history of mathematics - National STEM Centre

104
Descartes
In the next section you will go deeper into the method for finding the normal to a
curve and see how this process is a forerunner of modern differential calculus.
For now, take a look at another part of the text, the beginning of Book III.
On the Construction of Solid and Supersolid Problems
31 While it is true that every curve that can be described by a continuous
motion should be recognised in geometry, this does not mean that we
should use at random the first one that we meet in the construction of a
given problem. We should always choose with care the simplest curve that
can be used in the solution of a problem, but it should be noted thatthe
simplest means not merely the one most easily described, nor the one that
leads to the easiest demonstration or construction of the problem, but
rather the one of the simplest class that can be used to determine the
required quantity.
32 ... On the other hand, it would be a blunder to try vainly to construct
a problem by means of a class of lines simpler than its nature allows.
Descartes says that, for every construction, you have to use a curve from a class that
is as simple as possible. But take care not to choose too simple a class. If you want
to construct the solution of a second-order equation, do not take a parabola or an
ellipse. Take a circle and a line. However, the construction cannot be performed
only with lines.
He goes on to say that, before you start, you should ensure that the algebraic
equation is in its simplest form. Descartes writes in detail about this in Book III.
He starts by saying:
Activity 8.5 Reflecting on Descartes, 23
33 Before giving the rules for the avoidance of both these errors, some
general statements must be made concerning the nature of equations. An
equation consists of several terms, some known and some unknown, some
of which are together equal to the rest; or rather, all of which taken
together are equal to nothing; for this is often the best form to consider.
1 According to Descartes, what is the best form in which to write
jc 3 =3x 2 +2x-61
In Book III, he explores the theory of algebraic equations, with the aim of
simplifying an equation and if possible lowering its order. This results in several
ideas that were remarkably original. These can be seen in paragraphs 34 and 35.
34 Every equation can have as many distinct roots (values of the
unknown quantity) as the number of dimensions of the unknown quantity in
the equation. Suppose,for example, x = 2 or ;c-2 = 0, and again, x = 3
or jc-3 = 0. Multiplying together the two equations x-2 = 0 and
jt-3 = 0, we have x 2 - 5jc + 6 = 0, or x 2 = 5*-6.This is an equation in
which x has the value 2 and at the same time x has the value 3. If we next