Archimedes and Circles

Monday, April 24th, 2006

Most people know the figuring of the area of a circle is A = pi x r squared. It was Archimedes in the 3rd century BC who did the figuring in Measurement of the Circle. But what I find important about this is not the formula but the kind of thinking that proved the point. Archimedes writes

Since then the area of the circle is neither greater than nor less that [the area of the triangle], it is equal to it.

The suppositions go: A< T or A=T or A>T. If the first and third don’t work, then the second must be true, double reductio. For kicks: 1st case

A – Area (inscribed polygon) < A - T leads to T < Area (inscribed polygon) Area (inscribed polygon) = 1/2hQ < 1/2rC = T where h = apothem, Q = the perimeter of the polygon and C = the circumference of the circle Since here T < the area of the inscribed polygon and the area of the inscribed polygon is also < T then the supposition of A < T contradicts itself

and so forth. Here I’ve paraphrased Will Dunham’s Journey through Genius: The Great Theorums of Mathematics. Archimedes proves the area and approximates pi in the same text but does so by a wonderful bit of questioning and analysis. The area is important, but without the skill of bringing it all together, the mathematician is guessing.


2 responses to “Archimedes and Circles”

  1. Helen Coffield says:

    Hey Steve,
    I didn’t know where else to write you a note. I’m not that familiar with blogs…anyway, for some reason, late at night when I’m thinking about things, you always pop into my mind and I thought I’d just write you and let you know. As a nursing major, I was not too thrilled about taking brit lit. I convinced myself that I was no good at nor did I care for the liberal english overanalyzation that goes on in lit classes. however, I thouroughly enjoyed your class and the mind-opening it did for me. I’m sure that’s really poor english by the way….
    Now I miss the freedom and expression that comes with lit and other similar classes. I don’t always want to know what metabolic cycle occurs when no oxygen is present or how the kidney filters toxins. I want to talk about how Milton used satan as a protaganist….
    not really, but I want to feel the freedom and release that comes with that type of class. I thouroughly believe that I am a classic middle of the road american. I like all subjects and all sports and art and music and everything else…and I’m average at all of them….not great at one, no passion for one particular area or subject, just mediocore at everything. where do you go with that? where does one get from being decent at a variety of things? people seem to excel when they have a true passion for one thing, and they’re good at it. people say, well, what’s your passion? well…everything! I like everything….I don’t want to be limited to one thing. I just seems like life is so pressured towards finding out what you want to do and making money with that. I just want to do everything and be my mediocore self… Thanks for being an inspirational teacher…there are not many of you left.
    take care,
    helen*

  2. Steve says:

    Great to hear from you, Helen.

    I think there’s still lots more to think about with Milton, such as his conception of time–when, for Milton, or as a consequence of his thought, does time begin?

    Here’s another question: when people ask about your passion, why not wonder what you don’t like or what gets your goat?

    Yet another Milton related question: it’s fine to know the fuction of the kidney. But do you do enough with what to do with that knowledge?