history of mathematics - National STEM Centre

Descartes
Activity 6.1 Head about Descartes
Activity 6.2
Figure 6.1
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1 Read a book, such as A concise history of mathematics by D J Struik, and
summarise in about 10 lines the most important details about Descartes's life.
The appendix of Descartes's book, La Geometric, became a major influence on
mathematics after the Leiden professor of mathematics, Frans van Schooten,
published a Latin translation with extensive commentaries and explanations.
But why was this work so important? What revolution came about when
mathematicians began to understand and elaborate on Descartes's ideas?
The main ideas, which now seem simple, are these.
• Descartes used algebra to describe geometrical objects and to solve geometrical
problems.
• He also used geometry and graphical methods to solve algebraic problems.
Describing geometrical objects and ideas in the language of algebra is now
commonplace. Every student thinks of a' as the volume of a cube with edge length
a, and can verify that you can make a larger cube, of edge la, with eight of these
smaller cubes. It feels quite normal to prove this geometric statement by using
algebra. For example, eight smaller cubes have volume 8a 3 ; one cube of edge 2a
has volume (2a)'; and the proof amounts to observing that (Id)' = 8a 3 .
This algebraic thinking was a major step forward. It is difficult for us to understand
the intellectual boldness of this step in 1637, when the usual way of thinking about
geometry was that of the ancient Greeks. To recognise the ingenuity of Descartes's
work you must remind yourself about the mathematics before Descartes.
The difficulty of geometric constructions
In Chapter 3 in The Greeks unit you learned about the Greeks' classical geometrical
construction problems and the strict rules that their solutions had to obey. In the
next two activities you will re-visit two problems to re-create some of these difficul­
ties for yourself. You will then begin to understand what it was that frustrated
Descartes to such an extent that he created a new way of solving these problems.
Two geometrical construction problems
If you can, work in a group, especially for question 4.
Figure 6.1 shows two points, A and B, in a plane.
1 Use a straight edge and a pair of compasses to construct the set of points such
that the distance of each point X of the set from A is equal to its distance from B.
2 Show that each point X of your set satisfies XA = XB. Show also that no other
point can be a member of your set. If you can, prove that your solution is correct.