Archimedes Integration of the Circle

A lot of people have asked me to write something about “Archimedes Integration”, and I’m finally getting around to fulfilling that request.
As most of you already know, Archimedes was a philosopher in ancient Greece who, among other things, studied mathematics. He invented a technique for computing areas that’s the closest thing to calculus before Newton and Leibniz. Modern mathematicians call Archimedes technique “the method of exhaustion”.
The basic idea of the method of exhaustion is to take the figure whose area you want to compute, and to divide it into pieces whose area you already know how to compute; and to make the divisions smaller and smaller, *exhausting* the area not included.

For a simple example, we can start with a circle, and inscribe it with *equilateral* triangles that run from the center to the edge of the circle. For each of those triangles, the length of the base is the radius *r* of the circle; the height of the triangle is *r×(sqrt(3)/2)*. So the area of one triangle is *(1/2)(sqrt(3)/2)r2; and the approximate area of the circle with six triangles inscribed is *6(1/4)(sqrt(3)r2 = (3sqrt(3)/2)r2* or roughly 2.6r2. Not bad, but not great. Now, Archimedes wouldn’t have said it quite like that. You see, the Greeks didn’t use equational reasoning; they worked in terms of comparisons and ratios. So Archimedes wouldn’t say “1/4 r2“; he’d figure out a single triangle whose area was equivalent to the sums of the 6 triangles – a triangle whose height was the same as the radius of the circle, and whose base was *3sqrt(3)r*. The following diagram shows the circle with triangles inscribed, and the triangle with an area equal to the sum of the inscribed triangles area drawn over the triangle with area equal to the circle.
Now, moving on, suppose we made the triangles smaller. Cut the angle at the center of the circle in half, so that we had twice as many triangles, each with a central angle of 30 degrees rather than 60. Then our approximation would be (1/4)r2; with 12 triangles, that would give us 3r2; or an area equal to a triangle whose height was the radius, and whose width was 6, as in the following diagram:
Suppose we kept making the triangles smaller. Each time we reduce the size of the triangles, we remove (exhaust) more of the uncovered part of the circle; and length of the triangle whose area equals the sum of the areas of the inscribed triangles gets closer and closer to *2πr*. Or in the terms Archimedes would have used: the length of the base gets closer and closer to the radius of the circle. So the area of the circle gets closer and closer to the area of a triangle whose height is the radius of the circle, and whose width is circumference of the circle. So in equational terms, we end up with *(1/2)(2πr)(r) = πr2*.
The Greek mathematicians had no *explicit* concept of limits or how to work with them. But you can clearly see the origin of the concepts of limits here. And Archimedes didn’t stop there. He went on
to improve the method – to take a figure, and *enclose* the figure in triangles as well as inscribe triangles inside the figure. The area of the triangles *enclosing* the figure is an upper bound; the area *inscribed* in the figure is a lower bound; and the average of the two is a pretty good estimate.
The way that Archimedes did it – basing it on triangles and ratios – is obviously a very limited technique. But it’s very impressive for reasoning without the tools of algebra, and it *does* work on a good number of very regular figures.

0 thoughts on “Archimedes Integration of the Circle

  1. Blake Stacey

    Actually, it looks like the method of exhaustion was first used by Eudoxus (408–355) and later picked up by Archimedes (287–212). The notion of computing an area or volume by adding up lots of tiny pieces may have stemmed from Democritus (c. 460–c. 370), who might naturally have thought along those lines since he espoused the idea that all matter is made of atoms. To the best of current knowledge, Democritus was the first to state that (a) the volume of a cone is one third that of a cylinder having the same height and base, and (b) that the volume of a pyramid is one third that of a rectangular prism with the same base and height. He may have deduced these formulas by treating the cone and the pyramid as being made of innumerably many thin slices, which is a clear precursor to integration. (Johannes Kepler did the same thing with wine barrels, centuries later.)
    Lately, I have been reading with great pleasure the works of George Sarton. His Ancient Science Through the Golden Age of Greece is full of information on Democritus and all that merry lot.

  2. Mark C. Chu-Carroll

    The URL limit doesn’t trash things, it just tosses them into the moderation queue. I already published your comment out of the queue.
    On the subject of people who write about ancient science, have you read anything by Paul Keyser?

  3. Paul Carpenter

    Hmmm, life must have been hard before calculus. It’s strange to think what a brainwave it was and how much of a difference it made to start explicitly thinking about limits and stuff.

  4. Blake Stacey

    @Paul Carpenter:
    I’ve always liked what Alfred North Whitehead said in his Introduction to Mathematics (1911). “Civilization advances by extending the number of important operations which we can perform without thinking about them.”
    @Mark C. Chu-Carroll:
    Thanks for rescuing my comment! I have known a few moderation queues which more resembled /dev/null, so I really appreciate it. To answer your question, no, I haven’t read Keyser. (My reading habits are always more erratic than I’d like; I discovered Sarton by finding a book of his crammed onto a shelf at a used-book store.) Is he good?

  5. Mark C. Chu-Carroll

    I do my best to check the mod queue often. In general, things shouldn’t sit in the queue for more than a couple of hours, unless I don’t notice it. If you think you might have something stuck, just shoot me an email and I’ll release it.
    As for Paul Keyser… To be entirely honest, I haven’t read any of his books (yet). But he’s a friend of mine, and we’ve worked together for a couple of years. He’s a really interesting, fun guy, and a great speaker, so I expect his writing is probably good. He’s got two PhDs, one in physics, and the other in classics, but he now works as a programmer to make a living, and does classics work writing papers and books about ancient greek science in the evenings/weekends.
    Since you mentioned something about greek science, I immediately thought of Paul.

  6. Jianying

    To be complete I think the Archimedes palimpsest and “The Method” should be mentioned. Especially archimedes’s physical based integration ideas, which combined democritus’s ideas with center of mass calculations.

  7. MiguelB

    Thanks, Mark (I was one of the original people asking for this).
    I recently read Lucio Russo’s “The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to Be Reborn”, and found it to be very interesting, although not without flaws. Anyway, it includes an appendix explaining how Archimedes calculated the area under a parabolic segment. It’s essentially the same method Mark explained, except the parabola’s shape adds a couple of difficulties. Recommended reading.

  8. Blake Stacey

    @Mark C3:
    Thanks again! If all blogs functioned as smoothly, we’d be two-thirds of the way to Paradise. 🙂
    I’ll add Paul Keyser to the list of authors I mean to read “Real Soon Now”. My deadline at work will soon whoosh past, and mere days after that, Thomas Pynchon’s Against the Day will hit the stores. Once (if) I have survived that, I’ll see about reading through my queue.
    Lots of odd things are in that queue, of course. Back in the spring, I decided to re-teach myself all the Latin I had forgotten so I could read the first Harry Potter book in Latin translation. You know, so I could be that guy who read J. K. Rowling in Latin before he ever read her stuff in English.

  9. DouglasG

    While this method demonstrates a way to get the correct answer, it would have been totally unacceptable to the people of Archimedes day (and before). Thus, Archimedes would use this method to determine the answer, and then work up an explanation that would convince his peers or just give the solution without explaining how he came up with it.

  10. Jonathan Lubin

    Mark, I really think you’re wrong to refer to Archimedes as a philosopher who also did mathematics: I never heard any such thing, and all his extant works are mathematical.

  11. bn

    I’m so glad somebody else appreciates this method of calculation, Archimedes was absolutely brilliant. The devices he was able to construct with the limited knowledge he had is fascinating. I think it’d be safe to say that we can put him ahead of isaac asimov as the father of modern robotics. It’s also interesting to note that al-jahzara(sp?) from the middle-east was the first to recognize the significance in archimedes’ ideas and then built upon them to construct the first “programmable” timing device.


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