But Cæsar's assumption of 365¼ days as the length of the year was erroneous, being about 11 min. 14 sec. longer than the real tropical year. In the sixteenth century this error had accumulated to such a degree that the months were becoming seriously disjointed from the seasons with which they had been customarily associated. In consequence, Pope Gregory XIII, assisted by the astronomer Clavius, introduced a slight reform of the Julian calendar. The accumulated days were dropped, and a new start taken, and the rule for leap year was changed so as to read that “all years, whose date-number is divisible by four without a remainder are leap years, unless they are century years (such as 1800, 1900, etc.). The century years are not leap years, unless their date number is divisible by 400, in which case they are.” And this is the rule as it prevails to-day, although there is now (1912) serious talk of undertaking a new revision. But the Gregorian calendar is so nearly correct that more than 3000 years must elapse before the length of the year as determined by it will differ by one day from the true tropical year. The subject of the reform of the calendar is a very interesting one, but, together with that of the rules for determining the date of Easter, its discussion must be sought in more extensive works. There is one other measure of time, depending upon the motion of a heavenly body, which must be mentioned. This is the month, or the period required for the moon to make a revolution round the earth. Here we encounter again the same difficulty, for the month also is incommensurable with the year. Then, too, the length of the month varies according to the way in which it is reckoned. We have, first, a sidereal revolution of the moon, which is measured by the time taken to pass round the earth from one conjunction with a star to the next. This is, on the average, 27 days, 7 hours, 43 minutes, 12 seconds. Next we have a synodical revolution of the moon, which is measured by the time it takes in passing from the phase of new moon round to the same phase again. This seems the most natural measure of a month, because the changing phases of the moon are its most conspicuous peculiarity. (These will be explained in Part III.) The length of the month, as thus measured, is, on the average, 29 days, 12 hours, 44 minutes, 3 seconds. The reason why the synodical month is so much longer than the sidereal month is because new moon can occur only when the moon is in conjunction with the sun, _i.e._ exactly between the earth and the sun, and in the interval between two new moons the sun moves onward, so that for the second conjunction the moon must go farther to overtake the sun. It will be observed that both of the month measures are given in average figures. This is because the moon's motion is not quite regular, owing partly to the eccentricity of its orbit and partly to the disturbing effects of the sun's attraction.