THE SPIRAL OF ARCHIMEDES 
231 
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