The periodic time of the moon is the interval required for one circuit of its orbit. This interval is known with accuracy at the present day, and by means of the ancient eclipses the period of the moon's revolution two thousand years ago can be also ascertained. It had been discovered by Halley that the period which the moon requires to accomplish each of its revolutions around the earth has been steadily, though no doubt slowly, diminishing. The change thus produced is not appreciable when only small intervals of time are considered, but it becomes appreciable when we have to deal with intervals of thousands of years. The actual effect which is produced by the lunar acceleration, for so this phenomenon is called, may be thus estimated. If we suppose that the moon had, throughout the ages, revolved around the earth in precisely the same periodic time which it has at present, and if from this assumption we calculate back to find where the moon must have been about two thousand years ago, we obtain a position which the ancient eclipses show to be different from that in which the moon was actually situated. The interval between the position in which the moon would have been found two thousand years ago if there had been no acceleration, and the position in which the moon was actually placed, amounts to about a degree, that is to say, to an arc on the heavens which is twice the moon's apparent diameter. If no other bodies save the earth and the moon were present in the universe, it seems certain that the motion of the moon would never have exhibited this acceleration. In such a simple case as that which I have supposed the orbit of the moon would have remained for ever absolutely unchanged. It is, however, well known that the presence of the sun exerts a disturbing influence upon the movements of the moon. In each revolution our satellite is continually drawn aside by the action of the sun from the place which it would otherwise have occupied. These irregularities are known as the perturbations of the lunar orbit, they have long been studied, and the majority of them have been satisfactorily accounted for. It seems, however, to those who first investigated the question that the phenomenon of the lunar acceleration could not be explained as a consequence of solar perturbation, and, as no other agent competent to produce such effects was recognised by astronomers, the lunar acceleration presented an unsolved enigma.