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A brief introduction to Rational TrigonometryGreek approach to measurementN. J. WildbergerEuclid does not deal directly with distance. The ancient Greeks understood the meaning of saying two segments in the same direction were in the ratio `three to two', but they did not have a direct notion of distance, because that would have entailed an understanding of real numbers, which they did not have. Similarly, Euclid does not measure angle. To him, an angle was just the geometrical configuration consisting of two intersecting lines. In fact the modern notion of `radian measure' is only a little more than a hundred years old. Instead of distance and angle, the ancient Greeks believed that area is the fundamental quantity in planar geometry. To measure the separation of two lines, they measured the area of a square built on that line segment, a process called quadrature. A new theory of trigonometry, called rational trigonometry, was developed in 2005 by N. J. Wildberger (UNSW) in Divine Proportions: Rational Trigonometry to Universal Geometry, Wild Egg Books, Sydney, 2005, http://wildegg.com. Downloads from the book, supporting articles and links to reviews are available at http://wildegg.com/authors/. This note briefly motivates this new approach and lists the main laws, which require no transcendental circular functions. Pythagoras' theorem
There are two really important theorems about areas formed by three points.
One is well known, the other much less so. Proposition 47 of Book 1 of
Euclid's Elements states that for a right angled triangle, the
square on the hypotenuse is the sum of the squares on the
other two sides. This version of Pythagoras'
theorem uses area, not distance. Area is an affine concept, in the
sense that proportions between areas are invariant under affine
transformations, which include linear transformations. To measure a line
segment, Euclid measures the area of a square on that
segment, a number which we call quadrance, being close
to the word quadrature. If
Triple quad formula
Pythagoras' theorem has a sister theorem that Euclid does not mention, but
which is implicit in work of Archimedes, and fundamental for rational
trigonometry. It concerns the case when the three points
The figure shows a case where
Main laws of rational trigonometry
An angle is a circular distance, that is distance
measured along a circular arc, and this is too
complicated a concept to qualify as fundamental for measuring the
separation of two lines. To define angles properly you require calculus, a
logical point that is rarely acknowledged by educators. Teachers of
trigonometry constantly rely on
The true separation between lines
is
ratio is clearly independent of the choice of
When lines are expressed in Cartesian form, the spread becomes a rational
expression in the coefficients of the lines. It therefore makes sense over
arbitrary fields, although there is the possibility of null lines for which
the denominator involved in the spread is zero. Note that in the triangle
In diagrams a spread
So now a triangle
The other main laws of rational trigonometry are
The relationship between the three spreads of the triangle is the
Triple spread law:
The Triple quad formula, Pythagoras'
theorem, the Spread law, the
Cross law and the Triple spread
formula are the five main laws of rational trigonometry. They are
implicitly contained in the geometrical work of the ancient Greeks. As
demonstrated at some length in `Divine Proportions',
these formulas and a few additional secondary ones suffice to solve the
majority of trigonometric problems, usually more simply, more accurately and
more elegantly than the classical theory involving
Example
The triangle with vertices
Since the laws are purely algebraic, they hold over a general field.
Example
If we work over the field with
 There is a collection of videos at YouTube that explains in detail the novel view point:
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