A new mathematics book...
DIVINE
PROPORTIONS: Rational Trigonometry to Universal Geometry
by N J Wildberger
"A
unique and revolutionary text which establishes new
foundations for trigonometry and Euclidean geometry'
AVAILABLE NOW from Wild Egg Books
: http://wildegg.com
Order
here
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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. It develops a complete theory of planar
Euclidean geometry over a general field without any
reliance on `axioms'. 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. For example the new Spread
polynomials replace the closely related Chebyshev
polynomials of the first kind, but extend to arbitrary
fields.
Increased accuracy
Classical
trigonometry relies on transcendental functions, so
that the numbers output by calculators are only approximations
to the true values. This means round off errors are
ever present in classical trigonometry, and for complicated
computations these can become significant. Rational
trigonometry is more accurate, since ordinary arithmetic
carries along all the information as it goes. Decades
ago this would have been unwieldy for large scale applications,
but with the development of computer algebra systems,
such as MAPLE, MATHEMATICA, MUPAD and so on, it can
be made fast and accurate.
New areas of research
Divine Proportions: Rational Trigonometry
to Universal Geometry opens up new areas of research
not only in Euclidean geometry, but also in algebraic
geometry, number theory, combinatorics,
sprecial functions, Lie theory and
non-Euclidean geometries. It proposes
a return to the classical algebraic geometry of Fermat,
Newton, Euler and their contemporaries, where metrical
aspects are encoded in the algebraic concepts of quadrance
and spread.
Opportunities for mathematics
education
Trigonometry
has for centuries been difficult for students to learn.
Part of the reason is that the foundations of the subject
are problematic, relying on vague pictorial definitions
of the key concepts of angles and trigonometric functions.
That means students rarely come to grips with the logical
structure of the theory. Rational trigonometry is much
simpler, relying on the fundamental quadratic notions
of quadrance and spread,
and with basic laws that can be derived completely
rigorously using only elementary algebra.
Quality book
Divine
Proportions: Rational Trigonometry to Universal Geometry
is published by Wild Egg, an innovative new publishing
company specializing in ground breaking mathematical
titles. The first edition is hardcover, 320 pages on
100% acid-free paper, with section sewn binding. It
was printed by BPA, one of Australia's foremost quality
printers. The cover design is by Alex Snellgrove. It
is available online at http://wildegg.com.
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