Construction of a CLEPSYDRA. To divide any cylindrical vessel into parts to be emptied in each division of time; the time wherein the whole, and that wherein any part, is to be evacuated, being given.
Suppose, for example, a cylindrical vessel, whose charge of water thrown out in 12 hours, were required to be divided into parts to be evacuated each hour.
1. As the part of time 1 is to the whole time 12; so is the same time 12 to a fourth proportional, 144.
2. Divide the altitude of the vessel into 144 equal parts: here the last will fall to the last hour; the three next above to the last part but one; the five next to the tenth hour, &c.; lastly, the 23 last to the first hour. For since the times increase in the series of the natural numbers, 1, 2, 3, 4, 5, &c. and the altitudes, if the numeration be in retrograde order from the twelfth hour, increase in the series of the unequal numbers 1,
3, 5, 7, 9, &c. the altitude, computed from the twelfth hour, will be as the squares of the times, 1, 4, 9, 16, 25, &c. therefore the square of the whole time 144 comprehends all the parts of the altitude of the vessel to be evacuated. But a third proportional to 1 and 12 is the square of 12, and consequently it is the number of equal parts into which the altitude is to be divided, to be distributed according to the series of the unequal numbers, through the equal intervals of hours. Since in lieu of parts of the same vessel, other less vessels equal thereto may be substituted, the altitude of a vessel emptied in a given space of time being given, the altitude of another vessel to be emptied in a given time may be found; viz. by making the altitudes as the squares of the time. For a further description, see HYDRODYNAMICS Index.