, the circle
about P as centre and passing through the point C will touch but not cut
the curve CE; but if this point P be ever so little nearer to or farther from A
than it should be, this circle must cut the curve not only ate but also in
another point.
In short, point P, the centre of a circle with radius PC, has a fixed place on GA, as
the circle intersects with the ellipse in C. If PC were not perpendicular to the ellipse,
the circle would intersect with the ellipse at another point as well.
Descartes names various known and unknown quantities.
42 Suppose the problem solved, and let the required line be CP. Produce
CPto meet the straight line GA, to whose points the points of CE are to be
related. Then, let MA = CB = >•; and CM = BA = x. An equation must be
found expressing the relation between x and v. I let PC = s, PA = v,
whence PM = v~y.
The ellipse and the circle can be represented as two equations in x and y (the
parameters of the ellipse are known, so its equation can be found). The problem can
now be solved algebraically. You can eliminate x or y from the equations, leaving
one equation with one unknown quantity.
Suppose that C is the point of contact of the ellipse and the circle. What happens
then? Or, put in another way, what happens if the circle and the ellipse have two
coincident points of intersection?
I
43 ... and when the points coincide, the roots are exactly equal, that is
to say, the circle through C will touch the curve CE at the point C without
cutting it.
In that case the equation has a double solution.
In Activity 8.9, you will retrace this solution with a concrete example.
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