||"dawnbreak in the west"|
Monday, September 19, 2005
Moldy old Babylonian gods
This could be good news: a new trigonometry is arising. Australian mathematician Dr Norman Wildberger wants to rid us of our sines, cosines, and other such functions, replacing them with more intuitive concepts. His upcoming book is Divine Proportions: Rational Trigonometry to Universal Geometry and will be published by Wild Egg books.
Currently, our means of measuring the divergence between two lines is the obvious way: the fraction of a circle, alias "angle". Fortunately the common system of measuring angles is a fractional system. Unfortunately it's a Babylonian fractional system, with sexagesimals instead of decimals. When we write "900" we're expressing that the angle is 900/3600 of a circle. A Babylonian would be able to scratch this into his clay tablets easily as a second-order fraction, with the glyph for "15" in the fractional column for sixtieths. This Babylonian wouldn't even need to fill out the column for 3600ths. His readers would spot it instantly as one quarter the circle, even faster than we could spot a ".25" as one quarter of "1".
The Greeks and Romans kept their own notation for simple arithmetic and borrowed Babylonian notation for trigonometry (and therefore timekeeping, mapmaking, and astronomy). My guess for why they didn't move to a base-x notation is that they associated it with the Babylonians, and that the Babylonian multiplication tables were absurdly complicated: decimals require you to memorise a hard-enough 55 pairs, but sexigesimals demand an impractical 1830. Europeans gradually moved toward Arabic notation (decimals) for keeping accounts and taking measurements, but kept the vestiges of Babylon in the sciences.
I'd actually quit using Babylonian notation during college, when I realised that it wasn't helping with my calculus. Differentiation and integration of sine functions of "x" assume that x is expressed in "radians"; 2*pi of such radians filling out the circle. So I started using my own notation, the "pie-radian". I'd say that the aforementioned "900" angle was a "half-pi radian" angle. But there's still a lot of pi floating around that's hard to calculate with precision.
Wildberger's fractional "spread" is not an angle, although it is a function of one: it's the square of the opposite-over-hypotenuse (sine). This means he doesn't even need a protractor to calculate his stuff. So no more Babyloniana and, as Eric Cartman would say, no... more... pi.
I can only wish this cause the best, whether or not Wildberger's way proves the best way.
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