history of mathematics - National STEM Centre

8 An overview of La Geometrie
Descartes used a novel way of classifying curves to find which one he needed to
solve a geometrical construction problem. At the same time he formulated a
classification of construction problems.
In Book II, Descartes wrote extensively about curves but he did not derive equations
of curves. Equations of curves, however, were very new and were not the goal of
Descartes's study but merely a means to come to a classification. Yet, it was clear to
him that equations contain a great deal of information about curves.
Activity 8.3 Reflecting on Descartes, 21
28 When the relation between all points of a curve and all points of a
straight line is known, in the way I have already explained, it is easy to find
the relation between the points of the curve and all other given points and
lines; and from these relations to find its diameters, axes, centre and other
lines or points which have especial significance for this curve, and thence
to conceive various ways of describing the curve, and to choose the
easiest.
29 By this method alone it is then possible to find out all that can be
determined about the magnitude of their areas, and there is no need for
further explanation from me.
1 If you want to characterise a curve today, giving an equation will do. To
Descartes this was not (yet) acceptable. From which sentence in paragraphs 28 and
29 can you conclude this?
In paragraph 30, Descartes shows for the first time a method for determining the
normal to a curve. He regarded the normal as very useful for discovering a number
of properties of curves.
Activity 8.4 Reflecting on Descartes, 22
30 Finally, all other properties of curves depend only on the angles which
these curves make with other lines. But the angle formed by two
intersecting curves can be as easily measured as the angle between two
straight lines, provided that a straight line can be drawn making right
angles with one of these curves at its point of intersection with the other.
This is my reason for believing that I shall have given here a sufficient
introduction to the study of curves when I have given a general method of
drawing a straight line making right angles with a curve at an arbitrarily
chosen point upon it. And I dare say that this is not only the most useful
and most general problem in geometry that I know, but even that I have
ever desired to know.
1 From paragraph 30, it is clear that an important property of curves can be
determined with the help of normals. Which property is meant?
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