Computistics versus Phantom Time Thesis

Does the Computus Paschalis lead the phantom time thesis ad absurdum?

Karl-Heinz Lewin

At the Zeitensprünge annual meeting in Zurich (05.-08.05.2005) Ulrich Voigt astonished, after a detailed presentation of the late antique computistics and their calculations of the Easter cycles, the present guests with the statement that the agreement of the calculation method of Dionysius Exiguus (calculated in the era "ab incarnatione Domini", "a. i.D") with the calculation method used since the Middle Ages (calculated in the era "after Christ's birth") allows either only 0 or 532 inserted phantom years. Since 532 phantom years are indisputable (the year 1000 would correspond then to the year 468), there could have been no phantom time. [Voigt 2005a; see also Voigt 2005b in the magazine ZS 17 (2)]

A truly hard chunk for Zeitensprünge friends: If Voigt's assertion holds, this would force to have to fill the centuries laboriously purged in the last 16 years of fictitious Karls, fictitious Tassilos, doubled Alphonses, fictitious Vikings, Madjars doubled to late Avars etc. again with archaeologically not verifiable history!

In two in-depth conversations with Ulrich Voigt, I learned detailed reasons for his statement:

• In the Julian calendar, the same date (day and month) falls on the same day of the week every 28 years.

• The lunar cycle, which is the basis of the Easter calculation, makes the same calculated lunar phase (thus in particular the spring full moon) fall on the same (Julian) date every 19 years.¹

• The calculated spring full moon² falls regularly on the same day of the week every 19×28 = 532 years, therefore Easter Sunday falls regularly on the same (Julian) date every 532 years.³

• The calculation of Dionysius Exiguus does not only exist on perhaps deceptive parchment, but is carved in stone in the Museo Arcivescovile in Ravenna. The stone slab (hereafter “calendar stone”) with the 95-year Easter cycle attributed to him is dated to the 6^th century.

Therefore the year 532 a.i.D. (Dionysius Exiguus) == 532 CE.⁴

Detailed evidence and critical assessment

In the following, the author retraces Voigt's argumentation and elaborates it in such detail that possible conclusions become obvious.

Let us assume that when King Otto III and Pope Sylvester introduced the modern calendar, N phantom years had been inserted. Then the following problems arise:

• If N is not divisible by 28, then either the days of the week or the dates of the new calendar do not match those of the old calendar.

• If N results in the remainder 11 or 17 when divided by 28 (e.g. 291 = 10×28 + 11 or 297 = 10×28 + 17), then the dates agree in three of four years each, but in one of four years there are discrepancies by one day. In the first case (remainder 11) the leap year in the "new" calendar comes one year too late (or three years too early), in the second case (remainder 17) one year too early (or three years too late).⁵

• If N is not divisible by 19, then the calculated moon phases do not match, so especially not the calculated spring full moon (297÷19 gives 15 remainder 12).

• So if N is not divisible by 532, then the Easter dates calculated according to the “old” calendar and the dates calculated according to the “new” calendar do not match continuously.

For example, an insertion of 291 or 297 years would result in the following discrepancies in the weekdays of March 21 (for the calculation see [Voigt 2003, 19-35]):⁶

The deviations by one weekday shown in red bold italics would repeat every four years. It is also possible to subtract or add 28 from the year numbers chosen as an example in each column without changing anything in the other data.

Here, it does not matter whether the years of the “old counting” were counted in the specified counting or in another era.

The lunar cycle now plays a role insofar as it is decisive for the earliest possible Easter date. Its influence is, so to speak, somewhat leveled, however, by the weekdays to be waited up to the following Sunday in each case. With an insertion of 297 years, however, the dates of the “old” calendar do not fit in any case with those of the “new” one:

Furthermore, 8.4.1000 is a Monday.

Let’s assume that the exact date of the Easter moon doesn’t matter at all, that its calculated date may well be one to three days earlier or later, as long as the Easter date calculated from it matches in both calendars. Perhaps one must work for the “new” calendar only with new tables, in order to receive the “correct” Easter date? Unfortunately, this hope cannot be confirmed. The best correspondences with the years important for the new year counting provide the following year numbers:

However, there are still the years starting from 467, whose dates correspond continuously with those of the years starting from 999 ...

This proof is based on assumptions as presuppositions, which I will discuss in the following. I classify them in advance under the headings “Self-evident prerequisites” and “Assumptions worthy of discussion”. Under the former I subsume those assumptions which seem to me personally unquestionable, but which I nevertheless want to name explicitly in order to leave them open to criticism. The second contains assumptions for which it is questioned whether they may be taken for granted as facts.

Fig.1: Copper engraving of the calendar stone in Henricus Noris’
Dissertatio de latinorum paschali cyclo 1691, taken from [Voigt 2003, 114].⁷

Self-evident prerequisites

1. When the modern calendar was introduced, the days of the week ran unchanged: A Monday was followed by a Tuesday, etc.

2. The months and their respective durations were already firmly established when the modern calendar was introduced: January 1 in the “old” calendar was January 1 in the “new” calendar, etc. up to and including December 31.

3. The introduction of the new yearly counting obviously did not lead to renewed controversies (“Easter controversy”) about the date of Easter.

Assumptions worthy of discussion

4. Leap years were the same in both eras.

• However, I wonder how leap years can be determined without a continuous year count. On the other hand, if the popes did not want to join the new year count until the 13^th century, the new count had to be leap year compatible with the papal year count (or that of their computists).

5. The calendar stone in Ravenna was made in the 6^th century (first half).

• That is for the art historians to judge.

6. The calendar stone really says what is claimed: an Easter cycle that coincides with the later Easter calculation.

• It would be at least conceivable that the calendar stone contains systematic errors, like for example every 4 years a date of Easter deviating by one day. In his writing “libellus de cyclo magno paschae” Dionysius Exiguus has “erred” at least twice: He dated “the incarnation of Jesus (VIII cl. Aprl. = 25. Mar. 1) on a Sunday and the birth of Jesus (VIII cl. Ian. = 25. Dec. 1) on a Tuesday”, but “according to the rules of the Julian calendar the 25. Mar. 1 was a Friday and the 25. Dec. 1 a Sunday”. [Voigt 2003, 143].⁸

Therefore, I will now turn to the examination of the calendar stone (see Fig.1 and 2).

Fig.2: Calendar stone (“Calendario Liturgico”) in the Museo Arcivescovile at Ravenna
[Photo found on the Internet on 19.11.2017, source can no longer be found in 2023]

Reading of the calendar stone

The reading was taken sector and line by line from the copper engraving in Dissertatio de latinorum paschali cyclo [Noris, cited after Voigt 2003], for lack of a complete illustration of the original (Fig. 1), a comparison with the section of the calendar stone illustrated in [Voigt 2003, 113], which shows at least three sectors completely and two further sectors approximately halfway, did not show any deviations. I was able to verify the data personally by eye in Ravenna in 2006.

The reading begins with the Sector marked with a small cross as Sector 1, which is also referred to in the original by the large cross in the center.

Striking is the use of the numeral sign ↅ (Unicode character U+2185, late Roman six, probably originating from the Greek letter and numeral sign digamma, stigma or episemon) instead of “VI”, “Xↅ” instead of “XVI”, “XↅI” instead of “XVII” etc. (however only predominantly, not consistently, as the “XVIII” in Sector 1, line 5 shows). Pridie, the day before, is written once as PR (Sector 6, line 3), otherwise as PD (cf. Sector 7, right next to it).

Fig.3: Calendar stone, Detail from Sector 12 [Photo KHL 2006]

Fig.4: Calendar stone, Detail from Sector 10 [Photo KHL 2006]

Fig.5: Calendar stone, Detail from Sector 16 [Photo KHL 2006]

Anticipating the interpretation, it should be noted that the entry in Sector 12, Cycle 3 (cf. Fig. 3), highlighted above with a yellow background, is incorrect. According to the Julian calendar it must read either “ↅII ID APR LU Xↅ” (6.4. Moon 16) or “ↅI ID APR LU XↅI” (7.4. Moon 17). In Sector 10, Cycle 5 (highlighted with cyan above) the “I” of “LU XXI” (3.4. Moon 21) is only faintly visible on the stone – apparently a correction (cf. Fig. 4) – and is probably therefore missing in the engraving. Since these are not systematic errors and the Easter dates will prove to be consistent, I assume that in the first case it is a “printing error” of the stonemason in indicating the lunar day (for the meaning see below), in the second case a “reading error” of the engraver, just as the spelling “CII” instead of the meant “ↅII” in line 7 of Sector 16, whereby the hook of the “ↅ” on the stone (Fig. 5) is only recognizable as a very short tick.

Interpretation of the calendar stone

The following information can be found in each of the 19 sectors:

Heading (sector 9 only): The meaning of “DIVISIO CYCLI II” at this point is unclear to me.

Line 1: Lunar year of the 19-year lunar cycle. The cycle begins with the lunar year 17; this corresponds to a Byzantine counting[Voigt, personal communication], but the number has no meaning for the further interpretation.

Line 2: Year number of the first 19-year cycle in the 95-year cycle and date of the Easter full moon (Luna 14). The year number is at the same time the “golden number” (= year number mod 19 + 1). The “Year 1” (AN I) represents therefore the Julian year in the used era modulo 532 + 1, so it can mean the year 532 of the “Era of Christ”, as Noris writes (see fig. 1), or the year 0 (!), that would mean the year 1 before the beginning of the era (1 B.C.), or also the years 1064, 1596 ...

The Easter full moon is the calculated spring full moon.⁹ Its date repeats every 19 years, so it is valid for the whole sector.

Line 3: Easter date and lunar day of Easter (in the first 19-year cycle). The lunar day is the consecutive day number in the lunar month.

Lines 4 and 5: Easter date and lunar day of Easter in the 2^nd 19-year cycle.

Lines 6 to 8: Easter date and lunar day of Easter in the 3^rd 19-year cycle.

Lines 9 to 11: Easter date and lunar day of Easter in the 4^th 19-year cycle.

Lines 12 to 14: Easter date and lunar day of Easter in the 5^th 19-year cycle.

Line 15: Indication whether the lunar year in question is a leap year (“EB” for “annus embolismalis”, a year with 13 lunar months.) or a normal year (“CM” for “anno communis”, a year with 12 lunar months). (This has nothing to do with the Julian leap years – solar years).

Conversion to modern dates

To convert the dates, I use the following small table for the relevant time period so that I don’t need to know all the intricacies of the calendar calculation:

This gives us the following table, whose rows correspond to the sectors of the calendar stone:

The indication of the lunar leap year was taken from the last line of the original as “(L)” into the second column.

For each of the five cycles the Easter date and the lunar day are listed. The entry with two values for the lunar days (17/16 in year 12 of the 3^rd cycle) shows as first value the incorrect value taken over from the calendar stone and after the slash the correct one (cf. fig. 3 and the assessment above).

Verification of the data

“According to tradition, the 3^rd, 6^th, 8^th, 11^th, 14^th, 17^th and 19^th years of the lunar cycle are leap years and have 13 lunar months each.” [Metz, transl. DeepL]. The aforementioned (Alexandrian) tradition thus counted the lunar years in the same way as the Dionysian AN(no) count and not according to the Byzantine LU(na) count.

The date of the Easter Moon is simply calculated from the Golden Number (=year in column 2):

OM = 21 + GN; if OM <= 31, then OM is the date in March, otherwise OM − 31 is the date in April.

The calculation of the Easter date is a little more complicated. [Voigt 2003] gives detailed instructions on how to calculate the date of Easter in one’s head (!). However, I am a mathematician and therefore avoid calculating as much as possible. Moreover, as a software developer, I prefer to leave the calculating to the machines and have therefore transferred Voigt’s formulas into a spreadsheet programme, which thereby confirmed all Easter dates in the above table.

To make sure that this is the correct calculation, I ran two more spreadsheets with the algorithms described by [Knuth] and the German [wikipedia] (search “Osterdatum”), with consistent results: The Easter dates of the calendar stone are correct and agree with our calendar exactly when the 1^st year on this stone (AN I) is the year 0 or 532 or another multiple of 532.

The algorithm from German Wikipedia (as of 2004-06-24 and 2005-12-20) was the shortest and is best suited for both programmable calculators and spreadsheet programmes. Here is the variant for the spreadsheet (with correction from 2024-02-28) :

If you want to check it yourself, you should copy line 2 into lines 3 to 97 and enter the number 531 in A1; then all the Easter dates from the above table will appear in columns E and F.

Conclusion

Voigt's claim cannot be refuted on the basis of the dates on the calendar stone. If this stone was indeed made in the 6^th century, then the year 1 (AN I) depicted on it is firstly a year in the 6^th century and secondly equal to the year 532 of our calendar.

It would only be different if it could be proven that this stone dates from the 11^th century. (Then AN I = 1064.) Alas, this cannot be my task.

The proof, however, remains a mathematical one. Historically, it is a strong indication against a phantom time. On the other hand, there are numerous historical indications for a phantom period, as they have been shown in this journal ("Zeitensprünge") and other publications by the authors writing here. After studying Trier, the hometown of my school days, I myself can prove that neither building nor burial (and therefore no living) took place in Trier during the phantom period, and I will report on this.

But for the time being, it remains a mystery how times with no evidence can be reconciled with the mathematically correct assumption of a continuous yearly count since antiquity.

Footnotes

Literature

The author is a mathematician and worked as a software developer.

Karl-Heinz Lewin, Haar: Karl-Heinz.Lewin@t-online.de

Copyright © Karl-Heinz Lewin, 2005, 2023

First published in German in: Zeitensprünge 17 (2) 455-464

Copyright © Mantis Verlag Dr. Heribert Illig, 2005

Translated from an updated German version by DeepL with authoritative assistance by the author.