## They all laughed (2): al-Biruni

In Part 1 we saw how Eratosthenes, working within the ancient Greek mathematical tradition, obtained the first (maybe) (roughly) accurate estimates of the size of the Earth. We now pick up the story almost a thousand years later…

In ancient Greece, the principal branch of mathematics had been geometry. Greek geometry, encapsulated in the work of Eratosthenes’s predecessor Euclid, was an elaborate system of formal proofs, to be developed with minimal tools (an unmarked ruler and a pair of compasses). The Islamic mathematicians extended this with the systematic use of trigonometric ratios, and they also made the first steps toward the system of symbolic manipulation that became algebra. A major motivation for their mathematics was astronomy, which also drove the development of tools for making astronomical measurements. In particular, they brought to a very high degree of perfection a tool called the astrolabe, used among other things for measuring the elevation of a point above or below the horizontal.

At this point, Abu Rayhan al-Biruni (973–1048) enters the story. Another remarkable polymath, who was employed as an adviser by Sultan Mahmud of Ghazni, al-Biruni made significant contributions to mathematics, astronomy and political history, as well as being one of the first Westerners to engage seriously with Indian culture and philosophy. He also seems to have had a sharp sense of humour and an accurate notion of his own considerable abilities. It’s not clear whether al-Mam’un’s surveyors’ methods were too complicated for his tastes or whether he simply didn’t trust anybody else to do the job, but in his book Determination of the Coordinates of Cities, he announced that he had developed “another method for the determination of the circumference of the Earth. It does not require walking in circles.”

Al-Biruni’s approach not only avoids walking in circles: it also doesn’t require a friend with a conveniently positioned well, or even a sight of the sun. (Thus it could be carried out successfully even in Scotland.) It rests on the basic principle that the distance we can see depends on our elevation: the higher we are, the further away the horizon gets. To carry it out one requires simply an astrolabe, a working knowledge of trigonometry and a convenient mountain. As an astronomer, al-Biruni was expert in the use of the first; as a mathematician he was familiar with the Persian mathematician Abu’l-Wafa al-Buzjani’s recently calculated trig tables; and he found his convenient mountain overlooking the flat plains of Punjab during Mahmud’s campaigns in India in the first few decades of the 11th century.

Schematic of the horizon method. A is the observer; B is the horizon as seen by the observer; C is the centre of the Earth; D is the base of the tower or mountain; AH is a horizontal line (perpendicular to the vertical AC).

The figure above illustrates the geometry of the problem that al-Biruni set himself. We need to calculate the distance $R = |AC|$. The simplest way to proceed is to measure the distances $|AD|$ and $|AB|$ and apply Pythagoras’s theorem:

$|BC|^2 + |AB|^2 = |AC|^2 = (|AD|+|CD|)^2 = |AD|^2 +2|AD|\cdot|CD| + |CD|^2;$

then since $|BC| = |CD| = R$, we have

$R = \dfrac{|AB|^2-|AD|^2}{2|AD|}.$

The problem with this is the need to measure $|AB|$ accurately. Even assuming that the angle $\theta$ is small enough that $|AB|$ can be approximated reasonably by the arc-length from B to D along the surface of the Earth, we require a measurement over a considerable distance to get a reasonable estimate of $R$. (This is because $R$ is expressed as the difference of two lengths; unless the two terms are very different in magnitude, the relative errors will be large.)

Alternatively, if we have access to trigonometry and can measure the angle $\theta$, we can get away without measuring $|AB|$. We know that

$\cos(\theta) = \dfrac{|AB|}{|AC|} = \dfrac{R}{R+|AD|},$

and solving for $R$ then gives

$R = |AD|\dfrac{\cos(\theta)}{1-\cos(\theta)}.$

Only a single measurement of length is now required, but the accuracy of this estimate will now depend on how accurately we can measure angles and evaluate trig functions. (Since $\cos(\theta) \sim 1-\frac{1}{2}\theta^2$, we have $R \sim 2|AD|/\theta^2$ for small $\theta$, so small errors in measurement will be crucial.)

Each of these approaches requires us to measure, at the very least, the vertical length $|AD|$ (the altitude of the observer). If the observation point is a building then this may already be known, or it may at least be possible to measure it directly by hanging a plumbline from the top. Especially in the ancient world, though, there were not many buildings of sufficient height and with a clear view to the horizon. (One of the few that might have been suitable was the Pharos lighthouse of Alexandria, and this has led to speculation that this method was in fact what Eratosthenes used rather than the solar method he described.) More suitable would be a mountain overlooking either the sea or an extensive flat plain, but this gives the extra task of measuring the height of the mountain.

Schematic of how to measure the height of a mountain. A is the top of the mountain; D is the point at “ground level” directly below A (and thus inside the mountain); E and F are the locations of points from which the elevations $\alpha$ and $\beta$ are measured.

If a means of measuring angles is available, then we can again use some basic trigonometry to determine $|AD|$. As shown in the figure above, we take measurements of the angle of elevation of the mountain from each of two points E and F which are colinear with D. We have

$\tan(\alpha) = \dfrac{|AD|}{|DE|}$  and $\tan(\beta) = \dfrac{|AD|}{|DE|+|EF|}.$

Although we can’t measure $|DE|$ directly, we can measure $|EF|$ since both points lie outside the mountain. We now rearrange to eliminate $|DE|$ and obtain an expression for $|AD|$:

$|DE| = \dfrac{|AD|}{\tan(\alpha)} = \dfrac{|AD|}{\tan(\beta)} - |EF|,$ and thus $|AD| = |EF|\dfrac{\tan(\alpha)\tan(\beta)}{\tan(\alpha)-\tan(\beta)}.$

Again, accurate measurements of the angles, and accurate values for the tangents of these angles, are essential.

Without walking in circles, then, al-Biruni carried out his measurements and calculations, and obtained an estimate of 12 803 337 cubits, corresponding to roughly 6 340 km in modern units — accurate to within 0.3%. Perhaps to his disappointment, this turned out to be consistent with the earlier work of al-Ma’mun’s surveyors, but he must have had some satisfaction from showing that a skilful observer with a good grasp of mathematics could still do as well as a major government-sponsored research team. (Oh, happy days!)

Further refinement to al-Biruni’s estimate, in fact, was not to come for another seven centuries, when rival claims about the shape of the Earth — largely driven by disputes between the Newtonian (English) and Cartesian (French) schools of mechanics — prompted the first detailed measurements of the equatorial and polar radii. These measurements showed that the neat Pythagorean belief in a spherical Earth wasn’t quite true, but trumped it by showing that the shape of the Earth was consistent with Newton’s laws of motion. But that’s a topic, maybe, for another time…

(Article adapted by DP from an Honours project by Tara Hirshall.)