Archimedes' Perpetual-Motion Engine

Settled by conservation of energy / careful hydrostatic force accounting over the whole closed loop.

Buoyancy-driven perpetual motion machines are among the oldest and most persistent perpetual-motion schemes, reappearing in patent filings for centuries (a well-known example is the so-called 'Bishop's Wheel' using chained floats). Each design invariably hides an unaccounted work cost in returning floats to the low-pressure entry point. Lange's Chapter 3 puzzle uses Archimedes' name precisely because it dresses up an old fallacy in the language of a genuine, correct physical principle (buoyancy) to teach readers to check energy accounting on a full cycle, not just part of it.

It shows how a completely true physical law (Archimedes' principle) can be misapplied to manufacture an apparently free source of energy, and that finding the hidden cost requires following the whole cycle, not just the flashy part.

Explanations that circulated

• Counting only the buoyant force on the submerged floats as a 'free' upward push, while ignoring the work required to push floats down into the water (or through a seal/valve) against the surrounding pressure as they re-enter the loop.
• Analyzing the buoyant force on the submerged section of the loop in isolation, instead of tracking the net work over the ENTIRE closed cycle, including the often-overlooked chamber where floats must be forced back underwater — that work exactly cancels the apparent buoyant gain, leaving zero net energy output, consistent with the first law of thermodynamics.

References

• Perpetual Motion: The History of an Obsession, Arthur W. J. G. Ord-Hume, St. Martin's Press, 1977
• Wikipedia article (verified live, fetched during research), Wikipedia, 2026