Archimedes principle:
Any object, wholly or partially immersed in a stationary fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.
I get that this can be shown with force calculations on an object in a fluid. But how come this fact was suddenly so obvious to someone taking a bath thousands of years ago?
The bath is a later legend. The principle can be discovered and proved by the following argument which requires no calculation. Suppose that a body is submerged by water. The force with which water acts on it is the pressure on the surface of the body. Now replace the inside of the body by water. Then we have just water in water and nothing moves (equilibrium). Which means that the weight of the water inside the surface is equal to the pressure of the water outside of the surface. But this last pressure evidently does not depend on what's inside the surface. This proves the Archimedes law.
This was Archimedes's own argument, given in his book "On Floating bodies". All stories about bath and crown and bull's sacrifice and Eureka were invented by later biographers.
Fortunately there is some documentary evidence about Archimedes' discovery, in his work "On Floating Bodies", which survives in manuscript forms (https://en.wikipedia.org/wiki/On_Floating_Bodies). From the available sources it also translated into English by Thomas L Heath in the 1890s. It is included in his 1897 commentary and set of translations 'Works of Archimedes' (https://archive.org/details/worksofarchimede029517mbp), the translation can be read (with Heath's notes) at pages 253-300.
Just as Alexandre Eremenko's earlier answer states, the treatment in the book is by 'argument which requires no calculation'. The style is theoretical and geometrical, making no appeal to experiment. On the other hand, that is a standard ancient Greek approach to presentation, and can not really conclude the question whether there may also have been some experimental background to the discovery.
The anecdotes about experiments that have been handed down may indeed be partly or wholly the inventions of later commentators. But there has always been an almost irresistible urge (and not an entirely unreasonable one), upon commentators from the ancient to the modern, to suppose that such a practical matter may very probably have been grounded in some practical experience. Unfortunately the evidence that has come down is too meager to allow evidence-based conclusions about what any such experiences by Archimedes may have been. The translator Heath's own ideas are contained in his notes, and each reader will judge the plausibility of Heath's and the other anecdotes and conjectures.
