Price $28 (including postage)

## Description

This book shows how to predict eclipses, solve Kepler's Equation, solve spherical triangles, predict planetary positions etc.

## 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.