The Math Assignment
When I was at University, I recall working with a friend to help him develop a computer programme for a math assessment. At the time, Fortran was the programming language in vogue but for this assignment, he was asked to use a Commodore PET computer with BASIC as the programming language.
His lecturer set him the task of constructing an algorithm that would predict which day in the week it would be for any date in any year (within a defined window of several thousand years) that he cared to query. It took me and my friend about a month to research the history of calendars, write the algorithm in BASIC and submit his assignment. For my friend, this was an assessment but for me, it was one of those fun projects that captured my imagination.
In this post I will share what I learnt. And I will also declare how my friend fared in his Math assignment at the end of this post.
In Our Beginning ...
The history of calendars is linked very much to our thought about time. Researching the origins of time is to explore the very beginning of creation – something I tried to cover in a previous post (Big Bang Theory: The longest playing reality show in history). This post may be found at:
Ever since human beings existed on this planet, we have measured time by observing the natural world: the changing of the seasons or the migration of heavenly bodies across the sky. We now know that more than 30,000 years ago, humans in what is now central Europe tracked the Moon and stars across the sky by carving notches into the tusks of the mammoth. In doing so, they were following the movement of celestial bodies, visible to the naked eye, probably trying to determine if there was any order in the activity that was being observed.
The Moon has much to do with calendars. Indeed, the etymology of the word “month” originates from the word “Moon”. A “month” initially measured the duration it took for the Moon to complete a cycle around the Earth. So, the words 'Moon' and 'month' come from the same root.
The World's Oldest Calendars
The Moon seems to be connected to calendars in many respects. In 2013, British archaeological experts announced that they discovered what they believe to be the world's oldest 'calendar'. This calendrical monument, created by hunter-gatherer societies and dating back to around 8,000 BC, was discovered at Warren Field, Crathes in Aberdeenshire, Scotland. After several years of study, researchers concluded that this structure may have functioned as a luni-solar device for keeping track of the seasons by observing the Sun and Moon, some 5,000 years before formal time-measuring devices were developed in the Near East.
Before the British discovery in Aberdeenshire, the first formal calendars were thought to have been created in Mesopotamia circa 5000 years ago (i.e. 3000 BC). In the Near East, the Sumerian (Babylonian) Calendar was the earliest known to be used. This was a lunisolar calendar (based on the combined effects of the Sun and the Moon), with years consisting of twelve lunar months, each beginning when a new crescent Moon was first sighted low on the western horizon at sunset. In this calendar system, an intercalary month (intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or Moon phases) would be inserted as needed, by decree.
In later years, this Sumerian Calendar system was the basis of many other calendar systems that followed in the Near East (Egyptian, Assyrian, Elamite, Zoroastrian and Hebrew). It also gave rise to the Attic calendar – one of many Greek calendar systems. It is important to note that calendar systems had only local utility - to inform local citizens of domestic-related matters. Multiple forms of calendar systems were used contemporaneously, and even neighbouring regions did not follow a universal system of keeping track of time. Thus, the Attic calendar, like many other calendars, was an exclusively local phenomenon used to regulate the internal affairs of the Athenians, with little relevance to the outside world.
The Roman Calendar
The Roman calendar, also referred to as the Roman Republican Calendar, was probably an evolution of one of the Greek lunar calendars. According to legend, King Romulus, the founder of Rome, instituted this calendar in about 738 BC. The Romans themselves described their first organised year as one that began in March and consisted of 10 months, six of 30 days, and four of 31 days, making 304 days. It thus ended in December, followed by what seems to have been an uncounted winter gap. The 10 months were named Martius, Aprilis, Maius, Junius, Quintilis, Sextilis, September, October, November, and December. This system ran well short of the solar year, and it needed constant intercalation to keep religious festivals and other periodic activities within the time windows of their respective seasons.
Legend claims that around 713 BC, Numa Pompilius, the second king of Rome, reformed the Roman calendar significantly. The calendar became increasingly essential and had broader applications than just agriculture, so assigning the roughly 60 “monthless” days to two new months was necessary. Thus, January and February were added to the existing 10-month calendar. Initially, January was placed at the beginning of the year and February was given a position at the end. This situation prevailed until 452 BC, when February was placed between January and March. This would be just one of several reforms made to this calendar.
In 509 BC, a political revolution took place in ancient Rome and resulted in the expulsion of the last king of Rome, Lucius Tarquinius Superbus. Thus, was established the Roman Republic with the citizens after that electing two consuls annually to rule the city. After installing the Roman Republic, the years began to be dated by consulships and control over intercalation was granted to the pontifices (the most critical positions in ancient Roman religion). Eventually, these pontifices abused their power by lengthening the years controlled by their political allies and shortening them when their rivals held office.
In 60 BC, a powerful political alliance emerged, comprising Gaius Julius Caesar, Marcus Licinius Crassus, and Gnaeus Pompeius Magnus (Pompey the Great). They formed the First Triumvirate. From 60 BC to 51 BC, Caesar rose to become one of the most powerful politicians in the Roman Republic through a string of military victories in the Gallic Wars. Through his conquests, he significantly extended Roman territory. Crassus died in 53 BC, and the First Triumvirate was now left only with Caesar and Pompeius.
The Roman Senate ordered Caesar to step down from his military command and return to Rome as the Gallic Wars had concluded. Caesar openly defied the Senate's authority and led an army towards Rome. Thus, began Caesar’s Civil War (where he faced Pompeius, who had aligned himself with the Roman Senate). Caesar won this war, leaving him in a position of near unchallenged power and influence.
Hail, Caesar: The Advent of the Julian Calendar
After assuming control of the government, Caesar began a programme of social and governmental reforms, including the creation of the Julian Calendar. Julius Caesar proposed the Julian calendar in 46 BC. It took effect on January 1, 45 BC, by edict. It was a reform of the Roman calendar and was designed by Greek mathematical and astronomical scholars. Caesar and his scholars, mainly the philosopher, Sosigenes of Alexandria, made one crucial modification: instead of relying on the stars and making inconsistent adjustments to the calendar, they would add a day to every fourth year. In keeping with the Roman tradition of adjusting with the length of the month of February, that day would fall in the second month of the year. Thus, the “Leap Year” was born.
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Emperor Julius Caesar reformed the Roman Calendar. On 1 January 45 BC, the Julian Calendar took effect. Leap years were also introduced as part of this new calendar system. |
Julius Caesar was assassinated by a group of Senators, led by Marcus Junius Brutus and Gaius Cassius Longinus on 15th March 44 BC (the Ides of March). He was stabbed approximately twenty-three times but whilst the Emperor perished, the Julian Calendar, prevailed to became the predominant calendar in a growing Roman Empire. It remained the reference calendar for most of the Western World for more than 1,600 years, until 1582.
The Final Correction
In October 1582, Pope Gregory XIII introduced a minor modification to the Julian Calendar. He reduced the average year from 365.25 days to 365.2425 days. In doing so, he created the Gregorian Calendar.
There were two reasons to establish the Gregorian calendar. Firstly, the Julian calendar assumed incorrectly that the average solar year is exactly 365.25 days long, an overestimate of a little under one day per century. The Gregorian Reform shortened the average (calendar) year by 0.0075 days to stop the drift of the calendar with reference to the equinoxes. In addition, in the years since the First Council of Nicaea, in AD 325, the excess leap days introduced by the Julian algorithm had caused the calendar to drift such that the spring equinox in the northern hemisphere was occurring well before its nominal March 21st date. This date was essential to the Christian churches because it was fundamental to the calculation of the Easter celebration. To reinstate the correlation, the Gregorian Reform advanced the date by 10 days: Thursday, October 4, 1582 was followed by Friday, October 15, 1582. Thus, the Gregorian Reform practically made all those living at the time ten days older, one day later!
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Above: Stamp from Germany commemorating the 400th anniversary of the introduction of the Gregorian Calendar Below: First Day Cover from the Vatican, issued in 1982 commemorating the 400th anniversary of the introduction of the Gregorian Calendar |
Global adoption of this revised calendar, which became known as the Gregorian Calendar, gradually took place over the subsequent centuries, first in Catholic countries and subsequently in Protestant nations.
The AD Dating System
Whilst it is now accepted that by January 1, 45 BC, it had been established that there were 365.25 days in a year and a year comprised 12 months, the “Anno Domini” dating system was only devised in 525 AD by Dionysius Exiguus (Dionysius the Humble). Dionysius was a monk and a learned member of the Roman Curia (the administrative institution of the Holy See which is the under the purview of the Bishop of Rome, who is also the Pope).
During those historical times, the dates of Easter (the most important religious celebration of the Roman Catholic Church commemorating the resurrection of Jesus Christ) were set out in a format called the “Easter Tables”. These Easter Tables recorded the future dates of Easter based on the Diocletian numbering system. In 525, at the request of Pope John I, Dionysius prepared a table of 95 future dates of Easter (from 532-626) and a set of rules ("argumenta") explaining their calculation(“computus”). It seems that Dionysius used this opportunity to resolve two other issues that needed some attention.
The first of his issues concerned the “Diocletian era”, a method of numbering the years used by the Church of Alexandria from the 4th century AD. This method was named after the Roman Emperor, Diocletian, who reigned between 284 AD and 305 AD. In the Diocletian counting system, the beginning of the Emperor’s reign in 284 was used as the first year of that calendar. The early years of the power of Emperor Diocletian stabilised a Roman Empire that was on the brink of collapse. Still, he is more notoriously associated in the annals of history for the era of the Diocletianic Persecution (303 – 312). This was the Empire's most enormous, bloodiest and final official persecution of Christianity. Many Christians were martyred, causing this dark period for Christianity (also remembered as the “Era of the Martyrs”).
It disturbed Dionysius that a matter as prestigious as the naming of an eternal numbering system of the years had been named after an Emperor who had mercilessly persecuted the Christians for over a decade. This needed to be addressed and rectified as he did not wish to continue the memory of an Emperor who had been a tyrant.
There was also another issue. At the time, there was a wide-held belief that the end of the world would occur 500 years after the birth of Jesus Christ. It was believed that (based on the Anno Mundi calendar - a calendar era based on the biblical accounts of the creation of the world and its subsequent history), Jesus was born in the year 5500 (or 5500 years after the world was created) with the year 6000 of the Anno Mundi calendar marking the end of the world. Evidence exists of Dionysius’ desire to replace the Diocletian years with a calendar based on the incarnation of Christ to influence people from believing that the end of the world was imminent.
Dionysius thus selected the year of birth of Jesus Christ as the first year of his calendar system and when he added the new 95 Easter dates in the Tables as instructed by Pope John I in 525, he notated the newly inserted dates with the suffix, “Anni Domini Nostri Jesu Christi” (Years of our Lord Jesus Christ). Even though he made this insertion an estimated 525 years after the birth of Christ, he only started his numbering of the years from 532 as the existing Easter Tables had already fixed the Easter date for the next six years (i.e. until 531). This "error" would, many centuries later be challenged by Pope Benedict XVI (as explained in later in this post).
Prior to the AD system being devised by Dionysius, his table, calendar years were identified by naming the Roman Consuls who held office that year. Dionysius himself stated that the "present year" when he developed his system was "the consulship of Probus Junior", which was 525 years "since the incarnation of our Lord Jesus Christ".
Gradually, the Dionysian tables and his notation became widely used. There is evidence that most of the British Church had accepted the Dionysian Tables by AD 664, whilst on the European continent, Anno Domini (or AD) was only introduced as part of the dating system in the late eighth century. The usage of AD gradually became more common in Catholic countries from the 11th to the 14th centuries, with Portugal becoming the last Western European country to adopt a system initiated by Dionysius. In AD 1700, Russia did the same with some Eastern Orthodox countries, only adopting this approach in the 19th and 20th centuries.
Indeed, in selecting the birth year of Jesus Christ as the first year of his system of numbering the years, Dionysius achieved one of his objectives. He successfully put in place the primary building block that eventually resulted in the Diocletian system of numbering the years being confined to the footnotes of history.
BC, CE and BCE
Although “Anno Domini” was in reasonably widespread use by the 9th century, the term "Before Christ" (or its equivalent) did not become common until much later. "Anno ante Christi nativitatem" (in the year before the birth of Christ) is a phrase found in a AD 1474 work by a German monk. In AD 1627, the French Jesuit theologian Denis Pétau, in his work, “De Doctrina Temporum”, popularised the usage “ante Christum” (Latin for "Before Christ") to mark the years before AD. Thus, commenced another long-held convention.
Since 1856, the alternative abbreviations of CE and BCE (sometimes written C.E. and B.C.E.) are sometimes used in place of AD and BC. CE is the acronym for “Common Era” and BCE for “Before Common Era”. The use of these replacements was to secularise the calendar systems and not associate dates with religion.
The BC - AD Time Interface
In the AD year numbering system, AD 1 is immediately preceded by 1 BC, with nothing in between them (i.e. there was no year zero). The concept of zero being a digit had prevailed since Sumerian and later, Babylonian times. But the acceptance of zero, being a number in its own right, with a value of nothing that preceded the number 1; that concept seems to have been suggested by Hindu astronomer and mathematician Brahmagupta, only in AD 628. A circle inscribed on a temple wall in Gwalior in India dates back to the ninth century. According to the University of Oxford, this is the oldest recorded example of zero as a numeral. The numeral can also be seen on an ancient Indian scroll called the “Bhakehali Manuscript”. Discovered in AD 1881, the scroll was assumed to have been a contemporary of the symbol found in the temple in Gwalior, but modern carbon dating reveals its origin in the third or fourth century. Thus, many scientists believe that zero was a discovery made in India as a number in its own right. Clearly, this numerical conceptualization of zero did not feature in the thinking of Dionysius and others who developed the early calendars.
Astronomical Year Numbering
Modern science recognises the concept of zero as a number. So, for computational reasons, astronomical year numbering and standards established by the International Standards Organisation (ISO 8601) designate the years so that AD 1 = year 1, 1 BC = year 0 and 2 BC = year −1, etc. It is also important to note that ISO 8601 uses the Gregorian Calendar system.
Birth Date of Jesus Christ
As our current dating system pivots on the birth year of Jesus Christ, it would be critical to understand how accurately we know that date to be. The reality is that the date of birth of Jesus of Nazareth is not stated in the Gospels or any secular text. Most scholars assume a date of birth between 6 BC and 4 BC. The historical evidence is too fragmented to allow definitive dating.
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First Day Cover for Eire celebrating the birth of Jesus Christ. This event of the birth of Jesus Christ was first used as the event to denote Year 1 of the the "Anno Domini" calendar dating system by Dionysius Exiguus in 525 AD. |
The year of the birth of Jesus Christ has however been estimated through two different approaches. Firstly, by analysing references to known historical events mentioned in the Nativity accounts in the Gospels of Luke and Matthew. Secondly, by working backwards from the estimation of the start of the ministry of Jesus (i.e. the period that begins with the baptism of Jesus Christ in the countryside of Roman Judea, near the River Jordan and which ends in Jerusalem following the Last Supper with his disciples).
In 2012, in his book, "Jesus of Nazareth: The Infancy Narratives," Pope Benedict XVI asserted that the Christian calendar was based on a miscalculation because Jesus was born sometime between 7 BC and 2 BC. "The calculation of the beginning of our calendar - based on the birth of Jesus - was made by Dionysius Exiguus, who made a mistake in his calculations by several years," Pope Benedict XVI writes, "The actual date of Jesus' birth was several years before."
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Stamp honouring Pope Benedict XVI. In a publication in 2012, His Holiness challenged the belief that Jesus Christ was born in 1 AD. |
The First Day of the World
The Church has always had an interest in the creation of the world. In the summer of 1650, a Protestant bishop in the Catholic land of Ireland, James Ussher, published a monumental work titled “The Annals of the Old Testament” (The Annals). Bishop Ussher is forever remembered for a date that appears in the first paragraph of the first page of The Annals. Ussher wrote: “In the beginning, God created heaven and earth, which beginning of time, according to this chronology, occurred at the beginning of the night which preceded October 23rd in the year 710 of the Julian period.” In the right margin of the page, Ussher computes the date in “Christian” time as 4004 B.C.
Ussher began his calculation by adding the ages of the 21 generations of people of the Hebrew-derived Old Testament, starting with Adam and Eve. If the Bible is to be believed, they were an exceptionally long-lived group of humans. Genesis, for example, tells us that “Adam lived 930 years, and he died.” Adam’s great-great-great-great-great-grandson, Methuselah, claimed the longevity record, living a life of 969 years. It was believed that healthier living conditions contributed to the long life spans of the early generations of the Bible. Josephus, a Jewish theologian, writing in the first century, explained it this way: “Their food was fitter for the prolongation of life…and besides, God afforded them a longer lifespan on account of their virtue.”
Bishop James Ussher was considered an accomplished scholar. Indeed, such was his academic reputation that when he died, The Lord Protector of the British Isles, Oliver Cromwell, had his remains buried in the Chapel of St. Erasmus in Westminster Abbey.
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Oliver Cromwell led the forces of the English Parliament against King Charles I during the English Civil War. He honoured Bishop James Ussher on his death, by having Ussher's remains buried in the Chapel of St. Erasmus in Westminster Abbey. Sadly, Cromwell was not offered such state recognition. Cromwell died from natural causes in 1658 and was initially buried in Westminster Abbey but after the Restoration of King Charles II in 1660, Cromwell's body was exhumed, hung in chains and beheaded. |
The work of Bishop Ussher was considered a critical piece of scholarship at the time, and for several centuries afterwards, it was considered a credible reference for the beginning of time. In 1701, the Church of England adopted Ussher’s dates for use in its official Bible. For the next two centuries, Ussher’s dates so commonly appeared in Bibles that his dates “practically acquired the authority of the word of God.”
Some Final Timely Thoughts
The are many calendar systems in use. More common examples would be the Islamic, Hindu, Hebrew, Chinese, Persian, Mayan etc. Their origins primarily served religious and/or agricultural purposes. Some years ago, the Mayan Calendar shot to prominence as it suggested that the end of the world would occur on 21 December, 2012. Clearly, that did not take place.
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A specimen stamp from Switzerland issued between 1961 and 1970. It celebrates the Mayan Calendar. In the foreground there is an image of David, a masterpiece of Renaissance sculpture, created in marble between 1501 AD and 1504 AD by Michelangelo. |
Over the years, the Gregorian Calendar became the accepted dating system mainly because it considered precise astronomical observations (the rotation of the Earth around the Sun and the orbit of the Moon around the Earth) and considered critical religious events (Easter, the Birth of Jesus Christ). Once this calendar system became accepted in Europe, the European colonisation of the East and the Americas made this Gregorian Calendar system the global standard.
It has taken more than a thousand years to develop the Gregorian Calendar system. To conclude this post, I will leave you with this mental model about astronomical and geological timescales. Let us think of the entire history of Earth, compressed into a single 365-day year. As the clock strikes midnight, on December 31, when we would typically be raising our glasses of champagne and singing the “Auld Lang Syne”, the Earth and the Solar System are formed. On this timescale, sometime around February, about two months after our New Year’s Eve celebration, rocks appear on the planet as the Earth’s surface starts to cool down.
Around November 20th that same year, the first plants appear, and their flowers begin to bloom. By about mid-December, dinosaurs are roaming the planet. Still, due to a giant asteroid striking the Earth about 60 million years ago, they would disappear by December 26th and allow other forms of species to thrive.
Unbelievably, there are only about 4.5 days to go until the end of our hypothetical year, and humankind has yet to make an appearance!
The oldest known hominid (i.e. human-like) fossils that have been discovered date back to about midday of 31st December but our species, “Homo Sapiens”, only appear about three minutes before midnight at the end of our hypothetical year. Today, we know that Homo Sapiens have inhabited the Earth for about 150,00 to 200,000 years.
The Gregorian Calendar only covers about 1,500 years of that time!
How much we do not know!
Post Script:
Returning to the problem set by the Math lecturer at University in 1979, which prompted some of my early research into calendar systems …..
To confirm that the algorithm that had been developed was correct, the Math lecturer asked my friend to query our computer programme and provide him with the day of Nelson Mandela’s birthday in the year 5000 BC (if President Mandela was living at that time).
As a second part of the assignment, he then asked my friend to provide him with the day of Nelson Mandela’s birthday in the year 5000 AD (if President Mandela was to live that long).
This Math assignment taught me three new facts that autumn morning in 1979 at Bath University in England.
Firstly, I share the same birthday as Nelson Mandela (i.e. 18th July).
When querying the computer: 18 July 5000 BC ? The programme responded, stating that date was a Tuesday.
It was then time to input the second test date into our computer algorithm: 18 July AD 5000 ? In a flash the Commodore PET computer responded, stating that the date was a Friday.
The Math lecturer allowed himself a brief smile and then said to my friend, “Looks like you have passed!”
Hope you enjoyed this post.
All stamps and First Day Covers shown in this post are from my personal collection.
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