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Methods Of Exhaustion

  1. Jan 12, 2008 #1
    Can anyone give some historical examples of methods of exhaustion being used to solve problems. One popular example is the method Archimedes used to find the lower and upper bounds of the area of a circle (and therefore Pi) by inscribing circles inside and outside a circle?

    In particular i'm looking for early precursors to calculus/infinite series and the idea of limits. Most info points towards the Ancient Greeks as a starting points but surely there must be earlier developments in the field? Maybe early Chinese mathematics or Islamic/Arabic mathematics?
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  3. Jan 12, 2008 #2


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    There is of course no Islamic mathematics earlier than Archimedes, since Archimedes predated Muhammad by some nine centuries. There was little in the way of mathematics in the Arab world in the time of Archimedes, as their nomadic lifestyles were ill-suited to mathematical research.

    I do imagine the ancient Chinese discovered the method of exhaustion, and likely before the time of Archimedes. Unfortunately I was not able to find anything in my online searches -- perhaps hindered by the Burning of the Books and Burial of the Scholars which would have happened around the time of Archimedes' death.
  4. Jan 12, 2008 #3
    Book XII (12) of Euclid's elements has about 6 propositions which showcase the method of exhaustion. Prop. 2 is an easy one to digest and you can find lots of writing on it as most histories (eves, boyer, edwards) cover this as a representative example of the use of the method of exhaustion in greek geometry. In proposition 2 it is shown that the area of two circles is in the same ratio as the squares on the diameters.
    The method of exhaustion gets its effectiveness from book X (10) proposition one (1), which shows from the assumption that no infinitesimal magnitudes exist (so called axiom of archimedes) it follows that given a pre assigned magnitude, any magnitude may be reduced to less than the given magnitude by showing it may be reduced by half indefinitely.

    relevant propositions:

    sources I used for a paper on the method of exhaustion:
    Boyer(2), Carl B., The History of the Calculus and Its Conceptual Development, Dover,
    Boyer, Carl B. and Merzbach, Uta C., A History of Mathematics, second edition,
    John Wiley & Sons, 1989.
    Coolidge, Julian Lowell, A History of Geometrical Methods, Dover, 1963.
    Edwards Jr., C.H., The Historical Development of the Calculus, Springer-Verlag, 1979.
    Eves, Howard, An Introduction to the History of Mathematics, sixth edition, Saunders
    Colege Publishing, 1990.
    Heath, Sir Thomas L., The Thirteen Books of Euclid’s Elements, second edition
    Vols. II and III, Dover, 1956.
    Kline, Morris, Mathematical Thought from Ancient to Modern Times,
    Oxford University Press, 1972.
    Mueller, Ian, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements,
    The MIT Press, 1981.
    Stillwell, John, Mathematics and Its History, second edition, Springer, 2002.
  5. Jan 12, 2008 #4

    Gib Z

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    Archimedes used Riemann Sums to determine the area under a parabola.
  6. Jan 12, 2008 #5


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    it seems archimedes used the concept of center of gravity to discover the area and volume formulas of figures, then used exhaustion , i.e. limits, to prove them. just look at archimedes' works in heath's translation, published by dover. or search on google.
  7. Jan 13, 2008 #6

    Yeah, Islamic mathematics definitely came after the Greeks, lol but in any case it would be interesting to know if they used any similiar methods. I do recall reading somewhere about exhaustion methods being used by the Chinese but I've lost the link and can't verify it.
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