The studies of which the present paper is the first are intended to accumulate observations and discussions on all necessary points connected with exact time-keeping, such as the maintenance of motion, air-resistance, barometric error, escapement error, temperature compensation, and so forth. It may be taken that no clock so far made behaves uniformly so as to keep regular time. The clock is one of the fundamental measuring instruments of astronomy, but astronomical practice has kept in advance of the precision reached by its clocks by determining the clock error in the midst of each set of observations and so dispensing with the need for carrying forward for any length of time a doubtful clock rate. But there are many reasons why we should not be content with this, which amounts to an elimination of the clock. In the first place, it is not always feasible.