"If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving , a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane."
The above is the opening definition of a spiral in Archimedes's treatise On Spirals. Here is a topic of considerable interest in and of itself, which never gets mentioned until one goes looking, but it deserves just as much attention as its creator. Unfortunately, knowledge of his life is not as great as the reader would like(Plutarch appears to discuss him only because of his involvement with Marcellus in the siege of Syracuse by the Romans) but several stories do survive.
First, some definitions as befits the axiomatic method. The origin is the center of the spiral. At the end
of one complete revolution, the line segment from the spiral to the origin is
called the primary radius. The line containing the primary radius is called
the initial line. The length of the primary radius is called the first
distance. The circle centered at the origin with a radius equal to the first
distance is the first circle. A line which is tangent to the spiral only
intersects the spiral in one point, without cross to the other side of the
spiral. The best way to look at the spiral mathematically is through the use
of polar coordinates. The basic equation of the spiral becomes 
where
a=1 in the cases to be consider in this paper. Archimedes, however, used only
geometric methods in his analysis. In this case, the curve can be approximated
in a method similar to that used for the Quadratrix of Hippias. When the ray
rotates to pi/2(90deg.), the distance between the intersection of the spiral
and ray and the origin should be 1/4th of the first distance. At pi(180) is
should be half, ... If continued, the picture ends up like the one at the top of this page.
Now comes the cool part. Draw a line(CD) tangent to a spiral and a circle centered at the origin(A) containing the tangent point(C). Then draw a radius(AC) between the tangent point(C) and the origin and a line(AD) through the origin perpendicular to radius AC. Given this situation, Archimedes proved that the length of AD is equal to the length of the arc(CE) between tangent point C and the primary radius(AB).
The drawing below is a special case because the tangent line intersects the spiral at the end of the primary radius. Thus, AZ is equal to the circumference of the circle. Then, according to the theorem stated at the end of Squaring the Circle the area of triangle AZB is equal to the area of the primary circle!
