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Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by Af.

These are real analytic periodic functions with period T.

In the following paper it is shown that in a certain definite sense, exactly an odd number of squares can be inscribed in every such curve which does not contain an infinite number of inscribed squares.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

The latter theorem has been generalized by Yamabe and Yujobo, and Cairns to show that in Af there are families of such cubes.

Here, for the case of squares inscribed in plane curves, we remove the restriction to convexity and give certain other results.