🔭 astrophysics

Determination of the longitude difference between Baghdad and Khwarezm using a lunar eclipse (the method of Abu Rayhan al-Biruni and Abu al-Wafa al-Buzjani)

This paper examines the tenth-century collaborative effort by Iranian scholars Abu Rayhan al-Biruni and Abu al-Wafa al-Buzjani to determine the longitudinal difference between Baghdad and Khwarezm through the simultaneous observation of a lunar eclipse, analyzing their methodology, instruments, calculations, and the historical significance of this pioneering achievement in astronomy.

Original authors: Rizoi Bakhromzod

Published 2026-01-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rizoi Bakhromzod

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out how far apart two cities are, but you don't have a map, a GPS, or even a car to drive between them. You only have the night sky and a very specific plan. This is exactly what two brilliant scholars from the 10th century, Abu Rayhan al-Biruni and Abu al-Wafa al-Buzjani, managed to do.

Here is the story of how they solved a massive puzzle using a lunar eclipse, explained simply.

The Big Problem: The "Time Zone" Mystery

Back in the 10th century, scientists knew how to find latitude (how far north or south you are). It was easy: just look at how high the North Star is in the sky.

But longitude (how far east or west you are) was a nightmare. To know your longitude, you need to know the exact time difference between your location and a "home" location.

The Catch: In those days, there were no phones, no internet, and no accurate portable clocks. If you were in Baghdad and your friend was in Khwarezm (about 1,500 km away), you couldn't call them to say, "Hey, it's 8:00 PM here, what time is it there?"
The Result: Without synchronized time, calculating longitude was nearly impossible.

The Solution: The Sky as a Giant Clock

The scholars realized they needed a "signal" that happened at the exact same moment for everyone on Earth, regardless of where they stood. They found the perfect candidate: a lunar eclipse.

Think of a lunar eclipse like a giant, slow-motion shadow play. When the Earth's shadow hits the Moon, it happens at the exact same instant for everyone who can see the Moon. It's like a cosmic "flash" that everyone sees simultaneously.

The Experiment: A Long-Distance Teamwork

Here is how the two scholars pulled off this scientific magic trick:

The Setup: Al-Biruni was in Khwarezm (modern-day Uzbekistan), and Al-Buzjani was in Baghdad (modern-day Iraq). They wrote letters to each other beforehand and agreed to watch a specific lunar eclipse together.
The Plan: They decided to watch for a specific moment during the eclipse—like when the Earth's shadow first started "biting" the Moon or when the eclipse was at its darkest.
The Execution: On the night of the eclipse (May 997 CE), both men watched the sky.
- Al-Buzjani in Baghdad noted the time on his local clock when the shadow hit.
- Al-Biruni in Khwarezm did the exact same thing with his local clock.
- Crucially, they didn't talk to each other during the event. They were working alone, miles apart.
The Reveal: After the eclipse, they sent their notes to each other via a messenger (a caravan). When they compared the notes, they realized something amazing: The eclipse happened at different times on their clocks.

The Math: Turning Time into Distance

Because the Earth spins, the sun and moon appear to move across the sky. The scholars knew a simple rule:

The Earth spins 360 degrees in 24 hours.
That means it spins 15 degrees every hour.

When they compared their notes, they found that the eclipse happened about one hour later in Khwarezm than it did in Baghdad.

The Logic: If the event happened one hour later in the east, it means the Earth had to spin for one more hour to bring Khwarezm into the "view" of that event.
The Calculation: 1 hour × 15 degrees/hour = 15 degrees.

So, they concluded that Khwarezm was about 15 degrees east of Baghdad.

How Good Was Their Guess?

Let's check their work against modern technology:

Modern Reality: The actual difference is about 16.35 degrees (or roughly 1 hour and 5 minutes).
Their Result: They calculated 15 degrees (exactly 1 hour).
The Verdict: They were off by only about 1.35 degrees (roughly 120 kilometers).

Considering they were doing this over 1,000 years ago without satellites, this was an incredible feat. They were about 92% accurate, which is a huge success for medieval science.

Why Does This Matter?

This paper highlights a moment where human ingenuity beat the limitations of technology.

No Tech Needed: They didn't need a machine to send a signal; they used the universe itself as the signal.
Trust and Math: It required trust in each other's observations and a deep understanding of geometry and time.
A New Way to Map the World: This experiment helped them create better maps and understand the true size and shape of the Earth, correcting old, wrong ideas from ancient Greek scholars.

In short, two friends in different cities looked at the same moon, compared their pocket watches, and successfully measured the distance between their homes using nothing but the laws of physics and a little bit of patience.

Technical Summary: Determination of the Longitude Difference between Baghdad and Khwarezm via Lunar Eclipse

Problem Statement
The determination of geographical longitude has historically been a fundamental challenge in Earth sciences, particularly during the 10th and 11th centuries. While latitude could be readily calculated through the altitude of the Pole Star or the Sun's maximum elevation, longitude required knowledge of the time difference between two locations. In the medieval Islamic world, the absence of rapid communication and accurate portable clocks made synchronizing time between distant observers a significant technical barrier. Consequently, existing longitude data, often inherited from Ptolemy, contained substantial errors. The specific problem addressed by this paper is how scholars overcame the lack of instantaneous communication to measure the longitude difference between Baghdad and Khwarezm (specifically the city of Kath) with high precision.

Methodology
The paper details a collaborative experiment conducted by two prominent scholars, Abu Rayhan al-Biruni (based in Kath, Khwarezm) and Abu al-Wafa al-Buzjani (based in Baghdad), on May 24, 997 CE. Their methodology relied on the simultaneous observation of a partial lunar eclipse, which served as a natural, globally visible "time signal."

Coordination: The scholars coordinated via correspondence to observe the same specific phase of the eclipse (likely the beginning of the partial phase or the moment of maximum eclipse) using their local timekeeping instruments.
Observation:
- Instruments: Observers utilized water clocks (clepsydrae), astrolabes, and potentially sandglasses. In Baghdad, a large wall-mounted quadrant was available; in Kath, al-Biruni relied on his own instruments and stellar observations for calibration.
- Event Selection: The chosen eclipse (Saros series 85) was ideal because it was visible across the entire nighttime hemisphere, ensuring simultaneity. It was a partial eclipse with a umbral magnitude of 0.551, providing a clear shadow boundary for timing, and lasted long enough (approx. 157 minutes for the partial phase) to allow for precise recording.
Data Exchange: Since real-time communication was impossible, the observers recorded their local times independently. The data were exchanged weeks later via a caravan or courier.
Calculation: The difference in local time ( $\Delta t$ ) recorded for the same eclipse phase was converted to a difference in longitude ( $\Delta \lambda$ ) using the standard astronomical relationship: $\Delta \lambda = 15^\circ \times \Delta t$ (where $\Delta t$ is in hours).

Key Contributions

Biographical Context: The paper highlights the roles of al-Biruni and al-Buzjani as leading figures in medieval mathematics and astronomy. Al-Biruni is noted for his geodetic work and the Mas'udi Canon, while al-Buzjani is credited with systematizing trigonometry (introducing tangent and cotangent) and establishing a major observatory in Baghdad.
Innovation in Synchronization: The paper emphasizes the novelty of organizing a "simultaneous observation session" over a distance of approximately 1,500 km without direct contact during the event. This represented a shift from isolated observations to a coordinated, multi-site scientific program.
Methodological Rigor: The scholars addressed potential sources of error, including the calibration of timekeeping devices and the selection of an eclipse phase with high visibility and distinct timing markers.

Results

Measured Value: The scholars determined that the local time difference between Kath and Baghdad was approximately 1 hour.
Calculated Longitude: Applying the conversion factor, they derived a longitude difference of approximately $15^\circ$ (with Kath being east of Baghdad).
Comparison with Modern Data: Modern coordinates place the longitude difference at approximately $16.35^\circ$ (a time difference of ~1 hour 05 minutes).
Accuracy Assessment: The medieval result deviated by roughly $1.35^\circ$ (or ~5 minutes 24 seconds in time). This represents a relative error of approximately 8.3% in the longitude difference, or roughly 7–8% in terms of the physical distance between the regions (an error of ~120 km). The paper notes that for the 11th century, this level of accuracy is "truly remarkable," correctly identifying the order of magnitude and providing a close absolute value.

Significance
The paper concludes that this experiment stands as an outstanding achievement in the history of science and astronomy. It demonstrates the high scientific culture of the medieval Islamic world, which successfully synthesized classical heritage with practical innovation. By utilizing the sky as a "precise clock," al-Biruni and al-Buzjani overcame the era's primary technological limitation (lack of time synchronization) to achieve a geodetic measurement that remains worthy of admiration. The work serves as a vivid example of how theoretical insight and practical implementation can combine to solve complex geographical problems long before the advent of modern technologies.