19 average Julian years make 6939 days 18 hours.
235 average lunations make 6939 days 16 hours 31 minutes.
So that successive cycles of golden numbers, supposing the first to start right, amount to making the new moons fall too late, gradually, so that the mean moon of this cycle gains 1 hour 29 minutes in 19 years upon the mean moon of the heavens, or about a day in 300 years. When the Calendar was reformed, the calendar new moons were four days in advance of the mean moon of the heavens so that, for instance, calendar full moon on the 18th usually meant real full moon on the 14th.
7. If the difference above had not existed, the moon of the heavens (the mean moon at least), would have returned permanently to the same days of the month in 19 years; with an occasional slip arising from the unequal distribution of the leap years, of which a period contains sometimes five and sometimes four. As a general rule, the days of new and full moon in any one year would have been also the days of new and full moon of a year having 19 more units in its date. Again, if there had been no leap years, the days of the month would have returned to the same days of the week every seven years. The introduction of occasional 29ths of February disturbs this, and makes the permanent return of month days to week days occur only after 28 years. If all had been true, the lapse of 28 times 19, or 532 years, would have restored the year in every point: that is, A.D. 1, for instance, and A.D. 533, would have had the same almanac in every matter relating to week days, month days, sun, and moon (mean sun and moon at least). And on the supposition of its truth, the old system of Dionysius was framed. Its errors are, first, that the moments of mean new moon advance too much by 1h. 29m. in 19 average Julian years; secondly, that the average Julian year of 36514 days is too long by 11m. 10s.
8. The Council of Trent, moved by the representations made on the state of the Calendar, referred the consideration of it to the Pope. In 1577, Gregory XIII. submitted to the Roman Catholic Princes and Universities a plan presented to him by the representatives of Aloysius Lilius, then deceased. This plan being approved of, the Pope nominated a commission to consider its details, the working member of which was the Jesuit Clavius. A short work was prepared by Clavius, descriptive of the new Calendar: this was published<ref>The title of this work, which is the authority on all points of the new Calendar, is 'Kalendarium Gregorianum Perpetuum. Cum Privilegio Summi Pontifis Et Aliorum Principum. Romæ, Ex Officina Dominici Basæ. MDLXXXII. Cum Licentia Superiorum' (quarto, pp. 60)./ref> in 1582, with the Pope's bull (dated February 24, 1581) prefixed. larger work was prepared by Clavius, containing fuller explanation, and entitled 'Romani Calendarii a Gregorio XIII. Pontifice Maximo restituti Explicatio.' This was published at Rome in 1603, and again in the collection of the works of Clavius in 1612.
