In this little problem we attempted to show how,
reasoning, it may apparently be proved that the diagonal of a square is
of precisely the same length as two of the sides.
The puzzle was to
discover the fallacy, because it is a very obvious fallacy if we admit
that the shortest distance between two points is a straight
where does the error come in?
Well, it is perfectly true that so long as our
zigzag path is
"steps" parallel to the sides of the square that path must be of the
length as the two sides.
It does not matter if you have to use the most
powerful microscope obtainable; the rule is always true if the path is
made up of steps in that way.
But the error lies in the assumption that
such a zigzag path can ever become a straight line.
You may go on
increasing the number of steps infinitely—that is, there is
whatever theoretically to the number of steps that can be
can never reach a straight line by such a method.
In fact it is just as
much a "jump" to a straight line if you have a billion steps as it is at the very outset to pass from
the two sides to the diagonal.
It would be
just as absurd to say we might go on dropping marbles into a basket
they become sovereigns as to say we can increase the number of our
until they become a straight line.
There is the whole thing in a