Details

141 + v pages.
Size: 23cm by 15cm.
Paperback. 2010
Author: Kenneth Williams
ISBN 978-1-902517-22-3.

Reviews

I can not put your brilliant book down. May Vedic maths sweep the world. I spread the good news to all I can reach . . . Orm Girvan

Preface

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

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
 I     Inclination of Planetary Orbits  118
II    Derivation of the Formula for c(nu) in terms of c(M)
III   Calculation of the Radius Vector  122
IV   Proofs of Spherical Triangle Formulae  124

PLANET FINDER DIAGRAM
ANSWERS TO EXERCISES
REFERENCES
GLOSSARY
VEDIC SUTRAS
INDEX

Back Cover

The extraordinary power and versatility of Vedic Mathematics leads to new and extremely efficient methods in Astronomy.

In this self-contained book a few applications are given including Prediction of Eclipses, the digit by digit solution of Kepler’s Equation, the Prediction of Planetary Positions and the Solution of Spherical Triangles.

Normally the solution to such problems would not be attempted without the aid of a mechanical calculating device. But the ultra-efficient methods of Vedic Mathematics lead to easy and fast solutions without the need for such aids.

Many of the ideas introduced in this book can be developed further and also suggest other lines of research.