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 three-to-the-second, or
nine, represents the number of square inches in a Square whose side is
three 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 three-to-the-third 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 three; 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 three-to-the-second. xxx 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 three; and if a straight Line of three
inches, moving parallel to itself, makes a Square of three inches every
way, represented by three-to-the-second; it must be that a Square of
three inches every way, moving somehow parallel to itself (but I don't
see how) must make Something else (but I don't see what) of three
inches every way--and this must be represented by three-to-the-third."

"Go to bed," said I, a little ruffled by this interruption: "if you
would talk less nonsense, you would remember more sense."

So my Grandson had disappeared in disgrace; and there I sat by my
Wife's side, endeavouring to form a retrospect of the year 1999 and of
the possibilities of the year 2000; but not quite able to shake of the
thoughts suggested by the prattle of my bright little Hexagon. Only a
few sands now remained in the half-hour glass. Rousing myself from my
reverie I turned the glass Northward for the last time in the old
Millennium; and in the act, I exclaimed aloud, "The boy is a