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,
represented by 3^2; 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 3^3.
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