In this chapter, we use formulas from Section 1.12 to cast a number of the calendars presented in Part I into a unified framework. Years must be determined by the occurrence of some “critical” mean annual event, like a mean equinox or mean solstice. Months must also follow a uniform pattern. In single-cycle calendars, new years begin on the day the critical annual event happens before (or possibly at) some critical time of day. In double-cycle calendars, months begin on the day of a critical mensual event, and years begin with the month associated with the critical annual event.

Single Cycle Calendars

There are four “single-cycle” paradigms for calendars, as we independently allow

1. the determining critical annual event to occur strictly before, or to occur at or before, some critical time of day, and

2. the pattern of months to follow a fixed yearly pattern, according to equations (1.78)–(1.81) or to follow a mean monthly pattern, in tune with the yearly pattern.

The calendars in the second part of this book are based on accurate astronomical calculations. This chapter defines the essential astronomical terms and describes the necessary astronomical functions. Fuller treatment can be found in the references—an especially readable discussion is given in [9].

We begin with an explanation of how positions of locations on Earth and of heavenly bodies are specified, followed by an examination of the notion of time itself.

The French Revolutionary calendar (Le Calendrier Républicain)was instituted by the National Convention of the French Republic in October 1793. Its epoch is R.D. 654,415, that is, Saturday, September 22, 1792 (Gregorian), the day of the autumnal equinox of that year and also the first day following the establishment of the Republic. The calendar went into effect on Sunday, November 24, 1793 (Gregorian) and wasused by the French until Tuesday, December 31, 1805 (Gregorian); on Wednesday, January 1, 1806 (Gregorian), the Revolutionary calendar was abandoned by Napoleonic edict and France reverted to the Gregorian calendar, but the Revolutionary calendar was used again during the “Paris Commune” of May 6–23, 1871 (Gregorian), an insurrection that occurred after the collapse of Napoleon III's Second Empire.

Calendrical calculations are ubiquitous. Banks need to calculate interest on a daily basis. Corporations issue paychecks on weekly, biweekly, or monthly schedules. Bills and statements must be generated periodically. Computer operating systems need to switch to and from daylight saving time. Dates of secular and religious holidays must be computed for consideration in planning events. Most of these calculations are not difficult because the rules of our civil calendar (the Gregorian calendar) are straightforward.

Calendar

Several calendars are in use in Tibet. In this chapter we discuss the official Phugpa or Phukluk version of the Kālacakra (“Wheel of Time”) calendar, derived from the Kālacakra Tantra, translated into Tibetan from the Sanskrit in the eleventh century, used by the majority of Tibetans today, and sanctioned by the Dalai Lama. (The other widely used version is the Tsurphu.) The calendar is similar to the Hindu lunisolar calendars, somewhere between the arithmetic simplicity of the old Hindu version, and the astronomical complexity of the modern Hindu. There are also regional variants, because the calculated astronomical events are in terms of local time. The Bhutan, Mongolian, and Sherpa calendars are very similar.

The modern Persian calendar, adopted in 1925, is a solar calendar based on the Jalālī calendar designed in the eleventh century by a committee of astronomers, including a young Omar Khayyām, the noted Persian mathematician, astronomer, and poet. The Jalālī calendar had 12 months of 30 days each, followed by a 5-day period (6 in leap years), just like the Coptic/Ethiopic calendar described in Chapter 4. In addition to the Jalālī calendar, the Zoroastrian calendar, whose structure is described in Section 1.9, was also used historically in Persia.

Today, numerous calendars are used in India for different purposes. The Gregorian calendar is used by the government for civil matters; the Islamic calendar is used by Moslems; the Hindus employ both solar and lunisolar calendars. Indeed, there are over 30 variations of the Hindu calendar in active use. In March 1957, an attempt was made to revise the traditional calendar to follow the pattern of the Gregorian leap year structure [1]. The proposed reform has not, however, been widely accepted, though the new, National Calendar dates appear in published calendars.

The best known of several related systems used on the Indian subcontinent is the classical Hindu calendar of the (Present) Sūrya-Siddhānta (circa 1000), said to have been revealed to Asura Maya the Assyrian at the end of the last ‘Golden Age,’ in the year 2,163,154 B.C.E.

The Chinese calendar is a lunisolar calendar based on astronomical events, not arithmetical rules. Days begin at civil midnight. Months are lunar, beginning on the day of the new moon and ending on the day before the next new moon. Years contain 12 or 13 such months, with the number of months determined by the number of new moons between successive winter solstices. The details of the Chinese calendar have varied greatly—there have been more than 50 calendar reforms—since its inception in the fourteenth century B.C.E.; some of its history, in particular its effect on the development of mathematics in China, is described in [13]; other historical details can be found in [5] and [16].

The calculation of the date of Easter has a fascinating history, and algorithms and computer programs abound (for example, [1], [2], [9], [10], [13], and [14]). Many of the computations rely on the formulas of Gauss [5], [6] (see also [8]). Our fixed-date approach allows considerable simplification of “classical” algorithms.

The history of the establishment of the date of Easter is long and complex; good discussions can be found in [3] and [7]. The Council of Nicsa convened in 325 C.E. by Constantine the Great, was concerned with uniformity across various Christian groups. At the time of Nicæa, almost everyone in the official Church agreed to the definition that Easter was the first Sunday after the first full moon occurring on or after the vernal equinox [3] (a rule promulgated by Dionysius Exiguus and the Venerable Bede, who attributed it to the Council of Nicæa).

Structure

The Bahá'í (or Badī') calendar begins its years on the day of the vernal equinox. Theoretically, if the actual time of the equinox occurs after sunset, then the year should begin a day later [2]. Current practice in the West, however, is to begin on March 21 of the Gregorian calendar, regardless. The theoretical, astronomical version of the Bahá'í calendar is described in Section 15.3. The calendar is based on the 19-year cycle 1844–1863 of the Bāb, the martyred forerunner of Baha'u'llāh and co-founder of the Bahá'í faith.

As in the Islamic calendar, days are from sunset to sunset. Unlike the Islamic calendar, years are solar; they are composed of 19 months of 19 days each with an additional period of 4 or 5 days after the eighteenth month. Leap years in the West follow the same pattern as in the Gregorian calendar.