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I think Archimedes had some great non-infinitesimal methods for discovering the area and volume of shapes. Some very visual methods involving his method of exhaustion for the volume of a sphere for example. These ideas were transcribed from some Palimpsest discovered with various drawings on them too but almost hidden. These methods are known now so why aren't they taught in high school (a way to introduce High School student to Calculus)?

Rodrigo de Azevedo
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201044
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I think I answered this question here:

Are there any theorems that become "lost" and discarded over time?

Mathematics is very large. There are many beautiful things we could teach. But the time available is limited. Therefore we teach what is considered most important. Geometry (parts of Euclid) and algebra in high school. Calculus and linear algebra in the College. This is a very small part of mathematics.

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Alexandre Eremenko
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  • This doesn't answer my question about Archimedes methods and how they should be taught in high school ( with modern terminology). – 201044 Dec 30 '14 at 04:20
  • Thew short answer to your question is that Archimedes methods should NOT be taught at high school. However reading Archimedes himself is highly recommended to a young student with a strong interested in mathematics. – Alexandre Eremenko Dec 30 '14 at 20:09
  • Why shouldn't Archimedes methods be taught in High school to encourage a fascination with mathematics? – 201044 Jan 01 '15 at 06:21
  • Archimedes methods where visual and involved physics style thought experiments like finding the center of mass in an object the shape of a triangle ( actually a thin triangular prism) and balancing it , or something like that ( if I got that right). So relating visual images and physics thought experiments to find out a geometric idea is a fascinating mix of math techniques that could be taught in high school. – 201044 Oct 22 '15 at 04:07
  • Didn't Archimedes using his method of exhaustion and other arguments determine the ratio between the volume of a cylinder with a circular base and a sphere of the same diameter fitting snugly in it. So indirectly discovering the formula for the volume of a sphere way before 'pi' and modern terminology developed? These arguments the forerunner of calculus might be very useful to students.. – 201044 Feb 27 '16 at 00:13
  • This should be in schools with the story of the palimpsest is gripping and should appeal to most students. I am not are aware of any intuitive explanation of the integral of a cubic, ie x^3 -> x^4/4. The reverse power rule is not intuitive. Archimedes method of mechanical theorem uses a unit see-saw as explained on wiki which shows why its divided by 4. I have made 3D prints of this see-saw with polynomial sails at 90 degrees and explain more here: https://hsm.stackexchange.com/questions/12494/archimedes-method-of-mechanical-theorems-centroid-of-hemisphere?noredirect=1#comment24759_12494 – rupert Nov 25 '20 at 15:52