On trisection, a humble contribution

Because Trilema really does contain everything.

So recent discussion got me to reviewing old texts, wherein look what we find :

Let an arbitrary angle be BOA. Extend OA. Marc AC and CD of same length as OA. Draw parallel to OA through B. Draw the arc DE with center C. Drop a perpendicular from E to OA, call the foot F. Draw the arc FT, with center O. The segment OT will trisect the angle BOA.

You can draw it if it helps, but if you don't need this drawn you probably realise, as Underwood Dudley correctly points out (and unlike other things he incorrectly points out), that it is equivalent to the proposition that sin (A/3) = sin A / (2 + cos A).

This, it must be said, is actually a trisection of the angle. Not in the theoretical sense, of course, but very much in the practicalⁱ. Here's what I mean :

In the above graph, the sine of X is drawn in green ink, the sine of X/3 is drawn in blue ink, and the sine of the constructed angle is drawn in red ink. As you can see it lays at variance with the correct value both quantitatively and qualitatively (it's negative when it shouldn't be etc), but nevertheless it's not that far off.

Then again, just going by the unaided eye is also not that far off. Not to mention a protractor (do you know what a protractor is ?) or, for that matter, scissors.ⁱⁱ And besides, the author (which is neither Dudley nor Gardner) will be sure to point out that "the trisection is not intended to be applied to obtuse angles". Perhaps if he were the sort that'd be admitted in TMSR he'd add "or by obtuse people", but regardless, he would have a point : between -0.25 and +0.25 his approximation is actually very close.ⁱⁱⁱ

So then, you have learned an escher-and-compass method for trisection of the angle from Trilema. What now ?

And is it a cult yet ?

1. Which is exactly what the word actually denotes [↩]
2. Someone apparently proposed to cut out the fucking angle out of the paper, turn it around in three dimensions such as it makes a cone, trim this cone so as it is a right cone, and then trisect the circular base. Because you see, the resistence of the medium here implemented by the piece of paper will come to our rescue! And the fundamental difference between math and etsy crafts is taught nowhere -- today as in 1917. [↩]
3. And I very much recommend the combined numerical-visual analysis method here employed for general use in general practice in everyday life. Neither Dudley nor Gardner had easy access to numeric machines and so they never discovered this obvious point, but look how easy it makes our life! Just by looking at that graph, and noticing the inflection points, you should readily have three or four approaches to geometrically disprove the included fallacy. [↩]