Henry Ernest Dudeney/Modern Puzzles/209 - A Wheel Fallacy

Modern Puzzles by Henry Ernest Dudeney: $209$

A Wheel Fallacy

Here is a curious fallacy that I have found to be very perplexing to many people.

The wheel shown in the diagram makes one complete revolution in passing from $A$ to $B$.

It is therefore obvious that the line $AB$ is exactly equal in length to the circumference of the wheel.

Now the inner circle (the large hub in the diagram) also makes one complete revolution along the dotted line $CD$ and,

since the line CD is equal to the line $AB$, the circumference of the larger and smaller circles are the same.

This is clearly not true.

Wherein lies the fallacy?

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Wheel Paradox Problems: $209$. -- A Wheel Fallacy
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Geometrical Problems: Circle Puzzles: $293$. A Wheel Fallacy