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Odd analysis in an old book

July 29, 2025 at 11:28 PM by Dr. Drang

In retirement, I’ve taken to going through old engineering textbooks to fill in holes that weren’t covered in my classes. These aren’t necessarily the assigned texts, but they’re books I heard about. Recently, I’ve started on Mathematical Methods in Engineering by Theodore von Kármán and Maurice Biot, published in 1940.

Early in the book, in a section about using separation of variables to solve ordinary differential equations, the authors derive the equation for a catenary in a way that seems a little odd to me. Here it is, where I’ve put together into a single image an analysis that’s spread over two pages in the book:

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The first thing you may have noticed is that the last two equations,

$$\frac{dy}{dx}=\tan \theta =\frac{1}{2}(e^{\frac{wx}{H}}-e^{-\frac{wx}{H}})$$

and

$$y=\frac{H}{2w}(e^{\frac{wx}{H}}+e^{-\frac{wx}{H}})+C$$

don’t simplify their form by using hyperbolic trig functions. Most authors write these as

$$\frac{dy}{dx}=\tan \theta =\frac{1}{2}\sinh \frac{wx}{H}$$

and

$$y=\frac{H}{2w}\cosh \frac{wx}{H}+C$$

I don’t understand why they went to the trouble of manipulating the integral into the difference of two exponentials if they weren’t going to then use the definition

$$e^u-e^{-u}=\sinh u$$

It’s not wrong, just odd.

Even stranger, I thought, was the expression for

$${\int }_0^{\theta }\frac{d\theta }{\cos \theta }$$

I’m not complaining about how they should have used a dummy variable instead of $\theta$ inside the integral (although they should have). No, what’s strange is the expression they used for the answer:

$$\ln \tan (\frac{\pi }{4}+\frac{\theta }{2})$$

where I’m making it explicit that the logarithm is the natural log. Not only does the introduction of the $\pi /4$ term seem to complicate the expression unnecessarily, it leads to the use of this bizarre trig identity,

$$\tan (\frac{\pi }{4}+\frac{\theta }{2})-\cot (\frac{\pi }{4}+\frac{\theta }{2})=2\tan \theta$$

to get the integral in the form they want.

Now it’s true that I haven’t memorized a lot of trig identities, but that’s one I’m sure I’ve never even seen before. And the thing is, there’s a simpler expression for the antiderivative of $\sec \theta$:

$$\ln (\sec \theta +\tan \theta )$$

How is this simpler? Well, it’s obvious that

$$\sec \theta +\tan \theta =\frac{1+\sin \theta }{\cos \theta }$$

so it’s fairly easy to show that

$${(\sec \theta +\tan \theta )}^{-1}=\sec \theta -\tan \theta$$

as long as you remember that

$$(1+\sin \theta )(1-\sin \theta )=1-{\sin }^2\theta ={\cos }^2\theta$$

Therefore,

$$(\sec \theta +\tan \theta )-(\sec \theta -\tan \theta )=2\tan \theta =e^{\frac{wx}{H}}-e^{-\frac{wx}{H}}$$

which is where the authors want to go to get to Equation 3.11.

I now feel an obligation to look into books of similar or earlier vintage to see if von Kármán and Biot’s derivation technique was common back then.

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