## Triples

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.