http://www.mentalfloss.com/blogs/archives/8596 Feel Smart Again Sat, 20 Mar 2010 06:40:37 -0400 http://wordpress.org/?v=2.8.5 hourly 1 http://www.mentalfloss.com/blogs/archives/8596/comment-page-1#comment-28669 Leadhyena Mon, 08 Oct 2007 18:50:11 +0000 http://www.mentalfloss.com/blogs/archives/8596#comment-28669 This isn't necessarily that surprising. One of the coolest arguments in beginning geometry is the area of a sphere being equal to two cones subtracted from a cylinder. The reason this works is that if properly proportioned, each cross-section at equivalent heights gives the same area. This principal of cross-sectional equivalence is Archimedes's, on which he wrote about in length in a book called "On the Sphere and the Cylinder". The key here is that the argument is similar to a Riemannian integral, treating the volume as an infinite sum of planes. Archimedes was also familiar with infinite series, having used them to come up with a rough argument for the area under a parabola, and of course his method of approximating Pi which is in the linked article. I would say that given his already known texts' illustrations of the infinite, that finding a book that makes the next step is not as surprising as it is yet another illustration of his brilliance. This isn’t necessarily that surprising. One of the coolest arguments in beginning geometry is the area of a sphere being equal to two cones subtracted from a cylinder. The reason this works is that if properly proportioned, each cross-section at equivalent heights gives the same area. This principal of cross-sectional equivalence is Archimedes’s, on which he wrote about in length in a book called “On the Sphere and the Cylinder”. The key here is that the argument is similar to a Riemannian integral, treating the volume as an infinite sum of planes.
Archimedes was also familiar with infinite series, having used them to come up with a rough argument for the area under a parabola, and of course his method of approximating Pi which is in the linked article. I would say that given his already known texts’ illustrations of the infinite, that finding a book that makes the next step is not as surprising as it is yet another illustration of his brilliance.

]]> http://www.mentalfloss.com/blogs/archives/8596/comment-page-1#comment-28632 Miss Cellania Mon, 08 Oct 2007 12:21:09 +0000 http://www.mentalfloss.com/blogs/archives/8596#comment-28632 You got me! That should be 2,000 years. You got me! That should be 2,000 years.

]]> http://www.mentalfloss.com/blogs/archives/8596/comment-page-1#comment-28629 MathFreak Mon, 08 Oct 2007 11:38:14 +0000 http://www.mentalfloss.com/blogs/archives/8596#comment-28629 If we are talking about Archimedes here, that was well over 200 years before the whole Newton/Liebnitz thing... If we are talking about Archimedes here, that was well over 200 years before the whole Newton/Liebnitz thing…

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