THE SPIRAL OF ARCHIMEDES
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line) and revolves uniformly about the fixed extremity, while
a point also moves uniformly along the moving straight line
starting from the fixed extremity : (the origin) at the com
mencement of the straight line’s motion; the curve described
is a spiral.
The polar equation of the curve is obviously p — a6.
Suppose that the tangent at any point P of the spiral is
met at T by a straight line drawn from 0, the origin or pole,
perpendicular to the radius vector OP; then OT is the polar
subtangent.
Now in the book On Spirals Archimedes proves generally
the equivalent of the fact that, if p be the radius vector to
the point P,
0T= p*/a.
If P is on the nth turn of the spiral, the moving straight
line will have moved through an angle 2(n — 1)tt+ 0, say.
Hence p — a{2{n — 1)tt + 6},
and OT = p 2 /a = p {2(n — 1)tt + 6\.
Archimedes’s way of expressing this is to say (Prop. 20)
that, if p be the circumference of the circle with radius
OP (= p), and if this circle cut the initial line in the point K,
OT = (n — l)p + arc KP measured ‘ forward ’ from K to P.
If P is the end of the nth turn, this reduces to
OT — n (circumf. of circle with radius OP),
and, if P is the end of the first turn in particular,
OT — (circumf. of circle with radius OP). (Prop, 19.)
The spiral can thus be used for the rectification of any
circle. And the quadrature follows directly from Measure
ment of a Circle, Prop. 1.
(y) Solutions hy Apollonius and Carpus.
Iamblichus says that Apollonius himself called the curve by
means of which he squared the circle ‘ sister of the cochloid ’.
What this curve was is uncertain. As the passage goes on to
say that it was really ‘ the same as the (curve) of Nicomedes ’,
and the quadratrix has just been mentioned as the curve used