From dreams I proceed to facts.
It was the last day of the 1999th year of our era. The pattering of the rain had long ago
announced nightfall; and I was sitting in the company of my wife, musing on the events
of the past and the prospects of the coming year, the coming century, the coming
Millennium.
[Note: When I say "sitting", of course I do not mean any change of attitude such as you in
Spaceland signify by that word; for as we have no feet, we can no more "sit" nor "stand"
(in your sense of the word) than one of your soles or flounders.
Nevertheless, 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.
