An Herodotus machine, or bottom-source, up-stair, hybrid pump and step lock system, includes synchronized reciprocating cross-gate displacers with commonly synchronized gates and floating body movers; specifically including a sequence of ascending adjacent step locks separated by gates with reciprocating gears on axles aligned over respective gates. A chain over each reciprocating gear has a displacer on each end, with one displacer in the upper lock and the other displacer in the lower lock. Rotating first, the downstream displacer descends to raise the lower lock water level, while the upstream displacer rises to lower the upper lock water level, and vice versa on the reverse gear rotation. Displacers are sized such that water levels in alternate pairs of upper lock and lower lock are made the same. The gate is then opened and a watercraft can be transferred from the lower lock to the upper lock or vice versa.
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