The True Equant

The Indian astronomers could calculate the Manda Kendra ( The Equation of Center of Western Astronomy ) and the Manda Phala, but a problem presented itself when calculating the lunar longitude.

The Concentric Model and the Epicylic Model could not calculate Moon's longitude at quadrature, even though they could calculate the lunar longitude at the times of New Moon and Full Moon. There was a difference of 2.5 degrees between the longitude computed by the Concentric Equant and Epicyclic Models. So the ancients had to give a correction to the Equation of Center, which reached a maximum of 2.5 degrees.

So the Indian astronomers came out with a solution. They created a new Equant (E'), the true Equant, which moves on an epicycle, whose center is the Mean Equant, E. The epicycle has a radius e, equal to EoE., on the Line of Apsis, OA.

q1 = Equation of Center, first lunar inequality
q2 = Correction, second lunar inequality.

True Longitude = Mean long + Eq of Center + q2

The first lunar anomaly was the Evection and the second, the Variation. The first inequality was the Equation of Center and the Evection and the Variation became the second and third inequalities. Actually Indian Astronomy recognised 14 major perturbations of the Moon and 14 corrections are therefore given to get the Cultured Longitude of the Moon, the Samskrutha Chandra Madhyamam. Then Reduction to Ecliptic is done to get the true longitude of Luna !



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

The Concentric Equant Model of Aryabhata

Aryabhata developed a Concentric Equant Model, in the sixth century. The Sun moves on a circle of radius R, called a deferent, whose center is the Observer on Earth. The distance between the Earth and the Sun, the Ravi Manda Karna, is constant. The motion of the Sun is uniform from a mathematical point, called the " Equant", which is located at a distance R x e from the observer in the direction of the Apogee ( e = eccentricity ).

All Indian computations are based on this Concentric Equal Model. The normal equation for computing the Manda Anomaly is R e Sin M and resembles the Kepler Equation, M = E - e Sin E.


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

The Lunar Model Of Munjala

The Concentric Equant theory was developed by the Indian astronomer, Munjala ( circa 930 CE ).

The Geocentric theory of the ancient astronomers had the ability to produce true Zodiacal Longitudes for the Moon. But the perturbations of the Moon were so complex, that the early Indian and Greek astronomers had to give birth to complicated theories.

The simplest model is a concentric Equant Model to compute the lunar true longitude.

In the above diagram

M = Moon
O = Observer
Eo = Equant , located at a distance r from the observer , drawn on the Line of Apsis and the Apogee.

A = Apogee, Luna's nearest point to Earth






Angle Alpha = Angle between Position and Apogee ( Manda Kendra )
Angle q1 = Equation of Center . Angle subtended at Luna between Observer and Equant ( Manda Phala )
Equation in Astronomy = The angle between true and mean positions.


These diagrams are by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

Astronomy & Maths In India

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 )