Science Forums 2

[Don Blazys]

To: Ughaibu,

Galileo's "proof that all circles have the same circumference" is quite interesting, and in my opinion, requires "proper interpretation".

In Galileo's "cone in bowl" construct, the volume of the "top of the cone" is equal to the volume of the "top of the bowl", while the area of the "base of the cone" is equal to the area of the "base of the top of the bowl".

In the end, as the "plane" approaches the very top of the "cone in bowl" configuration, all that is left is the "tip of the cone" which is essentially a "point", and the "outer rim of the bowl", which is essentially a "curved line".

Now, the "volume or area" of the resulting "point" can still be viewed as being "equal to" the "volume or area" of the resulting "curved line" in that neither a "point" nor a "curved line" can actually posess any "volume" or "area"!

In other words, if the

(volume or area of any "point")=0

and the

(volume or area of any "curved line")=0

then with respect to volume and area,

(any "point")=(any "curved line" or "circumference")

because

0=0.
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That's one way of looking at it... here's another.

As the "plane" approaches the very top of the "cone in bowl" configuration, a final "point inside a circle" configuration can never actually be achieved without the plane and cone "losing contact" with each other and therefore "compromising" (rendering meaningless) the entire model or paradigm.

In other words, the definition of a "point" as "that which has no part" would actually require the abscence of a cone because clearly, the "tip" of a cone is a "part" of a cone!

Well, those are my thoughts on the subject.

However, if you "Google search" (Areas Explain Galileo's "Miraculous" Geometry Problem) then you will find some other thoughts on it, as well as some very profound commentary by that trancsendent genius Charles Arthur Mercier.

Don.