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Synopsis The unchanging laws of numbers have always been a source of delight and inspiration. Particularly attractive are the Pythagorean triples which have so many elegant and interesting properties. Pythagorean triples, like 3,4,5 have been a fascination for thousands of years. Now, for the first time, a simple and elegant system, based on these triples, have been developed which reveals unexpected applications in many areas of pure and applied mathematics. These include general applications in trigonometry, coordinate geometry, transformations, simple harmonic motion, astronomy etc. This book shows how the triples 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. Chapters on applied mathematics, applications, the triple method and hyperbolic functions are now included. Topics from previous edition, which have been expanded 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 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.