This spiral was studied by Conon, and later by Archimedes in On Spirals about 225 BC. Archimedes was able to work out the lengths of
various tangents to the spiral.
Archimedes' spiral can be used for compass and straightedge division of an angle into parts (including angle trisection)
and can also be used for circle squaring. In addition,
the curve can be used as a cam to convert uniform
circular motion into uniform linear motion (Brown 1923; Steinhaus 1999, p. 137).
The cam consists of one arch of the spiral above the x-axis
together with its reflection in the x-axis. Rotating
this with uniform angular velocity about its center will result in uniform linear
motion of the point where it crosses the y-axis.
![1/2a[thetasqrt(1+theta^2)+ln(theta+sqrt(1+theta^2))].](/images/equations/ArchimedesSpiral/Inline6.svg)
![a{theta+1/2sum_(k=3)^(infty)[P_(k-3)(0)+(k+1)/kP_(k-1)(0)]theta^k}](/images/equations/ArchimedesSpiral/Inline9.svg)
is a Legendre polynomial.
parts (including angle trisection)
and can also be used for circle squaring. In addition,
the curve can be used as a cam to convert uniform
circular motion into uniform linear motion (Brown 1923; Steinhaus 1999, p. 137).
The cam consists of one arch of the spiral above the x-axis
together with its reflection in the x-axis. Rotating
this with uniform angular velocity about its center will result in uniform linear
motion of the point where it crosses the y-axis.
