[Note: This revision, using the 93 flag & 0=Sun, and adding further parameters to cover certain Old Style Julian dates, adds two references to make it evident the original method was created as a user-friendly alternative to Mike Keith's in 1990. Since then it wasn't extended to calculate & show Old Style dates & calendars--until now via a newly defined Doomsdate parameter. Some awkward lines have been reworded. Please post comments, if any, via Usenet or ccnet. Thanks.] HOW TO CALCULATE THE DAY OF THE WEEK FOR JULIAN DATES AS OF 1.iii.IV - via Hans' E-Z Wondrous 2-Fold 3-Step 4-Function Calculator Method - This is a less involved method to calculate the day of the week for dates on Julian calendars from 1.iii.IV and as far into the future as workable using standard four-function calculators and certain mobile phones via a generalised formula requiring fewer key-clicks than any previous method (not relying on tables). Given: Day.Month.Year as a Date on the Julian Calendar (starting with 01.march.0004 as the earliest Date to be converted); & YLJLD as the 'Year in which the Last Julian Leap Day occurred' before the given Date for a Julian Calendar whose years begin on January 1st (starting at IV A.D, so that YLJLD is a positive multiple of 4); or Y2MB as the (historical) Julian 'Year 2 Months Before' the given Date whereby YLJLD/4 = Integer(Y2MB/4); one can convert the given Date into a day of the week after completing the 3 steps below using a standard four-function calculator with less than 3 dozen key-clicks (or at most 3 dozen using RPN & algebraic ones). STEP 1. Calculate the given Date's MonthIndex: Month * 2.56 + 93 (or 94 for certain old-style dates) and then drop the fraction, and the hundreds digit, if any, e.g. 123.72 becomes 23, to get an integer less than 100. STEP 2. Apply the generalised formula for converting the Date: (YLJLD/4 + Year + MonthIndex + Day) / 7 (If one knows YLJLD, one may use that for Y2MB, which in effect is identical to applying Hans' older formula of 1990's vintage.) 2.a. Calculate YLJLD/4 Divide Y2MB by 4. If the result has a decimal portion, re- -enter just the integer portion; 2.b. Continue with the remainder of the formula STEP 3. Apply Hans' keypad mapping: Take the first digit after the decimal point (if none, use 0) and map that to a day using the following patterns: +-----+-----+-----+ +-----+-----+-----+ | Fri | Sat | | | 1 | 2 | 3 | | 7 | 8 | 9 | | Mon | Tue | | +-----+-----+-----+ +-----+-----+-----+ | Wed | Thu | | | 4 | 5 | 6 | | 4 | 5 | 6 | | Wed | Thu | | +-----+-----+-----+ +-----+-----+-----+ | Mon | Tue | | | 7 | 8 | 9 | | 1 | 2 | 3 | | Fri | Sat | | +-----+-----+-----+ +-----+-----+-----+ | Sun | | 0 | | 0 | | Sun | +-----+ +-----+ (This is equivalent to assigning days to remainders of divisions by 7 as for: Sun=0 Mon=1 Tue=2 Wed=3 Thu=4 Fri=5 Sat=6.) The use of a value of IV A.D. for YLJLD in Hans' Method remains valid, even if IV A.D. was not a leap year as claimed by some historians, if it is used just for dates after February IV, whereby a value of IV A.D. for YLJLD is a boundary condition for valid conversions on later (historical) Julian dates. If IV A.D. is a leap year, Hans' Method is historically correct for Julian dates as far back as 1.iii.N by allowing, as do astronomers, a value of 0 for YLJLD as well as Julian year 'N'--the improvised Roman numeral for 0. EXAMPLE N. December 12th, 287 Y2MB = 287 12 * 2.56 + 93 = 123.72 (MonthIndex = 23) 287 / 4 = 71.75 71 + 287 + 23 + 12 Divide by 7 = 56.142857... first decimal = 1 Day of Week = Mon EXAMPLE I. October 13th, 1307 Y2MB = 1307 10 * 2.56 + 93 = 118.6 (MonthIndex = 18) 1307 / 4 = 326.75 326 + 1307 + 18 + 13 Divide by 7 = 237.714285... first decimal = 7 Day of Week = Fri EXAMPLE II. June 15th, 1215 Y2MB = 1215 6 * 2.56 + 93 = 108.36 (MonthIndex = 8) 1215 / 4 = 303.75 303 + 1215 + 8 + 15 Divide by 7 = 220.142857... first decimal = 1 Day of Week = Mon EXAMPLE III. February 8th, 1587 Y2MB = 1586 2 * 2.56 + 93 = 98.12 (MonthIndex = 98) 1586 / 4 = 396.5 396 + 1587 + 98 + 8 Divide by 7 = 298.428571... first decimal = 4 Day of Week = Wed EXAMPLE IV. January 30th, 1649 Y2MB = 1648 2.56 + 93 = 95.56 (MonthIndex = 95) 1648 / 4 = 412 (no need to re-enter) + 1649 + 95 + 30 Divide by 7 = 312.285714... first decimal = 2 Day of Week = Tue MODIFICATION FOR YEARS BEGINNING AFTER 1.i With just one modification, Hans' method is adaptable to Old Style dates too, such that it still is easy to do with a standard calculator, if not mentally. Because Julian & Old Style years overlap, Lag is zero from March 25th till December 31st; afterwards, Lag is +1 till the end of the Old Style year on March 24th for the common English usage of 'Old Style' as below. Including this lag into Step 1 replaces the Julian flag of 93 by 94. Y2MB remains defined in terms of the historical (Julian) calendar. (For the usage of 'Old Style' applicable to Anglo-Saxon times, include a lag of -1 after Step 1 is completed--an exercise left to the Reader.) EXAMPLE V. March 15th, 1751 Y2MB = 1752 3 * 2.56 + 94 = 101.68 (MonthIndex = 1) 1752 / 4 = 438 (no need to re-enter) + 1751 + 1 + 15 Divide by 7 = 315. first decimal = 0 Day of Week = Sun EXAMPLE VI. March 28th, 1752 Y2MB = 1752 3 * 2.56 + 93 = 100.68 (MonthIndex = 0) 1752 / 4 = 438 (no need to re-enter) + 1752 (+ 0) + 28 Divide by 7 = 316.857142... first decimal = 8 Day of Week = Sat EXAMPLE VII. February 8th, 1586 Y2MB = 1586 2 * 2.56 + 94 = 99.12 (MonthIndex = 99) 1586 / 4 = 396.5 396 + 1586 + 99 + 8 Divide by 7 = 298.428571... first decimal = 4 Day of Week = Wed EXAMPLE IIX. January 30th, 1648 Y2MB = 1648 2.56 + 94 = 96.56 (MonthIndex = 96) 1648 / 4 = 412 (no need to re-enter) + 1648 + 96 + 30 Divide by 7 = 312.285714... first decimal = 2 Day of Week = Tue (the day Charles I met his doom) And just when you thought that all's well, there lurks... AN OLD STYLE DOOMSDAY SLIDING CALENDAR a..mJ.o..aJ.sD.jF.mN.a..mJ.o..aJ.sD.jF u..aA.c..pU.eE.uE.aO.u..aA.c..pU.eE.uE g..yN.t..rL.pC.nB.rV.g..yN.t..rL.pC.nB su mo tu we th fr sa ......................1..2..3..4..5..6 .1..2..3..4..5..6..7..8..9.10.11.12.13 .8..9.10.11.12.13.14.15.16.17.18.19.20 15.16.17.18.19.20.21.22.23.24.25.26.27 22.23.24.25.26.27.28.29.30.31 29.30.31 This table will give calendars that are the same as for mar-dec of the historical Year, and for the following jan & feb of the Old Style year. Each Month is displayed vertically in its own column, usually in lower case, or in pairs, in which case the later month will be ALL CAPS. The column of dates underneath are those of all corresponding "Doomsdates"- -defined as all dates for which MonthIndex + Day is divisible by 7. Thus, all Old Style January 30ths are a Doomsdate. "Doomsday" is defined as the weekday on which Doomsdates occur in said twelve-month span. Because MonthIndex + Day is divisible by 7, one may simplify Step 2 by omitting both terms from that calculation, and so find the Doomsday, as in Step A below. This Doomsday is used in Steps B & C to display calendars just by sliding the row of weekdays--by adding or deleting spaces to the left of 'su'. Copy & paste the table into a text-editor with a monospaced font, e.g. Courier, and do as follows: A. Calculate Doomsday: For the given Year in the Old Style calendar, calculate the weekday of its Doomsdates using (YLJLD/4 + Year) / 7, and then applying Step 3's keypad mapping. B. Display the calendar for March & November: Slide the row of weekdays so that the weekday of Doomsday rests over 28 to show the calendar for the March that begins the Year with Lady Day. One has as well the calendar for the following November. C. Display calendars for the other 10 months: for any month other than March or the following November: slide the row of weekdays to put Doomsday under the month to be displayed. D. Display calendars for months in Century 21: if you're patient and can wait until 1.iii.MMC, the foregoing will also work for the remainder of that Century in the Gregorian calendar. (Why?) Q. How would you modify Step A for any earlier Gregorian Century? (There's more than one right answer.) -- REFERENCES: http://oz.ccnet.us/dayofweek/ http://www.merlyn.demon.co.uk/zel-like.htm#Keith http://www.angelfire.com/my/zelime/calendarslide.html http://en.wikipedia.org/wiki/Doomsday_rule#Overview_of_all_Doomsdays