One is a way that it is similar to some other works of fiction: In this story a male student falls in love with and attempts to "win the hand" of his math professor's beautiful daughter. This same subplot appeared in The Tachypomp (1873), Professor Morgan's Moon (1899), and of course Proof (2000). (See also The Einstein See-Saw (1932), The Fifth-Dimension Catapult (1931) , The Geometrics of Johnny Day (1941), and The Last Starship from Earth (1968) for variations on this theme.)

Also, rather than being a either spatial or temporal in nature (as "higher dimensions" usually are in these stories), dimension here seems to instead be about whether something is or isn't visible. This is explained in the following passage:

Of course, this argument does not actually make sense. This supposed "wave pattern" that is described really follows from optics (the fact that light moves along one-dimensional trajectories) and anatomy (that our retinas are two-dimensional objects onto which the light falls) and is not the start of an infinitely repeating sequence of "visible" and "invisible". Moreover, this seems to ignore the more interesting and useful idea that an extra spacial dimension would provide a place objects could exist regardless of whether it is seen.

In any case, the novel aspect of this story is that by discovering the fourth dimension, the professor (and the student who is the only other person that understands his theory) is able to become invisible. The plot revolves around the fact that the professor made an error in his calculations which leave his legs invisible. Consequently, he looks like a floating torso (though people can still hear it when he stomps his invisible foot.) This is a predicament from which the student could save him...but the student asks a favor in return.

Thanks to Vijay Fafat for sharing this pulp fiction story with me. It was originally published in the February 1934 issue of Astounding Stories.