Wednesday, December 23, 2015

The Roots of Calculus - Archimedes

Archimedes did a lot that nowadays looks like calculus...

He determined the value of pi very precisely, by starting with a hexagon inscribed in a circle, then a 12-sided polygon, then he kept doubling the number of sides until he got to a 96-gon. A procedure like this is called the 'method of exhaustion', and it looks a lot like what we do nowadays with limits.

I am embarrassed to admit that I couldn't figure out how he did it. (I think I was focusing on area, and that might be harder.) I just found a great video by David Chandler (whose youtube channel is Math Without Borders).

Here's a summary:
Start with a hexagon inscribed in a circle of radius 1 (giving diameter of 2). The perimeter of the hexagon will be 6. This gives a lower bound on pi, which is the ratio of circumference to  diameter. We know the circumferences is bigger than this perimeter of 6, so pi is bigger than 6/2 = 3.

If you cut one of the triangles that made the hexagon into two, you get a radius that crosses a side of the hexagon at right angles. You can use the Pythagorean Theorem (twice) to find the new side length. Repeat 3 times and you're at the 96-gon. Archimedes had none of our technology, and little or none of our algebraic symbolism, so the calculations were much harder for him. We can do all this on a spreadsheet, and up comes pi (if you have a column for the perimeter over the diameter). So satisfying!

If this doesn't make sense, watch this lovely video. Thank you, David!

Archimedes did a lot more than find a value for pi! What's your favorite bit of calculus that started out with Archimedes?


  1. I showed this method to Math Circle for Pi day a long time ago. We discovered that it converges very very slowly. (but I think it's still a bit faster than the arctan taylor series.)

  2. The cool thing is how easy to see why this is pi.

  3. I'm not sure this counts as starting with Archimedes, but your post made me wonder if he was credited with responses to any of Zeno's paradoxes. Wikipedia says yes: Proposed Solutions with the sum of an infinite geometric series.

    Also, there's always this song from Square 1 TV to enjoy: Archimedes Song

  4. Don't you mean the lower bound on pi is 3=6/2?

  5. Yes. I don't know what hiccup in my attention made me chop that up like that. I will fix it.

  6. Archimedes also used a circumscribed hexagon as well, so he established not just a lower bound, but an upper bound.

  7. Yes. And your comment got me back to playing in geogebra and on paper, to try to figure this all out. I'm now working on a handout for my students.

  8. The circumscribed looks lots harder to figure out. I'm not seeing the pattern yet.

  9. I showed the quadrature of the parabola to my math history students last year. It's astounding! (It's also pretty difficult, and you'll probably want to be able to have time to "follow along" on your own paper.) I liked this website, which has step by step diagrams to go with the text.


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