Chapter XXXIX: Part 39

7500 7600 7700 H 5 16 27 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 7800 8000 G 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 7900 8100 F 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 8200 8300 8400 E 2 13 24 5 16 27 8 19 * 11 22 3 14 25' 6 17 28 9 20 1500 1600 8500 D 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19

As an example of the use of the preceding tables, suppose it were required to determine the moon's age on the 10th of April 1832. In 1832 the golden number is ((1832 + 1) / 19)_r = 9 and the line of epacts belonging to the century is C. In Table III, under 9, and in the line C, we find the epact 28. In the calendar, Table IV., look for April, and the epact 28 is found opposite the second day. The 2nd of April is therefore the first day of the moon, [v.04 p.0997] and the 10th is consequently the ninth day of the moon. Again, suppose it were required to find the moon's age on the 2nd of December in the year 1916. In this case the golden number is ((1916 + 1) / 19)_r = 17, and in Table III., opposite to 1900, the line of epacts is B. Under 17, in line B, the epact is 25'. In the calendar this epact first occurs before the 2nd of December at the 26th of November. The 26th of November is consequently the first day of the moon, and the 2nd of December is therefore the seventh day.

_Easter._--The next, and indeed the principal use of the calendar, is to find Easter, which, according to the traditional regulation of the council of Nice, must be determined from the following conditions:--_1st_, Easter must be celebrated on a Sunday; _2nd_, this Sunday must _follow_ the 14th day of the paschal moon, so that if the 14th of the paschal moon falls on a Sunday then Easter must be celebrated on the Sunday following; _3rd_, the paschal moon is that of which the 14th day falls on or next follows the day of the vernal equinox; _4th_ the equinox is fixed invariably in the calendar on the 21st of March. Sometimes a misunderstanding has arisen from not observing that this regulation is to be construed according to the tabular full moon as determined from the epact, and not by the true full moon, which, in general, occurs one or two days earlier.

From these conditions it follows that the paschal full moon, or the 14th of the paschal moon, cannot happen before the 21st of March, and that Easter in consequence cannot happen before the 22nd of March. If the 14th of the moon falls on the 21st, the new moon must fall on the 8th; for 21 - 13 = 8; and the paschal new moon cannot happen before the 8th; for suppose the new moon to fall on the 7th, then the full moon would arrive on the 20th, or the day before the equinox. The following moon would be the paschal moon. But the fourteenth of this moon falls at the latest on the 18th of April, or 29 days after the 20th of March; for by reason of the double epact that occurs at the 4th and 5th of April, this lunation has only 29 days. Now, if in this case the 18th of April is Sunday, then Easter must be celebrated on the following Sunday, or the 25th of April. Hence Easter Sunday cannot happen earlier than the 22nd of March, or later than the 25th of April.

Hence we derive the following rule for finding Easter Sunday from the tables:--_1st_, Find the golden number, and, from Table III., the epact of the proposed year. _2nd_, Find in the calendar (Table IV.) the first day after the 7th of March which corresponds to the epact of the year; this will be the first day of the paschal moon, _3rd_, Reckon thirteen days after that of the first of the moon, the following will be the 14th of the moon or the day of the full paschal moon. _4th_, Find from Table I. the dominical letter of the year, and observe in the calendar the first day, after the fourteenth of the moon, which corresponds to the dominical letter; this will be Easter Sunday.

TABLE IV.--_Gregorian Calendar._

|-----------------------------------------------------|
|Days.| Jan. | Feb. |March. |April. | May. | June. |
|-----+-------+-------+-------+-------+-------+-------+
| | E |L| E |L| E |L| E |L| E |L| E |L|
|-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+
| 1 | * |A| 29 |D| * |D| 29 |G| 28 |B| 27 |E|
| 2 | 29 |B| 28 |E| 29 |E| 28 |A| 27 |C|25 26|F|
| 3 | 28 |C| 27 |F| 28 |F| 27 |B| 26 |D|25 24|G|
| 4 | 27 |D|25 26|G| 27 |G|25'26|C|25'25|E| 23 |A|
| 5 | 26 |E|25 24|A| 26 |A|25 24|D| 24 |F| 22 |B|
|-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+
| 6 |25'25|F| 23 |B|25'25|B| 23 |E| 23 |G| 21 |C|
| 7 | 24 |G| 22 |C| 24 |C| 22 |F| 22 |A| 20 |D|
| 8 | 23 |A| 21 |D| 23 |D| 21 |G| 21 |B| 19 |E|
| 9 | 22 |B| 20 |E| 22 |E| 20 |A| 20 |C| 18 |F|
| 10 | 21 |C| 19 |F| 21 |F| 19 |B| 19 |D| 17 |G|
|-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+
| 11 | 20 |D| 18 |G| 20 |G| 18 |C| 18 |E| 16 |A|
| 12 | 19 |E| 17 |A| 19 |A| 17 |D| 17 |F| 15 |B|
| 13 | 18 |F| 16 |B| 18 |B| 16 |E| 16 |G| 14 |C|
| 14 | 17 |G| 15 |C| 17 |C| 15 |F| 15 |A| 13 |D|
| 15 | 16 |A| 14 |D| 16 |D| 14 |G| 14 |B| 12 |E|
|-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+
| 16 | 15 |B| 13 |E| 15 |E| 13 |A| 13 |C| 11 |F|
| 17 | 14 |C| 12 |F| 14 |F| 12 |B| 12 |D| 10 |G|
| 18 | 13 |D| 11 |G| 13 |G| 11 |C| 11 |E| 9 |A|
| 19 | 12 |E| 10 |A| 12 |A| 10 |D| 10 |F| 8 |B|
| 20 | 11 |F| 9 |B| 11 |B| 9 |E| 9 |G| 7 |C|
|-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+
| 21 | 10 |G| 8 |C| 10 |C| 8 |F| 8 |A| 6 |D|
| 22 | 9 |A| 7 |D| 9 |D| 7 |G| 7 |B| 5 |E|
| 23 | 8 |B| 6 |E| 8 |E| 6 |A| 6 |C| 4 |F|
| 24 | 7 |C| 5 |F| 7 |F| 5 |B| 5 |D| 3 |G|
| 25 | 6 |D| 4 |G| 6 |G| 4 |C| 4 |E| 2 |A|
|-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+
| 26 | 5 |E| 3 |A| 5 |A| 3 |D| 3 |F| 1 |B|
| 27 | 4 |F| 2 |B| 4 |B| 2 |E| 2 |G| * |C|
| 28 | 3 |G| 1 |C| 3 |C| 1 |F| 1 |A| 29 |D|
| 29 | 2 |A| | | 2 |D| * |G| * |B| 28 |E|
| 30 | 1 |B| | | 1 |E| 29 |A| 29 |C| 29 |F|
|-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+
| 31 | * |C| | | * |F| | | 28 |D| | |
|------------------------------------------------------

|------------------------------------------------------|
|Days.| July. |August.| Sept. |October.| Nov. | Dec. |
|-----+-------+-------+-------+--------+-------+-------|
| | E |L| E |L| E |L| E |L | E |L| E |L|
|-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-|
| 1 | 26 |G|25 24|C| 23 |F| 22 |A | 21 |D| 20 |F|
| 2 |25'25|A| 23 |D| 22 |G| 21 |B | 20 |E| 19 |G|
| 3 | 24 |B| 22 |E| 21 |A| 20 |C | 19 |F| 18 |A|
| 4 | 23 |C| 21 |F| 20 |B| 19 |D | 18 |G| 17 |B|
| 5 | 22 |D| 20 |G| 19 |C| 18 |E | 17 |A| 16 |C|
|-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-|
| 6 | 21 |E| 19 |A| 18 |D| 17 |F | 16 |B| 15 |D|
| 7 | 20 |F| 18 |B| 17 |E| 16 |G | 15 |C| 14 |E|
| 8 | 19 |G| 17 |C| 16 |F| 15 |A | 14 |D| 13 |F|
| 9 | 18 |A| 16 |D| 15 |G| 14 |B | 13 |E| 12 |G|
| 10 | 17 |B| 15 |E| 14 |A| 13 |C | 12 |F| 11 |A|
|-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-|
| 11 | 16 |C| 14 |F| 13 |B| 12 |D | 11 |G| 10 |B|
| 12 | 15 |D| 13 |G| 12 |C| 11 |E | 10 |A| 9 |C|
| 13 | 14 |E| 12 |A| 11 |D| 10 |F | 9 |B| 8 |D|
| 14 | 13 |F| 11 |B| 10 |E| 9 |G | 8 |C| 7 |E|
| 15 | 12 |G| 10 |C| 9 |F| 8 |A | 7 |D| 6 |F|
|-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-|
| 16 | 11 |A| 9 |D| 8 |G| 7 |B | 6 |E| 5 |G|
| 17 | 10 |B| 8 |E| 7 |A| 6 |C | 5 |F| 4 |A|
| 18 | 9 |C| 7 |F| 6 |B| 5 |D | 4 |G| 3 |B|
| 19 | 8 |D| 6 |G| 5 |C| 4 |E | 3 |A| 2 |C|
| 20 | 7 |E| 5 |A| 4 |D| 3 |F | 2 |B| 1 |D|
|-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-|
| 21 | 6 |F| 4 |B| 3 |E| 2 |G | 2 |C| * |E|
| 22 | 5 |G| 3 |C| 2 |F| 1 |A | * |D| 29 |F|
| 23 | 4 |A| 2 |D| 1 |G| * |B | 29 |E| 28 |G|
| 24 | 3 |B| 1 |E| * |A| 29 |C | 28 |F| 27 |A|
| 25 | 2 |C| * |F| 29 |B| 28 |D | 27 |G| 26 |B|
|-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-|
| 26 | 1 |D| 29 |G| 28 |C| 27 |E |25'26|A|25'25|C|
| 27 | * |E| 28 |A| 27 |D| 26 |F |25 24|B| 24 |D|
| 28 | 29 |F| 27 |B|25'26|E|25'25|G | 23 |C| 23 |E|
| 29 | 28 |G| 26 |C|25 24|F| 24 |A | 22 |D| 22 |F|
| 30 | 27 |A|25'25|D| 23 |G| 23 |B | 21 |E| 21 |G|
|-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-|
| 31 |25'26|B| 24 |B| | | 22 |C | | |19'20|A|
|------------------------------------------------------|

_Example._--Required the day on which Easter Sunday falls in the year 1840? _1st_, For this year the golden number is ((1840 + 1) / 19)_r = 17, and the epact (Table III. line C) is 26. _2nd_, After the 7th of March the epact 26 first occurs in Table III. at the 4th of April, which, therefore, is the day of the new moon. _3rd_, Since the new moon falls on the 4th, the full moon is on the 17th (4 + 13 = 17). _4th_, The dominical letters of 1840 are E, D (Table I.), of which D must be taken, as E belongs only to January and February. After the 17th of April D first occurs in the calendar (Table IV.) at the 19th. Therefore, in 1840, Easter Sunday falls on the 19th of April. The operation is in all cases much facilitated by means of the table on next page.

Such is the very complicated and artificial, though highly ingenious method, invented by Lilius, for the determination of Easter and the other movable feasts. Its principal, though perhaps least obvious advantage, consists in its being entirely independent of astronomical tables, or indeed of any celestial phenomena whatever; so that all chances of disagreement arising from the inevitable errors of tables, or the uncertainty of observation, are avoided, and Easter determined without the [v.04 p.0998] possibility of mistake. But this advantage is only procured by the sacrifice of some accuracy; for notwithstanding the cumbersome apparatus employed, the conditions of the problem are not always exactly satisfied, nor is it possible that they can be always satisfied by any similar method of proceeding. The equinox is fixed on the 21st of March, though the sun enters Aries generally on the 20th of that month, sometimes even on the 19th. It is accordingly quite possible that a full moon may arrive after the true equinox, and yet precede the 21st of March. This, therefore, would not be the paschal moon of the calendar, though it undoubtedly ought to be so if the intention of the council of Nice were rigidly followed. The new moons indicated by the epacts also differ from the astronomical new moons, and even from the mean new moons, in general by one or two days. In imitation of the Jews, who counted the time of the new moon, not from the moment of the actual phase, but from the time the moon first became visible after the conjunction, the fourteenth day of the moon is regarded as the full moon: but the moon is in opposition generally on the 16th day; therefore, when the new moons of the calendar nearly concur with the true new moons, the full moons are considerably in error. The epacts are also placed so as to indicate the full moons generally one or two days after the true full moons; but this was done purposely, to avoid the chance of concurring with the Jewish passover, which the framers of the calendar seem to have considered a greater evil than that of celebrating Easter a week too late.

TABLE V.--_Perpetual Table, showing Easter._

--------------------------------------------------------------
| | |
| | Dominical Letter. |
|Epact.| For Leap Years use the SECOND Letter. |
| |-------------------------------------------------------|
| | A | B | C | D | E | F | G |
|------+-------+-------+-------+-------+-------+-------+-------|
| * |Apr. 16|Apr. 17|Apr. 18|Apr. 19|Apr. 20|Apr. 14|Apr. 15|
| 1 | " 16| " 17| " 18| " 19| " 13| " 14| " 15|
| 2 | " 16| " 17| " 18| " 12| " 13| " 14| " 15|
| 3 | " 16| " 17| " 11| " 12| " 13| " 14| " 15|
| 4 | " 16| " 10| " 11| " 12| " 13| " 14| " 15|
| 5 | " 9| " 10| " 11| " 12| " 13| " 14| " 15|
| 6 | " 9| " 10| " 11| " 12| " 13| " 14| " 8|
| 7 | " 9| " 10| " 11| " 12| " 13| " 7| " 8|
| 8 | " 9| " 10| " 11| " 12| " 6| " 7| " 8|
| 9 | " 9| " 10| " 11| " 5| " 6| " 7| " 8|
| 10 | " 9| " 10| " 4| " 5| " 6| " 7| " 8|
| 11 | " 9| " 3| " 4| " 5| " 6| " 7| " 8|
| 12 | " 2| " 3| " 4| " 5| " 6| " 7| " 8|
| 13 | " 2| " 3| " 4| " 5| " 6| " 7| " 1|
| 14 | " 2| " 3| " 4| " 5| " 6|Mar. 31| " 1|
| 15 | " 2| " 3| " 4| " 5|Mar. 30| " 31| " 1|
| 16 | " 2| " 3| " 4|Mar. 29| " 30| " 31| " 1|
| 17 | " 2| " 3|Mar. 28| " 29| " 30| " 31| " 1|
| 18 | " 2|Mar. 27| " 28| " 29| " 30| " 31| " 1|
| 19 |Mar. 26| " 27| " 28| " 29| " 30| " 31| " 1|
| 20 | " 26| " 27| " 28| " 29| " 30| " 31|Mar. 25|
| 21 | " 26| " 27| " 28| " 29| " 30| " 24| " 25|
| 22 | " 26| " 27| " 28| " 29| " 23| " 24| " 25|
| 23 | " 26| " 27| " 28| " 22| " 23| " 24| " 25|
| 24 |Apr. 23|Apr. 24|Apr. 25|Apr. 19|Apr. 20|Apr. 21|Apr. 22|
| 25 | " 23| " 24| " 25| " 19| " 20| " 21| " 22|
| 26 | " 23| " 24| " 18| " 19| " 20| " 21| " 22|
| 27 | " 23| " 17| " 18| " 19| " 20| " 21| " 22|
| 28 | " 16| " 17| " 18| " 19| " 20| " 21| " 22|
| 29 | " 16| " 17| " 18| " 19| " 20| " 21| " 15|
--------------------------------------------------------------

We will now show in what manner this whole apparatus of methods and tables may be dispensed with, and the Gregorian calendar reduced to a few simple formulae of easy computation.

And, first, to find the dominical letter. Let L denote the number of the dominical letter of any given year of the era. Then, since every year which is not a leap year ends with the same day as that with which it began, the dominical letter of the following year must be L - 1, retrograding one letter every common year. After x years, therefore, the number of the letter will be L - x. But as L can never exceed 7, the number x will always exceed L after the first seven years of the era. In order, therefore, to render the subtraction possible, L must be increased by some multiple of 7, as 7m, and the formula then becomes 7m + L - x. In the year preceding the first of the era, the dominical letter was C; for that year, therefore, we have L = 3; consequently for any succeeding year x, L = 7m + 3 - x, the years being all supposed to consist of 365 days. But every fourth year is a leap year, and the effect of the intercalation is to throw the dominical letter one place farther back. The above expression must therefore be diminished by the number of units in x/4, or by (x/4)_w (this notation being used to denote the quotient, _in a whole number_, that arises from dividing x by 4). Hence in the Julian calendar the dominical letter is given by the equation

L = 7m + 3 - x - (x/4)_w.

This equation gives the dominical letter of any year from the commencement of the era to the Reformation. In order to adapt it to the Gregorian calendar, we must first add the 10 days that were left out of the year 1582; in the second place we must add one day for every century that has elapsed since 1600, in consequence of the secular suppression of the intercalary day; and lastly we must deduct the units contained in a fourth of the same number, because every fourth centesimal year is still a leap year. Denoting, therefore, the number of the century (or the date after the two right-hand digits have been struck out) by c, the value of L must be increased by 10 + (c - 16) - ((c - 16) / 4)_w . We have then

L = 7m + 3 - x - (x/4)_w + 10 + (c - 16) - ((c - 16) / 4)_w;

that is, since 3 + 10 = 13 or 6 (the 7 days being rejected, as they do not affect the value of L),

L = 7m + 6 - x - (x/4)_w + (c - 16) - ((c - 16) / 4)_w.

x = 1839, (x/4)_w = (1839/4)_w = 459, c = 18, c - 16 = 2,
and ((c - 16) / 4)_w = 0.

((26 + 11(N - 6)) / 30)_r. But the numerator of this fraction becomes by reduction 11 N - 40 or 11 N - 10 (the 30 being rejected, as the remainder only is sought) = N + 10(N - 1); therefore, ultimately,

J = ((N + 10(N - 1)) / 30)_r.

On account of the solar equation S, the epact J must be diminished by unity every centesimal year, excepting always the fourth. After x centuries, therefore, it must be diminished by x - (x/4)_w. Now, as 1600 was a leap year, the first correction of the Julian intercalation took place in 1700; hence, taking c to denote the number of the century as before, the correction becomes (c - 16) - ((c - 16) / 4)_w, which [v.04 p.0999] must be deducted from J. We have therefore

S = - (c - 16) + ((c - 16) / 4)_w.

With regard to the lunar equation M, we have already stated that in the Gregorian calendar the epacts are increased by unity at the end of every period of 300 years seven times successively, and then the increase takes place once at the end of 400 years. This gives eight to be added in a period of twenty-five centuries, and x/25 in x centuries. But 8x/25 = 1/3 (x - x/25). Now, from the manner in which the intercalation is directed to be made (namely, seven times successively at the end of 300 years, and once at the end of 400), it is evident that the fraction x/25 must amount to unity when the number of centuries amounts to twenty-four. In like manner, when the number of centuries is 24 + 25 = 49, we must have x/25 = 2; when the number of centuries is 24 + 2 × 25 = 74, then x/25 = 3; and, generally, when the number of centuries is 24 + n × 25, then x/25 = n + 1. Now this is a condition which will evidently be expressed in general by the formula n - ((n + 1) / 25)_w. Hence the correction of the epact, or the number of days to be intercalated after x centuries reckoned from the commencement of one of the periods of twenty-five centuries, is {(x - ((x+1) / 25)_w) / 3}_w. The last period of twenty-five centuries terminated with 1800; therefore, in any succeeding year, if c be the number of the century, we shall have x = c - 18 and x + 1 = c - 17. Let ((c - 17) / 25)_w = a, then for all years after 1800 the value of M will be given by the formula ((c - 18 - a) / 3)_w; therefore, counting from the beginning of the calendar in 1582,

M={(c - 15 - a) / 3}_w.

By the substitution of these values of J, S and M, the equation of the epact becomes

E = ((N + 10(N - 1)) / 30)_r - (c - 16) + ((c - 16) / 4)_w + ((c - 15 -
a) / 3)_w.

It may be remarked, that as a = ((c - 17) / 25)_w, the value of a will be 0 till c - 17 = 25 or c = 42; therefore, till the year 4200, a may be neglected in the computation. Had the anticipation of the new moons been taken, as it ought to have been, at one day in 308 years instead of 312½, the lunar equation would have occurred only twelve times in 3700 years, or eleven times successively at the end of 300 years, and then at the end of 400. In strict accuracy, therefore, a ought to have no value till c - 17 = 37, or c = 54, that is to say, till the year 5400. The above formula for the epact is given by Delambre (_Hist. de l'astronomie moderne,_ t. i. p. 9); it may be exhibited under a variety of forms, but the above is perhaps the best adapted for calculation. Another had previously been given by Gauss, but inaccurately, inasmuch as the correction depending on ''a'' was omitted.

Having determined the epact of the year, it only remains to find Easter Sunday from the conditions already laid down. Let

P = the number of days from the 21st of March to the 15th of the
paschal moon, which is the first day on which Easter Sunday can fall;

p = the number of days from the 21st of March to Easter Sunday;

L = the number of the dominical letter of the year;

l = letter belonging to the day on which the 15th of the moon falls:

then, since Easter is the Sunday following the 14th of the moon, we have

p = P + (L - l),

which is commonly called the _number of direction_.

The value of L is always given by the formula for the dominical letter, and P and l are easily deduced from the epact, as will appear from the following considerations.

When P = 1 the full moon is on the 21st of March, and the new moon on the 8th (21 - 13 = 8), therefore the moon's age on the 1st of March (which is the same as on the 1st of January) is twenty-three days; the epact of the year is consequently twenty-three. When P = 2 the new moon falls on the ninth, and the epact is consequently twenty-two; and, in general, when P becomes 1 + x, E becomes 23 - x, therefore P + E = 1 + x + 23 - x = 24, and P = 24 - E. In like manner, when P = 1, l = D = 4; for D is the dominical letter of the calendar belonging to the 22nd of March. But it is evident that when l is increased by unity, that is to say, when the full moon falls a day later, the epact of the year is diminished by unity; therefore, in general, when l = 4 + x, E = 23 - x, whence, l + E = 27 and l = 27 - E. But P can never be less than 1 nor l less than 4, and in both cases E = 23. When, therefore, E is greater than 23, we must add 30 in order that P and l may have positive values in the formula P = 24 - E and l = 27 - E. Hence there are two cases.

When E < 24, P = 24 - E; l = 27 - E, or ((27 - E) / 7)_r,
When E > 23, P = 54 - E; l = 57 - E, or ((57 - E) / 7)_r.

By substituting one or other of these values of P and l, according as the case may be, in the formula p = P + (L - l), we shall have p, or the number of days from the 21st of March to Easter Sunday. It will be remarked, that as L - l cannot either be 0 or negative, we must add 7 to L as often as may be necessary, in order that L - l may be a positive whole number.

By means of the formulae which we have now given for the dominical letter, the golden number and the epact, Easter Sunday may be computed for any year after the Reformation, without the assistance of any tables whatever. As an example, suppose it were required to compute Easter for the year 1840. By substituting this number in the formula for the dominical letter, we have x = 1840, c - 16 = 2, ((c - 16) / 4)_w = 0, therefore

L = 7m + 6 - 1840 - 460 + 2
= 7m - 2292
= 7 × 328 - 2292 = 2296 - 2292 = 4
L = 4 = letter D . . . (1).

For the golden number we have N = ((1840 + 1) / 19)_r; therefore N = 17 . . . (2).

For the epact we have

((N + 10(N - 1)) / 30)_r = ((17 + 160) / 30)_r = (177 / 30)_r = 27;

likewise c - 16 = 18 - 16 = 2, (c - 15) / 3 = 1, a = 0; therefore

E = 27 - 2 + 1 = 26 . . . (3).

Now since E > 23, we have for P and l,

P = 54 - E = 54 - 26 = 28,

l = ((57 - E) / 7)_r = ((57 - 26) / 7)_r = (31 / 7)_r = 3;

consequently, since p = P + (L - l),

p = 28 + (4 - 3) = 29;

that is to say, Easter happens twenty-nine days after the 21st of March, or on the 19th April, the same result as was before found from the tables.

The principal church feasts depending on Easter, and the times of their celebration are as follows:--

Septuagesima Sunday } { 9 weeks }
First Sunday in Lent } is { 6 weeks } before Easter.
Ash Wednesday } { 46 days }

Rogation Sunday { 5 weeks }
Ascension day or Holy Thursday } { 39 days }
Pentecost or Whitsunday } is { 7 weeks } after Easter.
Trinity Sunday } { 8 weeks }

The Gregorian calendar was introduced into Spain, Portugal and part of Italy the same day as at Rome. In France it was received in the same year in the month of December, and by the Catholic states of Germany the year following. In the Protestant states of Germany the Julian calendar was adhered to till the year 1700, when it was decreed by the diet of Regensburg that the new style and the Gregorian correction of the intercalation should be adopted. Instead, however, of employing the golden numbers and epacts for the determination of Easter and the movable feasts, it was resolved that the equinox and the paschal moon should be found by astronomical computation from the Rudolphine tables. But this method, though at first view it may appear more accurate, was soon found to be attended with numerous inconveniences, and was at length in 1774 abandoned at the instance of Frederick II., king of Prussia. In Denmark and Sweden the reformed calendar was received about the same time as in the Protestant states of Germany. It is remarkable that Russia still adheres to the Julian reckoning.

In Great Britain the alteration of the style was for a long time successfully opposed by popular prejudice. The inconvenience, however, of using a different date from that employed by the greater part of Europe in matters of history and chronology began to be generally felt; and at length the Calendar (New [v.04 p.1000] Style) Act 1750 was passed for the adoption of the new style in all public and legal transactions. The difference of the two styles, which then amounted to eleven days, was removed by ordering the day following the 2nd of September of the year 1752 to be accounted the 14th of that month; and in order to preserve uniformity in future, the Gregorian rule of intercalation respecting the secular years was adopted. At the same time, the commencement of the legal year was changed from the 25th of March to the 1st of January. In Scotland, January 1st was adopted for New Year's Day from 1600, according to an act of the privy council in December 1599. This fact is of importance with reference to the date of legal deeds executed in Scotland between that period and 1751, when the change was effected in England. With respect to the movable feasts, Easter is determined by the rule laid down by the council of Nice; but instead of employing the new moons and epacts, the golden numbers are prefixed to the days of the _full_ moons. In those years in which the line of epacts is changed in the Gregorian calendar, the golden numbers are removed to different days, and of course a new table is required whenever the solar or lunar equation occurs. The golden numbers have been placed so that Easter may fall on the same day as in the Gregorian calendar. The calendar of the church of England is therefore from century to century the same in form as the old Roman calendar, excepting that the golden numbers indicate the full moons instead of the new moons.

_Hebrew Calendar._--In the construction of the Jewish calendar numerous details require attention. The calendar is dated from the Creation, which is considered to have taken place 3760 years and 3 months before the commencement of the Christian era. The year is luni-solar, and, according as it is ordinary or embolismic, consists of twelve or thirteen lunar months, each of which has 29 or 30 days. Thus the duration of the ordinary year is 354 days, and that of the embolismic is 384 days. In either case, it is sometimes made a day more, and sometimes a day less, in order that certain festivals may fall on proper days of the week for their due observance. The distribution of the embolismic years, in each cycle of 19 years, is determined according to the following rule:--

The number of the Hebrew year (Y) which has its commencement in a Gregorian year (x) is obtained by the addition of 3761 years; that is, Y = x + 3761. Divide the Hebrew year by 19; then the quotient is the number of the last completed cycle, and the remainder is the year of the current cycle. If the remainder be 3, 6, 8, 11, 14, 17 or 19 (0), the year is embolismic; if any other number, it is ordinary. Or, otherwise, if we find the remainder

R=((7Y+1) / 19)_r

the year is embolismic when R < 7.

The calendar is constructed on the assumptions that the mean lunation is 29 days 12 hours 44 min. 3-1/3 sec., and that the year commences on, or immediately after, the new moon following the autumnal equinox. The mean solar year is also assumed to be 365 days 5 hours 55 min. 25-25/57 sec., so that a cycle of nineteen of such years, containing 6939 days 16 hours 33 min. 3-1/3 sec., is the exact measure of 235 of the assumed lunations. The year 5606 was the first of a cycle, and the mean new moon, appertaining to the 1st of Tisri for that year, was 1845, October 1, 15 hours 42 min. 43-1/3 sec., as computed by Lindo, and adopting the civil mode of reckoning from the previous midnight. The times of all future new moons may consequently be deduced by successively adding 29 days 12 hours 44 min. 3-1/3 sec. to this date.

To compute the times of the new moons which determine the commencement of successive years, it must be observed that in passing from an ordinary year the new moon of the following year is deduced by subtracting the interval that twelve lunations fall short of the corresponding Gregorian year of 365 or 366 days; and that, in passing from an embolismic year, it is to be found by adding the excess of thirteen lunations over the Gregorian year. Thus to deduce the new moon of Tisri, for the year immediately following any given year (Y), when Y is

ordinary, subtract (10)(11) days 15 hours 11 min. 20 sec.,
embolismic, add (18)(17) days 21 hours 32 min. 43½ sec.

the second-mentioned number of days being used, in each case, whenever the following or new Gregorian year is bissextile.

Hence, knowing which of the years are embolismic, from their ordinal position in the cycle, according to the rule before stated, the times of the commencement of successive years may be thus carried on indefinitely without any difficulty. But some slight adjustments will occasionally be needed for the reasons before assigned, viz. to avoid certain festivals falling on incompatible days of the week. Whenever the computed conjunction falls on a Sunday, Wednesday or Friday, the new year is in such case to be fixed on the day after. It will also be requisite to attend to the following conditions:--

If the computed new moon be after 18 hours, the following day is to be taken, and if that happen to be Sunday, Wednesday or Friday, it must be further postponed one day. If, for an ordinary year, the new moon falls on a Tuesday, as late as 9 hours 11 min. 20 sec., it is not to be observed thereon; and as it may not be held on a Wednesday, it is in such case to be postponed to Thursday. If, for a year immediately following an embolismic year, the computed new moon is on Monday, as late as 15 hours 30 min. 52 sec., the new year is to be fixed on Tuesday.

After the dates of commencement of the successive Hebrew years are finally adjusted, conformably with the foregoing directions, an estimation of the consecutive intervals, by taking the differences, will show the duration and character of the years that respectively intervene. According to the number of days thus found to be comprised in the different years, the days of the several months are distributed as in Table VI.

The signs + and - are respectively annexed to Hesvan and Kislev to indicate that the former of these months may sometimes require to have one day more, and the latter sometimes one day less, than the number of days shown in the table--the result, in every case, being at once determined by the total number of days that the year may happen to contain. An ordinary year may comprise 353, 354 or 355 days; and an embolismic year 383, 384 or 385 days. In these cases respectively the year is said to be imperfect, common or perfect. The intercalary month, Veadar, is introduced in embolismic years in order that Passover, the 15th day of Nisan, may be kept at its proper season, which is the full moon of the vernal equinox, or that which takes place after the sun has entered the sign Aries. It always precedes the following new year by 163 days, or 23 weeks and 2 days; and Pentecost always precedes the new year by 113 days, or 16 weeks and 1 day.

TABLE VI.--_Hebrew Months._

----------------------------------
| |Ordinary |Embolismic|
|Hebrew Month.| Year. | Year. |
|-------------|---------|----------|
|Tisri | 30 | 30 |
|Hesvan | 29 + | 29 + |
|Kislev | 30 - | 30 - |
|Tebet | 29 | 29 |
|Sebat | 30 | 30 |
|Adar | 29 | 30 |
|(Veadar) | (...) | (29) |
|Nisan | 30 | 30 |
|Yiar | 29 | 29 |
|Sivan | 30 | 30 |
|Tamuz | 29 | 29 |
|Ab | 30 | 30 |
|Elul | 29 | 29 |
|----------------------------------|
|Total | 354 | 384 |
|----------------------------------|

The Gregorian epact being the age of the moon of Tebet at the beginning of the Gregorian year, it represents the day of Tebet which corresponds to January 1; and thus the approximate date of Tisri 1, the commencement of the Hebrew year, may be otherwise deduced by subtracting the epact from

Sept. 24 after an ordinary Hebrew year.
Oct. 24 after an embolismic Hebrew year.

[v.04 p.1001]

The result so obtained would in general be more accurate than the Jewish calculation, from which it may differ a day, as fractions of a day do not enter alike in these computations. Such difference may also in part be accounted for by the fact that the assumed duration of the solar year is 6 min. 39-25/57 sec. in excess of the true astronomical value, which will cause the dates of commencement of future Jewish years, so calculated, to advance forward from the equinox a day in error in 216 years. The lunations are estimated with much greater precision.

The following table is extracted from Woolhouse's _Measures, Weights and Moneys of all Nations_:--

TABLE VII.--_Hebrew Years._

Jewish Number Commencement Jewish Number Commencement
Year. of (1st of Tisri). Year. of (1st of Tisri).
Days. Days.
296 Cycle. 302 Cycle.
5606 354 Thur. 2 Oct. 1845 5720 355 Sat. 3 Oct. 1959
07 355 Mon. 21 Sept. 1846 21 354 Thur. 22 Sept. 1960
08 383 Sat. 11 Sept. 1847 22 383 Mon. 11 Sept. 1961
09 354 Thur. 28 Sept. 1848 23 355 Sat. 29 Sept. 1962
10 355 Mon. 17 Sept. 1849 24 354 Thur. 19 Sept. 1963
11 385 Sat. 7 Sept. 1850 25 385 Mon. 7 Sept. 1964
12 353 Sat. 27 Sept. 1851 26 353 Mon. 27 Sept. 1965
13 384 Tues. 14 Sept. 1852 27 385 Thur. 15 Sept. 1966
14 355 Mon. 3 Oct. 1853 28 354 Thur. 5 Oct. 1967
15 355 Sat. 23 Sept. 1854 29 355 Mon. 23 Sept. 1968
16 383 Thur. 13 Sept. 1855 30 383 Sat. 13 Sept. 1969
17 354 Tues. 30 Sept. 1856 31 354 Thur. 1 Oct. 1970
18 355 Sat. 19 Sept. 1857 32 355 Mon. 20 Sept. 1971
19 385 Thur. 9 Sept. 1858 33 383 Sat. 9 Sept. 1972
20 354 Thur. 29 Sept. 1859 34 355 Thur. 27 Sept. 1973
21 353 Mon. 17 Sept. 1860 35 354 Tues. 17 Sept. 1974
22 385 Thur. 5 Sept. 1861 36 385 Sat. 6 Sept. 1975
23 354 Thur. 25 Sept. 1862 37 353 Sat. 25 Sept. 1976
24 383 Mon. 14 Sept. 1863 38 384 Tues. 13 Sept. 1977
----------------------------------- -----------------------------------
297 Cycle. 303 Cycle.
5625 355 Sat. 1 Oct. 1864 5739 355 Mon. 2 Oct. 1978
26 354 Thur. 21 Sept. 1865 40 355 Sat. 22 Sept. 1979
27 385 Mon. 10 Sept. 1866 41 383 Thur. 11 Sept. 1980
28 353 Mon. 30 Sept. 1867 42 354 Tues. 29 Sept. 1981
29 354 Thur. 17 Sept. 1868 43 355 Sat. 18 Sept. 1982
30 385 Mon. 6 Sept. 1869 44 385 Thur. 8 Sept. 1983
31 355 Mon. 26 Sept. 1870 45 354 Thur. 27 Sept. 1984
32 383 Sat. 16 Sept. 1871 46 383 Mon. 16 Sept. 1985
33 354 Thur. 3 Oct. 1872 47 355 Sat. 4 Oct. 1986
34 355 Mon. 22 Sept. 1873 48 354 Thur. 24 Sept. 1987
35 383 Sat. 12 Sept. 1874 49 383 Mon. 12 Sept. 1988
36 355 Thur. 30 Sept. 1875 50 355 Sat. 30 Sept. 1989
37 354 Tues. 19 Sept. 1876 51 354 Thur. 20 Sept. 1990
38 385 Sat. 8 Sept. 1877 52 385 Mon. 9 Sept. 1991
39 355 Sat. 28 Sept. 1878 53 353 Mon. 28 Sept. 1992
40 354 Thur. 18 Sept. 1879 54 355 Thur. 16 Sept. 1993
41 383 Mon. 6 Sept. 1880 55 384 Tues. 6 Sept. 1994
42 355 Sat. 24 Sept. 1881 56 355 Mon. 25 Sept. 1995
43 383 Thur. 14 Sept. 1882 57 383 Sat. 14 Sept. 1996
----------------------------------- -----------------------------------
298 Cycle. 304 Cycle.
5644 354 Tues. 2 Oct. 1883 5758 354 Thur. 2 Oct. 1997
45 355 Sat. 20 Sept. 1884 59 355 Mon. 21 Sept. 1998
46 385 Thur. 10 Sept. 1885 60 385 Sat. 11 Sept. 1999
47 354 Thur. 30 Sept. 1886 61 353 Sat. 30 Sept. 2000
48 353 Mon. 19 Sept. 1887 62 354 Tues. 18 Sept. 2001
49 385 Thur. 6 Sept. 1888 63 385 Sat. 7 Sept. 2002
50 354 Thur. 26 Sept. 1889 64 355 Sat. 27 Sept. 2003
51 383 Mon. 15 Sept. 1890 65 383 Thur. 16 Sept. 2004
52 355 Sat. 3 Oct. 1891 66 354 Tues. 4 Oct. 2005
53 354 Thur. 22 Sept. 1892 67 355 Sat. 23 Sept. 2006
54 385 Mon. 11 Sept. 1893 68 383 Thur. 13 Sept. 2007
55 353 Mon. 1 Oct. 1894 69 354 Tues. 30 Sept. 2008
56 355 Thur. 19 Sept. 1895 70 355 Sat. 19 Sept. 2009
57 384 Tues. 8 Sept. 1896 71 385 Thur. 8 Sept. 2010
58 355 Mon. 27 Sept. 1897 72 354 Thur. 29 Sept. 2011
59 353 Sat. 17 Sept. 1898 73 353 Mon. 17 Sept. 2012
60 384 Tues. 5 Sept. 1899 74 385 Thur. 5 Sept. 2013
61 355 Mon. 24 Sept. 1900 75 354 Thur. 25 Sept. 2014
62 383 Sat 14 Sept. 1901 76 385 Mon. 14 Sept. 2015
----------------------------------- -----------------------------------
299 Cycle. 305 Cycle.
5663 355 Thur. 2 Oct. 1902 5777 353 Mon. 3 Oct. 2016
64 354 Tues. 22 Sept. 1903 78 354 Thur. 21 Sept. 2017
65 385 Sat. 10 Sept. 1904 79 385 Mon. 10 Sept. 2018
66 355 Sat. 30 Sept. 1905 80 355 Mon. 30 Sept. 2019
67 354 Thur. 20 Sept. 1906 81 353 Sat. 19 Sept. 2020
68 383 Mon. 9 Sept. 1907 82 384 Tues. 7 Sept. 2021
69 355 Sat. 26 Sept. 1908 83 355 Mon. 26 Sept. 2022
70 383 Thur. 16 Sept. 1909 84 383 Sat. 16 Sept. 2023
71 354 Tues. 4 Oct. 1910 85 355 Thur. 3 Oct. 2024
72 355 Sat. 23 Sept. 1911 86 354 Tues. 23 Sept. 2025
73 385 Thur. 12 Sept. 1912 87 385 Sat. 12 Sept. 2026
74 354 Thur. 2 Oct. 1913 88 355 Sat. 2 Oct. 2027
75 353 Mon. 21 Sept. 1914 89 354 Thur. 21 Sept. 2028
76 385 Thur. 9 Sept. 1915 90 383 Mon. 10 Sept. 2029
77 354 Thur. 28 Sept. 1916 91 355 Sat. 28 Sept. 2030
78 355 Mon. 17 Sept. 1917 92 354 Thur. 18 Sept. 2031
79 383 Sat. 7 Sept. 1918 93 383 Mon. 6 Sept. 2032
80 354 Thur. 25 Sept. 1919 94 355 Sat. 24 Sept. 2033
81 385 Mon. 13 Sept. 1920 95 385 Thur. 14 Sept. 2034
----------------------------------- -----------------------------------
300 Cycle. 306 Cycle.
5682 355 Mon. 3 Oct. 1921 5796 354 Thur. 4 Oct. 2035
83 353 Sat. 23 Sept. 1922 97 353 Mon. 22 Sept. 2036
84 384 Tues. 11 Sept. 1923 98 385 Thur. 10 Sept. 2037
85 355 Mon. 29 Sept. 1924 99 354 Thur. 30 Sept. 2038
86 355 Sat. 19 Sept. 1925 5800 355 Mon. 19 Sept. 2039
87 383 Thur. 9 Sept. 1926 01 383 Sat. 8 Sept. 2040
88 354 Tues. 27 Sept. 1927 02 354 Thur. 26 Sept. 2041
89 385 Sat. 15 Sept. 1928 03 385 Mon. 15 Sept. 2042
90 353 Sat. 5 Oct. 1929 04 353 Mon. 5 Oct. 2043
91 354 Tues. 23 Sept. 1930 05 355 Thur. 22 Sept. 2044
92 385 Sat. 12 Sept. 1931 06 384 Tues. 12 Sept. 2045
93 355 Sat. 1 Oct. 1932 07 355 Mon. 1 Oct. 2046
94 354 Thur. 21 Sept. 1933 08 353 Sat. 21 Sept. 2047
95 383 Mon. 10 Sept. 1934 09 384 Tues. 8 Sept. 2048
96 355 Sat. 28 Sept. 1935 10 355 Mon. 27 Sept. 2049
97 354 Thur. 17 Sept. 1936 11 355 Sat. 17 Sept. 2050
98 385 Mon. 6 Sept. 1937 12 383 Thur. 7 Sept. 2051
99 353 Mon. 26 Sept. 1938 13 354 Tues. 24 Sept. 2052
5700 385 Thur. 14 Sept. 1939 14 385 Sat. 13 Sept. 2053
----------------------------------- -----------------------------------
301 Cycle. 307 Cycle.
5701 354 Thur. 3 Oct. 1940 5815 355 Sat. 3 Oct. 2054
02 355 Mon. 22 Sept. 1941 16 354 Thur. 23 Sept. 2055
03 383 Sat. 12 Sept. 1942 17 383 Mon. 11 Sept. 2056
04 354 Thur. 30 Sept. 1943 18 355 Sat. 29 Sept. 2057
05 355 Mon. 18 Sept. 1944 19 354 Thur. 19 Sept. 2058
06 383 Sat. 8 Sept. 1945 20 383 Mon. 8 Sept. 2059
07 354 Thur. 26 Sept. 1946 21 355 Sat. 25 Sept. 2060
08 385 Mon. 15 Sept. 1947 22 385 Thur. 15 Sept. 2061
09 355 Mon. 4 Oct. 1948 23 354 Thur. 5 Oct. 2062
10 353 Sat. 24 Sept. 1949 24 353 Mon. 24 Sept. 2063
11 384 Tues. 12 Sept. 1950 25 385 Thur. 11 Sept. 2064
12 355 Mon. 1 Oct. 1951 26 354 Thur. 1 Oct. 2065
13 355 Sat. 20 Sept. 1952 27 355 Mon. 20 Sept. 2066
14 383 Thur. 10 Sept. 1953 28 383 Sat. 10 Sept. 2067
15 354 Tues. 28 Sept. 1954 29 354 Thur. 27 Sept. 2068
16 355 Sat. 17 Sept. 1955 30 355 Mon. 16 Sept. 2069
17 385 Thur. 6 Sept. 1956 31 383 Sat. 6 Sept. 2070
18 354 Thur. 26 Sept. 1957 32 355 Thur. 24 Sept. 2071
19 383 Mon. 15 Sept. 1958 33 384 Tues. 13 Sept. 2072

_Mahommedan Calendar._--The Mahommedan era, or era of the Hegira, used in Turkey, Persia, Arabia, &c., is dated from the first day of the month preceding the flight of Mahomet from Mecca to Medina, _i.e._ Thursday the 15th of July A.D. 622, and it commenced on the day following. The years of the Hegira are purely lunar, and always consist of twelve lunar months, commencing with the approximate new moon, without any intercalation to keep them to the same season with respect to the sun, so that they retrograde through all the seasons in about 32½ years. They are also partitioned into cycles of 30 years, 19 of which are common years of 354 days each, and the other 11 are intercalary years having an additional day appended to the last month. The mean length of the year is therefore 354-11/30 days, or 354 days 8 hours 48 min., which divided by 12 gives 29-191/360 days, or 29 days 12 hours 44 min., as the time of a mean lunation, and this differs from the astronomical mean lunation by only 2.8 seconds. This small error will only amount to a day in about 2400 years.

To find if a year is intercalary or common, divide it by 30; the quotient will be the number of completed cycles and the remainder will be the year of the current cycle; if this last be one of the numbers 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29, the year is intercalary and consists of 355 days; if it be any other number, the year is ordinary.

Or if Y denote the number of the Mahommedan year, and

R = ((11 Y + 14) / 30)_r,

the year is intercalary when R < 11.

[v.04 p.1002] Also the number of intercalary years from the year 1 up to the year Y inclusive = ((11 Y + 14) / 30)_w; and the same up to the year Y - 1 = (11 Y + 3 / 30)_w.

To find the day of the week on which any year of the Hegira begins, we observe that the year 1 began on a Friday, and that after every common year of 354 days, or 50 weeks and 4 days, the day of the week must necessarily become postponed 4 days, besides the additional day of each intercalary year.

Hence if w = 1 | 2 | 3 | 4 | 5 | 6 | 7
indicate Sun. | Mon. | Tue. | Wed. | Thur. | Frid. | Sat.

the day of the week on which the year Y commences will be

w = 2 + 4(Y / 7)_r + ((11 Y + 3) / 30)_w (rejecting sevens).

But, 30 ((11 Y + 3) / 30)_w + ((11 Y + 3) / 30)_r = 11 Y + 3

gives 120((11 Y + 3) / 30)_w = 12 + 44 Y - 4((11 Y + 3) / 30)_r,

or ((11 Y + 3) / 30)_w = 5 + 2 Y + 3((11 Y + 3) / 30)_r (rejecting
sevens).

So that

w = 6(Y / 7)_r + 3((11 Y + 3) / 30)_r (rejecting sevens),

the values of which obviously circulate in a period of 7 times 30 or 210 years.

Let C denote the number of completed cycles, and y the year of the cycle; then Y = 30 C + y, and

w = 5(C / 7)_r + 6(y / 7)_r + 3((11 y +3) / 30)_r (rejecting sevens).

From this formula the following table has been constructed:--

TABLE VIII.

Year of the Number of the Period of Seven Cycles = (C/7)_r
Current Cycle (y) 0 1 2 3 4 5 6
0 8 Mon. Sat. Thur. Tues. Sun. Frid. Wed.
1 9 17 25 Frid. Wed. Mon. Sat. Thur. Tues. Sun.
*2 *10 *18 *26 Tues. Sun. Frid. Wed. Mon. Sat. Thur.
3 11 19 27 Sun. Frid. Wed. Mon. Sat. Thur. Tues.
4 12 20 28 Thur. Tues. Sun. Frid. Wed. Mon. Sat.
*5 *13 *21 *29 Mon. Sat. Thur. Tues. Sun. Frid. Wed.
6 14 22 30 Sat. Thur. Tues. Sun. Frid. Wed. Mon.
*7 15 23 Wed. Mon. Sat. Thur. Tues. Sun. Frid.
*16 *24 Sun. Frid. Wed. Mon. Sat. Thur. Tues.

To find from this table the day of the week on which any year of the Hegira commences, the rule to be observed will be as follows:--

_Rule._--Divide the year of the Hegira by 30; the quotient is the number of cycles, and the remainder is the year of the current cycle. Next divide the number of cycles by 7, and the second remainder will be the Number of the Period, which being found at the top of the table, and the year of the cycle on the left hand, the required day of the week is immediately shown.

The intercalary years of the cycle are distinguished by an asterisk.

For the computation of the Christian date, the ratio of a mean year of the Hegira to a solar year is

Year of Hegira / Mean solar year = 354-11/30 / 365.2422 = 0.970224.