226
form of the progression in question arising from the fact thata clock, which, in consequence of such a cause, might become acertain fraction of a second wrong in one day, would the nextday show an error of double the magnitude, of treble themagnitude the third day, and so on.
It may also he calculated that in 2,500 years the error pro-duced would be about 625 times as great in its total mag-nitude as in one century ; so that the exceedingly minuteretardation of —I—, of a second in one day would involvea displacement of two hours of time in the place of the Moon in 2,500 years, which is equivalent to a space of one degree,or to about twice the Moon ’s diameter as seen upon the celestialsphere. When this apparent effect is added to the very nearlyequal amount by which the Moon ’s place is changed by thereal acceleration of its movement, the most ancient observa-tions are brought into excellent accordance with those of thepresent day.
Here, then, once more, we have an interesting example of theimportant results which astronomy teaches us may arise fromcauses not only apparently trivial, but which are such as mightbe supposed to be utterly unconnected with one another.
Since the Moon travels across the apparent width of its owndiameter in about an hour, we may put the above statementinto a slightly different form, and say ;—that a diminution ofthe length of the day caused by tidal friction only amountingday by day to . r(l( J OOM of a second, and which, in 2,500 years, or913,125 days, would only make the actual length of each dayless by of a second ( i.e ., by less than ^nd part of a
second), would by its accumulative effect upon the timereckoned by a clock, during those twenty-five centuries, causethe Moon to he seen in a given position two hours sooner thanwould otherwise be the case. It would also, in the samelength of time, alter the bounding longitudes upon the Earth from which an eclipse would be seen by 30°, inasmuch as adifference of one hour by the clock corresponds to a differenceof 15° of longitude.*
* In the above explanation we have closely followed that given bySir E. Beckett in “Astronomy without Mathematics,” pp. 177 and 182,