In this little problem we attempted to show how, by sophistical
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 line. But
where does the error come in?
Well, it is perfectly true that so long as our zigzag path is formed of
"steps" parallel to the sides of the square that path must be of the same
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 no limit
whatever theoretically to the number of steps that can be made--but you
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 until
they become sovereigns as to say we can increase the number of our steps
until they become a straight line. There is the whole thing in a
nutshell.