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).
Leading Features of the Gregorian Reform
The Gregorian Reform is an ingenious and, indeed, brilliant practical solution of the problems presented by the condition of the Kalendar at the close of the sixteenth century. The characteristic features of the Gregorian system will now be described.
1. It was known that the true vernal equinox was at this date (1582) about ten days earlier than March 21 as marked in the Kalendar. Should the equinox be fixed as at March 11? It was resolved to keep the equinox at the nominal date of March 21, and to bring the date into conformity with facts by the simple process of striking out ten nominal days. It was decreed that the day following Oct. 4, 1582 (when what is known as the New Style was to make its beginning), should be counted, not as Oct. 5, but as Oct. 15. And thus in the following year, 1583, the true vernal equinox would fall on March 21, as it was supposed to have fallen in A.D. 325, the date of the Council of Nicaea.
2. But how was it to be provided that in the future the same errors which had vitiated the old Kalendar should not come in time to vitiate the new?
It will be remembered that the time of the old Kalendar had gained on true solar time at the rate, almost precisely, of one day in every 130 years. If the counting of one day could be suppressed in every 130 years, the end would be obtained. For purposes of practical convenience the reformers of the Kalendar assumed that 133 years should be taken as the period in which the Kalendar time exceeded the solar time by one day. The difference, for the purpose in hand, was insignificant; and, as will be seen hereafter, this deliberately chosen error will not affect the Kalendar to the extent of one day till A.D. 5200, while it makes calculations much simpler.
Now the plan adopted to prevent the accumulation of the error in the old Kalendar was as follows: if one day could be withdrawn in every 133 years, or, what is the same thing, three days in every 399 years, the object would be attained.
In the Old Style, every year of an exact century—every centurial (or, as it was sometimes called, secular) year—was a leap-year of 366 days. What would be the effect of treating every centurial year as a common year of 365 days? We should have suppressed four days at the end of four centuries when we ought to suppress only three in 399 years. So it was suggested that while three successive centurial years should be regarded as common years, the fourth centurial year should be treated as a leap-year. Thus, in both Old and New Style the years 1600 and 2000 are leap-years; but 1700, 1800, and 1900, which in the Old Style were leap-years, are in the New Style treated as common years of 365 days. And the rule laid down in the Gregorian system was that if the number expressed by the first two figures of the century was exactly divisible by 4 it should be a leap-year, but if not exactly divisible by 4 it should be treated as a common year. The numbers 16 and 20 are exactly divisible by 4, but 17, 18, and 19 are not so divisible. The years 1600 and 2000 are in the New Style leap-years, but the years 1700, 1800, and 1900 are in the New Style common years.
It is true that the adoption of 133 years, instead of 130 years, as the time in which in the Old Style one day was gained by the Kalendar on the sun, imports an error into the system, which causes the Kalendar to fall behind the sun. This error, as has been said, will accumulate to the extent of one day in A.D. 5200. It may be thought that, if men be on the earth at that date, they will know how to deal with the case. Yet it is suggested for the instruction of our remote posterity that they will have only to make A.D. 5200 a common year, instead of a leap-year, to bring things back to correctness[166].