A Budget of Paradoxes Volume I Part 31

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 {362} 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 1 h. 29 m. in 19 average Julian years; secondly, that the average Julian year of 365 days is too long by 11 m. 10 s.

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[756] submitted to the Roman Catholic Princes and Universities a plan presented to him by the representatives of Aloysius Lilius,[757]

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 {363} was published[758] in 1582, with the Pope's bull (dated February 24, 1581) prefixed. A larger work was prepared by Clavius, containing fuller explanation, and ent.i.tled _Romani Calendarii a Gregorio XIII. Pontifice Maximo rest.i.tuti Explicatio_. This was published at Rome in 1603, and again in the collection of the works of Clavius in 1612.

9. The following extracts from Clavius settle the question of the meaning of the term _moon_, as used in the Calendar:

"Who, except a few who think they are very sharp-sighted in this matter, is so blind as not to see that the 14th of the moon and the full moon are not the same things in the Church of G.o.d?... Although the Church, in finding the new moon, and from it the 14th day, _uses neither the true nor the mean motion of the moon_, but measures only according to the order of a cycle, it is nevertheless undeniable that the mean full moons found from astronomical tables are of the greatest use in determining the cycle which is to be preferred ... the new moons of which cycle, in order to the due celebration of Easter, should be so arranged that the 14th days of those moons, reckoning from the day of new moon _inclusive_, should not fall two or more days before the mean full moon, but only one day, or else on the very day itself, or not long after. And even thus far the Church need not take very great pains ... for it is sufficient that all should reckon by the 14th day of the moon in the cycle, even though sometimes it _should be more than one day before or after_ the mean full moon.... We have taken pains that in our cycle the new moons should _follow_ the real new moons, so that the 14th of the moon should fall either the day before the mean full moon, or on that day, or not long after; and this was done on purpose, for if the new moon of the cycle fell on the same day as the mean new moon of the {364} astronomers, it might chance that we should celebrate Easter on the same day as the Jews or the Quartadeciman heretics, which would be absurd, or else before them, which would be still more absurd."

From this it appears that Clavius continued the Calendar of his predecessors in the choice of the _fourteenth_ day of the moon. Our legislature lays down the day of the _full moon_: and this mistake appears to be rather English than Protestant; for it occurs in missals published in the reign of Queen Mary. The calendar lunation being 29 days, the middle day is the _fifteenth_ day, and this is and was reckoned as the day of the full moon. There is every right to presume that the original pa.s.sover was a feast of the _real full moon_: but it is most probable that the moons were then reckoned, not from the astronomical conjunction with the sun, which n.o.body sees except at an eclipse, but from the day of _first visibility_ of the new moon. In fine climates this would be the day or two days after conjunction; and the fourteenth day from that of first visibility inclusive, would very often be the day of full moon. The following is then the proper correction of the precept in the Act of Parliament:

Easter Day, on which the rest depend, is always the First Sunday after the _fourteenth day_ of the _calendar_ moon which happens upon or next after the Twenty-first day of March, _according to the rules laid down for the construction of the Calendar_; and if the _fourteenth day_ happens upon a Sunday, Easter Day is the Sunday after.

10. Further, it appears that Clavius valued the celebration of the festival after the Jews, etc., more than astronomical correctness. He gives comparison tables which would startle a believer in the astronomical intention of his Calendar: they are to show that a calendar in which the moon is always made a day older than by him, _represents the heavens better than he has done, or meant to do_. But it must be observed that this diminution of the real moon's age has {365} a tendency to make the English explanation often practically accordant with the Calendar. For the fourteenth day of Clavius _is_ generally the fifteenth day of the mean moon of the heavens, and therefore most often that of the real moon. But for this, 1818 and 1845 would not have been the only instances of our day in which the English precept would have contradicted the Calendar.

11. In the construction of the Calendar, Clavius adopted the ancient cycle of 532 years, but, we may say, without ever allowing it to run out. At certain periods, a s.h.i.+ft is made from one part of the cycle into another.

This is done whenever what should be Julian leap year is made a common year, as in 1700, 1800, 1900, 2100, etc. It is also done at certain times to correct the error of 1 h. 19 m., before referred to, in each cycle of golden numbers: Clavius, to meet his view of the amount of that error, put forward the moon's age a day 8 times in 2,500 years. As we cannot enter at full length into the explanation, we must content ourselves with giving a set of rules, independent of tables, by which the reader may find Easter for himself in any year, either by the old Calendar or the new. Any one who has much occasion to find Easters and movable feasts should procure Francoeur's[759] tables.

12. _Rule for determining Easter Day of the Gregorian Calendar in any year of the new style._ To the several parts {366} of the rule are annexed, by way of example, the results for the year 1849.

I. Add 1 to the given year. (1850).

II. Take the quotient of the given year divided by 4, neglecting the remainder. (462).

III. Take 16 from the centurial figures of the given year, if it can be done, and take the remainder. (2).

IV. Take the quotient of III. divided by 4, neglecting the remainder. (0).

V. From the sum of I, II, and IV., subtract III. (2310).

VI. Find the remainder of V. divided by 7. (0).

VII. Subtract VI. from 7; this is the number of the dominical letter

1 2 3 4 5 6 7 (7; dominical letter G).

A B C D E F G

VIII. Divide I. by 19, the remainder (or 19, if no remainder) is the _golden number_. (7).

IX. From the centurial figures of the year subtract 17, divide by 25, and keep the quotient. (0).

X. Subtract IX. and 15 from the centurial figures, divide by 3, and keep the quotient. (1).

XI. To VIII. add ten times the next less number, divide by 30, and keep the remainder. (7).

XII. To XI. add X. and IV., and take away III., throwing out thirties, if any. If this give 24, change it into 25. If 25, change it into 26, whenever the golden number is greater than 11. If 0, change it into 30. Thus we have the epact, or age of the _Calendar_ moon at the beginning of the year. (6).

_When the Epact is 23, or less._

XIII. Subtract XII., the epact, from 45. (39).

XIV. Subtract the epact from 27, divide by 7, and keep the remainder, or 7, if there be no remainder. (7)

_When the Epact is greater than 23._

XIII. Subtract XII., the epact, from 75.

XIV. Subtract the epact from 57, divide by 7, and keep the remainder, or 7, if there be no remainder.

XV. To XIII. add VII., the dominical number, (and 7 besides, if XIV. be greater than VII.,) and subtract XIV., the result is the day of March, or if more than 31, subtract 31, and {367} the result is the day of April, on which Easter Sunday falls. (39; Easter Day is April 8).

In the following examples, the several results leading to the final conclusion are tabulated.

======================================================== GIVEN YEAR | 1592 | 1637 | 1723 | 1853 | 2018 | 4686 -------------------------------------------------------- I. | 1593 | 1638 | 1724 | 1854 | 2019 | 4687 II. | 398 | 409 | 430 | 463 | 504 | 1171 III. | --- | 0 | 1 | 2 | 4 | 30 IV. | --- | 0 | 0 | 0 | 1 | 7 V. | 1991 | 2047 | 2153 | 2315 | 2520 | 5835 VI. | 3 | 3 | 4 | 5 | 0 | 4 VII. | 4 | 4 | 3 | 2 | 7 | 3 VIII. | 16 | 4 | 14 | 11 | 5 | 13 IX. | --- | --- | 0 | 0 | 0 | 1 X. | 0 | 0 | 0 | 1 | 1 | 10 XI. | 16 | 4 | 24 | 21 | 15 | 13 XII. | 16 | 4 | 23 | 20 | 13 |0 say 30 XIII. | 29 | 41 | 22 | 25 | 32 | 45 XIV. | 4 | 2 | 4 | 7 | 7 | 6 XV. | 29 | 43 | 28 | 27 | 32 | 49 Easter Day |Mar.29|Apr.12|Mar.28|Mar.27|Apr.1 | Apr.18 --------------------------------------------------------

13. _Rule for determining Easter Day of the Antegregorian Calendar in any year of the old style._ To the several parts of the rule are annexed, by way of example, the results for the year 1287. The steps are numbered to correspond with the steps of the Gregorian rule, so that it can be seen what augmentations the latter requires.

I. Set down the given year. (1287).

II. Take the quotient of the given year divided by 4, neglecting the remainder (321).

V. Take 4 more than the sum of I. and II. (1612).

VI. Find the remainder of V. divided by 7. (2).

VII. Subtract VI. from 7; this is the number of the dominical letter

1 2 3 4 5 6 7 (5; dominical letter E).

A B C D E F G

VIII. Divide one more than the given year by 19, the remainder (or 19 if no remainder) is the golden number. (15).

XII. Divide 3 less than 11 times VIII. by 30; the remainder (or 30 if there be no remainder) is the epact. (12).

{368}

_When the Epact is 23, or less._

XIII. Subtract XII., the epact, from 45. (33).

XIV. Subtract the epact from 27, divide by 7, and keep the remainder, or 7, if there be no remainder, (1).

_When the Epact is greater than 23._

XIII. Subtract XII., the epact, from 75.