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Vedic mathematicians in Ancient India (part III)

and the Celestial Timekeepers

Kosla Vepa Ph.D

1.Introduction

The Ancient Vedics seemed to have an obsession for precision as well as a fascination for large numbers. A combination such as this makes an excellent prerequisite for time keeping and for devising a useful and practical calendar. So, they turned to the sky and began to decipher the meaning behind the various cycles they observed. Let us see how they went about developing a calendar that would convey a lot of information merely by knowing the day of the month after constant observation of the sky both during the day and the night over centuries.

The basic information they used for purposes of time keeping were the motions of the sun and the moon relative to the earth. So far nothing unusual, as did all the other ancients. The cycles they used apart from the day, the week, the fortnight, and the month are shown in Table 1.

2. Some Definitions

Let us establish the coordinate systems first. Everyday the celestial sphere appears to turn as the earth rotates, causing the daily rising and setting of the sun, stars and other celestial objects. (vide Figure 1)

Table 1

Figure 1 The celestial sphere showing the ecliptic and its inclination to the celestial equator

ecliptic  ?????????????? (Kranthivruth )

(eklIp´tIk, I-) , the great circle on the celestial sphere that lies in the plane of the earth's orbit (called the plane of the ecliptic). Because of the earth's yearly revolution around the sun, the sun appears to move in an annual journey through the heavens with the ecliptic as its path. The ecliptic is the principal axis in the ecliptic coordinate system . The two points at which the ecliptic crosses the celestial equator are the equinoxes. The obliquity of the ecliptic is the inclination of the plane of the ecliptic to the plane of the celestial equator, an angle of about 23 1/2 °. The constellations through which the ecliptic passes are the constellations of the zodiac .

equinox  ???? ???? (Vasanth Sampat) Vernal equinox

(e´kwInoks) , either of two points on the celestial sphere where the ecliptic and the celestial equator intersect. The vernal equinox, also known as “the first point of Aries,” is the point at which the sun appears to cross the celestial equator from south to north. This occurs about Mar. 21, marking the beginning of spring in the Northern Hemisphere. At the autumnal equinox, about Sept. 23, the sun again appears to cross the celestial equator, this time from north to south; this marks the beginning of autumn in the Northern Hemisphere. On the date of either equinox, night and day are of equal length (12 hr each) in all parts of the world; the word equinox is often used to refer to either of these dates. The equinoxes are not fixed points on the celestial sphere but move westward along the ecliptic, passing through all the constellations of the zodiac in 26,000 years. This motion is called the precession of the equinoxes . The vernal equinox is a reference point in the equatorial coordinate system

equatorial coordinate system

the most commonly used astronomical coordinate system for indicating the positions of stars or other celestial objects on the celestial sphere . The celestial sphere is an imaginary sphere with the observer at its center. It represents the entire sky; all celestial objects other than the earth are imagined as being located on its inside surface. If the earth's axis is extended, the points where it intersects the celestial sphere are called the celestial poles; the north celestial pole is directly above the earth's North Pole, and the south celestial pole directly above the earth's South Pole. The great circle on the celestial sphere halfway between the celestial poles is called the celestial equator; it can be thought of as the earth's equator projected onto the celestial sphere. It divides the celestial sphere into the northern and southern skies. An important reference point on the celestial equator is the vernal equinox , the point at which the sun crosses the celestial equator in March. To designate the position of a star, the astronomer considers an imaginary great circle passing through the celestial poles and through the star in question. This is the star's hour circle , analogous to a meridian of longitude on earth. The astronomer then measures the angle between the vernal equinox and the point where the hour circle intersects the celestial equator. This angle is called the star's right ascension and is measured in hours, minutes, and seconds rather than in the more familiar degrees, minutes, and seconds. (There are 360 degrees or 24 hours in a full circle.) The right ascension is always measured eastward from the vernal equinox. Next the observer measures along the star's hour circle the angle between the celestial equator and the position of the star. This angle is called the declination of the star and is measured in degrees, minutes, and seconds north or south of the celestial equator, analogous to latitude on the earth. Right ascension and declination together determine the location of a star on the celestial sphere. The right ascensions and declinations of many stars are listed in various reference tables published for astronomers and navigators. Because a star's position may change slightly (see proper motion and precession of the equinoxes ), such tables must be revised at regular intervals. By definition, the vernal equinox is located at right ascension 0 h and declination 0°.

 Another useful reference point is the sigma point, the point where the observer's celestial meridian intersects the celestial equator. The right ascension of the sigma point is equal to the observer's local sidereal time . The angular distance from the sigma point to a star's hour circle is called its hour angle ; it is equal to the star's right ascension minus the local sidereal time. Because the vernal equinox is not always visible in the night sky (especially in the spring), whereas the sigma point is always visible, the hour angle is used in actually locating a body in the sky.

3. Calendars and Tithis

Like most Asian calendars Indian calendars do not employ the solar year and day (i. e. tropical year and solar day) but the sidereal year, and the Synodic month(29.5306 days). Thus, the calendric year based on the sidereal year is defined as the time between two successive passes of the sun through a certain star's circle of declination. Lunar days and sidereal months are also used, and in certain lunisolar calendars lunar year and lunar month are taken into account, too.

Astronomical knowledge of Ancient India was written down in scientific treatises, called Siddhantas. In them, values for the lengths of months and years were given representing the latest knowledge at the time the Siddhanta was written. The values range from 365.258681 days in the Âryabhatiya to 365.258756 days in the Surya Siddhanta and are all too long compared with the modern sidereal year length of 365.25636 days. Nevertheless they are still in use for Indian calendars today.

The sidereal month is about two day shorter (27.3217) than the Synodic month

4. Meaning of Tithi 

According to the Indian calendar or Panchanga, Tithi is a lunar date based on the rotation of the moon around the earth, and is one of the five important aspects of an Indian almanac (Panchanga – Panch means five and anga means parts). Most of the Indian social and religious festivals are celebrated on a date corresponding to the original Tithi.

The current calendar “date” that we are so familiar with in our daily life is heliocentric and is based on the rotation of the earth around the sun.  It takes the earth approximately 365 ¼ days to complete its rotation around the Sun. The calendar that most of us use today divides the 365 days of earth’s period of rotation around the Sun in twelve months.  The leap year, which occurs once every four years, accounts for ¼ day per year. 

Similar to the solar calendar, the lunar calendar is also popular and widely used in the Asian countries such as China, Pacific-rim countries, Middle East countries, and India. The lunar calendar, which is believed to have originated in India, has been around for a very long time, even long before the solar calendar. 

The lunar calendar is geocentric and is based on the moon’s rotation around the Earth.  The lunar month corresponds to one complete rotation of the Moon around the Earth.  Since this period of rotation of moon around the earth varies, the duration of lunar month also varies.  On average, the lunar month has about 29 ½ days, the period of the lunar Synodic orbit.  In addition to moon’s rotation around the earth, the lunar year is based on earth’s rotation around the Sun.  In general, the lunar year has twelve lunar months of approximately 354 days (29.5 *12 ), thus making it shorter by about 11 days than the solar year.  However, the lunar calendar accounts for this difference by adding an extra lunar month about once every 2 ½ years.  The extra lunar month is commonly known as “Adhik Mas” in India (Adhik means extra and the Mas means month).  The concept of this extra month is similar to the “Blue Moon” in the West, which occurs almost with the same frequency of 2 ½ years. 

The Indian lunar year begins on the new moon day that occurs near the beginning of the Spring season. The twelve lunar months are: 

As mentioned earlier, to account for the difference between the solar and lunar year an extra lunar month occurs about every 2 ½ years as “Adhik Mas”.[1][1]

According to the Moslem calendar which is widely followed in Middle East and in other Moslem countries the lunar year is strictly based on twelve lunar months of 354 days per year.  That’s why their holy month of Ramadan occurs by approximately 11 to 12 days earlier than that in the preceding year.

The solar day (commonly referred as the “the date” in western calendar) has a fixed length of 24 hours. The change of date occurs at midnight as per local time or standard time of a given local time zone.  Thus, the date changes from midnight to midnight.  Similarly the day (as in weekdays) changes from midnight to midnight as per local or standard time for that location.  In other words, as per the western (or English) calendar the length of day and date is exactly 24 hours, and there is a definite correspondence between the date and the corresponding day of the week.

A lunar day usually begins at sunrise, and the length of lunar day is determined by the time elapsed between the successive sunrises.  As per the Jewish calendar their lunar day begins at the sunset, and lasts through the next sunset.  A lunar day is essentially the same as a weekday. In India the lunar day is commonly referred as “War”.  Just as the English calendar has seven days for a week, the Indian calendar has seven wars for a week. Thus,

The lunar day, however, varies approximately between 22 to 26 hours based on the angular rotation of moon around the earth in its elliptical orbit.  In the Indian calendar, the lunar date is referred as “Tithi”.  The basis for the length of a lunar date is geocentric and is defined as the angular distance between the sun and the moon as seen from the earth.  As the moon rotates around the earth, the relative angular distance between the sun and the moon as seen from the earth increases from 0 degrees to 360 degrees.  It takes one lunar month or about 29 ½ solar days for the angular distance between the sun and the moon to change from 0 to 360 degrees.  When the angular distance reaches zero, the next lunar month begins. Thus, at the new moon a lunar month begins, at full moon, the angular distance between the sun and the moon as seen from the earth becomes exactly 180 degrees.

The lunar cycle begins with crescent moon and the crescent phase lasts till that phase culminates in the full moon, typically lasting for about 15 days.  Then the moon enters in the waning phase until it disappears from the sky by lining up with the Sun.  The waning phase also lasts for about 15 days.  According Indian lunar month, the crescent lunar phase fortnight is called as “Shudha or Shukla Paksha” and the waning phase of the lunar cycle fortnight as “ Krishna Paksha”.  Thus, during Shudha (or Shukla) Paksha the angular distance between the moon and the sun varies from 0 degrees to 180 degrees while that during the  Krishna Paksha from 180 to 0 degrees.  If we divide 180 degrees into 15 equal parts, then each part becomes of 12 degrees in length.  Thus, this each twelve-degree portion of angular distance between the moon and the sun as it appears from the earth is the lunar date or Tithi.  Tithis or lunar dates in Shudha (or Shukla) Paksha begin with Prathama (first), Dwitiya (second), etc. till we reach the Poornima, the lunar date for full moon day.  Similarly for the waning fortnight lunar cycle or Wadya (or Krushna) Paksha, tithis begin again with Prathama (first), Dwitiya (second), etc. till we arrive Amavasya or a day before the new moon.  Thus when we refer to Ramnavami (the birthday of Rama), it’s the Navami (ninth lunar day) of Shudha Paksha of the lunar month Chaitra, or Chaitra Shudha Navami.  Similarly, the Gokulashtmi (also called as Janmashtami, the birthday of Krishna) occurs on Shrawan Wadya Ashtami (eighth lunar day of Wadya Paksha of the lunar month Shrawan).  

The angular velocity of moon in its elliptical orbit around the earth varies continuously as it is affected (according to Kepler’s Law) by the relative distance between the earth and the moon, and also by the earth’s relative distance from the sun.  As a result, the daily angular speed (the speed of the angular change between the moon and the sun as seen from the earth) varies somewhere between 10 to 14 degrees per  day. Since the length of a Tithi corresponds to 12 such degrees, the length of a Tithi also varies accordingly.  Therefore, a Tithi can extend over one day (24 hour period) or it can get sorteneded if two Tithis occur in one 24 hour day.

Since the angular distance between the moon and the sun as referred here is always relative to the entire earth, a lunar day or Tithi starts the same time everywhere in the world but not necessarily on the same day. Thus, when a certain Tithi starts at 10:30 PM in India it also begins in New York at the same time, which is 12 PM (EST) on the same day.  Since the length of a Tithi can vary between 20 to 28 hours, its correspondence to a War (a weekday) becomes little confusing.  

As per the Indian calendar, the Tithi for a given location on the earth depends on the angular distance between the moon and the sun relative to the earth at the time of sunrise at that location. Thus, for instance, assume on a November Monday sunrise in New York city occurs 8:30 AM (EST).  Further assume that at 9 AM (EST) on Monday the angular distance between the sun and moon is exactly 12 degrees just following the new moon of the Indian lunar month Kartik.  Since the length of a tithi is 12 degrees, the tithi, Kartik Shudha Dwitiya (second day) begins exactly at 9 AM on Monday of that November in New York.  However, at the time of sunrise on that Monday the tithi Dwitiya has not begun.  Therefore, the tithi for that Monday for city of New York is Kartik Shudha Prathama (first day).  

On the same Monday morning the sunrise in Los Angeles occurs well past 9 AM (EST).  Since the Tithi Dwitiya occurs everywhere in the world at the same instant, therefore, for Los Angeles, the Tithi for that Monday would be Karthik Shudha Dwitiya.

For the same Monday at 9 AM (EST), it would be 7:30 PM in Mumbai or New Delhi.  Thus, Tithi for that Monday for city of New York, Mumbai, and New Delhi is Karthik Shudha Prathama (the first day of Indian lunar month Karthik) while for most of the regions west of Chicago or St. Louis the Tithi for that Monday is Dwitiya.  In other words, the Tithi Karthik Shudha Prathama for regions west of Chicago or St. Louis should occur on the preceding day, the Sunday.  

Karthik Shudha Prathama (the first day of Indian lunar month Karthik) also happens to be the first day after Diwali.  Most of the Indians celebrate this as their New Year ’s Day. Indians living in India, Europe, and eastern part of the United States thus should celebrate their New Year on that Monday while regions west of Chicago should celebrate on the preceding day, the Sunday. (Based on description by  Jagdish C. Maheshri) October 12, 2000

[1] Adhik Mas occurs only when two amavasyas (no

5. The Clock, the Sidereal Zodiac, Nakshatras, and the Precession of the

 Equinoxes

The basis of the Hindu calendar calculation is Vedic[2]. This calendar has been modified and elaborated, but because it is based on the stars (Nakshatras) visible to the naked eye, and on the visible Lunar phases, it is more accurate than any others of the past. The actual moments when Lunar months begin can easily be checked by the regular appearances of Solar eclipses, and the middle moment of a Lunar month -- Poornima or full moon -- can similarly be verified by the more frequent Lunar eclipses. Hence the Hindu calendar, not requiring special instruments for its rectification, has maintained great accuracy for thousands of years.

The oldest calendar is probably the Vedic among the languages referred to as IE languages; at first lunar, later with solar elements added to it. The sister Avesta calendar is similarly first Lunar, but later only Solar. Both these calendars (the oldest in the IE universe) are influenced by the prehistoric calendars of the first and second root races at the North Pole and its surroundings, as they reckon with days and nights lasting six months. 

For untold ages, the Hindus have observed the motion of the moon, the sun and the seven planets along a definite path that circles our sky and is marked by fixed clusters of stars. The moon afforded the simplest example. These early astronomers observed that the moon, moving among these fixed star constellations which they called Nakshatras, returned to the same Nakshatra in 27.32166 days, the exact quantity determined by Aryabhatta, thus completing one Nakshatra month. They found it convenient to divide these groups of stars into 27 almost equal sections, or the 27 Nakshatras. By this method of reckoning, instead of giving the date of a month, as Western calendars do, the Hindus gave the name of the Nakshatra in which the moon was to be seen. (The moon is in each of these Nakshatras for approximately one day plus eighteen minutes.)

This scheme fitted nicely with the sun's cycle, for the Hindus noted that the sun traversed the same circle through the sky, but that it returned to its starting place only after 365.258756481 days, or what we call a Solar Sidereal Year. (Modern figures based on this Hindu figure quote 365.2596296 days -- a distinction without a difference, for ordinary purposes.) Now, having already divided the month into the 27 Nakshatras for the convenience of reckoning the moon's voyage through the heavens, what more natural than that these same Nakshatras should serve for the study of the Sun's course? Being in a circle of 360 degrees, each Nakshatra takes up 13  1/3 degrees of that circle. The Sun, moving about 1 degree in a day, is seen for 13  1/3 days in each Nakshatra. The system of reckoning according to the moon Nakshatras is current today that of the sun's being uncommon.

During the course of one day, the earth has moved a short distance along its orbit around the sun, and so must rotate a small extra angular distance before the sun reaches its highest point. The stars, however, are so far away that the earth's movement along its orbit makes a generally negligible difference to their apparent direction (see, however parallax), and so they return to their highest point in slightly less than 24 hours. A mean sidereal day is about 23h 56m in length. Due to variations in the rotation rate of the Earth, however, the rate of an ideal sidereal clock deviates from any simple multiple of a civil clock. The actual period of the Moon's orbit as measured in a fixed frame of reference is known as a Sidereal month, because it is the time it takes the Moon to return to the same position on the celestial sphere among the fixed stars (Latin: sidus): 27.321 661 days (27 d 7 h 43 min 11.5 s) or about 27 ? days. This type of month has appeared among cultures in the Middle East, India, and China in the following way: they divided the sky in 27 or 28 lunar mansions or Nakshatras, characterized by asterisms (apparent groups of stars), one for each day that the Moon follows its track among the stars.

The basis of the Hindu calendar calculation is Vedic. This calendar has been modified and elaborated, but because it is based on the stars (Nakshatras) visible to the naked eye, and on the visible Lunar phases, it is more accurate than any others of the past. The actual moments when Lunar months begin can easily be checked by the regular appearances of Solar eclipses, and the middle moment of a Lunar month -- Purnima or full moon -- can similarly be verified by the more frequent Lunar eclipses. Hence the Hindu calendar, not requiring special instruments for its rectification, has maintained great accuracy for thousands of years.

The oldest calendar is probably the Vedic among the languages referred to as IE languages; at first lunar, later with solar elements added to it. The sister Avesta calendar is similarly first Lunar, but later only Solar. Both these calendars (the oldest in the IE universe) are influenced by the prehistoric calendars of the first and second root races at the North Pole and its surroundings, as they reckon with days and nights lasting six months. 

For untold ages, the Hindus have observed the motion of the moon, the sun and the seven planets along a definite path that circles our sky and is marked by fixed clusters of stars. The moon afforded the simplest example. These early astronomers observed that the moon, moving among these fixed star constellations which they called Nakshatras, returned to the same Nakshatra in 27.32166 days, the exact quantity determined by Aryabhatta, thus completing one Nakshatra month. They found it convenient to divide these groups of stars into 27 almost equal sections, or the 27 nakshatras. By this method of reckoning, instead of giving the date of a month, as Western calendars do, the Hindus gave the name of the Nakshatra in which the moon was to be seen. (The moon is in each of these Nakshatras for approximately one day plus eighteen minutes.)

This scheme fitted nicely with the sun's cycle, for the Hindus noted that the sun traversed the same circle through the sky, but that it returned to its starting place only after 365.258756481 days, or what we call a Solar Sidereal Year. (Modern figures based on this Hindu figure quote 365.2596296 days -- a distinction without a difference, for ordinary purposes.) Now, having already divided the month into the 27 nakshatras for the convenience of reckoning the moon's voyage through the heavens, what more natural than that these same Nakshatras should serve for the study of the Sun's course? Being in a circle of 360 degrees, each Nakshatra takes up 13  1/3 degrees of that circle. The Sun, moving about 1 degree in a day, is seen for 13  1/3 days in each nakshatra. The system of reckoning according to the moon Nakshatras is current today, that of the sun's being uncommon.

In brief, then, the earliest method, the Vedic, of counting, was to name the moon through the various Nakshatras -- the circle or cycle repeating itself each Sidereal-Star-Month. Later the sun's place in the same Nakshatras was noted, the year ending when the Sun returned to the same Nakshatra. Then came the noting of the Solar and Lunar eclipses, and the observance of the New and Full Moons