Given that the book is intended for a mathematically mature audience, the development of the material, the calculations, and the accompanying discussions are well organized, clear, and well-written. I found no flaws in the mathematics although I did not read every argument in depth. The book presents an interesting and creative approach to the development of trigonometry and geometry. This could provide fertile ground for further investigation. The intuitive geometric clarity of distance and angle measure does not carry over to quadrance and spread. Moreover, additivity for distance between collinear points and for angle measure between adjacent angles is natural, expected, and useful in the real world; this property does not hold for quadrance and spread.

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About this title This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field.

This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus.

Examples over finite fields are also included. His main research interests are Lie theory, representation theory and hypergroups.

He has also done work in number theory, combinatorics and mathematical physics. He is married, and lives in Sydney with his wife Kim and daughter Ali. This is his first book.

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## Divine Proportions: Rational Trigonometry to Universal Geometry

This is a dramatic new approach to a beautiful subject, avoiding the logical difficulties that have beset it since the time of Newton and Leibniz by relying only on computationally explicit and concrete algebra, and avoiding all "infinite processes" and "arguments". It is a fully online course hosted at Open Learning. We first introduce the integral calculus, starting with a linear algebraic definition of signed areas for triangles and then splines, and moving progressively to de Casteljau Bezier curves. There will be 10 Chapters, opening up month by month.

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## Divine Proportions: Rational Trigonometry To Universal Geometry

About this title This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus. Examples over finite fields are also included.

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## ISBN 13: 9780975749203

Chapter27 New posting Revolutionary content This text introduces a new and simplified approach to trigonometry and a major restructuring of Euclidean geometry. It replaces cos, sin, tan and all those other transcendental trig functions with rational functions and elementary arithmetic. And it shows how to apply this new theory to a wide range of practical problems from engineering, physics, surveying and calculus. New formulas and theorems Divine Proportions: Rational Trigonometry to Universal Geometry contains dozens of new and important formulas, and almost a hundred theorems. The formulas are mostly polynomial or rational, and the basic ones, such as the Triple Quad formula, the Spread Law, the Cross law and the Triple Spread formula are quadratic in any variable. The theorems are often based on existing results, but they hold in much greater generality, and their formulations are sometimes unusual.

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## Rational trigonometry

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