Tantrasa graha, composed by the renowned Kerala astronomer N laka tha Somayaji (c. 1444 1545 CE) ranks along with ryabhat ya of ryabhata and Siddh nta iromani of Bh skar c rya as one of the major works that significantly influenced further work on astronomy in India. One of the distinguishing features of this text is the introduction of a major revision of the traditional planetary models which includes a unified theory of planetary latitudes and a better formulation of the equation of centre for the interior planets (Mercury and Venus) than was previously available.Several important innovations in mathematical technique are also to be found in Tantrasa graha, especially related to the computation of accurate sine tables, the use of series for evaluating the sine and cosine functions, and a systematic treatment of the problems related to the diurnal motion of the celestial objects. The spherical trigonometry relations presented in the text applied to a variety of problems such as the computation eclipses, elevation of the moon 's cusps and so forth are also exact.In preparing the translation and explanatory notes, the authors have used authentic Sanskrit editions of Tantrasa graha by Suranad Kunjan Pillai and K V Sarma. The text consists of eight chapters mean londitudes, true longitues, gnomonic shadow, lunar eclipse, solar eclipse, vyat p ta, reduction to observation and elevation of the moon 's cusps and 432 verses. All the verses have been translated into English and are supplemented with detailed explanations including all mathematical relations, figures and tables using modern mathematical notation.This edition of Tantrasa graha will appeal to historians of astronomy as well as those who are keen to know about the actual computational procedures employed in Indian astronomy. It is a self-contained text with several appendices included, enabling the reader to comprehend the subject matter without the need for further research.
Common terms and phrases:
ecliptic Moon prime vertical trijya
Cover
Sources and Studies in the History of Mathematics and Physical Sciences
Tantrasangraha of Nilakantha Somayaji
ISBN 9780857290359
Foreword
Preface
Contents
List of Figures
List of Tables
Introduction
Chapter 1 Mean longitudes of planets
1.1 Invocation
1.2 Measurement of civil and sidereal day
1.3 Measurement of smaller units of time
1.4 Lunar reckoning of time
1.5 Solar reckoning of time
1.6 Definition of an intercalary month
1.7 Nature of the intercalary month
1.8 Days of the God etc.
1.9 Number of revolutions of planets in a Mahayuga
1.10 Number of civil days in a Mahayuga etc.
1.11 Finding the number of days elapsed since an epoch
1.12 Finding the mean positions from Ahargana
1.13 Correction due to difference in longitude
1.14 Duration corresponding to difference in longitude
1.15 Initial positions at the beginning of the Kaliyuga
1.16 Introducing an alternative yuga
1.17 The Mandoccas at the beginning of the Kaliyuga
Chapter 2 True longitudes of planets
2.1 Definition of the anomaly and the quadrant
2.2 Computation of the Rsines and the arcs
2.3 Computation of the tabular Rsines
2.4 Another method for obtaining the Rsines
2.5 Obtaining the desired Rsines and Rcosines
2.6 Determining the length of the arc from the corresponding Rsine
2.7 Finding more accurate values of the desired Rsine
2.8 Computation of the Rsine value of a small arc
2.9 Computation of the desired Rsine
2.10 True longitude of the Sun
2.11 Pranas of the ascensional difference
2.12 Longitude of the planets at sunrise at the observer’s location
2.13 Durations of the day and the night
2.14 Obtaining the true Moon
2.15 Finding the arc corresponding to cara etc.
2.16 Obtaining the manda and s ghra hypotenuses
2.17 Obtaining the iterated hypotenuse
2.18 Another method of obtaining the iterated hypotenuse
2.19 Correcting the Sun using the iterated hypotenuse
2.20 Obtaining the mean Sun from the true Sun
2.21 Another method for getting the mean planet from the true planet
2.22 Another method for getting the manda-hypotenuse
2.23 Instantaneous velocity of a planet
2.24 Finding naksatra and tithi
2.25 The scheme of correction for the planets
2.26 The correction for Mars, Jupiter and Saturn
2.27 The correction for Mercury
2.28 The correction for venus
2.29 The daily motion of the planets
Chapter 3 Gnomonic shadow
3.1 Positioning the gnomon
3.2 Finding the east–west points
3.3 Correcting the east–west points
3.4 Fixing the directions in one’s own place
3.5 Equinoctial shadow
3.6 Relation between the gnomon, its shadow and the hypotenuse
3.7 Rsine and Rcosine of the latitude
3.8 More accurate values of the Rsine and Rcosine of the latitude
3.9 The prime vertical, the celestial equator and the amplitude at rising
3.10 The rising time of rasis at Lanka and one’s own place
3.11 Big gnomon and the gnomonic shadow at a desired time
3.12 Correction to be implemented in the direct process of finding the shadow
3.13 Time elapsed or to be elapsed from the mahasanku
3.14 Determination of the true Sun from the shadow of the gnomon
3.15 Motion of equinoxes
3.16 Latitude of the place from the zenith distance and declination
3.17 Determination of the directions from the shadow of the gnomon
3.18 Drawing the locus of the tip of shadow of the gnomon
3.19 Another method for finding the Rsine of the shadow
3.20 Gnomon when the Sun is on the prime vertical
3.21 True longitude of the Sun from the samasanku
3.22 Hypotenuse of the shadow from the samasanku in angulas
3.23 Obtaining the hypotenuse of the shadow by a different method
3.24 The duration elapsed and yet to elapse from the samamandala-sanku
3.25 Hour angle from the samamandala-sanku
3.26 Hour angle by another method
3.27 Ksitijya from the samamandala-sanku
3.28 The ten problems
3.29 Determination of the zenith distance and hour angle from the declination, amplitude and latitude (Problem 1)
3.30 Determination of the zenith distance and declination from the hour angle, amplitude and latitude (Problem 2)
3.31 Determination of the zenith distance and amplitude from the hour angle, declination and latitude (Problem 3)
3.32 Determination of the zenith distance and latitude from the hour angle, declination and amplitude (Problem 4)
3.33 Determination of the hour angle and declination from the zenith distance, amplitude and latitude (Problem 5)
3.34 Determination of the hour angle and amplitude from the zenith distance, declination and latitude (Problem 6)
3.35 Determination of the hour angle and latitude from the zenith distance, declination and amplitude (Problem 7)
3.36 Determination of the declination and amplitude from the zenith distance, hour angle and latitude (Problem 8)
3.37 Determination of the declination and latitude, and the amplitude and latitude, from the rest of the three (Problems9, 10)
3.38 Shadow along any direction
3.39 Shadow when the amplitude is 45 degrees
3.40 Obtaining the orient ecliptic point
3.41 Inaccuracy in determining the praglagna
3.42 Determination of the kalalagna
3.43 Determination of the zenith distance of the vitribhalagna
3.44 Exact determination of the ecliptic point that is rising or setting
3.45 Determination of the madhyalagna
3.46 Determining the madhyalagna without iteration
Chapter 4 Lunar eclipse
4.1 Time of conjunction of the Moon and the Earth’s shadow
4.2 Determination of the exact moment of conjunction by iteration
4.3 Radii of the orbits of the Sun and the Moon in yojanas
4.4 Radii of the Sun, Moon and the Earth in yojanas
4.5 Actual distances of the Sun and the Moon in yojanas
4.6 Second approximation to the radii of the orbits of the Sun and the Moon yojanas
4.7 Angular diameters of the orbs of the Sun and the Moon in minutes
4.8 Length of the Earth’s shadow
4.9 Angular diameter of the Earth’s shadow in minutes
4.10 Moon’s latitude and true daily motion
4.11 The occurrence and non-occurrence of an eclipse
4.12 The condition for the occurrence of a total eclipse
4.13 The time of half-duration, the first and the last contact
4.14 Iteration for obtaining the half-duration, and the time of the first and the last contact
4.15 The time of the first and the last contact from the half-duration
4.16 The visibility or otherwise of the the first and the last contact at sunrise and sunset
4.17 The visibility or otherwise of sparsa and moksa
4.18 Accurate distance of separation between the orbs
4.19 State of the eclipse being invisible
4.20 Shift of the instant of maximum obscuration from the instant of opposition
4.21 Deflection due to latitude and that due to declination
4.22 Graphical representation of the eclipse
Chapter 5 Solar eclipse
5.1 The possibility or otherwise of lambana and nati
5.2 Finding the drkksepa and the drggati
5.3 Parallax in longitude and its application for finding the instant of conjunction
5.4 Parallax in latitude of the Sun in minutes
5.5 Parallax in latitude of the Moon in minutes
5.6 The possibility of a solar eclipse
5.7 Application of lamabana in finding the half-duration
5.8 Time of sparsa by an iterative process
5.9 Time of moksa by an iterative process
5.10 Half-duration of obscuration and the time of submergence and emergence
5.11 The drkkarna of the Sun
5.12 The drkkarna of the Moon
5.13 Transformation to the observer-centred celestial sphere
5.14 Determination of the middle of the eclipse
5.15 Distance of separation between the Sun and the Moon
5.16 Announcement of the visibility of the eclipse
5.17 Graphical representation of the eclipse
Chapter 6 Vyat pata
6.1 The possibility of vyatipata
6.2 Finding the declination of the Sun and the Moon
6.3 Speciality in the determination of the desired declination of the Moon
6.4 Determination of the declination of the Moon by another method
6.5 The occurrence or non-occurrence of vyat pata
6.6 The criterion for the non-occurrence of vyat pata
6.7 Determining whether vyat pata has occurred or is yet to occur
6.8 The middle of vyatipata
6.9 The beginning and the end of vyat pata
6.10 Inauspiciousness of the later half of viskambhayoga and others
6.11 Inauspiciousness of the three vyat patas
Chapter 7 Reduction to observation
7.1 The two visibility corrections–due to the latitude of the observer and due to the position on the ecliptic
7.2 The desired latitude of the planets
7.3 Reduction to observation of the true planets
7.4 Alternate method for reduction to observation
7.5 Kalalagna and the divisions of time
7.6 Visibility or otherwise of the planets during their rising and setting
Chapter 8 Elevation of lunar horns
8.1 Correcting the distance of separation between the Earth and the Moon
8.2 The true motion of the Moon
8.3 Application of the true motion of the Moon etc. obtained earlier
8.4 The latitude and the zenith distance
8.5 Finding the distance of separation between the solar and lunar discs
8.6 The correction to the angular separation for finding the Moon’s phase
8.7 The measure of the phase
8.8 Deflection of the horn (phase of the Moon)
8.9 Graphical representation of the srngonnati
8.10 Time of moonrise etc. after the sunset
8.11 Obtaining the dimensions of the orbits of Mars and other planets
8.12 Verifying the measures of the discs with the observed values
8.13 Concluding words
Appendix A Representation of numbers
A.1 Katapayadi system
A.2 Bhutasankhya system
Appendix B Spherical trigonometry
B.1 Great and small circles
B.2 Spherical triangles
Appendix C Coordinate Systems
C.1 Celestial sphere
C.2 Locating an object on the celestial sphere
C.3 Precession of equinoxes
C.4 Equation of time
Appendix D Solution of the ten problems: a couple of examples from Yuktibhasa
D.1 Problem one: to derive the sanku and nata from the other three quantities
D.2 Problem two: the sanku and apakrama
Appendix E Derivation of the maximum declination of the Moon
E.1 Occurrence of Vyat pata
E.2 Derivation of declination of the Moon
E.3 Viksepa
E.4 Viksepa-calana
E.5 Karnanayana
Appendix F The traditional Indian planetary model and its revision by N lakantha Somayaj 1
F.1 The traditional Indian planetary model: Manda-samskara
F.1.1 Epicyclic and eccentric models
F.1.2 Calculation of manda-sphuta
F.1.3 Avisista-manda-karna: iterated hypotenuse
F.1.4 Madhava’s formula for the avisista-manda-karna
F.1.5 Manda-samskara for the exterior planets
F.1.6 Manda-samskara for interior planets
F.2 S ghra-samskara
F.2.1 Exterior planets
F.2.2 Interior planets
F.2.3 Four-step process
F.2.4 Computation of planetary latitudes
F.3 Geometrical picture of planetary motion according to Paramesvara
F.4 N lakantha’s revised planetary model
F.4.1 Identifying the mean Mercury and Venus
F.4.2 Computation of planetary longitudes
F.4.3 Planetary latitudes
F.4.4 Rationale for the revised planetary model
F.5 Geometrical picture of planetary motion according to Nilakantha
F.5.1 Geometrical picture of the motion of the exterior planets
F.5.2 Geometrical picture of the motion of the interior planets
F.6 N lakantha’s cosmological model
F.7 The problem of planetary distances
F.7.1 Planetary distances in traditional Indian astronomy
F.7.2 N lakantha on planetary distances
F.8 Annexure: Keplerian model of planetary motion
F.8.1 Elliptic orbits and the equation of centre
F.8.2 Geocentric longitude of an exterior planet
F.8.3 Geocentric longitude of an interior planet
F.8.4 Heliocentric and geocentric latitudes of a planet
Glossary
Bibliography
Index
Index of Half-verses
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