history of mathematics - National STEM Centre

Activity 9.1
Figure 9.2
This is an example of
fiproof by contradiction.
Figure 9.1
9 The beginnings of calculus
• a definition of tangent which they could use only for simple curves
• a limited number of fairly simple curves that they could study.
Their method for proving that a line was a tangent to a curve involved:
• specifying a construction for a tangent line at any given point P on the curve
• proving that the line so constructed does not meet the curve anywhere else.
Euclid's definition of tangents
Euclid, in Book III, proposition 16, proved that the line PT in Figure 9.2 is a tangent
to the circle at P if angle OPT is a right angle.
The straight line drawn at right angles to the diameter of a circle at its
extremity will fall outside the circle, and into the space between the
straight line and the circumference another straight line cannot be
interposed; further the angle of the semicircle is greater, and the
remaining angle less, than any acute rectilineal angle.
1 Complete Euclid's argument by filling in the gaps.
Suppose that PT is a tangent to the circle. Then suppose that PT meets the circle at
another point G.
ThenOG = ..........
This implies that angle OGP = ..........
Activity 9.2 Archimedes's spiral
Figure 9.3
This means that the triangle ...... contains ...... right angles, which is impossible.
Therefore the line PT cannot cut the circle anywhere else, and is proved to be a
tangent, according to the definition prevailing in Euclid's time.
Archimedes gave a construction for drawing a tangent in his book Tlept e?UK(ov',
which reads 'peri helikon' and means 'about spirals'
A spiral was defined in terms of motion. Archimedes wrote:
If a straight line of which one extremity remains fixed be made to revolve
at a uniform rate in a plane until it returns to the position from which it
started, and if, at the same time as the straight line revolves, a point move
at a uniform rate along the straight line, starting from the fixed extremity,
the point will describe a spiral in the plane.
Figure 9.3 shows such a spiral, called an Archimedes spiral, following these rules.
1 a The straight line starts in the position of the jc-axis, and rotates at 1 rad/s. The
point starts at the origin and moves out along the line at a speed of 1 m/s. Show that
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