Millennium Mistake
© 2003 Jan Zuidhoek 2009
www.millenniummistake.net
1 introduction
On 1-1-1801 the Italian astronomer Giuseppe Piazzi
discovered the planetoid Ceres. At the time that day was generally considered
by scientists as the first day of the first year of the nineteenth century,.
Precisely one century later, on
When people hear someone assert that the year 2000 was the last year of the previous millennium then they often react by saying something like: “oh no, the year 2000 was the first year of the new millennium, because the year zero was the first year of our era”. At first sight it possibly looks as if the logic of such a reaction leaves little to be desired, for a millennium is by definition a period of exactly one thousand years. But what is meant by “the year zero”? To be able to answer that question, and with this the tricky question when exactly the third millennium began, we have to find out which is exactly the structure of our era (the term ‘era’ here of course in the meaning of a linear system of numbered calendar years). For that purpose we will enter the field of chronology, which, as the science of locating historical events in time, is part of the discipline of history. Chronology is the backbone of history.
After having taken note of the history of the coming into existence of our era (in Section 2 and Section 5) we will establish that there is no year zero in our era (in Section 5) and explore why there is no year zero in our era (in Section 6); after having thus established which is the connection between the moment zero (i.e. the beginning moment) of our era and the millennium question the solution to this question (see Section 8), as well as the justification of the term ‘millennium mistake’ (see Section 10), is there for the taking. Not surprisingly it is just those sections which together stand for the original core of this website. Clarifying remarks in reply to the standpoint with regard to the millennium question taken in this website and sceptical reactions to it led to reformulation of passages or were included among the deductions of Section 7 or incorporated into the objections of Section 9.
Besides the millennium question still some other subjects of chronology (but not being of vital importance for the solution of the millennium question) are treated in this essay, e.g. in Section 3 calendars, in Section 4 Paschal tables, in Section 11 Anni Domini. By making an inquiry into Anni Domini a table can be obtained which us provides not only with the most proable two possible dates of Jesus’ dying day (see Section 11), but also with a clear indication that originally (around the year 320) there was generally a relatively large difference between the date of Alexandrian Paschal full moon (see Section 4) and the date of the fourteenth day of Nisan (see Section 3), which difference does not agree with the original formula “Paschal full moon = 14 Nisan” and will be examined more closely in Section 12. After the establishment that Jesus as good as certain was crucified a few hours before celebration of Pesach (see Section 3) began (see Section 13), an investigation into the way in which the sequence of dates of Alexandrian Paschal full moon might be constructed (see Section 14), an analysis of the meaning of two interesting remarks made by Beda Venerabilis (see Section 4) around het jaar 720 with regard to the famous total solar eclipse which was observed in the year 664 in Britain and Ireland (see Section 15) and a consideration devoted to possible coincidences of the date of the fourteenth day of Nisan with the date of Paschal Sunday in the fourth century (see Section 16) this essay is ended with an epilogue in which the reasons are summarized why this website exists (see Section 17) and a concise profile of the author (see Section 18).
This website is provided with an index and a concise bibliography.
2 incomplete era
Our era is the complete Christian era (see also Section 5), nowadays in combination with the Gregorian calendar (see also Section 3) the most widespread dating system on earth. The founder of that era is the learned monk Dionysius Exiguus, who, originating from a region in or near the Danube delta area, settled in Rome around the year 500. In the year 525 he finished his Pachal table (see Table 1), which forms a continuation of the Paschal table which is attributed to bishop Cyril of Alexandria (in Egypt) but probably was composed by a computist in his entourage (around het jaar 440). An important detail of Dionysius Exiguus’ Paschal table (see also Section 4) is that the calendar years according to the Roman calendar (see also Section 3) herein (see column A) are not numbered according to the era of the emperor Diocletianus, as still was the case in the Paschal table attributed to Cyril, but according to his new era, which was intended to have begun with Jesus’ incarnation.
Now the dating of Jesus’ birth is an impossible task, even for modern historians (see also Section 11). So it is not so surprising that Dionysius Exiguus was not able to that either. Be that as it may, he chose indirectly (via the era of the emperor Diocletianus) the Roman calendar year 754, i.e. the year 754 of the Anno Urbis Conditae (literaly ‘in the Year of the Foundation of the City’) era, as the starting year of his new era (see also Section 11). Then he took the successive Roman calendar years from and including that starting year and numbered them 1, 2, 3, ……. With the duration of a year as unit of time, the incomplete Christian era thus obtained, better known as Anno Domini (literaly ‘in the Year of the Lord’) era, boils down to our first timeline (Figure 1):
time * year 1 1 year 2 2 year 3 3 …… (time in years)
in which (modern) picture the moment * = the moment zero (i.e. the beginning moment) of our era, i.e. the midnight point in time at which the first day of our era began, and year 1 = the year 1 (of our era) = the Roman calendar year 754 and e.g. year 10 (this calendar year began at moment 9 and ended at moment 10) = the year 10 (of our era) = the Roman calendar year 763. The first day of our era is not the day of Jesus’ birth, but simply 1-1-1.
About a moment zero or about a year zero Dionysius Exiguus, who used no other numerals than Roman ones in his Paschal table and in his calculations, never worried. Though he understood very well that dividing (which in this case boiled down to repeated subtraction, for in his time in Europe division algorithms were not available yet) a positive integer by e.g. 19 sometimes produces no remainder, the number zero, being an (extremely important) mathematical concept, was not known to him. That is the reason why in our first timeline (see Figure 1) the place of the moment zero of our era has been marked by means of an asterisk (*).
Zero is a name of our tenth digit as well as of the unique number 0 with the property that x + 0 = x for any number x. The digit zero in our decimal positional system as well as the number zero is usually indicated with the symbol 0. For centuries before the invention of the number zero (see also Section 5) precursors of the number zero were used (e.g. in Egypt and in Mesopotamia), i.e. words or symbols which represented literally ‘nothing’ or an empty spot in a positional system and were not considered by their users as (abstract) numbers with which abstract calculations could be carried out actively.
Why must the digit 0 be considered (historically seen) as our tenth digit? Counting precedes calculating, personally as well as (pre)historically. From time immemorial one counts by means of the cardinals one, two, three, …… (in words, and without zero). In order to create a complete decimal positional system we need nine symbols for the first nine positive integers (e.g. the digits 1, 2, 3, 4, 5, 6, 7, 8, 9) and next a tenth symbol (e.g. the digit 0) to make it possible to compose a symbol (e.g. the symbol 10) for the tenth positive integer. And thus it has gone. Gerbert, the French mathematician who became pope Sylvester II in the year 999, knew of the first nine digits belonging to our decimal positional system, but it is certain that he did not know the real significance of the digit 0. It is the digit 0 which has enabled us to construct our decimal positional system. As inventing the number zero did not precede the discovery of the positive integers, inventing the digit 0 did not precede the formation of symbols for the first nine positive integers.
The number zero is a relatively modern concept, which could jell only after one had got sufficient experience with the use of its precursors. The last phase of that development was the phase in which one became definitely familiar with carrying out abstract calculations with all ten digits (including the digit zero) in the decimal positional system (this explains that the invention of the number zero happened so long after the discovery of the positive integers). In early medieval Europe one used no other numerals than Roman ones and one had to make do, just like in ancient Rome, with abacuslike aids and simple calculations in which neither a numeral zero nor the number zero was used. In that Europe nobody was acquainted with a numeral zero or the number zero. Neverless, the presence of the Latin word “nulla” (meaning ‘nothing’) in the third column of his Paschal table (see Table 1) creates the impression that Dionysius Exiguus did know the number zero. However, by analyzing the text accompanying his Paschal table, we can convince ourselves of the fact that that impression is false (see also Section 4).
Of course ‘the year 1’ means simply ‘the first calendar year of our era’, as ‘the king William I’ means nothing else than ‘the first king named William’. Numbering of tickets begins at 1, for the counting of any kind of things (contrary to for the measuring of the length of whatever) we do not need the number zero at all. So the counting of years is not different from the counting of any other kind of things (even though sometimes for a moment one could think that the counting of months actually ought to start from the number 0 instead of the number 1, for javascript planners thought to do science a service by assigning in their system the number 0 instead of the number 1 to the first month of the year). Someone born on 1-1-1 will (assuming that the day on which he was born was not looked upon as his first birthday) have celebrated his ninth birthday probably (as usual) on the day he completed the ninth year of his life, so probably on 1-1-10, his tenth birthday probably on the day he completed the tenth year of his life, so probably on 1-1-11.
Not relevant to the solution of the millennium question but illustrative for the fact that it is absolutely not obvious indeed that in case of the introduction of a new era one begins with a year zero, is the example of the French revolutionary era. When on 22-9-1792 French revolutionaries proclaimed the first French republic they also resolved to begin a new era on this special day; this day was regarded as the first day of the first month of the year 1 of their new era. They also had no need for a year zero, although in France as early as in the course of the eighteenth century the number zero had been generally accepted (see also Section 5). Apart from that it is interesting to remark that the introduction of the era of the French revolution, unlike the introduction of the Anno Domini era, was accompanied by a drastic reform of the calendar. Each calendar year of the French revolutionary era began close by the September equinox and consisted of twelve months of thirty days each and five or six single days. The French revolutionary era was in use only until 1-1-1806.
In Roman antiquity calendar years were frequently counted from any supposed year of foundation of the city of Rome. However, in reality the Anno Urbis Conditae era, like the Anno Domini era, did not exist yet in antiquity, for it was used systematically for the first time not before the beginning of the fifth century, namely, though in a rather careless way, by the Iberian historian Orosius (AUC 1 = the Roman calendaryear 1). Though probably Dionysius Exiguus was acquainted with (but never used) the Anno Urbis Conditae era, pope Boniface IV (around the year 600) seems to have been the first who recognized the connection between those two important eras (i.e. AD 1 = AUC 754). However, the complete Christian era (see also Section 5) came actively into use as a coherent system for dating historical and current events only about the year 730, thanks to the great English scholar Beda Venerabilis. Only in the tenth century our era was used for the first time for the dating of a papal document (namely in the year 967), and only about the year 1060 the church of Rome put this era definitely into use. Although drastically adapted by pope Gregory XIII in the year 1582 (see also Section 7), our era was never definitively replaced with another.
The next section which is directly related to the millennium question is Section 5.
3 calendars
Besides the millennium question still some other subjects of chronology are treated in this essay, e.g. in this section calendars and in Section 4 Paschal tables; however, for the solution of the millennium question these subjects are not essential. The next section which is directly related to the millennium question is Section 5.
Relatively shortly before he was assassinated, Julius Caesar had modernized the then not yet adequate Roman calendar, at which he had decreed not only that the Roman calendar year, to begin with the Roman calendar year being tantamount to the year 709 of the Anno Urbis Conditae era, henceforth should begin on 1 January and once every four years there should be a leap year, but also that this regulation was considered to apply (retrospectively) to the Roman calendar years then gone by. However, in the first half century of its existence the leap year regulation according to the Julian calendar thus obtained did not function well (see also Section 7). For that reason the emperor Augustus made (around the beginning of our era) a regulation according to which henceforth every fourth calendar year after the Roman calendar year being tantamount to the year 757 of the Anno Urbis Conditae era should be a leap year; this regulation was tantamount to the rule that henceforth any calendar year of the Anno Domini era after the year 4 should be a leap year if and only if its number is divisible by 4 (see also Section 11). Only in the sixteenth century the Roman calendar was adjusted once again, namely by pope Gregory XIII in the year 1582, which resulted in the (nowadays mondially used) Gregorian calendar provided with the current leap year regulation (see also Section 7). Although the Julian calendar was no ideal calendar, it functioned precisely and unbrokenly from 1-3-4 up to and including 4-10- (see also Section 7). Not surprsingly the dates given in Dionysius Exiguus’ Easter table are Julian calendar dates.
In the first four centuries of our era besides the Julian calendar still another solar calendar was in general use in the Roman empire, namely the Alexandrian calendar, which just like the Julian calendar was equipped with a leap year proportion of one to four (each Alexandrian calendar year consisted of twelve months of thirty days each and five or six single days). Though those two calendars were mutually convertible, the conversion of dates from one calendar to the other was no sinecure. Contrary to those two calendars, the Egyptian calendar (the calendar without leap year regulation of which the Alexandrian calendar was an improved version) was only used for agricultural and practical astromical purposes. It is a matter of course that with respect to historical events after the year 1582 we normally make use of Gregorian calendar dates and with respect to historical events before the year 1582 we normally make use of Julian calendar dates.
Unlike the Julian and the Alexandrian calendar, the Jewish calender is a lunar calendar, in which each new month begins more or less simultaneously with an (actual) Newmoon (i.e. point in time of conjunction of sun and moon). But since its coming into being, far before the beginning of our era, until the moment (about the year 360) at which a beginning was made with its definitive fixation the beginning of the new Jewish calendar month and of the new Jewish calendar year depended not only on astronomical but indirectly also on local circumstances (among other things meteorological circumstances under which in Palestine the first appearance of the moon crescent after Newmoon was searched for). In fact the Jewish calendar can be related to the Julian calendar just insofar as only time after the moment at which the Jewish calendar had been completely fixed (in the beginning of the ninth century) is taken into consideration. Every Jewish calender year consisted then (and still consists now) either of twelve (mostly) or of thirteen calendar months of 29 or 30 days each. In that time Nisan was the first, Iyyar the second and Adar the twelfth month of the Jewish calendar year and Pesach, i.e. Passover, i.e. the Jewish Paschal feast (which lasted seven days), was always prepared in the morning and afternoon of the fourteenth day of Nisan. In that time Pesach began always with the sunset of the fourteenth day of Nisan and the meal in which the Paschal lambs slaughtered in the afternoon of this day were eaten usually with the rise of the full moon usually roughly an hour after this sunset.
Since the coming into being of the Jewish calendar until the moment at which it had been completely fixed, the beginning of each new Jewish calendar month was determined in Palestine at a special moment (about half an hour after sunset) at the beginning of the thirtieth night after the sunset with by which the first day of the expiring Jewish calender month had begun. If at such a special moment the first appearance of the moon crescent after Newmoon was confirmed by the Jewish authorities in Palestine (this happened roughly once every two months) then this meant that the first day of the new Jewish calender month had just begun with the sunset having happened about half an hour ago; otherwise the first day of the new Jewish calender month began at the moment of the then next sunset (hence all Jewish calender months thus defined consisted of 29 or 30 days). Because usually a waxing moon is visible with the naked eye not earlier than 24 hours after Newmoon, at the time the first day of a new Jewish calender month usually began with the second sunset in Jerusalem after Newmoon. For the same reason at the time the (actual) Fullmoon (i.e. point in time of opposition of sun and moon) of a Jewish calender month fell on average roughly near the midnight point in time between the thirteenth and the fourteenth day of this Jewish calendar month.
Since the coming into being of the Jewish calendar until the moment at which it had been completely fixed, in Palestine at set times not only a decision had to be taken with respect to the point in time at which a new month of the Jewish calendar had to begin (once a month) but also a one concerning the beginning of a new year of this calendar (once a year). In that time the Jewish authorities in Palestine possessed the competence to interfere once a year, at the end of Adar, with the current Jewish calendar year (they did this about once every three years) by extending this Jewish calendar year by an extra month consisting of thirty days. In that time the Jewish authorities in Palestine could (by applying that competence carefully) not only prevent that the Jewish calendar year would become on average too short or too long but also that Pesach would be celebrated too early (i.e. entirely or partially still in winter) or too late. As a matter of fact, at the time the principle that Pesach should be celebrated as early as possible in spring was the only not opportunistic criterion they used as part of the exercise of that competence. At the time they must have been familiar with the growing of the days in winter and the phenomenon of the March equinox, which marks the beginning of spring on the northern hemisphere of the earth, although they then (being familiar neither with the Julian nor with the Alexandrian calendar) were not yet acquainted with any, either alleged or correct, date of the March equinox.
After the destruction of Jerusalem in the year 135, too, there were always Jewish communities in Palestine. In the course of the first millennium their number fluctuated greatly, around the third turn of the century their total size numbered probably no more than ten percent of their total size in the first half of the first century. At a particular moment (about the year 360) a beginning was made with the definitive fixation of the jewish calendar. Between the years 135 and 360 the beginning of the new month and the beginning of the new year of the Jewish calendar were determined officially still in Palestine and in principle still in the same way as formerly, however not in Jerusalem.
Unlike the Jewish communities in Palestine, the Jewish community in third century Alexandria (Egypt) felt compelled (because it wanted to celebrate its festivities as much as possible at the same time as the Jewish communities in Palestine) to make use of a lunar calendar scheme adjusted to the Alexandrian calendar by means of which the Alexandrian calendar dates of forthcoming Jewish festivities could be determined independently of the Jewish authorities in Palestine (usually accurate to a day). That (unfortunately unknown) lunar calendar scheme, constructed by Alexandrian Jewish calculators in the beginning of and perfected in the course of the third century, supposedly with the help of tables of lunar phases calculated by themselves or by Alexandrian astronomers, was a system according to which successive time intervals each with a total duration of 19 Alexandrian calendar years were subdivided always as much as possible in the same way into 235 Alexandrian lunations (each consisting of 29 or 30 days) as precisely as possible consonant with Jewish calendar months. The possibility thereto rests on the fact known in Mesopotamia as early as in the fifth century before Christ and rediscovered by the Greek astronomer Meton that time intervals consisting of 19 solar calendar years contain on average nearly as many days as a time interval consisting of 235 synodic months (both about 6940 days), which is a result of the astronomical fact that the synodic period of the moon is on average approximately 29.53059 days (this implies that the moon is taking about 6939.689 days to pass through all of its phases 235 times). Although the Julian calendar was no ideal calendar, it functioned unbrokenly regularly and precisely from 4 to 1582 (see also Section 7). All that time every century lasted 36525 days; so a time interval of 19 calendar years lasted an average of 6939.75 days then.
Although neither the dates of the fourteenth day of Nisan of third century Palestine nor the dates of the fourteenth day of the Alexandrian lunation of Nisan (this lunation did not always precisely coincide with Nisan) are exactly calculable, any of these dates can be estimated separately, usually accurate to a day, with the help of lunar phase tables of Newmoon and the (rather rough) rule that at the time the first day of Nisan began usually at the moment of the second sunset in Jerusalem after the Newmoon of Nisan. Not surprisingly, it is in a similar way that the Alexandrian lunations being part of the lunar calendar scheme which was used by the Jewish community in third century Alexandria must have been obtained. By the way, that lunar calendar scheme can certainly have been a source of inspiration for the Alexandrian computists (i.e. practioners of computus, i.e. calculating dates of Easter) who around the middle of the third century, on behalf of their Paschal tables (see also Section 4), began to experiment with sequences of dates with a period of 19 years. Those (Christian) computists as well as the (Jewish) producers of that lunar calendar scheme must then have been, probably independently of each other, on the lookout for a suitable and preferably as regular as possible sequence of dates of Paschal full moon with a period of 19 years standing in for a not exactly calculable sequence of dates of the fourteenth day of Nisan. In any of both cases (in order to be able to succeed in that endeavour) one must have been aware of the phenomenon of the March equinox (because Pesach should in principle be celebrated as early as possible in spring) and that endeavour must have led around the year 260 to the construction of a sequence of dates of Paschal full moon with a metonical structure. A metonically structured sequence of dates is by definition a sequence of dates with a period of 19 years such that every following date of the sequence can be obtained by subtracting from the last preceding date either 11 days modulo 30 days (normally) or 12 days modulo 30 days (only in the case of the so called saltus lunae occurring once every nineteen times); this definition rests on the congruence 18 · 11 + 1 · 12 ≡ 0 modulo 30. It is not so difficult to check that the sequence of Julian calendar dates of column F of Table 1 has a metonical structure with 21 March as earliest possible date and a saltus lunae at the transition from 550 to 551.
The (unfortunately unknown) lunar calendar scheme the Jewish community in third century Alexandria used, must have been as early as before the middle of the third century of such a quality that its sequence of dates of the fourteenth day of the Alexandrian lunation of Nisan must have had something of a metonical structure. The discovery, made in Alexandria by Jewish calculators and, independently of them, by Christian computists (see also Section 3), that it is possible (thanks to the astronomical fact that time intervals consisting of 19 solar calendar years contain on average nearly as many days as a time interval consisting of 235 synodic months) to approximate sequences of successive dates of the fourteenth day of Nisan by means of metonically structured sequences of substituting dates would later turn out to be the key to the solution of the great problem of the calculation of the date of Paschal Sunday.
Around the year 90 the (real) March equinox fell on 22 March, around the year 220 on 21 March, around the year 350 on 20 March, around the year 1500 on 11 March. However, for centuries, until the year 381, the date 25 March was considered by the church of Rome to be the date of the March equinox. The Jewish authorities in Palestine must have been then, because of the decision they had to make then once a year with regard to the beginning of the new year of the Jewish calendar, intuitively more familiar with the phenomenon of the March equinox than the Roman civil authorities. According to the Alexandrian astronomer of Greek descent Ptolemy the March equinox in his time (around the year 140) fell on 22 March. As a consequence, in the second half of the third century that date was considered by the church of Alexandria to be the date of the March equinox. Around the year 270 the Alexandrian scholar Anatolius, who was bishop of Laodicea from around the year 270 until his death around the year 290, made an attempt to reconcile the discrepant viewpoints of the churches of Rome and Alexandria with respect to the date of the March equinox by conceiving the moment of the March equinox not as a point in time or as a date but as a time interval consisting of four consecutive dates (22 up to and including 25 March). Shortly after the third turn of century the church of Alexandria definitely decided to consider the date 21 March so familiar to us (at the time and nowadays once again in fact usually the date of the first day after the date of the March equinox) as the date of the March equinox. The church of Rome took that step only in the year 381.
We will show by means of a table in which way Alexandrian Jewish calculators, making use of dates of Newmoon (probably calculated by themselves or by Alexandrian astronomers), could have obtained their metonically structured sequence of dates of the fourteenth day of the Alexandrian lunation of Nisan. That (unfortunately unknown) metonically structured sequence of (for convenience called) dates of Jewish Paschal full moon must have been constructed by Alexandrian Jewish calculators around the year 260 on the basis of dates of Newmoon insofar as belonging to any obvious substantial time interval I round the middle of the third century, e.g. the time interval consisting of the time between the years 220 and 260. As a matter of fact an adequate reconstruction of the sequence of dates of Jewish Paschal full moon will yield no more but also no less than a metonically structured sequence of “most probable” (i.e. approximate) dates of Jewish Paschal full moon. In order to be able to reconstruct the sequence of dates of Jewish Paschal full moon it is necessary to discount the variable position of Nisan in the Julian calendar (i.e. to allow for the principle that Pesach had to be celebrated as early as possible in spring) by establishing a suitable lower and upper limit (which of course must have a difference of roughly the synodic period of the moon) between which in all probability (ideally) all local Jerusalem points in time of Newmoon of Nisan belonging to the time interval I in question will occur. Because round the year 240 the real date of the March equinox was sometimes 20 but mostly 21 March, we may take our departure e.g. from an obvious lower limit 5 March 12:00 and upper limit 3 April 24:00 (we note that their difference is 29.5 days) for all those local Jerusalem points in time of the Newmoon of Nisan (we note that adding 2 + 13 days to 5 March gives indeed 20 March).
Generally speaking, in order to be able to obtain dates of the fourteenth day of Nisan we will firstly have to obtain dates of the first day of Nisan from points in time of Newmoon of Nisan. Hence in Table 2 (with dates according to the Julian calendar) we see for each indicated calendar year (in the primary column A) mentioned in column B the best possible estimated point in time for Jerusalem of the (actual) Newmoon of Nisan, in column C the most probable date of the first day of Nisan estimated on the basis of column B (using the fact that at the time the first day of Nisan usually began with the second sunset in Jerusalem after the Newmoon of Nisan), in column D the most probable date of the fourteenth day of Nisan estimated on the basis of column C, in column E the most probable date of Jewish Paschal full moon estimated on the basis of column D (the dates of column E must be chosen such that the sequence of dates of column E is an as good as possible metonically structured approximation of the sequence of dates of column D).
The best way to obtain the
(metonically structured) sequence of dates of column E of Table 2 from
column D seems doing this via reconstruction of the position of its
saltus lunae. Once the position of its saltus lunae has been determined, the
rest is easy. Seeking a lead in column D we see, after having
established that in this column four times a recurrence of a similar “regular
metonic little piece” consisting of two or more dates occurs, that that saltus
lunae must be located between
4 paschal tables
Besides the millennium question still some other subjects related to our era (but not being of vital importance for the solution of the millennium question) are treated in this essay, e.g. in Section 3 calendars and in this section Paschal tables. The next section directly relating to the millennium question is Section 5.
At the end of the first century the Christian Paschal feast was mostly celebrated on the fourteenth day of Nisan (see Section 3), at the end of the second century in principle on the first Sunday after the fourteenth day of Nisan. Around the second turn of century the moment of the beginning of Nisan was still not exactly computable. In order to need not be dependent on the not exactly predictable way in which in that time the beginning of Nisan was determined in Palestine, in the beginning of the third century computists of some churches, among which the church of Rome and the one of Alexandria (Egypt), began to calculate their own dates of Paschal full moon adjusted to one of the two (mutually convertible) solar calendars prevailing then in the Roman empire (see Section 3), and it is periodic sequences of this sort of substitutes for dates of the fourteenth day of Nisan that were used to generate Julian or Alexandrian calendar dates of Paschal Sunday. That led in the first half of the third century to the construction of Paschal tables which mutually often differed in their dates of Paschal full moon and in their dates of Paschal Sunday. Around the middle of the third century the church of Rome began to experiment with sequences of Julian calendar dates of Paschal full moon with a period of 84 years, the church of Alexandria with sequences of Alexandrian calendar dates of Paschal full moon with a period of 19 years. The sequence of dates of Paschal Sunday which was generated by the sequence of dates of Roman Paschal full moon constructed by the church of Rome in the secomd half of the third century had also a period of 84 years. That the period of the sequences of dates of Paschal Sunday which were generated by the sequences of dates of Paschal full moon constructed by the church of Alexandria in the second half of the third century and the first quarter of the fourth century was no less than 532 years, was discovered around the fourth turn of century. Nevertheless the sequence of dates of Alexandrian Paschal full moon constructed in the first quarter of the fourth century would appear to be the key to the very best solution of the great problem how to calculate the date of Paschal Sunday.
At the first council of Nicaea, convened in the year 325 by the emperor Constantine I, was decided that henceforth Paschal Sunday should be celebrated every year early in spring by all Christians on the very same Sunday after the fourteenth day of Nisan, the day on which the last preparations were made for the celebration of Pesach (see Section 3). At that important council one came also to the conclusion that anyhow it was necessary to be always amply in advance well informed about dates being eligible for the celebration of Paschal Sunday, and that therefore, because of the then incalculability of the Jewish calendar (see Section 3), accurate Paschal tables adapted to the Julian calendar (see Section 3) or to the Alexandrian calendar (see Section 3) were required. The bishops who were together in the year 325 in Nicaea, were agreed about that Easter Sunday always ought to be preceded by “the full moon of Nisan” as well as by the March equinox (see Section 3). However, they could not reach agreement with regard to the way in which the date of Paschal Sunday had to be calculated, owing to the fact that they remained in disagreement about the date of the March equinox and about the way in which “the Paschal full moon” and subsequently from this the date of Paschal Sunday had to be calculated.
As early as around the middle of the third century the church of Alexandria started using the date 22 March, which date she then considered as the date of the March equinox, as a lower limit of her dates of Paschal full moon. The first by name known Alexandrian computist who applied that principle to sequences of dates with a period of 19 years was Anatolius (see Section 3). Supposedly he was one of the Alexandrian computists who around the year 260, so still before his episcopal consecration, constructed their first sequences of dates of Paschal full moon provided with a metonical structure (see Section 3); it is plausible that a decade later Anatolius started from one of these sequences, its (unfortunately unknown) dates called for convenience dates of preanatolian Paschal full moon, to construct his famous Paschal cycle with a period of 19 years. Anatolius’ Paschal cycle (see also Section 14), constructed around the year 270, can be considered as a brave attempt to overcome almost irreconcilable differences of opinion between different churches; it was a rather impractical Paschal table, which, if ever actually used, must have gone out of use long before the end of the third century.
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As a matter of fact an adequate reconstruction of the metonically structured sequence of dates of preanatolian Paschal full moon, constructed around the year 260 and not to be confused with the (definitive) metonically structured sequence of dates of (postanatolian) Alexandrian Paschal full moon constructed about sixty years later, will yield no more than a sequence of most probable dates of preanatolian Paschal full moon, which could in principle be identical with the sequence of dates of preanatolian Paschel full moon. We realize that the Alexandrian Christian computists around the year 260 being busy constructing their sequence of dates of preanatolian Paschal full moon, it is plausible that they were able to make use of up to date tables of approximate dates of lunisolar conjunction constructed by themselves or by other Alexandrians, i.c. Greek astronomers or Jewish calculators just like the Alexandrian Jewish calculators about simultaneously being busy constructing their metonically structured sequence of dates of Jewish Paschal full moon (see Section 3), took hardly, if at all, the trouble to observe the moon but simply made use of Alexandrian calendar dates of Newmoon probably calculated by themselves or by Alexandrian astronomers. However, in order to reconstruct in the same way the sequence of dates of preanatolian Paschel full moon we must take into account the fact that around the year 260 the church of Alexandria considered 22 March as the date of the March equinox. We can do that by, instead of discounting the variable position of Nisan in the Julian calendar, doing this with the variable position of a particular substitute for Nisan, for convenience referred to as the computistical month Nisan^, namely the one which implicitly in the place of Nisan must have been applied to the construction of their sequence of dates of preanatolian Paschal full moon because of the fact that at the time the church of Alexandria considered 22 March as the date of the March equinox (with the result that the earliest possible date of preanatolian Paschal full moon had to be either 22 or 23 March). Hence column E of the table thus obtained, i.e. Table 3, differs in two horizontal rows, namely the ones concerning the years 235 and 254, from column E of Table 2. We note that for those years the most probable date of Jewish Paschal full moon is 21 March but the one of preanatolian Paschal full moon 20 April. By the way, it is not impossible that the sequence of dates of preanatolian Paschal full moon was simply obtained from the sequence of dates of Jewish Paschal full moon by adding thirty days to the earliest possible day of the sequence of dates of Jewish Paschal full moon. We still note that the points in time mentioned in column B of Table 2 present local Jerusalem points in time, on the other hand the points in time mentioned in column B of Table 3 local Alexandria points in time.
For the sake of completeness we still give a description of the construction of Table 3 (with dates according to the Julian calendar). The metonically structured sequence of dates of preanatolian Paschal full moon must have been constructed by Alexandrian Christian computists around the year 260 on the basis of dates of Newmoon insofar as belonging to any obvious substantial time interval I* round the middle of the third century, e.g. the time interval consisting of the time between the years 220 and 260. In order to be able to reconstruct the sequence of dates of preanatolian Paschal full moon it is necessary to discount the variable position of the computistical month Nisan* in the Julian calendar (i.e. to allow for the fact that at the time the church of Alexandria considered 22 March as the date of the March equinox) by establishing a suitable lower and upper limit (which must have a difference of roughly the synodic period of the moon) between which in all probability (ideally) all local Alexandria points in time of Newmoon of the computistical month Nisan* belonging to the time interval I* in question will occur. Because the earliest possible date of the fourteenth day of the computistical month Nisan* must be either 22 or 23 March, we may depart e.g. from an obvious lower limit 7 March 6:00 and upper limit 5 April 18:00 (we note that their difference is 29.5 days) for all those local Alexandria points in time of Newmoon of the computistical month Nisan* (we note that adding 2 + 13 days to 7 March gives indeed 22 March).
It is in essence the structure of Table 3 that reflects the simple and as formalistic as possible way in which third century Alexandrian computists succeeded in constructing their metonically structured sequence of dates of preanatolian Paschal full moon. We see in that table for each indicated calendar year (in the primary column A) mentioned in column B the as good as possible estimated point in time for Alexandria of the (actual) Newmoon of the computistical month Nisan*, in column C the most probable date of the first day of Nisan* estimated on the basis of column B (of course in just the same way as in Section 3), in column D the most probable date of the fourteenth day of Nisan* estimated on the basis of column C, in column E the most probable date of preanatolian Paschal full moon estimated on the basis of column D (the dates of column E must be chosen such that the sequence of dates of column E is an as good as possible metonically structured approximation of the sequence of dates of column D). It turns out (see column E) that there are two equally best metonically structured approximations of the sequence of dates of column D (one with 24 march and the other one with 25 march at two places in column E), each of them characterized by the position of its saltus lunae, and both an (on the face of it) ideal metonically structured sequence of dates of preanatolian Paschal full moon, with 23 March as earliest and 20 April as latest possible date.
Although the church of Alexandria always respected the principle that (what is called) the “age” of the moon on any of her dates of Paschal full moon always had to be 14 days, the adaptation of Nisan to the Alexandrian calendar she brought about would ultimately (around the year 320) result in such substantial changes of position of these dates with respect to Nisan that they almost all would land on or near the twelfth day of Nisan (see also Section 12) instead of on or near the fourteenth day of Nisan.
Relatively shortly after the third turn of century the church of Alexandria decided to consider henceforth 21 March as the date of the March equinox. Hence the classical metonically structured sequence of dates of (postanatolian) Alexandrian Paschal full moon constructed by Alexandrian Christian computists around the year 320 on the basis of dates of Newmoon insofar as belonging to any substantial time interval round the third turn of century, has 21 March as its earliest possible date. It has 18 April as its latest possible date. As early as in the year 410 the Alexandrian monk and computist Annianus succeeded in using that famous sequence of dates in order to create a complete (i.e. containing a periodic sequence of dates of Paschal Sunday) Paschal cycle, in the year 525 it was used by Dionysius Exiguus (see Section 2) in constructing his Paschal table so important considered from a chronological perspective. It is especially due to Annianus on the one hand (in the east) and by the agency of Dionysius Exiguus’ great follower Beda Venerabilis (see section 2) on the other (in the west) that the sequence of dates of Alexandrian Pachal full moon became crucial for celebrating Easter simultaneously by all churches for a very long time (from the eighth to the sixteenth century).
Fortunately the sequence of dates of Alexandrian Paschal full moon is fully known (see e.g. column F of Table 1). However, it is not quite clear in which way it came about (see also Section 14). Be that as it may, it is a matter of fact that the classical sequence of dates of Alexandrian Paschal full moon forms the backbone of all Easter tables composed in Alexandria around the year 320 as well as of all Paschal tables which directly or indirectly evolved from such a Paschal table (by means of extrapolation). The metonical core of each of the Alexandrian Paschal tables composed around the year 320 covers the special time interval consisting of the years 285 up to and including 303 (the first saltus lunae occurs at the transition from 303 to 304), and it is the repetitions of this metonical core which are so characteristic for all classical Alexandrian Paschal tables. A famous example of a classical Alexandrian Paschal table is Annianus’ Paschal cycle; its metonical core covers the original special time interval consisting of the years 285 up to and including 303. Also Dionysius Exiguus’ Paschal table (see Section 2) is an example of a classical Alexandrian Paschal table; the metonical core of this Paschal table (see Table 1) covers the time interval consisting of the years 532 up to and including 550 (the first saltus lunae occurs at the transition from 550 to 551), which time interval not surprisingly is congruent modulo 19 years with the metonical core of the Alexandrian Paschal tables composed around the year 320.
Already since the beginning of the third century the church of Alexandria respected the principle “Paschal Sunday is the first Sunday after the Paschal full moon” for the determination of the date of Paschal Sunday. According to that principle the date of preanatolian Paschal Sunday is the date of the first Sunday after the date of preanatolian Paschal full moon. Julian calendar dates of Alexandrian Paschal Sunday determined according to that principle we can find in column G of Table 1. The earliest possible date of Alexandrian Paschal Sunday is 22 March, the latest possible 25 April.
In the fourth century in the western half of the Roman empire Roman Paschal tables were in use which were real Paschal cycles because not only their sequences of dates of Paschal full moon (these dates called for convenience dates of Roman Paschal full moon) but also their sequences of dates of Paschal Sunday (these dates called for convenience dates of Roman Paschal Sunday) had a period of 84 years. In that time dates of Roman Paschal Sunday were determined according to the principle “Paschal Sunday is the first Sunday after the first day after the Paschal full moon” as far as this yielded a date between (exclusively) 21 March and 22 April; this restriction sometimes led to problems (e.g. in the years 303 and 360). In the fourth century the earliest possible date of Roman Paschal full moon was 16 March (in the year 352) and the earliest possible date of Roman Paschal Sunday 22 March (in the years 330, 341, 352), in spite of the fact that at the time until the year 381 the church of Rome considered the date 25 March to be the date of the March equinox. The Roman Paschal tables used in the fourth century were of less quality than the classical Alexandrian Paschal tables used in the fourth century in the sense that their dates of Paschal full moon fell much earlier out of step with the rhythm of the real moon phases.
By the publication around the year 320 of the first generation of classical Alexandrian Paschal tables the church of Alexandria was the first church who opted definitely for 21 March as the earliest (and for 18 April as the latest) possible date of (Alexandrian) Paschal full moon. Thus she opted also definitely for 22 March as the earliest (and for 25 April as the latest) possible date of (Alexandrian) Paschal Sunday, because of the Alexandrian formula for the date of Paschal Sunday, which applies for all classical Alexandrian Paschal tables. In the fourth century the curches in the western half of the Roman empire used mainly Roman Paschal tables, the curches in the eastern half chiefly Alexandrian ones. It is plausible that at the curches in fourth century Palestine (among which the churches of Jerusalem and Caesarea) no other Paschal tables than classical Alexandrian ones were in use.
The Paschal table of bishop Theophilus of Alexandria, which was composed in the year 385, was the first classical Alexandrian Paschal table which contained Julian instead of Alexandrian calendar dates (of Alexandrian Paschal full moon and of Alexandrian Paschal Sunday). In the beginning of the fifth century Annianus composed his Paschal cycle, in which not only the sequence of dates of (Alexandrian) Paschal full moon is periodic (with a period of 19 years) but also the sequence of dates of (Alexandrian) Paschal Sunday (with a period of 532 years). Just like Theophilus’ Paschal table, Annianus’ Paschal cycle was obtained as a result of extrapolation from the original (composed around the year 320) classical Alexandrian Paschal table. The Paschal table attributed to Cyril (see Section 2), which was obtained as a result of extrapolation from Theophilus’ Paschal table and was intended for use in the western half of the Roman empire, was just like Theophilus’ Paschal table provided with Julian instead of Alexandrian calendar dates. Dionysius Exiguus obtained his Paschal table, which is also provided with Julian calendar dates, by extrapolation from the Paschal table attributed to Cyril. The Paschal table attributed to Cyril concerns the years 437 up to and including 531, Dionysius Exiguus’ Paschal table the years 532 up to and including 626. Because the Alexandrian formula for Paschal Sunday holds for all classical Alexandrian Paschal tables and in all these Paschal tables the age of the moon on the date of Paschal full moon is always 14 days, in all these Paschal tables the age of the moon on the date of Paschal Sunday is always an integral number of days between 14 and 22.
In Dionysius Exiguus’ Paschal table (see Table 1), in which all dates are Julian calendar dates, we see for each indicated calendar year (in the primary column A) mentioned in column C the (what is called) epact (i.e. the age of the moon on 22 March), in column D the (what is called) concurrent (i.e. the difference between 24 March and the date of the last Saturday before 24 March), in column F the date of Alexandrian Paschal full moon, in column G the date of Alexandrian Paschal Sunday, in column H the age of the moon on the date of Alexandrian Paschal Sunday. In fact, each epact in column C and each concurrent in column D represents a number of days; “nulla” in column C means ‘nothing’, which is equivalent to ‘no days’. Each date in column F can be obtained by subtracting the corresponding epact in column C from 5 April and reducing the outcome modulo 30 days to a date between 20 March and 19 April. Each date in column G can be obtained by subtracting the corresponding concurrent in column D from 25 March and reducing the outcome modulo 7 days to a date between the corresponding date in column F and the date obtained by adding 8 days to this date in column F, which boils down to the same as applying the principle “Paschal Sunday is the first Sunday after the Paschal full moon”. Each age of the moon in column H represents a number of days that can be obtained by adding 14 days to the number of days obtained by subtracting the corresponding date in column F from the corresponding date in coloumn G. The numbers in columns B and E are less important.
The sequence of concurrents occurring in Dionysius Exiguus’ Paschal table is periodic with a period of 28 years. The oldest classical Alexandrian Paschal table in which that sequence of concurrents occurs is the Paschal table of Theophilus constructed in the year 385. That sequence of concurrents which all later classical Alexandrian Paschal tables have in common with Theophilus’ Paschal table, has the structure of a solar cycle, i.e. has a period of 28 years and the additional property that every following concurrent of the sequence can be obtained by adding either 1 modulo 7 days (normally) or 2 modulo 7 days (once every four times) to the last preceding concurrent of the sequence (this definition rests on the congruence 21 · 1 + 7 · 2 ≡ 0 modulo 7). On the other hand, the sequence of epacts which all classical Alexandrian Paschal tables have in common, has the structure of a lunar cycle, i.e. has (more or less just like the classical sequence of dates of Alexandrian Paschal full moon) a metonical structure, i.e. has a period of 19 years and the additional property that every following epact of the sequence has a period of 19 years and the additional property that every following epact of the sequence can be obtained by adding either 11 modulo 30 days (normally) or 12 modulo 30 days (only in the case of the so called saltus lunae occurring once every nineteen times) to the last preceding epact of the sequence (this definition rests on the congruence 18 · 11 + 1 · 12 ≡ 0 modulo 30). Hence since the year 385 the classical Alexandrian Paschal tables were characterized not only by their lunar cycle of epacts with a period of 19 years but also by their solar cycle of concurrents with a period of 28 years (the periodicity of this solar cycle rests on the leap year proportion one to four of the Alexandrian calendar and the fact that a week holds seven days). That consequently it must be possible to extend Theophilus’ Paschal table to a Paschal cycle with a period of 19 · 28 = 532 years was diusvovered by Annianus around the fourth turn of century. He completed his Paschal cycle in the year 410. Dionysius Exiguus was not acquainted with Annianus’ Paschal cycle, and he had no proper understanding of the possibility to extend his Paschal table to a Paschal cycle, as little as of the fact that the concurrents in the fourth column of his Paschal table (see Table 1) form a solar cycle.
In the year 616 an anonymous extended Dionysius Exiguus’ Paschal table to an Paschal table concerning the years 532 up to and including 721, and it is this Paschal table which around the year 640 was accepted by the church of Rome, which from the third century up till then had given preference to go on using her own, relatively inadequate, Roman Paschal tables. In the year 725 Beda Venerabilis (see section 2) published a new extension of Dionysius Exiguus’ Paschal table to a Paschal cycle which is in fact a reinvention of Annianus’ Paschal cycle. Beda Venerabilis’ Easter cycle and Annianus’ Paschal cycle contain essentially just the same dates of Paschal full moon and of Paschal Sunday. Like in Annianus’ Paschal cycle in Beda Venerabilis’ Easter cycle the concurrents form a solar cycle (with period 28) and the dates of Paschal full moon a lunar cycle (with a period of 19 years), and consequently the dates of Paschal Sunday a sequence of dates with a period of 532 years. In the Byzantine empire thanks to Annianus’ Paschal cycle at all times the churches were acquainted with the “only correct” date of the next Paschal Sunday. It is Beda Venerabilis’ Easter cycle by means of which also the churches in the part of Europe outside the Byzantine empire got that possibility.
It is the classical Alexandrian Paschal tables composed in Alexandria around the year 320 from which (a century later) Annianus’ Paschal cycle, (two centuries later) Dionysius Exiguus’ Paschal table and (four centuries later) Beda Venerabilis’ Easter cycle would evolve. At the moment the western half of the Roman empire went down (in the year 476), in the eastern half classical Alexandrian Paschal tables were in use abundantly. In the Byzantine empire no other Paschal tables than classical Alexandrian Paschal tables were used. However, in the part of Europe outside the Byzantine empire it lasted as late as the eighth century, when Beda Venerabilis’ Easter cycle was accepted by the churches in Britain and Ireland and in the Frankish kingdom, before all Paschal tables being in use were substituted for classical Alexandrian Paschal tables. The thus realized general use of classical Alexandrian Paschal tables (by means of which at last the churches could realize their old ideal of celebrating Easter simultaneously) was continued for centuries, in the Byzantine empire until the fall of this empire in the year 1453, in the greater part of Europe until the year 1582, when Beda Venerabilis’ Easter cycle was replaced with Easter tables adjusted to the Gregorian calendar (see Section 3).
The presence of the Latin word “nulla” in the third column of his Paschal table creates the impression that Dionysius Exiguus must have known the number zero. However, that word means plainly ‘nothing’, which is equivalent to ‘no days’. There where we say that in the first year the epact is 0 days, he would have said “annus primus non habet epactas”, which means “the first year has no epacts”. Where people calculate with “epacts” (e.g. 18 days + 12 days = 30 days ≡ no days modulo 30 days) the way infants do with numbers of apples (e.g. 12 apples – 12 apples = no apples) we cannot speak yet of ‘being acquainted with the number zero’. There where Dionysius Exiguus sees simply a column of mutually related separate numbers of days (such as “12 days” and “no days”), it is our modernized brain which thinks to see a purely mathematical structure, a sequence of (abstract) nonnegative integers. In his calculations Dionysius Exiguus never made use of any symbol for ‘zero’. His number system contains only positive numbers, “nulla” in the third column of his Paschal table means simply ‘no days’, not 0. But to call an erudite person like Dionysius Exiguus stupid because he did not know the number zero (which some people do) that is really stupid. We establish that Dionysius Exiguus is no exception to the generally accepted rule that in early medieval Europe nobody knew the number zero (see Section 2). There is nothing from which we can deduce that Dionysius Exiguus was acquainted with the number zero. In medieval Europe one had to wait until as late as around the twelfth century before one got dispose of that important number (see also Section 5).
5 complete era
Dionysius Exiguus (see Section 2) presented his Paschal table, with his Anno Domini era (see Section 2) included in it, to official representatives of pope John I in or shortly after the year 525. However, eventually it would last still a bit more than two centuries before one got round to put that era into use actively as a coherent system for dating historical events. That happened only in the year 731 through Dionysius Exiguus’ great follower Beda Venerabilis (see Section 4).
In order to create the possibility of localizing on the new time scale historical events that happened before the beginning of our era as well, of course the (incomplete) Anno Domini era had to be extended to a complete era. For that purpose first the calendar years (according to the Julian calendar) preceding the year 1 were numbered further and further back into the past 1, 2, 3, ……, which sequence of calendar years then was joined together with the sequence of calendar years 1, 2, 3, …… to the complete sequence of calendar years ……, 3, 2, 1, 1, 2, 3, ……, where the year 1 = the year 1 before Christ = the Roman year 753, and e.g. the year 10 = the year 10 before Christ = the Roman year 744. Thanks to Beda Venerabilis the calendar years of our era were divided into calendar years after Christ and calendar years before Christ, which division essentially boils down to a division into positively numbered and negatively numbered calendar years without the number 0 being allocated to any calendar year. With the duration of a year as unit of time, the complete Christian era thus obtained, boils down to our second timeline (Figure 2):
time …… -3 year -3 -2 year -2 -1 year -1 0 year 1 1 year 2 2 year 3 3 …… (time in years)
in which (modern) picture year -1 = the year 1 = the year 1 before Christ and e.g. year -10 = the year 10 = the year 10 before Christ (this calendar year began at moment -10 and ended at moment -9). The running of things at extending the (incomplete) Anno Domini era to the complete Christian era can be roughly summarized in our observation that year -x = the year -x (of our era) = the year x = the year x before Christ, where, however, we have to realize that negative numbers became available only in the course of the second millennium.
We observe that our second timeline (see Figure 2) looks like a complete linear time scale (with the duration of a year as unit of time) supplemented with the positions of the positive numbered and of the negative numbered calendar years of our era. However, on closer inspection that timeline cannot represent a pure linear time scale, because two calendar years are not always precisely equally long. Usually the difference between the lengths of two calendar years is either nil or one day (see also Section 7). For instance, the difference between moment 11 and moment 12 (this difference is 366 days) is not the same as the one between moment 10 and moment 11 (this difference is 365 days). Nevertheless we may interpret our second timeline (provided that the year -x is taken as the year x before Christ) as a simple and as such consistent mathematical model of the complete Christian era. Likewise our first timeline (see Figure 1) is to be interpreted as a simple and as such consistent mathematical model of the (incomplete) Anno Domini era.
What strikes us most (perhaps even is bugging us) in our second timeline is of course that in here there is no room for a year zero. We will still see (in Section 6) why our era from the outset, to this very day, had to do without a year zero, even though the number zero is common property now for a long time. Modern historians who know their job (and of course we take these people serious) really let the year 1 come immediately after the year -1. It is moment 0, the unique point in time from which the calendar years of our era are counted and which is identical with the point in time [1-1-1 0:00] (in modern notation), which marks the direct transition (turn of year) from the year -1 to the year 1, just like it marks the direct transition (turn of century) from the first century before Christ to the first century (after Christ). Just like there is no zeroth century (and no zeroth millennium), there is also no year zero, thanks to Beda Venerabilis.
Beda Venerabilis calculated (just like Dionysius Exiguus) only with positive integers represented by means of Roman numerals (these are the letters i, v, x, l, c, d and m of the Latin alphabet). He did not feel the slightest need for a numeral zero; e.g. the sum of cc = 200 and i = 1 was noted in Roman numerals simply as cci. Division algorithms were not available yet in early medieval Europe; in this Europe division boiled down to repeated subtraction. There where Beda Venerabilis in his important book “De Temporum Ratione” about “reckoning of time” explains dividing 725 by 19 he says first that 19 times 30 makes 570 and that 19 times 8 makes 152 and then “remanent iii”, meaning that the remainder is 3. But he refrains from naming the number zero to tell us which remainder one obtains when dividing 910 by 7, for answering this question he says, after having noted that 7 times 100 makes 700 and that 7 times 30 makes 210, simply “nihil remanet” or the equivalent “non remanet aliquid”, meaning “there is nothing left over”. Calculating, he never uses any symbol or word for ‘zero’. And there where he enumerates Greek numerals, he does not observe that there is among them no symbol or word for some numeral zero. There is nothing from which we can deduce that Dionysius Exiguus was acquainted with a numeral zero or with the number zero (see Section 4); the same holds for Beda Venerabilis.
In the standard work about “De Temporum Ratione” written by the Canadian historian Faith Wallis we find a modern version of Beda Venerabilis’ Easter cycle (see Section 4), with our modern digits and with epacts (see Section 4) being 0 once every nineteen years, and even mentioning the year ‑1. But in original manuscripts written by Beda Venerabilis himself you will find no nonpositive numbers at all and you will see only the Latin word “nihil” (meaning nothing but ‘nothing’) or a Latin word like “nulla” or “nullae” (which means ‘none’) on the places where we would expect to meet the number 0. For our modern brain it is difficult to interprete “de octaua decima in nullam facere saltum” else than as “to make a jump from 18 to 0”. But even modern people use phrases such as “jump into nothingness”. It is our modernized brain which tries to hoax us into believing to see the number zero there where by early medieval scholars simply ‘nothing’ or ‘none’ was meant. There where Beda Venerabilis calculates with (abstract) positive integers, as soon as the number zero comes into sight (i.e. enters our field of vision) he lapses, just like Dionysius Exiguus, into a less abstract terminology. Dionysius Exiguus’ “nulla” and Beda Venerabilis’ “nulla” or “nullae” in their columns of epacts are typical examples of precursors of the number zero, they stand for “no epacts” or ‘no days’, which boils down indeed to ‘nothing’; but the term ‘nothing’ is, in contrast to the number zero, no mathematical concept. For Dionysius Exiguus and Beda Venerabilis as well as for us ‘adding nothing’ boils down to ‘doing nothing’. But to be able to conceive refraining from any action (‘adding nothing’) as a special case of adding something (‘adding zero’) it takes more than skill in carrying out calculations with positive integers.
Beda Venerabilis like Dionysius Exiguus knew no other numbers than positive ones, just like everyone in first millennium Europe. Even Boetius (around the year 500), the only somewhat important mathematician in early medieval Europe, and Gerbert (see Section 2) were anything but familiar with the number zero. Nowhere in European literature come down to us from the first millennium the number zero itself can be found. So there is no reason at all to abandon the current opinion that the number zero was unknown in early medieval Europe. So the opinion that Dionysius Exiguus en Beda Venerabilis should be acquainted with the number zero remains really without any rational basis. They were great scholars and skilled computists, but not mathematicians (and also not astronomers). One does not need to be a mathematician to be able, starting from the periodic sequence of Julian calendar dates of Alexandrian Paschal full moon (see Section 4) and making use of the leap year regulation according to the Julian calendar (see Section 3) and the Alexandrian formula for the date of Paschal Sunday (see Section 4), to determine really all Julian calendar dates of Alexandrian Paschal Sunday. And if you want to do that with the help of Dionysius Exiguus’ Paschal table then you can restrict yourself to the use of columns A, D, F of Table 1. By the way, that does not alter the fact that the very first construction (around the year 260) of a sequence of dates provided with a metonical structure (see Section 3) as an approximation for a sequence of dates of the fourteenth day of Nisan (see Section 3) was an impressive arithmetical finding, which we probably owe to Anatolius (see Section 3).
Ptolemy (see Section 3) handled the symbol o for a numeral zero in the (originally Babylonian) sexagesimal positional system. But that symbol was not actively used by him as a numeral zero in combination with the Greek numerals (these are the 24 letters of the Greek alphabet supplemented with the obsolete Greek letters digamma, koppa and sampi) he used in his calculations; e.g. the sum of s = 200 and a = 1 was noted in Greek numerals simply as sa. In the sixth century the decimal positional system being then already a few centuries in use in India, which was already provided with symbols for the digits 1 up to and including 9, was increased with the symbol 0 for the digit zero, due to which it became possible to carry out abstract calculations in an efficient manner (by means of convenient algorithms). The clarification of the concept of number ensuing from the introduction of the symbol 0 for the digit zero inspirered the great Indian mathematician Brahmagupta about the year 630 to the invention of the number zero; he was the first who made explicit the most important properties of the number 0 (for any number x we have x + 0 = x and x · 0 = 0). The dissemination of the number 0 across Asia took centuries, as did the dissemination of this number across Europe, which began to get into its stride only around the twelfth turn of century (in Italy, after a hesitant beginning around the eleventh turn of century in Spain). Fibonacci (whose important book “Liber Abaci” was finished in the year 1202) was the first Italian, Robert Recorde (“Ground of Artes” in the year 1543) the first Briton, Simon Stevin (“De Thiende” in the year 1585) the first Dutchman who was familiar with that utmost important number. Without the number zero there would be no modern mathematics, and without modern mathematics our technology would have been completely impossible.
If only because of the fact that in the early middle ages the number zero and the negative integers still were completely unknown in Europe, Dionysius Exiguus and Beda Venerabilis could not possibly have understood our second timeline. Dionysius Exiguus did not worry about that, because he did not at all need those nonpositive numbers for the setting up of his incomplete era (which actually was used by him only for the benefit of his Paschal table), and Beda Venerabilis too could manage very well without these “unusual” numbers. Although the (incomplete) Anno Domini era was used by the church of Rome only in the tenth century for the first time, the complete Christian era was brought into use as a coherent system for dating historical events by Beda Venerabilis as early as in the year 731. However, the modern concept of the bilateral linear time scale, necessary to be able to understand our second timeline, only could make its entry after people in Europe had got to hand the number zero (around the year 1200) and the negative numbers (around the year 1500). The nonpositive integers began to be common property only in the first half of the eighteenth century by the invention of the thermometer (which sometimes indicates degrees below zero). Restrictions with regard to the lowest or the highest possible temperature excepted the scale of Anders Celsius is a bilaterally symmetrical linear calibration; it is the bilateral symmetry which we besides in this calibration also see in Figure 2 (of which e.g. the second decade before Christ corresponds to the temperature interval consisting of the temperatures between -20ºC and -10ºC). The French astronomer Jacques Cassini was the first who explicitly availed himself of negatively numbered calendar years (see also Section 6).
In times of scarcity of reliable historical factual material the dating of historical events was no simple matter. So by Beda Venerabilis the coming into power of the emperor Diocletianus (which took place in November of the year 284 but still had been dated by Orosius in the Roman year 1041) was dated in the year 286, the capture of Rome by Visigothic troops (which took place in the year 410) was dated in the year 409, and the death of pope Gregory I (who starved in the year 604) was dated in the year 605. Beda Venerabilis was the first medieval historian who, making use of the complete Christian era, ventured to date the first landing of Julius Caesar in Britain; this military action, which took place in the year -55, was dated by Beda Venerabilis in the year 60 before Christ.
6 argumentation
If we have a look to our second timeline (see Figure 2) just a bit longer and abstract from the fact that two calendar years are not always precisely equally long then we observe that our era, i.e. the complete christian era (taken as a linear system of numbered calendar years), is in principle (namely restrictions with regard to the beginning or the end of times excepted) bilaterally symmetrical with respect to moment 0, the unique point in time which is identical with [1-1-1 0:00]. That symmetry is rather obvious, we think, as we take it for granted that every century consists of one hundred years (as every kilometre contains thousand metres), and that every (positively or negatively numbered) calendar year of our era belongs to exactly one (positively or negatively) numbered century of our era (e.g. the year -100 does not belong to both the first and the second century before Christ). Consequently, in our era there simply cannot be a year zero (provided that we want to preserve symmetry). For such a year zero would have to belong to the first century before or to the one after Christ, but then also (due to the symmetry) both to the first century before and to the one after Christ; but this is incompatible with the principle that every calendar year of our era belongs to exactly one numbered century of our era.
Our era is a bilaterally symmetrical era without a year zero. Both an alternative era with the year 1 as a year zero and a one with the year -1 as a year zero (in fact there are no other possibilities to be taken into consideration seriously) are necessarily not symmetrical with respect to moment 0. It is for that reason that none of those two alternative eras became common property, though sometimes a variant of the latter one is used sometimes for practical purposes by scientists (mainly astronomers and chronologists). That (nonsymmetrical) variant is the astronomical era; this era, defined on the basis of the Julian dating system (not to be confused with the Julian calendar), which in the year 1583, shortly after the introduction of the Gregorian calendar, had been proposed by the great chronologist Joseph Scaliger, was taken in use in its present form (including a year zero and negatively numbered calendar years) by Jacques Cassini (see Section 5) in the year 1740. With the duration of a year as unit of time the astronomical era boils down to our third timeline (Figure 3):
time …… -3 year -2 -2 year -1 -1 year 0 0 year 1 1 year 2 2 year 3 3 …… (time in years)
in which (modern) picture year 0 does not exactly coincide with the year -1 (of the Christian era), which began two days later and ended one day later, due to the initially (during almost half a century) inadequate functioning of the Julian calendar (see also Section 7).
It is just as well that the followers of Dionysius Exiguus did not saddle his and our (certainly for historians ideal) era with any year zero. When push comes to shove, everybody, either unconsciously or consciously, prefers symmetry. Astronomers never proposed seriously to replace our bilaterally symmetrical era with their astronomical era (which had been brought into use only for practical reasons). We owe our era to Dionysius Exiguus, its bilateral symmetry to Beda Venerabilis. The absence of a year zero in our era is not in the least a mistake of Dionysius Exiguus or of Beda Venerabilis; it is purely and simply a condition our era has to satisfy in order to preserve its bilateral symmetry. We can but we do not have to be sad about the absence of a year zero in our era; it is such a thing as the absence of “the king William zero” in a company of kings named William.
The next section being of importance for the solution of the millennium question is Section 8.
7 deductions
The fact that in the complete Christian era no year zero exists, has farreaching consequences, e.g. that the first decade (after Christ) can be nothing but the time interval consisting of the years 1 up to and including 10 and the first decade before Christ nothing but the time interval consisting of the years -10 up to and including -1; these two decades are separated not by means of a year zero but by means of a point in time, namely moment 0 (see Section 5).
Any person born in the year 1 must have been conceived in the year -1 or at moment 0 or in the year 1. And someone born in the year -1 will have celebrated his tenth birthday preferably on the day it was ten years ago that he was born, so in the year 10, and this seems to be (but is not) inconsistent with the mathematical fact that -1 + 10 = 9.
From (inclusive) the year -776 up to and including the year 389 the officially recognized ancient Olympic games were held at Olympia every four years. It is easy to check that the year -4 was the first calendar year of the 194th Olympiad, the year 1 the first calendar year of the 195th Olympiad.
The Julian calendar (see Section 3) was introduced in the year ‑46 by means of a drastic adaptation of the Roman calendar (see Section 3), which was accompanied by a single lengthening of the Roman calendar year by eighty days, which lenghening (by means of which was effected that the March equinox was in fact put to 23 March), however, was neutralized immediately by the provision that the rule that a calendar year consists of 365 or 366 days was considered to hold not only for future calendar years but for all calendar years, including the calendar year in which the Julian calendar was introduced and (retrospectively) all calendar years then gone by.
Unfortunately, in the first half century after Julius Caesar had died (in the year -44) the leap year regulation according to the Julian calendar did not function well. The fact is, after the leap year -45 there was until the year -8 (by mistake) a leap year every three years (instead of every four years). That implies that between the leap years -45 and -9 there were as a matter of fact three leap years too much, namely eleven instead of eight. The regulation made for that reason by the emperor Augustus (see Section 3), according to which every fourth calendar year after which is tantamount to the Roman calendar year 757 had to be a leap year, effected in addition that none (instead of three) of the fifteen calendar years between the leap years -9 and 8 was a leap year. That implies in particular that the year 4 was no leap year. It is only from that year to the year 1582 that the Julian calendar functioned unbrokenly regularly and precisely.
In the year 325 the Julian calendar was adopted as official calendar of the church (see Section 4). However, the leap year regulation according to the Julian calendar was not accurate enough to be suitable to be used just like that indefinitely (for instance around the year 1500 the March equinox fell in reality on 11 March). That is the reason why in the year 1582 the Julian calendar was replaced with the Gregorian calendar, on the understanding that the Julian calendar, implicitly inclusive of the regulation made by the emperor Augustius mentioned in the previous paragraph, remained holdong for all calendar years before the year 1582. In that year pope Gregory XIII ordered ten days to be dropped from the tenth month (in fact in that year Thursday 4 October was the last day of the Julian calendar, and Friday 15 October the first day of the Gregorian calendar) and decreed that any calendar year of our era after that year should be a leap year if and only if its calendar year number is divisible by 4 but not by 100 unless by 400. We establish that the year 1582 comprised only 355 days, and so is the only exception on the rule that a calendar year of the Christian era consists of 365 or 366 days, and that [4-10-1582 24:00] = [15-10-1582 0:00]. Thus all leap years (and so all calendar years) of our era from the far past until now have been fixed. However, with regard to that far past we must realize that from the fifty first to the twelfth century before Christ the March equinox still fell in April.
The leap year regulation according to the Gregorian calendar was brought into force for an indefinite (future) time, and for the time being it will not be necessary to adjust it. In order to keep the March equinox in its place (on or close to 20 March) for a very long time it will be sufficient to drop, around the year 5000 for the first time, any leap day once every 3300 years. Thus all leap years (and so all calendar years) of our era from the far past until the far future have been fixed.
It is in combination with the Gregorian calendar (valid for all its calendar years after the year 1582) that the complete Christian era has been the most widespread dating system on earth. That era was never abolished or replaced with the astronomical era (see Section 6), which is a variant of an alternative era with the year -1 as a year zero, as in our third time line (see Figure 3). The astronomical era was complemented not with a proleptic leap year regulation according to the Gregorian calendar holding for all its calendar years, but with the (proleptic) leap year regulation according to the Julian calendar holding without reservation for its calendar years before the year 1582 and the (nonproleptic) leap year regulation according to the Gregorian calendar holding for its calendar years after the year 1582. Because, moreover, by definition the year 1582 of the astronomical era and the year 1582 (of our era) are identical, the astronomical era and the Christian one coincide exactly where it concerns the calendar years after the year 4, which implies that the moments 2000 of these eras are exactly equal. For that reason a choice for the astronomical era instead of for the complete Christian era would not have led to an other point in time of the second turn of millennium than [1-1-2001 0:00] (see also Section 8). The fact that the year -1 (of our era) ended one day later than the year 0 of the astronomical era does not detract from that conclusion.
The year -1 (of our era) began two days later and ended one day later than the year 0 of the astronomical era. That is caused by the fact that the years 0 and 4 of the astronomical era are leap years but the years -1 and 4 (of our era) not. The fact that the year -4 of the astronomical era is a leap year but the year -5 (of our era) was not, implies that the leap year -9 (of our era) began three days later than the leap year -8 of the astronomical era. It is not difficult to check that the leap year -21 (of our era) began two days later than the leap year -20 of the astronomical era and that the leap year -33 (of our era) began one day later than the leap year -32 of the astronomical era, and that the leap year -45 (of our era) = (exactly) the leap year -44 of the astronomical era. That implies that Julius Caesar, who was murdered at 15-3--44, died on 15 March of the year -43 of the astronomical era as well as of the year -44 (of the Christian era). By the way, every year x (of our era) after the year 4 (of our era) is exactly equal to the year x of the astronomical era, but every year -x (of our era) before the year -42 (of our era) is exactly equal to the year (-x+1) of the astronomical era. It is also true that the year -40 (of our era) = (exactly) the year -39 of the astronomical era.
According to the Roman historian Titus Livius, who lived around the beginning of our era, Rome was founded in the Roman calendar year 1, the first year of the Anno Urbis Conditae era (see Section 2). Should Rome indeed be founded in that calendar year then this important historical event will be three thousand years ago not in the year 2247 but in the year 2248 (I am just saying it meanwhile), because the Roman calendar year 1 = the year -753 (of our era). Anyway, the 800th anniversary of the foundation of Rome was celebrated exuberantly in the year 47, the 1000th one in the year 248. However, according to modern historians, Rome was founded not earlier than in the seventh century before Christ.
8 conclusion
Now that we have given account of the fact that our era, i.e. the complete Christian era (see Section 5), is quite all right (see Section 6) and that 1-1-1 is the first day of our era (see Section 2), the millennium question can be settled rapidly and definitely.
Someone born on 1-1-1 will have celebrated his tenth birthday preferably on 1-1-11 (see Section 2); and likewise he would, if all was well, have celebrated his 1000th birthday preferably on 1-1-1001, and his 2000th birthday preferably on 1-1-2001. By analogy with that we recognize that, because every millennium consists by definition of one thousand years, the second millennium began on 1-1-1000, and the third millennium on 1-1-2001.
Millennium mistake 1 was made by medieval people who thought that the world would perish on 1-1-1000; what these people did not realise was that on that very day only 999 years of the first millennium had passed. However, the first turn of millennium took place one year later, namely at [31-12-1000 24:00] = moment 1000 = [1-1-1001 0:00].
Millennium mistake 2 was made by modern people who had been fooled by commerce and media and authorities that also did not know any better (and by many a historian who had completely forgotten for a while that our era has no year zero) into believing that, rather than the “dull” date 1‑1‑2001, the “magic” date 1-1-2000 (with its millennium problem and its millennium madness) had to be the first day of the new millennium. However, the second turn of millennium took place one year later, namely at [31-12-2000 24:00] = moment 2000 = [1-1-2001 0:00].
Because moment 0 is identical with [1-1-1 0:00] the year 1 is the starting year of our era, and so it is the opening year of the first century and of the first millennium. It is not difficult to check that the year 2000 is the last year of the last decade of the last century of the second millennium and that the year 2001 is the first year of the first decade of the first century of the third millennium. The “magic” year 2000 is the closing year of the previous millennium and of the previous century, the “dull” year 2001 is the opening year of the new millennium and of the new century. And of course the year 3000 is the closing year of the third millennium (just as the year 300 is the closing year of the third century and the year 30 the closing year of the third decade).
The reason why a choice for the astronomical era (see Section 6) instead of for the complete Christian era would not have led to a point in time of the second turn of millennium different from [1-1-2001 0:00] is that the moments 2000 of these two eras are exactly equal (see Section 7). A choice for an alternative era with the year 1 (of our era) instead of with the year -1 (of our era) as a year zero indeed would have yielded a moment 2000 coinciding with the turn of year with which the year 2000 of this alternative era began, but evidently also this turn of year would have been identical with [1-1-2001 0:00].
9 objections
“All well and good” someone still objects, “but after all the twentieth century does consist exactly of those calendar years of our era whose numbers start from 19? This implies that the year 1999 is the last year of the twentieth century!”. The calendar years of our era whose numbers end in 00 throw a spanner into the works. There is no year zero in our era (see Section 5); it follows that the year 100 is the last (closing) year of the first century, that the year 200 is the last (closing) year of the second century, that the year 300 is the last (closing) year of the third century, and so on. So the year 1600 is the last (closing) year of the sixteenth century. On closer inspection the at first sight interesting standpoint of Maarten Prak (university of Utrecht) that the battle of Nieuwpoort (which took place in the year 1600) is one of the rare real battles the army of the Dutch republic fought out in the seventeenth century, turns out to be something like the remark that New Year’s Eve is one of the rare really cosy days of the month of January.
“All well and good” someone still objects, “but who is really mistaken? After all, the nineties of the twentieth century had passed on 1-1-2000!”. Indeed that is true, but the last decade of the twentieth century had begun only on 1-1-1991, and so it had passed only on 1-1-2001. Likewise the book with the pretentious title “The complete History of the twentieth Century”, rashly (just before 1-1-2000) printed in a very big edition, which finishes off with the treatment of the nineties of the twentieth century, is no complete history of the twentieth century, because what happened in the last year of the twentieth century is not in there.
“All well and good” someone still objects, “but what about my odometer? After I have driven exactly 1000 kilometers, it clearly shows three zeros!”. That is right, but what we state here is not a similarity, but it is just a difference between era and odometer, because of the fact that during its first kilometer the odometer indicates 0000, not 0001. It is true, there is a similarity between odometer and age (so during its twentieth kilometer the odometer indicates 0019, and during the twentieth year of your life you are nineteen years of age), but this is beside the point.
“All well and good” someone still objects, “but when numbering the floors of a building surely it is logical and common practice to name the second floor floor 2, the first floor floor 1, the ground floor floor 0, and the successive basements floor -1, floor -2, floor -3, ……? When numbering the calendar years of our era we cannot do without the number 0 either!”. Because floors must not be taken as spaces but as horizontal dividing planes between spaces (e.g. the ground floor) numbering the floors of a building does not correspond to numbering the calendar years but to numbering the turns of year of our era, as in our second timeline (see Figure 2).
“All well and good” someone still objects, “but what does it matter? After all, the beginning of our era was chosen haphazardly!”. That more or less haphazardly but “once and for all” chosen moment is moment 0 (i.e. the moment zero of our era), the unique point in time which is asterisked (*) so suggestively in our first timeline (see Figure 1) and is identical with [1-1-1 0:00]; it is just this unique point in time from which the calendar years of our era are being counted, it is just the way it is. In the year 1582 for an indefinite time the number of days of any calendar year of our era was fixed (see Section 7). That makes all turns of year, turns of decade, turns of century and turns of millennium of our era fixed for an indefinite time.
“All well and good” someone still objects, “but what does it matter? After all, it is completely unknown when Jesus was born!”. It is not the (indeed unknown) date of Jesus’ birth that matters for the solution of the millennium question, but it is the first day of the Anno Domini era, i.e. 1-1-1, that is essential here (see Section 8). Strictly speaking “the first century before Christ” is not “the last century before the birth of Jesus”, but “the last century preceding 1-1-1”.
“All well and good” someone still objects, “but surely the millennium question can be solved much more simply? Because there exists no year zero the supposition of a turn of millennium at [1-1-2000 0:00] leads to the absurd conclusion that the first decade consisted of nine years (so the tenth birthday of someone born on 1-1-1 coincided officially with his ninth birthday)!”. That reasoning is correct and leads to the observation that the supposition of a turn of millennium at [1-1-2000 0:00] cannot be part of a consistent system. So that supposition is (scientifically) untenable. But the solution of the millennium question still requires a proof of the fact that our era is quite all right (see Section 6).
“All well and good” someone still objects, “but the fact that in the year 67 games were held at Olympia does not agree with the assertion that the officially recognized ancient Olympic games were held at Olympia every four years (see Section 7)!”. The games held in the year 67 were no Olympic games but games which were organised at Olympia, Delphi, Nemea and Isthmia specially on behalf of the emperor Nero.
“All well and good” someone still objects, “but what on earth was against the celebration of the second turn of millennium on 1-1-2000?”. Of course nothing is against it to celebrate any memorable event whatever at any moment whatever (e.g. a turn of year on 30 December or your twentieth birthday on your nineteenth birthday). But the question is here that we have to distinguish between the direct transition from the year 1999 to the year 2000 (the “magic” moment at which all four digits of the number of the calendar year at present changed at the same time) and the accompanying turn of millennium, i.e. the direct transition from the second to the third millennium, exactly one year later, and that at the supreme moment relatively few people realized this.
“But nevertheless the people have the last word!” someone still objects. That means in my opinion that the people have right to self determination, not that the people are always perfectly right. Something does not automatically become true if many people believe that it is true. The earth does not become less round if many people believe that the earth is flat. Nor does something automatically become true by deciding it just like that, not even when this happens in a democratic way. Of course it was possible to celebrate the second turn of millennium at [31-12-1999 24:00] = moment 1999 = [1-1-2000 0:00]; but it was impossible to finish the second millennium as early as at this “magic” point in time (see Section 8). It is a long time ago since our era was started off, and we cannot change the past.
Whether something is true, is prescribed neither simply by the people nor simply by some authority, not even by the queen of the Netherlands (even though sometimes one for a moment could think she does, for the fact that there exists a statistical connection between smoking and lung cancer seems to be determined by Royal Decree). However, in order to determine whether something is true, sometimes logical (watertight) reasoning is not only necessary but also sufficient. So the logical reasoning of Section 6 and Section 8 inevitably leads to the conclusion of Section 8.
10 justification
Thanks to Dionysius Exiguus (see Section 2) and Beda Venerabilis (see Section 4) we have at our disposal a bilateral symmetrical era without any year zero (see Section 5 and Section 6). The year 1 comes immediately after the year ‑1, just like the first century (after Christ) comes immediately after the first century before Christ; there is in our era no year zero, just like there is no century zero. This is the official standpoint of our modern historians, and with good reason (as we saw in Section 6). Because our era has no year zero, we have to count our decades (and likewise our centuries and millennia) from [1-1-1 0:00]. That implies that the third millennium began not before 1-1-2001 (see Section 8). Therefore we are rather allowed to indicate the phenomenon that around the year 2000 commerce, media and authorities were amply under the delusion that the year 1999 was the last year of the twentieth century and of the second millennium with the term ‘millennium mistake’.
People believe all sorts of things. And usually what once is believed, is not given up easily. Insights that are at right angles to what once is believed often hardly get a chance to be tested to reason. Hence that people resisted so long against the insight that our earth is not flat but round, that the sun is a star and the earth is a planet revolving round the sun instead of the sun revolving round the earth, that under special circumstances primitive life (extremely gradually) comes into being, that all higher developed biological species (including Homo sapiens) are evolved out of other biological species, that all life is only temporary, that God is a product of human imagination and exists only as such (man proposes, God does not exist), that it is a mistake to think that atheists think they can prove that God does not exist (as a matter of fact atheists believe that there exists no God outside of human imagination). But to make it possible to continue our mental growth it is sometimes necessary to recognize that we were wrong (this concerns each of us personnally as well as mankind as a whole). So I got round to it, inspired to this by critical pupils who wanted to know all the ins and outs, to find out why exactly 1-1-2000 could not be the first day of the third millennium. There are circumstances in which trying to argue that what is wrong is right is simply wrong.
By the way, what is the sense of education? By no means only to emancipate people. Stimulating clear thinking and careful formulating by way of joint attention to essentials is an at least equally important education objective. Pupils ought to be able to calculate without calculator what is the sum of -753 and 3000. But also they have to know, I think, what structure our era has, in order to be able to understand that the answer to the question in which year Rome, assuming that this eternal city was founded in the year -753 (see Section 7), three thousand years will exist is not the year 2247 but the year 2248. It is not all that difficult.
11 anni domini
Now that we solved the millennium question completely (see Section 8) and justified the term ‘millennium mistake’ (see Section 10), the still unanswered question concerning the precise connection between the Anno Domini era (see Section 2) and Anni Domini (literaly ‘the Years of the Lord’), in particular Jesus’ birth and death, goes on intriguing us. Likewise closely connected with the millennium question (and neither essential for the solution of it) is the interesting question concerning the connection between the starting year of the Anno Domini era chosen by Dionysius Exiguus (see Section 2), i.e. the year 1 (of our era) = the Roman calendar year 754 (see Section 2), and Annus Dominicae Incarnationis, i.e. the calendar year of Jesus’ incarnation in the view of Dionysius Exiguus; also on the answer to this question historians do not entirely agree yet. In the writings of Dionysius Exiguus himself no clarification can be found about this, and in the writings of Beda Venerabilis (see Section 4) we meet diverse arguments leading to contradictory deductions. But the majority of modern historians think that Dionysius Exiguus believed Jesus was born in or shortly before the year 1.
Peter Rietbergen (university of Nijmegen) is of the opinion that Dionysius Exiguus believed Jesus was born one week before the year 1, so in the year -1 (of the complete Christian era) = the Roman calendar year 753. That view agrees with the known historical fact that Charlemagne let himself crown emperor just on 25-12-800. The opinion of Robert Fruin (around the year 1900) that Annus Dominicae Incarnationis = the year 1 is supported by Peter Verbist (university of Leuven) and by Georges Declercq (university of Brussels); this opinion seems to be at least as plausible as the other one because of the analogy between the beginning of the Anno Domini era and the one of the Anno Urbis Conditae era (see Section 2): “just as Rome was founded (on 21 April?) in the course of the Roman calendar year 1, Jesus was conceived (on 25 March?) and born (on 25 December?) in the course of the year 1 (of the Anno Domini era)” Dionysius Exiguus might have thought.
One of the most influential figures of the first council of Nicaea (see Section 4) was Eusebius, the historian who had become bishop of Caesarea shortly after the year 313. He was the first who hit upon the idea of an era with as its starting year the year of birth of Jesus according the Roman calendar. However, he thought Jesus was born in the third calendar year of the 194th Olympiad (see Section 7), in accordance with the opinion of Orosius (see Section 2), a century later, that Jesus was born in the Roman year 752. Nevertheless, Dionysius Exiguus chose (indirectly) the Roman calendar year 754 as the starting year of his new era (see Section 2). It may be that he did so only with a view to effect that in his new era (just like in the era of the emperor Diocletianus) the rule should hold that leap year numbers are divisible by 4.
Probably Dionysius Exiguus did not know in which Roman calendar year Jesus was born, and we do not know either. Nobody believes that moment 0 (i.e. the moment zero of our era), the unique point in time which is asterisked (*) so suggestively in our first timeline (see Figure 1) and is identical with [1-1-1 0:00], could be the moment of Jesus’ birth. According to modern historians Jesus was born sometime between the years -9 and -1, so some time before the beginning of the Christian era, a remarkable paradox. On the year (let alone on the date) of Jesus’ birth authorities do not yet agree. We establish that in all probability Jesus was born about the year -5. Sometime in the nineties of the previous century the day on which it had been two thousand years since Jesus was born, slipped by.
Still more interesting than the question when precisely was the beginning of Anni Domini, is the question when precisely was its end. Neither the year in which nor the date on which Jesus died is known for certain. It is common knowledge that Jesus died about the year 30 in Jerusalem, on a Friday in the afternoon, namely on (according to the fourth canonical gospel) or on or one day after (according to the three synoptic gospels) a day on which Pesach (see Section 3) was prepared, so on a fourteenth or on a fifteenth day of Nisan (see Section 3). However, that Jesus would have been crucified on a fifteenth day of Nisan, we may exclude because the fifteenth day of Nisan was a feast day on which one did not administer justice in Jerusalem. The religious persuasion that Jesus was crucified a few hours before celebration of Pesach began agrees, anyway, with the fact that at the end of the first century the Christian Paschal feast was mostly celebrated on the fourteenth day of Nisan. It is certain that Jesus died during the reign of the emperor Tiberius (who reigned from 14 to 37) and during the procuratorship of Pontius Pilatus, who was procurator of Judea from 26 to 36.
Beda Venerabilis tried to find the date of Jesus’ dying day with the help of his Easter cycle (see Section 4), taking his departure from the centuries old idea that “Paschal full moon = 14 Nisan”. He hoped to arrive at 25-3-34, evidently partly due to the tradition dating back to the third century according to which Jesus would have died on a Friday 25 March (of an as yet unknown calendar year). Beda Venerabilis took it for granted that the validity area of his Easter cycle extended without fail to the beginning of the Christian era. However, making use of the columns of his Easter cycle corresponding with columns F and G of Dionysius Exiguus’ Paschal table (see Table 1) he had to establish to his disappoinment that in the year 34 (like in the year 566, for 34 ≡ 566 modulo 532) the date of the Alexandrian Paschal full moon (see Section 4) was Sunday 21 March and not the Thursday 24 March expected by him. Evidently his presuppositions were inconsistent.
The persuasion that Jesus died on 25 March is without any rational foundation. For quite a while one cherished the conviction, resting on the oldest known Roman Paschal table, namely the Paschal table of Hippolytus Romanus (around the year 220), afterwards proved to be unreliable, according to which Jesus should have died on 25-3-29. But the more one got dispose of Paschal tables which kept better step with astronomical reality (see Section 4), the more the perception grew that that thesis was untenable. Nevertheless in the course of the fourth century the idea that Jesus both was conceived on 25 March and died on 25 March came into being. Not only we may have doubts about the correctness of that vision (to which after all still two calendar year numbers are lacking) but also about the perfectness of Beda Venerabilis’ Easter cycle, which, accurate as it may be, could ultimately prove to be imperfect, still besides the fact that the dates of Alexandrian Paschal full moon included in this Easter cycle anyhow not all correspond exactly with the dates of the fourteenth day of Nisan whose substitutes they were (see Section 4). Nevertheless it remains an interesting question whether it is possible or impossible for us to trace Jesus’ dying day in the manner of Beda Venerabilis.
Because Julian calendar dates of Alexandrian Paschal full moon are defined meaningfully only insofar the Julian calendar (see Section 3) after intervention of the emperor Augustus functioned properly (see Section 7), the classical sequence of Julian calendar dates of (postanatolian) Alexandrian Paschal full moon, which forms the backbone of Beda Venerabilis’ Easter cycle, reaches in fact from 4 to 1582 (see Section 3). Because that sequence of dates is periodic with a period of 19 years, we may take that sequence of dates as a strictly regularly running (of course imaginary) clock with a dial of which the hour hand has been replaced with a year hand which takes without surcease 19 years (instead of 12 hours) to go round one time. Around the year 300 that clock, which can be supposed to have run precisely and unbrokenly from 4 to 1582, nearly kept time with the then astronomical reality (with regard to the real phase of the moon on the date of Alexandrian Paschal full moon), because the classical sequence of dates of Alexandrian Paschal full moon (defined around the year 320) may be expected to have been the final result of calculations which were made on the basis of lunar phase tables referring to a time interval round the third turn of century. But thereafter that clock more and more went to lose time, as a result of the fact that a time interval consisting of 235 synodic months and a time interval consisting of 19 years contain not precisely the same number of days (see Section 3).
There is no need to let the fact that (which I call for convenience) Beda Venerablis’ great clock, i.e. the sequence of Julian calendar dates of Alexandrien Paschal full moon indicated by Beda Venerabilis’ Easter cycle taken by us as a clock, only during a time interval around the year 300 nearly kept time with astronomical reality get us down. Namely, we can, in contrast to Beda Venerabilis, calculate how much time after the moment of exactly keeping time that clock lost a whole day, and even which was the astronomical reality at the moment of exactly keeping time (see also Section 12).
Although the Julian calendar was no ideal calendar, it functioned precisely and unbrokenly from 4 to 1582. All that time a time interval of 19 calendar years lasted an average of 6939.75 days but the moon was taking about 6939.689 days to pass through all of its phases 235 times (see Section 3). From that it follows that Beda Venerabilis’ great clock with the elapsing of time after the moment of exactly keeping time lost time more and more, namely after every new time interval consistong of 19 years approximately 6939.75 – 6939.689 = 0.061 days more, so after every new time interval consistong of 1 year approximately 0.0032 days more. That implies that that clock was taking about 310 years to get behind a whole day. We conclude that around the year 600 Beda Venerablis’ great clock lost roughly a whole day, and, by analogy with this, that in the time of the reign of the emperor Tiberius it gained roughly a day.
The dramatic confrontation between Jesus and the Roman procurator Pontius Pilatus must have taken place in Jerusalem after the first year of the procuratorship of Pontius Pilatus, so elsewhere between the years 26 and 37. Beda Venerabilis’ great clock did not keep time then with astronomical reality, but gained roughly a day. In order to make an attempt to determine the date of Jesus’ death we therefore correct the Julian calendar dates of Alexandrian Paschal full moon according to Beda Venerabilis’ Easter cycle holding for the years 27 up to and including 36 (these dates are the same as those of the years 559 up to and including 568 in column F of Table 1 and have been stated also in column B of Table 4) by adding one day to each of them and determine then (with the help of column D of Table 1 or normally with the help of the Julian calendar) for each of the dates thus obtained on which day of the week this date fell; the corrected dates of Alexandrian Paschal full moon thus obtained, are stated in column C of Table 4 (in which all dates are Julian calendar dates).
Actually, in order to determine Jesus’ dying day we would want to have at our disposal dates of the fourteenth day of Nisan (which are unfortunately not exactly calculable). However, we would make a serious mistake now if we took it for granted that only to a negligible extent there would be a difference between the sequence of corrected dates of Alexandrian Paschal full moon obtained in the last paragraph and the sequence of dates of the fourteenth day of Nisan represented by them. Therefore the corrected dates of Alexandrian Paschal full moon stated in column C of Table 4 may be regarded by way of precaution at most as very roughly estimated dates of the fourteenth day of Nisan. We establish that the method of Beda Venerabilis to determine Jesus’ dying day is certainly inadequate as long as we have no idea of the connection between the sequence of dates of the fourteenth day of Nisan and the sequence of dates of Alexandrian Paschal full moon representing them (see also Section 12).
Fortunately we can obtain estimated dates of the fourteenth day of Nisan along a different route as well, roughly in the same way as was originally done by Jewish calculators, for the obtaining of their dates of Jewish Paschal full moon (see Section 3), and by Christian computists, for the obtaining of their dates of preanatolian Paschal full moon (see Section 4) in third century Alexandria (Egypt). They operated with the help of lunar phase tables (with dates according to the Alexandrian calendar) of around the year 240. We will make use of modern lunar phase tables in order to obtain estimated points in time and Julian calender dates of Newmoon of Nisan and subsequently most probable dates of the first and of the fourteenth day of Nisan in sofar belonging to the time interval consisting of the time between 26 and 37. We will show that by means of Table 4.
During the time interval consisting of the time between 20 and 50 the real date of the March equinox was sometimes 23 but mostly 22 March. In order to be able to obtain estimated points in time and Julian calender dates of Newmoon of Nisan it is necessary to discount the variable position of Nisan in the Julian calendar (i.e. to allow for the principle that Pesach had to be celebrated as early as possible in spring) by establishing a suitable lower and upper limit (which in principle must differ a synodic period of the moon) between which in all probability (ideally) all local Jerusalem points in time of the Newmoon of Nisan belonging to that time interval will occur. Because around the year 30 the real date of the March equinox was sometimes 23 but mostly 22 March, we may take our departure e.g. from an obvious lower limit 7 March 0:00 and upper limit 5 April 12:00 (we note that their difference is 29.5 days) for all those local Jerusalem points in time of the Newmoon of Nisan (we note that adding 2 + 13 days to 7 March gives indeed 22 March).
In Table 4 (with dates according to the Julian calendar) we see for each indicated calendar year (in the primary column A) mentioned in column B the (uncorrected) date of Alexandrian Paschal full moon, in column C the corrected date of Alexandrian Paschal full moon, in column D the best possible estimated point in time for Jerusalem of the (actual) Newmoon of Nisan, in column E the most probable date of the first day of Nisan estimated on the basis of column D (in just the same way as in Section 3), in column F the most probable date of the fourteenth day of Nisan estimated on the basis of column E. The surprisingly large difference of roughly 2 days between columns C and F of this table does need a further investigation (see also Section 12).
We establish that Jesus died on a Friday, on a fourteenth day of Nisan. For that reason first of all the Fridays in columns C and F of Table 4 seem to be eligible to possibly be dates of Jesus’ dying day. Hence we see them back in column G. On further consideration we may still reject the first and the fourth of those four in principle possible dates of Jesus’ dying day, the first because it is virtually certain that Jesus was baptized not earlier than in January of the year 27 and manifested himself thereafter in any case more than a whole year, the fourth by reason of the fact that it is virtually certain that not later than in the year 35 the apostle Paul became a follower of Jesus (after Jesus’ death). The two remaining dates are the most obvious dates on which Jesus could have died. However, the possible dates of Jesus’ dying day mentioned in column G are not the only possible ones, and for this reason it is better to try first to discover the cause of the discrepancy between columns C and F. The dates of column C may be regarded at most as very roughly estimated dates of the fourteenth day of Nisan, after all, and the dates of column F are not all exact dates of the fourteenth day of Nisan either. Besides, it is also still possible that just in the year that Jesus died Pesach was celebrated, by mistake or for an opportunistic reason, a month consisting of thirty days “too early” or just a one “too late” (see Section 3).
For an attempted explanation of the discrepancy between columns C and F please refer to Section 12, for a completion of the subject of this section to Section 13.
12 full moons
Relatively shortly before the first council of Nicaea (see Section 4) the church of Alexandria (Egypt) decided to consider henceforth 21 March as the date of the March equinox (see Section 3) and to determine definitely the dates of Alexandrian Paschal full moon (see Section 4), so important to the history of Christianity. Because the sequence of dates of Alexandrian Paschal full moon was calculated on the basis of lunar phase tables constructed around the third turn of century we could expect (as much as Dionysius Exiguus and Beda Venerabilis did) that within a substantial time interval round the year 300 the dates of the day on which Pesach (see Section 3) was prepared, which day always coincided with the fourteenth day of Nisan (see Section 3), anyhow usually would differ no more than a day from their Alexandrian substitutes. If, however, we relate within such a time interval the dates of Alexandrian Paschal full moon as well as the most probable dates of the fourteenth day of Nisan, which can be obtained in the same way as in Section 3, to dates of (the actual) Newmoon and of (the actual) Fullmoon then it turns out, and this is relatively new (January 2005), that within the time interval in question the date of Alexandrian Paschal full moon was mostly at least two days earlier than the date of the fourteenth day of Nisan. We will show that by means of Table 5 (with dates according to the Julian calendar).
Around the third turn of century, and hereafter still until the moment at which the Jewish calendar (see Section 3) was fixed (about the year 360), the beginning of the new month and of the new year of this then not yet exactly calculable calendar was determined officially still in Palestine and still as in the first century of our era (see Section 3). In order to be able to obtain (approximate) dates of the fourteenth day of Nisan belonging to a substantial time interval J round the third turn of century, e.g. the time interval consisting of the time between the years 280 and 320, we will firstly have to obtain (approximate) dates of the first day of Nisan from points in time of Newmoon of Nisan, but in order to be able to do this it is necessary to discount the variable position of Nisan in the Julian calendar (i.e. to allow for the principle that Pesach had to be celebrated as early as possible in spring) by establishing a suitable lower and upper limit (which of course must have a difference of roughly the synodic period of the moon) between which in all probability (ideally) all local Jerusalem points in time of Newmoon of Nisan belonging to the time interval K in question will occur. Because round the year 300 the real date of the March equinox was sometimes 21 but mostly 20 March, we may take our departure e.g. from an obvious lower limit 5 March 0:00 and upper limit 3 April 12:00 (we note that their difference is 29.5 days) for all those local Jerusalem points in time of the Newmoon of Nisan (we note that adding 2 + 13 days to 5 March gives indeed 20 March). Hence in Table 5 we see for each indicated calendar year (in the primary column A) mentioned in column B the best possible estimated point in time for Jerusalem of the Newmoon of Nisan, in column C the most probable date of the first day of Nisan estimated on the basis of column B (in just the same way as in Section 3), in column D the date of Alexandrian Paschal full moon, in column E the best possible estimated point in time for Jerusalem of the Fullmoon of Nisan, in column F the most probable date of the fourteenth day of Nisan estimated on the basis of column C. One could still object that the most probable dates of the fourteenth day of Nisan mentioned in column F could possibly insufficiently fit with historical reality; however, it will turn out that it is not these dates which deviate for the major part at least one day from the dates of (the actual) Fullmoon mentioned in column E but the dates of Alexandrian Paschal full moon mentioned in column D.
Looking at columns D and F of Table 5 and comparing them with columns D and E of Table 2 and of Table 3 we establish first of all that the (definitive, classical) sequence of dates of Alexandrian Paschal full moon must have been obtained on the basis of principles of which at least one must have been essentially different from the corresponding principle of the principles on the basis of which the sequence of dates of Jewish Paschal full moon (see Section 3) and the sequence of dates of preanatolian Paschal full moon (see Section 4) were obtained. After all, if that would not have been the case then the sequence of dates of Alexandrian Paschal full moon would have looked more or less the same as the sequence of dates of Jewish Paschal full moon (see Table 2), because in each of these two cases either explicitly or implicitly a March equinox falling on 21 March was presupposed.
The second remarkable observation for which Table 5 gives cause, is that in each of the years 284, 292, 303 and 311 very probably the date of Alexandrian Paschal full moon, unlike the date of Roman Paschal full moon, fell outside Nisan, i.c. on or by the twelfth day of Iyyar (see Section 3).
We must realize that there are different phases of the moon which with the naked eye look like full moons indistinguishable from each other. A midnight pure full moon (i.e. nearly Fullmoon) is always preceded by a (still waxing) seemingly full moon one night earlier and followed by an (already waning) seemingly full moon one night later (see Figure 4). For example as shown in Table 5 at Jeruzalem in the year 308 during the night beginning with the sunset of the date of Alexandrian Paschal full moon (22 March) insofar as the meteorological circumstances were favourable one could see a waxing full moon (i.e. still waxing seemingly full moon) and during the night beginning with the sunset of the most probable date of the fourteenth day of Nisan (24 March) a waning full moon (i.e. already waning seemingly full moon), and during the night in between a pure full moon.
Comparing columns D, E, F of Table 5 to each other we see that columns E and F differ about half a day and that if we exclude the years of our era in which very probably the date of Alexandrian Paschal full moon fell outside Nisan the dates of column D fall on average one and a half days earlier than the dates of column E but about 1.9 days earlier than the dates of column F. That implies that around the third turn of century during the night immediately following the setting of the sun of the date of Alexandrian Paschal full moon usually the moon was present in the guise of a waxing full moon roughly one day before Fullmoon but during the night immediately following the setting of the sun of the date of the fourteenth day of Nisan (so during the crucial first night of Pesach) in the guise of a waning full moon roughly one day after Fullmoon. Roughly speaking, in that time dates of Alexandrian Paschal full moon usually belonged to the (what I call) ‘waxing full moon category’, dates of the Fullmoon of Nisan of course always to the (what I call) ‘pure full moon category’, dates of the fourteenth day of Nisan usually to the (what I call) ‘waning full moon category’. Although in column E of Table 2 as well as in column D of Table 5 the earliest possible date is 21 March and the latest possible date 18 April, the sequences of dates in question differ essentially because they belong to completely different astronomical categories.
Consideration of columns D and E of Table 5 leads to the observation that within the time interval in question the date of Alexandrian Paschal full moon insofar as not falling outside Nisan usually fell either one day or two days before the date of the Fullmoon of Nisan. At the time one usually saw in Palestine in the night beginning with the sunset of the date of Alexandrian Paschal full moon a waxing full moon roughly one day younger than Fullmoon which had been risen roughly an hour before sunset; we establish that this is the astronomical reality with whitch Beda Venerabilis’ great clock nearly kept time around the third turn of century.
The fact that the difference between corresponding dates of columns E and F of Table 5 is on average about half a day, is simply a consequence of the fact that around the third turn of century the Fullmoon of Nisan fell on average roughly near the midnight point in time of the night of the thirteenth to the fourteenth day of Nisan (see Section 3). In the first three centuries of our era and in the fourth century until the moment (about the year 360) at which the Jewish calendar was fixed the fourteenth day of Nisan coincided usually either with the date of the Fullmoon of Nisan or with the first day after the date of the Fullmoon of Nisan. At the time one usually saw in Palestine in the night of the fourteenth to the fifteenth day of Nisan a waning full moon roughly one day older than Fullmoon which had been risen with great splendour roughly an hour after sunset; it is this astronomical reality to which the Jewish tradition of the celebration of Pesach in Palestine was closely joined for centuries. We can quite well imagine that at the time the emergence of an impressive full moon roughly an hour after sunset must have been experienced as an ideal beginning moment for the most important meal of Pesach, which must have urged the Jewish authorities to take care to keep the Jewish calendar under control and particularly to be time and again careful with regard to the determination of the beginning of Nisan.
Consideration of columns D and F of Table 5 leads to the observation that within the time interval in question the dates of (postanatolian) Alexandrian Paschal full moon almost all fell on or near the twelfth day of Nisan instead of on or near the fourteenth day of Nisan. That confirms the obvious conjecture arising from the remarkable difference between columns C and F of Table 4 (see Section 11). That conjecture was of course that the Alexandrian computists who around the year 320 were composing the first classical Alexandrian Paschal tables in order to obtain their dates of Alexandrian Paschal full moon opted for dates which were almost all roughly two days “too early” (which implies that in the extremely important case of the dates of (postanatolian) Alexandrian Paschal full moon the of course not completely right formula “Paschal full moon = 14 Nisan”, which nevertheless was assumed to be approximately correct, is not right at all and therefore simply wrong). We wonder why they did that. In order to get an answer to that question, we will try to devise (in Section 14) a satisfactory hypothesis concerning the way in which around the year 320 Alexandrian computists could have constructed their dates of Alexandrian Paschal full moon.
13 crucifixion
Still in Section 11 we established that Jesus died somewhere between 27 en 36; so Table 4 can be shortened by leaving out its dates concerning the years 27 and 36. Moreover, by using the connection between columns D and F of Table 5 and the periodicity of the sequence of dates of (postanatolian) Alexandrian Paschal full moon (see Section 4) we can improve the quality of the table thus obtained by replacing the corrected dates of Alexandrian Paschal full moon in its column C with the corresponding estimated dates of the fourteenth day of Nisan (see Section 3) obtained by adding 1 or 2 days to each date in this column depending on the calendar year in question, namely to the fourth date in this column and to the fifth 1 day each and to each of the other dates in this column 2 days, and subsequently accomplishing its column G. In the table thus obtained, i.e. Table 6 (with dates according to the Julian calendar), columns C and F show no significant difference (we had not expected anything else). We establish that the dates 7-4-30 en 3-4-33, already found in Section 11, in column G of that table must be maintained but have to wonder which other dates still could be added to this column.
There are different manners in which from estimated dates of the fourteenth day of Nisan in colums C and F of Table 6 possible dates of Jesus’ dying day can be deduced. Not only when such an estimated date of the fourteenth days of Nisan is a Friday but also when it is a Thursday or a Saturday, it generates, because the estimated dates of the fourteenth day of Nisan in colums C and F usually deviate no more than a day from the real dates of the fourteenth day of Nisan, simply and immediately a possible date of Jesus’ dying day. As a matter of fact, it is in this direct manner that the possible dates of Jesus’ dying day 7-4-30 and 3-4-33 were obtained. However, it is also possible that just in the year that Jesus died Pesach (see Section 3) was celebrated, by mistake or for an opportunistic reason, a month consisting of thirty days “too early” or such a month “too late” (see Section 3). For example, it is possible that in the year 34 the fourteenth day of Nisan fell on Friday 23 April (in this case probably 30 days “too late”), or, which is still more improbable, in the year 29 on Friday 18 March (in this case probably 31 days “too early”). But because ultimately determining the date of Jesus’ dying day can not be more than a question of cancelling much less probable against much more probable eventualities we may neglect the probabilities of such exceptional eventualities, perhaps with the exception of the probability of the eventuality that Jesus would have been died on 23-4-34; it is exclusively for this reason that only this date still was added to column G (and not because Isaac Newton thought that this date was the right date of Jesus’ death).
We establish that it is very probable that Jezus died either on 7-4-30 or on 3-4-33. Apart from that, there are various arguments on the basis of which 3-4-33 could be considered as the most probable possible date of Jesus’ dying day. As early as in the year 1910 that date was presented as such by Friedrich Westberg (who was Oberlehrer at a German language public secondary school in Riga). He thought, rightly or mistakenly, that 6-4-30 was a day on which Pesach was prepared and so 7-4-30 a day on which one did not administer justice in Jerusalem. Apart from that, the fact that Pontius Pilatus (see Section 11) only from the year 31, in which year his patron Lucius Sejanus fell in disgrace with the emperor Tiberius, had no need to defy the Jewish authorities in Jerusalem, is a strong argument in favour of the opinion that Jesus died on 3-4-33.
14 reconstructions
It is chiefly a particular sequence of dates provided with a metonical structure (see Section 3) which we will need to explain why the rule “Paschal full moon = 14 Nisan”, at least in the singularly important case that we have to do with dates of Alexandrian Paschal full moon (see Section 4) in the time of their coming into existence, is wrong. That sequence of dates is the sequence of dates of preanatolian Paschal full moon (see Section 4). We wonder why the (definitive, “classical”) sequence of dates of Alexandrian Paschal full moon, which we will still reconstruct in this section, and the sequence of dates of preanatolian Paschal full moon constructed about six decades earlier, reconstructed in Section 4, differ as much as they in fact do. It is the difference between the sequence of dates of preanatolian Paschal full moon and the sequence of dates of Alexandrian Paschal full moon which underlies the conclusion formulated in Section 12 that between the years 280 and 360 the date of Alexandrian Paschal full moon roughly coincided with the date of the twelfth day of Nisan (see Section 3) instead of with the fourteenth. We still remark that it is the difference between the sequence of dates of Jewish Paschal full moon (see Section 3) and the sequence of dates of preanatolian Paschal full moon which underlies the conclusion formulated in Section 12 that in the years 330 modulo 19 between the years 280 and 360 the date of Alexandrian Paschal full moon (unlike the date of Roman Paschal full moon) mostly fell roughly on the twelfth day of Iyyar (see Section 3); this difference is exclusively a result of the stand taken by the church of Alexandria (Egypt) during the second half of the third century with regard to the date of the March equinox (see Section 3). We still note that in third and fourth century Alexandria both Jewish calculators and Christian computists made use of the Alexandrian calendar (see Section 3).
It is plausible that the church of Alexandria around the year 260, still before Anatolius (see Section 3) constructed his ingenious Paschal cycle and still before his episcopal consecration, could make use of the sequence of dates of preanatolian Paschal full moon with the matching sequence of dates of preanatolian Paschal Sunday (see Section 4) and that Anatolius for the construction of his Paschal cycle virtually started from the sequence of dates of preanatolian Paschal full moon. Towards obtaining their dates of Alexandrian Paschal full moon required for the calculation of their dates of Alexandrian Paschal Sunday (see Section 4) the Alexandrian computists who around the year 320 were composing the first classical Alexandrian Paschal tables (see Section 4) opted for dates which were almost all roughly two days “too early” (see Section 12). We have established (see Section 12) that the sequence of dates of Alexandrian Paschal full moon must have been obtained by them in a way different from the manner in which the sequence of dates of preanatolian Paschal full moon was obtained by their predecessors. There are basically two ways in which that could have happened: either by means of a direct intervention in at the time already existing Alexandrian dates of Paschal full moon, e.g. the dates of preanatolian Paschal full moon, or in the Paschal dates of Anatolius’ Paschal cycle (see Section 4) or by a radically new manner of determining dates of Paschal full moon owing to changed views of the church of Alexandria with regard to the way in which dates of Easter had to be calculated. Contrary to the sequence of dates of Alexandrian Paschal full moon the sequence of dates of preanatolian Paschal full moon is not exactly known, but fortunately we may make use of the (metonically structured) sequence of most probable dates of preanatolian Paschal full moon (see Table 3), and in addition, thanks to the Alexandrian formula for the date of Paschal Sunday (see Section 4), of the sequence of most probable dates of preanatolian Paschal Sunday.
Seven medieval manuscripts are known containing a more or less complete version of a text entitled “De Ratione Paschali”. Accoding to Daniel McCarthy (university of Dublin) and Aidan Breen (idem) that text is a translation (into Latin) of the unfortunately completely lost original Greek text of Anatolius’ Paschal cycle (see Section 4). All seven of those manuscripts contain one and the same sequence of subsequent Paschal dates without year indication, which sequence of dates has a period of 19 years but no metonical structure; in one of those seven manuscripts the text still contains in addition the original metonically structured sequence of epacts (see Section 4) being connected with this sequence of dates. At first sight the Paschal dates of “De Ratione Paschali”, each of them provided with an Anatolian lunar phase number (i.e. “age” of the moon according to Anatolius) being at least 14 but at most 20, seem to be Julian calendar dates, but according to Daniel McCarthy and Aidan Breen they are calendar dates belonging to a variant of the Julian calendar ingeniously invented by Anatolius, in which case we may call them dates of Anatolian Paschal Sunday, because within the framework of this (which I call for convenience) Anatolian calendar they indeed all fell on a Sunday. Within the framework of “De Ratione Paschali” the Paschal dates are one by one explicitly obtained from the corresponding epacts (in this case each epact is defined as the Anatolian lunar phase number of the first day of the Anatolian calendar year in question).
During a relatively short (about seven years lasting) time interval the Paschal dates of “De Ratione Paschali” must have been Anatolian as well as Julian calendar dates. Therefore it could be worth (e.g. in order to work out to what extent it is possible to relate that sequence of Paschal dates to the sequence of preanatolian or to the sequence of Alexandrian Paschal Sunday) making an inquiry into the sequence of dates that we obtain by taking those Paschal dates simply (by way of “first approximation”) as if they were Julian calendar dates, in which case it is preferable to speak of dates of Anatolian Paschal day (instead of of dates of Anatolian Paschal Sunday), because in the framework of the Julian calendar they did not all fall on a Sunday. In particular it could be interesting to inquire into the sequence of dates of (which I call for convenience) Anatolian Paschal full moon that can be obtained out of the sequence of dates of Anatolian Paschal day by replacing each date of Anatolian Paschal day with the matching date having an (Anatolian) lunar phase number being 14. In Table 7 we see, next to the metonically structured sequence of epacts originally being part of “De Ratione Paschali” in column A, the sequence of corresponding dates of Anatolian Paschal day with the matching sequence of lunar phase numbers in column B, and next to this the sequence of corresponding dates of Anatolian Paschal full moon with the matching sequence of lunar phase numbers in column C.
Please note not only the differences between the definitions of the dates of preanatolian, Anatolian and Alexandrian Paschal full moon, but also the ones between the definitions of the dates of Anatolian Paschal Sunday, Anatolian Paschal day and Anatolian Paschal full moon. Supposedly the sequence of dates of Anatolian Paschal full moon was implicitly used by Anatolius to construct the sequence of Paschal dates of “De Ratione Paschali”. The fact that 23 March (and not 22 or 21 March) is the earliest possible date of Anatolian Paschal full moon, is a decisive reason why the sequence of Pachal dates of “De Ratione Paschali” must go back to the second half of the third century and so indeed in all probability must originate from Anatolius.
Until recently (June 2009) there was a difficulty with regard to the anchoring of Anatolius’ Paschal cycle in the Christian era (see Section 5), but with the help of our sequence of most probable dates of preanatolian Paschal full moon and the sequence of dates of Anatolian Paschal full moon it is not so difficult to solve this problem. Comparing the sequence of dates of Anatolian Paschal full moon with our sequence of most probable dates of preanatolian Paschal full moon (both sequences have a period of 19 years, so there are no more than 19 possibilities here) for the case that the first Paschal date (16 April) of “De Ratione Paschali” was intended for any year 260 modulo 19, which case according to Daniel McCarthy and Aidan Breen is the very first possible case eligible for closer examination, we must establish that in this case only 5 times out of 19 the date of Anatolian Paschal full moon differs no more than one day from the corresponding most probable date of preanatolian Paschal full moon. If the first Paschal date of “De Ratione Paschali” really would have been intended for any year 260 modulo 19 then in all probability all or almost all 19 times out of 19 the difference in question should have had to be 0 or 1 day. So the year 260 cannot have been the initial year of “De Ratione Paschali”.
In an analogous manner for the case that the first Paschal date of “De Ratione Paschali” was intended for any year 263 modulo 19 there can be established that only 11 times out of 19 the difference in question is 0 or 1 day. In only one case, namely in case the first Paschal date of “De Ratione Paschali” was intended for any year 271 modulo 19, all 19 times out of 19 the difference in question turns out to be 0 or 1 day, as in Table 8 (compare columns B and C). Because in each other in principle possible case none times out of 19 the date of Anatolian Paschal full moon differs no more than one day from the corresponding most probable date of preanatolian Paschal full moon, only the year 271 can have been the initial year of “De Ratione Paschali” (this is not so surprising, after all, because it must have been around the year 270 that Anatolius constructed his Paschal cycle).
We can use the result of the previous paragraph to make a table which gives an impression of the relations between the sequences of dates of Anatolian Paschal full moon, Anatolian Paschal day and Anatolian Paschal Sunday in order to work out on which Julian calendar dates the Anatolian Paschal Sunday fell and when the Anatolian Paschal day was a (real) Sunday. Hence in Table 8 (with dates according to the Julian calendar) we see for each indicated calendar year (in the primary column A) mentioned in column B the date of Anatolian Paschal full moon with matching (Anatolian) lunar phase number, in column C the date of Anatolian Paschal day with matching (Anatolian) lunar phase number, in column D the date of Anatolian Paschal Sunday (on the basis of column C). We may conclude that only eight times the Anatolian Paschal day was a (real) Sunday, to wit in the years 264 up to and including 271. Seven of those eight times the date of that Sunday coincided with the most probable date of preanatolian Paschal Sunday (and also seven times out of eight with the date of Alexandrian Paschal Sunday). Remarkable is the concentration of Anatolian Paschal days falling on Sunday around or shortly before or after the moment at which Anatolius’ Paschal cycle was constructed.
In order to gain insight into the way in which the sequence of dates of Alexandrian Paschal full moon (with 21 March as earliest possible date) in the course of a time interval from around 260 to around 320 could have evolved, whether or not via Anatolius’ Paschal cycle, out of the sequence of dates of preanatolian Paschal full moon (with 23 March as most probable earliest possible date), we could bring together the restrictions of all sequences of dates of Paschal full moon which could have played a part in this development (these sequences of dates are all periodic with a period of 19 years) to (e.g.) the time interval consisting of the Julian calendar years between the years 270 and 290 in one table. Hence we can see in Table 9 (with dates according to the Julian calendar) for each indicated calendar year (in the primary column A) mentioned in column B the most probable date of preanatolian Paschal full moon (according to Table 3), in kolom C the date of Anatolian Paschal full moon (according to Table 8), in column D the “Anatolian date” of Paschal full moon suggested by the German historian Eduard Schwartz (around the year 1900), in column E the “Anatolian date” of Paschal full moon suggested by the American historian Alden Mosshammer (around the year 2000), in column F the date of Alexandrian Paschal full moon.
Looking at Table 9 first of all we see that there is surprisingly little difference (not to say a strong correlation) between columns B and C. It turns out that our favourite metonically structured approximation to the sequences of dates of preanatolian Paschal full moon, namely the one with its saltus lunae (see Section 3) at the transition from 288 to 289, differs (one day) in only 4 dates out of 19, namely just the latest possible 4 dates out of 19, from the sequence of dates of Anatolian Paschal full moon. Furthermore, these two essentially different but evidently yet very cognate sequences of dates appear to have the same earliest possible date 23 March (in the same year 281), being evidence of both their origin in the third century. On further consideration, our favourite metonically structured approximation to the sequences of dates of preanatolian Paschal full moon appears to be also just the very best metonically structured approximation to the not metonic sequence of dates of the Anatolian Paschal full moon, reason enough to take it as our hypothetical sequence of dates of the preanatolian Paschal full moon. In fact, we have difficulty escaping the impression that the (not metonic) sequence of dates of Anatolian Paschal full moon must have been obtained at the time, either directly or indirectly, from the metonically structured sequence of dates of preanatolian Paschal full moon by advancing the latest possible 4 dates out of 19 by a day. Anyway, it seems probable that the sequence of Paschal dates of “De Ratione Paschali” developed from the sequence of dates of preanatolian Paschal full moon via the sequence of dates of Anatolian Paschal full moon. The sequence of dates of Anatolian Paschal full moon seems to be the result of manipulating the sequence of dates of preanatolian Paschal full moon; it is here that we catch sight of the last missing link in the series “14 Nisan” ® “preanatolian Paschal full moon” ® “Anatolian Paschal full moon” ® “Anatolian Paschal day” ® “Anatolian Paschal Sunday”. The historical context of that series is formed by the churches of Alexandria and Laodicea around the seventh decade of the third century. We still note that with the exception of the second of the five all sequences of dates of Table 9 have a metonical structure and that any of the (mutually different) positions of the saltus lunae in the three last of the four metonically structured sequences presented in this table, unlike the position of the saltus lunae in our favourite metonically structured approximation to the sequences of dates of preanatolian Paschal full moon, differs from the position of the imaginary saltus lunae in the sequence of dates of Anatolian Paschal full moon induced by the position of the saltus lunae in the metonically structured sequence of epacts originally being part of “De Ratione Paschali” (see columns A and C of Table 7).
The remarkable correlation between columns B and C of Table 9 underlines not only the justness of our result that the year 271 must have been the initial year of “De Ratione Paschali” but also the relevance of the sequence of dates of preanatolian Paschal full moon as well as the one of the sequence of dates of Anatolian Paschal full moon and so also the great relevance of the sequence of Paschal dates of “De Ratione Paschali” itself. Moreover, that correlation supports in particular the justness of the assumption that in the second half of the third century the church of Alexandria considered not 21 but 22 March as the date of the March equinox (otherwise the dates in columns B and C concerning the year 273 would have been 21 March, not 20 or 19 April). On the other hand, we may safely assume that it is not later than around the year 310 that the church of Alexandria considered 21 March to be the date of the March equinox, although at the time the real date of this equinox was mostly 20 March.
By relating columns B, D, E of Table 9 to column C of this table we can establish that the sequence of “Anatolian dates” of Paschal full moon suggested by Eduard Schwartz and the one suggested by Alden Mosshammer, contrary to the sequence of dates of preanatolian Paschal full moon, differ to such an extent from the sequence of dates of Anatolian Paschal full moon that they cannot possibly have underlain Anatolius’ Paschal cycle. By the way, the date 21 March in the sequence of “Anatolian dates” of Paschal full moon suggested by Alden Mosshammer may be considered as an anachronism, because Anatolius was not acquainted with the real date of the March equinox in his time (sometimes 21 sometimes 20 March), which is evident from the fact that the earliest possible date of Anatolian Paschal full moon is 23 March.
At first sight a further analysis of the sequences of dates of Table 9 gives little more than the observation that there is an unbridgeable gap of roughly two days between on the one hand the sequence of dates of preanatolian and the sequence of dates of Anatolian Paschal full moon and on the other hand the sequence of “Anatolian dates” of Paschal full moon suggested by Alden Mosshammer and the sequence of dates of Alexandrian Paschal full moon and that even the sequence of “Anatolian dates” of Paschal full moon suggested by Eduard Schwartz cannot bridge this gap. So we must not exclude that the classical sequence of dates of Alexandrian Paschal full moon could have come in existence by means of any direct intervention in a then already existing sequence of Alexandrian dates of Paschal full moon, e.g. the sequence of dates of preanatolian or of Anatolian Paschal full moon. On the contrary, we must wonder which new views around the year 320 would have led to such a radically new manner of determining dates of Paschal full moon.
We call attention that each of the five sequences of dates presented in Table 9 belongs to one of three clearly different astronomical categories, namely the “waning full moon category”, “pure full moon category”, and “waxing full moon category” defined in Section 12. Most dates of preanatolian Paschal full moon as well as most dates of Anatolian Paschal full moon were, just like most dates of the fourteenth day of Nisan and most dates of Jewish Paschal full moon, marked by a sunset accompanied by a night waning full moon (first category). The “Anatolian dates” of Paschal full moon suggested by Eduard Schwarz were for the most part marked by a sunset accompanied by a night pure full moon (second category). The dates of Alexandrian Paschal full moon (and also the “Anatolian dates” of Paschal full moon suggested by Alden Mosshammer) were for the most part marked by a sunset accompanied by a night waxing full moon (third category).
The dates of Anatolian Paschal full moon belong for the most part to the waning full moon category, which is in perfect accordance with the fact that they were defined amply before the third turn of century. Because on the face of it both a waning full moon and a waxing full moon look like a pure full moon (see Figure 4), the church of Alexandria, which almost two centuries after the elimination of the Jewish community of Jerusalem possibly no longer wanted to have to do with the whims of the (then still not exactly calculable) Jewish calendar (see Section 3), which then still was determined in Palestine and used nearly exclusively in Palestine, found itself around the year 320 in a position to choose its metonically structured sequence of dates of Alexandrian Paschal full moon with no heed for distinction between the aforesaid three astronomical categories. As a matter of fact, at the time she wanted to distance herself as much as possible from Jewish principles.
It is columns B and F of Table 9 that reflect the difference between the old way in which before and the new way in which after the third turn of century Alexandrian computists constructed their metonically structured sequence of dates of Paschal full moon. Around the year 260 the old way resulted, as we have seen in Section 4, via adding 13 days to approximate dates of the first day of the computistical month Nisan* implicitly defined by third century Alexandrian computists, in the construction of the metonically structured sequence of dates of preanatolian Paschal full moon. Likewise around the year 320 the new way resulted, via adding 13 days to approximate dates of the first day of a still to be determined computistical month Nisan^ implicitly defined by fourth century Alexandrian computists, in the construction of the classical metonically structured sequence of dates of Alexandrian Paschal full moon; we realize that in the framework of the construction of this famous sequence of dates these Alexandrian computists must have been interested in Alexandrian calendar dates of Newmoon (see Section 3) insofar as belonging to a certain relevant sufficiently large time interval J^ round the third turn of century, e.g. the time interval consisting of the time between the years 280 and 320. It is plausible that around the third turn of century the church of Alexandria changed her view on the extent to which she, at least where the calculating of the dates of Easter was concerned, had to allow for the primeval Jewish principle of observing the moon and that it is this change of view that led up to an advance by roughly two days of her dates of Paschal full moon, e.g. by means of the replacement of the old (rather rough) principle ‘the first day of Nisan is the first day after the second sunset in Jerusalem after the Newmoon of Nisan’ with a new (rather rough as well) principle like ‘the first day of Nisan^ is the day of the first sunset in Alexandria after the Newmoon of the computistical month Nisan^’. Just like the computistical month Nisan* (see Section 4), such a computistical month Nisan^ cannot be Nisan itself but must differ somewhat (in this case perchance a few days) from Nisan, which difference in this case is mainly caused by the fact that in the first half of the fourth century Alexandrian computists had to bear in mind both such a new principle and the requirement that the earliest possible date of the fourteenth day of their computistical month Nisan^ had to be 21 or 22 March (certainly not 23 March), owing to the fact that around the third turn of century the church of Alexandria had gone to consider 21 March as the date of the March equinox.
We may test the new principle staged by way of example in the previous paragraph by trying to reconstruct the classical sequence of dates of Alexandrian Paschal full moon by means of taking our depature from the combination of this new principle with the new March equinox date 21 March. In order to be able to succeed therein first of all we will have to mark out the computistical month Nisan^ by determining a lower and an upper limit (differing roughly the synodic period of the moon) between which in all probability all local Alexandria points in time of Newmoon of the computistical month Nisan^ belonging to the time interval J^ in question occurred. We are able to realize that by taking care that the earliest possible date of the dates of the fourteenth day of the computistical month Nisan^ to be obtained will be 21 or 22 March by taking our departure (this time necessarily not from a lower limit 8 March 6:00 and an upper limit 6 April 18:00 but) e.g. from a lower limit 7 March 23:00 and an upper limit 6 April 11:00 for all those local Alexandria points in time of Newmoon of the computistical month Nisan^ (we note that adding 1 + 13 days to 7 March gives indeed 21 March).
It is the structure of Table 10 that reflects in essence the simple and as formalistic as possible way in which in the fourth century Alexandrian computists constructed their classical sequence of dates of Alexandrian Paschal full moon. In column B we see estimated local Alexandria points in time of Newmoon of the computistical month Nisan^. In column C we see approximate dates of the first day of the computistical month Nisan^, one by one obtained from the corresponding points in time of column B by determining the day of the first sunset in Alexandria after the Newmoon of the computistical month Nisan^. In column D we see approximate dates of the fourteenth day of the computistical month Nisan^, one by one obtained from the corresponding dates of column C by adding 13 days to each of them. It will turn out that there are four slightly different metonically structured sequences, one of them just the metonic sequence of dates of Alexandrian Paschal full moon, that are equally best of all metonically structured approximations of the sequence of dates of column D.
The best way to reconstruct the classical metonically structured sequence of dates of Alexandrian Paschal full moon seems to be the way in which in section 3 the metonically structurered sequence of dates of Jewish Paschal full moon and in section 4 the one of preanatolian Paschal full moon was reconstructed. So let us try to reconstruct the sequence of dates of Alexandrian Paschal full moon by means of a reconstruction of the position of its saltus lunae. Seeking a lead in column D of Table 10 we see, after having established that in this column three times a recurrence of a similar “regular metonic little piece” consisting of two or more dates occurs, that its saltus lunae must be located between 29-3-283 and 13-4-287 modulo 19 years. For any metonically structured sequence of dates it is evident that any transition in four steps from 29 march to 13 april must consist of two advances and two regressions, the difference of 15 days betraying that there must be a saltus in here somewhere; so either one of these two advances must be a step of 12 instead of 11 days or one of these two regressions must be a step of 18 instead of 19 days. Because with regard to the position of the saltus lunae between 29 march and 13 april in four steps obviously there are four possibilities and with regard to the remaining 15 out of 19 steps there is no more than one possibility, we may conclude that in the framework of our seeking the best metonically structured approximation of the sequence of dates of column D we must still weigh up four metonically structured sequences of dates (see column E). Analyzing the four intended sequences of dates in column E (by determining in each of the four cases the sum of the absolute values of the deviations with respect to the sequence of dates of column D), we may conclude that they are the equally best of all four metonically structured approximations of the sequence of dates of column D. Extending our investigation to the time interval consisting of the time between the years 240 and 320, we make no progress in our pursuit to explain why it is just the second one of those approximations (the one with its saltus lunae at the transition from 284 to 285) the church of Alexandria opted for. However, it is surprisingly by restricting our investigation to (e.g.) the time interval consisting of the time between the years 290 and 320 that we may reach that aim. The part in question of column D points unmistakebly to the classical sequence of dates of Alexandrian Paschal full moon.
Around the year 320 there were two stimuli that compelled the church of Alexandria to replace her metonically structured sequence of dates of preanatolian Paschal full moon with her classical sequence of dates of Alexandrian Paschal full moon, the first being the switch of her date of the March equinox from 22 to 21 March, the second her desire to distance herself as much as possible from Jewish principles, the latter resulting in an advance of her dates of Paschal full moon by roughly two days. As a matter of fact, that replacement of the old dates of preanatolian with the new dates of Alexandrian Paschal full moon made the dates in question fall mostly on or before the twelfth instead of mostly on or after the fourteenth day of Nisan. We remark that in years 292 modulo 19 the date of preanatolian as well as the date of Alexandrian Paschal full moon, contrary to the date of Roman Paschal full moon, fell outside Nisan.
After all, the sequence of dates of Jewish Paschal full moon defined in Section 3, just like the classical sequence of dates of Alexandrian Paschal full moon a sequence of dates with 21 March as earliest possible date and 18 April as latest possible date, but unlike this sequence of dates a real metonically structured approximation of “the” sequence of dates of the fourteenth day of Nisan, really would have been an ideal sequence of dates of Paschal full moon to generate dates of Paschal Sunday which between the years 310 and 1582 would as good as never have coincided with any date of the fourteenth day of Nisan (see also Section 16).
15 solar eclipse
Unlike the date of the day on which in Palestine traditionally early in spring Pesach (see Section 3) was prepared, which day always coincided with the fourteenth day of Nisan (see Section 3), the date of Alexandrian Paschal full moon (see Section 4) in the time it was defined (around the year 320) coincided mostly with the date of the twelfth day of Nisan. We justified that statement in Section 12 and not surprisingly no facts are known which are inconsistent with it (and as for the vague formula “Paschal full moon = 14 Nisan”, it is just the question to what extent this is a fact which is under discussion here). That does not exclude that it might be worth pointing out that the difference of mostly two days between date of Alexandrian Paschal full moon and date of the fourteenth day of Nisan is perfectly in line with the difference of on average exactly two days between the dates of Alexandrian and the dates of Anatolian Paschal full moon (see Section 14), which can be easily deduced from Table 9 (compare column C with column G). In addition there are two interesting remarks Beda Venerabilis (see Section 4) made around the year 720 as a result of his analysis of the moment at which the famous total solar eclipse observed in Britain and Ireland in the year 664 occurred, which lead to a conclusion which is illustrative for the relatively large shift with respect to Nisan by which the replacement of the dates of preanatolian with the dates of Alexandrijnse Paschal full moon was accompanied. The first of those remarks is the (correct) observation that “the moon sometimes appears older than its computed age”, the second concerns his (wrong) assessment of the date on which that solar eclipse occurred.
In the year 664 the date of Alexandrian Paschal full moon was 17 April and the date of the accompanying (actual) Fullmoon was 16 April; these dates were preceded by the unnoticed, after all invisible, (actual) Newmoon of 2-4-664 (in Britain early in the morning), and succeeded by the Newmoon of 1-5-664, which however did not go by unnoticed, for this Newmoon was accompanied by a total solar eclipse which was observed in Britain and Ireland in the afternoon. According to observers that solar eclipse indeed occurred on 1 May so already fourteen days after the date of Alexandrian Paschal full moon of 17-4-664, which was understanded then by no one, as little as by Beda Venerabilis half a century later. Beda Venerabilis did understand that the last Newmoon preceding that solar eclipse must have taken place 29 days earlier (in Britain early in the morning) than that solar eclipse. But assuming that on 17-4-664 the “age” of the moon was 14 days and that the Alexandrian lunar phase number of the day of the last Newmoon preceding that solar eclipse was 1, he thought to have to conclude that it was Newmoon on 4-4-664 and furthermore that that solar eclipse must have been happened on 3-5-664. He could impossibly accept that on 3-5-664 the moon could have been two days “older” in the heavens than in his tables.
The mistake Beda Venerabilis made with respect to the date of the solar eclipse observed in Britain and Ireland in the year 664 can be explained by remarking that he, though familiar with the fact that at the transition from 664 to 665 a saltus lunae (see Section 3) of the sequence of dates of Alexandrian Paschal full moon occurs, but being ignorant of the fact that in the second half of the seventh century his (thought up by us) great clock (see Section 11) lost already more than a whole day, neither he could know that in the year 664, due to astronomical causes mainly connected with the mutual motions of sun, earth and moon which were unknown to him, the date of Alexandrian Paschal full moon could very well coincide with the date of the fifteenth (instead of the implicitly presupposed thirteenth) day after the last Newmoon preceding this solar eclipse. Besides we establish that Beda Venerabilis, on the other hand, had no trouble, perhaps even was familiar, with the idea (stemming from the first quarter of the fourth century) of an Alexandrian Paschal full moon on a date at the same time thirteen days after the date of the last Newmoon and sixteen or seventeen days before the date of the next Newmoon; we have to do here with a phenomenon which was around the year 320 indeed a very usual thing (as a matter of fact occurred in a half of the first forty calendar years of the fourth century), but had occurred in the year 647 for the last time (owing to the more and more getting behind of Beda Venerabilis’ great clock). That phenomenon is well matched to our idea of dates of (postanatolian) Alexandrian Paschal full moon originally insofar as not falling outside Nisan preceding the date of the neighbouring Fullmoon with a difference of on average about 1.5 days (see Section 12). Of course the moment of Fullmoon itself is on average located midway between the two neighbouring moments of Newmoon; but it is just only in the eighth century that the date of Alexandrian Paschal full moon was on average located in this position.
Beda Venerabilis’ great clock lost after the moment of exactly keeping time (around the third turn of century) with the elapsing of time more and more time, namely roughly one day every three centuries (see Section 11). Originally (around the year 300) the (postanatolian) dates of Alexandrian Paschal full moon on average preceded the neighbouring Fullmoon with a difference of about 1.4 days, as in the years 284, 292, 303 and 311 the neighbouring Fullmoon fell on 17, 19, 18 and 19 April respectively. So, roughly speaking, originally (post-Anatolian) Alexandrian Paschal full moons were seemingly full but mostly waxing moons on average about 1.4 days younger than Fullmoon, and it is around the eighth century that they were usually pure full moons. Around the eleventh century they were mostly waning full moons on average about one day older than Fullmoon. However, long before the tenth, even (very exceptionally) in the seventh century, sometimes waning Alexandrian Paschal full moons one day older than Fullmoon occurred, but at the time nobody was aware of it.
Starting from the original (around the third turn of century) connection between the dates of Alexandrian Paschal full moon and the dates of Fullmoon (see Section 12) it is easy to understand why it still had to last so long before it became possible to establish by means of direct observation of the moon that the Alexandrian Paschal full moon clearly had become a waning moon. During more than eight centuries the Alexandrian Paschal full moon was usually a full moon, i.e. with the naked eye indistinguishable from Fullmoon (see Figure 4), and not surprisingly it is only after the twelfth century that it became clearly observable “too old”. That waning Alexandrian Paschal full moon and a much too early date of the March equinox (see Section 7) were only two out of many pressing problems with which the church of Rome was confronted in the sixteenth century; at least these two problems were solved in the year 1582 in a reasonable way by means of the replacement of the Julian calendar with the Gregorian calendar (see Section 3) and (of course at the same time) the replacement of Beda Venerabilis’ Easter cycle (see Section 4) with Easter tables adjusted to the new calendar.
16 coincidences
It is just the ultimate choice (around the year 320) of the church of Alexandria (Egypt) for her dates of Alexandrian Paschal full moon (see Section 4) which was the cause of the well known fact that between the years 320 and 360 the date of Easter must have coincided more than once with the date of the fourteenth day of Nisan (see Section 14). For example, in the year 323 the most probable date of the fourteenth day of Nisan was Sunday 7 April and the date of Alexandrian Paschal full moon Friday 5 April, which because of the principle “Paschal Sunday is the first Sunday after the Paschal full moon” implies that two years before the first council of Nicaea (see Section 4) the date of Alexandrian Paschal Sunday (see Section 4) probably coincided with the date of the day on which Pesach (see Section 3) was prepared.
At the first council of Nicaea was stated emphatically that henceforth the date of Paschal Sunday should never coincide with the date of the fourteenth day of Nisan, in the morning and afternoon of which in Palestine traditionally Pesach was prepared. One then thought to avert the “danger” of such a coincidence by equalizing any “Paschal full moon” whatever to the corresponding fourteenth day of Nisan. That (illusory) equalization, resulted in the Christian tradition which says that “Paschal full moon = 14 Nisan”, but not at all to a solution of the problem in question. By the fixation of the Jewish calendar (see Section 3) about the year 360 that problem was only partially solved.
Between the year 325 (in which year the first council of Nicaea took place) and the year 360 (about which year the Jewish calendar was fixed) there was indeed a real danger of coincidence of the date of the fourteenth day of Nisan with the date of Paschal Sunday. That emerges from Table 11 (with dates according to the Julian calendar).
Around the third turn of century, and hereafter still until the moment at which the Jewish calendar was fixed (about the year 360), the beginning of the new month and of the new year of this not exactly calculable calendar was determined officially still in Palestine and still as in the first century of our era (see Section 3). In order to be able to obtain (approximate) dates of the fourteenth day of Nisan belonging to a substantial time interval K round the year 340, like the time interval consisting of the time between the years 320 and 360, we will firstly have to obtain (approximate) dates of the first day of Nisan from points in time of the Newmoon of Nisan, but in order to be able to do this it is necessary to mark Nisan out beforehand by determining a lower and an upper limit (differing roughly the synodic period of the moon) between which by far the most of the local Jerusalem points in time of Newmoon of Nisan belonging to the time interval K in question occurred, allowing for the principle that Pesach had to be celebrated as early as possible in spring and the fact that around the year 340 the real date of the March equinox was 20 March. We are able to realize that lot by taking care that the earliest possible date of the dates of the fourteenth day of Nisan to be obtained will be 20 or 21 March by taking our departure (this time necessarily not from a lower limit 4 March 18:00 and an upper limit 3 April 6:00 but) e.g. from a lower limit 4 March 22:00 and an upper limit 3 April 10:00 for all those local Jerusalem points in time of Newmoon of Nisan (we note that adding 3 + 13 days to 4 March gives indeed 20 March). Hence in Table 11 we see for each indicated calendar year (in the primary column A) mentioned in column B the best possible estimated point in time for Jerusalem of the (actual) Newmoon of Nisan, in column C the most probable date of the first day of Nisan estimated on the basis of column B (in just the same way as in Section 3), in column D the date of Alexandrian Paschal full moon, in column E the most probable date of the fourteenth day of Nisan estimated on the basis of column C, in column F the date of the Alexandrian Paschal Sunday (on the basis of column D).
According to Table 11 between the years 325 and 360 the preparation of Pesach in Palestine could have taken place perchance still about three times on and about four times one day after Paschal Sunday as well. In each of those cases the cause was rather a “too early” Paschal full moon than a “too late” Pesach (and anyway no “too early” Pesach). It is certain that in the year 346 the Christian Paschal feast was celebrated, with consent of Athanasius, at the time bishop of Alexandria, not on the date of the Alexandrian Paschal Sunday but a week later (on 30 March, the date of the Roman Paschal Sunday). By the way, Table 11 confirms our conclusion (in Section 12) that between the years 280 and 360 the date of Alexandrian Paschal full moon mostly coincided with the date of the twelfth day of Nisan (compare column D to column E); also our conclusion (in Section 12) that in the years 330 modulo 19 between the years 280 and 360 the date of Alexandrian Paschal full moon mostly fell roughly on the twelfth day of Iyyar, is confirmed by this table.
In order to get insight into the consequences of the fixation of the Jewish calendar (about the year 360) for the date of the fourteenth day of Nisan related to the date of the (actual) Fullmoon of Nisan, which date formerly usually coincided with the date of the thirteenth or of the fourteenth day of Nisan (see Section 3), as well as for the date of the Alexandrian Paschal Sunday related to the date of the fourteenth day of Nisan during the part of the fourth century from the moment at which the Jewish calendar was fixed, we look at Table 12 (with dates according to the Julian calendar); in this table we see for each indicated calendar year (in the primary column A) mentioned in column B the date of Alexandrian Paschal full moon, in column C the best possible estimated point in time for Jerusalem of the Fullmoon of Nisan, in column D the date of the fourteenth day of Nisan according to the fixed Jewish calendar, in column E the date of Alexandrian Paschal Sunday (on the basis of column B). We still note that the sequence of dates of column D does not have a period of 19 years, let alone a metonical structure.
From Table 12 it appears that between the years 360 and 400, thanks to the fixation of the Jewish calendar about the year 360, the Alexandrian Paschal Sunday never preceded the fourteenth day of Nisan, but coincided with it still four times (compare columns D en E). We establish that the Alexandrian Paschal Sunday coincided with the fourteenth day of Nisan in the years 367, 370, 374, 394. In each of those years very probably the preanatolian Paschal Sunday would have fallen just a week after the fourteenth day of Nisan, whitch can easily be verified. Also appears from Table 12 that between the years 360 and 400 the date of the fourteenth day of Nisan fell on average about 0.3 days later than the date of the Fullmoon of Nisan, and that in the years 368, 379, 387, 398 the date of the Alexandrian Paschal full moon fell in Iyyar instead of in Nisan, in accordance with our initial impression (see Table 5) that in years 303 modulo 19 and in years 311 modulo 19 the Alexandrian Paschal full moon must have fallen outside Nisan many a time.
17 epilogue
It is moment 1999, i.e. [31-12-1999 24:00] = [1-1-2000 0:00], the “magic” moment at which all four digits of the number of the calendar year at present changed, which was at the same time the moment of the start of the countdown to what must pass for the next turn of millennium. So at present we are going with perfect precision straight to millennium mistake 3. It is to be hoped that towards the year 3000 people will know any better, for otherwise then once again we shall have to undergo, all over the world, how a crowd of singing and dancing people, made mad by commerce, media and authorities and one year too early, is waiting on the platform for the next millennium train, to get on then “all together” by mistake in the last year local preceding this millennium train. To be precise again: the last year train preceding the fourth millennium train will leave at [1-1-3000 0:00], the fourth millennium train itself will leave at [1-1-3001 0:00], for, do you remember (see Section 5), the first millennium train left at the moment zero of our era, i.e. at [1-1-1 0:00], in order to reach its final destination at [31-12-1000 24:00].
Around the year 2000 certainly more than six hundred websites were made in which attention was given to the millennium question. In most of those websites one declared oneself, like in the sextilingual website “Millennium”, in favour of the proposition that the year 2001 is the first year of the third millennium and related this rightly to the fact that in our era we have no year zero. But, and this is the first (and original) reason for being of this website, only on this website “Millennium Mistake” and on its (Dutch language) alter ego “Millenniumvergissing” one can find in addition the observation that the absence of a year zero is not in the least a mistake of Dionysius Exiguus (see Section 2) or Beda Venerabilis (see Section 4) but as a matter of fact purely and simply a condition the Christian era (see Section 5) must satisfy in order to preserve her bilateral symmetry (see Section 6). Our era has no year zero because we want, either conscious or intuitive, to keep our era symmetrical with respect to its moment zero, as in our second time line (see Figure 2).
A second reason for being of this website is the publication (since the year 2004) within the framework of this website of the result of my investigation into Anni Domini (see Section 11). It is Table 4 which played at that investigatioan an important part; this table provides us not only with the most probable two possible dates (of course according to the Julian calendar) of Jesus’ dying day, but also with a first indication that the ultimate substitution of “the” (not exactly calculable) sequence of successive dates of the fourteenth day of Nisan (see Section 3) for the classical sequence of dates of Alexandrian Paschal full moon (see Section 4), which substitution the church of Alexandria (Egypt) ultimately implemented on behalf of her classical Alexandrian Paschal tables composed around the year 320, must have gone hand in hand with rather substantial changes of position with respect to Nisan which are not in agreement with the principle “Paschal full moon = 14 Nisan”. As a consequence of those changes of position the formula “Alexandrian Paschal full moon = 12 Nisan” had in that time a higher degree of truth than the more natural formula “Alexandrian Paschal full moon = 14 Nisan”. Hence a third reason for being of this website is the publication (since the year 2005) within the framework of this website of the result of my further investigation into those changes of position (see Section 12), about which subject (to my knowledge) never was published before. Neither in the standard work about the important book “De Temporum Ratione” of Beda Venerabilis written by Faith Wallis (see Section 5) nor in the essay “Anno Domini” about early Christian chronology and origin and spread of the Christian era written by Georges Declercq (see Section 11), as little as in the extensive study recently written by Alden Mosshammer (see Section 14) which is largely about the calculation of Easter in the time of early Christianity, such changes of position come up. Thus unintentionally the impression is given as if in the fourth century there would be no great difference (certainly not greater than a day) between “the Paschal full moon” and “14 Nisan”, a misconception going back to the first half of the fourth century (see Section 16).
The ultimate replacement of (not exactly calculable) dates of the day on which Pesach (see Section 3) was prepared with dates of Alexandrian Paschal full moon which the church of Alexandria applied on behalf of the construction of her Paschal tables, involved for the years 330 and 349 of the time interval between the year 320 and the moment (about the year 360) on which the Jewish calendar was fixed a shift of roughly 28 days to roughly the twelfth day of Iyyar and for the other Julian calendar years of this time interval an advancing by on average two days to roughly the twelfth day of Nisan (see Section 12). It is that adaptation of Nisan to the Alexandrian calendar (see Section 3) which gave the impetus to the origin of the classical Alexandrian Paschal tables by means of which from the fourth to the eighth century the date of Alexandrian Paschal Sunday and from the eighth to the sixteenth century for all churches the date of Easter Sunday was determined. But that adaptation of Nisan to the Alexandrian calendar must in the fourth century also have resulted in some celebrations of Easter on or one day before a fourteenth day of Nisan (see Section 16); in each of these cases the cause was rather a “too early” Alexandrian Paschal full moon than a “too late” Pesach.
A fourth reason for being of this website is the publication (since the year 2006) within the framework of this website of an explanation from the perspective of the sequence of dates of preanatolian Paschal full moon (see Section 4) for the rather drastic manner in which around the beginning of the fourth century the church of Alexandria definitively put things in order with regard to her dates of Paschal full moon by means of the construction of her classical sequence of dates of Alexandrian Paschal full moon (see Section 14), a fifth the publication (since the year 2007) within the framework of this website of a consideration devoted to the possible coincidences of the date of Alexandrian Paschal Sunday with the date of the thirteenth or of the fourteenth day of Nisan in the fourth century (see Section 16), already touched on in the previous paragraph. Since the year 2009 Section 14 contains still another new first publication with regard to the connection between the sequence of preanatolian and the sequence of Anatolian Paschal full moon (see Section 14) and the anchoring of Anatolius’ Paschal cycle (see Section 4) in the Christian era.
18 author
Jan Zuidhoek (see Figure 5), the author of this English language website “Millennium Mistake”, was born in the year 1938, studied mathematics (with physics and astronomy) at the university of Utrecht from 1960 to 1969, and was a teacher of mathematics from 1970 to 2001 at the Gymnasium Celeanum in Zwolle. This website evolved from the (Dutch language) article “Millenniumvergissing” that he, inspired to this by critical pupils who wanted to know all the ins and outs, wrote in the year 2000 about the millennium question for Euclides, the organ of the Dutch association of teachers of mathematics. The aim of this website is to make a scientifically solid contribution to chronology. That applies also for his contribution to the third international conference in Galway on the history of computus (see Section 3), in the year 2010, where he presented his paper (see Article 3) about the initial year of “De Ratione Paschali” (see Section 14) and the relevance of its Paschal dates. In that paper both the important sequences of dates provided with a metonical structure (see Section 3), the one of preanatolian Paschal full moon (see Section 4) and the one of Alexandrian Paschal full moon (see Section 4), are reconstructed according to the global method of determining the first day of the first month of the Jewish calendar (see Section 3).