Measuring the sky in Earth radii

This lesson asks you to quantify distances in the solar system and the universe by applying an understanding of relative distances and sizes, using appropriate units. The museum offers one medieval astronomer's table of distances measured in Earth radii, and one method of finding the Earth's size from a mountain. Distances to the planets and stars as we measure them today, and the units for them, are for your class. ¹

The expectation asks for appropriate units. Think of how a unit that suits a mountain would fail for the Moon, and how a distance given in several units can be compared by turning each into the same one.

Two records show the same difficulty: a distance is only as clear as its unit. Al-Farghani gives the distances of the Moon, Sun and the five planets, and of the sphere of the fixed stars, in Earth radii and in miles, taking the Earth's diameter as 6500 miles ². The record says the figures are internally consistent with that radius, and that the least Moon distance is doubtful (OCR) ². In al-Biruni's measurement the height of the mountain is given in regional cloth-measuring cubits, and the record says it retains the numbers in their unit rather than selecting a modern cubit length to obtain an impressive answer ³. It adds that Maslikov's abstract warns against inflated accuracy and uncertain historical length conversion ⁴.

For al-Biruni the record preserves no sentence stating his motive; it shows the book's aim. The book places Earth measurement within correcting distances between cities, and the advantage the passage names is avoiding long desert traverses to obtain circumference ⁵. For al-Farghani too the records read here give no stated purpose and no preface stating a religious one. They show that al-Mas'udi counts him among those who engaged in the sciences of the heavens ⁶, that his chapters estimate the earth's circumference from a degree measured by the cubit in al-Ma'mun's days ⁷, and that Golius places his time in al-Ma'mun's era ⁸.

How accurate was al-Biruni's result? MacTutor offers a favourable assessment, with a comparison with the West ⁹. The record identifies that as the biography authors' assessment, set alongside Maslikov's caution about accuracy and units ^{9 4}. The record does not choose a number and announce a spectacularly tiny error, and it says that establishing what a cubit means and how ratios were approximated is part of the historical question ^{9 4}. The museum gives no verdict.

A short, rounded number can look like proof of great accuracy. The record says otherwise for al-Biruni: the angle is small, as is the difference between total sine and the other side, and rounded numbers are not automatic proof of exceptional accuracy ^{3 4}. The record also says a large number alone is not accurate to every digit, and asks what was measured, what was calculated and which unit was used ¹⁰.

For al-Farghani the museum reads a Latin edition printed at Amsterdam in 1669 with Golius's notes ¹¹. The record marks the least Moon distance as doubtful (OCR), and says the other figures are consistent with the radius it uses ². For al-Biruni the witness is an illustrated English translation by Jamil Ali, and the record states that the museum has read only the abstract of Maslikov's article and not inspected the original manuscript ^{12 4}.

The record gives a favourable assessment of al-Biruni's result from MacTutor and a caution about accuracy and units from Maslikov, whose full article the museum has not read, only its abstract. Which source deserves more trust on how accurate the result was, and why? What would you need to know about the cubit before you could decide?

• The distances of the heavenly bodies: al-Farghani's table in Earth radii and miles.
• Al-Biruni's mountain-and-horizon measurement: the numbers in their own units and the question of accuracy.
• al-Farghani: the author of the table and the Latin edition it is read in.
• Al-Biruni: his life and works.

Read about the distances of the heavenly bodies. Choose one body and write its distance in the units the record uses, then in a unit of your own.