As we approach the beginning of the Third Millennium there is uncertainty both about when to celebrate it and what to celebrate. This uncertainty seems to arise from a lack of understanding of the origins of our calendar and of our system of numbering. Historical information from astronomy and mathematics can help to shed some light on the subject.

The source of confusion is easy to identify. From the time we first learnt to write we have been dating documents with the prefix `19'. It seems obvious that when we change to `20' we have entered a new century. However, this assumption is false; the year 2000 is merely the last of the 20th century. Historians remind us that there is no Year Zero so that the first century ran from the beginning of AD 1 until the last day of AD 100. Similarly, the 20th century began on 1st January 1901 and will end on 31st December 2000. The Third Millennium will not begin until 1st January 2001. Those who plan to celebrate at the end of 1999 will be marking merely the 2000th anniversary of the start of 1 BC.

Our habit of referring to the decades as the 70s, 80s and 90s also adds to the confusion. There is no reason why we should not mark off the decades, centuries and millennia in this way except that it ignores the convention that the first year of the Christian Era was designated 'one' just as the first day of a month is labelled 'one'. This convention is derived from the almost universal use of Roman numerals in Christendom until the sixteenth century.

**The Origins of Our Calendar**

Attempts to devise satisfactory calendars go back to the dawn of civilization and involve precise astronomical observations. It is estimated that some forty types of calendar are in use worldwide of which six are widely known. The intervals of most importance in everyday life are the day, the month and the year. Some calendars use the year as the basic unit and others use the lunar month. The difficulty of constructing a calendar arises from the natural periods of time being incommensurable; for instance the length of the tropical year (equinox to equinox) is 365.242199 days or 12.368267 mean lunar months (phase to phase).

The Gregorian calendar is now accepted as a worldwide standard. It was derived from the Julian calendar, established by Julius Caesar in 46 BC by revising the ancient local calendar of the city of Rome. Caesar took the advice of the Alexandrian astronomer Sosigenes and adopted a year of 365.25 days that was known to the Egyptians from the first appearance at dawn of the bright star Sirius. The Julian calendar with a leap day inserted every fourth year reached its final form in AD 8 and was used throughout the length and breadth of the Roman Empire.

Since the length of the mean Julian calendar year exceeded the length of the tropical year by about 11min. 14sec. or by about three days in every 400 years, the calendar gradually got out of step with the seasons. This defect had a very noticeable effect on the date of Easter; by the 16th century the spring equinox had fallen back to about March 11th and Easter was on average ten days late. The Gregorian calendar was instituted in 1582 by Pope Gregory XIII in order to regulate the date of Easter and the ecclesiastical calendar. The reform consisted of omitting ten days from the calendar and adopting a new rule for leap years. By omitting the leap day in those years that are divisible by 100 but not by 400, the mean length of the Gregorian calendar year became 365.2425 days. The residual error amounts to only one day in over 3000 years. Ladies should note that the year 2000 will be a leap year!

When the ten days were dropped from the calendar in 1582, Thursday, October 4th of the Julian calendar was followed by Friday, October 15th of the Gregorian calendar and the cycle of weekdays was not disturbed. It seems likely that religious authorities have maintained without interruption the cyclic continuity of the week from its origin in biblical times to the present day. In Britain and its dominions the Gregorian calendar was not adopted until September 1752 when eleven days were omitted; the start of the year was also moved from March 25th to January 1st. Assigning exact historical dates to events can be a hazardous business!

The custom of reckoning dates from the start of the Christian Era was a by-product of the work of the scholarly monk, Dionysius Exiguus (Denis the Little) in compiling a set of Easter tables which appeared in AD 525. Previous tables had been reckoned from the accession of the Roman Emperor Diocletian who was notorious for persecuting many of the early Christians. Dionysius chose "not to link the memory of this ungodly persecutor to our new cycles". Like some others before him, he counted the years as "the years of Our Lord Jesus Christ". However, it was neither his intention nor his task to determine the birth year of Christ. According to Gustav Teres of Oslo University, Dionysius used St. Luke's gospel as the basis for his chronology; i.e. John the Baptist began to preach in the fifteenth year of Tiberius Caesar's reign and shortly afterwards Jesus started to teach, when he was "about thirty years old" (Luke 3, v1 and v23).

Dionysius calculated that AD 1 corresponded to A.U.C. 754, A.U.C. standing for ab urbe condita (from the foundation of Rome). Unfortunately he ignored the four years when the Emperor Augustus reigned under his family name of Octavian and two years when Tiberius ruled in Syria. The implication that Christ was born in 6 or 7 BC is supported by engravings on a stone tablet - the Monumentum Ancyranum, found in Caesar Augustus's temple in Ankara. The tablet records a census taken by Augustus in the years 7 and 8 BC - probably the census mentioned in St. Luke's gospel. Moreover, astronomers have identified the Star mentioned in St. Matthew's gospel with the triple conjunction of Jupiter and Saturn between May and December 7 BC. It seems likely that Jesus was born in 7 BC and certainly not at the conventional origin of the Christian Era.

The Anno Domini system devised by Dionysius was adopted by some scholars but did not gain wide acceptance until the eighth century when it was advocated by the Venerable Bede, the highly respected Anglo-Saxon historian from Jarrow. Bede also extended the system backwards in time by counting years before the conventional origin. Thus the day before 1st January AD 1 was 31st December 1 BC. A Year Zero was not included for the very good reason that the concept of zero was unknown at that time and the Roman numerals were used universally throughout Christendom. For convenience in estimating intervals that span the origin, astronomers reckon that AD 1 = +1, 1 BC = 0, 2 BC = -1 and so on. The astronomer Jacques Cassini introduced this system in 1740.

**The Origin of the Decimal System**

We take our numerals 0 1 2 3 4 5 6 7 8 9 for granted but they have an interesting history. We speak of these numerals as being `arabic' but in fact they were never used by the Arabs! The marvellous invention of zero was first employed not by the Egyptians nor the Greeks nor the Romans but by the Maya Indians of Yucat n (now part of Mexico). By the beginning of the Christian Era the Maya were using a zero sign and positional values of numbers reading from right to left. Quite independently and some five centuries later, the Hindus used these same inventions in India. The Hindu numerals reached Baghdad, the capital of the Islamic Empire about AD 800 but took a further 700 years to reach northern Europe.

Soon after the Moslem empire was founded with its capital in Baghdad, an Arab scholar called al-Khwarizmi (also known as Algorithmus) wrote the first book on algebra. He also wrote a small book commending the new Hindu arithmetic. The original was lost but an English monk had translated it into Latin in the 12th century and it was used by Moorish scholars in Spain. In 1202 Leonardo Fibonacci of Pisa wrote an excellent book on arithmetic, Liber Abacus, which used the "figurae Indorum" that he had learned as a boy from his Moorish teacher. His book was a success and merchants quickly adopted the new numerals for book keeping. A setback came in 1299 when the bankers of Florence outlawed the Hindu numerals as well as the writing of numbers in columns. The reason was the ease with which a 0 could be changed to a 6 or a 9; it was not so easy to falsify Roman numerals. However, the Hindu numerals gradually gained ground and the first printed arithmetic in German appeared in 1493 to be followed by many others. It is hard to believe that only 500 years ago practically everyone north of the Alps still calculated with the abacus and used Roman numerals! One can only wonder why the production dates of some television programmes are given nowadays in the Roman style.

When al-Khwarizmi wrote his book on algebra, one problem left unsolved was how to deal with negative numbers. Fibonacci also saw the need for negative numbers but a full understanding of them did not appear until 1545 when the Italian mathematician Cardano published a monumental treatise on equations, his Ars Magna (Great Art). Scholars then realized that negative numbers arose naturally from the solution of quadratic and cubic equations.

**Other Number Systems**

The problem of dealing with fractions in the decimal system was considered by several mathematicians in the 15th and 16th centuries but the decimal point as we know it did not appear until 1617 in a book by the Scot, John Napier who is better known for his invention of logarithms.

The base-ten decimal system is so ubiquitous that it is not generally realized that other number systems are useful. For instance, computers operate on the base-two binary system of ones and zeros. The representation of 2000 in binary form is the unremarkable sequence 11111010000. The numbers 1000 and 2000 seem significant only because we use the decimal system and its popularity was probably due to our ancestors finding it convenient to count on their fingers.

The transition from 1999 to 2000 will confront the computer industry with immense problems. In the past most computer programs have represented years by only two digits (e.g. 96 for 1996) but catastrophic problems may arise after 31/12/99 unless precautions are taken. February 29th, 2000 will be another stumbling block unless the code was written by a programmer familiar with the rules for leap years in the Gregorian calendar. It is estimated that the worldwide cost of correcting these problems will be about $1,200 billion! This alone is a persuasive argument for teaching something about the calendar and elementary astronomy in schools.

The days of the week constitute a base-seven system that is assumed to have originated in Babylonia where the number seven was considered sacred. The names of the days of the week are derived from the seven naked-eye 'planets': Mercury, Venus, Mars, Jupiter, Saturn, the Moon and the Sun. In English speaking countries the names of the days have been modified by Saxon usage.

Our custom of subdividing hours and minutes into sixty units and the circle into 360 degrees can be traced to the Babylonians and their predecessors, the Sumerians who flourished in Mesopotamia about 3000 BC. One advantage of using sixty as a base is its many factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) whereas 10 has only 1, 2, 5, 10 as factors. Spare a thought for the Mesopotamians when you next glance at your watch.

**The Jubilee Year**

The Vatican's announcement of a Jubilee Year has added to the confusion about the start of the new millennium. According to the Encyclopaedia Britannica, the institution of jubilee years dates from a bull of Pope Boniface VIII in February 1300. Near the close of 1299, a rumour spread through Rome that everyone visiting St. Peter's on January 1st would receive full absolution. This resulted in a large influx of pilgrims, to the profit of both clergy and citizens.

In 1343 Pope Clement VI decreed that the jubilee should recur every fiftieth year instead of every century as had been originally intended by Boniface. The 50-year jubilee was in harmony with the Hebrew law that after seven sabbaths of years the fiftieth year would be a time to "proclaim liberty throughout all the land" (Leviticus 25, v10). However, the interval was further reduced to 33 years in 1389 by Pope Urban VI who was badly in need of money, and Pope Paul II set it finally at 25 years in 1470. Pope Alexander VI introduced a special ritual in 1500 for the opening of the jubilee on Christmas Eve. The coming Jubilee Year will start on Christmas Eve 1999, one year and seven days before the beginning of the Third Millennium.

The BC/AD convention of counting years is unsatisfactory because of the lack of zero and the increase of numbers both forwards and backwards in time. Time intervals spanning the BC/AD boundary cannot be calculated easily - for example the interval between June 2 BC and June AD 2 is not four years but three years. The late Cesare Emiliani of the University of Miami suggested a solution to the problem by resetting the origin of the calendar to 1st January, 10,000 BC. Thus AD 1 would become 10,001 and the preceding year 1 BC would be 10,000. All AD dates would be increased by 10,000 and all BC dates subtracted from 10,001. For instance, the founding of Rome in 753 BC would become 9248 under the new system. What better time to make this reform than the start of the new millennium?

So what shall we celebrate at the beginning of 2001? Christians can mark the 2000th anniversary of the conventional time of Christ's birth but not his actual birth. As the last seconds of this millennium are counted out, some may like to remember the ancient Egyptians and Mesopotamians to whom we owe our system of hours, minutes and seconds. Others may like to honour the many scholars who have contributed to our ingenious calendar. Still others may celebrate the invention of the decimal system and that curious symbol zero - they truly will be celebrating nothing!

Dr Ian Elliott

Dunsink Observatory,

Dublin 15.