history of mathematics - National STEM Centre

7 Constructing algebraic solutions
On the nature of curves
In La Geometric, Descartes pushed back some of the frontiers which had restricted
mathematicians. You have already seen that the introduction of the unit and - more
or less related to this - the relaxing of the law of homogeneity, enabled a fruitful
interaction between geometry and algebra to take place. This cleared the way for a
generally applicable method for solving geometrical construction problems.
However, yet another step was needed. What use is the method of analysis if you
cannot then construct the solution? What about equations of third and higher
degree? In general, the solutions of these equations cannot be constructed with a
pair of compasses and a straight edge. But how can you construct them? Can you
use other curves and circles?
Here, too, Descartes proved himself to be original and creative. Before he supported
carrying out constructions with curves other than straight lines and circles, he
discusses the nature of geometrical curves in Book II.
The book begins as follows.
Geometry
Book II
On the Nature of Curved Lines
21 The ancients were familiar with the fact that the problems of
geometry may be divided into three classes, namely, plane, solid, and
linear problems. This is equivalent to saying that some problems require
only circles and straight lines for their construction, while others require a
conic section and still others require more complex curves. I am surprised,
however, that they did not go further, and distinguish between different
degrees of these more complex curves, nor do I see why they called the
latter mechanical rather than geometrical.
22 If we say that they are called mechanical because some sort of
instrument has to be used to describe them, then we must, to be
consistent, reject circles and straight lines, since these cannot be
described on paper without the use of compasses and a ruler, which may
also be termed instruments. It is not because other instruments, being
more complicated than the ruler and compasses, are therefore less
accurate, for if this were so they would have to be excluded from
mechanics, in which the accuracy of construction is even more important
than in geometry. In the latter, the exactness of reasoning alone is sought,
and this can surely be as thorough with reference to such lines as to
simpler ones. I cannot believe, either, that it was because they did not
wish to make more than two postulates, namely (1) a straight line can be
drawn through any two points, and (2) about a given centre a circle can be
described passing through a given point. In their treatment of the conic
sections they did not hesitate to introduce the assumption that any given
cone can be cut by a given plane. Now to treat all the curves which I mean
to introduce here, only one additional assumption is necessary, namely,
two or more lines can be moved, one upon the other, determining by their
intersection other curves. This seems to me in no way more difficult.
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