history of mathematics - National STEM Centre

Figure 6. U
Figure 6.15
6 The approach of Descartes
1 Figure 6.14 shows a straight line AB, with a point C on it. Produce AB to the
point D such that the rectangle with sides AD and DB, is equal (in area) to the
square with side CD.
Call the lengths AC, CB and BD, a, b and x.
1 Formulate an equation involving a, b and x.
2 Express x in terms of a and b.
Trisecting an angle
Activity 6.13 Trisecting an angle
Figure 6.16
A classical problem from Greek mathematics was the problem of trisecting an
angle.
Let O be an angle, shown in Figure 6.15. Construct two lines that divide the angle at
O into three equal parts.
Descartes, who did not know that this problem cannot be solved with a pair of
compasses and a straight edge, hoped to find a solution using his new method.
It was only in 1837 that a relatively unknown Frenchman, Pierre-Laurent I
Wantzel (1814-1848), proved that it was impossible to trisect an angle j
using only a straight edge and a pair of compasses. |
The algebraic analysis of this problem is given in Activity 6.13. The geometric
construction is left until Chapter 7.
Let O be the centre of a circle with radius 1. The points N, Q, T, P, R and M are as
shown in Figure 6.16.
With the help of the figure, you can re-formulate the problem: 'Let NP be a chord.
Find the length of chord NQ'.
1 By considering angles standing on the chord PQ, prove that ZNOQ = ZQNR.
2 Prove that triangle NOQ is similar to triangle QNR, and triangle ROM is similar
to triangle QNR.
3 Let NP = q and NQ = z. Prove that RM = q - 2z.
4 Use similar triangles to prove that z 3 = 3z - q.
Reflecting on Chapter 6
What you should know
• the major contributions made by Descartes and Viete to the development of
mathematics
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