Price $36 (including postage)

This book shows an original and highly effective way of unifying many branches of mathematics using Pythagorean triples. A simple, elegant method for combining these triples gives unexpected and powerful general methods for solving a wide range of mathematical problems. There are applications in trigonometry (you do not need any of those complicated formulae), coordinate geometry (2 and 3 dimensions), transformations (2 and 3 dimensions), simple harmonic motion, astronomy etc., etc.

173 + vii pages.
Size: 24cm by 17cm.
Paperback. 2010
Author: Kenneth Williams
ISBN 978-1-902517-19-3.

I feel strongly that the methods you have pioneered, such as "Triples", etc; will catch on all around the world....
Gary Adamson, U.S. mathematician.

I bought your books on Triples and Astronomical Applications. I am at the moment pursuing the book on triples. I like it very much. In fact I noticed the mathematics of addition , subtraction of angles using triples. It makes the laborious proofs and steps needed for the derivations using coordinate geometry (as is usually done nowadays) look superfluous. It is so simple and easy to understand. It is simply enjoyable. Thanks for all the ideas you have put forth because it has opened up vast vistas for imagination. Raji Sharma, Vedic Maths teacher, India.

An eye-opener!
If you are attracted to the beauty and simplicity of Mathematics, and if you are drawn to the elegant patterns within Pythagorean Triples then this book will take you on an amazing voyage of discovery. Following the introduction of a few simple arithmetic methods for working with triples, there follows a host of traditional mathematical topics worked using the new 'triple' method. The efficiency and elegance of the mathematics is often breathtaking. Solutions which traditionally may take many steps of complex algebra just fall out in a few straightforward applications of the triple method.

At the core of the book is Mathematics from the Indian Vedic culture (c2000 BC to c600 AD). This is a new territory for me (and I suspect for most) but is a departure that I found refreshing and stimulating. I thoroughly recommend this text for anyone with a knowledge of Mathematics at approximately A Level standard and a love for the beauty of the subject.
Des Prez (Ireland), 14 May 2009

The unchanging laws of number have always been a source of delight and inspiration. Particularly attractive are the Pythagorean triples which have so many elegant and interesting properties. But these triples are also of great practical use: through the theorem of Pythagoras the triples link the three main branches of mathematics: number, algebra and geometry.

This appears to be the first time that these triples have been developed into a useful structure, having applications in trigonometry, transformations in 2 and 3 dimensions, coordinate geometry in 2 and 3 dimensions, solution of triangles and equations, complex numbers, hyperbolic functions, simple harmonic motion, astronomy etc. Many more applications are also likely to appear. This book shows how the triples (and their 3-dimensional equivalent, quadruples) can be developed and applied and how they form a unifying thread linking many areas of mathematics.

Several chapters from previous editions of this book have been replaced or rewritten. Spherical trigonometry and prediction of planetary positions have gone and chapters on applied mathematics applications, the triple method and hyperbolic functions are now included. Topics which have been extended from previous editions include trigonometric equations and rotation of curves in 3-dimensional space. The chapter on angles in perfect triples has been improved and the chapter on sine, cosine, tangent and their inverses has been completely changed.

This book also serves as an illustration of Vedic Mathematics: a mathematical system which has been rediscovered by Sri Bharati Krsna Tirthaji (1884-1960) from ancient Vedic texts and is expounded in his book (see Reference 1). This system is based on sixteen formulae which are said to give one line answers to all mathematical problems. Being based on fundamental principles these Vedic formulae are therefore conspicuous in any structure that is developed in a simple and natural way. As the triples idea is introduced and extended in this book the operation of these formulae is evident. The formulae are expressed in word form (for example, By One More than the One Before) and as they arise in the text they are indicated by italic type. An index of these formulae will be found at the end of the book.

The diagram below gives a guide as to how the chapters in this book depend upon each other, so that Chapter 8 for example, can be understood by first reading only Chapters 1, 2 and 6.

 

 PREFACE vii

1 TRIPLES 1
The Triple Theorem 2
Some Historical Background 3
Notation for Triples 4
Equal, Prime and Complementary Triples 4
Some Perfect Triples 5

2 TRIPLE ARITHMETIC 6
Addition of Triples 6
Double Angle 8
Triple Subtraction 9
Quadrant Angles 11
Triple Geometry 13
Angles of 30°, 60°, 45° etc. 13
Half Angle 15
Simplifying Calculations 17
Summary 18

3 TRIPLE TRIGONOMETRY 19
Introduction 19
Inverse Functions 21
The General Triple 22
Solution of Trigonometrical Equations 25
Further Trigonometrical Equations: A 27
Further Trigonometrical Equations: B 28
Further Trigonometrical Equations: C 30

4 TRANSFORMATIONS IN A PLANE 32
Transposition of the Origin 33
Rotations 33
Spirals 37
Integration of cosx etc. 38
Rotation of Lines and Curves 39
Reflections 41

5 COORDINATE GEOMETRY 44
Length of Perpendicular 44
Foot of Perpendicular 45
Angle between Two Lines 46
Equation of a Line 48
Further Examples 49

6 CODE NUMBERS 51
Grouping Triples 51
Code Numbers 52
Geometrical Significance of the Code Numbers 54
Code Number Pairs 55
Code Numbers as Triples 55
Algebraic Formulation 56
Converting Code Numbers to Triples 57
Converting Triples to Code Numbers 57
Addition and Subtraction of Code
Numbers 58
Code Numbers of Code Numbers 59
Code Numbers of Complementary
Triples (CT) 59
Code Numbers of Supplementary
Triples (ST) 60
Code Numbers for 0°, 90°, 180°, 270° 61
Relation between Code Numbers and Angles 61
Further Examples 63
Summary 65

7 SOLUTION OF TRIANGLES 66
The Angle-Deficiency Formula 66
The Sine Formula 70
The Code-Number Formula 72

8 FURTHER APPLICATIONS OF TRIPLES 75
Solution of Equations 75
Complex Numbers 77
Conics 78
Difference and Sum of Two Squares 82
Incircles and Circumcircles 83
The Golden Triple 85

9 ANGLES IN PERFECT TRIPLES 86
Revision 86
Triples and Their Angles 87
Finding the Angle in a Given Triple 90
Further Applications of Code Numbers 94
Finding a Triple with a Given Angle 95
A Refinement 96

10 SINE, COSINE, TANGENT AND INVERSES 99
Sine, Cosine and Tangent 99
Inverse Cosine and Inverse Sine 101
Inverse Tangent 104

11 HYPERBOLIC FUNCTIONS 107
Addition and Subtraction 107
Double Angle 108
Equations 109

12 APPLIED MATHEMATICS APPLICATIONS 112
Simple Harmonic Motion 112
Projectiles 117
Forces in Equilibrium 120
Work Done by a Force and Moment of a Force 123

13 THE TRIPLE METHOD 124
Range of Application 125
Deriving the Conventional Formulae 126
Two Comparisons of the Conventional
and Triple Methods 128

14 QUADRUPLES 129
Introduction 129
Quadruple Generators 131
Obtaining the Code Numbers of a Perfect Quadruple 131
The Coordinate Axes 131
Quadruple Subtraction 132
Comparative Densities of Perfect Triples and Perfect Quadruples 132

15 APPLICATIONS OF QUADRUPLES 133
Coordinate Geometry 133
Work and Moment 136
Rotation about Coordinate Axis 137
3-Dimensional Rotation of Curves 138
Rotation 139
Conicoids 141

16 QUADRUPLES IN ASTRONOMY 144
1. Addition of Perpendicular Triples 144
2. Change of Coordinate System 146
3. Quadruples and Orbits 148
4. Quadruple for given i and A 148
5. Inclination of Orbit 149
6. Quadruple Subtraction 149
7. Quadruple Addition 149
8. Doubling and Halving a Quadruple 150
9. Code Number Addition and Subtraction 151
10. Angle in a Quadruple 152
11. Angular Advance 153
12. Relationship between d and A 154
13. To Obtain a Quadruple with a Given Inclination 155

PROOFS 158
ANSWERS TO EXERCISES 163
REFERENCES 166
INDEX OF VEDIC FORMULAE 167
INDEX 168

Pythagorean triples, like 3, 4, 5 have been a fascination for thousands of years. Now for the first time a simple elegant system, based on these triples, has been developed which reveals unexpected applications in many areas of pure and applied mathematics.

These include general applications in trigonometry, coordinate geometry (in 2 and 3 dimensions), transformations (in 2 and 3 dimensions), simple harmonic motion, astronomy etc.

The easy triple method links these areas and replaces large numbers of apparently unconnected formulae with a single device.

This book fully explains the various applications and most of it should be accessible to anyone with the basic understanding of mathematics which a school leaver should have.