Price $28 (including postage)
This book shows how to predict eclipses, solve Kepler's Equation, solve spherical triangles, predict planetary positions etc.
141 + v pages.
Size: 23cm by 15cm.
Author: Kenneth Williams
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
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.
1 INTRODUCTION TO VEDIC MATHEMATICS
1.1 PRODUCTS AND CROSS-PRODUCTS
1.2 THE VINCULUM
1.3 LEFT TO RIGHT CALCULATIONS
Using the Vinculum
1.4 MOVING MULTIPLIER
2 PREDICTION OF ECLIPSES
2.1 PREDICTION OF THE TIMES OF CONTACT OF THE MOON’S PENUMBRAL AND UMBRAL SHADOWS WITH THE EARTH
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
Solution of the Eclipse Equation
3 KEPLER’S EQUATION
3.1 A TRANSCENDENTAL EQUATION
3.2 SOLUTION OF KEPLER’S EQUATION
4 INTRODUCTION TO TRIPLES
4.1 NOTATION AND COMBINATION
The Half-Angle Triple
4.2 TRIPLE CODE NUMBERS
Addition and Subtraction of Code Numbers
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
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
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
Application of Determinants
The Cotangent Rule
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
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
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
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.