A revolutionary new text


DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry

by N J Wildberger (Assoc Prof UNSW Sydney)

provides a complete and powerful alternative to classical trigonometry and Euclidean geometry. Now available here at

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Review by Prof Jan Aarts (Delft Univ) for NAW (Nieuw Archief voor Wiskunde) 5/11 Sept. 2010

Review by Prof Michael Henle for The American Mathematical Monthly (Dec 07)

Review by Prof Robert Dawson for CMS Notes

Review for Newsletter of the European Mathematical Society (June 07)

Review by Dr Laura Wiswell for The Proceedings of the Edinburgh Mathematical Society

Review by Dr Rod Gover for The Gazette of the Australian Mathematical Society

Review by Prof James Franklin for The Mathematical Intelligencer

Review by Prof. S. L. Arlinghaus for Solstice: An Electronic Journal of Geography and Mathematics

Review by Prof. Michael Hardy for

Review by Prof. Joseph C. Varilly for


Free downloads (pdf) : Preface / Contents / Introduction / Chapter1

Book Description

Rational trigonometry

This new theory, described here for the first time, dispenses with all the usual complicated trigonometric functions and relations and establishes a trigonometry based entirely on elementary arithmetic and algebra. The essential quadratic aspects of geometry are brought to the fore, with the key concepts of quadrance and spread replacing distance and angle. The two subjects of geometry and periodic motion are cleanly separated; it is neither necessary nor desirable to have a theory which merges the two.

In Divine Proportions: Rational Trigonometry to Universal Geometry, Associate Professor N J Wildberger of the School of Mathematics at the University of New South Wales shows how to solve even complicated geometric problems without tables or calculators, with more accurate answers. Many engineering, surveying and physics problems are solved in this book, illustrating in detail the advantages of this new approach. For large scale problems, rational trigonometry harnesses modern computational power more efficiently.

Because of the rational form of the basic laws, the theory extends to general fields, opening up numerous new areas of investigation for number theory, algebraic geometry and combinatorics. A lovely new family of orthogonal polynomials called spread polynomials extends the theory of special functions to general fields.

Universal geometry

Using rational trigonometry as a basis, a new form of metrical Euclidean geometry is constructed, incorporating most of high school geometry, but extending in exciting new directions. This new Universal Geometry contains similiar triangles, Heron's formula, centroids, circumcenters, orthocenters, cyclic quadrilaterals and Brahmagupta's theorem, circles, parabolas, the Euler line, tangent lines and tangent conics, nine point circles, and much more---all over a general field!

Examples are taken not only from the usual decimal numbers, but also finite fields and the complex numbers.


Divine Proportions argues that there is a good reason why the standard high school mathematics curriculum turns so many people off mathematics, often permanently. The reason is that the mathematics being currently taught is logically weak and overly complicated for what it accomplishes. It confuses the study of triangles with the subject of uniform circular motion. These are separate areas of mathematics, requiring different concepts and techniques.

To understand the geometry of a triangle, one doesn't require transcendental trigonometric functions such as sin, cos, tan and their inverse functions. Until now, there has been no sensible alternative to focus the minds of educators, even though most students intuitively understand that the theory is too complicated. This book finally addresses the failure of modern geometry to win the minds of young people, and provides the mathematical foundation for a dynamic and elegant new approach to teaching trigonometry.


  • Two dimensional geometry problems
  • Surveying problems (Elementary)
  • Surveying problems (Advanced)- Resection and Hansen's problems
  • Boxes, pyramids, wedges and pagodas
  • Physics---Projectile motion, Snell's law, Algebraic dynamics, Lorentzian addition of velocities (as in Einstein's special theory of relativity)
  • Platonic solids
  • Folium of Descartes and Bernoulli's lemniscate
  • Five fold symmetry and regular polygons
  • Calculus problems involving rotational and spherical symmetry (volumes, surface areas, centroids, moments of inertia of spheres, caps, hyperboloids, toroidal rings etc.)
  • New formulas for classical curves using rational polar coordinates.

Features of the book

  • Introduction and overview chapter motivate and explain the key features of the new theory, along with some of the weaknesses of traditional trigonometry
  • Preliminary chapter provides review of necessary background material
  • More than one hundred theorems (most of them new) over four parts
  • More than one hundred and seventy illustrative diagrams
  • Index with separate table of theorems
  • Two appendices.

Intended audience

The book's content is largely elementary, but is presented concisely. It requires mathematical maturity and skill at algebraic manipulation, along with an interest in geometry and its applications. It will be especially valuable to

  • Professional mathematicians, especially those with an interest in geometry (including algebraic geometry and differential geometry), number theory, combinatorics and special functions
  • Scientists with an interest in mathematics, i.e. physicists, chemists
  • Engineers and some computer scientists
  • Mathematically talented high school students
  • Undergraduate mathematics or physics majors
  • High school and college mathematics teachers and lecturers
  • Amateur mathematicians with strong algebraic skills and an interest in geometry.

General members of the public who do not fit into one of these categories may well find the book too technical to be easily accessible. Dr Wildberger intends on writing a companion book at a more elementary level which explains the subject to the general public.


Dr N J Wildberger (BSc University of Toronto, PhD Yale University) has taught at the University of Toronto (as an undergraduate), Yale University (as a graduate student), Stanford University and the University of Toronto.

He is currently Associate Professor in the School of Mathematics and the University of New South Wales (UNSW) in Sydney Australia. His main research interests are Lie theory, hypergroups, representation theory and geometry. He has also done work in number theory, combinatorics and mathematical physics. This is his first book.

Wildberger is now using Rational Trigonometry as a springboard to redesigning metrical geometry in a large-scale way. He has recently completed a trilogy of papers on Universal Hyperbolic Geometry. Rational Trigonometry is also supported by the YouTube series WildTrig available at Wildberger's channel Insights into Mathematics.

First edition

This is a quality hardcover book of 320 pages. It uses 100% acid free paper, section sewn and printed by BPA group, one of Australia's foremost printers, with a striking cover design by designer Alex Snellgrove. This book is available here at this website, for the special low price of $49.95 (AUS) plus shipping.

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Divine Proportions: Rational Trigonometry to Universal Geometry is now available online here at for the special low price of $49.95 (AUS) plus shipping. Order now

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