A square tablecloth spread on a rectangular table will hang from the table edges same as before if the cloth is rotated ninety degrees lying on the table. A rectangular cloth not square—a cloth made longer in one length, shorter in the other—will require two rotations of ninety degrees to hang from the table edges same as before. We can open our geometry books and find that such facts are aspects of the rotational isometries of quadrilaterals.

Facts of pure geometry can have explanatory value to us for some facts about table cloths. I should distinguish two sorts of necessities in pure geometries. One sort is the plain necessity of geometric facts, such as one-hundred-eighty degrees being the sum of the angles in any triangle in the Euclidean plane or such as any square’s diagonal being not of any integer-ratio to the length of its sides. The other sort is the necessity between *if* and *then* in the inferences one makes in proving such results. Continue reading