history of mathematics - National STEM Centre

94
Descartes
23 It is true that the conic sections were never freely received into
ancient geometry, and I do not care to undertake to change names
confirmed by usage; nevertheless, it seems very clear to me that if we
make the usual assumption that geometry is precise and exact, while
mechanics is not; and if we think of geometry as the science which
furnishes a general knowledge of the measurement of all bodies, then we
have no more right to exclude the more complex curves than the simpler
ones, provided they can be conceived of as described by a continuous
motion or by several successive motions, each motion being completely
determined by those which precede;...
Descartes starts by mentioning three kinds of geometrical construction problems:
plane, solid and linear. The terms 'plane', 'solid' and 'linear' relate to the curves
that are needed to construct the solutions. This three-way classification dates from
antiquity (it is mentioned for the first time in the work of Pappus) and calls for some
explanation.
In Chapter 6 you carried out a number of constructions with lines and circles as
'curves'. These curves originated in antiquity in the plane (two-dimensional space)
and are therefore called plane curves. The geometrical construction problems solved
with these curves are consequently called plane problems.
The conic sections - the ellipse, the parabola and the hyperbola - were known and
used by the mathematicians of antiquity. The conic sections were found as the
'intersection curves' of a cone and a plane, shown in Figure 7.9. An elaborate theory
on this subject can be found in the famous work Conica from the Greek
mathematician Apollonius of Perga (262-190 BC). It is remarkable that almost all
properties of these curves, as we know them now from the analytical geometry, can
be found in Apollonius's work.
Figure 7.9
hyperbola
As these conic sections originated in a solid three-dimensional figure, the cone, they
were called solid curves. In Greek antiquity, constructions with these curves were
described, but the search for constructions in two dimensions that could replace
them was continued, since these were preferred. It was only in the 19th century that
it became clear that a number of geometrical construction problems including the
trisection of the angle could be solved only with solid curves.
Finally, there were other, more complicated, curves, such as the spiral, the
quadratrix, the cissoid and the conchoid. These curves were sometimes used to carry