circle/sphere formulas and calculus

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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?
ericqb

Re: circle/sphere formulas and calculus

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).

Re: circle/sphere formulas and calculus

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