This mystical story, as recounted by a lawyer, is about a brilliant mathematician ("an erratic genius who had written some articles in Mind on that dreary subject, the mathematical conception of infinity", "[his] tastes were on the borderlands of sciences, where mathematics fades into metaphysics and physics merges in the abstrusest kind of mathematics. ) who wonders why animals have a "homing instinct", why they are able to have an uncanny sense of direction. He theorizes about mathematical laws that govern absolutely empty space, a sort of fourth dimension that has involutions, is "not an empty homogeneous medium, but full of intricate differences, intelligible and real, though not with our common reality". A primal brain is able to comprehend these differences far better than a civilized mind. He ends up gaining a very deep insight into the nature of this plane governed by mathematical forms and figures, finally sensing "Presences" therein which terrify him. In the end, the mathematician falls off a cliff while climbing a mountain blindly, with a hint that only his body has perished, his mind may have merged with his new "Space".

While the author does not mention this, I think he is very likely describing the Platonic world of forms. Throughout the story are sprinkled sentences which point toward this interpretation: