history of mathematics - National STEM Centre

7 Constructing algebraic solutions
out a construction, more as a curiosity than as an accepted geometrical construction.
For example, the problem of doubling a cube could be solved with a cissoid as
carried out by Diocles in about 180 BC. These curves were described as linear curves
in the antiquity.
Activity 7.10 Reflecting on Descartes, 15
Figure 7.10
P(x,y)
Look back at paragraphs 21 to 23.
1 Descartes objects to the division of geometrical problems into three classes by
the mathematicians in antiquity. What is his opinion on the third class, the linear
problems?
2 How did the mathematicians of antiquity discriminate between 'geometrical
curves' and 'mechanical curves'? What is Descartes's opinion on this?
3 Descartes mentions two postulates from Euclid's Elements. Which are they?
4 Why does Descartes discuss these postulates here and not in another part of La
Geometrie?
5 In paragraph 22 Descartes suggests that if straight edge and compasses are
acceptable instruments for the construction of curves then there might be other
instruments that are equally acceptable. Which conditions must be met, according to
Descartes, on how these instruments work?
Instruments for the construction of curves
Figure 7.10 shows how you can make an instrument to draw an ellipse.
The pin at A moves 'horizontally' along the slot MON which lies along the jc-axis
while the pin at B moves 'vertically' in the slot KOL which lies along the y-axis.
The pointer P is a point along the arm AB so that the lengths AP and BP are fixed.
As A and B move, so P traces a path.
Activity 7.11 Reflecting on Descartes, 16
Let the point P have coordinates (x, y), and suppose that PA = a, PB = b and
AB = c so that a = b + c.
1 Show that: y = — y B and x = - — XA .
2 Show that the point P describes an ellipse.
3 Descartes accepts a curve under the condition that it has originated from a
movement that can be linked in a direct and clear way to a straight or a circular
movement. Does the construction with this instrument meet this requirement?
95