Stand on top of a mountain in Massachusetts this week and look straight out toward the horizon. It feels level, but it isn't. The far edge of the land sits a little below your eye, because the Earth curves away beneath you. About a thousand years ago, a Muslim scholar used that tiny drop, and a hill in the Punjab, to measure the size of the whole planet.

His name was Abu Rayhan al-Biruni. With fall foliage fading, the last week of the Mount Greylock summit road season ahead, and long clear evenings, it is a good moment to tell his story and to try a little of his math from our own hills.

Who was al-Biruni?

Al-Biruni was born in 973 in Kath, in Khwarazm, in what is now Uzbekistan, and died around 1048 to 1050 in Ghazni, in today's Afghanistan. He wrote in Arabic and Persian and worked across astronomy, mathematics, geography, history, medicine and the study of other religions. According to Wikipedia's summary of his works, 95 of his 146 books dealt with astronomy, mathematics and related subjects such as mathematical geography.

From 1017 he lived at the court of Mahmud of Ghazni and traveled with Mahmud's campaigns into the Indian subcontinent. His long book on India, finished around 1030, is still read by historians today. During those years he also carried out the measurement he is now best known for.

Faith was part of the reason this kind of work mattered. As Wikipedia's article notes, astronomy in his era had a religious dimension, because prayer requires knowing the direction of the Kaaba, and that can be found accurately only with astronomical and geographic data. The Encyclopaedia Iranica entry on his geography says his main concern in that field was locating places, finding their latitudes and longitudes, and computing the direction of the qibla.

The method: one hill, one angle

Earlier attempts to measure the Earth needed long journeys. Eratosthenes compared shadows in two Egyptian cities, and astronomers working for the Abbasid caliph al-Ma'mun measured a degree of latitude by surveying across a desert. Al-Biruni's idea, as the Paris Observatory's astronomy dictionary describes it, could be done in one place.

He worked at Nandana Fort, at the edge of the Salt Range in present-day Pakistan, where a hill overlooks a wide, flat plain. He did it in two steps.

First, he measured the height of the hill. He sighted the summit from two points on the plain in a straight line with it, measured the distance between those points and the angle up to the top from each one, and used trigonometry to get the height. His figure was about 652 cubits, roughly 321.5 meters, or a little over 1,050 feet.

Second, he climbed the hill and measured the dip of the horizon. From the top, he measured how far below a level line the edge of the plain appeared. He found 34 arcminutes, a little more than half a degree, which is about the width of the full moon in the sky.

Then came the geometry. Picture a right triangle with one corner at the center of the Earth, one at the point on the horizon, and one at the hilltop. The line from your eye to the horizon just touches the Earth, so it meets the radius there at a right angle. That gives a simple relationship: the Earth's radius equals the hill's height times the cosine of the dip angle, divided by one minus that cosine.

Al-Biruni reported a radius of 12,851,369.845 cubits, which the Paris Observatory dictionary converts to about 6,336 kilometers. The modern average radius is about 6,371 kilometers. How close he really came depends on how long his cubit was, and historians disagree on that, so his result is best described as remarkably close rather than exact.

What he got right, and what he couldn't know

The method is sound, but real measurements are messy. The Paris Observatory entry lists the problems: the plain was not as smooth as the sea, his instrument was probably not as precise as the answer suggests, and the air bends light near the horizon. That bending, called atmospheric refraction, lifts the horizon slightly and makes the dip look smaller than it should. Wikipedia, citing Harvard physicist John Edward Huth's book The Lost Art of Finding Our Way, notes that al-Biruni didn't know about refraction and that it can change the dip by about one sixth.

Here is one striking check. If you take today's value for the Earth's radius and work backward, a 321.5-meter hill should show a dip of about 34.5 arcminutes if you ignore refraction. That is essentially what al-Biruni wrote down. A paper on his measurement also points out that with modern arithmetic, his own height and angle would give a radius closer to 6,573 kilometers. Part of the difference came from the trigonometric tables he worked with, which were calculated by hand.

None of that takes away from the idea. A careful observer, with an instrument and a theorem, found the size of the world without leaving one hill.

Try the math from a Massachusetts summit

We ran the same geometry, without refraction, for two places Massachusetts families can reach easily. These are our own calculations, using the official summit heights.

Mount Greylock (3,491 feet). Mass.gov calls Greylock the highest point in Massachusetts and says that on a clear day you can see as far as 90 miles. From the summit, the true horizon over level ground would sit about 1 degree below eye level, nearly twice the dip al-Biruni measured. The geometric distance to that horizon is about 72 miles. The 90-mile views work because distant peaks rise above the curve, so you can see their tops from farther away. That is the same curve al-Biruni measured, just from the other side.

Wachusett Mountain (2,006 feet). In Princeton, Wachusett's summit has a 360-degree view, according to Mass.gov. There, the dip would be about 48 arcminutes, and the geometric horizon is about 55 miles away.

At the beach (about 6 feet). Standing on the sand at eye height, the dip is only about 2.6 arcminutes and the sea horizon is about 3 miles away. That is why ships seem to sink below the horizon so quickly from the shore.

A few honest caveats. Massachusetts hills are surrounded by other hills, not a flat plain, so the "edge" you see from Greylock or Wachusett is usually a far-off ridge, not the true horizon. A phone level app is not precise enough to measure an angle this small. The point is to understand what you're seeing, not to beat al-Biruni's number.

Plan your visit, and mind the dates

  • Greylock's summit roads close for the season on Friday, Oct. 30, at 3 p.m., according to Mass.gov and Explore Western Mass. The park itself is open dawn to dusk, and vehicles longer than 22 feet are not allowed on the summit roads. The visitor center at 30 Rockwell Road in Lanesborough stays open to all vehicles.
  • Wachusett's summit road runs from Memorial Day weekend to the last weekend of October, 9 a.m. to sunset, according to Mass.gov. The state lists a daily parking fee through Oct. 26 of $5 for Massachusetts plates and $20 for out-of-state plates, with the summit parkway included.
  • Pray on time. Sunset comes early in late October, and Maghrib may arrive while you are still on the mountain. Check your prayer app for the exact time at the summit, bring a compass or qibla app, and pack a mat. If you're heading out near dusk, bring a flashlight for the walk back.
  • Hunting season is on. Both reservations list hunting among their activities, so wear bright orange on the trails this time of year.

Why it still matters

The Qur'an invites people again and again to look at the heavens and the earth and reflect. Surah Al 'Imran says, "Indeed, in the creation of the heavens and the earth and the alternation of the night and the day are signs for those of understanding" (3:190). That verse doesn't contain a measurement, and al-Biruni didn't claim it did. What he did was take the invitation seriously, pick up an instrument and do the work.

So if you make it up Greylock or Wachusett before the roads close, take a minute at the top. Look out at the line where the land meets the sky, remember that it sits a little below you, and remember the scholar who turned that small drop into the size of the Earth.