Astronomical & Gregorian Easter date using Julian Day Numbers

1. Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15 Table-A Calculation Formula for Astronomical Easter Calculation, Gregorian Year JD# Method A B C D E F YR pEpact JD#1Jan JD#JanNM JD#PFM1 JD#Eqnx -4000 24.3 260090.45 260113.8 260187.8 260170.2 2019 6.0 2458484.8 2458489.8 2458563.8 2458563.4 G H I J JD# Easter 260188.8 2458566.8 IF(H=1,G+7,8- INT(H)+G) JD#PFM 260187.8 2458563.8 WkDy 7 5 D#ck 1 1 74.02+D F.3 or F.7 IF(E>=F,E,E+29.5) f.6(G) Yr pEPCT JD#Jan1.3 (B-1)+C f.6(i) The above Table-A is a tabulation of calculation steps for finding Astronomical date of Easter. The example calculations use formula-2 of Table-B to determine moment of January New Moon for selected year, expressed as astronomical Julian Day number, JD#. Alternatively one could use any number of resources to arrive at a value for column B. Just be sure to express the moment as an Astronomical formatted Julian Day Number, JD#. Julian calendar or Gregorian calendar dates may be verified by the day of week. The numbers in column F, JD#.Equinox, are for equinox moment by formula 4 of Table B. The month number and day-of-month may be determined in a spreadsheet by adding two more columns and using formula 8 and 9,Table B, given that Year, Yr, is stated as input in column-A. Alternatively, the Easter JD# may be converted back to Gregorian Date by several free programs. This routine was compared by 70 dates. If using JMT in place of GMT (UT), then add an offset of 0.098 days to step “I” and “J”. A check was made against the WCC Easter dates table for years 2025 to 2001. Against that Table this method using formula 2 and 4 reproduced their result. Other checked instances returned results that matched either the Catholic Easter dates or alternative astronomical calculation results. However the data of Ovidiu Vaduvescu did not confirm the astronomical values set forth in the WCC document nor results of this calculation. If Equinox and PFM dates are closer than one (1) day, verification by a more precise routine is advised on slide-9. NEXT N3 1

2. Astronomical Easter Comparison & Calculation by OP Armstrong 9.5/15 Name (Nu) TABLE-B Excel Astronomical Name Formula: Yr-year; JD#-Julian Day 29.09-MOD(MOD(Yr,19)*11-INT((Yr-1502.57-12*MOD(Yr,19))/228),29.983) pEpact.Cassidy.f1 29.5-MOD(MOD(Yr,19)*11-INT((Yr-1584-12*MOD(Yr,19))/228),30) pEpact.Cassidy.0.f1b IF((1+MOD((365.242454*(-4006-Yr)),29.5306))>=30,((1+MOD((365.242454*(-4006- Yr)),29.5306))-30),(1+MOD((365.242454*(-4006-Yr)),29.5306))) pEpact.Lunation#.f2 JD# Jan1.f3 257898.52-365.242454*(-4006-Yr) (2457102.448+(Yr-2015)*365.2422)+((-0.0005947871)*((Yr-2015)/1000)^4+(- 0.00392591)*((Yr-2015)/1000)^3+(0.013808751)*((Yr- 2015)/1000)^2+(0.1590901)*((Yr-2015)/1000)) JD# Equinox.f4 1stPage March 1stMoon.f5 JD#.Jan1 + pEpact + 59 Day of Week.f6 (1+INT(MOD((1.5+JD#),7))) one is Sunday and 7 is Saturday, etc 257978.00-365.242454*(-4006-Yr) JD#21March.f7 1+INT(MOD(((INT(MOD(((INT(JD#+0.5)+(- 37+INT(0.5+0.75*INT((INT(JD#+0.5)-4479.5)/36524.25))))- 59.25),365.25)))+0.5),30.6)) Day of Month.f8 1+MOD((2+INT(((INT(MOD(((INT(JD#+0.5)+(- 37+INT(0.5+0.75*INT((INT(JD#+0.5)-4479.5)/36524.25))))- 59.25),365.25)))+0.5)/30.6)),12) Month Number.f9 , 3=March, 4=April 2 Next page

3. Astronomical Easter Comparison & Calculation by OP Armstrong 9.5/15 Look at 135 different years between 4000BC and 2038AD. The average lunar conjunction age at 4pm Easter Sunday was about 19. days, with a minimum of 15.5 days and a max of 22.2 days. Without the postponement rule, then some Easter Sundays would land before a lunar age of 13.5 day This gives a very early Lunar Good Friday of just 12.5 days. Application following the Paschal Moon" keeps Good Friday in better alignment with the 14th lunar day. For at the minimum found above, Good Friday falls more closely on the 14thor 15thlunar day. Recapping the "raised up on 3rd day", Good Friday was day one by inclusive counting as Jesus was laid in grave before sunset on the Sabbath. Then day 2 was Friday after sunset to sunset Saturday, and day 3 inclusive count was Saturday after sunset unto about sunrise Sunday. Thus the reason for sunrise Easter service is to memorialize "Christ our Passover", "a feast by an ordinance for ever." To the end that others may wish to abolish this ".- feast by an ordinance for ever", they neglect the blessings of the Almighty and sadly invite the fires of judgment to blot their memory from off the earth! Adjustment to Jerusalem accomplished by adding 2.33 hours or 0.098 days to the “if” statements and day of week calculations. Why the Sunday after the Paschal Full Moon? Take a look at either Astronomical Easters for years 30 and 33AD. For these dates, the proleptic Gregorian Easter dates are April 7 and April 3, respectively. Next determine the Lunar Conjunction (astronomical new moon) age of 17.9 and 17.2 days, respectively, at 4pm Sunday. Given that Jesus died before Friday sunset and was buried before sunset, so 1 day back is, Saturday 4pm and another day back is Friday 4pm. Thus at time of Jesus death, the Lunar days are 15.9 and 15.2 respectively. Given that the visible new crescent or sighted moon is some where about 1 day after conjunction. So on Good Friday the lunar sighted moon of these years is 14 days in age. Thus Good Friday of years 30AD and 33AD would either be on Nissan 14 or 15. However, if this skip week, “after” were not used, then sometimes the Easter would come early to the lunar calendar. Why the Postponement Rule? From this it is seen that reckoning Easter by these methods speaks of "Christ our Passover", Jn18:39, 19:14, 1Cor5:7, Heb11:28, Num9:13, Ex12:14. Since the Catholic Easter concurs the Astronomical Easter in about than 95% of the dates, keeping Good Friday and Easter are keeping "Christ our Passover" as "a memorial; and .... a feast to the LORD throughout your generations; ... feast by an ordinance for ever." Ex12:14. The postponement rule importance is illustrated by calculation. NEXT the proleptic Catholic or of the rule "Sunday time can be 3

4. Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15 Frequency of Variation For 24 years, from 2001 to 2025, only once did the Astronomical Easter differ from the Catholic Easter. That being 2019, with Astro Easter on 24March, and Catholic Easter on 21April. Many of the tabulated dates of the table were evaluated for known difficult dates likely to offer discrepancy between the two methods. Even so, only 11 of 60 years show variance between the methods. So most likely, better than 95% of years will show the Catholic Easter and Lunar Easter are same. "And ye shall count unto you from the morrow after the Sabbath, from the day that ye brought the sheaf of the wave offering; seven Sabbaths shall be complete: Even unto the morrow after the seventh Sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the LORD." Lev 23: 15/16. The Fifty days were numbered from a Saturday, so the 50th day falls always on a Sunday, 49days or 7 weeks after Easter Sunday For Christians, the Easter Sunday sets the precedent to find the Sunday in the 7th month. The earliest Easter being 22 March. By this the 7thlunar month starts around the last week of September. The latest Easter falls on 25 April, by which the 7thlunar month looks to fall on last week of October. The 1stSunday after and a 3rdSunday would be a logical memorial for the Autumn Feasts. Effect of using Jerusalemas Principle Meridian Exercise Caution when the Universal Time new moon moment is on a Saturday night after 21:30 hours. As in 1998, when the Full Moon moment was Saturday at 22:23. The +2h21m offset between GMT and Jerusalem, gives a Paschal full moon on 00:44 Sunday 12 Apr. This astronomical Easter is delayed unto the following Sunday. Compared to a Catholic and/or Universal Time Easter of 12April. This shows the principle difference in the two methods. The Catholic calculation is stepped in days and weeks. The Astronomical method is a moment defined calculation. Thus it depends upon the details of complex astronomical calculations. NEXT Other Feasts: Pentecost, Tabernacles, and The Lord's Supper The other two of the three main feasts were Pentecost and Tabernacles. For Christians, Pentecost (7 weeks after Easter) celebrates the Holy Spirit. The feast of Tabernacles celebrates being freed from bondages of the flesh, the world, and eternal death, thru election by God the Father. By celebrating these 3 feasts, Passover (hope of a more better resurrection thru Jesus Christ), Pentecost (hope of continual renewal by Holy Spirit), and Tabernacles (hope of election and freedom from bondage thru grace of Father God), the Holy Trinity can be honored by Christians. Because Jesus initiated the Holy Communion, then communion on these days should be held in high esteem. The 1st of the first and seventh lunar months are seen as special days or Sabbath's. Also, the middle of these months are feasts of remembrance. Pentecost is by the definition, always on a Sunday by virtue of the week count, 7th Sunday after Easter. 4

5. Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15 stmonth, 7 thMonth Ex23:14, and Pentecost The 3 Mosaic Feasts of 1 Comparisonof ThreeMajor HebrewFeasts The Table at right compares the three feasts set forth by Moses. Two of these feast weeks are routinely celebrated in most churches. The first is Easter Holy week. That starts with Palm Sunday, followed by Good Friday, and lastly, Resurrection Sunday or Easter Sunday. The first of these two Holy Days of Palm Sunday and Good Friday, correspond approximately to the 10thand 15thdays of a lunar month. This provisioned that the 1stday of the lunar month falls upon a Friday, the 8thday being a Friday also, the 10thday is then a Sunday, and the 15thday being a Friday. It is provisioned that Jesus died on a Friday and His empty tomb found early Sunday; thus the term Holy Week. The feast of Pentecost is celebrated in most churches 7 weeks after Easter. These first two Mosaic Feasts being celebrated, begs the question? If two are celebrated, then why not the 3rdand if so, then how so? If one remembers the words of the blessed Savior, “When you Fast” they were not in the permissive sense but in the imperative instance. The 40 days of Lent could be taken as something dealing with fasting. However the Master was noted to have kept the Feast of Tabernacles. So then does it not seem proper for churches to also remember this 3rdFeast? The Sunday following Passover is typically Catholic Easter Sunday. As Easter is the Sunday following the first full moon after Spring Equinox, then the start of the lunar month is about 2 weeks prior. A lunar month is about 28.5 days in length. Six lunar months are then 171 days or 24.4 weeks. Pentecost, Du16:16-17 Reference Nissan, Lev23, Nu28:16 Trumpets -Tishari MonthDay/ date+\- March/April May/June Ex23:14 September/Oct. Sabbath & blow Trumpets 1stof 7th, New Moon & sin offering 1stDay-A Sabbath Sabbath, 1stDay of 1st Month 15thNissan + 49D, Easter + 49 days A Sabbath Lev23:8, Ex12:6 pick passover Lamb, Psalm Sunday Passover: sunset Deut16:6, kill lamb, Ex12:6 unleven bread 7 days, 15this Good Friday & 17thEaster evening end unleven 7thDay A Sabbath 9thevening to 10th evening day of atonement, fasting 10th 14th Tabernacles for 7 days, start with a Sabbath 15th 21st 22nd 23rd offering for 7 day Assembly Taking out the two and a fraction of weeks elapsed unto Easter, then 22 weeks after Easter corresponds to nearly the start of the seventh lunar month. A comparison of dates 22 weeks after Easter to 1 Tishrei shows the two dates agree within a few days. The odd exceptions being when Hebrew calendar postponement rules are applied. One possible church memorial to this time would be Eucharist 22 and 24 weeks after Easter. BACK NEXT 5

6. Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15 Easter by this astro 28-Mar 2465503.59 24-Mar 2458563.57 19-Apr 2450915.43 2-Apr 2439575.64 97 YR's Astro v Catholic Easter Variants 2038AD-1AD Only one application offers astronomical or ‘uniform’ Easter calculation. Kalendis has not been ported to mobile devices. This Excel spreadsheet will find the Astronomical Easter for years between 4007BC and 3027AD. In Table to right, are differences between this calc and others Astro-Easter. For these, exact values of full Moon and Equinox are input. In all evaluated points, this excel sheet was verified by Kalendis to be correct. The below chart shows variations of my Full Moon Date to Kalendis. The normal range is +/- 1 hour for about 7000 years. Beware, not all implementations of mobile Excel have the needed numerical accuracy to evaluate these complex expressions. Only IOS-Numbers and MS-Excel routines could accurately execute these complex calculations. UTC UTC Astro 28Mr 24Mr 19Ap 02Ap 25Mr 25Ap 28Mr 24Ap 16Ap 23Mr 19Ap 03Ap 22Mr 14Ap 18Ap 29Mr 09Ap 04Ap 26Ap me26Mr Cath 25Ap 21Ap 12Ap 26Mr 22Ap 18Ap 25Ap 17Ap 09Ap 30Mr 26Ap 10Ap 29Mr 21Ap 25Ap 05Ap 16Ap 11Ap 19Ap NoCalc YR Full Moon Equinox 2038 2019 1998 1967 1962 1954 1943 1927 1876 1845 1829 1825 1818 1805 1802 1744 1724 1700 550 -61 2465503.03 2458563.42 2450893.33 2439570.82 25-Mar2437744.83 25-Apr 28-Mar 2430805.42 24-Apr 9-Apr 2406353.32 30-Mar 2395014.35 26-Apr 2389167.08 10-Apr 2387719.77 29-Mar 2385151.09 21-Apr 2380425.49 25-Apr 2379333.61 29-Mar 9-Apr 4-Apr 26-Apr 1922022.20 26-Mar 1698859.401 1698859.395 2437744.60 2430805.00 2406333.76 2395011.24 2389167.36 2387706.39 2385149.70 2380401.54 2379305.83 1922022.43 First. Page 6

7. Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15 Excel Formula for Catholic Easter by Julian Day Number for Calendar Converter or direct Excel Cell 30 14 C var .f10 !. Single Input of Gregorian Year, if BC then 1-Yr.BC & find Easter in 7 steps, i.e. 30AD=> P' =MOD(-8-11*MOD(Yr,19)+INT((Yr-1600)/100)-INT((Yr-1600)/400)-INT((8*INT((Yr-1400)/100))/25),30) .f11 P =P'-IF(P'=29,1,IF((1+MOD(Yr,19))>11,IF(P'=28,1,0),0)) 14 .f12 D.1 =118+INT(365.25*(Yr+4712))-INT(0.75*INT(((Yr)/100)+49)) 1732097 .f13 D.2 =D.1+P 1732111 .f6 D.3 =1+INT(MOD((1.5+D.2),7)) 5 Next page .f14 D.4 =IF(D.3=1,D.2+7,8-D.3+D.2)&"Catholic Easter as Julian Day Number" 1732114 end Easter use D.4 to get Day of Month, f.9, &Month#, f.8, &day-of-Week,f.6 as =D&f.n&… D7M4wkd1 Above is Catholic Easter formulation using Gregorian Year as input to find Easter Date as a Julian Day Number. This simplifies several steps as compared to other methods. The resultant JD# may be used to rapidly find Pentecost Sunday or any number of Easter dependant days. The JD# can be rapidly changed to calendar dates via formula f.6, f.8, & f.9 for most, if not all, dates. Beware excel dates do not display for years prior to 1900. Thus the above work around. The Easter JD# can be ported to excel date system for years 1900 forward. Simply determine the offset to Excel date number and JD#, then subtract offset to other dates. Apply this method when a wide range of dates are to be reviewed. Most Easter routines, but not this one, are valid for a few hundred years after 1901. This is substantial since Christ our Passover was slain from the foundation of the world. Thus there has been a perpetual Easter since the day this world was founded. That date was when Adam sinned, 4000 years prior to Jesus Baptism at river Jordan by Saint John. The day of resurrection was hidden by God and revealed by Christ earthly Easter day in 30AD. The following list dates when Astronomical Easter is not same date as the Catholic Easter, about 1 in 12 years. Another estímate of astronomical PFM given at left. This uses Delaunay arguments for lunar and sun anomalies, expressed in Julian century, C. is 257973-365.242454*(-4006-Y)+MOD((365.242454*(-4006-Y)),29.5306)+CF CF= -0.40614*SIN(l')+ 0.01614*SIN(2l’) + 0.17302*SIN(l) - 0.17+ CFt CFt = - C^3/999999.45 - C^2/4028.335 - C/64.259 + 1/ 547.41 7

8. Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15 Variant Astro Easter Years to Catholic 2049 1700 1457 1206 915 543 254 2045 1693 1453 1199 895 536 242 2038 1685 1446 1182 881 519 235 2019 1666 1429 1162 861 516 218 1981 1629 1427 1155 854 509 216 1974 1622 1419 1142 846 496 191 1967 1609 1409 1135 837 489 188 1962 1598 1408 1128 827 482 171 1954 1590 1402 1111 817 474 168 1943 1582 1389 1108 810 469 144 1927 1578 1375 1101 783 438 137 1924 1571 1370 1088 736 411 124 1923 1362 1081 729 Variant Astro Easter Years to Catholic 1903 1558 1351 1074 716 1900 1552 1332 1061 685 1876 1551 1331 1030 634 1873 1527 1328 1003 614 1845 1514 1313 998 1829 1507 1311 979 1825 1503 1308 960 1818 1487 1294 959 1805 1484 1284 941 1802 1483 1277 935 1778 1473 1237 932 1744 1465 1226 922 1724 1463 1218 384 367 343 323 320 313 303 292 289 273 269 1998 262 73 66 42 22 15 -5 -29 -49 -53 -56 607 590 588 587 570 569 563 550 This tabulation covers from years 2050 back to 60BC. A span of about 2100 years. The above Tabulation shows that some centuries have less discrepancy than others. The cycle of 391-19 or 372 years can be seen in some skip sequences. The average is about 11 of 12 years agree and 1 in 12 years are not in agreement between the two Easter Methods. However the pattern is not uniform. Some centuries have fewer concurrences than others. Given the primitive nature of physics and astronomy at the time of the original formulation of the Easter Calendar, the Catholic method mostly agrees with the Astronomical formulation. That being stated, the Paschal Moon of the Catholic Easter is not a true moon. In fact there is similarity to ‘Molad’ of the calculated Hebrew Calendar. The Molad event is timed to start around the autumn equinox vs. around or about the Spring Equinox for Catholic Easter. Next page 8

9. Astronomical Easter Comparison & Calculation by OP Armstrong 10.25/15 Astro Easter Spreadsheet Logic: Find full moon using longitude routine and following logic : Moon age = (Solar less Moon Longitudes)*29.5306/360 Since Full moon is 180 degree, then full moon age is half of 29.530… Three approximations (February, March, April) are used to find Paschal full moon date. When solved yield average days between spring full moons for the year. The first guess when corrected, may not meet Paschal Criteria after correction. Thus 2 checks are used. Given the Equinox for a year, as EQ.jd, then the logics are: 1) IF(guess < EQ.jd, guess, then new is old guess+29.5) Guess-1 is 257973 -365.242454 *(-4006-Yr) +MOD(( 365.242454*(-4006-Yr)),29.5306) from which guess, an age is determied. This AGE is converted to full moon correction, CF, for an improved guess as follows: Cfi=29.5306/2 - IF(AGE>0, AGE, 29.5306+AGE) Guess2 = Guess1 + CF1, other iterations are made as Guess3 = Guess2 +CF2, the 4thand final value, Guess4 = Guess3 + CF3 is taken as the Full Moon date. A final check is used to ensure the Full Moon date falls after the Spring Equinox. Macro loops are avoided by using successive Full Moon dates: estimate, estimate +/- 29.5 days and logically select PFM. The day, 257973, is March FM of Gregorian year -4006. The file is embedded on page 6 of the MS document file. This file may be used to better see the method. A simplified method screened the prior table for validation by 70 term method. The Paschal Full Moon is defined as first moon, after the Spring or Vernal Equinox. This spreadsheet method of longitudes was corrected and adapted after T. Alonso Albi’s adaptation of P.Duffet but with additional terms. T. Albi’s original lambda terms were only about 24. Accuracy was improved by using more Lambda terms. The total Sine terms comes to about 70. Also evaluate the Delaunay Chapront solution ELP-S2001 was used to improve accuracy. The ELP-S2001 4thorder terms are reported to be accurate to within 10 seconds down to 1500BC if using the full series of 100’s of adjustments. Here only about 70 terms were employed. Thus an Equation of time is employed to extend the solution. The extended solution calculates Paschal Full moon dates to within an hour of NASA or Kalendis for years between 4000BC to 3000AD. Typically less than 60 minutes difference is found. The graph on page 6 shows the variation for 110 points over a 7100 year span. Some incorrectly apply angle reduction Mod360, without care, in cases produces backward results. The spreadsheet solution is as follows: reduce angle by A/360=rA, then find IF(ABS(rA)>1, 360*(ABS(rA)-INT(ABS(rA)))*SIGN(rA), 360*rA). This yields a correctly reduced angle for either positive or negative input value. matches by limiting the year range, to say 1 century. to & arguments, Chapront of negative value. This It is possible to get closer 9 Next page