ng 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:

"AN EXAMINATION OF SOME SUPPOSED MATHEMATICAL IMPOSSIBILITIES."

"There," he murmured, as he put the sheet back; "I think that such a
theme, adequately treated, will considerably astonish my learned friends
in general, and my esteemed critic, Van Huysman, in particular."

From which remark it will be gathered that Franklin Marmion had
certainly recrossed the dividing line between the two Planes of
Existence.

CHAPTER VIII

MISS BRENDA ARRIVES, AND PHADRIG THE EGYPTIAN PR