This afternoon on #perl, the topic was "Trig refactored to remove sin/cos/tan" with a pointer to the upcoming Divine Proportions: Rational Trigonometry to Universal Geometry by N J Wildberger.
The introduction makes an ambitious claim:
Rational trigonometry deals with many practical problems in an easier and more elegant fashion than classical trigonometry, and often ends up with answers that are demonstrably more accurate. In fact rational trigonometry is so elementary that almost all calculations may be done by hand. Tables or calculators are not necessary, although the latter certainly speed up computations. It is a shame that this theory was not discovered earlier, since accurate tables were for many centuries not widely available.
Because rational trigonometry uses only arithmetic and algebra, it allows the development of Euclidean geometry in a consistent and general way in an arbitrary field. This is universal geometry.
Emphasis Wildberger's, not mine. I look forward to reading more!