11. ASTRONOMICAL APPLICATIONS OF VEDIC MATHEMATICS - details

This book shows how to predict eclipses, solve Kepler's Equation, solve spherical triangles, predict planetary positions etc. 2003 (first published 2000). Author: K R Williams. 144 pages, size 16cm by 24cm, paperback. ISBN 81-208-1983-7. Price 9.75 pounds.

Preface to "Astronomical Applications of Vedic Mathematics"

Since the publication of the book “Vedic Mathematics” by Sri Bharati Krsna Tirthaji in 1960 many new applications of the Vedic system have been found. Vedic Mathematics contains many examples of striking methods of calculation and there is a remarkable coherence to the system which makes it very attractive. Also the Vedic system itself suggests a kind of approach that involves going directly to the answer.

Vedic mathematics is based on sixteen Sutras and some sub-Sutras which provide links through mathematics: the word ‘Sutra’ means ‘thread’. These Sutras are given in word form, for example Vertically and Crosswise and By Addition and By Subtraction, and where they arise in this text they are indicated by italics. The Sutras, and sub-Sutras, can all be related to natural mental functions. A full list of the Sutras and sub-Sutras will be found on Page 138.

Having had a keen interest in Astronomy for many years  I had the opportunity, when studying for a degree in this subject, to look into Astronomical Applications of Vedic Mathematics for a final year project (in 1981). The result was what is now the contents of Chapters 2 and 3 of this book. The left to right method of calculation, which has so many useful applications (see “Vertically and Crosswise”, Reference 3), was initially developed by the author in order to solve Kepler’s Equation. Being encouraged to publish this work and take it further by studying for a higher degree, more such applications were found.

The methods given in this book are not intended to be a complete or thorough treatment of the topics they deal with. All of the ideas can probably be developed further or applied in other areas, and all can doubtless be improved upon. The mathematician will also observe a certain lack of rigor as an attempt has been made to make the material intelligible to as wide a readership as possible. To this end a Glossary and an Index have been added. An attempt has also been made to make the book as self-contained as possible so that the first chapter introduces some of the Vedic methods of calculation which are used in the book and the fourth chapter introduces the arithmetic for Pythagorean triples which is used in the subsequent chapters.

K. R. W.

January 2000

Contents of "Astronomical Applications of Vedic Mathematics"

PREFACE
1  INTRODUCTION TO VEDIC MATHEMATICS
   1.1          PRODUCTS AND CROSS-PRODUCTS        
   1.2          THE VINCULUM            
   1.3          LEFT TO RIGHT CALCULATIONS         
                Addition  
                Subtraction  
                Multiplication  
                Using the Vinculum  
   1.4          MOVING MULTIPLIER     
   1.5          SQUARING                                                                                         
   1.6          DIVISION                     

2  PREDICTION OF ECLIPSES                                                                                            
    2.1         PREDICTION OF THE TIMES OF CONTACT OF THE MOON’S PENUMBRAL AND UMBRAL SHADOWS WITH THE EARTH                     
                   Partial Phase  
                   Total Phase 
   2.2         THE APPROXIMATE POSITION OF THE ECLIPSE PATH            
   2.3         TIME OF TOTAL ECLIPSE FOR AN OBSERVER ON THE EARTH  
               Comparison with Bessel’s Method  
                   Early Eclipse Prediction                                 
                   Notes  
               Solution of the Eclipse Equation 

3  KEPLER’S EQUATION                                                                                                      
   3.1         A TRANSCENDENTAL EQUATION         
   3.2         SOLUTION OF KEPLER’S EQUATION        
               Another Example  

4  INTRODUCTION TO TRIPLES                                                                                        
    4.1         NOTATION AND COMBINATION     
                Triple addition  
                Quadrant Triples  
                Rotations  
                Triple Subtraction  
                The Half-Angle Triple  
    4.2         TRIPLE CODE NUMBERS             
                Addition and Subtraction of Code Numbers  
                Complementary Triples  
    4.3         ANGLES IN PERFECT TRIPLES           
4.4 GENERAL ANGLES                                                 

5  PREDICTION OF PLANETARY POSITIONS
   5.1            HELIOCENTRIC POSITION    
                  The Mean Anomaly  
   5.2            GEOCENTRIC POSITION                        
                    Definition of the Reference Point and the Geocentric Longitude
                   finding the Geocentric Correction
   5.3         THE PLANET FINDER           
                Construction 
                Application 
                Table of Planetary Data 

6 SPHERICAL TRIANGLES                                                                                                                        
   6.1         SPHERICAL TRIANGLES USING TRIPLES                                  
                Triple Notation for Spherical Triangles  
                Standard Formulae of Spherical Trigonometry  
                The Cosine Rule to find an Angle, a Side 
                    The Sine Rule to find an Angle, a Side 
                    The Cotangent Rule to find an Angle, a Side 
                    The Polar Cosine Rule to find an Angle, a Side 
                    Further Illustrations  
    6.2          RIGHT-ANGLED SPHERICAL TRIANGLES                        
                    Solution of Scalene triangles using Right-angled Triangles  
    6.3          SPHERICAL TRIANGLES USING CODE NUMBERS                
                To find an Angle given Three Sides  
                    Given Two Sides and the Included Angle to find the Side Opposite  
                    To find a Side given Three Angles  
                    Given Two Angles and the Side between them to find the Angle Opposite  
                    Given Two Sides and an Angle Opposite to find the other Angle opposite  
                    Given Two Angles and a Side Opposite to find the other Side Opposite  
    6.4           DETERMINANTS
                    Application of Determinants  
                    The Cotangent Rule  
    6.5           SUMMARY          

7   QUADRUPLES                                                                                                                                                                7.1         INTRODUCTION       
    7.2       Addition of Perpendicular Triples                          
    7.3         ROTATION ABOUT A COORDINATE AXIS                     
                   Change of Coordinate System 
    7.4         QUADRUPLES AND ORBITS
                    Quadruple for a given i and A 
                    Inclination of Orbit 
                    Quadruple Subtraction 
                    Quadruple Addition 
                    Doubling and Halving a Quadruple 
                    Code Number Addition and Subtraction 
                    Angle in a Quadruple 
                    Relationship between d and A
    7.5         TO OBTAIN A QUADRUPLE WITH A GIVEN INCLINATION 
                    A Note on Continued Fractions 
    7.6         ANGLE BETWEEN TWO DIRECTIONS 
                    Spherical Triangles

APPENDICES

 PLANET FINDER DIAGRAM
ANSWERS TO EXERCISES
REFERENCES
GLOSSARY
VEDIC SUTRAS
INDEX