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 in- cluding 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 prolep- tic 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 12 October 2020, 17:39h 1 of 13 Solution to the Roman Leap Day Error, 45BC-08AD.docx 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.