Let (p + 13) / 4 = p'; then p = 4 p' - 13;
whence m' = 16 p' - 52 - p' = 15 p' - 52.
Now in this equation p' may be any number whatever, provided 15 p' exceed
52. The smallest value of p' (which is the one here wanted) is therefore 4;
for 15 × 4 = 60. Assuming therefore p' = 4, we have m' = 60 - 52 = 8; and
consequently, since x = 19 m' + 16, x = 19 × 8 + 16 = 168. The number
required is consequently 28 × 168 + 10 = 4714.
Having found the number 4714 for the first of the era, the correspondence
of the years of the era and of the period is as follows:--
Era, 1, 2, 3, ... x,
Period, 4714, 4715, 4716, ... 4713 + x;
from which it is evident, that if we take P to represent the year of the
Julian period, and x the corresponding year of the Christian era, we shall
have
P = 4713 + x, and x = P - 4713.
With regard to the numeration of the years previous to the commencement of
the era, the practice is not uniform. Chronologists, in general, reckon the
year preceding the first of the era -1, the next preceding -2, and so on.
In this case
Era, -1, -2, -3, ... -x,
Period, 4713, 4712, 4711, ... 4714 - x;
whence
P = 4714 - x, and x = 4714 - P.
But astronomers, in order to preserve the uniformity of computation, make
the series of years proceed without interruption, and reckon the year
preceding the first of the era 0. Thus
Era, 0, -1, -2, ... -x,
Period, 4713, 4712, 4711, ... 4713 - x;
therefore, in this case
P = 4713 - x, and x = 4713 - P.
_Reformation of the Calendar._--The ancient church calendar was founded on
two suppositions, both erroneous, namely, that the year contains 365¼ days,
and that 235 lunations are exactly equal to nineteen solar years. It could
not therefore long continue to preserve its correspondence with the
seasons, or to indicate the days of the new moons with the same accuracy.
About the year 730 the venerable Bede had already perceived the
anticipation of the equinoxes, and remarked that these phenomena then took
place about three days earlier than at the time of the council of Nicaea.
Five centuries after the time of Bede, the divergence of the true equinox
from the 21st of March, which now amounted to seven or eight days, was
pointed out by Johannes de Sacro Bosco (John Holywood, _fl._ 1230) in his
_De Anni Ratione_; and by Roger Bacon, in a treatise _De Reformatione
Calendarii_, which, though never published, was transmitted to the pope.
These works were probably little regarded at the time; but as the errors of
the calendar went on increasing, and the true length of the year, in
consequence of the progress of astronomy, became better known, the project
of a reformation was again revived in the 15th century; and in 1474 Pope
Sixtus IV. invited Regiomontanus, the most celebrated astronomer of the
age, to Rome, to superintend the reconstruction of the calendar. The
premature death of Regiomontanus caused the design to be suspended for the
time; but in the following century numerous memoirs appeared on the
subject, among the authors of which were Stoffler, Albert Pighius, Johann
Schöner, Lucas Gauricus, and other mathematicians of celebrity. At length
Pope Gregory XIII. perceiving that the measure was likely to confer a great
_éclat_ on his pontificate, undertook the long-desired reformation; and
having found the governments of the principal Catholic states ready to
adopt his views, he issued a brief in the month of March 1582, in which he
abolished the use of the ancient calendar, and substituted that which has
since been received in almost all Christian countries under the name of the
_Gregorian Calendar_ or _New Style_ The author of the system adopted by
Gregory was Aloysius Lilius, or Luigi Lilio Ghiraldi, a learned astronomer
and physician of Naples, who died, however, before its introduction; but
the individual who most contributed to give the ecclesiastical calendar its
present form, and who was charged with all the calculations necessary for
its verification, was Clavius, by whom it was completely developed and
explained in a great folio treatise of 800 pages, published in 1603, the
title of which is given at the end of this article.
It has already been mentioned that the error of the Julian year was
corrected in the Gregorian calendar by the suppression of three
intercalations in 400 years. In order to restore the beginning of the year
to the same place in the seasons that it had occupied at the time of the
council of Nicaea, Gregory directed the day following the feast of St
Francis, that is to say the 5th of October, to be reckoned the 15th of that
month. By this regulation the vernal equinox which then happened on the
11th of March was restored to the 21st. From 1582 to 1700 the difference
between the old and new style continued to be ten days; but 1700 being a
leap year in the Julian calendar, and a common year in the Gregorian, the
difference of the styles during the 18th century was eleven days. The year
1800 was also common in the new calendar, and, consequently, the difference
in the 19th century was twelve days. From 1900 to 2100 inclusive it is
thirteen days.
The restoration of the equinox to its former place in the year and the
correction of the intercalary period, were attended with no difficulty; but
Lilius had also to adapt the lunar year to the new rule of intercalation.
The lunar cycle contained 6939 days 18 hours, whereas the exact time of 235
lunations, as we have already seen, is 235 × 29.530588 = 6939 days 16 hours
31 minutes. The difference, which is 1 hour 29 minutes, amounts to a day in
308 years, so that at the end of this time the new moons occur one day
earlier than they are indicated by the golden numbers. During the 1257
years that elapsed between the council of Nicaea and the Reformation, the
error had accumulated to four days, so that the new moons which were marked
in the calendar as happening, for example, on the 5th of the month,
actually fell on the 1st. It would have been easy to correct this error by
placing the golden numbers four lines higher in the new calendar; and the
suppression of the ten days had already rendered it necessary to place them
ten lines lower, and to carry those which belonged, for example, to the 5th
and 6th of the month, to the 15th and 16th. But, supposing this correction
to have been made, it would have again become necessary, at the end of 308
years, to advance them one line higher, in consequence of the accumulation
of the error of the cycle to a whole day. On the other hand, as the golden
numbers were only adapted to the Julian calendar, every omission of the
centenary intercalation would require them to be placed one line lower,
opposite the 6th, for example, instead of the 5th of the month; so that,
generally speaking, the places of the golden numbers would have to be
changed every century. On this account Lilius thought fit to reject the
golden numbers from the calendar, and supply their place by another set of
numbers called _Epacts_, the use of which we shall now proceed to explain.
_Epacts._--Epact is a word of Greek origin, employed in the calendar to
signify the moon's age at the beginning of the year. [v.04 p.0995] The
common solar year containing 365 days, and the lunar year only 354 days,
the difference is eleven; whence, if a new moon fall on the 1st of January
in any year, the moon will be eleven days old on the first day of the
following year, and twenty-two days on the first of the third year. The
numbers eleven and twenty-two are therefore the epacts of those years
respectively. Another addition of eleven gives thirty-three for the epact
of the fourth year; but in consequence of the insertion of the intercalary
month in each third year of the lunar cycle, this epact is reduced to
three. In like manner the epacts of all the following years of the cycle
are obtained by successively adding eleven to the epact of the former year,
and rejecting thirty as often as the sum exceeds that number. They are
therefore connected with the golden numbers by the formula (11 n / 30) in
which n is any whole number; and for a whole lunar cycle (supposing the
first epact to be 11), they are as follows:--11, 22, 3, 14, 25, 6, 17, 28,
9, 20, 1, 12, 23, 4, 15, 26, 7, 18, 29. But the order is interrupted at the
end of the cycle; for the epact of the following year, found in the same
manner, would be 29 + 11 = 40 or 10, whereas it ought again to be 11 to
correspond with the moon's age and the golden number 1. The reason of this
is, that the intercalary month, inserted at the end of the cycle, contains
only twenty-nine days instead of thirty; whence, after 11 has been added to
the epact of the year corresponding to the golden number 19, we must reject
twenty-nine instead of thirty, in order to have the epact of the succeeding
year; or, which comes to the same thing, we must add twelve to the epact of
the last year of the cycle, and then reject thirty as before.
This method of forming the epacts might have been continued indefinitely if
the Julian intercalation had been followed without correction, and the
cycle been perfectly exact; but as neither of these suppositions is true,
two equations or corrections must be applied, one depending on the error of
the Julian year, which is called the solar equation; the other on the error
of the lunar cycle, which is called the lunar equation. The solar equation
occurs three times in 400 years, namely, in every secular year which is not
a leap year; for in this case the omission of the intercalary day causes
the new moons to arrive one day later in all the following months, so that
the moon's age at the end of the month is one day less than it would have
been if the intercalation had been made, and the epacts must accordingly be
all diminished by unity. Thus the epacts 11, 22, 3, 14, &c., become 10, 21,
2, 13, &c. On the other hand, when the time by which the new moons
anticipate the lunar cycle amounts to a whole day, which, as we have seen,
it does in 308 years, the new moons will arrive one day earlier, and the
epacts must consequently be increased by unity. Thus the epacts 11, 22, 3,
14, &c., in consequence of the lunar equation, become 12, 23, 4, 15, &c. In
order to preserve the uniformity of the calendar, the epacts are changed
only at the commencement of a century; the correction of the error of the
lunar cycle is therefore made at the end of 300 years. In the Gregorian
calendar this error is assumed to amount to one day in 312½ years or eight
days in 2500 years, an assumption which requires the line of epacts to be
changed seven times successively at the end of each period of 300 years,
and once at the end of 400 years; and, from the manner in which the epacts
were disposed at the Reformation, it was found most correct to suppose one
of the periods of 2500 years to terminate with the year 1800.
The years in which the solar equation occurs, counting from the
Reformation, are 1700, 1800, 1900, 2100, 2200, 2300, 2500, &c. Those in
which the lunar equation occurs are 1800, 2100, 2400, 2700, 3000, 3300,
3600, 3900, after which, 4300, 4600 and so on. When the solar equation
occurs, the epacts are diminished by unity; when the lunar equation occurs,
the epacts are augmented by unity; and when both equations occur together,
as in 1800, 2100, 2700, &c., they compensate each other, and the epacts are
not changed.
In consequence of the solar and lunar equations, it is evident that the
epact or moon's age at the beginning of the year, must, in the course of
centuries, have all different values from one to thirty inclusive,
corresponding to the days in a full lunar month. Hence, for the
construction of a perpetual calendar, there must be thirty different sets
or lines of epacts. These are exhibited in the subjoined table (Table III.)
called the _Extended Table of Epacts_, which is constructed in the
following manner. The series of golden numbers is written in a line at the
top of the table, and under each golden number is a column of thirty
epacts, arranged in the order of the natural numbers, beginning at the
bottom and proceeding to the top of the column. The first column, under the
golden number 1, contains the epacts, 1, 2, 3, 4, &c., to 30 or 0. The
second column, corresponding to the following year in the lunar cycle, must
have all its epacts augmented by 11; the lowest number, therefore, in the
column is 12, then 13, 14, 15 and so on. The third column corresponding to
the golden number 3, has for its first epact 12 + 11 = 23; and in the same
manner all the nineteen columns of the table are formed. Each of the thirty
lines of epacts is designated by a letter of the alphabet, which serves as
its index or argument. The order of the letters, like that of the numbers,
is from the bottom of the column upwards.
In the tables of the church calendar the epacts are usually printed in
Roman numerals, excepting the last, which is designated by an asterisk (*),
used as an indefinite symbol to denote 30 or 0, and 25, which in the last
eight columns is expressed in Arabic characters, for a reason that will
immediately be explained. In the table here given, this distinction is made
by means of an accent placed over the last figure.
At the Reformation the epacts were given by the line D. The year 1600 was a
leap year; the intercalation accordingly took place as usual, and there was
no interruption in the order of the epacts; the line D was employed till
1700. In that year the omission of the intercalary day rendered it
necessary to diminish the epacts by unity, or to pass to the line C. In
1800 the solar equation again occurred, in consequence of which it was
necessary to descend one line to have the epacts diminished by unity; but
in this year the lunar equation also occurred, the anticipation of the new
moons having amounted to a day; the new moons accordingly happened a day
earlier, which rendered it necessary to take the epacts in the next higher
line. There was, consequently, no alteration; the two equations destroyed
each other. The line of epacts belonging to the present century is
therefore C. In 1900 the solar equation occurs, after which the line is B.
The year 2000 is a leap year, and there is no alteration. In 2100 the
equations again occur together and destroy each other, so that the line B
will serve three centuries, from 1900 to 2200. From that year to 2300 the
line will be A. In this manner the line of epacts belonging to any given
century is easily found, and the method of proceeding is obvious. When the
solar equation occurs alone, the line of epacts is changed to the next
lower in the table; when the lunar equation occurs alone, the line is
changed to the next higher; when both equations occur together, no change
takes place. In order that it may be perceived at once to what centuries
the different lines of epacts respectively belong, they have been placed in
a column on the left hand side of the table on next page.
The use of the epacts is to show the days of the new moons, and
consequently the moon's age on any day of the year. For this purpose they
are placed in the calendar (Table IV.) along with the days of the month and
dominical letters, in a retrograde order, so that the asterisk stands
beside the 1st of January, 29 beside the 2nd, 28 beside the 3rd and so on
to 1, which corresponds to the 30th. After this comes the asterisk, which
corresponds to the 31st of January, then 29, which belongs to the 1st of
February, and so on to the end of the year. The reason of this distribution
is evident. If the last lunation of any year ends, for example, on the 2nd
of December, the new moon falls on the 3rd; and the moon's age on the 31st,
or at the end of the year, is twenty-nine days. The epact of the following
year is therefore twenty-nine. Now that lunation having commenced on the
3rd of December, and consisting of thirty days, will end on the 1st of
January. The 2nd of January is therefore the day [v.04 p.0996] of the new
moon, which is indicated by the epact twenty-nine. In like manner, if the
new moon fell on the 4th of December, the epact of the following year would
be twenty-eight, which, to indicate the day of next new moon, must
correspond to the 3rd of January.
When the epact of the year is known, the days on which the new moons occur
throughout the whole year are shown by Table IV., which is called the
_Gregorian Calendar of Epacts_. For example, the golden number of the year
1832 is ((1832 + 1) / 19)_r = 9, and the epact, as found in Table III., is
twenty-eight. This epact occurs at the 3rd of January, the 2nd of February,
the 3rd of March, the 2nd of April, the 1st of May, &c., and these days are
consequently the days of the ecclesiastical new moons in 1832. The
astronomical new moons generally take place one or two days, sometimes even
three days, earlier than those of the calendar.
There are some artifices employed in the construction of this table, to
which it is necessary to pay attention. The thirty epacts correspond to the
thirty days of a full lunar month; but the lunar months consist of
twenty-nine and thirty days alternately, therefore in six months of the
year the thirty epacts must correspond only to twenty-nine days. For this
reason the epacts twenty-five and twenty-four are placed together, so as to
belong only to one day in the months of February, April, June, August,
September and November, and in the same months another 25', distinguished
by an accent, or by being printed in a different character, is placed
beside 26, and belongs to the same day. The reason for doubling the 25 was
to prevent the new moons from being indicated in the calendar as happening
twice on the same day in the course of the lunar cycle, a thing which
actually cannot take place. For example, if we observe the line B in Table
III., we shall see that it contains both the epacts twenty-four and
twenty-five, so that if these correspond to the same day of the month, two
new moons would be indicated as happening on that day within nineteen
years. Now the three epacts 24, 25, 26, can never occur in the same line;
therefore in those lines in which 24 and 25 occur, the 25 is accented, and
placed in the calendar beside 26. When 25 and 26 occur in the same line of
epacts, the 25 is not accented, and in the calendar stands beside 24. The
lines of epacts in which 24 and 25 both occur, are those which are marked
by one of the eight letters b, e, k, n, r, B, E, N, in all of which 25'
stands in a column corresponding to a golden number higher than 11. There
are also eight lines in which 25 and 26 occur, namely, c, f, l, p, s, C, F,
P. In the other 14 lines, 25 either does not occur at all, or it occurs in
a line in which neither 24 nor 26 is found. From this it appears that if
the golden number of the year exceeds 11, the epact 25, in six months of
the year, must correspond to the same day in the calendar as 26; but if the
golden number does not exceed 11, that epact must correspond to the same
day as 24. Hence the reason for distinguishing 25 and 25'. In using the
calendar, if the epact of the year is 25, and the golden number not above
11, take 25; but if the golden number exceeds 11, take 25'.
Another peculiarity requires explanation. The epact 19' (also distinguished
by an accent or different character) is placed in the same line with 20 at
the 31st of December. It is, however, only used in those years in which the
epact 19 concurs with the golden number 19. When the golden number is 19,
that is to say, in the last year of the lunar cycle, the supplementary
month contains only 29 days. Hence, if in that year the epact should be 19,
a new moon would fall on the 2nd of December, and the lunation would
terminate on the 30th, so that the next new moon would arrive on the 31st.
The epact of the year, therefore, or 19, must stand beside that day,
whereas, according to the regular order, the epact corresponding to the
31st of December is 20; and this is the reason for the distinction.