millennium mistake

© 2000 Jan Zuidhoek 2014

> www.janzuidhoek.net

 

 

 

 

0 prologue

Each of the six versions differing in languuage of this sextilingual website (called millennium) consists of two chapters, in this (english language) version called millennium question and millennium mistake, 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 is written in six languages (Dutch, English, German, French, Italian, and Spanish), the second in Dutch (millenniumvergissing) and in English (millennium mistake), and only partly in German (millenniumirrtum) and in French (erreur de millénaire), but not yet in Italian (errore di millennio) and also not yet in Spanish (error de milenio). These two chapters contain the same subjects, but in the second these subjects are treated much 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), the Julian and the Jewish calendar in antiquity (in Section 5), Alexandrian Paschal full moons (in Section 6), the date of Jesus’ crusifixion (in Section 7), and Metonic structure (in Section 8). Insofar as any paragraph of this chapter has not yet been realised, this is indicated by the symbol &.

When someone asserts that the year 2000 was the last year of the previous millennium then people often react with a denial, such as: “oh no, the year 2000 was the first year of the new millennium, because the year zero was the first year of our era”. Perhaps at first sight a similar reaction seems not 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 use this to solve the tricky millennium question, we must pay attention to the structure of our era (the term ‘era’ here of course in the meaning of a linear system of numbered calendar years). For this purpose we will enter the field of chronology, which, as the science of locating historical events in time, is part of the field of history (chronology is the backbone of history). The millennium question is a question of chronology.

In practice locating an event in time boils down to placing the moment of this event in the framework of our era. Our era is the Christian era, which however did not begin on the day Jesus was born. Dates (of events) are in preinciple dates of the Christian era. 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. 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), in the sixteenth century explicitly. In all probability Jesus was born some years before moment zero.

At moment zero it was midnight in Greenwich. At that moment the year 1, i.e. the year 1 of our era, i.e. the starting year of the Christian era, began. Therefore moment zero 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]. 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.

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 former websites “Millenniumvergissing” and “Millennium Mistake” (in different languages but with the same content) of the same author. Clarifying remarks in reply to the standpoint with regard to the millennium question taken in this website and sceptical reactions to it led to emendations or were included among the deductions of Section 3 or incorporated into the objections of Section 4.

 

1 moment zero

Our era is the Christian era (see Section 0), which did not begin on the day Jesus was born but at moment zero (see Section 0). Jesus was born some years before moment zero. The Christian era, since the year 1582 in combination with the Gregorian calendar by far the most widespread chronological system on earth, was originally (until the year 1582) coupled to the Julian calendar, which had been introduced by Julius Caesar as early as before the beginning of our era, but was replaced with the Gregorian calendar in the year 1582 by a decree of pope Gregorius XIII. These two calendars differ solely in their leap year regulation (see also Section 5). The calender years of our era before the year 1582 are Julian calendar years, the calender years of our era after the year 1582 are Gregorian calendar years. The dates of our era before the year 1582 are Julian calendar dates, the dates of our era after the year 1582 are Gregorian calendar dates. The year 1582 comprised only 355 days (see also Section 5).

The Julian calendar was a drastically improved version of the primeval Roman calendar, with calendar years beginning and ending in winter. In the first four centuries of our era there was besides the Julian calendar still another solar calendar which was generally used in the Roman empire, namely the Alexandrian calendar, with calendar years beginning and ending in summer.

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 respectively tenth century. Dionysius Exiguus’ Paschal table is a continuation of the Paschal table which has been attributed (probably wrongly) to bishop Cyrillus of Alexandria (Egypt). The Paschal table attributed to Cyrillus was composed about het jaar 440; this Paschal table contained Julian instead of Alexandrian calendar dates and probably it was the first Alexandrian Paschal table with this property. The Julian calendar dates in this Paschal table are numbered according to the era of the emperor Diocletianus, in Dionysius Exiguus’ Paschal table, on the other hand, according to Dionysius Exiguus’ new era, which had the intention to have begun with Jesus’ incarnation.

Until now our historians did not succeed in determining the date of the birth of Jesus. So it is not so surprising that Dionysius Exiguus was not able to do that either. Be that as it may, he chose indirectly (via the era of the emperor Diocletianus) the Roman year 754, i.e. the Julian calendar year 754 of the (incomplete) Anno Urbis Conditae (literaly ‘in the Year of the Foundation of the City’) era, as the starting year of his new era. Then he put the Roman years, i.e. the Julian calendar years of the Anno Urbis Conditae era, from this Roman year 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 (so this Roman year 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 (so this Roman year began at the moment 9 and ended at the moment 10). 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 the Roman years were counted from a supposed year of foundation of the city of Rome. In spite of this, in reality the Anno Urbis Conditae era, like the Anno Domini era, did not exist yet in antiquity, for it was systematically used for the first time not before the beginning of the fifth century, namely by the Iberian historian Orosius (although in a rather careless way). Though in all probability Dionysius Exiguus was acquainted with (but never used) the Anno Urbis Conditae era, pope Bonifatius IV (around the year 600) seems to have been the first who recognized the connection between these two important incomplete eras (i.e. AD 1 = AUC 754).

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 boiling down 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 perhaps inconsequential looking but extremely important mathematical concept, 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 more than ‘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).

Why must the digit 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 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 next 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 in combinstion 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). So it must have gone, in the India of around the year 600. More than three centuries later Arab merchants brougt an Arabic version of the positional system in question 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 spread of the Arabic prototype of our modern decimal positional system over 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 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 been familiar with the number zero. However, there where we would say that the epact (see also Section 6) 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’. There where computists 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 being familiar with the number zero. There where Dionysius Exiguus sees simply a column of numbers of epacts (such as ‘12 epacts’ and ‘no epacts’), it is our modernized brain which 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 only (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 really stupid. We establish that he is no exception to the generally accepted rule that in early medieval Europe nobody knew the number zero. It is only about 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 jell only after one had got 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 carrying out abstract calculations in the decimal positional system with all its ten digits (including the digit zero). This explains why 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 first king who was designated by his name George’. Numberings of admission tickets begin at 1; for the counting of any things whatsoever (unlike for the measuring of the length of any things whatsoever), we do not need the number zero at all. Consequently the counting of years does not differ from the counting of any things whatsoever, and therefore someone born on 1‑1‑1 will have celebrated his first birthday (not being the day 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 accepted 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 this table also outside the framework of this table. However, the first who did this, was not the church of Rome but Beda Venerabilis, the first English historian. It is by the agency of this great scholar that as early as about the year 730 the Christian era was actively taken into use as a coherent system for dating 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 definitely 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

In order to create the possibility of localizing 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 calendar years thus obtained was joined together in the most obvious way with the sequence of calendar years 123…… to the complete sequence of calendar 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 = the Roman year 744. 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, which division essentially boils down to a division into (strictly) positively numbered and strictly) negatively numbered calendar years without the number 0 being allocated to any calendar year. With the duration of a year as unit of time, the (complete) Christian era (see Section 1) 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 = the year 1 before Christ (consequently this calendar year began at the moment -1 and ended at the moment 0) and e.g. year -10 = the year 10 = the year 10 before Christ (consequently this calendar year began at the moment -10 and ended at the moment -9). The running of things at extending the (incomplete) Anno Domini era to the (complete) Christian era can be simply summarized in our observation that the year -x (of our era) = year -x = the year x = the year x before Christ, though until in the thirteenth century negative numbers were entirely unknown in Europe.

The most important property of the Julian calendar (see Section 1), which after far reaching precautionary measures took off with the beginning of the year -45, is its proleptic leap year regulation (a leap year once every four years). In principle this leap year regulation holds for all calendar years of our era before the year 1582. However, owing to the initially inadequate functioning of this regulation, there were between the leap years -45 and -9 three leap years too much (namely a leap year once every three instead of once 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 calendar 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 (see also Section 5). Thus all leap years and hence all calendar years of our era, from the farthest past until very far into the 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, on closer inspection that timeline cannot represent a pure linear time scale, because two of those calendar years are not always precisely equally long. Usually the difference between the lengths of two of those calendar years is either nil 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 (provided that year -x is taken as the year -x = the year x before Christ) as a simple and as such consistent mathematical model of the (complete) Christian era. Likewise our first timeline (see Figure 1) can be interpreted as a simple and as such consistent mathematical model of the (incomplete) Anno Domini era.

What strikes us most (perhaps even is bugging us) in our second timeline is of course that in here there is no room for a year zero. We will still see (in Section 3) why our era from the outset, and to this very day, had to do without a year zero, even though the number zero is common property for a long time. Modern historians who know their job, let the year 1 be preceded by the year -1 really 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 with 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, just like it marks the direct transition (turn of century) from the first century before Christ to the first century (after Christ). Just like there is no zeroth century (and no zeroth millennium) in our era, there is also no year zero, thanks to Beda Venerabilis (see Section 1).

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. There where Beda Venerabilis in his book “De temporum ratione” about “reckoning of time” explains dividing 725 by 19, he says first that 19 times 30 makes 570 and that 19 times 8 makes 152 and then “remanent iii”, 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’). There where he is calculating, he never uses any symbol or word for (the number or a numeral) zero. And there 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 infer that Beda Venerabilis was acquainted with zero; the same holds for Dionysius Exiguus.

In the standard work about “De temporum ratione” written by the Canadian historian Faith Wallis we find a modern version of Beda Venerabilis’ Easter cycle (see also Section 6), with our modern digits and with epacts (see also Section 6) 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 there 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 us believe that we see the number zero there where early medieval scholars only had thought of ‘nothing’ or ‘none’. There 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 number zero remains really 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 Julian calendar 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 formula for the date of Easter Sunday (see also Section 6), to determine really all Julian calendar 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 calendar years are Julian calendar years and all dates Julian calendar dates). By the way, that does not alter the fact that the very first construction (about the year 260) of a Metonic sequence of dates (see also Section 8) of whitch the dates acted as substitutes for dates of the fourteenth day of Nisan (see also Section 5) was an impressive arithmetical finding, which can be attributed to the Alexandrian computist Anatolius (see also Section 5).

The great Alexandrian astronomer Ptolemaios used the symbol o as a numeral zero in the (originally Babylonian) sexagesimal positional system. But the 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, 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 the symbol o for the digit zero, due to which it became possible to carry out abstract calculations efficiently, i.c. by means of handy algorithms. Around the year 600 the clarification of the concept of number which ensued from the introduction of the symbol o for the digit zero (in modern notation 0) led to the invention of the number zero (in modern notation 0 as well). The great Indian mathematician Brahmagupta was the first who, about the year 630, made explicit the most important properties of this unique number (for any number x we have x + 0 = x and 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 into ones stride in the first half of the thirteenth century (in Italy, after a hesitant beginning 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, though as early as around the year 720 the 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 got to hand 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 only 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.

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 the emperor 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, making use of the (complete) Christian era, ventured to date the first landing of Julius Caesar in Britain; this military action, which took place in the year 55, was dated by Beda Venerabilis in the year 60 before Christ.

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 moment zero, the unique point in time which is identical with [31‑12‑ -1; 24:00] = [1‑1‑1; 0:00]. This symmetry is a property 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). Therefore, our era cannot contain a year zero (assumed that we want to preserve the symmetry of our era). For such a year zero would have to belong to the first century before or to the one after Christ, but then also (due to the symmetry) both to the first century before and to the one after Christ; but this is incompatible with the principle that every calendar year of our era belongs to exactly one numbered century of our era.

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 became common property, although sometimes a variant of the latter one is used sometimes 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 Julian calendar. The Christian and the astronomical era differ only in their calendar years before their first common year 5; by definition they show no difference in their calendar years after the year 4. 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 a proleptic Julian leap year regulation 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). Although the moments 0 of the astronomical and the Christian era differ one day, the moments 2000 of these two eras coincide exactly (they are both identical to [31‑12‑2000; 24:00] = [1‑1‑2001; 0:00]).

In order to keep the March equinox, i.e. the moment at which in the northern hemisphere of the earth spring begins, in its place (since the year 1582 on or close to 20 March) it is for the time being sufficient to drop, about the year 4915 for the first time, a leap day per four centuries once every 3333 years. That way it will last as late as Millennium 325, this is the millennium of which 1‑1‑324001 is the first day, before all our leap days are used up. We do not know whether there still will be human beings then. The present century is the first century of Millennium 3. Therefore, there is not the slightest reason to in the short term replace our era with an other one.

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 absolutely not obvious indeed to let begin a new era by starting from 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 this 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 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.

 

3 conclusions

The Christian era (see Section 1) has a bilaterally symmetric structure (see Section 2), and it is just as well 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 never proposed seriously 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 not in the least a mistake of Dionysius Exiguus or of Beda Venerabilis. More than that, 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 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 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 at [31‑12‑10; 24:00] = [1‑1‑11; 0:00], i.e. the moment 10 of our era.

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 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 happened 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. Hence Olympiad 194 ended in the summer of the year 1, and Olympiad 291 was the last ancient olympiad, which 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 that 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.

Millennium mistake 1 was made by medieval people who believed that on 1‑1‑1000 the first millennium would expire and the world perish. These humans did not realise that on that date only 999 years of the first millennium had passed. The first turn of millennium took place exactly one year later, namely in [31‑12‑1000; 24:00] = [1‑1‑1001; 0:00], i.e. the moment 1000 of our era.

Millennium mistake 2 was made by modern people who had been fooled by commerce and media and authorities that also did not know any better (and by many a historian who had entirely forgotten that our era contains no year zero) into believing that not the “dull” date 1‑1‑2001 but the “magic” date 1‑1‑2000 (which was accompanied by its mikllennium question, its millennium mistake, its millennium problem, and its 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 (Gregorian) calendar year changed simultaneously, i.e. 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. the point of time [31‑12‑2000; 24:00] = [1‑1‑2001; 0:00].

Summarizing we can say that the millennium mistake is by definition a mistake resting on the misunderstanding that numbered millennia would end not with the end but with the beginning of their thousandth year.

Because moment zero is identical with [1‑1‑1; 0:00], the year 1 is the starting year of our era, and so it is the opening year of the first decade, of the first century and of the first millennium. It is not difficult to check that the year 2000 is the last year of the last decade of the last century of the second millennium and that the year 2001 is the first year of the first decade of the first century of the third millennium. The “magic” year 2000 is the closing year of the previous millennium and of the previous century, the “dull” year 2001 is the opening year of the new millennium and of the new century. And of course the year 3000 is the closing year of the third millennium (just as the year 300 is the closing year of the third century and the year 30 the closing year of the third decade).

The reason why an option 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). 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 indeed would have yielded a moment 2000 coinciding with the turn of year with which the year 2000 of this alternative era began, but evidently also this turn of year would have been identical with [1‑1‑2001; 0:00].

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 Anno Urbis Conditae era (see Section 1). If Rome indeed was founded in the Roman year 1 then it will be three thousand years ago that this important historical event took place not in the year 2247 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. By the way, according to modern historians Rome was not founded in the eighth but in the seventh century before Christ.

 

4 objections

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. This section contains a little anthology therefrom.

“All well and good” someone still objects, “but after all the twentieth century does consist exactly of those calendar years of our era whose numbers start from 19? This implies that the year 1999 is the last year of the twentieth century!”. The calendar years of our era whose numbers end in 00 throw a spanner into the works. There is no year zero in our era (see Section 2); it follows that the year 100 is the last (closing) year of the first century, that the year 200 is the last (closing) year of the second century, that the year 300 is the last (closing) year of the third century, and so on. So the year 1600 is the last (closing) year of the sixteenth century. On closer inspection the at first sight interesting standpoint of Maarten Prak (university of Utrecht) that the battle of Nieuwpoort (which took place in the year 1600) is one of the rare real battles the army of the Dutch republic fought out in the seventeenth century, turns out to be something like the remark that New Year’s Eve is one of the rare really cosy days of the month of January.

“All well and good” someone still objects, “but who is really mistaken? After all, the nineties of the twentieth century had passed on 1‑1‑2000!”. Indeed that is true, but the last decade of the twentieth century had begun only on 1‑1‑1991, and so it had passed only on 1‑1‑2001. Likewise the book with the pretentious title “The complete History of the twentieth Century”, rashly (just before 1‑1‑2000) printed in a very big edition, which finishes off with the treatment of the nineties of the twentieth century, is no complete history of the twentieth century, because what happened in the last year of the twentieth century is not in there.

“All well and good” someone still objects, “but what about my odometer? After I have driven exactly 1000 kilometers, it clearly shows three zeros!”. That is right, but what we state here is not a similarity, but it is just a difference between era and odometer, because of the fact that during its first kilometer the odometer indicates 0000, not 0001. It is true, there is a similarity between odometer and age (so during its twentieth kilometer the odometer indicates 0019, and during the twentieth year of your life you are nineteen years of age), but this is beside the point.

“All well and good” someone still objects, “but when numbering the floors of a building surely it is logical and common practice to name the second floor floor 2, the first floor floor 1, the ground floor floor 0, and the successive basements floor -1, floor -2, floor -3, ……? When numbering the calendar years of our era we cannot do without the number 0 either!”. Because floors must not be taken as spaces but as horizontal dividing planes between spaces (e.g. the ground floor) numbering the floors of a building does not correspond to numbering the calendar years but to numbering the turns of year of our era, as in our second timeline (see Figure 2).

“All well and good” someone still objects, “but 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 1582the 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 can be solved much more simply, after all! 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 not [1‑1‑2000; 0:00] but [1‑1‑2001; 0:00] was the second turn of millennium (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 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 the celebration of the second turn of millennium on 1‑1‑2000?”. Of course nothing is against it to celebrate any memorable event whatever at any moment whatever (e.g. a turn of year on 30 December or your twentieth birthday on your nineteenth birthday). But the question is here that we have to distinguish between the direct transition from the year 1999 to the year 2000 (the “magic” moment at which all four digits of the number of the current calendar year changed at the same time) and the accompanying turn of millennium, i.e. the direct transition from the second to the third millennium, exactly one year later, and that at these two striking moments relatively few people realized this.

“But nevertheless the people have the last word!” someone still objects. That means in my opinion that the people have right to self determination, not that the people are always perfectly right. Something does not automatically become true if many people believe that it is true. The earth does not become less round if many people believe that the earth is flat. Nor does something automatically become true by deciding it just like that, not even when this happens in a democratic way. One can decide to go over to summer time but not that henceforth the sun will rise one hour later. It was possible to decide to celebrate the second turn of millennium at moment 1999 of our era, so just 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 to decide that that was not one year too early.

Whether something is true, is prescribed neither simply by the people nor simply by some authority, not even by the king or the queen of the Netherlands (even though sometimes one for a moment could think she does, for the fact that there exists a statistical connection between smoking and lung cancer seems to be determined by Royal Decree). However, in order to determine whether something is true, sometimes logical (watertight) reasoning is necessary and sufficient. So the logical reasoning of Section 2 inevitably leads to the conclusion that an adequate era, being a linear system of numbered calendar years, is bilaterally symmetrical if and only if it has no year zero.

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

People believe all sorts of things. And usually what once is believed, is not given up easily. Insights that are at right angles to what once is believed often hardly get a chance to be tested to reason. Hence that people resisted so long against the insight that our earth is not flat but round, that the sun is a star and the earth is a planet revolving round the sun instead of the sun revolving round the earth, that under special circumstances primitive life (extremely gradually) comes into being, that all higher developed biological species (including Homo sapiens) are evolved out of other biological species (inclusively Homo sapiens), that all life is only temporary (dust thou art and unto dust shalt thou return), that God is a product of human imagination and exists only as such (man proposes but God does not dispose), that it is a mistake to think that atheists think they can prove that God does not exist (as a matter of fact atheists believe that there exists no God outside of human imagination). Many atheists are humanists; they try to believe in the ultimate goodness of man, and in his vocation to create a humane human society. That implies a belief in mental growth. As a consequence, we have to be willing to test the tenability of our insights and redeem our mistakes (this concerns each of us personnally as well as mankind as a whole). That is why I felt compelled to find out, inspired to this by critical pupils who wanted to know all the ins and outs, why in fact 1‑1‑2000 could not have been the first day of the third millennium.

By the way, what is the sense of education? By no means only to emancipate people. Stimulating clear thinking and careful formulating by way of joint attention to essentials is an at least equally important education objective. Pupils ought to be able to calculate the sum of -753 and 3000 by heart. But also they have to know, I think, what structure our era has, in order to be able to understand that the answer to the question in which year Rome, assuming that this eternal city was founded in the year -753 (see Section 3), three thousand years will exist is not the year 2247 but the year 2248; this is not all that difficult.

 

5 calendars

The Julian calendar (see Section 1) was the result of the proleptic calendar reform decreted in the year -46 by Julius Caesar. In the year 1582 pope Gregorius XIII replaced the Julian calendar with the Gregorian calendar (see Section 1). These two calendars differ solely in their leap year regulation. The calender years of our era before the year 1582 are Julian calendar years, the calender years of our era after the year 1582 are Gregorian calendar years. The dates of our era before the year 1582 are Julian calendar dates, the dates of our era after the year 1582 are Gregorian calendar dates.

The Julian calendar was introduced by Julius Caesar in the year -46 by means of a drastic adaptation of the then hopelessly superseded Roman calendar (see Section 1), until then not provided with any leap year regulation. The adaptation in question boils down to the measure according to which the then current Roman calendar year was shifted eighty days in the direction of the future (owing to which the March equinox, which marks the beginning of spring on the northern hemisphere of the earth, was moved 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 Roman calendar years (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 month of the Roman calendar year) and to consist of 366 instead of of 365 days once every four years, to begin with the then next Roman calendar year (boiling down to the year -45).

Unfortunately, in the first half century after the death of Julius Caesar (in the year -44) the leap year regulation according to the Julian calendar would not come to much. The fact is, after the leap year -45 there was until the year -8 (by mistake) a leap year every three years (instead of every four years). That implies that between the leap years -45 and -9 there were as a matter of fact three leap years too much, namely eleven instead of eight. About the year -8 this problem was solved by the emperor Augustus by replacing all Roman leap years between the leap years -9 and 8 with ordinary Roman calendar years consisting of 365 days. That implies in particular that the year 4 was no leap year. Julian calendar years betwecn 4 and 1582 are leap years only if their year numbers are 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 Julian calendar dates.

It is 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 according to 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 regulation made by the emperor Augustius mentioned in the previous paragraph, remained holdong 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 (as a matter of 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 calendar year number was divisible by 4 but not by 100 unless by 400. We establish that the year 1582 comprised only 355 days, and so is the only exception on the rule that a calendar year of the Christian era consists of 365 or 366 days, and that [4‑10‑1582; 24:00] = [15‑10‑1582; 0:00]. Thus all calendar years of our era from the farthest past into the farthest 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).

The leap year regulation according to the Gregorian calendar was brought into force in the year 1582 for an indefinite (future) time, and for the time being it will not be necessary to adjust this regulation. In order to keep the March equinox in its place (about 20 March) it is for the time being sufficient to drop, about the year 4915 for the first time, a leap day per four centuries once every 3333 years. That way it will last as late as Millennium 325, this is the millennium of our era of which 1‑1‑324001 is the first day, before all leap days of the Gregorian calendar are used up. But it is not certain whether there still will be human beings then (the present century is only the first century of Millennium 3).

It is in combination with the Gregorian calendar (valid for all its calendar years after the year 1582) that the Christian era has been 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 complemented with a proleptic leap year regulation according to the Gregorian calendar holding for all its calendar years, but with the pure leap year regulation according to the Julian calendar holding for its calendar years before the year 1582 and the leap year regulation according to the Gregorian calendar holding for its calendar years after the year 1582. Because, moreover, by definition the year 1582 of the astronomical era and the year 1582 (of our era) are identical, the restrictions of the astronomical and the Christian era to the calendar years after the year 4 coincide exactly, which implies that the moments 2000 of these eras are identical. For that reason the choice for the astronomical era instead of for the Christian era would not have led to an other point in time of the second turn of millennium than [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 does not detract from that conclusion.

The year -1 (of our era) began two days later and ended one day later than the year 0 of the astronomical era. That is caused by the fact that the years 0 and 4 of the astronomical era were leap years but the years -1 and 4 (of our era) not. The fact that the year -4 of the astronomical era was a leap year but the year -5 (of our era) was not, implies that the leap year 9 (of our era) began three days later than the leap year -8 of the astronomical era. It is not difficult to check that the leap year -21 (of our era) began two days later than the leap year -20 of the astronomical era and that the leap year -33 (of our era) began one day later than the leap year -32 of the astronomical era, and that the leap year -45 (of our era) = (exactly) the leap year -44 of the astronomical era. That implies that Julius Caesar, who was murdered 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 identical with the year x of the astronomical era, but every year -x (of our era) before the year -42 (of our era) is identical with 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.

In the year -30 the primeval Egyptian calendar, which was not provided with any leap year regulation, was replaced with the Alexandrian calendar (see Section 1). The emperor Augustus let the first calendar year of the Alexandrian calendar begin on 29‑8‑ -30. Each Alexandrian calendar year consisted of twelve months of thirty days and five or six single days. Just like the Julian calendar, the Alexandrian calendar was provided with a leap year regulation with a leap year proportion of one to four. But just like the introduction of the Julian calendar the introduction of the Julian calendar was accompanied by one and the same wrong application of its leap year regulation. As a matter of fact, each of the nine leap days of the Julian calendar between the years -30 en -8 (always in February) was connected with 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. 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; converting dates from the one to the other calendar must already have been known in first century Alexandria. At the time Thoth was the first, Phamenoth the seventh, Pharmouthi the eighth, and Pachon the ninth month of the Alexandrian calendar year, and fell the fifth day of Phamenoth always on 1 March and the fifth day of Pachon always on 30 April. Contrary to the Alexandrian calendar, the Egyptian calendar was only used for agricultural and practical astromical purposes.

Unlike the four calendars already mentioned in this section, the Jewish calender is a lunar calendar, in which each new month begins more or less simultaneously with a Newmoon (proper), i.e. point of time of lunisolar conjunction (i.e. conjunction of sun and moon). But since the coming into being of the Jewish calender, far before the beginning of our era, until the beginning (about the year 360) of the long time interval during which it was piecemeal definitely fixed, the beginning of any new Jewish calendar month whatsoever depended not only on astronomical but (indirectly) also on local circumstances (in particular meteorological circumstances under which in Palestine the first appearance of the moon crescent after Newmoon was searched for). Every Jewish calender year consisted then (and still consists now) either of twelve (mostly) or of thirteen calendar months each consisting of 29 or 30 days. At the time Nisan was the first, Iyyar the second, and Adar the twelfth month of the Jewish calendar year and Pesach, i.e. Pesah, i.e. the Jewish Paschal feast (which in principle 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 the meal in which the Paschal lambs slaughtered in the afternoon of 14 Nisan were eaten, began usually with the rise of the full moon ideally an hour after this sunset.

Since the coming into being of the Jewish calendar until the moment of its complete fixation, the beginning of each new Jewish calendar month was in principle determined in Palestine, at a special moment about half an hour after the beginning of the thirtieth night after the sunset with by which the first day of the expiring Jewish calender month had begun. If at the time at such a special moment the first appearance of the moon crescent after Newmoon was confirmed by the Jewish authorities in Palestine (this happened roughly once every two months) then this meant that the first day of the new Jewish calender month had just begun with the sunset having taken place in Jerusalem about half an hour ago; if not, then the first day of the new Jewish calender month began at the moment of the next sunset taking place in Jerusalem. That is why at the time all Jewish calender months thus defined consisted of 29 or 30 days. Because usually a waxing moon is visible with the naked eye not earlier than 24 hours after Newmoon, at the time the first day of a new Jewish calender month usually began with the second sunset taking place in Jerusalem after Newmoon. For the same reason at the time (the actual) Fullmoon, i.e. point of time of lunisolar opposition (i.e. opposition of sun and moon) of a Jewish calender month differed on average not so much from the midnight point of time of the fourteenth day of this Jewish calendar month.

Since the coming into being of the Jewish calendar until the moment of its complete fixation, in Palestine at set times not only a decision had to be taken with respect to the point in time at which a new month of the Jewish calendar had to begin (once a month) but also a one concerning the beginning of a new year of this calendar (once a year). In that time the Jewish authorities in Palestine possessed the competence to intervene once a year, at the end of Adar, in the current Jewish calendar year (they did this roughly once every three years) by extending this Jewish calendar year by an extra month consisting of thirty days. In that time the Jewish authorities in Palestine were not only able (by handling that competence carefully) to prevent that the Jewish calendar year would become on average too short or too long but also that Pesach was celebrated too early (i.e. entirely or partially still in winter) or too late. As a matter of fact, at the time the principle that Pesach had to be celebrated as early as possible in spring was the only not opportunistic criterion they whether or not applied within the scope of that competence. At the time they must have been familiar with the growing of the days in winter and spring and the phenomenon of the March equinox (see section 2), although they then (being familiar neither with the Julian nor with the Alexandrian calendar) were not yet acquainted with any, either alleged or correct, date of the March equinox. However, in actual practice the rule of the equinox in question was often ignored, and in consequence Pesach celebrated one month too early many a time. As a matter of fact, the Jewish calendar can only be related to the Julian calendar insofar as the former agrees with the current fixed Jewish calendar, the completion of which took place in the second quarter of the ninth century.

Around the year 90 the (real) March equinox fell on 22 March, around the year 220 on 21 March, around the year 600 on 18 March, around the year 1500 on 11 March. Nevertheless, for centuries, until the fourth century, the date 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 in his time (around the year 140) fell 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 Rome and Alexandria 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) conception that the moment of the March equinox is not a question of a point in time or a date, e.g. 22 or 25 March, but a one of the time interval consisting of the four consecutive dates 22 up to and including 25 March. Shortly after the third turn of century the church of Alexandria decided to consider henceforth 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 somewhere in the course of the second half of the fourth century.

 

6 paschal full moons

At the end of the first century Easter, i.e. the Christian Paschal feast, was mostly celebrated on the fourteenth day of Nisan (see Section 5), at the end of the second century in principle on the first Sunday after the fourteenth day of Nisan. Around the second turn of century the moment of the beginning of Nisan was still not exactly computable. In order not to remain dependent on the not exactly predictable way in which in that time the beginning of Nisan was determined in Palestine, in the beginning of the third century calculators of some churches, among which the church of Rome and the one of Alexandria (Egypt), began to calculate, of course with the help of tables of lunar phases, their own so called dates of Paschal full moon adjusted to one of the two (mutually convertible) solar calendars prevailing then in the Roman empire (see Section 5). These dates of Paschal full moon served as substitutes for dates of the fourteenth day of Nisan which were not known in advance in the third century. Each such date of Paschal full moon can be considered as the date of  the fourteenth day of Nisan of a substitute month Nisan^ consisting of thirty days, of which the first day got assigned beside its date the lunar phase number 1, et cetera, and the thirtiest day got assigned beside its date the lunar phase number 30. Thus were by a number of churches beside each relevant real (Jewish but not beforehand known) month Nisan a number of (Christian but beforehand known) substitute months Nisan^ created. The lunar phase number of a day being part of such a substitute month was referred to as “age of the moon” on the date in question, which expression must of course not be confused with the actual age of the moon (about 5 biljon years). The “age of the moon” on a date of Paschal full moon was always 14.

In the first half of the third century the activities of computists, i.e. practioners 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 via the construction of their own (substitute) months Nisan^ to the construction of periodic sequences of dates of Paschal full moon of consecutive calendar years of either the Alexandrian calendar (see Section 5) or the Julian calendar (see Section 1), which sequences of dates of Paschal full moon however not only could be different between themselves, but moreover not at all always led to one and the same Sunday for the celebration of Easter.

The sole criterion for first new moon visibility with which third and fourth century Alexandrian computists were acquainted, is the primeval Babylonian rule that every new moon will be visible (to the unaided 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 used 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 around the midnight point in time of the fourteenth day of Nisan, because from the middle of the local Jerusalem ‘from sunset to sunset’ day of the Newmoon in question to the midnight point in time of the fourteenth (‘from sunset to sunset’) day of Nisan it is just about half a synodic period of the moon.

About the middle of the third century the church of Rome began to experiment with sequences of dates of Paschal full moon of consecutive Julian calendar years with a period of 84 years, the church of Alexandria with sequences of dates of Paschal full moon of consecutive Alexandrian calendar years with a period of 19 years. Then the church of Alexandria began also to use the date of the twenty sixth day of Phamenoth (see Section 5), that is 22 March, which date she then considered as the date of the March equinox (see Section 5), as a lower limit of her dates of Paschal full moon. The first by name known Alexandrian computist who applied this principle to 19-year periodic sequences of dates of Paschal full moon was Anatolius (see Section 5). Supposedly around the year 260 (before his episcopal consecration) he took an actieve part in the construction of the sequence of dates of the proto Alexandrian Paschal full moon, the until recently unknown sequence of dates of Paschal full moon of consecutive Alexandrian calendar years with a period of 19 years from which some years later Anatolius must have started in order to construct his famous 19 year Paschal cycle, here referred to as the Anatolian Paschal cycle. The limit dates of the sequence of dates of the proto Alexandrian Paschal full moon were (the Alexandrian equivalents of) 23 March and 20 April, and so its dates were dates between 22 March and 21 April.

The for ages thought lost Anatolian Paschal cycle must have been part of a Paschal table composed about the year 270. This Paschal table was rather impractical, through the fact that its dates of Paschal Sunday were no dates of the Alexandrian or of the Julian calendar but dates of the Anatolian calendar, i.e. the variant of the Julian 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 has survived in the medieval text “De ratione paschali”. In this Latin text we find the Anatolian Paschal cycle again in the form of a sequence of dates of Paschal Sunday of consecutive Anatolian calendar years with a period of 19 years, however without any calendar year indication. The 19 year Paschal cycle included in “De ratione paschali” is not what it at first sight seems to be (an enigmatic sequence of Paschal dates of the Julian calendar). This sequence of Paschal dates of the Anatolian calendar is really the Anatolian Paschal cycle, which has been convincingly demonstrated by the Irish scientists Daniel McCarthy and Aidan Breen in the year 2003.

In the second half of the third century besides the sequence of dates of the proto Alexandrian Paschal full moon another important sequence of dates of Paschal full moon with a period of 19 years must have been constructed, namely the sequence of dates of the Anatolian Paschal full moon constructed about the year 270, which was, unlike the sequence of dates of the proto Alexandrian Paschal full moon, a sequence of dates of Paschal full moon 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 Julian calendar years 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 Julian instead of as an Anatolian calendar date, and transforming it, making use of the accompanying lunar phase number mentioned in “De ratione paschali”, into the corresponding Julian calendar date with lunar phase number 14. Its limit dates were 23 March and 19 April.

Unfortunately, there is no historical source at all from which the sequence of dates of the proto Alexandrian Paschal full moon can be deduced. But in the year 2009 the Julian equivalent (see Section 5) of this sequence of dates of Paschal full moon of consecutive Alexandrian calendar years was reconstructed by the author of this website. It was possible to succeed in it by making use of modern tables of Newmoon (see Section 5) concerning the time interval between the years 220 and 260. By comparing the sequence of dates of the Anatolian with the one of the proto Alexandrian Paschal full moon he could establish, in addition, that the initial year of “De ratione paschali”, i.e. the Julian calendar year to which the first date of the 19 year Paschal cycle included in “De ratione paschali” (16 April) belonged, must have been the year 271. In the year 2010 he wrote an article about this which will appear in print in the year 2014 (see also Section 9).

About the third turn of century the church of Alexandria decided to consider henceforth 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, probably the sequence of dates of the proto Alexandrian Paschal full moon, with a new one, a one which had 21 March as its earliest possible date, namely the sequence of dates of the (classical) Alexandrian Paschal full moon, which was, like the sequence of dates of the proto Alexandrian Paschal full moon, a sequence of dates of Paschal full moon of consecutive Alexandrian calendar years with a period of 19 years. The dates of the proto Alexandrian Paschal Sunday as well as the dates of the (classical) Alexandrian Paschal Sunday were calculated according to the third century Alexandrian Paschal principle “Paschal Sunday is the first Sunday after the Paschal full moon”. That the sequence of dates of the (classical) Alexandrian Paschal Sunday is a sequence of dates of consecutive Alexandrian calendar years with a period of as much as 532 years, was discovered in the second half of the fourth century. Anyway, it is the sequence of dates of the (classical) Alexandrian Paschal full moon which ultimately would appear to be the key to an adequate solution of the great problem how to calculate the date of Easter (see also Section 8). We see this sequence of dates of Paschal full moon in column F of Table 1; in column G of this table we see dates of the (classical) Alexandrian Paschal Sunday. As a matter of fact, the limit dates of the sequence of dates of the (classical) Alexandrian Paschal full moon were 21 March and 18 April, the ones of the sequence of dates of the (classical) Alexandrian Paschal Sunday 22 March and 25 April.

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 should be celebrated every year early in spring by all Christians on the very same Sunday after the fourteenth day of Nisan, on which day traditionally the last preparations were made for the celebration of Pesach (see Section 5). The bishops who were together in the year 325 in Nicaea, came also to the conclusion that anyhow it was necessary to be always amply in advance well informed about dates being eligible for the celebration of Easter (since the second century always on Sunday), and that therefore, because of the then incalculability of the Jewish calendar (see Section 5), accurate Paschal tables adapted to the Julian or to the Alexandrian calendar were required. They were agreed about that Paschal Sunday not only ought to be preceded by “the full moon of Nisan” but also by the March equinox (see Section 5). However, they could not reach agreement with regard to the way in which the date of Paschal Sunday had to be calculated, owing to the fact that they were in disagreement about the date of the March equinox and about the way in which “the Paschal full moon” and subsequently from this the date of Paschal Sunday had to be calculated.

In Section 8 we will relate the three sequences of dates of Paschal full moon with a period of 19 years mentioned in this section to each other. The chronologically first and third of these three sequences of dates, namely the one of the proto Alexandrian Paschal full moon, also referred to as proto Alexandrian cycle, and the one of the (classical) Alexandrian Paschal full moon, also referred to as (classical) Alexandrian cycle, respectively, have both a so called Metonic structure (see also Section 8) and were due to this, from an astronomical point of view, ideal sequences of dates of Paschal full moon. There exists a close relation between the chronologically first and second out of these three sequences of dates: the first is the Alexandrian equivalent of the best Metonically structured approximation to the sequence of dates of the Anatolian Paschal full moon.

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 Julian calendar years, 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 consecutive dates of Paschal full moon of consecutive Julian calendar years with a period of 84 years. As late as in the course of the second half of the fourth century, after the church of Rome had decided to consider henceforth the date 21 March as the date of the March equinox instead of 25 March, these attempts gradually led to the construction of such a sequence of dates of Paschal full moon, namely the sequence of dates of the Roman Paschal full moon, ideally with limit dates 18 March and 16 April. The sequence of dates of the Roman Paschal Sunday being the sequence of dates determined according to the Roman Paschal principle “Paschal Sunday is the first Sunday after the first day after the Paschal full moon” was, like the sequence of dates of the Roman Paschal full moon, a sequence of dates of consecutive Julian calendar years with a period of 84 years, however ideally with limit dates 21 March and 23 April. Solely in the years 303, 333, 360 modulo 84 this sequemce of dates of Paschal Sunday provided the church of Rome with an (according to herself) unsuitable (according to her own secondary Paschal criteria) date for Easter, either a too early date (21 March) or a too late (22 or 23 April).

As we have seen in the previous paragraph, from time to time the sequence of dates of the Roman Paschal Sunday provided the church of Rome with an unsuitable date for Easter. Initially innocent but in the long term much more serious the problems were which arised from the fact that with each new period of 84 years the difference of the Roman Paschal full moon with the accompanying Fullmoon (see Section 5) increased on average by about 1.29 days, and the difference with the 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 Roman Paschal Sunday did not coincide with the date of the Alexandrian Paschal Sunday, in the first half of the fifth century five times (namely in the years 401, 406, 425, 431, 448), in the second half of the fifth century 11 times, in the first half of the sixth century 16 times, in the second half of the fifth century 21 times, in the first half of the sixth century 35 times. The fact that in the first half of the fourth century the number in question was much higher than in the second (19 and 2 respectively) implies that the sequence of dates of the Roman Paschal full moon definitely took shape as late as around the fourth turn of century. The sequence of dates of the Roman Paschal full moon, also denoted as Roman cycle, as well as the one of the Roman Paschal Sunday, also denoted as Roman Paschal cycle, has a period of 84 years.

As early as in the year 410 the great Alexandrian computist Annianus succeeded, on the basis of the (classical) Alexandrian cycle, in extending the until then existing sequence of dates of the (classical) Alexandrian Paschal Sunday to the famous (classical) Alexandrian Paschal cycle, which is a sequence of dates of Paschal Sunday of consecutive Alexandrian calendar years with a period of 532 years. In the year 525 the Julian equivalent of the classical Alexandrian cycle was used by Dionysius Exiguus (see Section 1) to construct his Paschal table so important from a chronological point of view. In the year 725 Dionysius Exiguus’ great follower Beda Venerabilis (see Section 1) published an extension of Dionysius Exiguus’ Paschal table containing an extension of Dionysius Exiguus’ sequence of dates of Paschal Sunday to a 532 year Paschal cycle which as a matter of fact is the Julian equivalent (and as such a reinvention) of the (classical) Alexandrian Paschal cycle. Simultaneous celebration of Easter became possible not until (at the earliest somewhere in the second half of the eighth century) all churches had been familiar with either the original Alexandrian version (Annianus) or the Julian version (Beda Venerabilis) of the (classical) Alexandrian Paschal cycle.

Although the (classical) Alexandrian cycle is fully known (in column F of Table 1 we see its Julian equivalent) it is only recently that we have some idea how it originated (see also Section 8). It or its equivalrnt forms the backbone of the classical Alexandrian Paschal tables thus defined. By the publication about the year 310 of the first generation of classical Alexandrian Paschal tables the church of Alexandria was the first church who opted definitely for 21 March as the earliest (and for 18 April as the latest) possible date of her Paschal full moon. Thus she opted also definitely for 22 March as the earliest (and for 25 April as the latest) possible date of her Paschal Sunday, because of the Alexandrian Paschal principle, which applies for all classical Alexandrian Paschal tables. It is plausible that around the fourth turn of century the churches in the eastern half of the Roman empire, among which the churches in Palestine, used for the most part classical Alexandrian Paschal tables, the curches in the western half for the most part Roman Paschal tables provided with the Roman cycle.

 Annianus’ Paschal table and the Alexandrian equivalent of the Paschal table attributed to Cyril (see Section 1) were obtained by extrapolation from a foregoing generation of classical Alexandrian Paschal tables. The Paschal table attributed to Cyril was intended for use in the western half of the Roman empire, and it is for this reason that this table was provided with Julian instead of Alexandrian calendar dates. 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 applies for all classical Alexandrian Paschal tables and in the “age of the moon” on a date of Paschal full moon is always 14, in all Paschal tables mentioned in this paragraph the “age of the moon” on a date of Paschal Sunday is always an integral number between 14 and 22.

In Dionysius Exiguus’ Paschal table (see Table 1), in which all calendar years are Julian calendar years and all dates Julian calendar dates, we see for each indicated calendar year (in the primary column A) mentioned in column C the epact being the “age of the moon” on 22 March, in column D the concurrent being the difference between 24 March and the date of the last Saturday before 24 March, in column F the date of the Alexandrian Paschal full moon, in column G the date of the Alexandrian Paschal Sunday, in column H the “age of the moon” on the date of the Alexandrian Paschal Sunday. As a mater of 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 (see Table 1) emerges 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 in column F can be obtained by subtracting the corresponding epact in column C from 5 April and reducing the outcome modulo 30 days to a date between 20 March and 19 April. Each date in column G can be obtained by subtracting the corresponding concurrent in column D from 25 March and reducing the outcome modulo 7 days to a date between the corresponding date in column F and the date obtained by adding 8 days to this date in column F. This calculation boils down to the same as applying the Alexandrian Paschal principle to each date 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 in column F from the corresponding date in coloumn G. Dionysius Exiguus’ Paschal table is provided with two independent numerations of 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 every following epact of the sequence can be obtained by adding either 11 modulo 30 days (normally) or 12 modulo 30 days (only in the case of the so called saltus occurring once every nineteen times) to the last preceding epact of the sequence (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 of the classical Alexandrian cycle (see also Section 8).

Not only the sequence of epacts but also the sequence of concurrents occurring in Dionysius Exiguus’ Paschal table (see column D of Table 1) has a particular structure. The oldest classical Alexandrian Paschal table in which that sequence of concurrents occurs is the Paschal table of bishop Theophilus of Alexandria constructed in the year 385. That sequence of concurrents, which all classical Alexandrian Paschal tables constructed after the fourth century 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 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 constructed after the fourth century 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. It is Annianus who around the fourth turn of century discovered that it must be possible to extend the classical Alexandrian Paschal tables until then constructed 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 in the year 410. Dionysius Exiguus was not acquainted with Annianus’ Paschal table, 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 up till then had given preference to go on using her own, relatively inadequate, Roman Paschal tables. Moreover, this newer Paschal table was the source of inspiration which led to the invention and construction (by extrapolation) of the complete Julian version of the (classical) Alexandrian Paschal cycle which was published by Beda Venerabilis in the year 725. In the Byzantine empire thanks to Annianus’ Paschal table at all times the churches were acquainted with the right date of the next Paschal Sunday. In the eighth century thanks to Beda Venerabilis’ Paschal table also the churches in the part of Europe outside the Byzantine empire got that possibility. It is only in the eighth century that all churches became the possibility to celebrate Easter on the same day.

It is the Alexandrian Paschal tables composed in Alexandria (Egypt) 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 abundantly. In the Byzantine empire no other Paschal tables than classical Alexandrian Paschal tables were used. However, in the part of Europe outside the Byzantine empire it was only in the eighth century, when Beda Venerabilis’ Easter table was accepted by the churches in Britain and Ireland and in the Frankish kingdom, that classical Alexandrian Paschal tables became generally accepted. The thus realized general use of classical Alexandrian Paschal tables (by means of which at last the churches could realize their old ideal of celebrating Easter simultaneously) was continued for centuries, in the Byzantine empire until its fall in the year 1453, in the greater part of Europe until after the year 1582, when Beda Venerabilis’ Paschal table was replaced with Easter tables adjusted to the Gregorian calendar (see Section 5).

 

7 anni domini

Now that we have completely solved the millennium question (see Section 3) and justified the term ‘millennium mistake’ (see Section 4), the still unanswered question concerning the precise connection between the Anno Domini era (see Section 1) and Anni Domini (literaly ‘the Years of the Lord’), in particular Jesus’ birth and death, goes on intriguing us. Likewise closely connected with the millennium question (and neither essential for the solution of it) is the interesting question concerning the connection between the starting year of the Anno Domini era chosen by Dionysius Exiguus, i.e. the year 1 (of our era) = the Roman year 754 (see Section 1), and Annus Dominicae Incarnationis, i.e. the Roman year of Jesus’ incarnation in the view of Dionysius Exiguus; also on the answer to this question historians do not entirely agree yet. In the writings of Dionysius Exiguus himself no clarification can be found about this, and in the writings of Beda Venerabilis we meet diverse arguments leading to contradictory deductions. But the majority of modern historians think that Dionysius Exiguus believed Jesus was born in or shortly before the year 1.

Peter Rietbergen (university of Nijmegen) is of the opinion that Dionysius Exiguus believed Jesus was born one week before the year 1, so in the year -1 (of our era) = the Roman year 753. That view agrees with the known historical fact that Charlemagne let himself crown emperor just on 25‑12‑800. 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 seems to be at least as plausible as the other one because of the analogy between the beginning of the Anno Domini era and the one of the Anno Urbis Conditae era: “just as Rome was founded (on 21 April?) in the course of the Roman year 1, Jesus was conceived (on 25 March?) and born (on 25 December?) in the course of the year 1 (of the Anno Domini era)” Dionysius Exiguus could have thought.

One of the most influential figures of the first council of Nicaea was Eusebius, the historian who had become bishop of Caesarea shortly after the year 313. He was the first who hit upon the idea of an era with as its starting year the year of birth of Jesus according to the Julian calendar. He thought Jesus was born in the third Roman year of Olympiad 194 (see Section 3), in accordance with the opinion of Orosius (see Section 1), a century later, that Jesus was born in the Roman year 752. However, Dionysius Exiguus chose (indirectly) the Roman year 754 (instead of the Roman year 752) as the starting year of his new era (see Section 1). Perhaps he felt compelled to do so in order to effect that in his new era (just like in the era of the emperor Diocletianus) the rule should hold that leap year numbers are divisible by 4.

In all probability Dionysius Exiguus did not know in which Roman year Jesus was born, and we do not know either. Nobody believes that moment zero, the unique point in time which is asterisked (*) so suggestively in our first timeline (see Figure 1) and is identical with [1‑1‑1; 0:00], could be the moment of Jesus’ birth. According to modern historians Jesus was born sometime between the years -9 and -1, so some time before the beginning of the Christian era, a remarkable paradox. On the year (let alone on the date) of Jesus’ birth scholars do not yet agree. We establish that in all probability Jesus was born about the year -5. Sometime in the nineties of the previous century the day on which it had been two thousand years since Jesus was born, slipped by.

At least as interesting as the question when precisely was the beginning of Anni Domini, is the question when precisely was its end, the crucifixion which was the occasion to the coming into being of Christianity. Neither the year in which nor the date on which Jesus died is known for certain. It is common knowledge that Jesus died about the year 30 in Jerusalem, on a Friday in the afternoon, namely on (according to the fourth canonical gospel) or one day after (according to the three synoptic gospels) 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, that Jesus would have been crucified on a fifteenth day of Nisan, may be doubted, because the fifteenth day of Nisan was a feast day on which one did not administer justice in Jerusalem. The religious persuasion that Jesus was crucified a few hours before celebration of Pesach began agrees, anyway, with the fact that at the end of the first century the Christian Paschal feast was mostly celebrated on the fourteenth day of Nisan. It is certain that Jesus died during the reign of the emperor Tiberius (who reigned from 14 to 37) and during the procuratorship of Pontius Pilatus, who was procurator of Judaea from 26 to 36.

Beda Venerabilis has tried to find the date of the day of Jesus’ death with the help of his Paschal table (see Section 6), evidently taking his departure from the principle “Paschal full moon = 14 Nisan”. He hoped to arrive at 25‑3‑34, evidently partly due to the tradition dating back to the third century according to which Jesus would have died on a Friday 25 March (of an as yet unknown calendar year). Beda Venerabilis took it for granted that the validity area of his Paschal table extended without fail to the beginning of the Christian era. However, making use of the columns of his Paschal table corresponding with columns F and G of Dionysius Exiguus’ Paschal table (see Table 1) he had to establish to his disappoinment that in the year 34 (like in the year 566, for 34 ≡ 566 modulo 532) the date of the (classical) Alexandrian Paschal full moon (see Section 6) was Sunday 21 March and not the Thursday 24 March expected by him. Apparently he believed that Jesus had died on a fifteenth day of Nisan, and evidently his presuppositions were inconsistent.

The conviction that Jesus died on 25 March is without any rational foundation. For quite a while one cherished the conviction, resting on the oldest known Roman Paschal table, namely the Paschal table of Hippolytus Romanus (around the year 220), afterwards proved to be unreliable, according to which Jesus should have died on 25‑3‑29. But the more one got dispose of Paschal tables which kept much better step with astronomical reality (see Section 6), the more the perception grew that this proposition was untenable. Nevertheless in the course of the fourth century the idea that Jesus both was conceived on 25 March and died on 25 March came into being. Not only we may have doubts about the correctness of that vision (to which after all still two calendar year numbers are lacking) but also about the correctness of Beda Venerabilis’ evident presupposition that all dates of the (classical) Alexandrian Paschal full moon included in his Paschal table perfectly agreed with the dates of the fourteenth day of Nisan whose substitutes they were (see Section 6). Nevertheless it is an interesting question whether it is possible to trace the date of the day of Jesus’ death in the manner of Beda Venerabilis, i.e. by applying the principle “Paschal full moon = 14 Nisan” 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 elsewhere between the years 26 and 36. In order to be able to make a serious attempt to determine the date of Jesus’ crucifixion in the manner of Beda Venerabilis, we consider the Julian calendar dates of the (classical) Alexandrian Paschal full moon according to Beda Venerabilis’ Paschal table holding for the nine years 27 up to and including 35 (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 the weekdays (Saturday and Sunday inclusive) of these dates (with the help of column D or of column G of Table 1); column B of Table 2 (in which all calendar years are Julian calendar years and all dates Julian calendar dates) shows the result. None of these dates appears to fall on a Thursday, which implies that none of these dates can be the date of the last day preceding the day that Jesus was crucified. So it is impossible to determine in the manner of Beda Venerabilis a possible date of Jesus’ crucifixion which is in accordance with the three synoptic gospels. The only one of the dates in question which (perhaps) could be a date of Jesus’ crucifixion, in accordance with the fourth canonical gospel, is 15‑4‑29, because this date is the only one of the dates in question which fell on a Friday; however, Beda Venerabilis took no thought for it.

Because Julian calendar dates of the (classical) Alexandrian Paschal full moon are defined meaningfully only insofar the Julian calendar after intervention of the emperor Augustus functioned properly (see Section 5), the sequence of Julian calendar dates of the (classical) Alexandrian Paschal full moon, which forms the backbone of Beda Venerabilis’ Paschal table, reaches in fact from 4 to 1582 (see Section 5). Because that sequence of dates is periodic with a period of 19 years, we may take that sequence of dates as a strictly regularly running clock (of course imaginary) with a dial of which the hour hand has been replaced with a year hand which takes without surcease 19 years (instead of 12 hours) to go round one time. In the time that Alexandrian computists were constructing the original sequence of Alexandrian calendar dates of the (classical) Alexandrian Paschal full moon, around the third turn of century, by definition that imaginary clock, which can be supposed to have run precisely and unbrokenly from 4 to 1582, kept time with the then astronomical reality of an average lunar phase of about 1.4 days before Fullmoon (see Section 5), which can be established by comparing dates of the (classical) Alexandrian Paschal full moon with dates of Fullmoon (of course with the help of a suitable lunar phase table). But thereafter our imaginary clock went more and more to lose time, 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 (both 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 calendar years lasted an average of 19 × 365.25 = 6939.75 days but the moon was taking about 235 × 29.53059 ≈ 6939.689 days to pass through all its phases 235 times. It follows that with the passage of time after its moment of keeping time our imaginary clock went to lose time more and more, namely with each new period of 19 years about 6939.75 − 6939.689 = 0.061 days more, so with each new time interval consisting of 1 year about 0.0032 days more. That implies that after its moment of keeping time our imaginary clock was always taking about 310 years to get further behind a whole day. This implies that, compared to the astronomical reality of the third turn of century, around the year 600 our imaginary clock lost about a whole day, but at the time of the reign of the emperor Tiberius it gained almost a whole day. Not surprisingly, 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 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 cycle (see Section 6) are far more suitable ingredients than the dates of the (classical) Alexandrian cycle (see Section 6), because around the year 30 proto-Alexandrian Paschal full moons were usually full moons on average about 0.1 days younger than Fullmoon. If Beda Venerabilis had been acquainted (but of course he was not) with the proto-Alexandrian instead of with 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 fourth gospel, could have been the date of Jesus’ crucifixion (see columns C and D of Table 2).

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 unknown. However, in order to obtain all possible dates of such a fourteenth day of Nisan, we may make use of the simple rule that Nisan usually began with the second sunset in Jerusalem after a Newmoon (see Section 5), of course with the help of accurate lunar tables. The fact that a new Jewish calender month usually began with the second sunset in Jerusalem after a Newmoon, is an obvious consequence of the primeval Babylonian rule that usually, if weather permits, a new moon will be visible (to the unaided eye) for the first time (for a while) relatively shortly after sunset between 24 and 48 hours after Newmoon.

Applying the rule concerning the beginning of Nisan means that for each Newmoon which heralded the beginning of a month Nisan, the date (of the daylight part) of the first day of this month Nisan can be estimated quite accurately by simply adding 2 or 3 days to the local Jerusalem date of this lunisolar conjunction, depending on whether the local Jerusalem point of time of this lunisolar conjunction fell before or after 18:00, respectively. It is possible to locate the possible months Nisan in question in the Julian calendar by making use of the only not opportunistic Jewish principle in question, namely that the day of preparation of Pesach (this is the last day preceding the celebration of Pesach) had to fall as early as possible in spring, i.e. on or as soon as possible after the day of the March equinox (see Section 5), and in addition taking into account the fact that in actual practice the Jewish authorities in Jerusalem often ignored this rule and in consequence let begin their (i.c. the real) month Nisan and with this also the celebration of their Paschal feast in fact a month too early (i.e. before the day of the March equinox) many a time. Hence the thing is to present two local Jerusalem points of time of Newmoon for each of the (nine) calendar years in question, in concreto a one in column B of Table 3 (in which all calendar years are Julian calendar years and all dates Julian calendar dates) and a one in column B of Table 4 (idem), of which the former generates a possible date of the fourteenth day of Nisan after the March equinox (in column D of Table 3) via a possible date of the first day of Nisan (in column C of Table 3) and the other a possible date of the fourteenth day of Nisan before the March equinox (in column D of Table 4) via a possible date of the first day of Nisan (in column C of Table 4).

During the time interval consisting of the time between the years 20 and 40 the real date of the March equinox was sometimes 23 March sometimes 22 March (equally often). In order to be able to obtain (of course with the help of a suitable lunar phase table) for each of the (nine) calendar years in question at least two possible Nisan heralding local Jerusalem points of time of Newmoon it is necessary and sufficient to establish a suitable lower and upper limit which in principle must differ about 59 days (being about twice the synodic period of the moon) between which limit dates possible Nisan heralding local Jerusalem points of time of Newmoon 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 achieve our aim by taking our departure from a lower limit 6 February 18:00 and upper limit 5 April 18:00, because 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 3 which contains for each of the calendar years mentioned in column A of this table the corresponding as good as possible estimated local Jerusalem point in time of the second Newmoon between those limits. It is column B of Table 4 which contains for each of these calendar years the corresponding as good as possible estimated local Jerusalem point in time of the first Newmoon between those limits. That is why column C of Table 3 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 4 for each of these calendar years a possible date of the first day of Nisan before 9 March. That is why column D of Table 3 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 4 for each of these calendar years a possible date of the fourteenth day of Nisan before the March equinox.

Because the dates mentioned in column C of Table 3 and the dates mentioned in column C of Table 4 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 tables. Therefore it is only the Thursdays, Fridays and Saturdays in columns D of these tables which are of importance to us. We conclude that to begin with there are six dates which could be the date of the day of Jesus’ death. Assumed Jesus was crusified on a fourteenth and not on a fifteenth day of Nisan, that is of course in the first place (because at first sight they are the most probable possible dates of the day of Jesus’ death) the dates of the Fridays in columns D of Table 3 and Table 4, namely the Fridays 11‑4‑27, 7‑4‑30, 3‑4‑33, and only in the second place (because at first sight they are less probable possible dates of the day of Jesus’ death) the dates of the Fridays immediately following any Thursday in these columns or immediately preceding any Saturday in these columns, namely the Fridays 18‑3‑29, 14‑3‑32, 6‑3‑33. The six Fridays in question are mentioned in columns E of Table 3 and Table 4.

The first of the three dates coming to the fore in the previous paragraph seems to be too early to be able to be the right date of the day of Jesus’ death, since it must be considered as certain that Jesus was baptized not earlier than in the early part of the year 27 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 the day of Jesus’ death 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 diminished self assurance due to the decrease of the support from Rome which was a consequence of the fact that in the year 31 his patron Lucius Sejanus had fallen into disgrace with the emperor Tiberius. That implies that 3‑4‑33 has the greatest chance to be the right date of the day of Jesus’ death. The English monk and scholar Roger Bacon, who lived in the thirteenth century, was the first who substantiated that 3‑4‑33is the most probable date for Jesus’ crucifixion.

 

8 metonic structure

It is interesting to relate the three sequences of dates of Paschal full moon with a period of 19 years mentioned in Section 6 to each other. Their Julian calendar dates fall between 20 March and 21 April, their Alexandrian calendar dates between the twenty fourth day of Phamenoth (see Section 5) and the twenty sixth of Pharmouthi (see Section 5). Furthermore, they 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 over each period of 19 years 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 the ones 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 (taking the years to consist of 365 ¼ days and synodic months of 29.53059 days, we get 19 × 365 ¼ 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; hence such particular sequences of dates besides their particular structure, like the lunar cycle in question, are called Metonic. Summarizing, we can say that a Metonic sequence of dates is understood to be 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 famous sequence of dates of the (classical) Alexandrian Paschal full moon (see Section 6), dating from the fourth century, that would turn out to be the key to the solution of the great problem of the calculation of the date of Easter. Contrary to the sequence of dates of the Anatolian Paschal full moon (see Section 6), also the sequence of dates of the proto Alexandrian Paschal full moon (see Section 6), which was a precursor of the sequence of dates of the (classical) Alexandrian Paschal full moon, had a Metonic structure.

By an Anatolian Paschal day is meant a Paschal day which according to the Anatolian calendar (see Section 6) was a Sunday (but perhaps no Sunday according to the Julian calendar). This implies that each of the dates of the Anatolian Paschal full moon can be obtained by reducing by means of the corresponding lunar phase number the Anatolian calendar date of the corresponding Anatolian Paschal day taken as Julian calendar date to the corresponding date with lunar phase number 14 (see Section 6). The sequence of dates of the Anatolian Paschal day is like the sequence of dates of the Anatolian Paschal full moon a sequence of dates of consecutive Julian calendar years with a period of 19 years.

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 (see Section 6), and the classical Alexandrian cycle (see Section 6). In columns B, C, D, E of Table 5 (in which all calendar years are Julian calendar years and all dates Julian calendar dates) we see not only the restrictions of these four sequences of dates with a period of 19 years to the time interval consisting of the time between the years 270 and 290, but also the four accompanying sequences of lunar phase numbers. Although the year 285 was considered to be the initial year of the classical Alexandrian cycle (because it was the first year of consulate of the emperor Diocletianus), 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 5 with each other, that the proto Alexandrian cycle not only differs (one day) in no more than 4 out of 19 dates with the sequence of dates of the Anatolian Paschal full moon, but is even the best Metonically structured approximation to it. This observation was the final piece of the reconstruction of the Julian equivalent (see Section 5) 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 all these three sequences of dates. There is every appearance 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 subdivided into two types: the ones of the first type, characterized by 11 ordinary advances of 11 days, 1 saltus advance of 12 days and 7 ordinary regressions of 19 days, and the ones of the second type, characterized by 12 ordinary advances 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 5 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 in column B the immediate successor of 1 April is 20 April, but in column E it is 21 March.

By comparing columns B and E of Table 5 with each other, we can establish that the difference between proto Alexandrian and classical Alexandrian Paschal full moon 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, supposedly 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 classical Alexandrian cycle and thst 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 substitute month Nisan^ (see Section 6) as the moment of the last sunset in Alexandria preceding the Newmoon (see Section 5) in question instead of as such a thing 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, cannot possibly 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 were the dates of the Anatolian Paschal Sunday (Sunday by definition) between the years 250 and 290. This can be seen in Table 6 (in which all calendar years are Julian calendar years and all dates Julian calendar dates). In this table column B shows dates of the proto Alexandrian Paschal full moon, column C dates of the Anatolian Paschal day, column D dates of the Anatolian Paschal Sunday, and column E dates 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 two times that the date of the Anatolian Paschal Sunday did not coincide with the date of the proto Alexandrian Paschal Sunday.

In the first three and a half century of our era the Fullmoon (see Section 5) of Nisan (see Section 5) on average roughly coincided with the midnight point of time of the fourteenth day of Nisan (see Section 5), 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 its Fullmoon.

Somewhere in the beginning of the fourth century the church of Alexandria definitely opted for the classical Alexandrian cycle. At that time the date of the proto Alexandrian Paschal full moon fell on average about 0.8 days after the date of its Fullmoon, on the other hand the date of the classical Alexandrian Paschal full moon on average about 1.4 days before the date of its Fullmoon. The proto Alexandrian and the classical Alexandrian cycle have the same (Metonic) structure. The former of these sequences of dates functioned less than half a century, whereas the other acted from the beginning of the fourth century until the year 1582. The 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 date of the classical Alexandrian Paschal full moon and the corresponding date of Fullmoon had been decreased by a day. Only around the middle of the eighth century this average distance was minimal. Only against the end of the twelfth century it reached its initial value (1.4 days) once again; up to then Alexandrian Paschal full moons had always more or less the appearance of a pure full moon (i.e. about Fullmoon) had. By nature a midnight pure full moon is always preceded by a waxing full moon one night earlier and followed by a waning full moon one night later (see Figure 4). It is only since the first half of the thirteenth century that Alexandrian Paschal full moons were for the most part no pure full moons. They will be pure full moons again during circa nine centuries around the first half of Millennium 12.

 

9 epilogue

The turn of year [31‑12‑1999; 24:00] = [1‑1‑2000; 0:00] is the most recent “magic” moment at which all four digits of the number of the current calendar year changed simultaneously. However, this 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 shall have to undergo how a dancing and jumping crowd of frenzied people, made mad by commerce, media and authorities, one year too early is waiting on the platform for the next millennium train, in order to 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 remember (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 at [31‑12‑1000; 24:00].

Around the year 2000 certainly more than six hundred websites devoted to the millennium question were made. On most of those websites one declared oneself, like on this website millennium, in favour of the proposition that the year 2001 is the first year of the third millennium and related 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 a mistake 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 preserve her 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, as in our second time line (see Figure 2), always has outweighed the practical advantage of the 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 7) and a 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 July of the year 2010) on the history of the computus paschalis (see Section 6) was an explanation of the way in which he succeeded in determining the first year of this famous 19 year Paschal cycle. The accompanying article, which will appear in print in the year 2014 contains among other things a reconstruction of the proto Alexandrian and a one of the classical Alexandrian cycle (see Section 6).

 

 

 

 

> millennium question

> velox@janzuidhoek.net

> curriculum vitae