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_(n-3)(0)+(n+1)/nP_(n-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.
