Wednesday, 8 July 2009

Finding the area of a circle

Having made an abortive attempt to get a degree in mechanical engineering, I did get far enough advanced in pure mathematics to be able to appreciate how beautiful some of it can be, even if—or perhaps especially if—you don't quite understand much of what's going on.

For example, I remember how awed I was—silent, upon a peak in Darien, you might say—when I was shown one of the ways of calculating the area of a circle using calculus. At that time I did have a hazy notion of what calculus was about; it's all gone, of course, and nowadays I couldn't tell the difference between an integral coefficient and a packet of soda mints.

No great disgrace in this. Ever since the Rhind papyrus, thought to be based on a document dating from 1800 BC, the greatest minds have wrestled with these concepts. Archimedes and others had a shot at squaring the circle, but it took Newton or maybe Leibnitz to make calculus really useful.

Anyway, never mind about that. What I do remember is that the particular method of finding the area of a circle that I was shown consisted of only a few lines of ineffable elegance. It went like this:

You take a slice across the circle, of thickness x and width y. Then, and this is the bit which will cause anyone who knows any serious mathematics to fall about with uncontrollable laughter, you can say that the area of the strip is d(x) times d(y) and when d(x) tends to zero and d(y) is twice the radius of the circle then the area of the circle is obtained by giving the whole thing a thorough differentiation and then integrating between the limits of something or other. The answer, as everybody knows, then works out as πr(squared). (How do you do superscript in Blogger?)

All the above is of course meaningless gibberish, but it should give enough of a hint to enable someone with a bit of maths (or math) to identify the particular method that I so vaguely remember and then to spell out the few elegant lines which have swum out of my ken.

If there is a Senior Wrangler out there who can remind me of them then I shall be happy to reward him for his trouble by sending a cheque for twenty pounds sterling to the Save the Children Fund.

[Oh no, I see that the last Senior Wrangler, who was admitted a hundred years ago, died in 1946, so he won't be much help. Anyone else, then?]


Froog said...

I wonder if the term Senior Wrangler is still in use, informally. I'm sure people in the Cambridge mathematical community must know who tops the class lists each year, particularly if they are an outstanding talent, conspicuously outperforming their contemporaries.

When you say "the last Senior Wrangler died in 1946", I suppose you mean the one who graduated in 1909, the last year the University chose to publish a full ranking of results. The poor chap wasn't all that old at his demise, probably not yet 60. I would imagine the last surviving Senior Wrangler might have been around quite a bit longer than that: surely some of the last ten or so holders of the accolade must have lived into the 1960s?

I never really grasped calculus, I'm afraid. Maths was my weakest subject, so I cannot assist with the full formula/explanation you are seeking.

Did you see a TV documentary a decade (or two) ago - BBC2's Horizon, I suspect - which suggested that Archimedes might have invented calculus a thousand years before anyone else, but that the crucial notebooks had been lost?

Tony said...

Well, OK.

Percy Daniell was the last person to become Senior Wrangler but not necessarily the last SW to to die. I suppose the DNB would tell us but as you raised this quibble I will leave the necessary research to you.

I really did grasp calculus, briefly, but like so many other once important parts of my life it is like some insubstantial pageant faded and left not a wrack behind.

I rather think I did see the documentary you mention but that's one more thing of which I have little recollection. It's a terrible thing when your memory starts to go (this is the punch line of a very funny story which I'm sure you know).

I've had no replies yet and I think my £20 is fairly safe; your top mathematician may be a fast man with a Bessell-Clifford function but is rarely much good at writing.

Elizabeth said...

Why on earth did the ancient Greeks want to know?
Look where it has got us.

Tony said...

Dolmades, feta, ouzo, tzatziki, keftethes, retsina, non-stick saucepans, hovercraft.... the list is endless. Oh, and don't forget those little bits of stuff wrapped up in thin pastry.

Tony said...

The response to my query has been very disappointing: only one real mathematician has answered. Johnf has very kindly send me a clear explanation of a method of calculating the area of a circle using calculus.
But he uses the method based on first finding the area of a quadrant; there are several variations of this listed on the internet, of which this is one. That's fine, but it isn't the one I was seeking, which takes a chord slice and must be simpler. Or perhaps there is no such method and I just dreamt the whole thing.
Anyway, Johnf should not feel that his effort has been wasted and to express my appreciation I am awarding a second prize by sending a cheque for £10 to the Save the Children Fund.

johnf said...


Using the tags 'area of circle' and 'history' goole brought up this approach by archimedes

which was a method of successive approximations using inscribed polygons of increasing numbers of sides.

If there had been a simpler method not using the calculus, Archimedes presumably would have found it - or Euclid!

Tony said...

Yes, I daresay.

Sal said...

i've seen a CV recently with Senior Wrangler writ proud (by a living person).

>(How do you do superscript in Blogger?)

<super>wibble</super> should work.

altho i note i'm unable to use it in this comment box.

Tony said...

Then he's a pretentious idiot.

I've remembered now, 2 squared can be written ^2