Wednesday, 17 June 2009

Trisecting the Angle

It can't be done... You can not trisect an angle using the classic tools of straightedge and compass. But trisecting angles, like squaring the circle, solving quintics is one of those magnets that amateur mathematicians often think they have solved. In April of 1912, Nevil Maskelyne (more about whom later) sent a note to the Philosophical Society with the directions for just such a construction. The first paragraph explains that he was aware of the mathematical view:
"The Trisection of an angle, irrespective of its magnitude, by means of any geometrical construction, has been regarded as impossible. The problem has been associated with perpetual motion, the transmutation of metals, 'zetetic' astronomy (the term zetetic was associated then, and perhaps now, with a psychic approach to knowing), and similar absurdities. Mathematicians have spoken in no measured terms of the folly displayed by those who, in the face of proved impossibility, attempt to provide a solution.
And yet, he goes on to supply what he thinks is a geometric trisection.
His directions, accompanied by my illustrations(click to enlarge) are as follows:
"Take any angle BAC and at regular distances from A, describe the arcs FG, DE and BC with A as center.

Join, respectively BG, CF, DG, and EF. The line AIH, of course, bisects the angle BAC.
With radius AH, and A as center, describe the arc JK cutting DG and EF at points J and K. The lines J and K, trisect the angle BAC (St. George's Hall, W; Dec 11, 1911)

I have created here a geogebra page which will reproduces his (very nearly) trisection of an angle.

In the following month, the first post illustrating the error in his method was presented, and in the following months and years, a number of people sent mechanical means of approximating the trisection of an angle. Because It amuses me, I will try to followup as soon as possible with geogebra illustrations of a couple of the methods.

The first, written on 16 April of 1912, begins, "Gentlemen, It seems unnecessary to give any formal proof that Mr. Maskelyne's construction cannot trisect an angle with exactness; but it may be of some interest to estimate the degree of accuracy in his approximation." The post then goes on to estimate that for a small angle, the error would be about the cube of the angle. It then gives a listing of the errors for angles of 30o (about .10 degrees), up to 150o (almost 5degrees error)..

The Nevil Maskelyne who wrote the communication was not just a typical mathematical kook. He was a Fellow of the Royal Astronomical Society and had participated in and contributed to the observation of the corona of the sun in the US and was apparently a descendant of either the Astronomer Royal, or the brother of the Astronomer Royal of that same name; the Nevil Maskelyne who had been the nemesis of Harrison in his quest for the Longitude prize.

His father (pictured at the top) was a famous magician and spiritualist who invented many tricks that are still performed by modern stage magicians. Interestingly, his father was also the inventor of the pay toilet mechanism. One historical reference lists, "was the best known magician in late 19C England and also a notable mechanic and inventor. He invented the coin lock as used in pay toilets, several proportionally spaced typewriters and he built numerous automata. His whist-playing 'Psycho' [see image at top] of 1875 should be in the London Museum.... There seem to have been two versions - the earlier could do arithmetic, but the later smoked a cigarette instead. The Maskelyne family presented it to the London Museum in 1934. The Director of the Museum recently told me that Psycho is not working. I have been told that a version has been restored to working order by a collector in Los Angeles"
If anyone has information on this, I would love to know.
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