# Euler’s Discovery and Resolution of D’Alembert’s Paradox

• William W. Hackborn
Conference paper

## Abstract

This article makes a case for Euler as the first discoverer of what has come to be known as d’Alembert’s paradox. Suppose a body is immersed in an unbounded fluid and moves with constant velocity relative to the fluid, which is otherwise undisturbed: d’Alembert’s paradox asserts that, contrary to experimental evidence, the fluid exerts no drag force on the body (in the direction opposite to the body’s motion) if the fluid is inviscid and incompressible. Euler demonstrates this, for a two-dimensional body or an axisymmetric body whose axis aligns with its motion, in his extensive 1745 commentary on New Principles of Gunnery, a book published in 1742 by Benjamin Robins. After a rigorous analysis, Euler recognizes that the absence of a drag force conflicts with experience for fluids like air and water, and he uses Robins’ experiments with musket balls to explain this anomaly as a consequence of greater fluid pressure fore of the body than aft of it, due to a corresponding fore–aft asymmetry in the density of the fluid. Essentially, he resolves the apparent paradox by removing the assumption of the fluid’s incompressibility.

## Notes

### Acknowledgements

The author gratefully acknowledges that all the images used herein were scanned by and are used at the courtesy of, the University of Calgary, Military Museums Library and Archives.

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