Has anybody noticed or written about this quaint fact?

The n-1 surface of an n-dimensional sphere is equal to the cross-product of
the formula for the surface of a circle (2*pi*r) times the formula for the volume
of an n-2 dimensional sphere at least up to n=8 and by induction beyond that.
For instance 2*pi*r times 8/15*pi^2*r^5 = 16/15*pi^3*r^6.
That is the formula for the surface of a square times the formula for the volume
of a 5-dmensional sphere is equal to the (7-1) surface of a 7-dimensional
sphere. Similarly the cross product of the formula for a circle times the volume of a circle is equal to the (4-1) surface of a 4-dimensional sphere.

What I really want to know is whether that is an original observation or not. I have not been able to find a discussion of it.