The Spirals...
Archimedes of Syracue (287-212 B.C.E.) was an engineer, a physicist, and a mathemaician, some people, like Dudley, has called him one of the mens with the "greatest minds humanity has produced". Archimedes comes from the line of Greek mathematicisans such as Thales and Pythagoras. Archimedes had many contributions to the world of mathematics; one of the most significant contribution is the sprial. The sprial of Archimedes was publish in one of his works On Spirals. Although Archimedes had not been the first to study the sprials he was the first one to come up with a the use of spirals to trisect an angle.
The most significant feature of the method is that the ratio of the distance between the vertex and any point on the spiral to the measure of the angle formed needs to be constant. This means that the only remaining thing to do in order to trisect the angle is to trisect the segment from the vertex to the spiral. Then by rotating the resulting segment until it meets the spiral so that we have that the angle has been trisected. Another of the great properties of the Archimedes spiral is that every ray from the center intersects every turn of the spiral at a constant angle.
The spiral of Archimedes improved the ancient Greek method of calculating the area of the circle because the spiral allowed them to obtain better measurements of the circle's circumference. This of course help them calculate the area. Nevertheless, later on Archimedes found out a more accurate value of pi, which ended the use of the spiral to calculate the area of the circle. Yet, Archimedes spiral is widely used in the mathematical world. Even nowadays mathematicians continue studying Archimedes' discovery.