Discovery of the Original Boundaries of Nakshatras

Zero Points of Vedic Astronomy

Part 1 of 8 — A Tale of two Coordinate Systems

It was little over three years ago that I came across a paper by Pingree and Morrissey about Indian astronomy in which they pronounce that all Indian astronomers were incompetent and did not know how to measure the position of the stars. Here are some quotes from this technical paper [1]:

We hope thereby to demonstrate three things; that there is no basis for identifying the stars included in the Vedic nakṣatras, and therefore no grounds for comparing them, for example, with the Chinese lunar mansions; that the catalogue of stars found in Paitāmahasiddhānta, which is almost exclusively the basis of the rest of the Indian tradition, since it is at the beginning of the Indian attempts to provide coordinates and uses a coordinate system derived from Greek astronomy, is more likely to be an Indian adaptation of a Greek star catalogue than to be based on observations that were made in India; and that the ineptitude with which Indians historically tried to ‘correct’ these coordinates militates against any theory that is founded upon the idea that the Indians of medieval period were experts in astronomical observation. … Our apparent success in finding “identifications” for Lalla’s star catalogue, wherein the coordinates are so clearly a mixture of ecliptic and polar values, shows the futility of attaching any credence to them. … Whichever stars the author of Sūryasiddhānta intended to indicate, he was incapable of determining their coordinates accurately, … It is most astonishing to see an astronomer convert λ* into λ and call the latter λ*; even more astonishing is to see him take λ to be λ*, convert it on that assumption into another λ, and to assert that this wrongly derived λ is λ*! There is no excuse for Āryabhaṭa’s coordinates. … The impression of incompetence does not disappear when we examine our last star catalogue, that which Gaṇeṣa incorporated into his Grahalāghava (XI 1–5) in 1520. … Therefore, either Gaṇeṣa was also incompetent, or he intended to give the coordinates of a different set of stars. … We must conclude from this survey that the Indians did not observe the positions of the stars with accuracy; by implication, they also did not observe those of the planets with accuracy.

As you can see, the interpretation of coordinates given in Indian astronomical texts as polar coordinates has resulted in Indian astronomers being called incompetent. When I read this paper, I knew that something is seriously wrong. I refused to believe that Indian astronomers were incompetent. I knew that there is much more to this than meets the eye. My first suspicion was that the colonial era historians have deliberately misinterpreted the data and used it to discredit Indian astronomers and the line of reasoning has been continued by scholars like late David Pingree. Could it be possible that the data was really in ecliptic coordinates that do not change with time and that is why Indian astronomers were not updating the coordinates of the stars? To understand this better, we will need to know the difference between polar and ecliptic coordinates. Let’s start by understanding how the coordinates of stars are defined.

There are many ways to define the coordinates of the stars. The principle is similar to how we define the longitude and latitude of any place on earth. Just as the longitude and latitude specify a place on earth, the equivalents of longitude and latitude specify the position of a star. Just as earth is considered a sphere (bhūgola), the sky is also considered a sphere called celestial sphere (khagola) of arbitrarily large radius. Its radius does not matter as we only deal with angles. To specify coordinates on a sphere, you need to specify a fundamental plane that passes through the centre of the sphere, which defines its poles which are at 90° from the fundamental plane. In case of earth, the choice of fundamental plane is obvious, its equator, and the poles are North Pole and South Pole. The latitude is defined as the angular distance north or south from the fundamental plane, the equator. To measure the longitude, a point is arbitrarily selected on the fundamental plane. Now on a sphere, all points are same, so how do you choose a point for measurement? This is simply by convention. In ancient times, that point was the intersection of meridian passing through Ujjain and the equator. This point in astronomical texts was called Laṅkā, which is different from the Laṅkā of Rāmāyaṇa. To distinguish these two Laṅkās, I will call the theoretical point Laṅkā of coordinate system as astronomer’s Laṅkā and the Laṅkā of Rāmāyaṇa as geographical Laṅkā. A meridian is a circle of constant longitude passing through the poles. The meridian that is used as the zero longitude is called Prime Meridian. In recent times the Prime Meridian passes through Greenwich, while in ancient India it passed through Ujjain.

When we come to defining the coordinates of stars or any astronomical body, the situation is far different than earth. While on earth there is obvious choice of the equator as fundamental plane, multiple choices exist for choosing a fundamental plane on the celestial sphere. The coordinate systems are named after the choice of fundamental plane and zero point of longitude. The most widely used celestial coordinate systems are Horizontal, Equatorial, and Ecliptic coordinate systems.

1. Horizontal coordinates

Horizontal coordinates are specified by providing altitude and azimuth. Figure 1 illustrates the horizontal coordinate system. In this picture, O is the location of the observer, ON is the north direction, OE is the east direction, OS is the south direction, OW is the west direction, Z is the zenith, S′ is the location of a star, and Z′ is the nadir. Z is the point on celestial sphere directly overhead the observer and Z′ is the point on celestial sphere directly underneath the observer. The great circle connecting N, E, W, and S is the horizon. The vertical circle passing through N, Z, and S is called the meridian. Altitude of the star is given by S′A and the azimuth of the star is given by NA. In horizontal coordinate system, the altitude is determined by measuring the angular distance from the horizon to the star along the great circle passing through the star and the zenith. Azimuth is determined by measuring the angular distance along the horizon from the north direction going eastward. In the horizontal coordinate system, the fundamental plane is the observer’s horizon and the zero point is the observer’s north point on the horizon, point N in Figure 1.

Figure 1: Illustration of horizontal coordinate system

2. Equatorial coordinates

Equatorial coordinate system measures the coordinates from and along the celestial equator, which is the projection of the earth’s equator on the celestial sphere. The poles of the celestial equator are called North Celestial Pole (NCP) and South Celestial Pole (SCP), which are the projections of the North Pole and South Pole on the celestial sphere respectively.

Equatorial coordinates are specified by providing declination and right ascension. Figure 2 illustrates the equatorial coordinate system. In this picture P is the North Celestial Pole (NCP), ♈ is the first point of Aries, S is the location of a star, and P′ is the South Celestial Pole (SCP). Declination of the star is given by SA and the right ascension of the star is given by ♈A. In equatorial coordinate system, the declination is determined by measuring the angular distance from the celestial equator to the star along the great circle passing through the star and the North Celestial Pole. Right ascension is determined by measuring the angular distance along the celestial equator from the first point of Aries to the intersection of the celestial equator and the great circle passing through the star and the North Celestial Pole.

In the equatorial coordinate system, the fundamental plane is the celestial equator and the zero point is first point of Aries, point ♈ in Figure 2.

Figure 2: Illustration of equatorial coordinate system

3. Ecliptic coordinates

Ecliptic coordinate system measures the coordinates from and along the ecliptic, which is a great circle on the celestial sphere representing the sun’s apparent path during a year. The poles of the ecliptic are called North Ecliptic Pole (NEP) and South Ecliptic Pole (SEP).

Ecliptic coordinates are specified by providing ecliptic latitude and ecliptic longitude. Figure 3 illustrates the ecliptic coordinate system. In this picture K is the North Ecliptic Pole (NEP), ♈ is the first point of Aries, S is the location of a star, and K′ is the South Ecliptic Pole (SEP). Ecliptic latitude of the star is given by SA and the ecliptic longitude of the star is given by ♈A.

In ecliptic coordinate system, ecliptic latitude is determined by measuring the angular distance from ecliptic to the star along the great circle passing through the star and the North Ecliptic Pole. Ecliptic longitude is determined by measuring the angular distance along the ecliptic from the first point of Aries to the intersection of the ecliptic and the great circle passing through the star and the North Ecliptic Pole.

In the ecliptic coordinate system, the fundamental plane is the ecliptic and the zero point is first point of Aries, point ♈ in Figure 3.

Figure 3: Illustration of ecliptic coordinate system

First point of Aries (♈) shown in Figures 2 and 3 is the point of intersection of celestial equator and ecliptic. This point is constantly changing in the background of stars due to the phenomenon of precession. So unlike GPS on earth, where the longitudes are always measured from Greenwich, in ecliptic coordinate system the zero point of longitude is constantly changing. It is therefore necessary to specify the year for ecliptic coordinate system when talking about ecliptic coordinates. Currently year 2000 CE is used to specify the ecliptic coordinates and the ecliptic coordinates are denoted J2000.0. This system is more specifically called tropical ecliptic coordinate system. Ancient Indians used an ecliptic coordinate system in which the ecliptic longitude was measured from a fixed star. This system is called sidereal ecliptic coordinate system. In this system the coordinates of stars do not keep changing similar to GPS on earth. If the Indian astronomers were using sidereal ecliptic coordinate system, then they would not have needed to update the coordinates of the stars. As we saw from the quote in the beginning of this article, it is believed that Indian astronomers were using polar coordinates, which change with time. Since they were not updating the coordinates, they are being called incompetent. So what exactly are polar coordinates?

4. Polar coordinates

Many Indian astronomy texts give the coordinates of stars, most prominent of them being Sūrya Siddhānta. It is currently accepted that the coordinates of yogatārās or junction stars given in Sūrya Siddhānta are polar longitudes and latitudes. The terms polar longitude and polar latitude were coined by Burgess in his translation of Sūrya Siddhānta [2], which uses the term Dhruvaka for longitude and Vikṣepa for latitude. Burgess has identified Dhruvaka and Vikṣepa as polar longitude and polar latitude respectively. The concept of polar coordinates of stars as illustrated by Burgess is shown in Figure 4. To determine the polar longitude and latitude of a star (S or S′), a segment of circle of declination (PSca or Pc′a′S) is drawn from North Celestial Pole (P) passing through the star up to the ecliptic. Polar latitude is the angular distance of the star (Sa or S′a′) from the ecliptic along the circle of declination. Polar longitude is the angular distance (La or La′) from reference point (L) on the ecliptic and the point of intersection of the ecliptic with the circle of declination passing through the star (a or a′).

Figure 4: Illustration of polar longitude and latitude of stars by Burgess [2]

A simpler illustration of polar coordinate system is shown in Figure 5. In this diagram, just like Figure 2, P is the North Celestial Pole (NCP), ♈ is the first point of Aries, S is the location of a star, and P′ is the South Celestial Pole (SCP). Just like Figure 3, K is the North Ecliptic Pole (NEP) and K′ is the South Ecliptic Pole (SEP). Polar latitude of the star is given by SA and the polar longitude of the star is given by ♈A.

Figure 5: Illustration of polar coordinate system

It should be noted that this whole geometrical construction for determining polar longitude and latitude is very artificial. In every coordinate system, the latitude is measured respective to the corresponding pole. In the artificial construct of polar latitude, the angular distance is measured from the ecliptic along the great circle that does not pass through the pole of ecliptic (North Ecliptic Pole), but passes through North Celestial Pole instead. Burgess has justified this artificial construction by taking the meaning of Dhruvaka as pertaining to Dhruva or pole star, and therefore he has drawn great circle passing through North Celestial Pole. In accordance with Dhruvaka, Burgess has postulated that Vikṣepa means polar latitude. However, this method of determining coordinates is not described in any Indian astronomical text.

So is it possible that coordinates given by Indian astronomers are sidereal ecliptic coordinates but are being interpreted as polar coordinates? If this were so, won’t it be obvious to anybody with knowledge of astronomy? I decided to investigate this further during the winter break of 2017. I started by assuming that these coordinates are sidereal ecliptic coordinates and with this assumption I could see in those numbers what nobody has seen before.

References:

1. Pingree, D. and Morrissey, P. (1989). On the identification of the “Yogatārās” of the Indian Nakṣatras, Journal for the History of Astronomy, 20.2: 99–119.

2. Burgess, E. (1860). Translation of the Surya-Siddhanta: A Text-Book of Hindu astronomy, with notes, and an appendix. Journal of the American Oriental Society, 6: 141–498 (information on page 319).

Email: rajarammohanroy108@gmail.com

More about the author

I am a seeker in search of the true history and heritage of India. I have strong scientific background (B.Tech. in Metallurgical Engineering from Indian Institute of Technology, Kanpur and Ph.D. in Materials Science and Engineering from The Ohio State University, USA) and a deep interest in ancient Indian texts. My work on Indology spans three different fields: cosmology, astronomy, and history.

Next: Zero Points of Vedic Astronomy: Part 2 of 8 — A Tale of two Yogatārās

Vedic Scholar, Materials Scientist, Author of books on Vedic Astronomy, Jain Astronomy, and Ancient Indian History

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