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Our last lecture ended with the question: Why did people invent number systems? Was it the curiosity of the human mind, or was it the desire to improve the conditions for daily life?
Civilization had made life easier, because it made society less exposed to the vagaries of nature. Floodwaters were regulated and large food reserves stored in public granaries by good administrators. But harvests still depended on good or bad seasons. To understand the procession of the seasons better and to be able to plan ahead was clearly the major need of early societies. To respond to that need the priests turned to the first true science: astronomy.
The rise of astronomy can explain why people had a need for a number system, but it does not explain the interest of mathematicians in the nature of infinity. Early Indian mathematical texts contain long discussions of the infinite, which they divide into infinity in one, two and three dimensions. What would have been the practical use for such a discussion?
The answer to the problem of living in tune with the seasons is the calendar. The history of the invention and improvement of the calendar demonstrates how the needs of society, the results of scientific study and the influences from religion and politics all act together to produce a central element of each civilisation: the method to measure time.
Our discourse on numbers began with the question: Is there a natural number system? Let us begin our discourse on calendars with a similar question: Is there a natural way to measure time? We are used to a calendar of days, weeks, months and years. Are these divisions suggested by nature, or are they arbitrary?
The basic unit of one day is obviously a natural time measure. It regulates periods of activity and sleep and is a regular experience for everyone. The year likewise is an obvious experience, although its duration is not as easy to determine. Its great length requires subdivisions which are even more difficult to define. Most are essentially artificial and defined by convention.
A phenomenon that can be used to define a natural time unit shorter than a year but longer than a day is the time from one New Moon to the next (the lunation). This period, about 29.5 days, is particularly attractive as a time measure because it is close to the average menstrual period of women, and keeping track of it can thus help in controlling fertility. This makes the month a natural unit of time, at least to some degree.
Other time units are set by convention. The length of the week varied greatly between different societies, and the subdivisions of the day were also not fixed. Most (but not all) civilisations divided the day into 12 daylight hours and 12 hours of night, and the length of each hour varied over the year as the Sun rose and set earlier or later. The first recorded Indian calendar from 1000 BC divided the day into 30 hours.
The concept of seasons existed in all societies but varied depending on climatic conditions. Tropical climates often display only a dry and a rainy season; temperate climates allow the identification of more seasons. Before the development of calendars people kept track of the seasons by observing natural phenomena such as the flight of migratory birds, the mating ritual of cranes, the arrival of flying fish and similar events.
The principal scientific problem for every calendar designer is the fact that day, month and year - the natural measures of time - are not fractions or multiples of a common unit. The day is the result of the Earth's rotation around its axis. The month uses the rotation of the Moon around the Earth as its unit; it is thus a lunar phenomenon. If we use the day as the fundamental unit of time one lunation or synodic month equals 29.53059 days. The year is determined by the Earth's movement around the Sun; it is a solar phenomenon with a period of 365.242199 days (the tropical year). The art of calendar making consists of dividing the year, a solar phenomenon, into manageable smaller pieces with the help of the Moon’s movement.
For practical reasons a calendar cannot include fractions of a day, so whatever number of days is used to define a month or a year, the calendar will invariably get out of step with the real year and its seasons as time goes on. A year of 12 synodic months, for example, consists of 354.36706 days and is thus more than 10 days short of the tropical year. It is therefore necessary to insert additional days, known as intercalations, at appropriate intervals.
Early societies without much mathematical and astronomical knowledge measured time in lunar months of equal length (either 29 or 30 days) and simply added or skipped an entire month when their calendar got too far out of step with the solar year. The decision whether this was necessary was made through comparison with bird flight, the onset of tree flowering and other events of nature. These events were of course not necessarily fixed in relation to the solar year, and their occurrence across a country could vary as well. As a result each city had more or less its own calendar, and there was no fixed relationship between calendars of cities even within a single civilization. This situation changed only slowly with the introduction of scientific measurement and mathematical principles.
To understand the task a calendar maker faces it is necessary to appreciate some facts about the solar system. For the purpose of the calendar the stars are at fixed positions in space and form the celestial sphere. The rotation of the Earth around its axis produces an apparent rotation of the celestial sphere around the Earth's axis in the opposite direction. At the north and south poles the stars thus rotate around the zenith; at any other place on Earth they rise in the east and set in the west. The length of the day can be measured by observing the setting of a star on successive days. This is known as the sidereal day.
The path of the Earth around the Sun on the ecliptic describes an ellipse that is inclined against the Earth's equator by 23.44°. The Sun's position as a star is fixed. Seen from the Earth, however, the Sun is therefore observed north of the equator for half a year and south of the equator for the other half. It rises and sets at different locations during the year. The time from noon to the next noon (the passage of the sun through the zenith) can also be used to measure the length of the day and is known as the solar day. It is nearly 4 minutes longer then the sidereal day.
The Sun does not only exhibit apparent daily movement across the sky but an apparent change in relation to the stars as well. This is because an observer on Earth projects the Sun against the celestial sphere and sees it from different positions as a year passes. The sequence of constellations through which the Sun apparently passes over a year is known as the Zodiac.
Early attempts to link the measurement of time to the regular path of the stars across the celestial sphere lead to the construction of dedicated observatories, in which buildings and other objects were used to mark points along the horizon and allow observations of star risings at exact locations. Stonehenge in England is a well-known example. Fajada Butte in the Chaco Canyon of today's New Mexico was an observation site of the Pueblo civilizations of pre-Columbian America.
The Maya calendar is not the oldest known calendar but the most accurate one. Lecture 5 had some unfinished business with regard to Maya numbers that are related to the calendar, so the Maya calendar is discussed first.
The basis for the impressive accuracy of the Maya calendar were astronomical observations over long time intervals. The Maya astronomers knew that small errors can accumulate to significant discrepancies and that very long time series are required to keep errors small. The astronomers of the cities of Copán and Palenque measured the duration of the lunar cycle over many years; their result compares extremely well with today's value:
| measurement | value (days) | observation |
| Copán | 29.5302 | 149 lunations = 4,400 days |
| Palenque | 29.53086 | 81 lunations = 2,392 days |
| modern | 29.53059 |
Similarly accurate measurements were made for the Sun and some planets. The Dresden Codex contains tables of positions of Venus which give its orbital period as 584 days; today's value is 583.92 days. The length of the year determined by the Maya was more accurate than that of today's Gregorian calendar:
| measurement | value (days) | error (days) |
| Maya calendar | 365.242 | 1.98 in 10,000 |
| Gregorian calendar | 365.2425 | 3.02 in 10,000 |
| scientific | 365.242198 |
We are beginning to see the answer to the question why people were interested in large numbers. To establish accurate astronomical measurements you have to be able to count well beyond a few thousand. The Maya number system is the best example that numbers were invented for a purpose: To match the measurement of time the Maya scientists did not adopt a place-value system based on multiple powers of a base but modified it to allow easier counting of days and years.
We know from Lecture 5 that the Maya number system used 20 as its base. An exact place-value system based on 20 would use the powers of 20: 1, 20, 202 (= 400), 203 (= 8000), 204 (= 160,000) etc. The Maya system had an irregularity in the third place; it used the sequence 1, 20, 18x20 (= 360), 18x202 (= 7,200), 18x203 (= 144,000) etc.
The use of 18 instead of 20 in the third place destroys the convenience of the place-value system for multiplication and division. But it brings its value 360 close to the number of days in the year. It is remarkable that the Maya allowed the calendar problem to determine the properties of their number system.
Maya timekeeping employed three different measures of time. The civil or administrative calendar had 18 months of 20 days each. This accounted for 360 days of the year. 5 days were added at the end of the 360 days. They were considered "bad" days, and anyone born on such a day could expect lifelong misfortune.
The religious calendar consisted of a combination of the 20 days with the sequence from 1 to 13. The number 13 is a prime number, and if the two sequences 1 ... 13 and 1 ... 20 are run in parallel they repeat after 13 x 20 = 260 days. The religious calendar thus had a length of 260 days, and each day in it was uniquely defined by its name day and number between 1 and 13.
The reason for the adoption of 13 is unclear. The Maya believed that above the earth were 13 spheres, each inhabited by a number of gods, so 13 must have been a sacred number. But whether this is the reason for using it in a calendar, or whether 13 was the result of mathematical analysis of longer astronomical cycles cannot be assessed today.
When the civil and religious calendar are employed in parallel, a simple calculation shows that the same combination of civil and religious date occurs every 52 years (18,980 days). It is thus possible to unequivocally fix any day within a 52 year period by giving its civil and religious date. If the movement of Venus is also recorded, this period can be extended to 104 years, since two periods of 52 years correspond nearly exactly to an integer number (65) of Venus years.
Unlike most other early civilizations, which did not show much interest in history and therefore did not employ a continuous year count to identify dates, the Maya civilization was well aware of its history and recorded all its achievements and major events. A system in which dates repeat themselves every 52 or 104 years is obviously not sufficient for keeping accurate historical records. Maya priests therefore used a third method of timekeeping, the continuous day count from a defined starting date. The reason for the choice of the starting date is not known, but it is generally believed to be the Gregorian date 12 August 3113 BC.
The Maya civilization flourished for several hundred years, so the continuous day count soon grew to very large numbers. Maya astronomer-priests therefore had good reason to reflect on infinity. Their day count was the very reason for their place-value number system:
| name | factors | value (days) |
| kin day |
1 | 1 |
| uinal = 20 kin month |
20 | 20 |
| tun = 18 uinal year |
18 x 20 | 360 |
| katun = 20 tun 20 year cycle |
18 x 202 | 7,200 |
| baktun = 20 katun 400 year cycle |
18 x 203 | 144,000 |
| pictun = 20 baktun 8,000 year cycle |
18 x 204 | 2,880,000 |
| calabtun = 20 pictun 160,000 year cycle |
18 x 205 | 57,600,000 |
| kinchiltun = 20 calabtun 3,200,000 year cycle |
18 x 206 | 1,152,000,000 |
| alautun = 20 kinchiltun 64,000,000 year cycle |
18 x 207 | 23,040,000,000 |
We can now adjust the inaccurate depiction of the Maya date on the stelae of Quiriguá and establish that it was erected on 24 January 771 AD.
Did the Maya priests really expect the Maya empire to last 23 billion days (63 million years)? It would surprise me. It is much more likely that, having developed a number system that could take them into the tens of thousands, they would have felt curious to see how that system continues into larger and larger numbers. The Dresden Codex does indeed contain some very long numbers that go well beyond the normal day count and have not been fully explained. There is no doubt that Maya scientists explored the world of mathematics further than required for their daily needs. But there can also be no doubt that their number system arose from the need of society to manage agriculture. Scientific curiosity often expands a capability of science, developed initially in response to a need, into areas of no immediate use.
The Maya calendar was the prototype for all Central American civilizations. The Aztecs had an inferior number system and no script, but their famous calendar stone depicts the same arrangement of religious and civil calendars and day counts.
The civil calendar and the day count have been abandoned, but the religious calendar is still in use. In the same way as Roman Catholic parents often give their children the names of the saints associated with their children's birthdays, Maya parents in present-day Guatemala name their children after the day of the religious calendar on which their children are born.
The calendar of Babylon is the oldest of all known calendars, and its development is a good demonstration of the movement away from empirical observations towards scientific calculation. Before the invention of the place-value number system Mesopotamia used a lunar calendar in which every month began with the sighting of the New Moon. In the city kingdoms declaring the beginning of a new month was the task of the court astronomer, who had to report every first sighting of the New Moon to the king.
The development of civilization meant regular annual grain deliveries as taxes to support the administration and the need to establish a financial year for bookkeeping and accounting purposes. Thus, the structure 30 x 12 = 360 days was already in office use before 2,400 BC, but everyone else relied on the announcements of the New moon from the court.
Shortly after 2,100 BC some cities replaced the regulation of the calendar through direct observation of the New Moon by a lunar calendar of alternating 29 and 30 days. This calendar had 354 days in a year and had to be somehow aligned with the agricultural year of 365 days. This was achieved by adding ("intercalating") an additional month when the difference between the solar year and the lunar calendar became too large.
The decision for intercalation was the prerogative of the people in power and did not occur uniformly across the region, so that different cities operated on different calendars. As late as 541 BC the declaration of an intercalation was done by royal edit. The importance of the prerogative to declare or not declare an intercalation should not be underestimated. Agricultural accounts were settled annually, and debts had to be repaid in a certain month after the harvest. Intercalating a month thus defers debt repayments by one month, and attempts to obtain the declaration of an intercalation through bribery occurred regularly.
By about 430 BC the Greek astronomer Meton, who worked in Athens, Europe's intellectual centre at the time, discovered that 235 lunar months correspond very closely to 19 solar years (the so-called Metonic cycle). Some fifty years later the astronomers of the Persian kings, who used the Babylonian calendar, heard of his discovery and used it to regulate the intercalations, thus synchronizing the lunar clendar with the solar year. Since 19 years of 12 months amount to 228 months, the additional 7 months required for synchronization were intercalated in years 3, 6, 8, 11, 14, 17 and 19. This turned the occurrence of intercalations into an objective event and ended all attemps at extending credit by bribing the court astronomer.
The lunar calendar survives in the Jewish religious calendar. The Jews adopted the Babylonian calendar during the period of Babylonian exile in the 6th century BC. Although a succession of foreign occupation brought new civil calendars to Judaea (the Macedonian calendar under Alexander the Great and possibly the Roman calendar later), religious holidays continued to be regulated through the Babylonian lunar calendar, which required observations of the New Moon and the stand of the crops. Jewish communities around the Mediterranean Sea and elsewhere had to be instructed about the dates of festivities for the coming year by courier from Jerusalem.
This method was in operation from before 143 BC until about 200 AD, when the synchronization of the lunar and solar calendar was finally done through astronomical calculation. The method of calculation was kept secret, however, and couriers were still required to inform the Diaspora. When people in the Diaspora grew tired of waiting for couriers and began to set their own festival dates, the "secret" formula was made public in 359 AD - and turned out to be the Babylonian 19 year cycle of 380 BC with some modifications determined by religious needs.
Today's Jewish calendar is still based on the Babylonian 19 year cycle with intercalated months in years 3, 6, 8, 11, 14, 17 and 19. This places it in competition with the Hindu calendar as the most complicated calendar in use today and means that Jewish religious holidays move significantly back and forth with respect to the civil calendar.
Egypt also used lunar months but based its regulation with the seasons not on the sun but on the star Sirius (called Sothis in Egypt). The year determined by the seasonal appearance of Sirius is very close to the solar year. (It is 12 minutes shorter.)
The Nile was and is the lifeline of Egypt, and its annual floods required a calendar based on the seasons. The Egyptians therefore introduced a year of 12 months, each of 30 days, plus 5 intercalated days before the year's end, giving a total of 365 days. Such a calendar is of course neither controlled by the lunar months nor by the sun but an approximation to both and will fall behind the true solar year by one day every four years. It was used for administration, while the lunar calendar continued to be used in agriculture and daily life.
A loss of one day every four tropical years equates to the loss of one whole Egyptian calendar year every 1,460 tropical years, known as the Sothic cycle. The Egyptian astronomers recognized this very early. They tried to overcome the problem by introducing a second lunar calendar, which was based on control by the solar cycle. At the time of the Roman occupation in the first century BC Egypt was thus using three parallel calendars.
This rather complicated state of affairs bore, however, one important lesson: That any calendar should be synchronized with the tropical year and that the lunar month should be abandoned. This lesson was learnt by Julius Caesar and is Egypt's contribution to our own calendar. It could not be reached without the advanced astronomical skills of the Egyptians, on which Caesar relied when he designed the Julian calendar.
One might argue that when Caesar embarked on his calendar reform synchronization with the solar year had already been achieved in Persia. But the Metonic cycle achieves synchronization through the intercalation of months, which is a much cruder method than the Egyptian intercalation of single days. The Egyptian calendar laid the basis for a calendar where every year has the same number of months and differs in length from the previous and following years by no more than a single day.
The development of the Indian number system suggests that Indian astronomy was very advanced. In the context of the history of science and civilization the traditional Hindu calendar nevertheless stands out as an example of time division governed by religious control. Documents from about 1000 BC testify that the Hindu calendar consisted of 12 lunar months with an intercalation of an additional month every 60 months. According to the Vedas, India's early religious scriptures written between 1500 BC and 800 BC, the length of each month was 27 days. (The Vedas differ on this point; the Atharvaveda, the fourth of the Vedas, gives 28 days.) Keeping track of true time was achieved through observations of the passage of the Sun and the Moon through the constellations of the ecliptic.
Although the system suggests the application of advanced mathematics and astronomy, it was supported by an elaborate system of horoscopes and astrology and gave rise to an associated division of time into lunar and solar units that operate parallel to each other. The solar day is divided into 30 muhurta (equivalent to 48 minutes), each muhurta into two ghatik (24 minutes) and each ghatik into 3600 vipala (0.4 seconds).
At the same time the lunar month (which has about 29.5 solar days) is divided into 30 lunar days called tithi which are thus shorter (by about 24 minutes) than the ordinary day and not tied to daylight and night; a tithi can begin at any time of day. To keep solar and lunar days together, occasionally a tithi is eliminated from the month. Determining the date in the Hindu calendar is therefore quite complicated and requires the knowledge of the time of sunrise on that day.
Another cycle, introduced about 600 AD, is based on the movement of Jupiter around the sun. Jupiter's movement with respect to the stars has a period of nearly 12 years. This phenomenon was used to count "decades" of 12 years and "centuries" of 60 years.
Modern India has of course adopted the Gregorian calendar for administration and commerce. But the traditional Hindu calendar is still in use for horoscopes and governs all religious festivities. It is a living demonstration how advanced science can be sidelined by other interests of society.
Chinese astronomers had established the solar year as 365.25 days and the lunation as 29.5 days well before 1300 BC. Their calendar used 12 lunar months of 29 or 30 days with an intercalation of 7 months of 29 or 30 days over a 19 year cycle (the Metonic cycle). The scientific description of this cycle appeared in Chinese texts between 770 and 476 BC and predates its discovery by Meton by at least 100 years.
Intercalation of a month at the end of years synchronizes the lunar and solar years but does not prevent individual months from moving back and forth against the true (climatological) seasons. To achieve better synchronization with the seasons the Chinese trialed a system to insert the additional month between any suitable months of the year as early as under the Shang dynasty (about 1760 to 1120 BC). But their astronomical knowledge or mathematical skills were not good enough to do this, and the trial was abandoned.
At about 300 BC science had progressed sufficiently to allow another attempt of the system. Astronomers determined 24 season points on the ecliptic. It takes the sun 15.2 days to travel from one season point to the next. The months of the Chinese calendar were lunar months and alternated between 29 or 30 days, so most months contain two season points but some only one, which triggers the intercalation of an additional month.
This method eliminated the arbitrariness of the earlier trial while greatly reducing the movement of the months against the season points. It effectively established an accurate solar control over a lunar calendar through scientific methods and is a testament to the advanced state of Chinese astronomy, which was superior to astronomy practiced in other parts of the world until at least 1200 AD. The Chinese yin-yang li (lunar-solar calendar) remained the official calendar of China until 1912 (when its use in government was replaced by the Gregorian calendar) and is still used by the general public today.
Many traditional Chinese scholars tried to improve the synchronization of the lunation and the solar year and published astronomical almanacs to that effect. At least 102 kinds of almanacs were known, and some were used regularly. The position of calendar master at the Imperial court was a highly regarded post. It was also a risky employment, because mistakes in the calendar master's almanac usually resulted in punishment, including death.
The calendar that is now the international standard in commerce, administration and science developed from the Roman civilization, but its scientific roots are in Egypt. The original local Roman calendar that existed at Rome's foundation in the 8th century BC was a crude lunar calendar with 10 months, six of which had 30 days and the other four 31, giving a total of 304 days. The year started with March and ended in December, followed by a winter gap without name and order. The months January and February were introduced around 700 BC and the number of days redistributed among the months, and the new year of 12 months had 355 days. An additional 22 or 23 days were intercalated every second year.
In theory this arrangement should result in 4 x 355 + 22 + 23 = 1,465 days over 4 years, or an average 366.25 days per year. This was one day longer than the tropical year and demonstrates the poor state of Roman science before it came into closer contact with the Greek world. In practice the situation was much worse because the length and frequency of intercalations was controlled by a board of the chief magistrate (the Pontifices), who kept the reasons for their decision secret. Commercial and political interests thus had ample opportunity to influence the calendar to their advantage through bribery and corruption, and by 50 BC the vernal equinox occurred in mid-May instead of late March.
To eliminate the possibility of political interference and improve the length of the calendar year emperor Julius Caesar decided to establish a mathematical rule for intercalations and employed the Greek astronomer Sosigenes to implement the calendar reform. Sosigenes had worked in Alexandria and was thus familiar with the solar-based Egyptian calendar. He recommended that the lunar calendar be abandoned, that the months be arranged on the basis of the seasons and that the Egyptian year length of 365.25 days be adopted.
Sosigenes' recommendation was accepted, and through imperial decree Caesar brought the calendar back into line with the seasons and allocated alternately 30 and 31 days to the months. February received 29 months, and an additional day was intercalated every fourth year (now known as the "leap year"). The new calendar became known as the Julian calendar.
The new system got off to a bad start. The Pontifices misunderstood Caesar's directives and declared every third year a leap year. The error went unnoticed for 36 years, when Emperor Augustus corrected it by skipping some leap years. By that time the Senate of Rome, who had already honoured Julius Caesar by renaming the month of Quinctilis into Julius, proposed to rename the following month (Sextilis) into Augustus. The emperor did not like the idea that Caesar's month should have 31 days and his month only 30, so he changed the allocation of days to some months and gave both July and August 31 days. This arrangement is still in use today.
The Julian calendar became the standard calendar in Europe for several centuries. Despite its improved length of 365.25 days it is still longer than the tropical year (365.242199 days), and the calendar again slipped against the seasons as time went on. By the 16th century the error had reached 10 days and affected the determination of Christian holidays such as Easter. Discussions about an improved calendar were held in the Catholic Church for quite some time, and in 1545 a church council authorized Pope Paul III to take action.
The task to determine the length of the tropical year with the necessary accuracy was a challenge for medieval science, and it took over 30 years before Pope Gregory XIII could issue a papal bull in 1582 that introduced what became known as the Gregorian calendar. This calendar, which has become the world standard today, is based on a year of 365.2422 days. It differs from the Julian calendar by 3.12 days every 400 years. As a consequence, the Gregorian calendar skips 3 of the Julian leap years for every 400 year period. This is achieved by ruling that centennial years can only be leap years if they are fully divisible by 400. (Thus, 2000 was a leap year but 1900 was not.)
From the scientific point of view the introduction of the Gregorian calendar was the latest development in the calendar problem. Our discussion demonstrated that its design was not without political and religious interference, and the irregular sequence of the number of days in each month in our present-day calendar is entirely the result of a political edict. The credit for its major scientific strength - exclusive reliance on the solar year - has to go to Egypt: It is the solution to the problem that the Egyptian astronomers had recognized but had been unable to solve.
Other calendars are the result of religious or political decisions but did not add new scientific insight. The Muslim calendar introduced a new year count by setting the beginning of year 1 of the Muslim era to 16 July 622, the year when Prophet Mohamed moved from Mecca to Medina. Originally the Muslim calendar was a lunar calendar with 12 months of alternately 29 and 30 days, making a year length of 354 days. Because there were no intercalations the months moved through the seasons once every 32.5 years. This calendar is still used in Islamic countries to regulate family life and religious affairs and is the official calendar in Saudi Arabia. As other calendars with better scientific foundation developed, some Muslim countries adjusted their calendar by introducing the solar month but keeping the Muslim year count. Turkey introduced the Julian months and leap years in 1677 and the Gregorian months and leap years towards the end of the 19th century but kept the Muslim year count and the beginning of the year at March. In the 20th century it changed to the Gregorian calendar. Iran adopted the Gregorian months and leap years around 1925 - 1940 but kept the Muslim era and the beginning of the year at 21 March. Thus the Iranian year 1382 began on 21 March 2003 of the western year count.
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