Rational trigonometry

Rational trigonometry is a proposed reformulation of metrical planar and solid geometries (which includes trigonometry) by Canadian mathematician Norman J. Wildberger, currently a professor of mathematics at the University of New South Wales. His ideas are set out in his 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry. According to New Scientist, part of his motivation for an alternative to traditional trigonometry was to avoid some problems that he claims occur when infinite series are used in mathematics. Rational trigonometry avoids direct use of transcendental functions like sine and cosine by substituting their squared equivalents. Wildberger draws inspiration from mathematicians predating Georg Cantor's infinite set-theory, like Gauss and Euclid, who he claims were far more wary of using infinite sets than modern mathematicians. To date, rational trigonometry is largely unmentioned in mainstream mathematical literature.The line AB has the general form:Construct a line AD dividing the spread of 1, with the point D on line BC, and making a spread of 1 with DB and DC. The triangles △ABC, △DBA and △DAC are similar (have the same spreads but not the same quadrances). Rational trigonometry is a proposed reformulation of metrical planar and solid geometries (which includes trigonometry) by Canadian mathematician Norman J. Wildberger, currently a professor of mathematics at the University of New South Wales. His ideas are set out in his 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry. According to New Scientist, part of his motivation for an alternative to traditional trigonometry was to avoid some problems that he claims occur when infinite series are used in mathematics. Rational trigonometry avoids direct use of transcendental functions like sine and cosine by substituting their squared equivalents. Wildberger draws inspiration from mathematicians predating Georg Cantor's infinite set-theory, like Gauss and Euclid, who he claims were far more wary of using infinite sets than modern mathematicians. To date, rational trigonometry is largely unmentioned in mainstream mathematical literature. Rational trigonometry follows an approach built on the methods of linear algebra to the topics of elementary (high school level) geometry. Distance is replaced with its squared value (quadrance) and 'angle' is replaced with the squared value of the usual sine ratio (spread) associated to either angle between two lines. (The complement of Spread, known as cross, also corresponds to a scaled form of the inner product between line segments taken as vectors). The three main laws in trigonometry – Pythagoras's theorem, the sine law and the cosine law – are given in rational (square-equivalent) form, and are augmented by two further laws – the triple quad formula (relating the quadrances of three collinear points) and the triple spread formula (relating the spreads of three concurrent lines) –, giving the five main laws of the subject. Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with a point defined as an ordered pair of rational numbers and a line as a general linear equation with rational coefficients a, b and c. By avoiding calculations that rely on square root operations giving only approximate distances between points, or standard trigonometric functions (and their inverses), giving only truncated polynomial approximations of angles (or their projections) geometry becomes entirely algebraic. There is no assumption, in other words, of the existence of real number solutions to problems, with results instead given over the field of rational numbers, their algebraic field extensions, or finite fields. Following this, it is claimed, makes many classical results of Euclidean geometry applicable in rational form (as quadratic analogs) over any field not of characteristic two. The book Divine Proportions shows the application of calculus using rational trigonometric functions, including three-dimensional volume calculations. It also deals with rational trigonometry's application to situations involving irrationals, such as the proof that Platonic Solids all have rational 'spreads' between their faces. Rational trigonometry (RT) is mentioned in only a modest number of mathematical publications besides Wildberger's own articles and book. Divine Proportions was dismissed by reviewer Paul J. Campbell, in the Mathematics Magazine of the Mathematical Association of America (MAA): 'the author claims that this new theory will take 'less than half the usual time to learn'; but I doubt it. and it would still have to be interfaced with the traditional concepts and notation.' Reviewer William Barker, Isaac Henry Wing Professor of Mathematics at Bowdoin College, also writing for the MAA, was more approving: 'Divine Proportions is unquestionably a valuable addition to the mathematics literature. It carefully develops a thought provoking, clever, and useful alternate approach to trigonometry and Euclidean geometry. It would not be surprising if some of its methods ultimately seep into the standard development of these subjects. However, unless there is an unexpected shift in the accepted views of the foundations of mathematics, there is not a strong case for rational trigonometry to replace the classical theory' New Scientist's Amanda Gefter described the approach of Wildberger as an example of finitism. James Franklin in the Mathematical Intelligencer argued that the book deserved careful consideration. An analysis by Michael Gilsdorf of the example problems given by Wildberger in an early paper disputed the claim that RT required fewer steps to solve most problems, if free selection of classical methods (such as the 'shoelace formula' for the area of a triangle from the coordinates of its vertices or applying a special case of Stewart's theorem directly to a triangle with a median) is allowed to optimize the solution of problems. Concerning pedagogy, and whether using the quadratic quantities introduced by RT offers real benefits over traditional learning, the author observed that classical trigonometry was not initially based on use of Taylor series to approximate angles at all, but rather on measurements of chord (twice the sine of an angle) and thus with a proper understanding students could reap continued advantages from use of linear measurement without the claimed logical inconsistencies when circular parametrization by angle is subsequently introduced.