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From: JEMebius on 6 May 2010 20:55
calvin wrote:
> Why do you call it volume? In two dimensions,
> it's area. In three dimensions, it's volume.
> In four dimensions, it would be something else.
> In four dimensions, volume would be to that
> something else as cross-sectional area is to
> volume in three dimensions.

Then it is up to you to devise informative, yet completely new, names for "4D volume, "5D
volume" etc.

Please recall the origin of the word "volume": volumen (Latin) = roll, scroll.
Perhaps the meaning shifted from "scroll" via "book" to "separately bound part of a large
book" (e.g. encyclopedia), and from the object: a book, to its contents. In a next stage
the physical-geometrical concept of "contents" and the abstract "contents" concept got
confused, so to say. A result of all this is that the word "volume" is being used in
several different meanings, among which the absolute value of an N-form in an
N-dimensional space.

I am open for any and every correction and improvement, even if it looks like a
hair-splitting exercise.

Good luck: Johan E. Mebius
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