When at school I learnt Archimedes's theorem that the area of a zone of a sphere between two latitudes is equal to the corresponding area on a circumscribing cylinder. I think that's all he needed to see the result you mention. R. On Wed, 2 Jun 2004, R. William Gosper wrote:
The volume formula for the perpendicular intersection of two cylinders of radius r can be found by replacing pi by 4 in the volume of the inscribed sphere. It is said that Archimedes found this thirteen centuries before Newton's calculus. Did he also know that the same 4/pi trick gives the surface area? If not, perhaps he noticed the universal prismatoid volume formula
A + 4 A + A t m b V = h --------------, 6
(= one panel of Simpson's rule, with A:=area, t:=top, m:=middle, b:=bottom, h=height), which works on all the elementary solids, and deduced that it also works for the "biroller". (Is there a classical name for this solid?)
I've said this before, but a fairly dense, fairly hard biroller (say, lathed from 2" aluminum bar stock) makes an intriguing chaotic toy, when rolled diagonally down a shallow, inclined trough (like a bathtub, but longer and shallower). The roller oscillates with gradually decaying amplitude until a downswing pauses on one of the two poles, whereupon it rocks with renewed amplitude, downslope on a perpendicular arc, taking a completely unpredictable time to reach the bottom. This was shown to me by an adolescent David Silver ca. 1970. --rwg PS, a Macsyma command to draw a biroller is block([plotnum1:2,f],[sin(t),sin(t),cos(t)],f:(1-u)*%%+u*[-1,1,1]*%%, makelist(f:part(f*[-1,1,1],[2,1,3]),k,1,4),plotsurf(%%,t,0,%pi,u,0,1))
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