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Mathematical Models in Indian AstronomyAstronomy In India(500-1200 CE)

Geocentric Astronomy:Astronomical parameters and computation of mean celestial motions and positions.Trignometric methods for finding true celestial positions.Computing the apparent direction, place and time of celestial phenomenon as seen from a location.Calculations for lunar and solar eclipses. Astronomical calculations in Siddhantas:Computation of eclipses played a crucial part of an astronomers work.Even today, astrochronological studies consider mention of eclipses as a major parameter to date the manuscripts.EclipsesSolar Eclipse:

Mathematics in India by Kim PlofkerBrahma-sputa-siddhantaImage Courtesy:BBCImage Courtesy(Manuscript):Jaina Vidya Sansthan,Jaipur.References:Super SunSolar SystemCycle of the 4 YugasTheprecession(caused by gradual rotation of the Earths axis) of the equinoxes is the period of time that it takes the Earths axis to pass through one complete cycle of the zodiac. It takes the planet 72 years to pass through one degree of the zodiac and 25,920 years to complete one full circle of 360 degrees. One half of the journey takes 12,960 years and covers the four yugas. Satya Yuga lasts 5184 years. Treta Yuga lasts 3888 years. Dwapara Yuga lasts 2592 years. Kali Yuga lasts 1296 years. These four yugas taken together come to a total of 12,960 years91296 yrsKali YugaDwaparaYugaTretaYugaSatYugaTretaYugaSatYugaDwaparaYugaKali Yuga2592 yrs3888 yrs5184 yrsCycle of the 4 YugasTheprecession(caused by gradual rotation of the Earths axis) of the equinoxes is the period of time that it takes the Earths axis to pass through one complete cycle of the zodiac. It takes the planet 72 years to pass through one degree of the zodiac and 25,920 years to complete one full circle of 360 degrees. One half of the journey takes 12,960 years and covers the four yugas. Satya Yuga lasts 5184 years. Treta Yuga lasts 3888 years. Dwapara Yuga lasts 2592 years. Kali Yuga lasts 1296 years. These four yugas taken together come to a total of 12,960 years10Geometric Models in AstronomySubject of gola is separate from ganitaThe spherical cosmology underlying sidhanta seemed controversial at that time.

golaganita(popular and scared ideas involved a flat earth)11Geometric Models (contd.)

The sphere of the earth [made of] earth, water, fire and air, in the middle of the cage of the constellations [formed of] circles, surrounded by the planetary orbits, in the center of heavens is everywhere circular.

In the center of the heavens: The earth is not at all above [the center], and not below, hence it is not falling Now other think [that] the earth is supported by [the cosmic serpent] Sesa or [something] else: that is not rational Now if they [i.e. Sesa, etc can] stay fixed by their own power, why cannot this power be assumed for the earth?12ReferencesSri Yukteswar Giri: Sri Yukteswar was a Kriya yogi, a Jyotisha (Vedic astrologer), a scholar of the Bhagavad Gita and the Bible, an educator and an astronomerIsha Foundation: ishafoundation.org/blogPlofker, K. (2009). Mathematics in India. Princeton Press.

CONCEPTS OF COSMIC TIME IN THE PURANASDIVISION OF COSMIC TIME1 Kalpa = 4.32 billion years = Day of BrahmaA Kalpa is divided into a thousand Mahayugas each lasting for 4.32 million years, while a Mahayuga is further divided into 4 stages (Satya Yuga, Treta Yuga, Dwapara Yuga and Kali Yuga) in a 4:3:2:1 duration ratio. The world decays from good to bad through the stages.The last and shortest phase of a Mahayuga is the Kali Yuga the worst phase.It is currently believed that we are in the Kali Yuga!Pg 68, Plofker16There was also another division of time which was prevalent.In this case, a kalpa was divided into longer periods of time called a Manvantara which encompassed 71 Mahayugas.It was believed that each Manvantara was ruled by a Manu, the father of man. A Manus lifespan was 71 Mahayugas at the end of which the Manu would die and a new Manu was created by Brahma. Thus, each Manvantara begins with the advent of a new Manu to produce a new race of human beings.There were 14 Manvantaras in all. It was believed that there was a gap period between the death of a Manu and the creation of the next i.e. between two Manvantaras. The kalpa is made up of the 14 Manvantaras along with all the gap periods between any two successive Manvantaras.LIFE CYCLES OF THE UNIVERSEIt was believed that the Universe is subject to the same processes of birth, death and rebirth much like in the case of human beings or any other mortal creature.At the beginning of each Brahma day, Brahma would create the Universe with a lifespan of 4.32 billion years. At the end of that period, the Universe would be destroyed and night falls on Brahma, who would then rest for the duration of the night which was equal to the length of the day (4.32 billion years).The next day, Brahma would create the Universe afresh and the cycle continues.LIFE CYCLES OF BRAHMABrahma himself was thought to be a creation of the Supreme God know as the Para Brahman or simply the Brahman.Brahma was created with a lifespan of 100 Brahma years at the end of which he would die.This is followed by 100 Brahma years at the end of which the next Brahma is created (with a lifespan of 100 Brahma years) and the cycle continues.SOME BASIC TERMSCelestial Equator: The great circle on the celestial sphere in the plane of the earths equator.

Ecliptic: The great circle on the celestial sphere in the plane of the suns apparent annual orbit about the earth.

Equinox: One of the two points where the celestial equator and the ecliptic intersect.THREE QUESTIONSProblems in the Siddhantas which dealt with the computation of time, place (latitude, longitude, etc.) and direction for an observer located anywhere on the surface of the earth.Local latitude (tropical latitude) of a place using a gnomon on the day of the equinox.

Solution:At the equinox, the gnomon is [proportional to the Sine of solar] altitude, the shadow [to] the Sine of the latitude, [and] the square root of the sum of the squares of those is the equinoctial hypotenuse. - Brahmagupta, Brahma-sphuta-siddhanta (Kim Plofker)

REFERENCESMathematics in India by Kim Plofkerhttps://en.wikipedia.org/wiki/Hindu_cosmologyMaharishi Mahesh Yogion theBhagavad GitaTranslation and Commentary, Arkana, 1967 p. 253http://books.google.com/books?id=QIvnxhasdGoC&pg=PA6&dq=hindu+cosmology&lr=&ei=g23qSeXaOIG4M9Sz6OkN#PPA8,M1Time Scales and Environmental Change, Chapman and Driver, p.8

DISTANCE RATIOSThe diameter of the sun multiplied by 108 equals the distance between the sun and the earth.Mean distance between the Earth and the Sun ,x = 149,597,870.691 km .Diameter of the sun, y = 1,392,000 km.Ratio, x:y =107.46973469181034482758620689.The value is approximately equal to 108. 108108The diameter of the moon multiplied by 108 equals the distance between the moon and the earth.Mean distance between the Moon and the Earth, x = 384,400 km.Diameter of the moon, y = 3,474.8 km.Ratio, x:y = 110.625071946586853919650051.The value is approximately equal to 108.

108The diameter of the Sun is 108 times the diameter of the earth.The diameter of the Sun, S= 1,392,000 kmThe diameter of the Earth, E= 12,756.2 kmRatio, S:E = 109.12340665.The radius of the Moon is 108 miles.

How could the writers of old Sanskrit texts have known the unknowable?In searching for an explanation we first need to understand that these ancient scientists were not just intellectuals, they were practicing yogis. The very first lines of the Surya Siddhantha, for of the Golden Age a great astronomer named Maya desired to learn the secrets of the heavens, so he first performed rigorous yogic practices. Then the answers to his questions appeared in his mind in an intuitive flash.Does this sound unlikely?Yoga Sutra 3:26-28 states that through, samyama (concentration, meditation, and unbroken mental absorption) on the sun, moon, and pole star, we can gain knowledge of the planets and stars. Sutra 3:33 clarifies, saying: "Through keenly developed intuition, everything can be known." Highly developed intuition is called pratibha in yoga. It is accessible only to those who have completely stilled their mind, focusing their attention on one object with laser-like intensity. Those who have limited their mind are no longer limited to the fragments of knowledge supplied by the five senses.All knowledge becomes accessible to them.Referenceshttp://www.stephen-knapp.com/108_the_significance_of_the_number.htmhttp://www.khenpo.eu/108.htmlhttp://www.lovearth.net/108.htmProblem of origins:

No record of observational practices of astronomers.Construction of astronomical models. Revelation from a deity! Or observed positions of celestial bodies in the sky.Models in siddhantas have familiarity with Greek models. Curious case of Indian Observational Astronomy: