history of mathematics - National STEM Centre

Figure 9.4
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Calculus
you can write the Cartesian parametric equations of this Archimedes spiral in the
form x = t cos t, y = t sin t.
b Find the polar equation of the same Archimedes spiral.
2 Suppose that the straight line in question 1 rotates at co rad/s, and that the point
moves along the line with speed v m/s. Find its Cartesian parametric and polar
equations.
Here is the construction Archimedes gave for the tangent at the point P on an
Archimedes spiral (see Figure 9.4).
• Draw an arc with centre O and radius OP to cut the initial line at K.
• Draw OT at right angles to the radius vector OP and equal in length to the
circular arc PK.
• PT will be the tangent to the spiral at P.
3 a For the spiral in question 2, let P be the point reached after t seconds. Find
the coordinates of the point T according to Archimedes's construction.
b Find the gradient of the line PT and show that it has the same gradient as the
tangent at P.
There is some evidence that Archimedes knew the parallelogram rule for adding
vectors. The spiral is defined by two motions which you can treat as vectors: the
velocity v of the point P along OP, and the angular velocity (o of the line OP around
the point O, which gives rise to a velocity of r(0 perpendicular to OP.
Figure 9.5 shows how these motions are combined vectorially. The resultant motion
at P is along the tangent line at P.
Figure 9.5
o
4 Use similar triangles to show that OT = rcot. Show that this is the length of the
arc PK in Figure 9.4.
Some historians have argued that, since Archimedes was apparently relating motion
and tangents, his method ought be described as differentiation. Others believe that
the lack of generality does not justify this view. However, the relationship between
tangent and motion was a fruitful concept when mathematicians in the 17th century
began to work on the problem of finding tangents.
Tangents in the 17th century
Fermat, Descartes's contemporary, who was familiar with Archimedes's work on
spirals, used the Greek definition of tangent in a more general method based on a
way of finding maxima and minima for constructing tangents.