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 isuniform 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.

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

Angle q1 = Equation of Center . Angle subtended at Luna between Observer and
Equant

Equation in Astronomy = The angle between true and mean positions.

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,
theSiddhanta 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.