Talk:Method of exhaustion

area in a spiral
"The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length" - doesn't this depend on the type of spiral? —Preceding unsigned comment added by 193.172.19.20 (talk) 18:36, 28 November 2007 (UTC)
 * Yes, this depends on the type of spiral. Archimedes used the spiral of r = aθ, which a is a constant and (r, θ) means a polar coordinates system.  cf.  --Ttwo (talk) 14:10, 3 August 2010 (UTC)

Computer applications?
Are there occurrences/applications using computers, now that sums can be done much more easily? Or are they all done using Riemann sums? —Preceding unsigned comment added by 128.100.71.134 (talk) 14:25, 19 September 2008 (UTC)


 * One of the basic numerical integration methods (as used by computers) is the Trapezoidal Rule which is analogous to Method of Exhaustion (except it occurs in one dimension). Added the reference to See Also section... Cowbert (talk) 09:21, 18 October 2009 (UTC)

Area/Volume Descriptions
"   * The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height;    * The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes;    * The volume of a sphere is 4 times that of a cone having a base and height of the same radius;    * The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter;    * The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length;    * Use of the method of exhaustion also led to the successful evaluation of a geometric series (for the first time)."

There needs to be images associated with these area/volume formulas. Also, the volume of a cone compared to a sphere is not exactly clearly stated.

Vol_sphere=4 * Vol_cone, if b=h=r A beginner would likely assume that the radius of the base of the cone, and the height are the same, but still wouldn't be sure this was correct? This could be a lot clearer with a diagram... Are there any already on the site? —Preceding unsigned comment added by JWhiteheadcc (talk • contribs) 02:36, 17 December 2007 (UTC)

Formula for Exhaustion
The point of exhaustion is to give a method for rigorous proofs for areas and volumes, not to give a chopping up approximation scheme. The simple chopping up schemes are trivialities now, and always were trivialities. Exhaustion allows you to prove exact results by showing that the error term is small, by bounding it above by an arbitrarily small quantity. It can't be summed up by a formula.Likebox (talk) 18:18, 8 January 2009 (UTC)

Apparently it can. The formula wasn't put there for no reason and I won't let it be removed for no reason. If the infinity in the first sigma was finite, that would be another story. srn347 —Preceding unsigned comment added by 68.7.25.121 (talk) 03:45, 28 January 2009 (UTC)


 * Uh.. I don't know where you learned math, but geometry can be done with a formula. That's how things work. Don't just remove things like that because you don't understand how it applies. I put it back in. Matt (talk) 00:49, 8 November 2009 (UTC)

Some remarks on this formula. The source of this formula is (as claimed in the link) the ''Int. Jour. of Math. and Math. Sci.'', a fee-based peer review journal published by HPC. In any case, the formula is correct. It is a quadrature formula for continuous functions on the interval [a,b]. Details: a continuous function f can be uniformly approximated interpolating it by piecewise affine functions fn (linear splines). In particular, here the nodes of the interpolation correspond to the uniform dyadic subdivision of [a,b]. By the uniform continuity of f, the sequence fn converges uniformly to f, so the integrals &int;ab fn(x)dx converge to &int;abf(x)dx. The special choice of the nodes gives &int;abfn(x)dx the form of partial sums of a series. Thus, the formula follows by a very common approximation argument. That said, is has to be remarked that this formula is just one quadrature formula among hundreds of others, and not particularly relevant from the point of view of numerical analysis. It is not didactically relevant either, as compared with the semplicity and generality of the traditional Riemann integral, though it may provide a nice exercise in a course on elementary analysis. From a historical point of view, it is of no interest, of course, and reporting it here seems to me a bit misleading for the readers. It is not quite correct to call it a new form of the method of exhaustion: in a strict acceptance, this is not the method of exhaustion (if the function is not positive and concave, the triangles may even exceed the subgraph). In a generalized sense it is, but then all modern analysis may be considered a form of the method of exhaustion as well. As mentioned above, we can't say it's "new" method, either. Notice that filling with triangles is exactly how Archimedes computes the area of a segment of a parabola, in the statement 24 of his work "Quadrature of the parabola" (by the way: this is the information a reader would possibly like to find, and that we should provide in the article). I think we better remove this formula from this article that is devoted to history of science, and possibly find another place for it, maybe as an example of quadrature approximation scheme. --pma (talk) 12:31, 8 November 2009 (UTC)


 * No, the formula is indeed exhaustion. It's just more generalized. It belongs on this article. The mathematician that derived it (his paper is here, described it as exhaustion and proved how it was such geometrically. Saying it isn't is not only incorrect but also could be considered to be original research So the formula should stay.Matt (talk) 08:39, 9 November 2009 (UTC)


 * Well, whether this formula is, or is not, to be considered "method of exhaustion", it depends on how generalized is your definition of "method of exhaustion". If you generalize enough the definition as to include this formula, then you should include a huge amount of mathematical analysis as well. But if we want this article to be an article on history of mathematics (which has to be decided), then formulas like that have no place in it. In particular, notice that a characteristic feature of the exhaustion method is the monotonicity: recall that ethymologically exhaustion is the process of taking away completely (elementary pieces from the shape to be measured)  --today we describe it as "filling" (the shape with the pieces), which is essentially  the same idea. Moreover, the exhaustion method is not aimed to an approximation scheme, but to a proof by contradiction, as Likebox explained to you. However, the approximations in this formula do converge (see my previous post), but not monotonically, in general. It is however, reminiscent of the triangularization procedure of Archimedes, that's true.
 * On a side remark, the article from arxiv.org you quoted has no proof about the validity of this formula, in particular, it makes no use of the method of exhaustion in whatever form. The only mention to convergence is the sentence : This procedure is continued indefinitely to “exhaust” the remaining area (why should the area converge to the integral of f as n goes to infinity? Why shouldn't it became much bigger, or remain much smaller forever?)  Also notice that the expression "remaining area" is uncorrect or misleading, as the triangles may even exceed the shape, where the function is not concave. The subsequent sentence The method of exhaustion will converge to the value of the integral at least as fast as a geometric series, because when each new triangle is small enough so that the local curvature between intersection points on f(x) is slowly varying, is just wrong, because the function was only assumed to be continuous or piecewise continuous. Indeed, for a generic continuous function the rate of convergence may be arbitrarily slow.  As to the originality of the formula, it's not the point to be discussed here. --pma (talk) 09:56, 9 November 2009 (UTC)

I deleted the section again. The formula may be true and somewhat intersting but it does not belong to this article. It is no more connected with the method of exhaustion than any other quadrature formula. MathHisSci (talk) 21:29, 7 June 2011 (UTC)

I agree. --pm a 14:02, 21 June 2011 (UTC)

Inaccurate description?
Based on my readings elsewhere on the 'net, this passage seems incorrect:

''The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region (which can be “exhausted” so that its area becomes arbitrarily close to the true area). The proof involves assuming that the true area is greater than the second area, and then proving that assertion false, and then assuming that it is less than the second area, and proving that assertion false, too.''

I believe that the method of exhaustion actually consisted of finding two regions: one that was provably larger than the the region that you wanted to find the area of, and one that was provably smaller. See http://personal.bgsu.edu/~carother/pi/Pi3a.html

The only way you could do it the way the article currently says is by finding a region with exact same area as the one you want to find the area of (if x is our first region, and a is our approximation, then (area of x <= area of a) ^ (area of x >= area of a) => (area of x == area of a).--76.102.149.170 (talk) 04:22, 18 May 2009 (UTC)


 * The word typical can mean anything. Typical for what kind of proofs? Green blue proofs? The best would be if the paragraph had a reference to some credible work that gives the required evidence. The evidence runs rather against the claim. For example I guess Archimedes parabola bisection is rather trivial to proof without contradiction. The bisection uses rational numbers, and it shouldn't be so hard to do the necessary rational number comparisons, even in a proof with variables, without contradiction.
 * Jan Burse (talk) 23:16, 18 October 2017 (UTC)

The description is inaccurate. The method of exhaustion relies solely on the Archimedean property and an argument by contradiction. 71.219.100.40 (talk) 14:22, 28 July 2018 (UTC)

image + box
In the section Used by Archimedes (now called Application by Archimedes), the positioning of the images, pi box and list is very awkward. I removed the pi box and replaced with a "main article" link as it is excessive emphasis on links to pi, which are far less important than the diagram itself and the actual content of the section. That one link to pi, from which a reader can find out much more on pi, is plenty. Also the box extends too far down the page into the See also and Notes and references sections... F = q(E+v×B) ⇄ ∑ici 14:05, 19 May 2012 (UTC)

The illustration which appears in many places in wikipedia is misleading. Archimedes calculated pi by successively doubling the sides of a polygon, not by successively adding one to the number of sides. 93.80.54.16 (talk) 14:59, 14 July 2015 (UTC)

Where's the Work!?!
Um…has anyone noticed that none of the work that give the results mentioned on the page via the method of exhaustion is shown in the article? Don't you think that showing all of the work involved here would help people understand what the article is trying to convey?

—&thinsp;RandomDSdevel (talk) 21:13, 29 September 2013 (UTC)