ertheless, we perfectly well recognize the different mental states
of volition implied in "lying", "sitting", and "standing", which are to
some extent indicated to a beholder by a slight increase of lustre
corresponding to the increase of volition.

But on this, and a thousand other kindred subjects, time forbids me to
dwell.]

My four Sons and two orphan Grandchildren had retired to their several
apartments; and my wife alone remained with me to see the old
Millennium out and the new one in.

I was rapt in thought, pondering in my mind some words that had
casually issued from the mouth of my youngest Grandson, a most
promising young Hexagon of unusual brilliancy and perfect angularity.
His uncles and I had been giving him his usual practical lesson in
Sight Recognition, turning ourselves upon our centres, now rapidly, now
more slowly, and questioning him as to our positions; and his answers
had been so satisfactory that I had been induced to reward him by
giving him a few hints on Arithmetic, as applied to Geometry.

Taking nine Squares, each an inch every way, I had put them together so
as to make one large Square, with a side of three inches, and I had
hence proved to my little Grandson that--though it was impossible for
us to SEE the inside of the Square--yet we might ascertain the number
of square inches in a Square by simply squaring the number of inches in
the side: "and thus," said I, "we know that 3^2, or 9, represents the
number of square inches in a Square whose side is 3 inches long."

The little Hexagon meditated on this a while and then said to me; "But
you have been teaching me to raise numbers to the third power: I
suppose 3^3 must mean something in Geometry; what does it mean?"
"Nothing at all," replied I, "not at least in Geometry; for Geometry
has only Two Dimensions." And then I began to shew the boy how a Point
by moving through a length of three inches makes a Line of three
inches, which may be represented by 3; and how a Line of three inches,
moving parallel to itself through a length of three inches, makes a
Square of three inches every way, which may be represented by 3^2.

Upon this, my Grandson, again returning to his former suggestion, took
me up rather suddenly and exclaimed, "Well, then, if a Point by moving
three inches, makes a Line of three inches represented by 3; and if a
straight Line of three inches, moving parallel to itself, makes a
Square of three inches every way, represe