"No," he said in the peculiar low, musing tone which was habitual to him during these monologues, "no; after all, I do not see that there would be any harm in that. Wrong, nay, sinful it would undoubtedly be to prove to demonstration that religious, social, and physical laws, may, under certain changing circumstances, be both true and false at the same time. I am, or was--or whatever it is--perfectly right in considering that to deliberately produce such a chaos as that would do would be the most colossal crime that a man could commit against humanity, as far as this plane is concerned, but there can be no harm in making a few mathematical experiments." He took a few more turns up and down the room, pulling slowly at his pipe, and with his mind not wholly unoccupied with speculations as to what Professor Van Huysman's feelings might be if he were watching the said experiments. Then he began again: "At the worst I shall only be carrying certain investigations a few steps farther, and developing theories which have been seriously discussed by the hardest-headed scholars in the world. Both the Greek and the Alexandrian philosophers speculated on the possibility of a state of four dimensions; and didn't Cayley, before this very Society, deliberately say that at the present rate of progress in the Higher Mathematics, the eye of Intellect might ere long see across the border of tri-dimensional space? "Surely I cannot do any very great harm by carrying his arguments to their logical conclusions--if I can. Of course, physical demonstrations would never do: I should frighten my brilliant and learned audience out of its seven senses; but, as for mere mathematics--well, I may make them stare, and set a good many highly-respected brains--my gifted friend Huysman's, among them--working pretty hard. Of course, he will be especially furious, but there's no harm in that either. Yes, I shall certainly do it. If he can't understand my demonstrations, that's not my concern." He went and sat down at his desk, still smiling, and went very carefully through the notes he had already made, and then through Professor Hartley's letter, and his speculations on the Forty-Seventh Proposition. This done, he plunged into a fresh vortex of figures, and symbols, and diagrams, in which he remained for the next two hours, his mind hovering, as it were, over the borderland which at once divides and unites the higher and the lower planes. When he returned to earth, the dreamy, abstracted look faded away from his face; his eyes lit up, and the pleasant smile came back. He opened the middle drawer in his desk, and took out the first page of the fair copy of his notes, which Nitocris had made for him--thinking the while how easy it would have been for him in the state of N4 to take it out without opening the drawer at all--and looked at it. It was headed: "RECENT PROGRESS IN THE HIGHER MATHEMATICS." He crossed the title out carefully, and wrote above it: