Curious results of an incorrect “theorem”

I hope all readers of this blog (all three of you – you know who you are!) have heard of, and maybe even read, Underwood Dudley‘s hugely enjoyable book: “A Budget of Trisections” or its later version “The Trisectors“. Dudley has made it a lifetime hobby to collect the stories, and the writings, of people he refers to as “cranks”. For a lovely introduction to this weird world, you may read his article What to do when the trisector comes.

It turns out there are many people out there, with dimly recalled snippets of school mathematics, who fail completely to understand what it means in mathematics to be impossible: for example, that a general angle cannot be trisected by using only Euclidean straight-edge and compass. Augustus de Morgan, the eminent 19th century British mathematician, also collected such cranks, whose writings he called “paradoxes”, in the old sense of meaning a statement contrary to accepted opinion. His own book: A Budget of Paradoxes vol I and vol II (1872) is now online, and I can heartily recommend it. De Morgan quotes a “paradoxer” (his happy name for persons producing such paradoxes) as claiming his work to be “The consequence of years of intense thought”. De Morgan’s terse comment is “very likely, and very sad.”

Well, I have a paradoxer for you. This one did not come to me in person, I found this in a published journal, a journal with a board of reviewers and all the trimmings. The article, published in 2013, is called ARC LENGTH of an ELLIPTICAL CURVE and the journal is the nobly named International Journal of Scientific and Research Publications. The author claims to have “a new patent rule for computing ARC LENGTH of an ELLIPTICAL CURVE. It is based on Geometrical Theorems.” Indeed. I don’t know if this rule has in fact been patented, but it would be in keeping with the spirit of mathematical cranks if it was – apparently patenting their wonderful results is all part of the behaviour of the species.

So what is this rule? It says (I am slightly paraphrasing here) that:

The circumference C of an ellipse given by the equation \displaystyle{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1} is given by the formula

    \[ C=2\pi\sqrt{\frac{a^2+b^2}{2}}. \]

We don’t need no steenking elliptic integrals!

As far as I can tell, the author arrived at this splendid result by assuming that the arc length of an ellipse is linear in the subtended angle at the centre – which is true only if the ellipse is a circle.

The exact circumference of an ellipse, as you know, can be expressed as an integral:

    \[ 4\int^{\pi/2}_0\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}\,d\theta \]

so that if, for example, (a,b)=(4,3), then the arc length is

    \[ 4\int^{\pi/2}_0\sqrt{16\cos^2\theta+9\sin^2\theta}\,d\theta=16\int^{\pi/2}_0\sqrt{1-\frac{7}{16}\sin^2\theta}\,d\theta =16E(\sqrt{7}/4) \]

where E(k) (sometimes written E(\pi/2,k) is the complete elliptic integral of the second kind. Here \displaystyle{k^2=1-\frac{b^2}{a^2}}. The values of E(k) can be determined numerically, and most computer algebra systems will have this function built in. Note that sometimes the function E is defined in terms of k^2; thus E(k^2). For the values given, we have

    \[ E=1.318472107994621 \]

and so the total circumference is 21.09555372791393 to those decimal places. According to the new patent rule,

    \[ 2\pi\sqrt{\frac{16+9}{2}}=22.21441469079183 \]

which is out by quite a considerable margin.

We can write the new patent rule as an integral

    \[ 2\pi\sqrt{\frac{a^2+b^2}{2}}=4\int^{\pi/2}_0\sqrt{\frac{a^2+b^2}{2}}\,d\theta. \]

Comparing this with the elliptic integral above, we find that

    \[ \frac{a^2+b^2}{2}=a^2\cos^2\theta+b^2\sin^2\theta=b^2+(b^2-a^2)\cos^2\theta. \]

Solving this equation for \cos\theta we find that

    \[ \cos\theta=\frac{1}{\sqrt{2}} \]

or that \cos\theta is a constant function. Betcha didn’t know that!

As a side issue, you’ve got to wonder about a journal with any pretensions to legitimacy publishing this sort of thing.

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