# How to find Easter

The following lecture was delivered to the British Astronomical Association by A M W Downing, who was, at that time, Vice-President of the Royal Astronomical Society.

How to find Easter.

On the page of the "Nautical Almanac" for 1893, which is headed "Principal Articles of the Calendar" will be found the following particulars for the current year, viz., Dominical Letter A and Golden Number 13. I hope to show you in a few words how these "Articles of the Calendar" may be utilised for the purpose of finding Easter.

First, however, it will be necessary to explain briefly the difference between the Julian, or Old Style, and the Gregorian, or New Style Calendars. The mean Julian year consists of 365.25 mean solar days; and as a year suitable for everyday purposes cannot contain fractions of a day, the rule adopted was that three years in succession should consist of 365 days, and that every fourth year should consist of 366 days. Thus the average length of each of the four years was 365.25 days. The year of 366 days is called" bissextile," because the additional, or intercalary day, was inserted after February 24, and every school-girl knows, now-a-days, that in the Roman method of reckoning, February 24 is the sixth day before the Kalends of March. So that in every fourth year there were two "sixth days" before the Kalends of March, and hence the name "bissextile." "Leap-year," the other and more familiar name for the year of intercalation, is so called because, as the Prayer Book of 1604 has it, "On every fourth year the Sunday Letter leapeth," and therefore, as we shall see presently, the day of the week corresponding to any particular day of the month, after the intercalary day, advances two places with reference to its position the preceding year, instead of one place as in ordinary cases. Thus, January 1, 1892, was a Friday; but, since 1892 was a leap-year, January 1, 1893, was a Sunday, instead of being a Saturday, as it would have been had 1892 been a common year.

The Julian Calendar has thus the merit of great simplicity, but the demerit of considerable inaccuracy; and in the middle of the 16th century it was found necessary to effect a reformation. By that time it appeared that the spring equinox, which ought to have happened on March 21, actually occurred on March 11. Luigi Lilio, a native of Calabria, found the error in the Julian year to amount to about three days in 400 years, and his scheme, submitted to Pope Gregory XIII, was that 10 days should be dropped, so as to bring the equinox up to March 21 again, and that a more accurate mean length of the year should be adopted. The Pope referred the matter to a commission, the principal member of which was a German Jesuit, named Schlüssel, better known by his Latinised name of Clavius. It is to him, chiefly, that we owe the Gregorian Calendar in its present form. It was decided, in order to bring up the spring equinox to what was considered the proper date, that the day after October 4, 1582, should be called October 15, and in order to correct for the assumed error in the length of the Julian year, of three days in 400 years, that the centennial years should be counted as leap-years only when the number of centuries is divisible by 4. Thus the years 1700, 1800, and 1900, which in the Julian style are leap-years, are common years in the reformed calendar, whilst the year 2000 is a leap-year in both styles.

The Gregorian Calendar was immediately adopted in Roman Catholic countries, but the old style remained in force in England until 1752. In that year the day after September 2 was called September 14, the accumulation of error in the Julian reckoning having amounted by that time to 11 days. The old style is continued in Russia and Greece up to the present time. It will be seen that the mean length of the Gregorian year is 365.2425 days. The actual length of the tropical year being 365.24222 days, the error in the Gregorian year amounts to about one day in 4000 years. This can be allowed for by making the year 4000 (which according to the reformed calendar is a leap-year) a common year, and every succeeding multiple of 4000 also a common year.

I may remark in passing that it is very unfortunate that Pope Gregory did not adopt the alternative scheme of assigning the spring equinox to March 11, instead or dropping 10 days of the year. Had he done so, all the trouble and confusion arising from the difference of style, which those who have to take account of it know so well, would have been avoided, and many chronological difficulties which now beset us would have been smoothed away. But the idea that the spring equinox had been assigned to March 21 by a Church Council was rooted in men's minds too firmly to be disregarded, and the opportunity of effecting a simple and natural reformation of the calendar was lost for ever. If, in addition, Pope Gregory had restricted the limits of Easter to a fixed week, decreeing, in fact, that Easter Day should be (for example) the last Sunday in March or the first Sunday in April, he would, I think, have greatly strengthened his claim to the gratitude of mankind throughout succeeding generations.

I will now invite your attention to the definition of Easter given in the Book of Common Prayer, which has been handed down to us from the time of the Council of Nice, A.D. 325, and which is designed to preserve, as nearly as possible, the same relation between the times of celebration of Easter and of the Passover as obtained at the time of the Resurrection, and especially that the former should not be celebrated before, or on the same day as, the latter; hence the second clause of the definition. "Easter Day is always the first Sunday after the full moon which happens upon, or next after, the 21st day of March; and if the full moon happens upon a Sunday, Easter Day is the Sunday after." To find Easter, then, we must know what days of the year are Sundays, and on what days of the year the full moons occur. The Dominical Letter gives us the required information with regard to the Sundays, and the Golden Number gives us the required information with regard to the full moons. And here it must be particularly noted that the moon referred to is the Calendar, not the real moon, and that the Calendar moon is considered to be full on the 14th day, i.e., 13 days after new moon; and the Calendar moon is evidently more appropriate for such a purpose than the real moon would be, dependent as the latter is on meridians and local times, circumstances which would cause doubt and confusion in the time of celebration of any festival which depended on them.

The Dominical, or Sunday, letters are the first seven letters of the alphabet attached to the several days of the year; $A$ to January 1, $b$ to January 2 , $c$ to January 3, and so on, over and over again, throughout the year, as is shown in the Prayer Book "Calendar with the Table of Lessons." No letter is attached to February 29, the intercalary day in the English Ecclesiastical and Civil Calendar. To find the Sundays throughout the year (for a common year) it is then only necessary to note what letter is attached to the first Sunday in the year, and every day throughout the year, to which that letter is attached, is a Sunday, and the letter is called the Dominical or Sunday letter for the year. Thus in 1893, January 1 was a Sunday, therefore $A$ is the Sunday Letter, and every day to which the letter $A$ is attached in the Calendar is a Sunday. In leap years the same letter ($d$) is attached to February 29 and to March 1, so that after February 29 the Sunday Letter for the year retrogrades one place. There are thus two Sunday Letters in a leap-year; one from the beginning of the year up to February 29, and the other for the remainder of the year. For example, in 1892 the Sunday Letters were $c b$, As a common year consists of 52 weeks + 1 day, and a leap-year of 52 weeks + 2 days, it is evident that from one common year to the next, the Sunday Letter retrogrades one place, whilst after a leap-year the Sunday Letter retrogrades two places. It appears therefore, that knowing the Sunday Letter for any year, the letter for any other year of the Julian Calendar may be found from an expression of the form $\large\frac{1}{7}\normalsize (N + \large\frac{1}{4}\normalsize N)w + k$, where $N$ is the year, $w$ indicates that the whole number alone resulting from the division $\large\frac{1}{4}\normalsize N$ is to be taken, and $k$ is a constant. Now, everybody knows that January 1, A.D. 1 was a Saturday, and adopting the Prayer Book scale, i.e., that when the remainder, after performing the operations indicated in the above formula is 0, the Sunday Letter is $A$, when remainder is 1, the letter is $g$, when 2, $f$, etc., and, allowing for change of style, we find that, for the period 1800 to 1899, $k = 0$. In 1900, however, a change occurs in the formula, as that year is a common year in the Gregorian Calendar (the formula being, of course, adapted to the Julian style), therefore, we subtract 1 from or add 6 to the previous value of k, but as the year 2000 is a leap-year in both styles, the same value of k is applicable up to the year 2099. The formula for the Sunday Letters for the period 1800-1899 is, therefore $\large\frac{1}{7}\normalsize (N + \large\frac{1}{4}\normalsize N)w$, and for the period 1900-2099, it is $\large\frac{1}{7}\normalsize (N + \large\frac{1}{4}\normalsize N)w + 6$. In leap-years the Sunday Letter so found will be the second letter, the first being the preceding one in the above scale. These formulae agree with the precepts for finding the Sunday Letter in the Prayer Book "Table to find Easter Day."

Now, with regard to the Golden Numbers. These are the numbers attached to each year of a cycle of 19 years, after which the Calendar new moons fall on the same days of the Julian year. Thus, if a new moon falls on January 1 in any year, it will again fall on January 1 after a lapse of 19 years, and to each of these years the same Golden Number would be attached. This cycle is said to have been discovered by Meton, a celebrated Athenian astronomer, about the year B.C. 433, and was called from him the Metonic cycle; and the successive years of the cycle, with the dates of the full moons corresponding to each year, were inscribed in characters of gold upon the walls of the Temple of Minerva. Hence the origin of the name Golden Numbers. In the distribution of the Golden Numbers over the successive years of the Metonic cycle, it was assumed (as was indeed an actual fact), that a new moon fell on January 1 in the third year of the cycle. The year 0 of our era is reckoned the first year of the cycle, therefore, the Golden Number for any year is the remainder obtained from the expression $\large\frac{1}{19}\normalsize (N + 1)$ if there is no remainder the Golden Number is 19. This is in accordance with the precept in the Prayer Book Tables.

With regard to the accuracy of the cycle as a practical means of representing the dates of new moon, it is assumed that 235 Calendar lunations (of 29 or 30 days' duration, combined in a certain proportion) = $6,939\large\frac{3}{4}\normalsize$ days = 19 mean Julian years; whence a mean Calendar lunation = $29\large\frac{1}{2}\normalsize$ days + 44 m 25.5 s. But in adapting the cycle to the Gregorian style we have to take account of the assumed error of the mean Julian year, viz., three days in 400 years, and so (allowing for the centennial years not made bissextile in the new style) we find that the time of Calendar new moon will advance (i.e., fall later), three days in 400 years. Also it must be noted that $6,939\large\frac{3}{4}\normalsize$ days is $1\large\frac{1}{2}\normalsize$ hours greater than 235 mean astronomical lunations, and, therefore (on account of this error in the adopted length of the calendar lunations), the Calendar new moons occur $1\large\frac{1}{2}\normalsize$ hours too late at the end of each cycle of 19 years, or one day too late in 308 years. In the Calendar it is assumed that the error from this cause amounts to eight days in 2,500 years. And the correction necessary to keep the Calendar new moons in fair agreement (i.e., with an error of not more than ± 2 days), with the actual new moons is applied by subtracting one day from the date of calendar new moon whenever the error amounts to this quantity.

If we now examine the Prayer Book Tables (which were drawn up by Bradley, and extend to the year 8500 of our era), we shall see that the plan adopted is to affix the Golden Numbers to the dates of the Paschal full moons, i.e., the full moons which occur either on, or next after March 21 up to April 18 inclusive; so that knowing the Sunday Letter and the Golden Number, the date of Easter is immediately determined. It will be noticed also that the Golden Numbers are affixed to different days at different periods of time, e.g., the first Prayer Book table holds good until the year 1899, and after that a readjustment is required. This readjustment is really the application to the cycle of Golden Numbers of the two corrections referred to above. The first, i.e., that depending on the difference between the Gregorian and Julian year, consists in adding one day to the date of full moon, or shifting the Golden Numbers to a position one day later in each of the years 1700, 1800, 1900, 2100, etc., which are leap-years in the Julian calendar, but are common years in the Gregorian style. The second correction referred to, i.e., that depending on the error in the assumed length of the Calendar lunation, consists in subtracting one day from the dates of full moon, or shifting the Golden Numbers to a position one clay earlier in each of the years 1800, 2100, etc. So that the same system of Golden Numbers holds good from 1700 to 1899, and another system holds good from 1900 to 2199.

Easter Day, then may fall on any one of the 35 days between March 22 and April 25 inclusive, and the Golden Numbers are affixed, in the Prayer Book tables, to the days of Paschal full moons in the different years of the 19-year cycle to which they correspond. The process of finding Easter Day (once the requisite tables are computed), is thus a perfectly simple one. For example, in the current year the Golden Number = remainder of $\large\frac{1}{19}\normalsize (1893 + 1) = 13$. Then looking in the "Table to find Easter Day," now in force, we find that April 1 is the date of the Calendar full moon corresponding to Golden Number 13. Also the Sunday Letter is found from remainder of $\large\frac{1}{7}\normalsize (1893 + \large\frac{1}{4}\normalsize 1893)w$ = remainder of $\large\frac{1}{7}\normalsize (1893 + 473) = 0$. Therefore, the Sunday Letter is $A$. And looking in the table we see that the first date, after April 1, to which $A$ is opposite (i.e., the Sunday after April 1), is April 2, which is Easter day.

This is "How to find Easter" for 1893.

Last Updated June 2021