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The klepsydra is a water clock consisting of a vessel with a hole at the bottom. The height of the water level is used to measure time. For this level to be proportional to time, the vessel has to have the shape of a quartic parabola: $h=cr^4$, where $h$ is the height and $r$ is the radius of the cross-section at this height.

Question: did the ancients know this? To discover this fact, mathematics and physics of Archimedes would be sufficient, in principle. But I am not aware of any ancient literature discussing the theory of water clocks.

It is possible that they could somehow discover the required shape empirically. It is hard to judge from the pictures since one needs to know the shape of the inner surface which is usually not seen in the pictures.

But even more interesting question is: if they tried to discover the correct shape by experiment, what kind of standard time could they use for comparison? Before the modern era, essentially only sundials and klepsydras were available to measure time.

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  • $\begingroup$ Merely speculation, but, I'd suspect that some sort of pendulums, and hour-glasses, were available for measuring relatively short time intervals... Good question, though, indeed! :) $\endgroup$ Mar 15 at 21:16
  • $\begingroup$ @paul garrett: The discovery of the fact that pendulum oscillations are approximately isochronous is due to Galilei. The ancients did not use pendulum. Neither it was used in mechanical clocks before him. If by hour glass you mean sand glass, this is just a poor version of a water clock, regarding accuracy. $\endgroup$ Mar 16 at 13:10
  • $\begingroup$ @Conifold, Can you enter your comment as an answer, to remove this question from the "unanswered" list? $\endgroup$ Mar 17 at 13:01

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At least, in ancient Greece and Rome I am not aware of the vessel shape being used as a way to uniformize the flow of water. Instead, "correct" clepsydras were constructed using a different mechanism. The water level in a reservoir (of any shape) was kept constant by inflow source and overflow pipes, and time was measured by floats rising with the (uniform due to constant pressure) outflow from the reservoir. Some of those are described by Vitruvius, see McTutor, Water-clocks.

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