Spotting sensitivity in an equation

The new book Heavenly Mathematics describes in the first chapter how the medieval scholar Abū Rayḥān al-Bīrūnī calculated the earth’s radius. The derivation itself is interesting, but here I want to expand on a parenthetical remark about the calculation.

The earth’s radius r can be found by solving the following equation.

\cos\theta = \frac{r}{r + 305.1m}

The constant in the denominator comes from a mountain which is 305.1 meters tall. The angle θ is known to be 34 minutes, i.e. 34/60 degrees. Here is the remark that caught my eye as someone more interested in numerical analysis than trigonometry:

There is a delicate matter hidden in this solution however: a minute change in the value of θ results in a large change in the value of r.

How can you tell that the solution is sensitive to changes (i.e. measurement errors) in θ? That doesn’t seem obvious.

Think of r as a function of θ and differentiate both sides of the equation with respect to θ. We’ll convert θ to radians because that’s what we do. (Explanation at the bottom of this post.) We get

-\sin\theta = \frac{305.1m}{(r + 305.1m)^2} \frac{dr}{d\theta}


\frac{dr}{d\theta} = -\sin\theta \frac{(r + 305.1m)^2}{305.1m}

Now let’s get a feel for the size of the terms in this equation. θ is approximately 0.01 radians, and so sin θ is approximately 0.01 as well. (See explanation here.) The radius of the earth is about 6.4 million meters. So the right side of the equation above is about 1.3 billion meters, i.e. it’s big.

A tiny increase in θ leads to a large decrease in r. For example, if our measurement of θ increased by 1%, from 0.01 to 0.0101, our measurement of the earth’s radius would decrease by 130,000 meters.

I’d like to point out a couple things about this analysis. First, it shows how it can be useful to think of constants as variables. After measuring θ we could think that we know its value with certainty and treat it as a constant. But a more sophisticated analysis takes into account that while θ might not change, our measurement of θ has changed from the true value.

Second, we used the radius of the earth to determine how sensitive our estimate of the earth’s radius is to changes in θ. Isn’t that circular reasoning? Not really. We can use a very crude estimate of the earth’s radius to estimate how sensitive a new estimate is to changes in its parameters. You always have some idea how big a value is before you measure it. If you want to measure the distance to the moon, you know not to pick up a yard stick.

4 thoughts on “Spotting sensitivity in an equation

  1. It’s worth noting that its not just the radius which is sensitive but also the log of the radius. That is, the derivative normalized by the radius (1/r)dr/dtheta is still proportional to something big (r/300m or so). This is the more important metric, as it is dimensionless.

  2. Also note the using simple second- and first-order approximations (resp. on the left and right-hand sides) leads to the simple equation
    r = 305.1 (2/theta^2)
    giving r=6.23824*10^6 (accurate to more than 0.005% if compared to the true solution r=6.23799*10^6).

  3. Your analysis seems to be saying that a 1% error in θ leads to a 2% error in r.

    That shouldn’t be a surprise should it? If you knew nothing about numerical analysis then you might guess that a 1% error in the input made a 1% error in the output, so 2% doesn’t seem so surprising.

    Or is the surprise that 2% * a big number is still a big number?

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