We acknowledge
that we have not brought this result under the canon, but look upon it
as indicating the necessity of another canon to somewhat this
effect,--that in the laws of form direction is a more important element
than distance.
We have said that a curved line is one in which every point is subject
to one and the same law of position. Now it may be easily proved, that,
in a series of points in a plane, each of which fulfils one and the
same condition of position, any three, if taken sufficiently near each
other, lie in one straight line. A fourth point near the third lies,
then, in a straight line with the second and third,--a fifth with the
third and fourth, and so on. The whole series of points must, in short,
form a line. But it may also be easily proved that any four of these
points, taken sufficiently near each other, lie in the arc of a circle.
How strange the paradox to which we are thus led! Every law of a curve,
however simple, leads to the same conclusion; a curve must bend at
every point, and yet not bend at any point; it must be nowhere a
straight line, and yet be a straight line at every part. The
blacksmith, passing an iron bar between three rollers to make a tire
for a wheel, bends every part of it infinitely little, so that the
bending shall not be perceptible at any one spot, and shall yet in the
whole length arch the tire to a full circle.
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