Did Archimedes really shout Eureka in the bath?
Almost certainly not as told — the story appears two centuries after his death, in a source not otherwise noted for rigour, and the physics as described does not work well.
The story. Hiero II of Syracuse suspected a goldsmith of adulterating a votive crown with silver. He asked Archimedes to determine whether it was pure gold without damaging it. Archimedes, stepping into a bath and noticing the water rise, realised he could measure volume by displacement, and ran naked through the streets shouting Heureka — "I have found it".
Where it comes from. Vitruvius, the Roman architect, writing in De architectura around the first century BC — roughly 200 years after Archimedes died. Archimedes himself, whose surviving works are extensive and precise, never mentions it. Nor do earlier writers.
Why the physics is questionable. The simple displacement method requires measuring an extremely small difference in water level. Silver is less dense than gold, so an adulterated crown of the same weight would be only slightly larger — the water-level difference would be a fraction of a millimetre in any plausible vessel, well beyond the measurement precision available, and easily swamped by water clinging to the object, surface tension and evaporation.
What Archimedes actually could have done. His genuine work On Floating Bodies sets out the principle of buoyancy — that a body immersed in fluid experiences an upward force equal to the weight of fluid displaced. That gives a far better method: weigh the crown in air, then weigh it submerged, and compare. The difference is measurable with a good balance, and Archimedes had the theory to do it. Galileo proposed exactly this reconstruction.
So the plausible core is that Archimedes solved the problem using buoyancy, and that a memorable but garbled version reached Vitruvius.
What is not in doubt is Archimedes' actual achievement — the buoyancy principle, work on levers, the approximation of pi, and the method of exhaustion anticipating integration.