Indian Astronomy Original

The Physics Professor of Florida State University, Dennis Duke remarks

"The planetary models of ancient Indian mathematical astronomy are described in several texts. These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true longitudes by adding corrections to mean longitudes, the obvious choices for these latter models are those from the Greco-Roman world. In order to investigate if there is any connection between Greek and Indian models, we should therefore focus on the oldest Indian texts that contain fully described, and therefore securely computable, models. We shall see that the mathematical basis of the Indian models is the equant model found in the Almagest, and furthermore, that analysis of the level of development of Indian astronomy contemporary to their planetary schemes strongly suggests, but does not rigorously prove, that the planetary bisected equant model is pre-Ptolemaic. "

The mutli step algorithms of Indian Astronomy never approximated any Greek geometrical model. Ptolemy's Almagest was the first book, according to Western Astronomy. We have now the information that Ptolemy did not invent the equant.

Bhaskara II was an astronomer-mathematician par excellence and his magnum opus, the Siddhanta Siromani (" Crown of Astronomical Treatises") , is a treatise on Astronomy and Mathematics. His book deals with arithemetic, algebra, computation of celestial longitudes of planets and spheres. His work on Kalana ( Calculus ) predates Liebniz and Newton by half a millenium.

The Siddanta Siromani is divided into four parts


1)The Lilavati - ( Arithmetic ) wherein Bhaskara gives proof of c^2 = a^ + b^2. The solutions to cubic, quadratic and quartic indeterminate equations are explained.

2)The Bijaganitha ( Algebra )- Properties of Zero, estimation of Pi, Kuttaka ( indeterminate equations ) , integral solutions etc are explained.

3)The Grahaganitha ( Mathematics of the planets ).


For both Epicycles

The Manda Argument , Mean Longitude of Planet - Aphelion = Manda Anomaly

The Sheegra Argument, Ecliptic Longitude - Long of Sun = Sheegra Anomaly

and the computations from there on are explained in detail.

4)The Gola Adhyaya ( Maths of the spheres )


Bhaskara is known for in the discovery of the principles of Differential Calculus and its application to astronomical problems and computations. While Newton and Liebniz had been credited with Differential Calculus, there is strong evidence to suggest that Bhaskara was the pioneer in some of the principles of differential calculus. He was the first to conceive the differential coefficient and differential calculus.

Khagola, the Celestial Coordinate System

( Above diagram by courtesy of www.wikipedia.org )

A 360 degree Circle is a coordinate System. And Khagola is the Celestial Coordinate System.

Like the Geographical Coordinate System, the Celestial Coordinate System is another coordinate system, which computes the coordinates of the Khagola, the Celestial Sphere.

The Ascending Sign is called the Ascendant ( Raseenam udayo lagnam ) and is the intersecting point of the Ecliptic ( at the East Point) with the Celestial Horizon. The Descending Sign is the Descendant ( Astha Lagna ) and lies 180 degrees West on the Khshithija, the Celestial Horizon.

Like the Geographical Meridien ( the Prime Meridien ) and the Geographic Equator, the Celestial Coordinate System has a Celestial Meridien ( Nadi Vritta ) and a Celestial Equator.

The Vernal Equinox and the Autumnal Equinox are two intersecting points of the Ecliptic with the Vishu vat Vritta, the Celestial Equator, known as Meshadi and Thuladi.

The Hindu Zero Point of the Ecliptic starts from 0 degrees Beta Arietis, Ashwinyadi, which is the beginning point of the Sidereal Zodiac. This is the Nirayana System, sidereal. The Tropical System, Sayana, also has its adherents in India and starts from Meshadi, 0 degree Aries.




The Galactic Center, the Vishnu Nabhi, lies in Sagittarius. NEP is the North Ecliptic Pole, NGP is the North Galactic Pole and NCP is the North Celestial Pole.

It is raining cats and dogs in Kerala

On Saturday, the heavens brimmed with pessimistic prophecies and then came the downpour. ( Today is 19th Jul 2011 )

The Sun has disappeared and it is now raining cats and dogs here. As a concomitant result, I got cold !

This SW Monsoon, defined as a failure this season, may perk up, compensating for the lack of rains during the earlier Mrigasira and Aridra Solar Periods ( Njattuvelas ). Now Punarvasu Njattuvela is on, as the Sun transits Beta Geminorum.

Now the paddy fields are full of water and it rained heavily at night day before yesterday. The ocean became hostile on Chavakkad Beach and surrounding areas, wreaking destruction.

The Double Epicyclic Model of India

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

We have the Double Epicyclic Model - that of Manda Epicycle and Sheegra Epicycles - in Indian Astronomy, which explain the Zodiacal and Solar anomalies. One Epicycle explains the Zodiacal Anomaly and the other the Solar Anomaly.

( Zodiacal Anomaly - That all planets move slower at Aphelion and faster at Perihelion.
Solar Anomaly - The astronomical phenomenon of Retrogression. Backward Motion. When a planet changes its course from perihelion to aphelion, it retrogrades in order to gain the Sun's celestial gravity )

Dennis Duke, of Florida State University, says " We have only to conclude that Ptolemy did not invent the equant. " If Ptolemy did not invent the equant, as Westerners widely believe, then who did ?

"The bisected Indian equant model is pre-Ptolemaic' says he. Other Greek books, prior to Ptoemy, may have influenced Indian Astronomy,says he. Then what are those books, prior to the Almagest, which had influenced the Indian system? The answer is "unknown sources".




Remarks Duke " Indeed, since the very earliest investigation of the Indian models by Western scholars it has been presumed that the models are somehow related to a double epicycle system, with one epicycle accounting for the zodiacal anomaly, and the other accounting for the solar anomaly (retrograde motion) This perception was no doubt reinforced by the tendency of some Indian texts to associate the manda and sighra corrections with an even older Indian tradition of some sort of forceful cords of air tugging at the planet and causing it to move along a concentric deferent . Since our goal in this paper is to investigate the nature of any connection with ancient Greek planetary models, it is only important to accept that the models appear in Indian texts that clearly pre-date any possible Islamic influences, which could, at least in principle, have introduced astronomical elements that Islamic astronomers might have derived from Greek sources. ( "The Equant in India: the Mathematical Basis of Ancient Indian Planetary Models" By Dennis Duke, Florida State University )

Computation of Geocentric Distance, Sheegra Karna

In the diagram above, the geocentric distance, EQ called X here , the distance of the planet from the Earth is calculated by the equation

X^2 = EQ^2(EP+PL)^2 + QL^2

or = EN^2 + QN^2

In a trignometric correction, called Sheegra Sphashteekarana, this equation is given by Bhaskara.

where

E = Earth
P = Planet in its Orbit
Q = Planet on the Epicycle
QL = Sin
PL = Cos

We have said that Sheegra Kriya reduces the heliocentric postions to the geocentric.

According to this oscillating Epicyclic Model of Bhaskara, EP = R ( Called Thrijya ), PQ is the Sheegra Phala, QL is the Bhujaphala and PL is Kotiphala.

The Hindu algorithms for the computation of mean and true celestial longitudes seems to be totally different from the Western, from the methods adopted by Kepler, Laplace and Co. Hence the Hindu Planetary Model is original and not influenced by Greco Roman sources, as some Western scholars believe.

Calculation of the geocentric longitude of Mercury

Different equations have been given for superior planets ( Mars, Jupiter and Saturn ) and inferior planets ( Mercury and Venus ) in Astronomia Indica.

In the case of Mercury, an inferior planet in the diagram above, the center of the Sheegra Epicycle is located on the straight line running through the Sun and the observer, on the geographical parallel of the observer.

The above diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The Sheegra Phalam, x, in the equation 1/2 Tan ( A -x ), where A is the Elongation or Sheegra Kendra, obtained is deducted from the Sun's longitude, to get the geocentric longitudes of Mercury and Venus.

Indian Astronomy Pre-Ptolemaic

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

In the above diagram, Saturn, a superior planet, is on the circumference of the Sheegra Epicycle, where it is met by a radius drawn parallel to the direction of the Sun from the observer.

To the Western scholars, Indian Astronomy is mysterious. Let us see what astro scholars have said about IA.

Dennis Duke, of Florida State University suggests that Indian Astronomy predates Greek Astronomy

"The planetary models of ancient Indian mathematical astronomy are described in several texts.1 These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true longitudes by adding corrections to mean longitudes, the obvious choices for these latter models are those from the Greco-Roman world. In order to investigate if there is any connection between Greek and Indian models, we should therefore focus on the oldest Indian texts that contain fully described, and therefore securely computable, models. We shall see that the mathematical basis of the Indian models is the equant model found in the Almagest, and furthermore, that analysis of the level of development of Indian astronomy contemporary to their planetary schemes strongly suggests, but does not rigorously prove, that the planetary bisected equant model is pre-Ptolemaic" says he.

The earliest Indian Planetary Models are two sets from the writer Aryabhata, both dating from 6th Century AD.

1) The Sunrise System , after the Epoch, which is taken from the sunrise of 18th Feb 3102 ( Arya Paksha ). It appears first in Aryabhatiya

2) The Midnight System, after the Epoch, which is taken from the midnight of 17/18 FEB 3102 ( Ardha Ratri Paksha ). It appears first in Latadeva's Soorya Siddhanta


The Local Meridien is taken as Lanka, Longitude 76 degrees, Latitude 0 degrees.