"
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.
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