question of chronology

© 2000 Jan Zuidhoek 2016

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0 prologue

This section is an extension of Section 0 of millennium question.

The millennium hype which preceded the second turn of millennium was the occasion for the coming into being of this website (called millennium). Since then it has gradually, as of oneself, grown into the sextilingual website that it is now. Its rather provisional character can be explained from this. In the year 2011 the former, much smaller, websites ‘Millenniumvergissing’ and ‘Millennium Mistake’ of the same author, whose ambitions, incidentally, are only of a scientific and not of a literary nature, were included in this website.

Each of the six versions differing in language of this sextilingual website consists of two chapters, in this (english language) version called millennium question and question of chronology, which both contain much more than the information needed so as to be able to solve the millennium question (at which point of time did the third millennium begin?) by removing the misunderstanding on which the millennium mistake (being the misconception that the third millennium began at the moment of the transition from 1999 to 2000) was based. The first of these two chapters has emanated from the very first version of this website and has been written in six languages (Dutch, English, German, French, Italian, and Spanish), the second has been written in Dutch (kwestie van chronologie), in English (question of chronology), in German (frage von chronologie), and in French (question de chronologie), and still only partly in Italian (questione di cronologia) and in Spanish (cuestión de cronología). The Dutch language version of the second has emanated from the former website ‘Millenniumvergissing’, the English language version of the second from the former website ‘Millennium Mistake’ of the same author.

The two chapters of which this website consists, contain the same subjects, but in the second these subjects are treated more extensively than in the first. These subjects are our era in its capacity of linear system of numbered calendar years (this subject is treated in Sections 1 and 2), the millennium question (in Sections 3 and 4), calendars in antiquity (in Section 5), Alexandrian Paschal full moons (in Sections 6), Paschal cycles (in Section 7), Metonic structure (in Section 8) and the date of Jesus’ crusifixion (in Section 9). They are connected with questions which we may encounter in the field of chronology, which, as the science of locating historical events in time, is part of the professional field of history, and can even be considered as the backbone of history.

In practice locating an event in time boils down to placing the moment of the event in question in the framework of our era, i.e. the Christian era, the most widespread chronological system on earth. Dates (of events) are in principle dates of the Christian era, which however did not begin on the day Jesus was born. This era is a (complete) linear system of numbered calendar years, and has as such a somewhat curious but nevertheless consistent structure. It is starting from its moment 0, i.e. its initial moment, i.e. the point of time from which its calendar years are counted, that we can solve the millennium question. That point of time, which is called moment zero, has been defined only afterwards: first, in the sixth century, only implicitly (see also Section 1), later, in the sixteenth century, explicitly (see also Section 2). In all probability Jesus was born some years before moment zero.

At moment zero it was midnight in Greenwich, by definition. At that moment the year 1, i.e. the year 1 of our era, i.e. the starting year of the Christian era, began. The year 1 ended at the moment 1, i.e. the first turn of year, of our era, precisely 365 days after moment zero. Summarizing we can say that moment zero, being the point of time from which the years 123……, i.e. the years 123…… of our era, are counted, is nothing else than the midnight point of time at which the first day of the first month of the year 1 in Greenwich began, in other words the point of time 0:00 of 1-1-1, in modern notation [1-1-1; 0:00]. It is this unique point of time that is denoted by the logo of this website (1-1-1, 00:00:00). In this way not only moment zero but each point of time of our era can be denoted accurately to a second. Thus all digital clocks that show the coordinated universal time, e.g. the digital clock that is part of the main page of this web site (see Figure 0), show, continuously and accurately to a second, points of time of our era.

Just on Christmas day of the year 800 Charlemagne let himself be crowned emperor. This implies that he believed that on that day, seven days before the beginning of the year 801, it was just eight centuries ago that Jesus was born.

In the month of December of the year 1799 the British newspaper ‘The Times’ must have received many letters on the question of when the eighteenth century would end, for in its edition of 26-12-1799 its editors refused all letters and any discussion on this question, stating that obviously the eighteenth century would not terminate before the year 1801.

The planetoid Ceres was discovered by the Italian astronomer Giuseppe Piazzi; this chanced to happen on 1-1-1801, the day at the time generally considered by scientists to be the first day of the nineteenth century. Although the German emperor Wilhelm II had proclaimed the opinion (on 1-1-1900) that the twentieth century had begun with the moment of the transition from 1899 to 1900, outside Germany celebration of the nineteenth turn of century took place for the most part precisely one year later (on 1-1-1901). However, against the end of the twentieth century, under the influence of the mass media, most people went to take the “magic” moment of the transition from 1999 to 2000 for the second turn of millennium, in fact a logical consequence of the early medieval conviction that the moment of the transition from IM999 to M1000 was the first (and the last). That is why the second turn of millennium was extensively celebrated all over the world on 1-1-2000.

To the remark that the year 2000 was the last year of the second millennium, around the year 2000 people often reacted with a denial, such as: “oh no, the year 2000 was the first year of the new millennium, for the year zero was the first year of our era”. Perhaps on the face of it such a reaction does not seem bad at all, for a millennium is by definition a time interval existing of exactly one thousand years. But what is meant by “the year zero”? In order to answer this question and to solve the millennium question, we must pay attention to the structure of our era. Evidently the millennium question is a question of chronology.

After having taken note of the history of the coming into existence of our era (in Section 1), we will establish that there is simply no year zero in our era and explore why our era contains no year zero (in Section 2). As soon as the connection between moment zero and the millennium question has been established, the solution of this problem (see Section 3), as well as the justification of the term ‘millennium mistake’ (see Section 4), is within easy reach. Not surprisingly it is just the sections mentioned in this paragraph which together represent the original core of each of the two former websites ‘Millenniumvergissing’ and ‘Millennium Mistake’ (in different languages but with the same content) of the same author. Clarifying remarks about and sceptical reactions to the explanation in question led to emendations in Section 1 or Section 2 or were incorporated in the deductions of Section 3 or in the objections of Section 4.

 

1 moment zero

This section is an extension of Section 1 of millennium question.

The calendar years of our era are counted from moment zero (zie Paragraaf 0). Moment zero is nothing else than [1-1-1; 0:00]; it is the in Greenwich midnight point of time from which not only the calendar years but also the numbered decades, centuries, millennia of our era are counted. The year 1 began with moment zero and ended with the first turn of year. Likewise the first decade began with moment zero and ended with the tenth turn of year. Therefore the year 10 is the last year of the first decade. We note that the first decade ended exactly one year after the moment of the transition from 9 to 10. This is nothing special: each moment at which the last digit of the number of the current calendar year suddenly becomes zero, is the presage of a turn of decade, always exactly one year later.

The Julian calendar is a drastically improved version of the old Roman calendar. In Roman antiquity sometimes years of the Roman calendar, which in principle began and ended in winter, were counted from any supposed year of foundation of the city of Rome. More than five centuries after the Roman year 754, i.e. the year 754 of this (incomplete) Ab Urbe Condita (literaly ‘From the Foundation of the City’) era, this inconspicuous year of the Julian calendar would be chosen as the starting year of our era.

Still before the beginning of our era the Julian calendar was introduced by Julius Caesar. In the year 1582 this calendar was replaced with the Gregorian calendar by decree of pope Gregorius XIII. The Julian calendar underlies the calendar years of the Christian era (see Section 0) before that year, the Gregorian calendar the ones after the year 1582. The year 1582, which comprised only 355 days (see also Section 5), is the only exception to the rule that a calendar year of the Christian era consists of 365 or 366 days (see also Section 5). The two calendars in question differ solely in their leap year regulation, i.e. regulation according to which it is determined which calendar years are leap years, i.e. consist of 366 instead of 365 days (see also Section 5). The calender years of our era before the year 1582 are years of the Julian calendar, the calender years of our era after the year 1582 are years of the Gregorian calendar. The dates of our era before the year 1582 are dates of the Julian calendar, the dates of our era after the year 1582 are dates of the Gregorian calendar.

The founder of our era is the monk and scholar Dionysius Exiguus, who, originating from a region in or near the Danube delta area, settled in Rome about the year 500. In or shortly before the year 526 he presented his Paschal table (see Table 1) at the request of a few officials of the papal chancery. Unfortunately neither this excellent Easter table nor his new era included in this table was accepted then by the church of Rome. This happened not earlier than in the seventh and tenth century respectively. Dionysius Exiguus’ Paschal table is a continuation of a Paschal table attributed to bishop Cyrillus of Alexandria (Egypt) which must have been composed in Alexandria about the year 440 and was provided with two interesting sequences of dates of the Julian calendar of which the dates were numbered according to the era of the emperor Diocletianus used by the church of Alexandria, according to which years of the Alexandrian calendar (see also Section 5) were counted from the one in which the consulate of this emperor began (the first day of this year of the Alexandrian calendar was 29-8-284). However, the dates of the corresponding two sequences of dates of the Julian calendar enclosed in Dionysius Exiguus’ Paschal table are numbered according to Dionysius Exiguus’ new era, which was intended to have begun with Jesus’ incarnation. This numeration begins with the year number 532 of his new era instead of with the year number 248 of the era of Diocletianus. All calendar years of Dionysius Exiguus’ Paschal table are years of the Julian calendar, all its dates are dates of the Julian calendar.

Until now our historians did not succeed in determining the date of the birth of Jesus. So it is not surprising that Dionysius Exiguus was not able to do this either. Be that as it may, he chose after careful consideration the Roman year 754 as the starting year of his new era. Then he put the years of the Julian calendar from this year of the Julian calendar in the right order and numbered them in this order 123……. The (incomplete) era thus obtained, which is known as Anno Domini (literaly ‘in the Year of the Lord’) era, is part of the (complete) Christian era. With the duration of a year as unit of time, the Anno Domini era boils down to our first timeline (Figure 1):

 

(time in years)                                                       *  year 1  1  year 2  2  year 3  3  …… 

 

in which the moment * = moment zero,  year 1  = the year 1 (of our era) = the Roman year 754 (this year of the Julian calendar began at the moment * and ended at the moment 1), and e.g.  year 10  = the year 10 (of our era) = the Roman year 763 (this calendar year began at the moment 9 and ended at the moment 10). We establish that the (incomplete) Anno Domini era contains only positively numbered calendar years (as the timeline of Figure 1) and is defined by the formula ‘the year x (of our era) is the Roman year (x+753)’. The first day of our era is not the day of the birth of Jesus, but simply 1-1-1. In all probability Jesus was born some years before the beginning of the Christian era.

In Roman antiquity sometimes Roman calendar years were counted from any supposed year of foundation of the city of Rome. However, in reality the Ab Urbe Condita era did not exist yet in antiquity, for it was used systematically for the first time only in the fifth century, namely by the Iberian historian Orosius. Though probably Dionysius Exiguus knew (but never used) the Ab Urbe Condita era, it was not he but pope Bonifatius IV (around the year 610) who seems to have been the first who recognized the connection (AD 1 = AUC 754) between this and the Anno Domini era. However, the Anno Domini era was used systematically for the first time only in the first half of the eighth century, but not by the church of Rome.

Neither about something like a numeral zero or the number zero nor about moment zero or something like a year zero, Dionysius Exiguus, who used no other numerals than Roman numerals in his Paschal table and in his calculations, has ever worried. Though he understood very well that sometimes the division (in his case being nearly tantamount to repeated subtraction of the divisor, for in his time in Europe division algorithms were not available yet) of a (strictly) positive integral number by (e.g.) 19 does not leave a (strictly) positive remainder, neither a numeral zero nor the number zero, a mathematical concept which perhaps seems inconsequential but is extremely important, was known to him. This is the reason why in our first timeline (see Figure 1) the place of moment zero has been marked by means of an asterisk (*).

Long before the invention of the number zero, precursors of this number were used (e.g. in Egypt and in Mesopotamia). These were words or symbols which initially represented nothing but ‘nothing’, namely empty spots in any positional system. The calculators in question did not consider them to be numerals or numbers. Our digit 0 has a record as a precursor of the number zero. In the sixth century it arose as a numeral zero, often represented by the symbol o, from the decimal positional system which was in use already in the fourth century (still without a numeral zero then) in India. It must have been in the India of around the sixth turn of century that the experience gained with this numeral o led gradually to the invention of the number zero, initially also often represented by the symbol o, with the characteristic property that x + o = x for any number x (see also Section 2). The modern symbol 0 for (both numeral and number) zero originated relatively early from the older symbol o for zero.

Why must the numeral 0 be considered, historically seen, to be our tenth digit? Counting precedes calculating, personally as well as (pre)historically. From time immemorial one counts by means of the cardinal numbers one, two, three, …… (initially only in words and not much farther than to hundred), without zero. In order to create a complete decimal positional system we need nine symbols (e.g. the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9) for the first nine (strictly) positive integers and subsequently a tenth symbol (e.g. the numeral 0) for the number zero (with an eye to the extension of the set consisting of these numbers downward), which however also must be applied to compound with the symbol (e.g. the numeral 1) for the first (strictly) positive integer a symbol (e.g. the compound 10) for the tenth (strictly) positive integer (with an eye to the extension of this set of numbers upward). Thus the modern notations for the numbers 0 and 10 began to take shape, in the India of around the year 600. More than three centuries later Arab merchants brought an Arabic version of the Indian decimal positional system with them to Spain. Gerbert, the French mathematician who became pope Sylvester II in the year 999, knew the first nine Arabic numerals, but not the real significance of the tenth. The dissemination of the Arabic prototype of our decimal positional system to Europe began in the Italy of around the twelfth turn of century. Not surprisingly it is in Europe that this Arabic prototype then evolved, in the course of four centuries, into our modern decimal positional system with its ten digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 and its decimal notation for all real numbers.

The presence of the Latin word “nulla” in the third column (C = epact) of his Paschal table (see Table 1) creates strongly the impression that Dionysius Exiguus must have known the number zero. However, where we would say that the epact (see also Section 7) of the first year is zero, Dionysius Exiguus must have said something like ‘annus primus non habet epactae”, which literally means ‘the first year has no epacts’. And so the Latin word “nulla” in the third column (C = epact) of his Paschal table must be interpreted as ‘no epacts’, just as the number 11 in this column must be interpreted as ’11 epacts’. Where computists (see also Section 6) like Dionysius Exiguus calculate with ‘no epacts’ (e.g. 18 epacts + 12 epacts ≡ no epacts modulo 30 epacts) instead of with zero (e.g. 18 + 12 ≡ 0 modulo 30), like infants with ‘no apples’ (e.g. 12 apples – 12 apples = no apples) instead of with zero (e.g. 12 – 12 = 0), we cannot speak yet of knowledge of the number zero. Where Dionysius Exiguus sees simply a column of numbers of epacts (such as ‘12 epacts’ and ‘no epacts’), our modernized brain thinks to see a purely mathematical structure in the form of a sequence of (abstract) integers. In his calculations he used no other numerals than Roman numerals and he never made use of any symbol for any zero. His number system contains no other numbers than (strictly) positive numbers, “nulla” in the third column of his Paschal table means ‘none’, not ‘zero’. But to call Dionysius Exiguus a dunce because he did not know the number zero (what some people do), that is what I call stupid. We establish that he is no exception to the generally accepted rule that in early medieval Europe nobody knew the number zero. It was not before around the year 1200 that medieval Europe was able to go to integrate this extremely important number, accompanied by the decimal positional system, in its culture (see also Section 2).

The number zero is a relatively modern concept, which could crystallize only after one had acquired sufficient experience with its precursors. The last phase of that development, which took place in the India of around the year 600, was the phase in which one became definitely familiar with performing abstract calculations in the decimal positional system with all its ten digits (among which the digit zero). This explains how it comes about that the invention of the number zero happened so long after the discovery of the (strictly) positive integers.

The first year of our eera is not any year zero but the year 1. And of course ‘the year 1’ means simply ‘the first calendar year of our era’, as ‘the king George I’ means nothing else than ‘the chronologically first king who is designated by the name George’. Numberings of admission tickets begin with 1; for the counting of   things whatever (unlike for the measuring of lengths of  things whatever), we do not need the number zero at all. Consequently the counting of years does not differ from the counting of things whatever, and therefore someone born on 1-1-1 will have celebrated his first birthday (not being the day on which he was born) probably (as usual) on the day he completed the first year of his life, on 1-1-2, and consequently his tenth birthday probably on the day he completed the tenth year of his life, on 1-1-11 (not on 1-1-10).

In or shortly before the year 526 Dionysius Exiguus granted a request to come and elucidate his Paschal table. This request was coming from official representatives of pope John I. Unfortunately the ecplanation of Dionysius Exiguus in question did not lead immediately to the acceptation of his Paschal table by the church of Rome. Only in the seventh century (about the year 640) the church of Rome decided to take in use a continuation of this Paschal table. In the tenth century she began to use the Anno Domini era included in Dionysius Exiguus’ Paschal table also outside the framework of a continuation of this Paschal table. However, the first who did this, was not the church of Rome but Beda Venerabilis, a great scholar and the first English historian, in the first quarter of the eighth century, two centuries after the invention of this era. It is thanks to him          that as early as about the year 730 the (incomplete) Anno Domini era was extended to the christian era,  and that this complete era, which essentially, because of its calendar years before Christ, contains also negatively numbered calendar years, was actually taken into use as a coherent system for the dating of historical and current events. Only in the tenth century (in the year 967) the Christian era was used for the first time for the dating of a papal document, and only about the year 1060 the church of Rome put this era definitively into use. Our era was drastically adapted to the seasons by pope Gregory XIII in the year 1582, and has never been replaced with another.

 

2 eras

This section is an extension of Section 2 of millennium question.

In order to create the possibility of locating on our timeline also historical events which have taken place before the beginning of our era, of course the Anno Domini era (see Section 1) had to be extended to a complete era. For that purpose first the Roman years (see Section 1) preceding the year 1 were numbered 123…… further and further in the direction of the past, after which the sequence of years of the Julian calendar (see Section 1) thus obtained was joined together in the most obvious way with the sequence of years 123……, which resulted in the complete sequence of years ……321123……, in which the year 1 = the year 1 before Christ = the Roman year 753, and e.g. the year 10 = the year 10 before Christ. It is since and thanks to Beda Venerabilis (see Section 1) that the calendar years of our era are divided into calendar years after Christ and calendar years before Christ. This division is essentially a division into (strictly) positively numbered and strictly) negatively numbered calendar years without that the number 0 has been allocated to any calendar year. With the duration of a year as unit of time, the (complete) Christian era (see Section 0) thus obtained, boils down to our second timeline (Figure 2):

 

(time in years)  ……  -3 year -3 -2 year -2 -1 year -1 0  year 1  1  year 2  2  year 3  3  …… 

 

in which year -1 = the year -1 (of our era) = the year 1 before Christ (this year of the Julian calendar began at the moment -1 and ended at the moment 0), and e.g. year -10 = the year -10 (of our era) = the year 10 before Christ (this year of the Julian calendar began at the moment -10 and ended at the moment -9). We note that moment zero (see Section 0) = the moment 0 of our era. We establish that the extension of the (incomplete) Anno Domini era to the (complete) Christian era is defined by the formula ‘the year -x (of our era) is the year x before Christ’, despite the fact that until in the thirteenth century negative numbers were entirely unknown in Europe.

The most important property of the Julian calendar, which after far reaching precautionary measures went ahead with the beginning of the year -45, is its proleptic leap year regulation, which means that that henceforth all years of the Roman calendar, in past, present, and future, were supposed to begin or to have begun on 1 January and to consist of 366 instead of of 365 days once every four years, to begin with that year of the Roman calendar, by way of a leap day in February (see also Section 5). In principle this leap year regulation applied to all years of the Julian calendar, and so to all calendar years of our era before the year 1582. However, owing to the initially defective functioning of this regulation, there were between the leap years -45 and -9 three leap years too much (namely a leap year every three instead of every four years) and between the leap years -9 and 8 no leap years instead of three ones. The year 1582, being the calendar year of our era in which the Julian calendar was replaced (for an indefinite future time) with the Gregorian calendar (see Section 1), comprised only 355 days. That calendar year is the only exception to the rule that a calendar year of the (complete) Christian era consists of 365 or 366 days. The (not proleptic) leap year regulation according to the Gregorian calendar (solely calendar years of which the year number is divisible by 4 but not by 100 unless by 400 are leap years) holds in principle for all calendar years of our era after the year 1582. Thus all leap years and hence all calendar years of our era from the furthest past into a far future have been fixed.

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 positively numbered and of the negatively numbered calendar years of our era. However, all things considered this timeline cannot possibly represent a pure linear time scale, because two calendar years of our era are not always precisely equally long. Usually the difference between the lengths of two of these calendar years is either zero or one day. For instance, in our second timeline the difference between the moments 11 and 12 (this difference is 366 days) is not the same as the one between the moments 10 and 11 (this difference is 365 days). Nevertheless we may interpret our second timeline as a simple and as such consistent mathematical model of the (complete) Christian era. Likewise our first timeline (see Figure 1) can be interpreted as a simple and as such consistent mathematical model of the (incomplete) Anno Domini era.

What attracts the most attention (perhaps even bothers us) in our second timeline, is of course that in here there is no room for a year zero. From the outset, and to this very day, our era has coped without a year zero, notwithstanding the fact that the number zero is common property for a long time. Modern historians who take their profession seriously, let the year 1 be preceded by the year -1 indeed without intermission. It is moment zero, the unique point of time from which the calendar years of our era are counted and which is identical to the midnight point in time [31-12- -1; 24:00] = [1-1-1; 0:00], which marks the direct transition (turn of year) from the year -1 to the year 1, precisely like it marks the direct transition (turn of century) from the first century before Christ to the first century (after Christ). Precisely like there is no zeroth century (and no zeroth millennium) in our era, there is also no year zero, thanks to Beda Venerabilis.

The presence of the Latin word “nulla”, which means ‘none’, in the third column (C = epact) of his Paschal table (see Table 1) creates the impression that Dionysius Exiguus (see Section 1) knew the number zero. However, in the explanation for his table he is talking about “nullae epactae”, which literally means ‘no epacts’, but the number zero does not occur in it. That extremely important number (without number zero our modern mathematics would not have been possible, and without our modern mathematics our technology would have been completely impossible), which was discovered only around the sixth turn of century in India after a long maturing process, was not part of his arithmetic nor of the one of his great follower Beda Venerabilis. They did not need the number zero, and neither a year zero. After all, in early medieval Europe nobody was acquainted with the number zero, let alone with one year zero or another.

Beda Venerabilis calculated (just like Dionysius Exiguus) only with (strictly) positive integers represented by Roman numerals (these are the letters i, v, x, l, c, d, m of the Latin alphabet). He had not 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. In early medieval Europe division algorithms did not exist yet and division boiled down to repeated subtraction. Where Beda Venerabilis in his book ‘De temporum ratione’ (see also Section 7) on “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”, which literally means ‘there remain 3’ (and not ‘there remains 3’). Likewise he refrains from naming the number zero in order 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 “non remanet aliquid”, which literally means ‘there does not remain something’, or its logical equivalent “nihil remanet”, which literally means ‘there remains nothing’ (and not ‘there remains 0’). Where he is calculating, he never uses any symbol or word for (the number or a numeral) zero. And where he enumerates Greek numerals, he does not observe that among them there is no equivalent of any numeral zero known to him. There is nothing at all from which we could deduce that Beda Venerabilis was acquainted with zero; the same applies to Dionysius Exiguus.

In the standard work about “De temporum ratione” written by the Canadian historian Faith Wallis we find a modern version of the 532 year Paschal cycle which was part of Beda Venerabilis’ Paschal table (see also Section 7), with our modern digits and with epacts (see also Section 7) being 0 once every nineteen years, and even mentioning the year -1. However, there is no Latin manuscript at all written before the thirteenth century which contains numbers which are not (strictly) positive, and not surprisingly one will find where in such a manuscript the number zero would have been in its place, only the Latin word “nihil” (meaning nothing but ‘nothing’) or a Latin word like “nulla” (see Section 1). 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 make believe that we see the number zero where early medieval scholars only had thought of ‘nothing’ or ‘none’. Where Beda Venerabilis makes calculations with (abstract) (strictly) 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. The terms “nulla” of Dionysius Exiguus and “nulla” or “nullae” of Beda Venerabilis in their columns of epacts are typical examples of precursors of the number zero, they mean literally ‘none’, namely ‘no epacts’, which boils down 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 to perform calculations with (strictly) positive integers.

Like Dionysius Exiguus, Beda Venerabilis knew no other numbers than (strictly) positive numbers, just like everyone in first millennium Europe. Even Boetius (around the year 500), in far the most important mathematician of early medieval Europe, and Gerbert were anything but familiar with the number zero. Nowhere in the preserved European literature of the first millennium the number zero 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. The idea that Dionysius Exiguus en Beda Venerabilis should be acquainted with the number zero really remains without any rational basis. They were great scholars and skilled computists (see also Section 6), but no mathematicians (and also no astronomers). One does not need to be a mathematician to be able, starting from the sequence of dates of the classical Alexandrian Paschal full moon (see also Section 6) and making use of the leap year regulation according to the Julian calendar (see also Section 5) and the Alexandrian Paschal principle (see also Section 6), to determine really all dates of the classical 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 ADF of Table 1 (in which all dates are dates of the Julian calendar). By the way, that takes nothing away from the fact that the very first construction (about the year 260) of a Metonic sequence of dates (see also Section 8) of which each date functioned as substitute of a (in principle unknown) date of the full moon of the fourteenth day of Nisan (see also Section 5) was an impressive arithmetical finding, which can be attributed to the Alexandrian scholar Anatolius (see also Section 5).

The great Alexandrian astronomer Ptolemaios, who lived around the first half of the second century, used a symbol o as a numeral zero in the (originally Babylonian) sexagesimal positional system. But this symbol o 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 (see Section 1), which then already for centuries, with its symbols for the digits 1 up to and including 9, was in use in India, was improved with a symbol o for the numeral zero in this modern positional system, due to which it became possible to carry out abstract calculations efficiently, i.c. by means of handy algorithms. Around the year 600 the elucidation of the concept of number which ensued from the introduction of a symbol o for the digit zero (in modern notation 0) led to the invention of the number zero (idem). The great Indian mathematician Brahmagupta was the first who, about the year 630, made explicit the most important properties of this unique number 0: this number is the only number 0 such that for each number x it holds that x + 0 = x and that x × 0 = 0. The spread of the number zero over Asia was a matter of centuries, as was its spread over Europe, which began to get under way after the beginning of the thirteenth century (in Italy, after a hesitant start in the second half of the tenth century in Spain). Fibonacci (whose important book ‘Liber Abaci’ was finished in the year 1202) was the first Italian, Robert Recorde (idem ‘The Grounde of Artes’ in the year 1543) the first Briton, Simon Stevin (idem ‘De Thiende’ in the year 1585) the first Dutchman who was familiar with this extremely important number. Our modern mathematics is unthinkable without the number zero, without our modern mathematics our technology would have been completely impossible.

Simply 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 would certainly not have been able to understand our second timeline (see Figure 2). Dionysius Exiguus did not worry about it, because he did not at all need these numbers to be able to give shape to his incomplete era (which he actually only used for the benefit of his Paschal table), and even Beda Venerabilis, to whom we owe the extension of the (incomplete) Anno Domini era to the (complete) Christian era, could get along excellently without the number zero and the negative integers. Only in the tenth century the church of Rome used the Anno Domini era for the first time also outside the framework of any continuation of Dionysius Exiguus’ Paschal table, though as early as around the year 720 the (complete) Christian era had been used by Beda Venerabilis as a coherent chronological system for dating historical events. The modern concept of the bilateral linear scale division, necessary to be able to understand our second timeline, only could go to function after people in Europe had obtained the disposition of the number zero (around the year 1200) and the negative numbers (around the year 1500). The number zero and the negative integers began to be common property in the first half of the eighteenth century as a result of the invention of the thermometer (which sometimes indicates degrees below zero). Setting aside restrictions with regard to the lowest or the highest possible temperature, the well known Celsius scale, i.e. the temperature scale obtained (in the year 1745) by reversing the original temperature scale of the swedish astronomer Anders Celsius (died in the year 1744), has the same structure as the complete bilaterally symmetric time scale we see in Figure 2. The French astronomer Jacques Cassini was the first who explicitly availed himself of negatively numbered calendar years.

Beda Venerabilis was the first (around the year 730) who used the (complete) Christian era as a coherent chronological system (as in the timeline of Figure 2 provided that year -x is taken as the year -x = the year x before Christ) for the dating of historical events. For this reason Beda Venerabilis can be considered as the great promoter of this (nowadays generally used) era. In times of scarcity of reliable historical evidence the dating of historical events was no simple matter. And so it is not surprising that Beda Venerabilis dated the coming into power of Diocletianus (which took place in November of the year 284 but still had been dated by Orosius in the Roman year 1041) in the year 286, the capture of Rome by Visigothic troops (which took place in the year 410) in the year 409, the death of pope Gregory I (who starved in the year 604) in the year 605. Beda Venerabilis was the first medieval historian who, by making use of the (complete) Christian era, hazarded a datation of the first landing of Julius Caesar (see Section 1) in Britain; this military action, which took place in the year -55, was dated by Beda Venerabilis in the year 60 before Christ.

If we take a look at our second timeline (see Figure 2) once more and abstract from the fact that two calendar years are not always precisely equally long then we observe that our era (taken as a linear system of numbered calendar years), i.e. the (complete) Christian era, is in principle (namely besides its restrictions with regard to the beginning and the end of times) bilaterally symmetrical with respect to its initial moment. It is this bilaterally symmetric structure of our era which we experience as obvious, as obvious as the fact that every century consists of hundred years (as every kilometre contains thousand metres), and as the fact 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). That is why our era cannot contain a year zero (presupposed that we want to preserve the symmetry of our era). Such a year zero would have to belong to the first century before or to the one after Christ after all, but then also (because of the symmetry) both to the first century before and to the one after Christ; but this is contrary to the principle that every calendar year of our era belongs to exactly one numbered century of our era.

A millennium is by definition a time interval consisting of a thousand years. The first millennium (after Christ) consists of the (thousand) years 1 up to and including 1000, the first millennium before Christ consists of the (thousand) years -1 down to and including -1000 (provided that the year x is taken as the year x before Christ). These two millennia are separated from each other by moment zero instead of by any year zero. Likewise the first century (after Christ) and the first century before Christ, the first decade (after Christ) and the first decade before Christ, and the years 1 and -1 are separated from each other by moment zero. None of the calendar years of our era has the number 0, i.e. the number zero. The (either positively or negatively numbered) calendar years of our era are symmetrically arranged with respect to moment zero. Insertion of a year zero into our era would disturb this structure. In the previous paragraph we have demonstrated this by logical reasoning.

As long as the Julian calendar functioned (from the second half of the first century before Christ to the year 1582), the March equinox, i.e. the moment at which in the northern hemisphere of the earth spring begins, shifted slowly but surely (about 0.78 days per century) more and more forward (ultimately from 23 to 10 March). This was the main reason why the Julian calendar was replaced with the Gregorian one (in the year 1582). In order to keep the March equinox in its place (from the year 1582 on or close to 20 March) until a very far future, it is sufficient (e.g.) to maintain the Gregorian calendar and to cancel the Gregorian leap day in the calendar years of our era of which the year number is divisible by 4000. So there is not the slightest reason to replace our era with an other one.

According to Ptolemaios in his time the March equinox fell on 22 March. Around the year 1500 the (real) March equinox fell on 11 March, around the year 220 on 21 March. Because of the prolepticity of the leap year regulation according to the Julian calendar, around the year -1190 the March equinox fell on 1 April. It is only since the eleventh century before Christ that the March equinox falls definitively in March. Somewhere in the first half of the eleventh century before Christ the March equinox fell in April for the last time, somewhere in the second half of the fiftieth century before Christ it fell in May for the last time. At the time of the neolithic revolution, i.c. the coming into being of agriculture, the March equinox fell in the second half of May (but of course at that time nobody was aware of this phenomenon because the Julian calendar was invented only about nine millennia later).

It is thanks to Beda Venerabilis that our era has a bilaterally symmetric structure and no year zero (as in the timeline of Figure 2). Both an alternative era with the year 1 as a year zero and an alternative era with the year -1 as a year zero (on close inspection there are no other possibilities) are necessarily not symmetric with respect to moment zero. It is for that reason that none of these two alternative eras has become common property, although a variant of the latter one is used for an obvious practical reason by scientists (mainly astronomers and chronologists). This (nonsymmetric) variant is the astronomical era, which around the seventeenth turn of century emanated from 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 chronologer Joseph Scaliger. Joseph Scaliger attached the name of Julius Caesar to his dating system to underline that with regard to the time before the year 1582 he would maintain the original Julian calendar. Our era has never been officially replaced with the astronomical era. Incidentally, they differ only in their calendar years before their common year 5. The astronomical era was taken into use in its present form, by definition including a year zero and negatively numbered calendar years and provided with the original proleptic Julian leap year regulation (a leap year once every four years) applying to its calendar years before the year 1582, in the year 1740 by Jacques Cassini. With the duration of a year as unit of time the astronomical era boils down to our third timeline (Figure 3):

 

(time in years)  ……  -3 year -2 -2 year -1 -1 year 0  0  year 1  1  year 2  2  year 3  3  …… 

 

in which year 0  does not exactly coincide with the year -1 (of the Christian era), which began two days later but ended one day later, which was a consequence of an initially (during half a century) inadequate functioning of the Julian calendar (see also Section 5). Contrary to the year 4 (of our era), the year 4 of the astronomical era was a leap year, which ended at [31-12-4; 24:00], but began at [31-12-3; 0:00] instead of at [1-1-4; 0:00]. Because the year 4 of the astronomical era began one day earlier than the year 4 (of our era), the year 0 of the astronomical era ended one day earlier than the year -1 (of our era). This implies that the moments 0 of the astronomical and the Christian era differ one day (see Figure 2 and Figure 3). However, their moments 2000 coincide exactly (they are both equal to [31-12-2000; 24:00] = [1-1-2001; 0:00]), because they show no difference in their calendar years after the year 4.

Though it is not relevant to the solution of the millennium question, the example of the era of the French revolution is illustrative for the fact that it is indeed not obvious at all that a new era must begin with a year zero. When on 22-9-1792 French revolutionaries proclaimed the first French republic (one day after they had abolished kingship), at the same time they resolved to let begin a new era on that particular day, which contained the September equinox, i.e. the moment at which in the northern hemisphere of the earth autumn begins, and was regarded by them 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, though in France as early as in the course of the eighteenth century the number zero had been generally accepted. 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 calendar reform. Each calendar year of the era of the French revolution consisted of twelve months of thirty days and five or six separate days; this era was in use until 1-1-1806.

The first century began with the moment 0 and ended with the moment 100 of our era. Therefore the year 100 is the last year of the first century. We note that the first century ended exactly one year after the moment of the transition from 99 to 100. This is nothing special: each moment at which the last two digits of the number of the current calendar year suddenly become zero, is the presage of a turn of century, always exactly one year later.

 

3 conclusions

This section is an extension of Section 3 of millennium question.

The Christian era (see Section 1) has a bilaterally symmetric structure (see Section 2), and it is good that the followers of Dionysius Exiguus (see Section 1), did not saddle his and our (certainly for historians ideal) era with any year zero. Ultimately everyone prefers symmetry, either unconsciously or consciously. Astronomers have never seriously proposed to replace our bilaterally symmetrical era with their astronomical era (see Section 2). We owe our era to Dionysius Exiguus, its bilateral symmetry, and with this its consistency, to Beda Venerabilis (see Section 1). The absence of a year zero in our era is by no means a mistake of Dionysius Exiguus or of Beda Venerabilis. What is more, it is a condition our era has to satisfy in order to preserve its bilateral symmetry. Grieving about the absence of a year zero in our era is such a thing as missing ‘king George zero’ in a company of kings named George. The seemingly insignificant fact that our era is not provided with a year zero is not only a good cause (and no mistake) but also the key to the solution of the millennium question.

The millennium question is a matter of chronology. We have ascertained that our era, i.e. the Christian era, is quite all right but contains no year zero. Therefore moment zero (see Section 0) is the beginning of the year 1 as well as the end of the year -1. This 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 that the first decade before Christ must be the time interval consisting of the years -10 up to and including -1. These two decades are separated from each other not by a year zero, but by a point of time, namely moment zero. This implies that the first turn of decade took place in the moment 10 of our era, i.e. in the point of time [31-12-10; 24:00] = [1-1-11; 0:00].

Everyone born in the year 1 must have been begotten in the year -1 or at moment zero 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.

The ancient Olympic games were organized at Olympia (Greece) in summer every four years, from the year -776 up to and including the year 389. At the time, olympiads were by definition time intervals of four years between two consecutive ancient Olympic games. For example, it was in the first year of Olympiad 95 that the great philosopher Socrates was condemned to death. This must have taken place in the year -399, because it was about the end of winter that this happened. Because Olympiad 1 began in the summer of the year -776, Olympiad 194 began in the summer of the year -4. And hence that Olympiad 194 ended in the summer of the year 1, and that Olympiad 291, which was the last ancient olympiad, ended in the summer of the year 389.

Now that we have given account of the fact that our era is quite all right and 1-1-1 is the date of the first day of our era (see Section 1), we can give short shrift to the millennium question. The date of the tenth birthday of someone born on 1-1-1, is 1-1-11. By analogy with this fact we establish that the second decade began on 1-1-11, the second millennium on 1-1-1001, the third millennium on 1-1-2001. The year 1000 was the last year of the first millennium, the year 2000 the last year of the second millennium, the year 2001 the first year of the third millennium. The last year of the third millennium is the year 3000.

Millennium mistake 1 was made by medieval people who believed that the first millennium would expire (and the world perish) on 1-1-1000. These humans did not realise that on that date no more than 999 years of the first millennium had passed. The first turn of millennium took place exactly one year later, namely in the moment 1000 of our era, i.e. in the pont of time [31-12-1000; 24:00] = [1-1-1001; 0:00], without the earth perishing.

Millennium mistake 2 was made by modern people who had no difficulty in being fooled by commerce and media and authorities that also did not know any better (and by many a historian who had entirely forgotten that our era contains no year zero) into thinking that not the “dull” date 1-1-2001 but the “magic” date 1-1-2000 (which was accompanied by the millennium question, the millennium problem, the millennium mistake, and the millennium madness) had to be the date of the first day of the new millennium. However, the second turn of millennium did not take place in the moment 1999 of our era in which all four of the digits of the number of the current calendar year changed simultaneously, i.e. in the point of time [31-12-1999; 24:00] = [1-1-2000; 0:00], but exactly one year later, namely in the moment 2000 of our era in which solely the last digit of the number of the current calendar year changed, i.e. in the point of time [31-12-2000; 24:00] = [1-1-2001; 0:00]: the second turn of millennium was nothing else than the transition from the year 2000 to the year 2001.

Summarizing, we can say that the millennium mistake is by definition the mistake resting on the misunderstanding that the numbered millennia of our era would end not with the end but with the beginning of their thousandth year. We note that the first millennium ended exactly one year after the “magic” moment of the transition from the year 999 to the year 1000. This is nothing special: each moment after moment zero in which the last three digits of the number of the current calendar year suddenly become zero, is the presage of a turn of millennium, always exactly one year later. For example, the “magic” moment of the transition from the year 1999 to the year 2000 was nothing else than the presage of the second turn of millennium, exactly one year later. Millennium mistake 3 still keeps us waiting, but is only a question of time.

The reason why a choice for the astronomical era instead of a one for the 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 2). Indeed a choice for an alternative era with the year 1 (of our era) instead of a one with the year -1 (of our era) as a year zero 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].

According to the Roman historian Titus Livius, who lived around the beginning of our era, Rome was founded in the Roman year 1, i.e. the first year of the Ab Urbe Condita era (see Section 1). If Rome indeed was founded in the Roman year 1 then it is not in the year 2247 that it will be three thousand years ago that this important historical event took place, but in the year 2248 (I say it in advance), because the Roman year 1 = the year -753 (of our era). The eighthundredth anniversary of the foundation of Rome was celebrated exuberantly in the year 47, the thousandth one in the year 248. Incidentally, according to modern historians Rome was not founded in the eighth but in the seventh century before Christ.

 

4 objections

This section is an extension of Section 4 of millennium question.

Innumerable objections have been raised against the idea that the first day of the third millennium was not 1-1-2000 but 1-1-2001 or against the argument underlying this idea. This section contains a little anthology therefrom.

“All well and good” someone still objects, “but the twentieth century, after all, consists precisely of the calendar years of our era whose year number starts from 19? This implies that the year 1999 was the last year of the twentieth century!”. The calendar years of our era whose year number ends in 00 spoil the feast. There is no year zero in our era (see Section 2); it follows that the year 100 was the last year of the first century, the year 200 the last year of the second century, the year 300 the last year of the third century, and so on. So the year 1600 was the last year of the sixteenth century. And so the at first sight interesting standpoint of Maarten Prak (university of Utrecht) that the battle of Nieuwpoort which took place in the year 1600 was one of the few real battles the army of the Dutch republic fought out in the seventeenth century, is worth no more than the assertion that New Year’s Eve is one of the few really cosy days of the month of January.

“All well and good” someone still objects, “but who is actually wrong? On 1-1-2000 the nineties of the twentieth century had passed!”. Of course this 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 Dutch book which was rashly printed in a big edition in the year 1999 and appeared under the pretentious title ‘De volledige Geschiedenis van de twintigste Eeuw’ shortly before 1-1-2000 is no complete history of the twentieth century, because what happened in the last year of the twentieth century, is not in this book.

“All well and good” someone still objects, “but what about my odometer then? After precisely 1000 kilometers this odometer shows three zeros!”. That is right, but what we state here is not a similarity, but just a difference between era and odometer. After all, during its first kilometer the odometer indicates 0000, not 0001. Incidentally, there is an analogy between odometer and age (so during its twentieth kilometer the odometer indicates 0019, and during your twentieth year of life you are nineteen years old).

“All well and good” someone still objects, “but when numbering the floors of a building nevertheless 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 zero either!”. Because floors must be considered not 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 it does not matter! After all, it is not known when Jesus was born!”. It is not the (indeed unknown) date of birth of Jesus that matters for the solution of the millennium question, but the first day of the Anno Domini era, i.e. 1-1-1, that is essential here (see Section 1). Strictly speaking, what we call the first century before Christ is not the last century preceding the day that Jesus was born, but the last (negatively) numbered century preceding moment zero.

“All well and good” someone still objects, “but it does not matter! After all, it is only haphazardly that the beginning of our era was chosen!”. That in retrospect and once and for all chosen moment is moment zero (see Section 0), the unique point of time which is asterisked (*) in our first timeline (see Figure 1) and is identical with [1-1-1; 0:00]. In the year 1582 the number of days of every calendar year of our era was fixed for an indeterminate time (see Section 2). Thus also all turns of year, turns of decade, turns of century and turns of millennium of our era have been fixed for an indefinite time.

“All well and good” someone still objects, “but the millennium question, after all, can be solved much more simply! Because our era contains no year zero, the supposition that [1-1-2000; 0:00] was the second turn of millennium leads to the absurd conclusion that the first decade of our era would have consisted of nine years (which would imply that the tenth birthday of everyone born on 1-1-1 would have coincided with his ninth birthday!)”. This reasoning is correct and confirms our conclusion that the second turn of millennium was not [1-1-2000; 0:00] but [1-1-2001; 0:00] (see Section 3).

“All well and good” someone still objects, “but the fact that in the year 67 Olympic games were held, does not agree with the assertion that the ancient Olympic games were held every four years (see Section 3)!”. The games held in Greece in the year 67 were no real ancient Olympic games but once only games in one and the same year organised at Olympia, Delphi, Nemea and Isthmia on behalf of the emperor Nero.

“All well and good” someone still objects, “but what on earth was against celebrating the second turn of millennium on 1-1-2000?”. Of course nothing is against it to celebrate any memorable event whatsoever in any moment whatsoever (e.g. a turn of year on a 30 December or your twentieth birthday on your nineteenth birthday). But here the whole point is that the direct transition from 1999 to 2000, the “magic” moment in which all four digits of the number of year of the current calendar year changed simultaneously, is something else than the accompanying turn of millennium, i.e. the direct transition from the second to the third millennium, exactly one year later, and that in these two striking moments relatively few people were aware of this.

“But finally it is the people who has the last word!” someone still objects. That means in my opinion that the people have right to self determination, not that the people are right in advance. An assertion does not automatically become true if there are many people who believe that this one is true. The earth does not become less round of it if there are many people who believe that the earth is flat. Nor does an assertion automatically become true by deciding that this one is true, not even when this happens in a democratic way. One can decide to go over to summer time, but not that henceforth the sun must rise one hour later. It was possible to decide to celebrate the second turn of millennium at the moment 1999 of our era, so one year too early (see Section 3). It was even possible to decide to do as if that was not one year too early, but not that that was not one year too early.

Whether or not something is true, is neither regulated by the people nor by some authority or other, not even by the king or the queen of the Netherlands (although sometimes one for a moment could think so, for the fact that there exists a statistical connection between smoking and lung cancer seems to be established by royal decree). To establish whether or not something is true, sometimes logical reasoning is necessary and sufficient, such as in Section 2 in order to establish that an adequate era, being a linear system of numbered calendar years, is bilaterally symmetrical if and only if it has no year zero, and in Section 3 to establish that the third millennium began on 1-1-2001.

Thanks to Dionysius Exiguus (see Section 1) and Beda Venerabilis (see Section 2) we have at our disposal a bilaterally symmetrical era without any year zero (see Section 2). The year 1 comes immediately after the year -1, precisely like the first century (after Christ) comes immediately after the first century before Christ; in the Christian era there is no year zero, precisely like there is no zeroth century in our era. This is the official standpoint of our modern historians, and with good reason (as we saw in Section 2). 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 did not begin before the year 2001 (see Section 3) and justifies the use of the term ‘millennium mistake’ for the phenomenon that around the year 2000 commerce, media en authorities were amply under the delusion that the year 1999 was the last year of the second millennium.

 Everyone believes in something, has his or her own belief. There are no unbelievers, even atheists believe in something (but not in God). However, most people are so attached to what they believe that insights which do not entirely seem to agree with this hardly get a chance to be taken in consideration. It is thereby 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 very special circumstances primitive forms of life (extremely gradually) originate from lifeless matter        , that all higher developed biological species (including Homo sapiens) have gradually developed from earlier biological species, that all life, i.e. all that lives, is only temporary (nobody has eternal life, for “dust thou art and unto dust shalt thou return”), that God is a product of the human imagination and exists only as such (man proposes but there is no God who disposes), that it is a misconception to think that atheists think they can prove that God does not exist (as a matter of fact atheists believe that there is no God outside of the human imagination). Many atheists are humanists. Humanists try to believe, like Anne Frank, in the inner goodness of man, and believe in the vocation of man to create a really humane society, i.e. a really democratic society of human beings who know how to live in harmony with each other and with the nature of our planet. This implies a belief in mental growth, and consequently that we have to be willing to revise our opinions and to redeem our mistakes (this concerns each of us personnally as well as mankind as a whole). It is thereby that I felt compelled to find out, inspired to it by critical pupils who wanted to know all the ins and outs, why the date 1-1-2000 could not have been the date of the first day of the third millennium.

An important objective of education is the stimulation of thinking clearly and formulating carefully by joint attention for the essence of a problem. Pupils must be able to calculate in their head the sum of -753 and 3000. But they must also know, I think, how our era is put together, in order to be able to understand that the reply to the question in which calendar year of our era Rome, assumed that this eternal city was founded in the year -753 (see Section 3), will exist three thousand years, is not the year 2247 but the year 2248; this is not so difficult after all.

 

5 calendars

This section is an extension of Section 5 of millennium question.

The Julian calendar (see Section 1) was the result of the proleptic calendar reform decreted in the year -46 by Julius Caesar (see Section 1). In the year 1582 pope Gregorius XIII (see Section 1) replaced the Julian calendar with the Gregorian calendar (see Section 1), which boiled down to an adaptation of the Julian calendar which existed of the measure according to which the calendar year of our era was shifted by ten days in the direction of the past, owing to which the March equinox (see Section 2) was abruptly displaced from 10 March (of the Julian calendar) to 20 March (of the Gregorian calendar), and an adaptation of the leap year regulation (of the Julian calendar). Essentially these two exceptionally important calendars differ solely in their leap year regulation. The calender years of our era before the year 1582 are years of the Julian calendar, the calender years of our era after the year 1582 years of the Gregorian calendar. The dates of our era before the year 1582 are dates of the Julian calendar, the dates of our era after the year 1582 dates of the Gregorian calendar.

The Julian calendar was introduced by Julius Caesar in the year -46 by means of a drastic adaptation of the in that time hopelessly out of date Roman calendar (see Section 1), which was not provided with any leap year regulation until then. The adaptation in question consisted of the measure according to which the current year of the Roman calendar was shifted eighty days in the direction of the future (owing to which the March equinox was displaced abruptly from 11 June of the previous to 23 March of the new Roman calendar, and the birthday of Julius Caesar, who was born in the summer of the year -100, from 1 October of the previous to 13 July of the new Roman calendar) and the provision that henceforth the years of the Roman calendar, in past, present, and future, would be supposed to begin or to have begun on 1 January instead of on 1 March (owing to which September became the ninth instead of the seventh month of the year of the Roman calendar) and to consist of 366 instead of of 365 days once every four years, to begin with the then next year of the Roman calendar (being the year -45), by means of a leap day in February.

Unfortunately, in the first half century after the death of Julius Caesar (in the year -44) the most important property of the Julian calendar, its leap year regulation (see Section 1), according to which a leap day had to be inserted once every four years, was badly applied. Namely, it is a fact that between the leap years -45 and -9 there was (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 in fact three leap years too much, namely eleven instead of eight. About the year -8 this problem was solved by the emperor Augustus by reducing the three Roman leap years between the leap years -9 and 8 to ordinary Roman years of 365 days. That implies in particular that the year 4 was no leap year. But each calendar year of our era betwecn 4 and 1582 satisfies the condition that it was a leap year if and only if its year number is integrally divisible by 4. Although the Julian calendar was no ideal calendar, it functioned perfectly from 4 to 1582, more precisely from 1-3-4 up to and including 4-10-1582. Not surprsingly the dates given in Dionysius Exiguus’ Easter table are dates of the Julian calendar.

Contrary to the years 40, -4 of the astronomical era (see Section 2), the years 4, -1, -5 (of our era) were no leap years. That implies that the year -1 (of our era) began one day later than the leap year 0 of the astronomical era, that the year -5 (of our era) began two days later than the leap year -4 of the astronomical era, and 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, 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 on 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.

It was under the influence of the emperor Constantinus I (Constantine the Great) that the Julian calendar was accepted as official calendar by the churches who were represented at the first council of Nicaea in the year 325. However, the leap year regulation of the Julian calendar was not accurate enough to be suitable to be used trouble free indefinitely; e.g. around the year 1500 the (real) March equinox fell on 11 March. That is the reason why in the year 1582 the Julian calendar was replaced with the (nowadays worldwide used) Gregorian calendar, on the understanding that the Julian calendar, inclusive of the unfortunate running of things between the years -45 and 8 with regard to its leap year regulation, remained holding for all calendar years of our era before the year 1582. In order to bring back the (real) March equinox to or near to 20 March, pope Gregorius XIII suppressed ten days of the tenth month of that calender year (in fact in that calender year Thursday 4 October was the last day of the Julian calendar and Friday 15 October the first day of the Gregorian calendar). Moreover, in that calendar year he decreed that every calendar year of our era after the year 1582 would be a leap year if and only if its year number was integrally 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 (complete) Christian era consists of 365 or 366 days, and that [4-10-1582; 24:00] = [15-10-1582; 0:00]. Thus all calendar years of our era from the farthest past into a vcry far future have been fixed. However, with regard to the far past we must realize that from the fiftieth to the twelfth century before Christ the March equinox fell in April (and from the ninetieth to the fiftieth century before Christ in May).

It is in combination with the Gregorian calendar (holding for the time after the year 1582) that the Christian era has become the most widespread chronological system on earth. Our era was never abolished or replaced with the astronomical era (see Section 2), 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 not completed with a proleptic leap year regulation according to the Gregorian calendar being effective for all times, but with the pure leap year regulation according to the Julian calendar holding for the time before the year 1582 and the leap year regulation according to the Gregorian calendar holding for the time 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 restrictions of the astronomical and the Christian era to their calendar years after the year 4 coincide exactly, which implies that the moments 2000 of these eras are identical. For that reason a choice for the astronomical era instead of for the Christian era would not have led to a point in time of the second turn of millennium different from [1-1-2001; 0:00]. The fact that the year -1 (of our era) ended one day later than the year 0 of the astronomical era takes nothing away from this conclusion.

In the first four centuries of our era, besides Rome there was another great centre of civilization in the region around the Mediterranean, namely Alexandria (Egypt). Likewise besides the Julian calendar another solar calendar was in general use then in the Roman empire, namely the Alexandrian calendar, however with calendar years beginning and ending in the summer. In the year -30 the Nile calendar, this is the oldest Egyptian calendar, which rested on the annual flooding of the Nile but was not provided with any leap year regulation, was replaced with the Alexandrian calendar. The emperor Augustus let the first year of the Alexandrian calendar begin on 29-8- -30. Still the Julian and the Alexandrian calendar are used, though not generally. Each year of the Alexandrian calendar consists of twelve months of thirty days and five or six epagomenal days at the end of this calendar year, between 23 and 29 August or between 23 and 30 August respectively. Contrary to the Alexandrian calendar, the Nile calendar was only used for agricultural purposes.

Just like the Julian calendar, the Alexandrian calendar is provided with the ordinary leap year regulation with leap year proportion of one to four. These two calendars are equivalent, which means that there exists a reciprocally univocal relation between these two calendars, which implies that they are mutually convertible. Each leap day of the Alexandrian calendar (in August) is six months later followed by a leap day of the Julian calendar (in February).

Just like the introduction of the Julian calendar, the introduction of the Alexandrian calendar was accompanied by a (the same) wrong application of its leap year regulation. As a matter of fact, each of the nine (instead of six) leap days of the Julian calendar between the years -30 en -8 (always in February) was preceded by a leap day of the Alexandrian calendar six months earlier (always in August). This holds also for each leap day of the Julian calendar after the year 7. But between the years -9 and 7 there were no leap years of the Julian calendar and also no leap years of the Alexandrian calendar. Still the Julian and the Alexandrian calendar are used, Thoth is the first, Phamenoth the seventh, Pharmouthi the eighth, and Pachon the ninth month of the Alexandrian calendar, and the fifth day of Phamenoth falls on 1 March, and the fifth day of Pachon on 30 April, of the Julian calendar. The first day (= 1 Thoth) of the year 1 of the era of the emperor Diocletianus (see Section 1) used by the church of Alexandria is 29-8-284.

Unlike the four calendars already mentioned in this section, the Jewish calender is a lunar calendar, of which each month begins relatively shortly (on average one and a half day) after a (proper) Newmoon, i.e. point of time of lunisolar conjunction (i.e. conjunction of sun and moon). But from the coming into being of the Jewish calender, far before the beginning of our era, on until the beginning (about the year 360) of the long time interval during which this calendar was gradually definitively fixed, the moment in which a new month of this calendar began generally depended not only on purely astronomical factors but (indirectly) also on local circumstances (in particular meteorological circumstances in which in Palestine once a year the first appearance of the crescent moon after Newmoon was searched for). As a consequence, it is impossible to reconstruct the development of the Jewish calendar during the time before the moment at which it was finally completely fixed (about the year 776). From the very first beginning every year of the Jewish calender consisted of twelve (mostly) or thirteen calendar months of 29 or 30 days. From the second half of the fifth century before Christ Nisan was the first, Iyyar the second, Shevat the eleventh, and Adar the last month of the Jewish calendar and Pesach, i.e. Pesah, i.e. the Jewish Paschal feast (which in Palestine lasted seven days), was always prepared in the morning and afternoon of the fourteenth day of Nisan. At the time Pesach began always with the sunset with which 14 Nisan ended and 15 Nisan began, and with the meal in which the Paschal lambs slaughtered in the afternoon of 14 Nisan were eaten, and always more or less at the same time as the rise of a full moon.

From the fourth century before to the fourth century after Christ, once a month, relatively shortly after Newmoon, the Jewish authorities in Palestine responsible for the jewish calendar had to determine in which moment a new month of their calendar had to begin, though this was sometimes only pro forma, e.g. at the beginning of Iyyar, because even then Nisan consisted always of thirty days. At that time the beginning of a new month was determined by them as the moment of a sunset in Jerusalem which less than half an hour later was followed by the appearing of a first in principle visible new moon. If at that time about half an hour after the beginning of the thirtieth night after the sunset with which the first day of an expiring month of the Jewish calendar had begun, the first appearance of the moon sickle after Newmoon was confirmed by them (this occurred about once every two months) then this meant that the first day of the new month of this calendar had already begun with the sunset having taken place in Jerusalem about half an hour ago; if not, then the first day of the new month of this calendar began at the moment of the next sunset taking place in Jerusalem. It is thereby that at that time all months of the Jewish calender, thus defined, consisted either of 29 or of 30 days. Because mostly, if weather permits, a waxing moon is visible with the naked eye for the first time between 24 and 48 hours after Newmoon, at the time the first day of a new month of the Jewish calender usually began with the second sunset taking place in Jerusalem after Newmoon. For the same reason at the time the (proper) Fullmoon, i.e. point of time of lunisolar opposition (i.e. opposition of sun and moon) of a month of the Jewish calender differed on average little from the midnight point of time of the fourteenth day of this calendar month.

From the fourth century before to the fourth century after Christ at set times the authorities in Palestine responsible for this calendar had to take not only a decision with respect to the point in time at which a new month of the Jewish calendar must begin (once a month) but also a one concerning the beginning of a new year of their calendar (once a year). These wise men possessed the competence to intervene once a year, at the end of Shevat, in the current year of the Jewish calendar by inserting an extra month consisting of thirty days; this happened about once every three years. They were not only able, by handling that competence carefully, to prevent that the year of the Jewish calendar 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. In fact, the principle that Pesach should be celebrated as early as possible in spring was the only not opportunistic criterion they whether or not applied within the scope of exercising this competence. They must have been familiar with the growing of the days in winter and spring and the phenomenon of the March equinox, but did not keep strictly to their rule of the equinox, i.e. the rule that the fourteenth day of Nisan should fall on or as soon as possible after the March equinox, owing to which Pesach was celebrated actually a month too early many a time.

From the fourth century before to the fourth century after Christ the first day of a new month of the Jewish calender usually began with the second sunset taking place in Jerusalem after a Newmoon and the Fullmoon of this calendar month differed on average little from the midnight point of time of the fourteenth day of this calendar month.

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 600 on 18 March, around the year 1500 on 11 March. Nevertheless, from the first half of the third century to the second half of the fourth century the oldest estimated date of equinox 25 March was considered by the church of Rome to be the date of the March equinox. According to Ptolemaios (see section 2) the March equinox fell around the year 140 on 22 March. As a consequence, in the second half of the third century this date was considered by the church of Alexandria to be the date of the March equinox. About the year 270 the Alexandrian scholar Anatolius, who was bishop of Laodicea (Syria) around the seventies of the third century, made an attempt to reconcile the discrepant viewpoints of the churches of Alexandria and Rome with respect to the date of the March equinox by means of the construction of his famous 19 year Paschal cycle (see also section 6) on the basis of the (incorrect) idea that the moment of the March equinox would not be a question of a point in time or a date, e.g. of 22 or of 25 March, but of the time interval consisting of the four dates 22 up to and including 25 March. Shortly after the third turn of century the church of Alexandria decided to henceforth consider the date 21 March so familiar to us (at the time and nowadays once again 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 this step only in the course of the second half of the fourth century.

 

6 paschal full moons

This section is an extension of Section 6 of millennium question.

The two most important calendars of the first millennium, the Julian calendar (see Section 1) and the Alexandrian calendar (see Section 5), are equivalent (see Section 5).

Jesus was crucified at a Friday afternoon; according to the fourth canonical gospel this horrible event, which was the occasion of the coming into being of Christianity, took place on a fourteenth, according to the three synoptic gospels on a fourteenth or a fifteenth day of Nisan (see Section 5). At the end of the first century Easter, i.e. the Christian Paschal feast, was mostly celebrated at full moon on the evening directly following the fourteenth day of Nisan, at the end of the second century mostly on the first Sunday after the fourteenth day of Nisan. Around the second turn of century the moment of the beginning of Nisan, and therefore also the moment of the beginning of the fourteenth day of Nisan, was not exactly computable yet. In order not to remain dependent on the not entirely predictable way in which in that time in the framework of the Jewish calendar (see Section 5) the beginning of Nisan was determined in Palestine (see Section 5), in the beginning of the third century, calculators of some churches, among which the church of Alexandria (Egypt) and the one of Rome, began to construct, with the help of lunar phase tables, periodic sequences of dates called dates of Paschal full moon of successive years either of the Alexandrian or of the Julian calendar of which each date functioned as substitute for a (in principle unknown) similar date of the full moon of the fourteenth day of Nisan and served as a starting point for the determination of a possible date of Paschal Sunday. Each date of Paschal full moon was the date of the fourteenth day of a lunation which as substitute for Nisan was part of a system of lunations consisting of 29 or 30 days fixed in the respective calendar. The ordinal numbers of the days belonging to such lunations were lunar phase numbers as well. And so the ordinal number of a day belonging to such a lunation was designated as the “age of the moon” on the day in question, which term must of course not be confused with the actual age of the moon (about 4.5 billion years). All the time the “age of the moon” on the date of Paschal full moon was by definition equal to 14, and the one on the date of Paschal Sunday anyway an integral number between 13 and 23.

From the first quarter of the third century until well into the middle ages the activities of computists, i.e. practitioners of the computus paschalis, i.e. the science developed since the beginning of the third century on behalf of the determination of dates of Easter, led, always in the framework of a system of lunations specially devised for this purpose, to the construction of several periodic sequences of dates of Paschal full moon of consecutive years of either the Julian or the Alexandrian calendar. However, not only these sequences of dates of Paschal full moon were often essentially different, but moreover they did not at all always lead to one and the same Sunday for the celebration of Easter, which has led to discord between the churches of Alexandria and Rome many a time. It would last two centuries before a complete and satisfactory solution to the big problem of the computus paschalis was found.

The sole criterion for first new moon visibility with which third century Alexandrian computists were acquainted, is the old Babylonian rule that around the beginning of the spring every new moon will be visible (with the naked eye) for the first time, if weather permits, relatively shortly after sunset, between 24 and 48 hours after Newmoon (see Section 5). This rule implies not only the rule undoubtly applied by them that the first day of Nisan usually began with the second sunset in Jerusalem after the Newmoon of Nisan, but also that the Fullmoon (see Section 5) of Nisan fell on average about the midnight point in time of the fourteenth day of Nisan, because the time difference between the middle of the day of the Newmoon in question counted from sunset to sunset in Jerusalem and the midnight point of time of the fourteenth day of Nisan (likewise counted from sunset to sunset in Jerusalem) is just about a half synodic period of the moon.

About the middle of the third century the church of Rome began experimenting with sequences of dates of Paschal full moon of consecutive years of the Julian calendar with a period of 84 years, the church of Alexandria with sequences of dates of Paschal full moon of consecutive years of the Alexandrian calendar with a period of 19 years. Then the church of Alexandria began also to ply the date of the twenty sixth day of Phamenoth (see Section 5), this is 22 March, which date she then considered as the date of the March equinox (see Section 2), as a lower limit for her dates of Paschal full moon. The first Alexandrian computist known by name who applied this principle to 19-year periodic sequences of dates of Paschal full moon was Anatolius (see Section 5). Most probably he took about the year 260, still before his consecration to bishop, an active part in the construction of the sequence of dates of the proto Alexandrian Paschal full moon, i.e. the sequence of dates of Paschal full moon of consecutive years of the Alexandrian calendar with a period of 19 years until recently unknown to us whose Julian equivalent around the year 270 must have been used by him to construct his famous 19 year Paschal cycle, here designated as Anatolian Paschal cycle. The limit dates of the sequence of dates of the proto Alexandrian Paschal full moon were 27 Phamenoth = 23 March and 25 Pharmouthi = 20 April, and so its dates were dates between 26 Phamenoth = 22 March and 26 Pharmouthi = 21 April. This particular sequence of dates has lost long ago and is not known from historical sources. However, its Julian equivalent (see Section 5) was reconstructed in the year 2009 by the author of this website. It was possible to succeed in it by making use of modern tables of Newmoon concerning the time interval between the years 220 and 260.

The for centuries thought lost Anatolian Paschal cycle must have been a part of a Paschal table composed about the year 270. This Paschal table was not very practical, in that its nineteen Paschal dates were no dates of the Alexandrian or of the Julian calendar but dates of the variant of the Julian calendar here called Anatolian calendar, ingeniously devised by Anatolius just on behalf of this Paschal table, and must have gone out of use, if it has ever been in use, long before the end of the third century. The original (i.c. written by Anatolius) Greek text to which this Paschal table belonged, has been lost, but a translation of this text in Latin dating from the fourth century has been preserved including the curious but consistent structure of this Paschal table under the name of ‘De ratione paschali’ in a little number of medieval manuscripts. In this Latin text we find the Anatolian Paschal cycle again in the form of a sequence of dates of Paschal Sunday of consecutive years of the Anatolian calendar with a period of 19 years, however without any calendar year indication. The 19 year Paschal cycle contained in ‘De ratione paschali’ is not what it at first sight seems to be: an enigmatic sequence of dates of the Julian calendar. This Paschal cycle is a sequence of dates of Paschal Sunday of consecutive years of the Anatolian calendar and is as such really the famous (19 year) Anatolian Paschal cycle, which has been convincingly demonstrated by the Irish scientists Daniel McCarthy and Aidan Breen in the year 2003.

Two of the sequences of dates of Paschal full moon with a period of 19 years which were constructed in the second half of the third century, are of fundamental importance. The first is the sequence of dates of the proto Alexandrian Paschal full moon; the second is the sequence of dates of the Anatolian Paschal full moon constructed about the year 270, which is by definition a sequence of dates of the Julian calendar. We obtain the sequence of dates of the Anatolian Paschal full moon as a sequence of dates of Paschal full moon of consecutive years of the Julian calendar with a period of 19 years, without calendar year indication, by starting from the 19 year Paschal cycle contained in ‘De ratione paschali’, taking each date of this Paschal cycle simply as a date of the Julian instead of as a date of the Anatolian calendar, and transforming it, making use of the accompanying lunar phase number mentioned in ‘De ratione paschali’, into the corresponding date of the Julian calendar with lunar phase number 14. Its limit dates were 23 March and 19 April.

Unfortunately, there is no historical source at all which confirms the historicity of the sequence of dates of the proto Alexandrian Paschal full moon. But in the year 2009 the Julian version of it was reconstructed by the author of this website. By comparing the sequence of dates of the Anatolian Paschal full moon to the sequence of dates of the proto Alexandrian Paschal full moon he moreover was able to establish how these two sequences of dates of Paschal full moon relate to each other and that the initial year of ‘De ratione paschali’, i.e. the calendar year of our era to which the first Paschal date (16 April) of ‘De ratione paschali’ belonged, must have been the year 271. In the year 2010 he wrote an article about this subject which will appear in print soon (see also Section 10).

About the third turn of century the church of Alexandria decided to consider henceforth 25 Phamenoth = 21 March as the date of the March equinox. This is one of the reasons why, about the year 310, the church of Alexandria replaced her sequence of dates of Paschal full moon then in use, possibly the sequence of dates of the proto Alexandrian Paschal full moon, with a new one. It is this new sequence of dates of Paschal full moon, the sequence of dates of the proto classical Alexandrian Paschal full moon, with which Athanasius, bishop of Alexandria around the middle of the fifth century, was familiar. This new sequence of dates and the sequence of dates  of the proto Alexandrian Paschal full moon were sequences of dates of consecutive years of the Alexandrian calendar with a period of 19 years, and generated suitable dates for the celebration of Paschal Sunday by means of the Alexandrian Paschal principle handled by the church of Alexandria since the third century ‘Paschal Sunday is the first Sunday after the Paschal full moon’, e.g. the dates of the proto Alexandrian Paschal Sunday, which was thus defined as the first Sunday after the proto Alexandrian Paschal full moon.

About the year 410 the great Alexandrian computist Annianus discovered that the sequence of dates of Paschal Sunday generated by the sequence of dates of the proto classical Alexandrian Paschal full moon by means of the Alexandrian Paschal principle has a period of (no less than) 532 years and that the same applies to the sequence of dates of Paschal Sunday generated by the sequence of dates of the classical Alexandrian Paschal full moon well known to us, which was obtained by him by advancing one of the 19 dates (namely 11 Pharmouthi = 6 April) of the sequence of dates of the proto classical Alexandrian Paschal full moon by 1 day, by means of the same Paschal principle. With this discovery the great problem how to determine the date of Easter had been solved; the key to the solution was the sequence of dates of the proto classical Alexandrian Paschal full moon (see also Section 8)         . The sequence of dates of the proto classical Alexandrian Paschal Sunday, i.e. the first of the two sequences of dates of Paschal Sunday in question, functioned from the first quarter of the fourth century to the first quarter of the fifth century, the sequence of dates of the classical Alexandrian Paschal Sunday, i.e. the second of these two sequernces of dates, thereafter until ther year 1582.

The limit dates of the sequence of dates of the classical Alexandrian Paschal full moon are 25 Phamenoth = 21 March and 23 Pharmouthi = 18 April, the ones of the sequence of dates of the classical Alexandrian Paschal Sunday 26 Phamenoth = 22 March and 30 Pharmouthi = 25 April. Column F of Table 1 shows the Julian equivalents of the dates of the classical Alexandrian Paschal full moon, column G of this table the dates of the classical Alexandrian Paschal Sunday calculated by Dionysius Exiguus (see Section 1). The sequence of dates of the classical Alexandrian Paschasl full moon differs in only 1 of the 19 calendar years from the sequence of dates of the proto classical Alexandrian Paschal full moon (in each of these cases 10 Pharmouthi instead of 11 Pharmouthi), the sequence of dates of the classical Alexandrian Paschal Sunday in only 4 of the 532 calendar years from the sequence of dates of the proto classical Alexandrian Paschal Sunday (in each of these cases 11 Pharmouthi instead of 18 Pharmouthi). The earliest possible date of the proto Alexandrian Paschal full moon is 27 Phamenoth = 23 March, the one of the proto classical Alexandrian Paschal full moon and the one of the classical Alexandrian Paschal full moon are 25 Phamenoth = 21 March. There exists a striking difference between the sequence of dates of the proto Alexandrian Paschal full moon and the one of the classical Alexandrian Paschal full moon. This difference consists of differences of 2 or 3 days between the corresponding dates of these two sequences of dates of Paschal full moon (see also Section 8).

At the first council of Nicaea, convened in the year 325 by the emperor Constantinus I (Constantine the Great), it was decided that henceforth Easter would have to be celebrated every year early in spring by all Christians on the very same Sunday shortly after the full moon of the fourteenth day of Nisan, on which day traditionally the last preparations were made for the celebration of Pesach (see Section 5). Moreover, the bishops who were together in the year 325 in Nicaea, established that it was necessary to be well informed henceforth about dates suitable for the celebration of Easter amply before, and that therefore, because of the then incalculability of the Jewish calendar (see Section 5), Paschal tables based on the Julian or on the Alexandrian calendar were indispensable. They were agreed about that Easter had to be celebrated in the spring shortly after the fourteenth day of Nisan, and that consequently Paschal Sunday not only had to be preceded by the fourteenth day of Nisan but also by the March equinox (see Section 5). However, they could not reach agreement about the way in which the date of Paschal Sunday should be calculated, through the fact that they were in disagreement about the date of the March equinox, about the way in which the date of Paschal full moon should be calculated, and about the way in which subsequently from this the date of Paschal Sunday should be calculated. They entrusted the churches of Alexandria and Rome with the solution of this problem. It would still take about eighty years before the problem in question was solved satisfactorily, in total something more than three centuries before the church of Rome had accepted the Alexandrian solution, in total more than four centuries before all churches were acquainted with this solution. This solution was the discovery that the sequence of dates of the proto classical Alexandrian Paschal Sunday has a period of 532 years and that the same applies to the sequence of dates of the classical Alexandrian Paschal Sunday. About the year 410 Annianus composed a Paschal table which contained the very first complete description of the construction of the second of these two Paschal cycles. With the help of this Paschal table it was not difficult to univocally determine a suitable Paschal date for any year of the Alexandrian calendar whatever.

Three of the four sequences of dates of Paschal full moon with a period of 19 years mentioned in this section, namely the one of the proto Alexandrian Paschal full moon, also indicated as proto Alexandrian cycle, the one of the proto classical Alexandrian Paschal full moon, also indicated as proto classical Alexandrian cycle, and the one of the classical Alexandrian Paschal full moon, also indicated as classical Alexandrian cycle, have a so called Metonic structure (see also Section 8), and were consequently, from an astronomical point of view, ideal sequences of dates of Paschal full moon. There exists a close relationship between the chronologically second of the four, this is the sequence of dates of  the Anatolian Paschal full moon, which has no Metonic structure, and the chronologically first of the four: the proto Alexandrian cycle is the Alexandrian equivalent of the best Metonically structured approximation of the sequence of dates of the Anatolian Paschal full moon (see also Section 8).

In the first half of the second century computists of the church of Rome experimented with sequences of dates of Paschal full moon of consecutive years of the Julian calendar, subsequently with a period of 8 years, 112 years, 84 years. In the second half of the third century they continued their attempts to construct a usable sequence of dates of Paschal full moon of consecutive years of the Julian calendar with a period of 84 years. It was only in the course of the second half of the fourth century, after the church of Rome had decided to consider henceforth 21 March as the date of the March equinox instead of 25 March, that these attempts gradually led to the construction of such a sequence of dates of Paschal full moon; the result was the sequence of dates of the classical Roman Paschal full moon, in principle with limit dates 18 March and 15 April. The dates of the classical Roman Paschal Sunday were determined by the church of Rome according to the (third century) Roman Paschal principle ‘Paschal Sunday is the first Sunday after the first day after the Paschal full moon’. The sequence of dates of the classical Roman Paschal Sunday was, just like the sequence of dates of the classical Roman Paschal full moon, a sequence of dates of consecutive years of the Julian calendar with a period of 84 years, but in principle with limit dates 21 March and 23 April. The classical Roman Paschal Sunday fell before 25 March many a time, in spite of the fact that until well into the fourth century he church of Rome considered 25 March to be the date of the March equinox. Solely in the years 303, 333, 360 modulo 84 the sequence of dates of the classical Roman Paschal Sunday provided the church of Rome with a date which was unsuitable (according to herself) for the celebration of Easter, either a too early date (21 March) or a too late (22 or 23 April). Around the first half of the fifth century in the western half of the Roman empire for the determination of dates of Easter almost exclusively Roman Paschal tables provided with the classical Roman cycle, i.e. the sequence of dates of the classical Roman Paschal full moon, and the sequence of dates of the classical Roman Paschal Sunday were used. But about the middle of the fifth century the differences of the sequence of dates of the classical Roman Paschal Sunday with the one of the classical Alexandrian Paschal Sunday began to result in controversies between the churches of Rome and Alexandria.

 

7 paschal cycles

This section is an extension of Section 7 of millennium question.

The two most important calendars of the first millennium, the Julian calendar (see Section 1) and the Alexandrian calendar (see Section 5), are equivalent (see Section 5).

Four old Paschal cycles are historically very important. They are, in chronological order, the Anatolian Paschal cycle (see Section 6), for which the Anatolian calendar (see Section 6) served as a basis, the classical Roman Paschal cycle, i.e. the sequence of dates of the classical Roman Paschal Sunday (see Section 6), for which the Julian calendar served as a basis, the proto classical Alexandrian Pascha cycle, i.e. the sequence of dates of the proto classical Alexandrian Paschal Sunday (see Section 6), for which the Alexandrian calendar served as a basis, and the classical Alexandrian Paschal cycle, i.e. the sequence of dates of the classical Alexandrian Paschal Sunday (see Section 6), for which the Alexandrian calendar served as a basis. The first is a sequence of dates of Paschal Sunday of consecutive years of the Anatolian calendar with a period of 19 years which was created by Anatolius (see Section 5) about the year 270, the second is a sequence of dates of Paschal Sunday of consecutive years of the Julian calendar with a period of 84 years which was completed by computists of the church of Rome somewhere in the fourth quarter of the fourth century, the third and the fourth are sequences of dates of Paschal Sunday of consecutive years of the Alexandrian calendar with a period of 532 years which were defined and completed by Annianus (see Section 6) about the year 410.

As we have seen in the previous section, from time to time the sequence of dates of the classical Roman Paschal Sunday provided the church of Rome with an unsuitable date for the celebration of Easter. Much more serious were the problems which arose from the fact that with each new period of 84 years the difference of the claascal Roman Paschal full moon with the accompanying Fullmoon (see Section 5) increased on average by about 1.29 days, and the difference with the classical Alexandrian Paschal full moon on average even by about 1.55 days. In the second half of the fourth century no more than two times (namely in the years 368 and 387) the date of the claascal Roman Paschal Sunday did not coincide with the date of the classical alexandrian Paschal Sunday, in the first half of the fifth century 6 times (namely in the years 401, 406, 425, 428, 431, 448), in the second half of the fifth century 11 times. The fact that in the first half of the fourth century the number in question was much higher than in the second (18 and 2 times respectively) implies that the system consisting of the classical Roman cycle and the classical Roman Paschal cycle took a definite shape only about the middle of the second half of the fourth century, however not for a long time (until the sixth decade of the fifth century).

Roman Paschal tables used around the fourth turn of century show more or less accurately the relation between the classical Roman cycle and the classical Roman Paschal cycle, as in Table 2 (in which all dates are dates of the Julian calendar). In this table constructed by the author of this website by each calendar year of our era indicated in the primary column A we see in column B the corresponding epact being the “age of the moon” on 1 January of the calendar year in question, in column C the corresponding concurrent being the as of old defined weekdaynumber of 1 Januari of the calendar year in question, in column D the corresponding date of the classical Roman Paschal full moon, in column E the corresponding date of the classical Roman Paschal Sunday, in column F the corresponding “age of the moon” on the classical Roman Paschal Sunday. In fact, the numbers in columns B, D, H represent numbers of days.

The structure of Table 2 emerges from the coherence between columns B, C, D, E, F of this table, in concreto the manner in which successively column D from column B, column E from columns C and D, and column F from columns D and E can be obtained. Each date x in column D can be obtained by subtracting the corresponding epact (in column B) from 14 April and reducing the outcome modulo 29 days to a date between 17 March and 17 April. Each date y in column E can be obtained by subtracting the corresponding concurrent (in column C) from 20 February and reducing the outcome modulo 7 days to a date between the date 1 day after the corresponding date x in column D and the date 9 days after this date x. This calculation boils down to the same as applying the Alexandrian Paschal principle to each date x in column D. The number of days which is represented by the the number in column F can be obtained by adding 14 days to the number of days obtained by subtracting the corresponding date x in column D from the corresponding date y in coloumn E.

By the construction of the proto classical Alexandrian cycle (see Section 6), about the year 310, the church of Alexandria was the first church who chose definitively 21 March as the earliest (and 18 April as the latest) possible date for her Paschal full moon, and, because of the Alexandrian Paschal principle (see Section 6), 22 March as the earliest (and 25 April as the latest) possible date for her Paschal Sunday. Although the proto classical Alexandrian cycle is fully known (see Section 6), it is only recently that we have some idea in which way this cycle originated (see also Section 8). It was the direct successor of this cycle, the classical Alexandrian cycle, which ultimately would oust all other sequences of dates of Paschal full moon. This cycle or its equivalent forms the backbone of all classical Alexandrian Paschal tables thus defined. Each of these Paschal tables, of which the Paschal table of Annianus (see Section 6), the one of Dionysius Exiguus (see Section 1), and the one of Beda Venerabilis (see Section 1) are the best known, generated for each of the therein indicated years of the Alexandrian or of the Julian calendar simply and univocally a suitable date for the celebration of Easter.

It is plausible that around the middle of the fifth century the churches in the eastern half of the Roman empire, among which the churches in Palestine, for the most part availed themselves of classical Alexandrian Paschal tables, and that at the same time the churches in the western half for the most part used Roman Paschal tables provided with the classical Roman cycle.

Neither Dionysius Exiguus nor Dionysius Exiguus’ great follower Beda Venerabilis was acquainted with the existence of some old 532 year Alexandrian Paschal cycle. In the year 525 Dionysius Exiguus used the Julian equivalent of the classical Alexandrian cycle to construct his Paschal table, so important from a chronological point of view. In the year 725 Beda Venerabilis published his treatise “De temporum ratione” on the computus paschalis, and in the framework of this famous book his Paschal table, which was an extension of Dionysius Exiguus’ Paschal table, and contained an extension of Dionysius Exiguus’ sequence of dates of Paschal Sunday to a 532 year Paschal cycle which in fact was precisely the Julian equivalent, and as such a reinvention, of the classical Alexandrian Paschal cycle. Only when (at the earliest in the second half of the eighth century) all churches had become familiar with either the original Alexandrian version (Annianus) or the Julian version (Beda Venerabilis) of the classical Alexandrian Paschal cycle, simultaneous celebration of Easter had become possible.

 The classical Alexandrian Paschal table attributed to Cyril (see Section 1) was intended for use in the western half of the Roman empire, and it is for this reason that this table was provided with dates of the Julian calendar instead of dates of the Alexandrian calendar. The same applies to Dionysius Exiguus’ Paschal table. Anyway, Dionysius Exiguus obtained his Paschal table by extrapolation from the Paschal table attributed to Cyril. The Paschal table attributed to Cyril concerned the years 437 up to and including 531, Dionysius Exiguus’ Paschal table the years 532 up to and including 626. Because the Alexandrian Paschal principle applied for all classical Alexandrian Paschal tables and in the “age of the moon” (see Section 6) on a date of Paschal full moon was always 14, in all classical Alexandrian Paschal tables the “age of the moon” on a date of Paschal Sunday is always an integral number between 14 and 22.

In Table 1 (in which all dates are dates of the Julian calendar), which presents a modern version of Dionysius Exiguus’ Paschal table, by each calendar year of our era indicated in the primary column A we see in column C the corresponding epact being the “age of the moon” on 22 March of the calendar year in question, in column D the corresponding concurrent being the as of old defined weekdaynumber of 24 March of the calendar year in question, in column F the corresponding date of the classical Alexandrian Paschal full moon, in column G the corresponding date of the classical Alexandrian Paschal Sunday, in column H the corresponding “age of the moon” on the classical Alexandrian Paschal Sunday. In fact, the numbers in columns C, D, H represent numbers of days. The Latin word “nulla” in column C denotes “nulla epacta” (which literally means ‘no epact’), which is logically equivalent with “nullae epactae”, which means nothing else than ‘no epacts’.

The structure of Dionysius Exiguus’ Paschal table emerges (see Table 1) from the coherence between columns C, D, F, G, H of this table, in concreto the manner in which successively column F from column C, column G from columns D and F, and column H from columns F and G can be obtained. Each date x 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 y 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 x in column F and the date 8 days after this date x. This calculation boils down to the same as applying the Alexandrian Paschal principle to each date x in column F. The number of days which is represented by the the number in column H can be obtained by adding 14 days to the number of days obtained by subtracting the corresponding date x in column F from the corresponding date y in coloumn G. Dionysius Exiguus’ Paschal table is provided with two periodic numerations of its rows, namely one with a period of 15 years, represented in column B, and the other with a period of 19 years, represented in column E.

The sequence of epacts which all classical Alexandrian Paschal tables have in common (see for example column C of Table 1), has a period of 19 years and the additional property that each 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 saltus thus defined occurring once every nineteen times) to the previous one (we remark that 18 × 11 + 1 × 12 ≡ 0 modulo 30). It is this particular structure of this sequence of epacts which is reflected in the so called Metonic structure (see also Section 8) of the classical Alexandrian cycle.

Not only the sequence of epacts but also the sequence of concurrents contained in Dionysius Exiguus’ Paschal table (see column D of Table 1) has a particular structure. The oldest Paschal table in which this sequence of concurrents occurs, is the proto classical Alexandrian Paschal table of bishop Theophilus of Alexandria composed in the year 385. That sequence of concurrents, which all classical Alexandrian Paschal tables have in common with Theophilus’ Paschal table, 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 (we remark that 21 × 1 + 7 × 2 ≡ 0 modulo 7). The particular structure of that sequence of concurrents rests on the leap year proportion of one to four of both the Alexandrian and the Julian calendar and the fact that there are seven days in a week.

We conclude that all classical Alexandrian Paschal tables have one and the same sequence of epacts with a period of 19 years as well as one and the same sequence of concurrents with a period of 28 years in common. Annianus was one of the first to understand that it must be possible to extend the proto classical Alexandrian Paschal tables until then composed to a Paschal table containing a 532 year Paschal cycle, because of the fact that 19 × 28 = 532; he suited the action to the word about the year 410. Dionysius Exiguus was unaware of the existence of some old 532 year Alexandrian Paschal cycle, and he had no proper understanding of the possibility to extend the sequence of dates of Paschal Sunday contained in his Paschal table to a Paschal cycle.

In the year 616 an Irish anonymous extended Dionysius Exiguus’ Paschal table to a Paschal table concerning the years 532 up to and including 721, and it is this newer Paschal table which about the year 640 was accepted by the church of Rome, who from the third century until then had given preference to use her own, relatively imperfect, Roman Paschal tables, and subsequently also by the other churches in Italy. In Ireland and Britain this newer Paschal table became the source of inspiration for the reinvention of the classical Alexandrian Paschal cycle, of which the Julian version in the first quarter of the eighth century was constructed by means of extrapolation from this newer Paschal table, which in the year 725 resulted in the famous Paschal table of Beda Venerabilis. This Paschal table generated for each year of the Julian calendar after the year 531 the Julian equivalent of the date of Easter according to the Paschal table of Annianus. In the Byzantine empire, thanks to Annianus’ Paschal table at all times the bishops were up on the date of the next Paschal Sunday. It was only in the eighth century, when the churches in the Frankish kingdom accepted Beda Venerabilis’ Paschal table, that the churches received the possibility to celebrate Easter on the same day.

It is the Alexandrian Paschal tables composed in Alexandria about the year 310 from which (a century later) Annianus’ Paschal cycle, (two centuries later) Dionysius Exiguus’ Paschal table, and (four centuries later) Beda Venerabilis’ Paschal cycle evolved. 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, and this remained so in the Byzantine empire. However, it was for the first time in the eighth century, when Beda Venerabilis’ Easter table was accepted by the churches in the Frankish kingdom, that classical Alexandrian Paschal tables were used really by all churches. This lasted until the year 1582, when Beda Venerabilis’ Paschal table was replaced with Easter tables adjusted to the Gregorian calendar (see Section 5).

 

8 metonic structure

This section is an extension of Section 8 of millennium question.

The two most important calendars of the first millennium, the Julian calendar (see Section 1) and the Alexandrian calendar (see Section 1), are equivalent (see Section 5).

It is interesting to relate the four sequences of dates of Paschal full moon with a period of 19 years occurring in Section 6 to each other. Their dates, which were originally dates of the Alexandrian calendar, fell between the twenty fourth day of Phamenoth (see Section 5) and the twenty sixth of Pharmouthi (see Section 5). Moreover, the three sequences of dates in question have the property in common that each of their following dates can be obtained by advancing the immediate predecessor of this date 10 or 11 or 12 days modulo 30 days but such that sur chaque période the total number of advanced days amounts to 210 (e.g. 4 × 10 + 10 × 11 + 5 × 12 = 210 days). Among the sequences of dates in question, particularly those in which each following date can be obtained by advancing its immediate predecessor 11 or, once every 19 years, 12 days modulo 30 days (i.c. 18 × 11 + 1 × 12 = 210 days) are highly interesting, because they reflect in the most natural way the phenomenon of the 19 year lunar cycle, i.e. the fact that time intervals of 19 years contain on average nearly as many days as time intervals consisting of 235 synodic months: if we apply that tropical years consist on average of about 365,2422 days and synodic months on average of about 29.53059 days, we get 19 × 365,2422 and 235 × 29.53059 days respectively, in both cases just about 6940 days. The astronomical fact in question was known as early as in the fifth century before the beginning of our era in Mesopotamia, besides in Greece, where the Athenian astronomer Meton discovered or rediscovered it. Consequently, such particular sequences of dates besides their particular structure, like the lunar cycle in question, are called Metonic. Resuming, we can say that by a Metonic sequence of dates we mean a sequence of dates of consecutive calendar years of either the Julian or the Alexandrian calendar falling between 20 March = 24 Phamenoth and 21 April = 26 Pharmouthi which is provided with a Metonic structure, i.e. has a period of 19 years and the property that each of its following dates can be obtained by advancing the immediate predecessor of this date 11 or, once every 19 years, 12 days modulo 30 days.

It is the Metonic structure of the proto classical Alexandrian cycle (see Section 6) and the famous classical Alexandrian cycle (see Section 6) that would turn out to be the key to the solution of the great problem how one would have to calculate the date of Easter. Contrary to the sequence of dates of the Anatolian Paschal full moon (see Section 6), also the proto Alexandrian cycle (see Section 6), which underlay the sequence of dates of the Anatolian Paschal full moon as well as was a precursor of the proto classical Alexandrian cycle, had that same Metonic structure.

The Paschal dates contained in ‘De ratione paschali’ (see Section 6), each with an appurtenant lunar phase number between 13 and 21, are dates of the Anatolian calendar (see Section 6), and as such they are real Sundays. However, taken as dates of the Julian calendar some of the days in question did not fall on Sunday. Therefore, it makes sense to use for each of these special days the term ‘Anatolian Paschal day’ and to reserve the term ‘Anatolian Paschal Sunday’ for each of the Sundays coinciding with or as close as possible to such a special day. Each date of the Anatolian Paschal full moon can be obtained by determining the date of the Julian calendar with lunar phase number 14 going with the corresponding date of the Anatolian Paschal day. The sequence of dates of the Anatolian Paschal day as well as the sequence of dates of the Anatolian Paschal full moon is a sequence of dates of consecutive years of the Julian calendar with a period of 19 years but without a Metonic structure.

Thanks to the fact that the initial year of ‘De ratione paschali’ (see Section 6) is known, it is the year 271, we can relate the sequence of dates of the Anatolian Paschal day and the one of the Anatolian Paschal full moon to the proto Alexandrian cycle and to the classical Alexandrian cycle. All these four important sequences of dates have a period of 19 years. In Table 3 (in which all dates are dates of the Julian calendar), by the sequence of calendar years of our era indicated in the primary column A we see in columns B, C, D, E the corresponding restrictions of these four sequences of dates, each accompanied by its sequence of lunar phase numbers. Although the year 285 was considered to be the initial year of the classical Alexandrian cycle, because it was the starting year of the era of the emperor Diocletianus (see Section 1),   this sequence of dates had in fact not yet been defined in the third century.

It is easy to establish, by comparing columns B and C of Table 3 with each other, that the proto Alexandrian cycle not only differs (one day) from the sequence of dates of the Anatolian Paschal full moon in no more than 4 of the 19 dates, but is even its best Metonically structured approximation. This observation was the final piece of the reconstruction of the Julian equivalent of the proto Alexandrian cycle. We conclude that the sequence of dates of the Anatolian Paschal full moon can be considered to be the link between the proto Alexandrian cycle and the sequence of Paschal dates of ‘De ratione paschali’, which underlines the relevance of each of these three sequences of dates. Everything seems to indicate that the Anatolian Paschal cycle (see Section 6) was developed from the proto Alexandrian cycle via the sequence of dates of the Anatolian Paschal full moon.

Metonic sequences of dates can be divided into two types: those of the first type, characterized by 11 ordinary progressions of 11 days, 1 saltus progression of 12 days and 7 ordinary regressions of 19 days, and the ones of the second type, characterized by 12 ordinary progressions of 11 days, 6 ordinary regressions of 19 days and 1 saltus regression of 18 days. For example, it is easy to verify that the periodic sequence of dates with a period of 19 years defined by column B of Table 3 as well as the one defined by column E of this table is a Metonic sequence of dates of the first type. We note that the immediate successor of the date 1 April in column B is the date 20 April, but in column E the date 21 March.

By comparing columns B and E of Table 3 with each other, we can establish that the difference between proto Alexandrian Paschal full moon (see Section 6) and classical Alexandrian Paschal full moon (see Section 6) is all the time 2 or 3 days. To be able to explain this difference, we must realise that about the year 310 the church of Alexandria replaced the Metonic sequence of dates of Paschal full moon used by her around the third turn of century, probably the proto Alexandrian cycle or else perhaps the Metonic sequence of dates of Anatolius’ lost Paschal table (not to be confused with the until recently thought lost Anatolian Paschal cycle), with the sequence of dates of the proto classical Alexandrian Paschal full moon (see Section 6) and that this was a result of her decision to advance her date of the March equinox by 1 day (from 22 to 21 March) and her desire to define the beginning of the first day of her lunation replacing Nisan (see Section 5) as the moment of the last sunset in Alexandria preceding the Newmoon (see Section 5) in question instead of as something as the moment of the second sunset in Alexandria after the Newmoon in question.

The fact that the year 271 is the initial year of ‘De ratione paschali’, implies that the Metonic sequence of dates which according to Eduard Schwartz as well as the one which according to Alden Mosshammer could be considered to be the Anatolian Paschal cycle, contrary to the Metonic sequence of dates of the proto Alexandrian Paschal full moon, can certainly not have underlain the Anatolian Paschal cycle. Thanks to the fact that the initial year of ‘De ratione paschali’ is known (it is the year 271), we can also establish in which calendar years of our era the Anatolian Paschal day was a Sunday and which Sundays were Anatolian Paschal Sundays in the time of the episcopate of Anatolius (see Section 5) (around the seventies of the third century). This can be seen in Table 4 (in which all dates are dates of the Julian calendar). In this table, by each calendar year of our era indicated in the primary column A we see in column B the corresponding dste of the proto Alexandrian Paschal full moon, in column C the corresponding dste of the Anatolian Paschal day, in column D the corresponding dste of the Anatolian Paschal Sunday, and in column E the corresponding dste of the proto Alexandrian Paschal Sunday. We establish that only in the years 264 up to and including 271 the Anatolian Paschal day was a Sunday, and that between the years 250 and 272 it occurred only twice that the Anatolian Paschal Sunday did not coincide with the proto Alexandrian Paschal Sunday.

In the first three and a half centuries of our era the Fullmoon (see Section 5) of Nisan fell on average in the vicinity of the midnight point of time of the fourteenth day of Nisan, and in consequence at that time the date of the fourteenth day of Nisan was on average half a day later than the date of the Fullmoon of Nisan. In the second half of the third century the date of the proto Alexandrian Paschal full moon fell on average about 0,7 days after the date of its Fullmoon.

About the year 310 the church of Alexandria opted for the proto classical Alexandrian cycle, which was definitely replaced with the classical Alexandrian cycle about the year 410, thanks to Annianus (see Section 6). Around the year 310 the date of the proto classical Alexandrian Paschal full moon fell on average about 1,4 days before the date of its Fullmoon, around the year 410 the date of the classical Alexandrian Paschal full moon on average about 1,1 days before the date of its Fullmoon. The proto Alexandrian cycle functioned less than half a century, the proto classical Alexandrian cycle a century, the classical Alexandrian cycle almost twelve centuries, until the year 1582, when the Julian calendar was replaced with the Gregorian calendar (see Section 1).

The (Metonic) structure of the classical Alexandrian cycle was to such an extent a realistic reflection of the rhythmicity of the lunar phases that only after three centuries the average distance (i.e. the average absolute value of the difference) between the date of the classical Alexandrian Paschal full moon and the date of its Fullmoon, which originally (about the year 410) still had amounted about 1,1 days, was decreased by a day (see also Section 9). Only around the middle of the eighth century this average distance was minimal, only shortly before the end of the eleventh century it reached its initial value again. Up to then proto Alexandrian, proto classical Alexandrian, and classical Alexandrian Paschal full moons had had always more or less the appearance of a pure full moon (i.e. about Fullmoon). By nature a pure full moon is always preceded by a waxing full moon one night earlier and followed by a waning full moon one night later, which both look like pure full moons (see Figure 4). It is only since the thirteenth century that classical Alexandrian Paschal full moons for the most part do not look like pure full moons.

 

9 anni domini

This section is an extension of Section 9 of millennium question.

The first year of Anni Domini (literaly ‘the Years of the Lord’) is the calendar year of our era that Jesus was born, the last is the calendar year of our era that he was crucified.

Although we have solved the millennium question completely (see Section 3) and justified the term ‘millennium mistake’ (see Section 4), the question of the connection between the Anno Domini era (see Section 1) and Anni Domini (literally ‘the Years of the Lord’), in particular Jesus’ birth and death, has still remained unsolved. The same applies to the question of the connection between the starting year of the Anno Domini era chosen by Dionysius Exiguus (see Section 1), i.e. the year 1 (of our era) = the Roman year 754 (see Section 1), and Annus Dominicae Incarnationis (literally ‘the Year of the Incarnation of the Lord’) in the view of Dionysius Exiguus. In the writings of Dionysius Exiguus himself no clarification on this question can be found, while in the writings of Beda Venerabilis (see Section 1) some observations with regard to this question are found which lead to contradictory deductions. But the majority of modern historians think that Dionysius Exiguus believed that Jesus was born seven days before the beginning of the year 1 or that he believed that He was born on 25-12-1.

Peter Rietbergen (university of Nijmegen) is of the opinion that Dionysius Exiguus believed Jesus was born one week before the beginning of the year 1, so in the year -1 (of our era) = the Roman year 753. This view accords with the well known historical fact that Charlemagne let himself crown emperor just on 25-12-800 (see Section 0). The opinion of the Dutch archivist 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 does not seem to be less plausible than the other one because of the analogy between the beginning of the Anno Domini era and the one of the Ab Urbe Condita era (see Section 1): “as Rome was founded (on 21 April?) in the Roman year 1 (of the Ab Urbe Condita era), so Jesus was conceived (on 25 March?) and born (on 25 December?) in the year 1 (of the Anno Domini era)” Dionysius Exiguus could have thought.

One of the most influential figures of the first council of Nicaea (see Section 6) was Eusebius, the historian who had become bishop of Caesarea (Palestine) shortly after the year 313. He was the first who hit upon the idea of an era with the year of birth of Jesus as starting year. He thought that Jesus was born in the third year of Olympiad 194 (see Section 3). The view of Orosius (see Section 1), a century later, that Jesus would have been born on 25 December of the Roman year 752, is in accordance with that. However, Dionysius Exiguus chose (indirectly) the Roman year 754 (instead of the Roman year 752) as starting year for his new era (see Section 1). Perhaps he felt compelled to do that in order to effect that to his new era (just like to the era of the emperor Diocletianus) the rule should apply that the year number of a leap year is divisible by 4 (n.b. AD 532 = the year 248 of the era of Diocletianus).

Dionysius Exiguus did not know, and we also do not know, on which date of the Julian calendar or in which calendar year of our era Jesus was born. It is in principle not impossible that moment zero, the unique point in time asterisked (*) so suggestively on our first timeline (see Figure 1) and identical to [1-1-1; 0:00], could have been the moment of Jesus’ birth. However, it is as good as certain that Jesus was born in any moment between the years -9 and -1, so more than a year before the beginning of the Christian era, a remarkable paradox. According to modern historians Jesus was born about the year -4. Sometime in the nineties of the previous century the day on which it had been two thousand years since Jesus was born, slipped by.

At least as interesting as the question “when precisely was the beginning of Anni Domini?” is the question “when precisely was the end of Anni Domini?”. The end of Anni Domini is the crucifixion which gave occasion to the coming into being of Christianity. Neither the calendar year of our era in which nor the date of the day on which Jesus died, is known for certain. It is generally known that Jesus died about the year 30 on a Friday afternoon in Jerusalem, namely (according to the three synoptic gospels) on a day on which or one day after a day on which or (according to the fourth canonical gospel) on a day on which Pesach (see Section 5) was prepared, so on a fourteenth or on a fifteenth day of Nisan (see Section 5). However, this day must have been a fourteenth day of Nisan, because the fifteenth day of Nisan was a feast day on which one did not administer justice in Jerusalem. The conviction of faith that Jesus was crucified a few hours before the celebration of Pesach would begin, agrees, anyway, with the fact that at the end of the first century the Christian Paschal feast was mostly celebrated on the evening directly following the fourteenth day of Nisan (see Section 6). It is certain that Jesus died on a Friday at the time of the reign of the emperor Tiberius (who reigned from 14 to 37) and of the procuratorship of Pontius Pilatus, who was procurator of Judaea from 26 to 36.

Beda Venerabilis has tried to determine the death day of Jesus with the help of the 532 year Paschal cycle which was part of his Paschal table (see Section 7), evidently taking his departure from the rather inaccurate principle ‘Paschal full moon = 14 Nisan’. He hoped to arrive at the date 25-3-34, evidently among other things 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 for granted that his Paschal table would be valid for all calendar years of the Anno Domini era. He looked at the column of his Paschal table corresponding to column F of Dionysius Exiguus’ Paschal table (see Table 1) and saw to his disappointment that the day indicated by this column of his Paschal table for the year 566 (≡ 34 modulo 532) was a Sunday 21 March and not the Thursday 24 March expected by him. Evidently he believed not only that Jesus had died on a 25 March as well as on a fifteenth day of Nisan (in accordance with the three synoptic gospels), but also that He had died in the year 34. Evidently his presuppositions were mutually incompatible.

There is no rational ground for the conviction of faith that Jesus would have died on a Friday 25 March. For quite a while one cherished the conviction resting on the oldest Roman Paschal table, constructed around the year 220 by the Roman scholar Hippolytus, that Jesus would have died on 25-3-29. But the more Paschal tables which kept much better step with astronomical reality (see Section 6) became available, the more the understanding grew that this proposition was untenable. In the fourth century people continued to believe that Jesus had died on a 25 March; then they went to believe also that he was conceived on a 25 March and born on a 25 December. We may doubt the correctness of this attractive vision (to which after all still two calendar year numbers are lacking). As much we may cast doubt on the unconditional applicability of the principle ‘Paschal full moon = 14 Nisan’ to the dates of the classical Alexandrian Paschal full moon (see Section 6) included in Beda Venerabilis’ Paschal table. Nevertheless one can wonder whether it is possible to trace the date of Jesus’ death day in the manner of Beda Venerabilis, i.e. by insouciantly applying this principle to the dates of the classical Alexandrian Paschal full moon between the years 26 and 36.

The dramatic confrontation between Jesus and the Roman procurator Pontius Pilatus must have taken place in Jerusalem at any moment between the years 26 and 36. In order to be able to make a serious attempt to determine the death day of Jesus in the manner of Beda Venerabilis, we consider the dates of the classical Alexandrian Paschal full moon belonging to the nine years 27 up to and including 35 (of our era) according to Beda Venerabilis’ Paschal table (these dates are the same as those of the years 559 up to and including 567 in column F of Table 1) closer by means of an investigation into their as of old defined weekdaynumbers (with the help of column D or of column G of Table 1); column B of Table 5 (in which all dates are dates of the Julian calendar) shows the result. It turns out that there is no Thursday among the days in question but a Friday, which in principle (perhaps) could have been Jesus’ death day. But this Friday 15-4-29 was too early to be able to find favour in the eyes of Beda Venerabilis.

The classical Alexandrian cycle (see Section 6), which forms the backbone of Beda Venerabilis’ Paschal table, functioned actually from the fifth to the sixteenth century, but its theoretical domain consists by definition of the calendar years of our era between 4 and 1582, because only during the time interval consisting of these years of the Julian calendar the leap year regulation of the Julian calendar functioned perfectly (see Section 5). Because the period of this cycle is 19 years, we may take this cycle as a strictly regularly running imaginary clock with a dial of which the hour hand has been replaced with a year hand which takes all the time 19 years (instead of 12 hours) to go round one time. This imaginary clock has run precisely and unbrokenly from 4 to 1582. By comparing dates of the classical Alexandrian Paschal full moon with dates of Fullmoon (see Section 5) can be established that in the time of the Alexandrian computists who constructed the proto classical Alexandrian cycle (see Section 6), around the third turn of century, this imaginary clock, like the proto classical Alexandrian cycle, had a lead of about 1.4 days on the astronomical reality (see Section 8). But thereafter our imaginary clock went to lose its lead, as a result of the fact that a time interval consisting of 235 synodic months is a little shorter than a one consisting of 19 years of the Julian calendar, though both time intervals consist of about 6940 days.

Although the Julian calendar was no ideal calendar, it functioned precisely and unbrokenly from 4 to 1582. All that time a time interval consisting of 19 years of the Julian calendar lasted on average 19 × 365.25 = 6939.75 days, but the moon took on average about 235 × 29.53059 ≈ 6939.689 days to pass through all its phases 235 times, because the synodic period of the moon amounts to about 29,53059 days (see Section 8). Around the year 300 our imaginary clock still had a lead on the astronomical reality of about 1.4 days. After the third turn of century, that lead decreased every new period of 19 years by about 6939.75 − 6939.689 = 0.061 days, so every year by about 0.0032 days. That implies that that lead had decreased by a whole day only after about 310 years. This implies not only that from the third to the sixth turn of century our imaginary clock had lost almost a whole day, but also that from the third turn of century back to the time of the reign of the emperor Tiberius it had gained almost a whole day. And so it is not surprising that around the year 30 classical Alexandrian Paschal full moons were no full but waxing moons, on average about 2.3 days younger than Fullmoon. This implies that there is no sense in applying the principle “Paschal full moon = 14 Nisan”, such as Beda Venerabilis endeavoured to do, to the dates of the classical Alexandrian Paschal full moon between the years 26 and 36.

As a matter of fact, for dating Jesus’ crucifixion the dates of the proto Alexandrian Paschal full moon (see Section 6) are far more suitable ingredients than the dates of the classical Alexandrian Paschal full moon, because around the year 30 proto Alexandrian Paschal full moons were usually full moons on average only about 0.1 days younger than Fullmoon. If Beda Venerabilis had been acquainted with the proto Alexandrian cycle (but of course he did not know this cycle) instead of the classical Alexandrian cycle then he could easily have come to the conclusion that solely 7-4-30, according to the three synoptic gospels, or 3-4-33, according to the three synoptic gospels as well as the fourth canonical gospel, could have been the date of Jesus’ crucifixion (see columns C and D of Table 5).

It is the (nine) dates of the fourteenth day of Nisan between the years 26 and 36 which are essential for the determination of the date of the day of Jesus’ death. Unfortunately they are not known. However, to obtain all possible dates of such a fourteenth day of Nisan, in order to obtain all possible dates of the day of Jesus’ death in systematic way, we can (of course with the help of a suitable lunar phase table) make use of the rule concerning the beginning of Nisan, being the old rule that Nisan usually begins with the second sunset in Jerusalem after its Newmoon (see Section 5). This simple rule is a result of the old Babylonian rule that around the beginning of the spring (on the northern hemisphere of the earth), waxing moons are usually (in clear weather) visible (with the naked eye) for the first time (during some seconds or minutes) between 24 and 48 hours after Newmoon, namely in the west relatively shortly after sunset.

The rule concerning the beginning of Nisan implies that if the point of time of a Newmoon generating a past jewish month Nisan is given, it is possible to obtain a quite accurate estimate of the date of the Julian calendar whose daylight part coincided with the daylight part of the first day of this month Nisan by simply adding 2 or 3 days to the date reduced to (the geografic length of) Jerusalem of this lunisolar conjunction, depending on whether the point of time reduced to (the geografic length of) Jerusalem of this lunisolar conjunction fell before or after 18:00, respectively. It is possible to locate all possible months Nisan in question in the Julian calendar by making use of the only not opportunistic Jewish principle in question, namely that the first evening of Pesach had to be celebrated at full moon as early as possible in spring, i.e. as early as possible after the day of the March equinox (see Section 2) counted from sunset to sunset in Jerusalem, and in addition taking into account the fact that in actual practice the Jewish authorities in Jerusalem often did not apply this rule strictly and in consequence let begin their month Nisan and so also the celebration of their Paschal feast actually a month too early many a time. So the whole thing is to present two points of time of Newmoon reduced to Jerusalem for each of the (nine) calendar years of our era in question, in concreto one in column B of Table 6 (in which all dates are dates of the Julian calendar) and one in column B of Table 7 (idem), of which the former generates a possible date of the fourteenth day of Nisan after the March equinox (in column D of Table 6) via a possible date of the first day of Nisan (in column C of Table 6) and the other a possible date of the fourteenth day of Nisan before the March equinox (in column D of Table 7) via a possible date of the first day of Nisan (in column C of Table 7).

During the time interval consisting of the time between the years 20 and 40 the date of the March equinox was sometimes 23 March sometimes 22 March (nearly equally often). In order to be able to obtain (of course with the help of lunar phase tables) for each of the (nine) calendar years in question the two points of time reduced to Jerusalem of a Newmoon which (possibly) in any way could have generated a jewish month Nisan, it is necessary and sufficient to determine a lower limit and an upper limit with a difference of about 59 days (being about twice the synodic period of the moon) between which these two points of time reduced to Jerusalem must occur in order to guarantee that the corresponding possible dates of the fourteenth day of Nisan will be not only not earlier than just 22 February (this date being just 29 or 30 days earlier than 23 March) but also not later than just 20 April (this date being just 29 days later than 22 March).

It is not surprising that we will be able to achieve the purpose set in the previous paragraph by taking our departure from the lower limit reduced to Jerusalem 6 February 18:00 and the upper limit reduced to Jerusalem 5 April 18:00, for adding 3 + 13 days to 6 February gives 22 February and adding 2 + 13 days to 5 April gives 20 April. It is column B of Table 6 which contains for each of the calendar years in question the corresponding as good as possible estimated point in time reduced to Jerusalem of the second Newmoon between these limits. It is column B of Table 7 which contains for each of these calendar years the corresponding as good as possible estimated point in time reduced to Jerusalem of the first Newmoon between these limits. That is why column C of Table 6 contains for each of these calendar years a possible date of the first day of Nisan after 9 March (this date is just 13 days earlier than 22 March) and column C of Table 7 for each of these calendar years a possible date of the first day of Nisan before 9 March. And that is why column D of Table 6 contains for each of these calendar years a possible date of the fourteenth day of Nisan after the March equinox and column D of Table 7 for each of these calendar years a possible date of the fourteenth day of Nisan before the March equinox.

Because the dates mentioned in columns C of Table 6 and Table 7 may be assumed to deviate no more than a day from what they represent, this applies also to the dates mentioned in columns D of these two tables. And because the day on which Jesus was crucified must have been a Friday the fourteenth day of Nisan, we can conclude that solely the Thursdays, Fridays and Saturdays in columns D of these two tables are of importance to us. The only Thursday among them could have been the last day before Jesus’ death day, each of the three Fridays among them Jesus’ death day, and each of the two Saturdays among them the first day after Jesus’ death day. That implies that in the framework of the Julian calendar there are only six possibilities for the date of Jesus’ death day, with probabilities which are difficult to estimate. In principle the Fridays in columns D of Table 6 and Table 7, namely the Fridays 11-4-27, 7-4-30, 3-4-33, qualify more for that than the Fridays immediately following a Thursday in these columns or immediately preceding a Saturday in these columns, namely the Fridays 18-3-29, 14-3-32, 6-3-33. One of the six Fridays mentioned in columns E of Table 6 and Table 7 must be Jesus’ death day, but in principle the three Fridays in column E of Table 6 stand as such a greater chance than the three in column E of Table 7.

The first of the three Fridays coming to the fore in the previous paragraph (11-4-27, 7-4-30, 3-4-33) seems to be too early to have been the death day of Jesus, since it must be considered as certain that Jesus was not baptized before early in the year 27 at the earliest and manifested himself thereafter during more than a year. The third of the three dates in question seems to be a more probable possible date of Jesus’ death day than the second, because the evident fact that at the decisive moment Pontius Pilatus thought that he could not permit himself to defy the Jewish authorities in Jerusalem, seems to indicate his undoubtedly diminished self confidence due to the fact that in the year 31 his patron Lucius Sejanus had fallen into disgrace with the emperor Tiberius. This implies that (for the time being) 3-4-33 has the greatest chance to be the death day of Jesus. The English monk and scholar Roger Bacon, who lived in the thirteenth century, was the first to substantiate the opinion that Jesus was crucified on 3-4-33.

 

10 epilogue

This section is an extension of Section 10 of millennium question.

The turn of year [31-12-1999; 24:00] = [1-1-2000; 0:00] is the most recent moment at which all four digits of the year number of the current calendar year of our era changed simultaneously. However, that “magic” point of time was not the second turn of millennium but the moment 1999 of our era. The beginning of the third millennium was not the moment 1999 but the moment 2000 of our era, i.e. [31-12-2000; 24:00] = [1-1-2001; 0:00].

It is to be hoped that towards the year 3000 one will have become wiser, for otherwise then once again we will have to undergo that a dancing and jumping crowd of frantic people, made mad by commerce, media and authorities, one year too early is waiting on the platform for the next millennium train, to then get on together by mistake in the last year local preceding this millennium train. To be precise once more: 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 still know (see Section 3), the first millennium train left at moment zero, i.e. at [1-1-1; 0:00], in order to reach its final destination in [31-12-1000; 24:00].

Around the year 2000 more than six hundred websites devoted to the millennium question were made. On the majority of those websites one declared, like on this website millennium, for the proposition that the year 2001 is the first year of the third millennium and connected this proposition rightly to the fact that in our era we have no year zero. But, and this is the original reason for being of this website, on this website one can also find the observation that that fact was not in the least an error of Dionysius Exiguus (see Section 2) or of Beda Venerabilis (see Section 5) but purely a condition the Christian era (see Section 0) must satisfy in order to be able to maintain its bilaterally symmetric structure (see Section 2). There is no year zero in our era simply because from the very outset it contained no year zero. And never any year zero has been added to our era because through the centuries preservation of its symmetric structure with respect to moment zero (see Section 0), as in our second time line (see Figure 2), has always outweighed the (relatively slight) practical advantage of an introduction of a year zero. Between the first century before and the first century after the beginning of our era there is no place for a zeroth century, and, for the same reason, no place for a year zero.

Jan Zuidhoek (see Figure 5), the author of this sextilingual website called millennium, 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 Gymnasium Celeanum in Zwolle. This website has evolved from the (Dutch language) article ‘Millenniumvergissing’ which he, inspired thereto by critical pupils who wanted to know all the ins and outs, wrote in the year 2000 on the millennium question for Euclides, the organ of the Dutch association of teachers of mathematics. After he had also made a contribution to the discussion on the millennium question on Internet, among other things via Wikipedia and via the websites ‘Millenniumvergissing’ and ‘Millennium Mistake’ (now no longer existing), his further investigations in the field of chronology led via successively a systematic treatment of the question of the date of the crucifixion which was the occasion to the coming into being of Christianity (see Section 9) and his reconstruction of the proto Alexandrian cycle (see Section 6) to the discovery (in the year 2009) that the initial year of the Anatolian Pascal cycle (see Section 6) must have been the year 271. And so his contribution to the third international conference held in Galway (in the year 2010) on the history and science of the computus paschalis (see Section 6) was an explanation of the way in which he succeeded in determining the initial year of this famous 19 year Paschal cycle. The accompanying article, which contains, among other things, a reconstruction of the proto Alexandrian cycle, still awaits its publication.

Not only the proto Alexandrian cycle, but also the classical Alexandrian cycle (see Section 6) was reconstructed by the author of this website. A summary of his reconstruction of the classical Alexandrian cycle will be presented here as soon as this reconstruction has  appeared in print.

 

 

 

 

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