The space at our disposal will not allow of our treating in detail of the work of the computists, and of the ‘Sunday Letters,’ ‘Epacts,’ and other technical terms which appear in the old Church Kalendars. For these, as well as for such terms as ‘Indiction,’ ‘Lunar Regulars,’ ‘Solar Regulars,’ and ‘Concurrents,’ reference may be made to such books as Sir Harris Nicholas’ _Chronology of History_, and Giry’s fuller and lucid _Manuel de Diplomatique_. _The Gregorian Reform._ The defects of the Nineteen-Year Cycle became apparent after some lapse of time. There were two grave sources of error. First, the Kalendar proceeded on the assumption that the solar year consisted of 365¼ days; but the true solar year is 11 minutes and some seconds shorter than the Kalendar year, and the accumulation of this error gradually brought confusion into the system. In one hundred and thirty years the Kalendar will have gained on the true solar year by almost exactly one day. At the date of the Council of Nicaea (A.D. 325) the vernal equinox was placed at March 21, but in the year A.D. 450 the true vernal equinox would be on March 20. In A.D. 585 the equinox would be on March 19; in A.D. 715 on March 18, and so on. And thus it will be seen that in A.D. 1582, when the Kalendar was reformed, the real vernal equinox was about ten days earlier than the March 21 of the Kalendar. The second source of error lay in the assumption that at the close of a cycle of nineteen years there was an exact agreement of solar and lunar time. Nineteen solar years, of 365¼ days, make 6939 days and 18 hours; but 235 moons of 29 days, 12 hours, 44 minutes, and 3 seconds and a fraction make 6939 days, 16 hours, and a fraction over 31 minutes. So it comes about that the solar time in nineteen years is nearly 1½ hours in excess of the real lunar time. In other words, the moons in the second cycle of nineteen years make their changes nearly 1½ hours earlier than they did in the first cycle. It is easy then to show that in about 308 years this difference would amount to a whole day; and in A.D. 1582, when the Gregorian reform was effected, the moon in the heavens made its changes nearly four days before the time which was indicated for these changes in the Kalendar. We must omit any notice of the various schemes for reforming the Kalendar prior to the reformation of Gregory XIII. After he had consented to the general idea that a reformation should be undertaken, various schemes were proposed. Of these, that of Luigi Lilio, a physician and astronomer of the city of Rome, obtained the preference[165]. And it is on the lines suggested by Lilio that the work was accomplished, mainly by a German mathematician then resident at Rome, the Jesuit, Christopher Schlüssel (or, in the Latin form of his name, Clavius), who afterwards published at Rome, in folio, an exposition of the work done, under the title _Romani Calendarii a Gregorio XIII Pontifice Maximo restituti Explicatio_ (1603).