Celestial Timekeepers

May 15, 2006

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)

1. 60 year Jovian cycle/ 360 year ‘divine cycle

2. 2700 year cycle of the Sapta Rishi or the Ursa Major

3. 27000 year cycle of the asterisms called the Great Year or the

precession cycle

4. 432,000 year cycle called a yuga (= duration of Kaliyuga)

5. 4,320,000 year cycle known as the Maha Yuga

6. Kalpa, the cycle consisting of 4.32*10**9 years

Table 1

Cosmology and Numerology

A bit of trivia – all of the numbers in Table 1 are divisible by 9

except the 60 year cycle.The number 9 and its multiples have a

mystic significance in the Vedic tradition.

One way of visualizing the number 9 is as follows

The universe is constituted of 3 factors – time, space, and

causation

The universe is constituted of 3 Gunas (ingredients) – Sattva,

rajas, and tamas

The universe is constituted of the three functions – creation,

preservation, and destruction

Thus 3 times 3 makes 9

Two times 9 makes 18, the number of chapters in the Bhagavad Gita

The 18 portions (parvas ) of the Mahabharata epic define in

detail the career of man on earth.

Thee are 18 days of warfare in the Great Bharata War

There are a total of 18 divisions in the Mahabharata war. 7

divisions on the Pandava side and 11 on the Kaurava side.

The Mahabharata war is thus an exposition of the human

possibilities and achievements graded into eighteen categories, the

first multiple of 9

Figure1 The celestial sphere showing the ecliptic and its

inclination to the celestial equator

ecliptic क्तांतीव्रुत्त (Kranthivruth)

(ēklĬp´tĬk, Ĭ-) , 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 वसंत संपत (Vasanth Sampat) Vernal equinox

(ē´kwĬnŏks) , 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

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 Indian calendar 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:

Chaitra

Vaishakh

Jeshta

Ashadh

Shrawan(Sawan)

Bhadrapad(Bhado)

Ashwin

Kartik

Margshirsh

Paush

Magha

Falgoon (Fagan)

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,

English calendar weekdays

Indian calendar weekdays

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Raviwar

Somwar (Chandrawar)

Mangalwar

Budhwar

Guruwar

Shukrawar

Shaniwar

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

Sl.No

Krsna paksa

(dark fortnight)

Waning moon

Gaura or shukla paksa

(bright fortnight)

Lightening moon

Deity and properties

1

Pratipat

The presiding deity of the first lunar day in Brahma and is good

for all types of auspicious and religious ceremonies

2

Dvitiya

Vidhatr rules this lunar day and is good for the laying of

foundations for buildings and other things of a permanent nature.

3

Trtiya

Visnu is the lord of this day and is good for the cuttings of

one's hair and nails and shaving.

4

Caturthi

Yama is lord of the 4th lunar day, which is good for the

destruction of one's enemies, the removal of obstacles, and acts of

combat.

5

Pancami

The Moon rules this day, which is favourable for administering

medicine, the purging of poisons, and surgery.

6

Sasti

Karttikeya presides over this day and is favourable for

coronations, meeting new friends, festivities, and enjoyment.

7

Saptami

The 7th lunar day is ruled by Indra; one may begin a journey,

buy conveyances, and deal with other such things as a movable

nature.

8

Astami

The Vasus rule this day, which is good for taking up arms,

building of one's defenses, and fortification.

9

Navami

The Serpent rules this day, with is suitable for killing

enemies, acts of destruction, and violence.

10

Dasami

The day is ruled by Dharma and is auspicious for acts of virtue,

religious functions, spiritual practices, and other pious

activities.

11

Ekadasi

Rudra rules this day; fasting, devotional activities, and

remembrance of the Supreme Lord are very favourable.

12

Dvadasi

The Sun rules this day, which is auspicious for religious

ceremonies the lighting of the sacred fire, and the performance of

one's duties.

13

Trayodasi

The day is ruled by Cupid and is good for forming friendships,

sensual pleasures, and festivities.

14

Caturdasi

Kali rules this day suitable for administering poison and

calling of elementals and spirits.

15

Amavasya

(new moon)

Purnima

(full moon)

The Vasve-devas rule the New Moon suitable for the propitiation

of the Manes and performance of austerities.

Zodiac sign

Sanskrit Name

a

Sector end

Aries

Mesha

00

30

Taurus

Vrishabha

30

60

Gemini

Mithuna

60

90

Cancer

Karka

90

120

Leo

Simha

120

150

Virgo

Kanya

150

180

Libra

Tula

180

210

Scorpio

Vrishchika

210

240

Sagittarius

Dhanus

240

270

Capricorn

Makara

270

300

Aquarius

Kumbha

300

330

Pisces

Meena

330

360

The Tropical Zodiac

Ecliptic, Tropical Zodiac and the Sidereal Zodiac

9 degrees to either side of the Ecliptic is a belt of the Heavens

known as the Zodiac. (Dante called it the Oblique Line that beareth

all planets).

First 30 degrees of the Zodiac constitute the sign of Aries. The

next 30 degrees Taurus and so on. The Zodiac counted from the first

degree of Aries to the 360th degree of Pisces is called the Tropical

Zodiac.

These 12 signs are the limbs of the Cosmic Man or Time Eternal

(Kalapurusha- The Almighty Self as Time).

Aries is His head, Taurus His face, Gemini His neck, Cancer His

heart, Leo the place beneath, Virgo His belly, Libra His generative

organs, Scorpio the place beneath, Sagittarius His upper thigh,

Capricorn his lower thigh, Aquarius His leg and Pisces His feet!

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

⅓ 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.