history of mathematics - National STEM Centre

Figure 7.2
Descartes
Activity 7.1
i———
Figure 7.3
Figure 7.4
Solution
• Draw an arbitrary acute angle O, as shown in Figure 7.1.
• Mark off the distances b and c along one side of the angle such that OB = b and
OC = c, and mark off a along the other side so that OA = a.
• Draw BZ parallel to AC.
• Then OZ is the required line segment with length z.
Example 2
Let a be a line segment. Construct the line segment z such that z = Va •
Solution
• Take the length a and the length of the unit and draw them in line, as in Figure
7.2, so that OA = a and OE = 1.
• Draw a circle with diameter AE.
• At O, draw a line perpendicular to AE. Call the point where this line intersects
the circle, Z.
• Then OZ is the required line segment with length z.
In the next activity there are two constructions for practice. If you want to use a unit,
take 1 cm as the unit.
Practice with constructions
1 Let a and b be the two line segments shown in Figure 7.3. Construct the line
segment z for z = ^^ab
2 Look again at Van Schooten's problem in Activity 6.12. Figure 7.4 shows a
straight line AB, with a point C on it. The problem was to find the point D on AB
produced, such that the rectangle with sides AD and DB is equal (in area) to the
square with side CD.
In Activity 6.12, you analysed this problem algebraically, and should have found
i 2 i
that x = — — , or — = ——— . Construct x using one of these expressions as a
a — b b a — b
starting point.
Constructions requiring a circle
One of the 'rules of the game' applied to geometrical constructions from antiquity
was that only a pair of compasses and a straight edge were used to execute the
constructions. Descartes discusses this in Book I:
12 I shall therefore content myself with the statement that if the student,
in solving these equations, does not fail to make use of division wherever
possible, he will surely reach the simplest terms to which the problem can
be reduced.