history of mathematics - National STEM Centre

98
Descartes
It is clear that the points B, D, F and H describe curves of different complexity.
Point B describes a circle, which is acceptable as a 'construction curve'. The
movements of points D, F and H, however, are directly and clearly coupled to the
movement of point B, and, as Descartes argues, why would these curves not be
acceptable as well?
Activity 7.13 Reflecting on Descartes, 18
1 Let YA = a, YC = x and CD = y in Figure 7.13. Show that the curve AD can be
described by the equation x 4 = a 2 1 x + y 2 ).
Acceptable curves
The previous section shows that Descartes did not want to restrict the set of
acceptable construction curves to the line and the circle. He wanted to expand this
set under one condition: the curve must be described by a continuous movement or
by an acceptable sequence of continuous movements, in which every movement is
completely determined by its predecessor. The line and the circle remain the starting
points; they act as the original 'predecessor' of all curves.
Descartes knew that this condition does not directly lead to an immediately
recognisable set of curves. Read what he writes about this.
26 I could give here several other ways of tracing and conceiving a
series of curved lines, each curve more complex than any preceding one,
but I think the best way to group together all such curves and then classify
them in order, is by recognising the factthat all points of those curves
which we may call "geometric", that is, those which admit of precise and
exact measurement, must bear a definite relation to all points of a straight
line, and that this relation must be expressed by means of a single
equation. If this equation contains no term of higher degree than the
rectangle of two unknown quantities, or the square of one, the curve
belongs to the first and simplest class, which contains only the circle, the
parabola, the hyperbola, and the ellipse; but when the equation contains
one or more terms of the third or fourth degree in one or both of the two
unknown quantities (for it requires two unknown quantities to express the
relation between two points) the curve belongs to the second class; and if
the equation contains a term of the fifth or sixth degree in either or both of
the unknown quantities the curve belongs to the third class, and so on
indefinitely.
According to many this part of the text initiates the birth of analytical geometry: all
points of a curve are unambiguously related to points on a straight line, and this
relation can be described with one single equation with two unknown quantities.