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Another classical construction,
although in modern colors - a
square circumscribed around a
circle. Archimedes concluded
that the actual perimeter of the
circle must be less than the
perimeter of the circumscribed
square and greater than the
perimeter of the inscribed
square. It was an easy leap (for
him) to realize that the actual
perimeter of the circle could be
approached by increasing the
number of sides of the
circumscribed polygons.