1. Introductory
The principle of this Rule is due to Gauss, whose proof of it is given in Zach’s Monatliche Correspondenz, August 1800, Vol. II, pp. 221–230, whence it is quoted by Mr. W. W. Rouse Ball, in his Mathematical Recreations, published by Macmillan and Co. The only original feature, in my version of the Rule, consists in its greater simplicity. By my method the result can be calculated mentally, without much difficulty, in about half-a-minute: by the method of Gauss, it would certainly require a much longer time, and also much greater powers of mental calculation.
Before learning the Rule itself, the Reader should make himself expert in certain necessary Arithmetical Processes, as here specified.
2. Some Necessary Arithmetical Processes
(1) To add 15 to a given No.
Go up in two steps—10 and 5.
[Thus, if the given No. be 187, say “197, 202.”]
(2) To find the Remainder that would be left if a given No. were divided by 4
Divide the last 2 digits.
(3) To find the Remainder that would be left if a given No. were divided by 7
Name the successive dividends. This is all the mental soliloquy required. The remainder in each dividend (which of course serves as the tens-digit of the next dividend) is easily seen by mere inspection.
[Thus, if the given No. be 4325, say “43; 12; 55; 6.”]
It will be well to cast out sevens, whenever that can conveniently be done.
[Hence, if a dividend be a multiple of 7, say “goes out”, and ignore it.
Thus, if the given No. be 4225, say “42 goes out; 25; 4.” If it be 4769, say “47; 56 goes out; 9; 2.”]
(4) To find the Remainder that would be left if a given No. were divided by 19
If the given No. be not greater than 30, the Remainder is easily seen by mere inspection.
If it be greater than 30, take enough of its digits to make a number greater than “1.” If this number be even, halve it, and add the next digit; if odd, take its lesser half, and add the next digit, with a “1” prefixed. Imagine the result to be substituted for the digits so used, and proceed as before.
[Thus, if the given No. be 88, say “4 and 8 is 12.”
If it be 98, say ”4 and 18 is 22; 3.”
If it be 147, say “7 and 7 is 14.”
If it be 157, say ”7 and 17 is 24; 5.”
If it be 687, say “3 and 8 is 11; 5 and 17 is 22; 3.”]
Cast out nineteens, whenever you can.
[Thus, if the given No. be 1992, ignore the first 2 digits, and say “4 and 12 is 16.”
If it be 5749, say “2 and 17 is 19, which goes out; 2 and 9 is 11.”
If it be 998, say “4 and 19 gives 4; 2 and 8 is 10.”
If it be 7994, say “3 and 19 gives 3; 1 and 19 gives 1; 14.”]
If you have to add 18, or 17, &c, call it “19 minus 1,” or “19 minus 2,” &c, and ignore the “19.”
[Thus, if given No. be 789, say “3 and minus 1 is 2; 1 and 9 is 10.”
If it be 967, say “4 and minus 3 is 1; 17.”]
But this method must not be used if the number, to which the 18, &c, is to be added, is less than the number which would be deducted,
[Thus, if the given No. be 567, do not say “2 minus 3,” but say “2 and 16 is 18; 9 and 7 is 16.”]
(5) To multiply a given No., of 2 digits, whose sum is no greater than 9, by 11
Put the sum of the digits between them.
(6) To find the Defect of a given No. from the lowest multiple of 30 which contains it
The given No. must be either (α) a multiple of 30, or (β) not more than 10 off the lowest multiple of 30 which contains it, or (γ) more than 10 off it.
In case (α), or (β), the Defect may be seen by mere inspection.
[Thus, if the given No. be 180, say “Defect is 0.” If it be 208, say “Defect is 2.”]
In case (γ), take the Excess of the given No. above the next lower Multiple of 30, and deduct from 30.
[Thus, if the given No. be 189, say “9 above; Defect is 21.” If it be 192, say “12 above; Defect is 18.”]
3. Rule for Finding Easter-Day for Any Given Date till a.d. 2499
The phrase “4-Rem,” used with reference to a certain No., means “the Remainder that would be left it the No. were divided by 4;” and similarly for the phrases “7-Rem” and “19-Rem.”
Three Nos. are required, two of which are known, by memory, as soon as the Date is named; the third has to be calculated. Let us call them a, h, k.
The Rule may conveniently be divided into three parts, as follows:—
(1) Name the given Date, and then recall, by memory, the values of a and h belonging to it. For Old Style the values are always 15 and 6. For New Style, they are given in the following Table:—
| No. of hundreds in Date | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| Value of a | 8 | 8 | 7 | 7 | 6 | 6 | 6 | 5 | 4 | 5 |
| ” h | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 0 | 1 | 1 |
Picture this in your mind’s eye, and say over the Nos. of hundreds, till you reach the given one; then name the values of a and h in each column.
[Thus if the given Date be “a.d. 1582” (which, for our present purpose is O.S., as N.S. did not begin till October), say “1582; Old Style; a and h are 15 and 6.”
If it be “a.d. 1583, N.S.,” say “1583; New Style; 15; a and h are 8 and 2.”
If it be “a.d. 1583, O.S.,” say “1583; Old Style; a and h are 15 and 6.”
If it be “a.d. 1948, N.S.,” say “1948; New Style; 15, 16, 17, 18, 19; a and h are 6 and 5.”]
(2) Again name the given Date, and find its 4-Rem and its 7-Rem: then take “4-Rem plus twice 7-Rem”; double; add h; the 7-Rem of this result is k.
[Thus, if the given Date be “a.d. 1582,” say “1582; 4-Rem; 82; 2; 7-Rem; 15, 18, 42; 0; 2 and 0 is 2; 4; and 6 is 10; k is 3.”
If it be “a.d. 1583, N.S.,” say “1583; 4-Rem; 83; 3; 7-Rem; 15, 18, 43; 1; 3 and 2 is 5; 10; and 2 is 12; k is 5.”
If it be “a.d. 1583, O.S.,” say “1583; 4-Rem; 83; 3; 7-Rem; 15, 18, 43; 1; 3 and 2 is 5; 10; and 6 is 16; k is 2.”
If it be “a.d. 1948, N.S.,” say “1948; 4-Rem; 48; 0; 7-Rem; 19, 54, 58; 2; 0 and 4 is 4; 8; and 5 is 13; k is 6.”
If it be “a.d. 1948, O.S.,” say “1948; 4-Rem; 48; 0; 7-Rem; 19, 54, 58; 2; 0 and 4 is 4; 8; and 6 is 14; k is 0.”]
(3) Name a and k; name given Date; find its 19-Rem; multiply by 11; add a; find Defect of this from lowest multiple of 30 which contains it; find highest multiple of 7 contained in Defect, and add k. If this result falls short of Defect, then either deduct 2 and call it “April,” or (if this cannot be done) add 29 and call it “March.” If the result does not fall short of Defect, then either deduct 9 and call it “April,” or (if this cannot be done) add 22 and call it “March.”
[Thus, if the given Date be “a.d. 1582,” and if a and k be known to be 15 and 3, say “a and k are 15 and 3; 1582; 7 and minus 1 is 6; 3 and 2 is 5; 55; and 15 is 65, 70; 10 above; Defect is 20; 14 and 3 is 17, which falls short; deduct 2; April 15.”
If it be “a.d. 1583, N.S.,” and if a and k be known to be 8 and 5, say “a and k are 8 and 5; 1583; 7 and minus 1 is 6; 3 and 3 is 6; 66; and 8 is 74; 14 above; Defect is 16; 14 and 5 is 19, which does not fall short; deduct 9; April 10.”
If it be “a.d. 1583, O.S.,” and if a and k be known to be 15 and 2, say “a and k are 15 and 2; 1583; 7 and minus 1 is 6; 3 and 3 is 6; 66; and 15 is 76, 81; Defect is 9; 7 and 2 is 9, which does not fall short; add 22; March 31.”
If it be “a.d. 1948, N.S.,” and if a and k be known to be 6 and 6, say “a and k are 6 and 6; 1948; 19 goes out, 2 and 8 is 10; 110; and 6 is 116; Defect is 4; 0 and 6 is 6, which does not fall short; add 22; March 28.”
If it be “a.d. 1948, O.S.,” and if a and k be known to be 15 and 0, say “a and k are 15 and 0; 1948; 19 goes out, 2 and 8 is 10; 110; and 15 is 120, 125; 5 above; Defect is 25; 21 and 0 is 21, which falls short; deduct 2; April 19.”]
4. Aids to Memory, &c
(1) To remember the given Date, while working the Rule
If you are handy at making Memoria Technica words, you will find that a very useful method. Otherwise, you had better write it down, as it is certainly a trial, for the temper, to find, after carefully working out Easter-Day, that the Date, for which you have calculated it, has vanished from your memory!
[The Dates, worked out as examples in the next Section, may conveniently be remembered by the following words:—
853 “kilt”; 1654 “box-leaf”; 1881 “choke boy.”]
(2) To remember the -Table
The first six columns are the most useful. In them, note that the values for a are “2 eights, 2 sevens, 2 sixes,” and that, in the first 3 columns, a and h add up to 10, and in the other 3 to 11.
In the last 4 columns, the values for a and h are given in the third and fourth lines of the following Memoria Technica stanza:—
List my song to!
’Tis as wrong to
Save a flea
As rob a bee.
The values for the present Century, viz. 7 and 4, had better be fixed firmly in the memory as a separate item.
(3) To remember the value of k till it is wanted
This can be done with one hand. For “0” keep the hand open: for “1,” double in the 1st finger, and put the thumb on it; similarly for “2,” “3,” and “4;” for “5” double in the thumb, and put the fingers on it; for “6,” clench the fist, with the thumb outside.
(4)
If it should happen, in working the second part of the Rule, that “4-Rem plus twice 7-Rem” is a multiple of 7, this would make k equal to h; so go on at once to the third part.
[Thus, if the given Date be “a.d. 1731, N.S.,” say “15, 16, 17, a and h are 7 and 3; 4-Rem; 31; 3; 7-Rem; 17, 33, 51; 2; 3 and 4 is 7; a and k are 7 and 3; 1731; 8 and minus 6 is 2, 1 and 1 is 2; 22; and 7 is 29; Defect is 1; 0 and 3 is 3, which does not fall short; add 22; March 25.”]
(5)
There is one single Date (and only one, so far as I know) for which this Rule fails. In the year a.d. 1954, New Style, Easter-Day will fall on the 18th of April: the Rule gives it as the 25th. I cannot in the least account for this very curious anomaly.
5. Examples worked as Specimens
(1) a.d. 853
“Old Style; a and h are 15 and 6; 4-Rem; 53; 1; 7-Rem; 85, 13; 6; 1 and 12 is 13; 26; and 6 is 32; k is 4.
a and k are 15 and 4; 853; 4 and 5 is 9; 4 and 13 is 17; 187; and 15 is 197, 202; Defect is 8; 7 and 4 is 11, which does not fall short; deduct 9; April 2.”
(2) a.d. 1654. [N.S.]
“15, 16; a and h are 8 and 2; 4-Rem; 54; 2; 7-Rem; 16, 25, 44; 2; 2 and 4 is 6; 12; and 2 is 14; k is 0.
a and k are 8 and 0; 1654; 8 and 5 is 13; 6 and minus 5 is 1; 11; and 8 is 19; Defect is 11; 7 and 0 is 7, which falls short; deduct 2; April 5.”
(3) a.d. 1654. [O.S.]
“Old Style; a and h are 15 and 6; 4-Rem; 54; 2; 7-Rem; 16, 25, 44; 2; 2 and 4 is 6; 12; and 6 is 18; k is 4.
a and k are 15 and 4; 1654; 8 and 5 is 13; 6 and minus 5 is 1; 11; and 15 is 26; Defect is 4; 0 and 4 is 4, which does not fall short; add 22; March 26.”
(4) a.d. 1881. [N.S.]
“a and h are 7 and 4; 4-Rem; 81; 1; 7-Rem; 18, 48, 61; 5; 1 and 10 is 11; 22; and 4 is 26; k is 5.
a and k are 7 and 5; 1881; 9 and 8 is 17; 8 and 11 is 19; 0; and 7 is 7; Defect is 23; 21 and 5 is 26, which does not fall short; deduct 9; April 17.”
(5) a.d. 1881. [O.S.]
“Old Style; a and h are 15 and 6; 4-Rem; 81; 1; 7-Rem; 18, 48, 61; 5; 1 and 10 is 11; 22; and 6 is 28; k is 0.
a and k are 15 and 0; 1881; 9 and 8 is 17; 8 and 11 is 19; 0; and 15 is 15; Defect is 15; 14 and 0 is 14, which falls short; deduct 2; April 12.”
6. Examples for Practice
For any Reader who does not possess “The Book of Almanacs” by Professor De Morgan, here are 100 miscellaneous Dates, which he can work as examples: the answers are given in the next Section.