Why are arithmetic progression and geometric progression called arithmetic and geometric respectively?
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Without denying the truthfulness of Michael Hardy's answer, I was always under the impression that the reason for connecting products to geometry was due to the fact that the height of a straight edge triangle is the geometric mean of the projections of its other two sides onto the hypotenuse. This and the fact that geometric shapes such as rectangles provide a very intuitive understanding for why multiplication is commutative.
Lucian
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Imagine a line of length $1$, and the next of length $2$, and the next of length $4$, and then $8$, etc. The pair of lines of lengths $1$ and $2$ has the same geometric shape as the pair of lengths $2$ and $4$, and similarly $4$ and $8$, etc.
