INTRODUCTION 9 



ments have been exploring our three-dimensional 

 universe, our universe limited by length, breadth and 

 thickness. It was easy for them to conclude that 

 they knew all the space abysses of the Cosmos. The 

 mole may push his snout above ground, and in his 

 blindness not see the green out-reaching globe around 

 him and the star-filled heavens that expand prodigi- 

 ously above him. So we men-moles are doubtless 

 moving in the presence of unknown spaces and of un- 

 imagined mysteries. 



Nevertheless, dim intimations of a fourth dimen- 

 sion of space have ever been hovering over the world 

 of men ; and many awakening minds are beginning to 

 respond to them, even as the delicate instruments of 

 the astronomer respond to the vibrations of a new 

 planet that is swimming toward his lens. 



Poets and prophets, stirred by intuitions from 

 within and by inspirations from above, have ever de- 

 clared that there is an invisible spiritual world, a 

 world of the good and the true and the beautiful, a 

 world whereof our nature-sphere is only the hint, the 

 shell, the shadow. 



But Miss Williams comes to us, basing her faith 

 upon findings in the more exact provinces of thought. 

 Mathematicians have long made use of the concept 

 of a fourth dimension to visualise the formulas of al- 

 gebra. Chemists and physicists are finding that the 

 conduct of certain molecules and crystals is best ex- 

 plained as a fourth-dimensional activity. And for 



