Archimedes: Straightening Out Circle Math

suzuki

We humans have a tendency to want to credit individuals with inventions or deeds for which, in truth, it would be more appropriate to spread the credit over several people, or even to credit a whole continuum of baby-stepping improvers walking a path stretching for centuries. Sometimes, we even credit individuals who may not have existed… for instance, a man named Homer is given credit for writing both The Iliad and The Odyssey, whereas in reality these stories were certainly passed down orally for generations– even if there was a “Homer” who at some point wrote down one version of one of the tales (the tone is so different between the stories, by the way, that I suspect they were NOT cobbled together by the same man).

Of course, unless we’re greatly mistaken there really was a gifted mathematician named Archimedes. However, whether he deserved credit for all of the inventions attributed to him is a matter of debate. But as a shortcut way of speaking about the changes in mathematics brought about during the Archimedean era, I will use his name, if for nothing else than as a symbol for the genius of the time…

Suzuki states that “almost all of what we consider to be the geometry of circles, spheres, and cones is due to Archimedes.” If you saw my post on Greek numbers (or rather, lack of them), you’ll know that the Ancient Greeks barely used numbers at all, but carried out their geometrical reasonings by logical proofs (or disproofs) of congruence.

By the time of Archimedes, however, number-math had become more acceptable to Greek mathematicians. Thus, it seems to be no big thing when Archimedes speaks of the value of ratio of the Circumference to the Diameter of every circle as being 22-to-7… the number we today symbolize with the Greek letter, “pi,” valued close to 3.14. The ancient Greeks before Archimedes would have despised this fraction, 22/7, because it is “irrational” (incapable of being expressed completely in a ratio, or fraction). Archimedes, however, appears comfortable with the number.

Knowing and acceping “pi” as a usable fraction (22/7), Archimedes was then able– and he is said to be the first– to give us the formula for the area of a circle (assuming the measurements of Circumference and Diameter are obtainable). He also produced formulas for sphere and cone volumes, and other mathematical contributions too numerous to go into.

I found it interesting that Archimedes showed that the Area of a Circle is equal to the area of a Right Triangle with one leg set equal to the Circle’s circumference, and the other leg equal to the Circle’s Radius. I had never heard this, but I did the math and it works perfectly… like this…

The area of a triangle is 1/2*base*height. Substituting Circumference for base and radius for height, we get…

1/2 * (Circumference) * (radius)
= 1/2 (2 * pi * r) (r)
= 1/2 * 2 * pi * r^2
= 1 * pi * r^2
= pi * r^2 , which is the area for a Circle!

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