Solution to the Roman Leap Day Error, 45BC-08AD.Docx

Solution to the Roman Leap Day Error, 45BC-08AD.docx

Solution to the Roman Leap Day Error Caesar Augustus corrected the Julian Leap Day Error

1. History&Proposed Solutions to the Roman Leap Day Error For a sound summary of this problem: http://en.wikipedia.org/wiki/Julian_calendar.

Leap year error “Although the new calendar was much simpler than the pre-Julian calendar, the Pontifices ini- tially added a leap day every three years, instead of every four. According to Macrobius, the error was the result of counting inclusively, so that the four-year cycle was considered as including both the first and fourth years; perhaps the earliest recorded example of a fence post error. After 36 years, this resulted in three too many leap days. Augustus remedied this discrep- ancy by restoring the correct frequency. He also skipped three leap days over 12 years in order to realign the year. Once this reform was complete, intercalation resumed in every fourth year and the Roman calendar was the same as the Julian proleptic calendar.[32]

The historic sequence of leap years in this period is not given explicitly by any ancient source, though Scaliger established that the Augustan reform was instituted in 8 BC. Several solutions have been proposed, which are summarised in the following table. The table shows for each solution the implied proleptic Julian date for the first day of Caesar's reformed calendar (Kal. Ian. AUC 709) and the first Julian date in which the Roman calendar date matches the proleptic Julian calendar after the completion of Augustus' reform.

Scholar Date Triennial leap years First Julian First aligned Quadrennial (BC) day day leap year resumes Candidate solutions which may be correct 42, 39, 36, 33, 30, 27, Scaliger[33] 1583 2 Jan. 45 BC 25 Feb. AD 4 AD 8 24, 21, 18, 15, 12, 9 45, 42, 39, 36, 33, 30, Ideler[34] 1825 1 Jan. 45 BC 25 Feb. AD 4 AD 8 27, 24, 21, 18, 15, 12, 9 44, 41, 38, 35, 32, 29, Bennett[35] 2003 31 Dec. 46 BC 25 Feb. 1 BC AD 4 26, 23, 20, 17, 14, 11, 8 Candidate solutions proven to be incorrect 45, 42, 39, 36, 33, 30, Bünting[36] 1590 1 Jan. 45 BC 25 Feb. 1 BC AD 4 27, 24, 21, 18, 15, 12 Christ- 43, 40, 37, 34, 31, 28, 1590 2 Jan. 45 BC 25 Feb. AD 4 AD 8 mann[36][37] 25, 22, 19, 16, 13, 10 after 43, 40, 37, 34, 31, 28, Harriot[36] 1 Jan. 45 BC 25 Feb. 1 BC AD 4 1610 25, 22, 19, 16, 13, 10 43, 40, 37, 34, 31, 28, Kepler[38] 1614 2 Jan. 45 BC 25 Feb. AD 4 AD 8 25, 22, 19, 16, 13, 10 44, 41, 38, 35, 32, 29, Matzat[39] 1883 1 Jan. 45 BC 25 Feb. 1 BC AD 4 26, 23, 20, 17, 14, 11 45, 41, 38, 35, 32, 29, Soltau[40] 1889 2 Jan. 45 BC 25 Feb. AD 4 AD 8 26, 23, 20, 17, 14, 11 45, 42, 39, 36, 33, 30, Radke[41] 1960 1 Jan. 45 BC 25 Feb. 1 BC AD 4 27, 24, 21, 18, 15, 12

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Scaliger's proposal is the most widely accepted solution. It closely matches Macrobius' descrip- tion and results in a calendar year and leap year cycle which exactly matches the proleptic Julian calendar at the time of Caesar's reform, except for his belief that the first reformed year, 45 BC, was not a leap year. Although some scholars, including Mommsen, support Ideler's view that 45 BC was a leap year, Brind'Amour has proved that there was only one bissextile day before 41 BC (in the table above, this rules out the solution proposed by Radke in 1960).[42]

All proposals which end the triennial cycle before 9 BC are provably incorrect (keeping only the candidate solutions proposed by Scaliger, Ideler and most recently by Bennett in the table above). The Asian calendar reform[43] decreed by the proconsul Paullus Fabius Maximus aligned the calendar of the Asian province to the Roman calendar with a New Year falling on Augustus' birthday. It cannot have taken effect any earlier than 9 BC, and the decree states that the first reformed year was a leap year in a triennial cycle.

In 1999, an Egyptian papyrus was published that gives an ephemeris table for 24 BC with both Roman and Egyptian dates.[30] While the Egyptian and lunar synchronisms match the Roman dates on the proleptic Julian calendar, they do not match them on any previously proposed solution for the triennial cycle. One suggested resolution of this problem, which matches the data of the papyrus, is a new triennial sequence, in which the triennial leap years started in 44 BC and ended in 8 BC, with leap years resuming in AD 4 (this is the solution most recently proposed in 2003 by Bennett in the table above).”

NOTES WIKIPEDIA ARTICLE 30 ^ a b A. R. Jones, "Calendrica II: Date Equations from the Reign of Augustus", Zeitschrift fűr Papyrologie und Epigraphik 129 (2000) 159-166.

32 ^ Macrobius, Saturnalia 1.14.13-15 (Latin); Nautical Almanac Offices of the United Kingdom and the United States. (1961). Explanatory Supplement to the Ephemeris London: Her Majesty's Stationery Office. p. 410–1.

33 ^ J. J. Scaliger, De emendatione temporum (Paris, 1583), 159, 238.

34 ^ C. L. Ideler, Handbuch der mathematischen und technischen Chronologie (Berlin, 1825) II 130-131. He argued that Caesar would have enforced the bissextile day by introducing it in his first reformed year. T. E. Momm- sen, Die Römische Chronologie bis auf Caesar (Berlin, 1859) 282-299, provided additional circumstantial argu- ments.

35 ^ C. J. Bennett, "The Early Augustan Calendars in Rome and Egypt", Zeitschrift fűr Papyrologie und Epigraphik 142 (2003) 221-240 and "The Early Augustan Calendars in Rome and Egypt: Addenda et Corrigenda", Zeitschrift fűr Papyrologie und Epigraphik 147 (2004) 165-168; see also Chris Bennett, A.U.C. 730 = 24 B.C. (Egyptian papyrus).

36 ^ a b c For the list of triennial leap years proposed by Bünting, Christmann and Harriot, see Harriot's comparative table reproduced by Simon Cassidy (Fig. 6). The comparative table does not clearly state when the quadriennial leap years were resumed, but these AD years are implied by Augustus decision to not apply the quadriennial leap years for 12 years after his reform.

37 ^ J. Christmann Muhamedis Alfragani arabis chronologica et astronomica elementa (Frankfurt, 1590), 173. His argument proposed that Caesar had intended leap years to be accounted from 46 BC, the year of Caesar's decree, and not 45 BC.

38 ^ J. Kepler, De Vero Anno Quo Æternus Dei Filius Humanan Naturam in Utero Benedictæ Virginis Mariæ Assumpsit (Frankfurt, 1614) Cap. V, repub. in F. Hammer (ed.), Johannes Keplers Gesammelte Werke (Berlin, 1938) V 28.

39 ^ H. Matzat, Römische Chronologie I (Berlin, 1883), 13-18. His argument rested on Dio Cassius 48.33.4,

12 October 2020, 17:39h 2 of 13 Solution to the Roman Leap Day Error, 45BC-08AD.docx which mentions a leap day inserted in 41 BC, "contrary to the (i.e. Caesar's) rule", in order to avoid having a market day on the first day of 40 BC. Dio stated that this leap day was compensated "later". Matzat proposed this was done by omitting a scheduled leap day in 40 BC, rather than by omitting a day from an ordinary year.

40 ^ W. Soltau, Römische Chronologie (Freiburg, 1889) 170-173. He accepted Matzat's phase of the triennial cycle but argued that it was absurd to suppose that Caesar would have made the second Julian year a leap year and that the 36 years had to be accounted from 45 BC.

41 ^ G. Radke, "Die falsche Schaltung nach Caesars Tode", Rheinisches Museum für Philologie, Geschichte und griechische Philosophie 103 (1960) 178-185. He proposed that Augustus initiated the reform when he became pontifex maximus in 12 BC.

42 ^ P. Brind'Amour, Le calendrier romain (Ottawa, 1983), 45-46.

43 ^ OGIS 458 (Greek); U. Laffi, "Le iscrizioni relative all'introduzione nel 9 a.c. del nuovo calendario della provincia d'Asia", Studi Classici e Orientali 16 (1967) 5-99.

END OF QUOTE

2. Proposed Solution to the Roman Leap Day Error From studying the options mentioned in the table above, Bennet35 seems to be right:

Triennial leap years (BC) First Julian day First aligned Quadrennial day leap year resumes 44, 41, 38, 35, 32, 29, 31 Dec. 46BC, or, 25 Feb. 1 BC AD 4 26, 23, 20, 17, 14, 11, 8. 1, 2 Jan. 45BC.

Reasons being: - there was only one bissextile day before 41 BC (Brind'Amour42–see above), and: - the year 24BC was not a Julian leap year (Egyptian Papyrus found in 199930–above), and: - the year 9 or 8BC was a triennial leap year (Asian Calendar reform43–etc.) - Augustus scrapped 3 leap days32, restarting a 4-year leap cycle in 8BC - the first new leap day insertion restarted in 4AD or 8AD33.

The last sequence therefore would have been: no leap day in 8BC, 5BC and 1BC. 8BC the last triennial leap day would have been omitted. Starting the 4-years count with 8BC being year 1 in the new sequence, gives 5BC as first quadrennial leap year: 2nd omitted leap day. That leaves but 1 leap day in surplus between February 5BC and 1BC: 3rd omitted in 1BC.

We know for certain that from AD4 or AD8 the leap years followed every fourth year until our times. Counting back in this scheme gives indeed AD4, 1BC, 5BC, or 1BC, 5BC, 8BC, as the three missing leap years of Caesar Augustus’ reform. Either way this is perfect.

Bennet (2003) solved the Egyptian Papyrus find by restarting the 3-year leap day sequence the year after 24BC, in 23BC: is this correct?

A weak point in this sequence is, that it takes not the first year of Julius Caesar’s Reform 45BC to start the count of the first four years sequence, but the year 708 a.u.c. (46BC), and than puts the first leap day not in the fourth, but in the third year of the sequence:

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46BC: first year in three-year sequence, 44BC: third year in the sequence, the first leap year.1

The number of leap days accumulated in this sequence is 13, not 12 (see table). From Mac- robius we know, at the time of the Augustan calendar reform –in 8BC– there were 3 leap days too many; that’s why Augustus decreed to omit 3 leap days. In Bennett’s sequence, they would have included the year of reform, 8BC. The 4-year sequence should have been:

Quadrennial leap First Julian day First aligned Quadrennial years that should day leap year have been (BC) resumes 44, 40, 36, 32, 28, 31 Dec. 46BC, or, 25 Feb. 1BC AD 4 24, 20, 16, 12, 08. 1/2 Jan. 45BC.

This range occurs when counting back from the Year of Reform 8BC, which must have been done by Augustus’ assistants in this calendar reform, to arrive at the proleptic solution to their calendar problem of too many leap days. It results in the final leap year 8BC, which counts until 5BC, included, that must be 10 leap days. Results in 3 days too many [13–10].

Another weak point in this solution is, that it assumes Augustus, despite recognizing the mistake of a leap day every third year, instead of every fourth year, would have counted till the next triennial year once more, to omit: 8BC: leap year (only in Bennett’s solution), 7BC: first year next sequence; 5BC: third year in this sequence, leap year omission. This, of course, can hardly be true!

Therefore, we must reject Bennett’s solution to the problem. I propose another solution in the next paragraphs.

+ + +

1 The natural alternative is 45BC=1st and 43BC=3rd or 42BC=4th.

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3. Possible Solution to the Roman Leap Day Error On Scaliger’s list, in 9BC, in the triennial-leap-year sequence 12 leap days have been acquired:

42, 39, 36, 33, 30, 27, 24, 21, 18, 15, 12, 09, =12 Julian leap days.

A strong point is: 45BC=year 1 of the new era, 42BC=year 4 of the new era.

For, in that first 4-year sequence, there is no triennial leap year. No inclusive reckoning applies before 42BC. Only after this the triennial counting may have started: 42BC=year 1; 39BC=year 4, inclusive reckoning, the second leap year since Caesar’s reform.

In 9BC, on the basis of the original 4-year sequence, 9 leap days would have been acquired and indeed 3 leap days were in surplus:

42, 38, 34, 30, 26, 22, 18, 14, 10, =9 Julian leap days.

Another strong point of this sequence, ‘apart from its start’, is that it makes perfect sense: Augustus omitted the following leap days: 5BC, 1BC and 4AD. For, Augustus re-instated the need for inserting a leap day after 4-years, not after 3-years, and the first 4th year since the final triennial leap year 9BC is 5BC2, non-inclusive.

Indeed, all other solutions suffer from this problem: they do not connect the final triennial year to the first quadrennial year: - those proposals ending the triennial sequence with 12BC should have restarted with 8BC; - those proposals ending the triennial sequence with 11BC should have restarted with 7BC; - those proposals ending the triennial sequence with 10BC should have restarted with 6BC; - those proposals ending the triennial sequence with 08BC should have restarted with 4BC. The last one was Bennett’s proposal (2003), restarting with 5BC, not 4BC.

All these are plainly wrong: why would Augustus, from his moment in time, not act according to his own instruction to re-instate the correct 4-year leap day immediately? It makes hardly any sense to chose another year but YEAR 4 to correct this matter.

All options but those who end the triennial sequence in 9BC suffer from this inconsistence; for the Julian Calendar certainly has its leap year counting continuously from every 4th year in our own calendar since that time: AD 8, 12, 16, 20…. 1996, 2004, 2008, 2012, 2016 etc.

It is therefore that all proposed solutions in table 1 omit the 3 leap days in surplus in either one of these years: 5BC, 1BC, 4AD (no year 0). For this reason Christmann’s solution (1590) needs to be rejected as well: although his solution is equal to Scaliger, with only 1 year earlier (which makes sense in that it starts with 43=1st leap year, with 45=year 1 in first range), it fails to connect to 5BC as year 4, after the reform started in 8BC (=year 1 in corrected range).

2 With 8BC=year 1, 7BC=year 2, 6BC=year 3, 5BC=year 4.

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In Bennett’s solution, which ends in 8BC, we must assume that Augustus would have continued the triennial sequence one more time (8 to 5BC) which is contradicted by his choice to return to the correct Julian quadrennial sequence (9 to 5BC). Indeed: this is a weak point.

This alone, I say, is reason to reject all other solutions, but those ending the triennial sequence in the year 9BC.

With the final triennial leap day in 9BC, it also becomes more than obvious that the 3 leap days omitted in Augustus’ Reform must have been: 5BC (1st), 1BC (2nd), 4AD (3rd).

Therefore, the first aligned day in accordance with the proleptic (correct) Julian Calendar, is 25 February 4AD, and the first next leap day since 25-02-09BC is 25 February 08AD.

This is the first year in the Roman Calendar since 9BC, to have had 366 days: 9BC / 745 A.U.C. had 366 days; 8AD / 761 A.U.C. had 366 days; all years in between had 365 days.

Christ’s Year of Birth With 5BC being Christ’s possible Year of Birth, this was a year without leap day (365 days). But the Roman Julian Calendar had been influenced in the Triennial Leap Years before, in the period 42BC until 9BC, causing 3 leap days in advance by 8BC.

The first year to omit a leap day was 5BC; so before the possible Birth date of Jesus, still two leap days were in surplus (after 28-02-05BC/from 01-03-05BC onward).

This is the difference between the historical Roman date of Jesus’ birth and the date you will find through astronomy software, containing the proleptic Julian leap day calendar that was as such not in use at Rome at the time of this reform, but was at Alexandria, as described above. The proleptic date will therefore be 2 days off after 1-3-5BC and 3 days off before that date. By 1-3-1BC it will still be off by 1 day.

For example if Jesus was born on 25 Kislew in the Hebrew Calendar, He was born on, - 25 Kislew = 25 December 05BC, in the Alexandrian Julian calendar; - 25 Kislew = 23 December 05BC, in the Roman Julian calendar.3

With 25 Kislew –in the Alexandrian Julian calendar– beginning on 25 December at sunset, one could make a case for a connection of the ‘25th’ for Christ’s Birth in this year in both the Jewish&Roman calendar, which would have been known by the early Christians, making the celebration of Jesus’ Birth on 25 December as such a very old custom, based in history.

If in Judea at this time the Roman calendar was (still) in use, the Roman calendar was still in the process of being reformed. Only from 25-02-08AD the proleptic Julian calendar and the historical sequence of days are back in sync. But if in Judea during Augustus’ Reform indeed the Alexandrian calendar had been in use, Jesus Christ was (possibly) born on 25 Kislew = 25 December 05BC(!).

3 Careful: many calendars use a non-historical Jewish intercalation pattern, counted backward from present: I take early 5BC to have been intercalated with an Adar I, moving dates after that 30 days later: Tebeth>Kislew. How do I come to these dates? see: “Jewish–Julian Calendar in Jesus’ Time v25.xlsx”, year 05BC.

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I found two theoretical alternative birth dates: - 25 Kislew = 28 November 08BC, in the proleptic Julian calendar; - 25 Kislew = 25 November 08BC, in the historical Julian calendar. A possible connection! But in Julian November, not December.4

So, in the year 8BC, there indeed is a connection to be made between 25 Kislew and 25 November: in this option Jesus would have been born e.g. at the evening of 24 November, going into the night of Christmas, while the Feast of Chanukah had started in the Temple at Jerusalem with the First Sabbath of the Feast at sunset 24 November, the feast at 25/11/08.

But 25 Kislew fits other data as well. So: maybe Jesus was born at 23 December, historical Julian date, what may have been recounted through the corrected Julian Calendar at a later moment back to 25 December 5BC (proleptic Julian date), being 25 Kislew. Or the same with 25 November 5BC (proleptic Julian date).

Maybe Alexandria, Egypt, never accepted the ‘triennial mistake’ committed in Rome, and the Alexandrian Calendar is equal to the Proleptic Julian Calendar, and was used in Israel?

Indeed! As will be shown hereafter.

Egyptian Papyrus: 24BC Back to our quest at hand.

So, I come back to the solution Scaliger published in 1583. Only one problem remains: the Egyptian Papyrus, dated in 24BC. This was not a leap year, but in Scaliger’s solution it is (triennial sequence).

I propose this solution:

In Egypt, a correct 4-year sequence was followed since 45BC; they did not make the mistake that was made in Rome, to intercalate every 3rd year instead of every 4th.

This makes sense: Julius Caesar had received the knowledge for re-ordering of the Roman Calendar, including the 4-year leap day sequence, from Alexandria, Egypt; it had been well known in Egypt since the days of Ptolemy III (238BC, ‘Decree of Canopus’).

The Julian method of a quadrennial leap day sequence was borrowed from scientists in Al- exandria: they knew perfectly well how to do this! Certainly they would not follow an incorrect adaptation of the leap year sequence, having been the masters, Rome their student. No! In Egypt a correct leap year sequence was followed all along, resulting in:

45BC: year 1. 42BC: first leap year, followed by: 38, 34, 30, 26, 22, 18, 14 and 10BC, which shows, there was no leap day in 24BC.

This satisfies the Papyrus question addressed by Bennett.5

4 In her famous visions on the Lives of Mary and Jesus, Anne Catharine Emmerick states Jesus was indeed born on a 25th of November, but on 11 Kislew instead of 25 Kislew. The year is not entirely clear from her visions.

5 If true –and I believe it is– it is proven, the famous Alexandrian school not only were leading in the Easter Date debate with Rome since the 325AD Council of Nicaea, whereafter Rome continued to use wrong Pascal Tables, which were finally harmonised in 525AD by Dionysius Exiguus –an Orthodox Abbas visiting Pope John–, but also in this early Julian Leap Day debate, where Alexandria would have continued to correctly intercalate the

Julian leap day every 4 years, instead of the incorrect intercalation every 3 years in Rome.

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But, after 10BC, as decreed by Augustus, who came to calculate now from 9BC, not 10BC, the Egyptians somewhere may have joined the Julian leap year sequence, although corrected, still differing by 1 year:

it came 1 year later than the original one, A. Rome: no leap day in 5BC, 1BC, 4AD; cont. 8, 12, 16, 20AD. B. Egypt: cont. leap years: 10BC, and than changed to 5BC, etc.6

+ + +

I found an article on this same matter, giving detailed answers to questions I here raised: “Calendrica II: Date Equations from the Reign of Augustus”, from the ‘Zeitschrift für Papyrologie und Epigraphik 129 (2000), p.159-166, by Alexander Jones.7

In this scientific article, the same questions are posed and answered with the same solution as above: a. the Egyptians intercalated every 4th year, not 3rd year (not copying the Roman error); b. the date equivalence with the Julian calendar dates in 24BC not only agrees with a non- leap year, but it also aligns perfectly with the correct (proleptic) Julian Calendar as started in 45BC, and not with the historical Roman calendar, that was 2 days off in 24BC; they also do not agree with the unreformed Egyptian calendar of 365 days (without leap day).

As Professor Jones8 writes (page 164):

What I suppose had happened was, that people in Egypt who needed to work with dates in the Roman calendar, during the latter part of Cleopatra’s reign as well as immediately after the beginning of Roman rule, did not depend on bulletins from Rome to regulate the calendar, since they knew the rule according to which the intercalations were supposed to take place.

The Egyptians must at some point have become aware that the Roman dates that they assigned to particular days differed by one or two days from the dates according to the pontifices, but we should not assume that they would have immediately changed their reckoning to conform with the official version of the calendar. The calendar equation Roman July 19 = Egyptian Epeiph 27 discussed by Hagedorn indicates that conformity was imposed by 2 B.C.Article Note 16

In any event, if the Egyptians were operating with a correctly intercalated version of the Roman calendar in 24 B.C., it is scarcely to be believed that they would simultaneously have employed an irregularly intercalated version of the Egyptian calendar.

Article Note 16: Prof. Hagedorn suggests, as an alternative explanation for the presence of both the correctly and incorrectly intercalated Roman calendars in Egypt, that officials in Egypt originally introduced the correct intercalation, while people subsequently coming to Egypt from Italy continued to reckon by the calendar with which they were familiar. END OF QUOTE

This confirms my considerations earlier in this Essay.

6 we do not know if Egypt followed the Augustinian reform immediately (5BC, 1BC, 4AD without omission), or if Egypt continued its own leap years from 10BC (6BC, 2BC, 3AD) and joined Rome in 8AD? But in 2BC they seem to agree (Hagedorn). Therefore, in 2BC Alexandria may have decided to synchronize with Rome, by omitting the annual leap day in 2BC, and inserting it in 1BC, or they did so during an earlier cycle. 7 to be found here: http://www.uni-koeln.de/phil-fak/ifa/zpe/downloads/2000/129pdf/129159.pdf. 8 Professor of the History of the Exact Sciences in Antiquity, Institute for the Study of the Ancient World, New York University. Professor of Mathematics (Associated Faculty), Courant Institute, New York University.

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Julius Caesar copied the leap year method from Egypt

Here follows some historical information on Julius Caesar’s reform. It explains the Year of Confusion, 46BC. It also explains why researchers name February 25 the Julian leap day, not 28 as we do: from: https://en.wikipedia.org/wiki/Julian_calendar

Julian reform “In Egypt, a fixed year of 365 days was in use, drifting by one day against the sun in four years. An unsuccessful attempt to add an extra day every fourth year was made in 238 BC (Decree of Canopus). Caesar probably experienced the solar calendar in that country.

He landed in the Nile delta in October 48 BC and soon became embroiled in the Ptolemaic dynastic war, especially after Cleopatra managed to be "introduced" to him in Alexandria.

Caesar imposed a peace, and a banquet was held to celebrate the event.[8] Lucan depicted Cae- sar talking to a wise man called Acoreus during the feast, stating his intention to create a calendar more perfect than that of Eudoxus[8] (Eudoxus was popularly credited with having de- termined the length of the year to be 365¼ days).[9] But the war soon resumed and Caesar was attacked by the Egyptian army for several months until he achieved victory. He then enjoyed a long cruise on the Nile with Cleopatra before leaving the country in June 47 BC.[10]

Caesar returned to Rome in 46 BC and, according to Plutarch, called in the best philosophers and mathematicians of his time to solve the problem of the calendar.[11] Pliny says that Caesar was aided in his reform by the astronomer Sosigenes of Alexandria[12] who is generally considered the principal designer of the reform. Sosigenes may also have been the author of the astronomical almanac published by Caesar to facilitate the reform.[13] Eventually, it was decided to establish a calendar that would be a combination between the old Roman months, the fixed length of the Egyptian calendar, and the 365¼ days of the Greek astronomy. According to Macrobius, Caesar was assisted in this by a certain Marcus Flavius.[14]

Realignment of the year The first step of the reform was to realign the start of the calendar year (1 January) to the trop- ical year by making 46 BC (708 AUC) 445 days long, compensating for the intercalations which had been missed during Caesar's pontificate. This year had already been extended from 355 to 378 days by the insertion of a regular intercalary month in February. When Caesar decreed the reform, probably shortly after his return from the African campaign in late Quintilis (July), he added 67 more days by inserting two extraordinary intercalary months between No- vember and December.[15]

These months are called Intercalaris Prior and Intercalaris Posterior in letters of Cicero writ- ten at the time; there is no basis for the statement sometimes seen that they were called "Un- decimber" and "Duodecimber".[16] Their individual lengths are unknown, as is the position of the Nones and Ides within them.[17]

Because 46 BC was the last of a series of irregular years, this extra-long year was, and is, re- ferred to as the "last year of confusion". The new calendar began operation after the realignment had been completed, in 45 BC.[18]

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THE ABOVE NOTES IN THIS ARTICLE:

8 Lucan, Pharsalia: Book 10 9 Émile Biémont, Rythmes du temps, astronomie et calendriers, éd. De Boeck (Bruxelles), 2000, p. 224 10 Suetonius, Caesar 52.1 11 Plutarch, Lives of the Noble Grecians and Romans: Caesar 59. 12 Pliny, Natural History: (Book 18, LVII) 13 Encyclopædia Britannica Sosigenes of Alexandria. 14 Macrobius, Saturnalia I.14.2 (Latin).

15 It is not known why he decided that 67 was the correct number of days to add, nor whether he intended to align the calendar to a specific astronomical event such as the winter solstice. Ideler suggested (Handbuch der mathematischen und technischen Chronologie II 123-125) that he intended to align the winter solstice to a supposedly traditional date of 25 December. The number may compensate for three omitted intercalary months (67 = 22+23+22). It also made the distance from 1 March 46 BC, the original New Years Day in the Roman calendar, to 1 January 45 BC 365 days.

16 E.g. "... we have a sidelight on what was involved in "the year of confusion" as it was called. According to Dion Cassius, the historian, there was a governor in Gaul who insisted that, in the lengthened year, two months' extra taxes should be paid! The extra months were called Undecimber and Duodecimber." (P. W. Wilson, The romance of the calendar (New York, 1937), 112). The eponymous dating of the cited passage (Dio Cassius 54.21) shows that it actually refers to an event of 15 BC, not 46 BC.

17 J. Rüpke, The Roman Calendar from Numa to Constantine: Time, History and the Fasti, 117f., suggests, based on the ritual structures of the calendar, that 5 days were added to November and that the two intercalary months each had 31 days, with Nones and Ides on the 7th and 15th.

18 William Smith, Dictionary of Greek and Roman Antiquities: Year of Julius Caesar), following Ideler, inter- prets Macrobius, Saturnalia 1.14.13 (Latin) to mean that Caesar decreed that the first day of the new calendar began with the new moon which fell on the night of 1/2 January 45 BC. (The new moon was on 2 January 45 BC (in the Proleptic Julian calendar) at 00:21 UTC, according to IMCCE (a branch of the Paris Observatory): Phases of the moon (between −4000 and +2500).) However, more recent studies of the manuscripts have shown that the word on which this is based, which was formerly read as lunam, should be read as linam, meaning that Macrobius was simply stating that Caesar published an edict giving the revised calendar — see e.g. p.99 in the translation of Macrobius by P. Davies. Smith gives no source or justification for his other speculation that Caesar originally intended to commence the year precisely with the winter solstice.

+ + +

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I Conclude:

Augustus’ Reform: I. Change of leap years from every 3rd to every 4th year following: 4th after 9BC=5BC. II. Omitting 3 leap days in the leap years following; so, in 5BC, 1BC, 4AD.

The surplus of 3 leap days is omitted as follows: A. Surplus 3 days at 1 January 5BC B. Omitted 1st surplus day at 25 February 5BC (period 5BC to 1BC has +2 days) C. Omitted 2nd surplus day at 25 February 1BC (period 1BC to 4AD has +1 day) D. Omitted 3rd surplus day at 25 February 4AD (period 4AD to 8AD: no surplus).9

CONCLUSION With that, all related synchronisms have been brought into harmony:

1. Julius Caesar’s reformed the Roman calendar in 708AUC=46BC (445 days); 2. The Julian Calendar starts January 45BC / 709 AUC (1st / 2nd day of month); 3. The first leap day was intercalated in year 4 of the first cycle: 25-02-42BC; 4. Only 1 bissextile leap day was inserted before 41BC, indeed: 42BC (Brind'Amour); 5. The year 24BC was not a leap year in Egypt, but it was in Rome (explained above); 6. The year 09BC was a triennial leap year (Asian inscription, must be 9BC); 7. At the year of Augustus Caesar’s reform, 3 leap days had been acquired in surplus: indeed: 45BC to 09BC should have generated 9, but generated 12 Julian leap days; 8. After the last triennial leap day (9BC) Augustus restored a 4-year interval; therefore the first omission is to be expected 4 years after the last triennial leap year; indeed, after 9BC followed 5BC, the first year of the omission; almost all other proposals fail this requirement (see page 5 of this essay); 9. The repetition of Julian leap years must be in harmony with all Julian leap years following the Augustinian reform: indeed, 9BC is followed by 5BC, 1BC, 4AD. From 8AD onward, the Julian leap day is inserted every 4 years, correctly.

Scaliger’s proposal (1583) proofs to be the correct version.

chs.07.05/19.09.14/07.02.16/26.12.18/17.07.19/12.10.20. Easter/Pascha, 21 April 2014,

Marcel van Raaij.

Table: How were the 3 extra Julian leap days at 8BC accumulated?

Quadriennal: BC 45 42 4th: 38 34 30 26 22 18 14 10 08 06 02 AD 03 08 Triennial/3y: BC 45 42 3rd: 39 36 33 30 27 24 21 18 15 12 09 08 05 01 AD 04 08 count JLD’s: +/– 0 ✓ +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 –1 –1 = surplus: = 0 0 1 0 1 0 1 1 2 1 2 1 2 2 3 2 3 2 3 =3 2 1 0 =

45BC: start of Julian Calendar, following reform in 46BC under Julius Caesar. 08BC: start of Reformed Julian Calendar under Augustus Caesar.

9 See: “The Jewish Calendar 13BC-45AD.xlsx” by me for more detail.

12 October 2020, 17:39h 11 of 13 Solution to the Roman Leap Day Error, 45BC-08AD.docx

Calendar reform https://en.wikipedia.org/wiki/Decree_of_Canopus#Calendar_reform

The traditional Egyptian calendar had 365 days: twelve months of thirty days each and an additional five epagomenal days. According to the reform, the five-day "Opening of the Year" ceremonies would include an additional sixth day every fourth year.[4] The reason given was that the rise of Sothis advances to another day in every 4 years, so that attaching the beginning of the year to the heliacal rising of the star Sirius would keep the calendar synchronized with the seasons.

This Ptolemaic calendar reform failed, but was finally officially implemented in Egypt by Augustus in 26/25 BCE, now called the Alexandrian calendar,[5] with a sixth epagomenal day occurring for the first time on 29 August 22 BC.[6] Julius Caesar had earlier implemented a 365¼ day year in Rome in 45 BC as part of the Julian calendar.

Notes 4• Canopic reform 5• Marshall Clagett, Ancient Egyptian Science: A Source Book, Diane 1989, ISBN 0-87169-214-7, p.47 6• Egyptian Civil Calendar

MR: From this late acceptance of the 4 year leap day in Egypt, it would follow that from 29/8/22BC a leap day was inserted every 4 years as follows: 29/8/22, 29/8/18, 29/8/14, 26/8/10, 29/8/6, 29/8/2BC.

So in the Roman Years 22, 18, 14, 10, 6, 2BC. This agrees with the table I presented on the preceding page, except that this quadrenniel range started not in 45BC, but in 22BC. It seems hardly feasible the Alexandrians, although perfectly instructing the court of Julius Caesar, were not able to introduce in due time the correct leap year system they knew so well in their own country, Egypt.

In any case – for our analysis – the establishment from the Egyptian leap year range from 22BC onward, suffices.

CONCLUSION The Egyptian–Alexandrian range of leap years ran as follows: August 22BC – 18BC – 14BC – 10BC – 06BC – 02BC.

In 02BC the Roman and Alexandrian date would have already be in complaince with each other’s calendar (Hagedorn, acc. to Jones: Roman July 19, 2BC = Egyptian Epephi 27, 2BC). This occurs before the Egyptian leap day of 29/8/2BC (19/7). So, from 2BC omward it seems, the Egyptians postponed 29/8/2BC to 29/8/1BC thus conforming with the year that the Romans now intercalated.

Another option is: they moved 29/8/10BC to 29/8/9BC, the year of Augustus’ reform at his request. Or: they moved 29/8/6BC to 29/8/5BC? Anyway, in the last decade before the Christian Era both calendars in Egypt and Rome were conformed with each other.

My guess is, it was not the year 9BC (see page 1, 2 above): too early; it was not 1BC: too late (for there was agreement at the date 29/7/2BC, before the expected Egyptian leap day 29/8/2BC). Therefore it makes perfect sense to assume the year 05BC was chosen as the year to align the Egyptian leap year sequence with the Roman leap year sequence, also since this was the first year of the Augustian Reform where a triennial leap day was omitted (as in 1BC and in 4AD). This makes perfect sense!

Note: Remains one odd thing: why would the Romans have imposed onto the Egyptians to insert a 4 year’s leap day in 22BC, while they themselves were in another range, without inserting the leap day in 22? Indeed, the original line ‘Quadriennal-1’ correctly designated 22BC as Julian leap year; but the Augustian Reform was necessitated for Rome not having correctly intercalated. At this moment in time,

12 October 2020, 17:39h 12 of 13 Solution to the Roman Leap Day Error, 45BC-08AD.docx

Rome had intercalated in 24BC and was intercalating again in 02/21BC. Was it because the start of the Egyptian year was 30/8/22 (1 Toth), while the Roman year started with the 1st of March? MR.11.07.19.

Final Table:

Quadriennal-41 BC 45 4th: 41 37 33 29 25 21 17 13 09 08 05 01 AD 04 08 Quadriennal-42 BC 45 4th: 42 38 34 30 26 22 18 14 10 06 02 AD 03 08 Triennial/3y: BC 45 3rd: 42 39 36 33 30 27 24 21 18 15 12 09 08 05 01 AD 04 08 count JLD’s: +/– 0 ✓ +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 –1 –1 = surplus: = 0 0 1 0 1 0 1 1 2 1 2 1 2 2 3 2 3 2 3 =3 2 1 0 =

Alexandrian: BC 22 4th: 22 18 14 10 06 05 01 AD 04 08 JLD’s Israel? = ? ? ? ? ? ? ? ? ? 22 18 14 10  05 01 AD 04 08

Remarks:

Quadriennal-41 The range that makes sense, but that would have supposed a leap day in 45BC, which did not happen.

Quadriennal-42 The range as it was meant to be by the Roman chronographers. It never happened.

Triennial The true historic range of Julian leap years, as it was executed. Historical.

Alexandrian The supposed historical range of leap years in Egypt, started with 29/08/22BC. Historical. The Alexandrian range:

29/8/22 – 29/8/18 – 29/8/14 – 29/8/10 – 29/8/5BC – 29/8/1BC – 29/8/4AD – 29/8/9AD, etc.

11.07.2019/12.10.2020.

Marcel van Raaij, author/researcher.

12 October 2020, 17:39h 13 of 13