Friday 4 February 2011

A Geometrical Theorem

I came across the following beautiful result in Clifford A  Pickover's Wonderful Math Book.
It got me interested, and so I set out to see what I could find.

The article below appeared in The Quarterly Journal of Pure and Applied Mathematics: Volume 2 - Page 38  in 1858. 

The simply titled article, by an almost unremembered Cambridge mathematician, must rank as one of the most unusual and interesting in all of math. Although demonstrated with a circle, the problem states that if you construct a chord on any closed curve which is smooth and convex, and divide the chord into two lengths p and q, with a point X; as the chord is moved around the circle the point will generate a path that will always cut out the same area between the locus and the outside boundary of the curve.... No matter what shape the original curve was..and that area is the area of an ellipse with semi-major axes of length and q..... yet there is no ellipse involved in the problem.  ....Or is there.

The motion is reminiscent of a drawing instrument called a trammel.  See Archimedes Trammel at Wikipedia.

In this image the trammel has the scribe, or drawing point, on the exterior of the two guide points where the theorem above presumes the scribing point is interior to the two guides, but that locus will also form an ellipse when drawn on two straight lines for the guides. 

Many students may already have seen that ellipse, or at least a fourth of one in illustrations of the Falling Ladder Problem which often shows up in related rates problem in calculus texts. For this illustration a fixed point on a ladder is followed as the foot of the ladder slides away from the wall and the top of the ladder slides down the wall.  An interactive image for those  who have never seen it is at the Geometers Sketchpad site.

But this means that in some cases, the idea that the outside curve must be smooth (differentiable?) is not true.  If you moved the chord along a square, or rectangle with sides greater than the length of the chord, then it would curve out four quater-ellipses around each corner of the rectangle, with a total area of    
I'm trying to think of a illustrate this theorem with an irregular curve on Sketchpad or Geogebra, but  the task has overwhelmed me to date... if you do it, send me a copy and I will link it here
History stuff

So anyway, it made me wonder about  this mysterious guy, Hamnet Holditch.  It turns out he must have been incredibly bright.  He was Senior Wrangler at Cambridge, and also won the Smith Prize.  In England, this is a big deal, and folks make a fuss about it.. they did then, they do now.

He went on to be a fellow of Gonville and Caius College and hold a number of positions.  But as the passage below indicates, he "held" them rather loosly

From Mr. Hopkins' Men: Cambridge Reform and British Mathematics in the 19th Century.

As it turns out he was from the area of King's Lynn in Norfolk, not far from where I live in  the UK.  I found a little more about him online.

Henry Hillen's e-book, "History of the Borough of King's Lynn".was born at Lynn (1800); he was the son of George Holditch,
described as pilot, beaconer and harbour-master, to whom the Society
of Arts awarded a gold medal for the invention of life-saving beacons
(1833); modifications of whose designs are still erected upon the
treacherous sands of the Lynn Deeps. The son was for three years
instructed by the Rev. L Coulcher, at the Grammar School.
When eighteen years of age, he was admitted pensioner
of Caius College, Cambridge (6th February 1818); he gained
his B.A. (senior wrangler and Smith's prizeman) in 1822,
when, as a mark of esteem, he was presented with the
freedom of Lynn; his M.A. in 1825, and was senior fellow from
1823 to his death. Probably, in view of a fellowship, he was
baptised late in life at St. Michael's church, Cambridge (17th March
1823). He held various responsible college offices, for example,
Hebrew and Greek lecturer, bursar (1828), etc. He was remarkable
for his extreme shyness. Owing to some slight, perhaps more
imaginary than real, he absented himself for many years from Hall
and Chapel, and was thus known to a few only. Hence a junior
fellow, taking him for an interested strnngcr, ])olitely shewed /lim
round the college ! He delighted in angling, and spent his summers
in Scotland or Wales. He died in college the 12th of December 1867, and was
buried at North Wootton.

No comments: