This short novel, originally published in the January 1931 ASTOUNDING, and republished by Damon Knight in SCIENCE FICTION OF THE 30'S (1975), involves a mathematical physicist whose theories get applied by an experimentalist who gets trapped, along with his beautiful daughter, in an alternate dimension. The theoretician must think out a rescue.

The story rolls in its simple-minded goshwowness, plotting, and characterizations, of the traditional pushbutton sort. Mathematics is mentioned frequently, including a nifty attempt to explain how higher-dimensional rotations can be induced from below: one attaches right-angled frames in tension, with the natural spring lengths corresponding to the lengths they will have in the desired folding.

The simple-mindedness of the characterizations becomes comic, even ridiculous, at times. Of mathematical interest is that the hero is so handsome in his appearance, and drives such a sporty car, that various characters are confused, and have a hard time believing he is the famed mathematical genius who has been called in for the rescue. Even the third-person narrator makes an effort to convince the reader that this peculiar mixture of looks, hipness, and brains is in fact the truth.

The writing is socially dated and not all that imaginative. What's mathematically interesting is Leinster's descriptions of how nD beings can interact with an (n+1)st spatial dimension: how 2D beings can create a 3D right angle using springs, how 3D beings can use mirrors to see into "extra" dimensions, and how 3D beings can use magnets to create pathways to "extra" dimensional spaces. (I'm putting "extra" in quotes because of course this isn't correct lingo mathematically.) Also cool: Leinster uses time as a 4th dimension and so goes to a 5th dimension in order to get more spacial dimensions. However, sometimes the narrative implies that the 4th dimension is spacial, but no one is perfect.