Henry Ernest Dudeney/Modern Puzzles/222 - A Mechanical Paradox/Solution

Modern Puzzles by Henry Ernest Dudeney: $222$

A remarkable mechanical paradox, invented by James Ferguson about the year $1751$, ought to be known by everyone, but, unfortunately, it is not.

It was contrived by him as a challenge to a sceptical watchmaker during a metaphysical controversy.

"Suppose," Ferguson said, "I make one wheel as thick as three others and cut teeth in them all,

and then put the three wheels all loose upon one axis and set the thick wheel to turn them,

so that its teeth may take into those of the three thin ones.

Now, if I turn the thick wheel round, how must it turn the others?"

The watchmaker replied that it was obvious that all three must be turned the contrary way.

Then Ferguson produced his simple machine, which anyone can make in a few hours,

showing that, turning the thick wheel which way you would,

one of the thin wheels revolved in the same way, the second the contrary way, and the third remained stationary.

Although the watchmaker took the machine away for careful examination, he failed to detect the cause of the strange paradox.

The machine shown in our diagram consist of two pieces of thin wood, $B$ and $C$, made into a frame by being joined at the corners.

This frame, by means of the handle $n$, may be turned round an axle $a$, which pierces the frame and is fixed in a stationary board or table, $A$,

and carries within the frame an immovable wheel.

This first wheel, $D$, when the frame revolves, turns a second and thick wheel, $E$,

which, like the remaining three wheels, $F$, $G$ and $H$, moves freely on its axis.

The thin wheels, $F$, $G$ and $H$, are driven by the wheel $E$ in such a manner that when the frame revolves

$H$ turns the same way as $E$ does,

$G$ turns the contrary way,

and $F$ remains stationary.

The secret lies in the fact that though the wheels may be all of the same diameter,

and $D$, $E$ and $F$ may ($D$ and $F$ must) have an equal number of teeth,

yet $G$ must have at least one tooth fewer, and $H$ at least one tooth more, than $D$.