Archimedes' Spiral
In the third century B.C., Archimedes of Syracuse created a special spiral-shaped curve by pulling the legs of a compass apart while turning it. By performing both actions at a steady rate, he found that the resulting spiral moved outward by the same amount with each turn of the compass. The groove in an old-style LP record is an example of such an Archimedean spiral.
The most significant mathematical use to which Archimedes tried to put his spiral was to create a better method of determining the area of a circle. Using a spiral to figure out the area of a circle seems a waste of energy today since anyone with a calculator can do so by pressing a few buttons. However, in ancient Greece either a physical measurement of the circumference of the circle had to be made or a critical factor in the still not widely known equation for determining the area had to be used. Measuring the length of a circle or any other curved shape is difficult and the area of the circle that is determined as a result can only be as accurate as the measurement. For these reasons, calculating the area of a circle presented a major problem for the mathematicians of Archimedes' time. Back then, they knew the area was related to the ratio of the circumference of the circle to its diameter, but this ratio, called pi, was (and even today still is) not known with complete accuracy.
Today we can calculate its value much more closely, but in ancient times mathematicians used a value for pi that was inaccurate enough that their determination of the area of a circle was unsatisfactory for many critical applications.
The Greeks and others before them had tried a number of methods for determining pi and figuring out the area of a circle. One of these involved constructing a right triangle that had one side with a length equal to the circumference of the circle and another with the length of the radius of the circle. Such a triangle has approximately the same area as the circle. The straight sides of the triangle can be measured accurately and the area of the triangle determined, but this method only gives an approximation of the circle's area because, again, it is dependent upon measurement of the circumference. Archimedes tried to use his spiral to improve upon this method. He started the drawing of the spiral at the center of a circle and rotated and opened the compass in such a way that the spiral reached the perimeter after one turn. This meant that the point where the spiral intersected the circle provided the point for one corner of the triangle. Since this triangle method can be carried out with equal accuracy with or without Archimedes' spiral, his method was really only of mathematical interest. However, Archimedes went on to determine a much more accurate value for pi, which advanced the determination of the area of circles in another way.
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