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circle/sphere formulas and calculus

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circle/sphere formulas and calculus


From ericqb
Did any one over the centuries before Newton and Leibniz remark on the parallel between the transition from 2 pi r to pi r squared (circumference of a circle to area of a circle)and the transition from 4 pi r squared to 4/3 pi r cubed (surface area of a sphere to volume of a sphere)? The geometric equivalence of these transitions, from the boundary to what is contained by that boundary, is obviously something that would prompt a mathematician to look for equivalence in the math.
I understand that the x,y system is how analytic geometry and calculus grew, but I would like to know if the transitions, integrations, I mention were recognized on record, and discussed as the same manipulation, before or after calculus was first described by Newton?

well Im not sure when there was the first mention of this idea , but what you are talking about it a special case of a bigger picture. What Im going to say is kind of surprising and anti intuitive. Let's talk about the volume of the unit balls in R^n .

For n =1 the 1-dim volume is 2
For n =2 the 2-dim volume is pi
For n =3 the 3 dim volume is 4/3 pi

Now for n = 4 the 4 dim volume is pi^2 / 2 !!

So the volume seems to increase as we increase the dimension , but...the volume of the general n-ball is given by the formula

V(n) = Pi^(n/2) / Gamma(n/2 + 1) where n is the dimension

Looking at the asymptotic behavior of this function , we see that as n tends to infinity , then V(n) tends to 0 !!! Therefore we see that the bigger the dimension , the less "that" dimensional stuff we can fit in it. So what about the surface area ?

The formula for the surface area of a unit ball in R^n is

w(n) = 2pi^(n/2) / Gamma( n/2)

So what's happening to the surface area as n tends to inifinity? Well I'll leave this to you to see whether your proposition remains true if you integrate w(n) trying to get v(n).

I'd say it's highly likely that if someone did observe that,
then they probably considered this relation for
circles and balls a corollary of the corresponding
relations for triangles and pyramids (area is base
times height over 2, but volume is base times height
over 3). The pre-calculus proofs for these kinds of
formulae typically employed reasoning about
(more-or-less) infinitesimals to a much higher degree
than the average modern mathematician might think.


Yes, infact Archimedes did the thing on the circle; basically a primitive form of integration. His idea of taking limits ovber the infintesimal was also known by the Egyptians who used this method to calculate the volume of a sphere; and of course there is also Eudoxus (who was a student with Euclid) who wrote on the method of Exhaustion, which is basically integration before analysis (ie Descartes) came along.

For further details on any of these topics, please feel free to write to me.

have fun with it!

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