Chapter XII: Part 12

5. The mode of applying astronomy to chronology has always involved these two principles. First, the actual position of the heavenly body is not the object of consideration, but what astronomers call its _mean place_, which may be described thus. Let a fictitious sun or moon move in the heavens, in such manner as to revolve among the fixed stars at an average rate, avoiding the alternate accelerations and retardations which take place in every planetary motion. Thus the fictitious (say _mean_) sun and moon are always very near to the real sun and moon. The ordinary clocks show time by the mean, not the real, sun: and it was always laid down that Easter depends on the opposition (or full moon) of the mean sun and moon, not of the real ones. Thus we see that, were the Calendar ever so correct {361} as to the _mean_ moon, it would be occasionally false as to the _true_ one: if, for instance, the opposition of the mean sun and moon took place at one second before midnight, and that of the real bodies only two seconds afterwards, the calendar day of full moon would be one day before that of the common almanacs. Here is a way in which the discussions of 1818 and 1845 might have arisen: the British legislature has defined _the moon_ as the regulator of the paschal calendar. But this was only a part of the mistake.

6. Secondly, in the absence of perfectly accurate knowledge of the solar and lunar motion (and for convenience, even if such knowledge existed), cycles are, and always have been taken, which serve to represent those motions nearly. The famous Metonic cycle, which is introduced into ecclesiastical chronology under the name of the cycle of the golden numbers, is a period of 19 Julian[755] years. This period, in the old Calendar, was taken to contain exactly 235 _lunations_, or intervals between new moons, of the mean moon. Now the state of the case is:

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 {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 3651/4 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 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.

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 God?... 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 291/2 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 passover 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 nobody 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 shift 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.

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 the result is the day of April, on which Easter Sunday (old style) falls. (37; Easter Day is April 6).

These rules completely represent the old and new Calendars, so far as Easter is concerned. For further explanation we must refer to the articles cited at the commencement.

The annexed is the table of new and full moons of the Gregorian Calendar, cleared of the errors made for the purpose of preventing Easter from coinciding with the Jewish Passover.

The second table (page 370) contains _epacts_, or ages of the moon at the beginning of the year: thus in 1913, the epact is 22, in 1868 it is 6. This table goes from 1850 to 1999: should the New Zealander not have arrived by that time, and should the churches of England and Rome then survive, the epact table may be continued from their liturgy-books. The way of using the table is as follows: Take the epact of the required year, and find it in the first or last column of the first table, in line with it are seen the calendar days of new and full moon. Thus, when the epact is 17, the new and full moons of March fall on the 13th and 28th. The result is, for the most part, correct: but in a minority of cases there is an error of a day. When this happens, the error is almost always a fraction of a day much less than twelve hours. Thus, when the table gives full moon on the 27th, and the real truth is the 28th, we may be sure it is early on the 28th.

{369}

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|Jan.|Feb.|Mar.|Apr.|May |June|July|Aug.|Sep.|Oct.|Nov.|Dec.|
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1 | 29 | 27 | 29 | 27 | 27 | 25 | 25 | 23 | 22 | 21 | 20 | 19 | 1
| 14 | 13 | 14 | 13 | 12 | 11 | 10 | 9 | 7 | 7 | 5 | 5 |
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2 | 28 | 26 | 28 | 26 | 26 | 24 | 24 | 22 | 21 | 20 | 19 | 18 | 2
| 13 | 12 | 13 | 12 | 11 | 10 | 9 | 8 | 6 | 6 | 4 | 4 |
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3 | 27 | 25 | 27 | 25 | 25 | 23 | 23 | 21 | 20 | 19 | 18 | 17 | 3
| 12 | 11 | 12 | 11 | 10 | 9 | 8 | 7 | 5 | 5 | 3 | 3 |
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4 | 26 | 24 | 26 | 24 | 24 | 22 | 22 | 20 | 19 | 18 | 17 | 16 | 4
| 11 | 10 | 11 | 10 | 9 | 8 | 7 | 6 | 4 | 4 | 2 |2,31|
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5 | 25 | 23 | 25 | 23 | 23 | 21 | 21 | 19 | 18 | 17 | 16 | 15 | 5
| 10 | 9 | 10 | 9 | 8 | 7 | 6 | 5 | 3 | 3 | 1 |1,30|
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6 | 24 | 22 | 24 | 22 | 22 | 20 | 20 | 18 | 17 | 16 | 15 | 14 | 6
| 9 | 8 | 9 | 8 | 7 | 6 | 5 | 4 | 2 |2,31| 30 | 29 |
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7 | 23 | 21 | 23 | 21 | 21 | 19 | 19 | 17 | 16 | 15 | 14 | 13 | 7
| 8 | 7 | 8 | 7 | 6 | 5 | 4 | 3 | 1 |1,30| 29 | 28 |
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8 | 22 | 20 | 22 | 20 | 20 | 18 | 18 | 16 | 15 | 14 | 13 | 12 | 8
| 7 | 6 | 7 | 6 | 5 | 4 | 3 |2,31| 30 | 29 | 28 | 27 |
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9 | 21 | 19 | 21 | 19 | 19 | 17 | 17 | 15 | 14 | 13 | 12 | 11 | 9
| 6 | 5 | 6 | 5 | 4 | 3 | 2 |1,30| 29 | 28 | 27 | 26 |
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10 | 20 | 18 | 20 | 18 | 18 | 16 | 16 | 14 | 13 | 12 | 11 | 10 | 10
| 5 | 4 | 5 | 4 | 3 | 2 |1,31| 29 | 28 | 27 | 26 | 25 |
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11 | 19 | 17 | 19 | 17 | 17 | 15 | 15 | 13 | 12 | 11 | 10 | 9 | 11
| 4 | 3 | 4 | 3 | 2 |1,30| 30 | 28 | 27 | 26 | 25 | 24 |
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12 | 18 | 16 | 18 | 16 | 16 | 14 | 14 | 12 | 11 | 10 | 9 | 8 | 12
| 3 | 2 | 3 | 2 |1,31| 29 | 29 | 27 | 26 | 25 | 24 | 23 |
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13 | 17 | 15 | 17 | 15 | 15 | 13 | 13 | 11 | 10 | 9 | 8 | 7 | 13
| 2 | 1 | 2 |1,30| 30 | 28 | 28 | 26 | 25 | 24 | 23 | 22 |
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14 | 16 | 14 | 16 | 14 | 14 | 12 | 12 | 10 | 9 | 8 | 7 | 6 | 14
|1,31| -- |1,31| 29 | 29 | 27 | 27 | 25 | 24 | 23 | 22 | 21 |
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15 | 15 | 13 | 15 | 13 | 13 | 11 | 11 | 9 | 8 | 7 | 6 | 5 | 15
| 30 | 28 | 30 | 28 | 28 | 26 | 26 | 24 | 23 | 22 | 21 | 20 |
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16 | 14 | 12 | 14 | 12 | 12 | 10 | 10 | 8 | 7 | 6 | 5 | 4 | 16
| 29 | 27 | 29 | 27 | 27 | 25 | 25 | 23 | 22 | 21 | 20 | 19 |
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17 | 13 | 11 | 13 | 11 | 11 | 9 | 9 | 7 | 6 | 5 | 4 | 3 | 17
| 28 | 26 | 28 | 26 | 26 | 24 | 24 | 22 | 21 | 20 | 19 | 18 |
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18 | 12 | 10 | 12 | 10 | 10 | 8 | 8 | 6 | 5 | 4 | 3 | 2 | 18
| 27 | 25 | 27 | 25 | 25 | 23 | 23 | 21 | 20 | 19 | 18 | 17 |
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19 | 11 | 9 | 11 | 9 | 9 | 7 | 7 | 5 | 4 | 3 | 2 |1,31| 19
| 26 | 24 | 26 | 24 | 24 | 22 | 22 | 20 | 19 | 18 | 17 | 16 |
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20 | 10 | 8 | 10 | 8 | 8 | 6 | 6 | 4 | 3 | 2 |1,31| 30 | 20
| 25 | 23 | 25 | 23 | 23 | 21 | 21 | 19 | 18 | 17 | 16 | 15 |
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21 | 9 | 7 | 9 | 7 | 7 | 5 | 5 | 3 | 2 |1,31| 29 | 29 | 21
| 24 | 22 | 24 | 22 | 22 | 20 | 20 | 18 | 17 | 16 | 15 | 14 |
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22 | 8 | 6 | 8 | 6 | 6 | 4 | 4 | 2 |1,30| 30 | 28 | 28 | 22
| 23 | 21 | 23 | 21 | 21 | 19 | 19 | 17 | 16 | 15 | 14 | 13 |
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23 | 7 | 5 | 7 | 5 | 5 | 3 | 3 |1,31| 29 | 29 | 27 | 27 | 23
| 22 | 20 | 22 | 20 | 20 | 18 | 18 | 16 | 15 | 14 | 13 | 12 |
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24 | 6 | 5 | 6 | 5 | 4 | 3 | 2 |1,30| 29 | 28 | 27 | 26 | 24
| 21 | 19 | 21 | 19 | 19 | 17 | 17 | 15 | 14 | 13 | 12 | 11 |
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25 | 5 | 4 | 5 | 4 | 3 | 2 |1,31| 29 | 28 | 27 | 26 | 25 | 25
| 20 | 19 | 20 | 19 | 18 | 17 | 16 | 15 | 13 | 13 | 11 | 11 |
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26 | 4 | 3 | 4 | 3 | 2 |1,30| 30 | 28 | 27 | 26 | 25 | 24 | 26
| 19 | 18 | 19 | 18 | 17 | 16 | 15 | 14 | 12 | 12 | 10 | 10 |
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27 | 3 | 2 | 3 | 2 |1,31| 29 | 29 | 27 | 26 | 25 | 24 | 23 | 27
| 18 | 17 | 18 | 17 | 16 | 15 | 14 | 13 | 11 | 11 | 9 | 9 |
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28 | 2 | 1 | 2 |1,30| 30 | 28 | 28 | 26 | 25 | 24 | 23 | 22 | 28
| 17 | 16 | 17 | 16 | 15 | 14 | 13 | 12 | 10 | 10 | 8 | 8 |
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29 |1,31| -- |1,31| 29 | 29 | 27 | 27 | 25 | 24 | 23 | 22 | 21 | 29
| 16 | 15 | 16 | 15 | 14 | 13 | 12 | 11 | 9 | 9 | 7 | 7 |
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30 | 30 | 28 | 30 | 28 | 28 | 26 | 26 | 24 | 23 | 22 | 21 | 20 | 30
| 15 | 14 | 15 | 14 | 13 | 12 | 11 | 10 | 8 | 8 | 6 | 6 |
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|Jan.|Feb.|Mar.|Apr.|May |June|July|Aug.|Sep.|Oct.|Nov.|Dec.|
-------------------------------------------------------------------------

{370}

=======================================================
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
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185 | 17 | 28 | 9 | 20 | 2 | 12 | 23 | 4 | 15 | 26
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186 | 7 | 18 | 30 | 11 | 22 | 3 | 14 | 25 | 6 | 17
-------------------------------------------------------
187 | 28 | 9 | 20 | 1 | 12 | 23 | 4 | 15 | 26 | 7
-------------------------------------------------------
188 | 18 | 30 | 11 | 22 | 3 | 14 | 25 | 6 | 17 | 28
-------------------------------------------------------
189 | 9 | 21 | 1 | 12 | 23 | 4 | 15 | 26 | 7 | 18
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190 | 29 | 10 | 21 | 2 | 13 | 24 | 5 | 16 | 27 | 8
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191 | 19 | 30 | 11 | 22 | 3 | 14 | 26 | 6 | 17 | 29
-------------------------------------------------------
192 | 10 | 21 | 2 | 13 | 24 | 5 | 16 | 27 | 8 | 19
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193 | 30 | 11 | 22 | 3 | 14 | 26 | 6 | 17 | 29 | 10
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194 | 21 | 2 | 13 | 24 | 5 | 16 | 27 | 8 | 19 | 30
-------------------------------------------------------
195 | 11 | 22 | 3 | 14 | 26 | 6 | 17 | 29 | 10 | 21
-------------------------------------------------------
196 | 2 | 13 | 24 | 5 | 16 | 27 | 8 | 19 | 30 | 11
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197 | 22 | 3 | 14 | 26 | 6 | 17 | 29 | 10 | 21 | 2
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198 | 13 | 24 | 5 | 16 | 27 | 8 | 19 | 30 | 11 | 22
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199 | 3 | 14 | 26 | 6 | 17 | 29 | 10 | 21 | 2 | 13
=======================================================

For example, the year 1867. The epact is 25, and we find in the table:

J. F. M. AP. M. JU. JL. AU. S. O. N. D.
New 5+ 4 5+ 4 3+ 2 1,31 29 28- 27 26 25
Full 20 19- 20 19- 18 17 16 15 13- 13 11+ 11

When the truth is the day after + is written after the date; when the day before, -. Thus, the new moon of March is on the 6th; the full moon of April is on the 18th. {371}

I now introduce a small paradox of my own; and as I am not able to prove it, I am compelled to declare that any one who shall dissent must be either very foolish or very dishonest, and will make me quite uncomfortable about the state of his soul. This being settled once for all, I proceed to say that the necessity of arriving at the truth about the assertions that the Nicene Council laid down astronomical tests led me to look at Fathers, Church histories, etc. to an extent which I never dreamed of before. One conclusion which I arrived at was, that the Nicene Fathers had a knack of sticking to the question which many later councils could not acquire. In our own day, it is not permitted to Convocation seriously to discuss any one of the points which are bearing so hard upon their resources of defence--the cursing clauses of the Athanasian Creed, for example. And it may be collected that the prohibition arises partly from fear that there is no saying where a beginning, if allowed, would end. There seems to be a suspicion that debate, once let loose, would play up old Trent with the liturgy, and bring the whole book to book. But if any one will examine the real Nicene Creed, without the augmentation, he will admire the way in which the framers stuck to the point, and settled what they had to decide, according to their view of it. With such a presumption of good sense in their favor, it becomes easier to believe in any claim which may be made on their behalf to tact or sagacity in settling any other matter. And I strongly suspect such a claim may be made for them on the Easter question.

I collect from many little indications, both before and after the Council, that the division of the Christian world into Judaical and Gentile, though not giving rise to a sectarian distinction expressed by names, was of far greater force and meaning than historians prominently admit. I took _note_ of many indications of this, but not _notes_, as it was not to my purpose. If it were so, we must admire the discretion of the Council. The Easter question was the {372} fighting ground of the struggle: the Eastern or Judaical Christians, with some varieties of usage and meaning, would have the Passover itself to be the great feast, but taken in a Christian sense; the Western or Gentile Christians, would have the commemoration of the Resurrection, connected with the Passover only by chronology. To shift the Passover in time, under its name, _Pascha_, without allusion to any of the force of the change, was gently cutting away the ground from under the feet of the Conservatives. And it was done in a very quiet way: no allusion to the precise character of the change; no hint that the question was about two different festivals: "all the brethren in the East, who formerly celebrated this festival at the same time as the Jews, will in future conform to the Romans and to us." The Judaizers meant to be keeping the Passover _as_ a Christian feast: they are gently assumed to be keeping, _not_ the Passover, _but_ a Christian feast; and a doctrinal decision is quietly, but efficiently, announced under the form of a chronological ordinance. Had the Council issued theses of doctrine, and excommunicated all dissentients, the rupture of the East and West would have taken place earlier by centuries than it did. The only place in which I ever saw any part of my paradox advanced, was in an article in the _Examiner_ newspaper, towards the end of 1866, after the above was written.

A story about Christopher Clavius, the workman of the new Calendar. I chanced to pick up "Albertus Pighius Campensis de aequinoctiorum solsticiorumque inventione... Ejusdem de ratione Paschalis celebrationis, De que Restitutione ecclesiastici Kalendarii," Paris, 1520, folio.[760] On the title-page were decayed words followed by ".._hristophor.. C..ii_, 1556 (or 8)," the last blank not entirely erased by time, but showing the lower halves of an _l_ and of an _a_, and {373} rather too much room for a _v_. It looked very like _E Libris Christophori Clavii_ 1556. By the courtesy of some members of the Jesuit body in London, I procured a tracing of the signature of Clavius from Rome, and the shapes of the letters, and the modes of junction and disjunction, put the matter beyond question. Even the extra space was explained; he wrote himself Cla_u_ius. Now in 1556, Clavius was nineteen years old: it thus appears probable that the framer of the Gregorian Calendar was selected, not merely as a learned astronomer, but as one who had attended to the calendar, and to works on its reformation, from early youth. When on the subject I found reason to think that Clavius had really read this work, and taken from it a phrase or two and a notion or two. Observe the advantage of writing the baptismal name at full length.

A COUPLE OF MINOR PARADOXES.

The discovery of a general resolution of all superior finite equations,
of every numerical both algebraick and transcendent form. By A. P.
Vogel,[761] mathematician at Leipzick. Leipzick and London, 1845, 8vo.

This work is written in the English of a German who has not mastered the idiom: but it is always intelligible. It professes to solve equations of every degree "in a more extent sense, and till to every degree of exactness." The general solution of equations of _all_ degrees is a vexed question, which cannot have the mysterious interest of the circle problem, and is of a comparatively modern date.[762] Mr. Vogel {374} announces a forthcoming treatise in which are resolved the "last impossibilities of pure mathematics."

Elective Polarity the Universal Agent. By Frances Barbara Burton,
authoress of 'Astronomy familiarized,' 'Physical Astronomy,' &c.
London, 1845, 8vo.[763]

The title gives a notion of the theory. The first sentence states, that 12,500 years ago [alpha] Lyrae was the pole-star, and attributes the immense magnitude of the now fossil animals to a star of such "polaric intensity as Vega pouring its magnetic streams through our planet." Miss Burton was a lady of property, and of very respectable acquirements, especially in Hebrew; she was eccentric in all things.

1867.--Miss Burton is revived by the writer of a book on meteorology which makes use of the planets: she is one of his leading minds.[764]

SPECULATIVE THOUGHT IN ENGLAND.

In the year 1845 the old _Mathematical Society_ was merged in the Astronomical Society. The circle-squarers, etc., thrive more in England than in any other country: there are most weeds where there is the largest crop. Speculation, though not encouraged by our Government so much as by those of the Continent, has had, not indeed such forcing, but much wider diffusion: few tanks, but many rivulets. On this point I quote from the preface to the reprint of the work of Ramchundra,[765] which I superintended for the late Court of Directors of the East India Company.

{375}

"That sound judgment which gives men well to know what is best for them, as well as that faculty of invention which leads to development of resources and to the increase of wealth and comfort, are both materially advanced, perhaps cannot rapidly be advanced without, a great taste for pure speculation among the general mass of the people, down to the lowest of those who can read and write. England is a marked example. Many persons will be surprised at this assertion. They imagine that our country is the great instance of the refusal of all _unpractical_ knowledge in favor of what is _useful_. I affirm, on the contrary, that there is no country in Europe in which there has been so wide a diffusion of speculation, theory, or what other unpractical word the reader pleases. In our country, the scientific _society_ is always formed and maintained by the people; in every other, the scientific _academy_--most aptly named--has been the creation of the government, of which it has never ceased to be the nursling. In all the parts of England in which manufacturing pursuits have given the artisan some command of time, the cultivation of mathematics and other speculative studies has been, as is well known, a very frequent occupation. In no other country has the weaver at his loom bent over the _Principia_ of Newton; in no other country has the man of weekly wages maintained his own scientific periodical. With us, since the beginning of the last century, scores upon scores--perhaps hundreds, for I am far from knowing all--of annuals have run, some their ten years, some their half-century, some their century and a half, containing questions to be answered, from which many of our examiners in the universities have culled materials for the academical contests. And these questions have always been answered, and in cases without number by the lower order of purchasers, the mechanics, the weavers, and the printers' workmen. I cannot here digress to point out the manner in which the concentration of manufactures, and the general diffusion of education, have affected the {376} state of things; I speak of the time during which the present system took its rise, and of the circumstances under which many of its most effective promoters were trained. In all this there is nothing which stands out, like the state-nourished academy, with its few great names and brilliant single achievements. This country has differed from all others in the wide diffusion of the disposition to speculate, which disposition has found its place among the ordinary habits of life, moderate in its action, healthy in its amount."

THE OLD MATHEMATICAL SOCIETY.

Among the most remarkable proofs of the diffusion of speculation was the Mathematical Society, which flourished from 1717 to 1845. Its habitat was Spitalfields, and I think most of its existence was passed in Crispin Street. It was originally a plain society, belonging to the studious artisan. The members met for discussion once a week; and I believe I am correct in saying that each man had his pipe, his pot, and his problem. One of their old rules was that, "If any member shall so far forget himself and the respect due to the Society as in the warmth of debate to threaten or offer personal violence to any other member, he shall be liable to immediate expulsion, or to pay such fine as the majority of the members present shall decide." But their great rule, printed large on the back of the title page of their last book of regulations, was "By the constitution of the Society, it is the duty of every member, if he be asked any mathematical or philosophical question by another member, to instruct him in the plainest and easiest manner he is able." We shall presently see that, in old time, the rule had a more homely form.

I have been told that De Moivre[766] was a member of this {377} Society. This I cannot verify: circumstances render it unlikely; even though the French refugees clustered in Spitalfields; many of them were of the Society, which there is some reason to think was founded by them. But Dolland,[767] Thomas Simpson,[768] Saunderson,[769] Crossley,[770] and others of known name, were certainly members. The Society gradually declined, and in 1845 was reduced to nineteen members. An arrangement was made by which sixteen of these members, who where not already in the Astronomical Society became Fellows without contribution, all the books and other property of the old Society being transferred to the new one. I was one of the committee which made the preliminary inquiries, and the reason of the decline was soon manifest. The only question which could arise was whether the members of the society of working men--for this repute still continued--were of that class of educated men who could associate with the Fellows of the Astronomical Society on terms agreeable to all parties. We found that the artisan element had been extinct for many years; there was not a man but might, as to education, manners, and position, have become a Fellow in the usual way. The fact was that life in Spitalfields had become harder: and the weaver could {378} only live from hand to mouth, and not up to the brain. The material of the old Society no longer existed.

In 1798, experimental lectures were given, a small charge for admission being taken at the door: by this hangs a tale--and a song. Many years ago, I found among papers of a deceased friend, who certainly never had anything to do with the Society, and who passed all his life far from London, a song, headed "Song sung by the Mathematical Society in London, at a dinner given Mr. Fletcher,[771] a solicitor, who had defended the Society gratis." Mr. Williams,[772] the Assistant Secretary of the Astronomical Society, formerly Secretary of the Mathematical Society, remembered that the Society had had a solicitor named Fletcher among the members. Some years elapsed before it struck me that my old friend Benjamin Gompertz,[773] who had long been a member, might have some recollection of the matter. The following is an extract of a letter from him (July 9, 1861):

"As to the Mathematical Society, of which I was a member when only 18 years of age, [Mr. G. was born in 1779], having been, contrary to the rules, elected under the age of 21. How I came to be a member of that Society--and continued so until it joined the Astronomical Society, and was then the President--was: I happened to pass a bookseller's small shop, of second-hand books, kept by a poor taylor, but a good mathematician, John Griffiths. I was very pleased to meet a mathematician, and I asked him if he would give me some lessons; and his reply was that I was more capable to teach him, but he belonged to a society of mathematicians, and he would introduce me. I accepted the offer, and I was elected, and had many scholars then to teach, as {379} one of the rules was, if a member asked for information, and applied to any one who could give it, he was obliged to give it, or fine one penny. Though I might say much with respect to the Society which would be interesting, I will for the present reply only to your question. I well knew Mr. Fletcher, who was a very clever and very scientific person. He did, as solicitor, defend an action brought by an informer against the Society--I think for 5,000l.--for giving lectures to the public in philosophical subjects [i.e., for unlicensed public exhibition with money taken at the doors]. I think the price for admission was one shilling, and we used to have, if I rightly recollect, from two to three hundred visitors. Mr. Fletcher was successful in his defence, and we got out of our trouble. There was a collection made to reward his services, but he did not accept of any reward: and I think we gave him a dinner, as you state, and enjoyed ourselves; no doubt with astronomical songs and other songs; but my recollection does not enable me to say if the astronomical song was a drinking song. I think the anxiety caused by that action was the cause of some of the members' death. [They had, no doubt, broken the law in ignorance; and by the sum named, the informer must have been present, and sued for a penalty on every shilling he could prove to have been taken]."

I by no means guarantee that the whole song I proceed to give is what was sung at the dinner: I suspect, by the completeness of the chain, that augmentations have been made. My deceased friend was just the man to add some verses, or the addition may have been made before it came into his hands, or since his decease, for the scraps containing the verses passed through several hands before they came into mine. We may, however, be pretty sure that the original is substantially contained in what is given, and that the character is therefore preserved. I have had myself to repair damages every now and then, in the way of conjectural restoration of defects caused by ill-usage. {380}

THE ASTRONOMER'S DRINKING SONG.

"Whoe'er would search the starry sky,
Its secrets to divine, sir,
Should take his glass--I mean, should try
A glass or two of wine, sir!
True virtue lies in golden mean,
And man must wet his clay, sir;
Join these two maxims, and 'tis seen
He should drink his bottle a day, sir!

"Old Archimedes, reverend sage!
By trump of fame renowned, sir,
Deep problems solved in every page,
And the sphere's curved surface found,[774] sir:
Himself he would have far outshone,
And borne a wider sway, sir,
Had he our modern secret known,
And drank a bottle a day, sir!

"When Ptolemy,[775] now long ago,
Believed the earth stood still, sir,
He never would have blundered so,
Had he but drunk his fill, sir:
He'd then have felt[776] it circulate,
And would have learnt to say, sir,
The true way to investigate
Is to drink your bottle a day, sir!

"Copernicus,[777] that learned wight,
The glory of his nation,
With draughts of wine refreshed his sight,
And saw the earth's rotation;
{381}
Each planet then its orb described,
The moon got under way, sir;
These truths from nature he imbibed
For he drank his bottle a day, sir!

"The noble[778] Tycho placed the stars,
Each in its due location;
He lost his nose[779] by spite of Mars,
But that was no privation:
Had he but lost his mouth, I grant
He would have felt dismay, sir,
Bless you! _he_ knew what he should want
To drink his bottle a day, sir!

"Cold water makes no lucky hits;
On mysteries the head runs:
Small drink let Kepler[780] time his wits
On the regular polyhedrons:
He took to wine, and it changed the chime,
His genius swept away, sir,
Through area varying[781] as the time
At the rate of a bottle a day, sir!

"Poor Galileo,[782] forced to rat
Before the Inquisition,
_E pur si muove_[783] was the pat
He gave them in addition:
{382}
He meant, whate'er you think you prove,
The earth must go its way, sirs;
Spite of your teeth I'll make it move,
For I'll drink my bottle a day, sirs!

"Great Newton, who was never beat
Whatever fools may think, sir;
Though sometimes he forgot to eat,
He never forgot to drink, sir:
Descartes[784] took nought but lemonade,
To conquer him was play, sir;
The first advance that Newton made
Was to drink his bottle a day, sir!

"D'Alembert,[785] Euler,[786] and Clairaut,[787]
Though they increased our store, sir,
Much further had been seen to go
Had they tippled a little more, sir!
Lagrange[788] gets mellow with Laplace,[789]
And both are wont to say, sir,
The _philosophe_ who's not an ass
Will drink his bottle a day, sir!

"Astronomers! what can avail
Those who calumniate us;
Experiment can never fail
With such an apparatus:
Let him who'd have his merits known
Remember what I say, sir;
Fair science shines on him alone
Who drinks his bottle a day, sir!

{383}
"How light we reck of those who mock
By this we'll make to appear, sir,
We'll dine by the sidereal[790] clock
For one more bottle a year, sir:
But choose which pendulum you will,
You'll never make your way, sir,
Unless you drink--and drink your fill,--
At least a bottle a day, sir!"

Old times are changed, old manners gone!

There is a new Mathematical Society,[791] and I am, at this present writing (1866), its first President. We are very high in the newest developments, and bid fair to take a place among the scientific establishments. Benjamin Gompertz, who was President of the old Society when it expired, was the link between the old and new body: he was a member of _ours_ at his death. But not a drop of liquor is seen at our meetings, except a decanter of water: all our heavy is a fermentation of symbols; and we do not draw it mild. There is no penny fine for reticence or occult science; and as to a song! not the ghost of a chance.

1826. The time may have come when the original documents connected with the discovery of Neptune may be worth revising. The following are extracts from the _Athenaeum_ of October 3 and October 17:

LE VERRIER'S[792] PLANET.

We have received, at the last moment before making up for press, the following letter from Sir John Herschel,[793] {384} in reference to the matter referred to in the communication from Mr. Hind[794] given below:

"Collingwood, Oct. 1.

"In my address to the British Association assembled at Southampton, on the occasion of my resigning the chair to Sir R. Murchison,[795] I stated, among the remarkable astronomical events of the last twelvemonth, that it had added a new planet to our list,--adding, 'it has done more,--it has given us the probable prospect of the discovery of another. We see it as Columbus saw America from the shores of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis, with a certainty hardly inferior to that of ocular demonstration.'--These expressions are not reported in any of the papers which profess to give an account of the proceedings, but I appeal to all present whether they were not used.

"Give me leave to state my reasons for this confidence; and, in so doing, to call attention to some facts which deserve to be put on record in the history of this noble discovery. On July 12, 1842, the late illustrious astronomer, Bessel,[796] honored me with a visit at my present residence. On the evening of that day, conversing on the great work of the planetary reductions undertaken by the Astronomer Royal[797]--then in progress, and since published,[798]--M. Bessel remarked that the motions of Uranus, as he had satisfied {385} himself by careful examination of the recorded observations, could not be accounted for by the perturbations of the known planets; and that the deviations far exceeded any possible limits of error of observation. In reply to the question, Whether the deviations in question might not be due to the action of an unknown planet?--he stated that he considered it highly probable that such was the case,--being systematic, and such as might be produced by an exterior planet. I then inquired whether he had attempted, from the indications afforded by these perturbations, to discover the position of the unknown body,--in order that 'a hue and cry' might be raised for it. From his reply, the words of which I do not call to mind, I collected that he had not then gone into that inquiry; but proposed to do so, having now completed certain works which had occupied too much of his time. And, accordingly, in a letter which I received from him after his return to Koenigsberg, dated November 14, 1842, he says,--'In reference to our conversation at Collingwood, I _announce_ to you (_melde_ ich Ihnen) that Uranus is not forgotten.' Doubtless, therefore, among his papers will be found some researches on the subject.

"The remarkable calculations of M. Le Verrier--which have pointed out, as now appears, nearly the true situation of the new planet, by resolving the inverse problem of the perturbations--if uncorroborated by repetition of the numerical calculations by another hand, or by independent investigation from another quarter, would hardly justify so strong an assurance as that conveyed by my expressions above alluded to. But it was known to me, at that time, (I will take the liberty to cite the Astronomer Royal as my authority) that a similar investigation had been independently entered into, and a conclusion as to the situation of the new planet very nearly coincident with M. Le Verrier's arrived at (in entire ignorance of his conclusions), by a young Cambridge mathematician, Mr. Adams;[799]--who will, I hope, {386} pardon this mention of his name (the matter being one of great historical moment),--and who will, doubtless, in his own good time and manner, place his calculations before the public.

"J. F. W. HERSCHEL."

_Discovery of Le Verrier's Planet._