Al-Biruni's mountain-and-horizon Earth measurement

How can a measurement on a mountain reveal Earth's size? Al-Biruni described a method combining known mountain height, the horizon's dip below the horizontal and tangent geometry, then reported applying it near Nandana fort. His Determination of the Coordinates of Positions for the Correction of Distances between Cities, or Tahdid, was composed in416 AH/1025 CE. That dates the book, not an independently established day of observation. ^{1 2}

The documented contribution is exposition, proof and numerical application, not the first recognition that Earth is spherical. The same text preserves an earlier account of Sanad ibn Ali, although the translator's preface calls the method the author's invention. Modern assessments of accuracy also differ. Visitors therefore see a contribution with its attribution limits, not a solitary hero without predecessors or questions. ^{4 3}

This is a method of Earth measurement, not a single surviving museum instrument. From an elevated viewpoint a distant horizon lies below the observer's horizontal plane: that dip is a small angle. With the observer's height known, it can be related to the radius of a spherical Earth. Al-Biruni explains the relation through similar triangles and sine ratios, not by walking around the world or seeing its centre. ⁵

Two stages are indispensable: surveying mountain height, then observing horizon dip from the summit. The mountain is not a ruler reaching Earth's centre; it supplies a small known length in a larger problem. A valid geometrical relation does not mean every visible horizon meets the conditions, or that a historical length has automatically become a modern metre measurement. ^{6 3}

The passage begins with a method in Ptolemy's Geography: a surveyed distance on Earth's surface, information about two positions and the direction of travel allow the circumference to be inferred. Al-Biruni then presents another method that avoids walking through deserts. The contrast is not between a flat Earth and a newly discovered sphere, but between ways to determine the size of the sphere already assumed in the calculation. ⁷

A specific predecessor matters more than a generic roll-call. The text reports Sanad ibn Ali accompanying al-Ma'mun on a campaign against the Byzantines. The caliph ordered him up a mountain near the sea to measure the sun's dip, from which he derived Earth's circumference. This is al-Biruni's transmitted account, not Sanad's field record independently inspected here; nevertheless, it prevents silently awarding the entire principle to al-Biruni. ⁸

Al-Biruni writes in the first person about living at Nandana fort in the land of India. A nearby mountain stood west of the fort, with a broad plain south of the mountain; he thought it suitable for testing the method. From its summit he observed the apparent meeting of earth and sky and surveyed the perpendicular representing mountain height. The account joins place, choice, observation and number, without supplying every detail of the observation day. ⁹

The origin record uses his Ghaznavid context rather than invented observation coordinates. Tümer mentions astronomical work at Nandana in1018, but that is not itself an explicit date for the radius experiment. The416/1025 date belongs to the book reporting it. Separating a written witness from the event prevents a plausible date becoming an unsupported certainty. ^{10 11}

Britannica explicitly identifies al-Biruni as a Muslim scholar. Günay Tümer discusses his commitment to Islam alongside disputed religious affiliation. The museum's Sunni frame is explicit: Tümer's reading challenging a settled Shia classification comes first, followed by the Shia and Zaydi classifications he reports from other researchers, without a museum verdict or inference from a patron's affiliation. The person record carries the detail. ^{12 13}

His role here is “describer”: author of a geometrical exposition and first-person application, not a title granting priority over the Sanad account he preserves. Jamil Ali translated the witness we read; he was not a participant in the fifth-century-AH experiment. Distinguishing author, transmitter and translator preserves the chain rather than collapsing it into one famous name. ¹⁴

The exposition calls for a high mountain near the sea or a broad level plain. It describes a ring with an alidade, a sighting rule, aligned to measure dip below the horizontal. For the general horizon observation, the observer looks through the alidade's two sights until seeing where earth and sky appear to meet. This is not the summit's elevation as seen from a valley; it is the horizon's depression as seen from above. ¹⁵

Mountain height has a separate procedure: a square plate one cubit on each side, equal divisions, pegs and an alidade, with a dropped stone establishing a vertical projection. Similar-triangle ratios yield a distance and then the required height. This is the square-plate procedure on187–188; we do not silently replace it with a familiar illustration using two separate ground stations. ¹⁶

In modern notation let radius be r, height h and dip alpha. A radius to the horizon contact is perpendicular to the tangent sightline, producing a right triangle whose hypotenuse is the centre-to-summit distance, r+h. Thus cos(alpha)=r/(r+h), giving r=h*cos(alpha)/(1-cos(alpha)). These are our explanatory symbols for the text's relation, not a claim that its page uses modern algebra. Omitting the cosine in the numerator changes the calculated quantity. ¹⁷

Read the figures carefully: the dip is 0;34 degrees, meaning34 arcminutes, not0.34 degrees. Height is 652;3,18 regional cloth-measuring cubits:652 plus3/60 plus18/3600, or652.055 cubits. On a scale with total sine60, the text gives the radius-side as 59;59,49, the difference as 0;0,11, and Earth's radius as 12,803,337;2,9 cubits. We retain the numbers in their unit rather than selecting a modern cubit length to obtain an impressive answer. ⁹

Composition, transmission and verification of an experiment are different questions. Tümer identifies the book's manuscript in the Fatih collection,3386, and Bulgakov's1962 Cairo edition; Ali's preface says his translation used that edition. This explains a route by which a modern reader encounters the description, not how many medieval surveyors adopted the method or where they practised it. ^{2 18}

Accordingly, we draw no arrows from Nandana to every later European school or navigational device. Similar measurement problems do not by themselves prove transmission: that needs evidence of reading, borrowing or teaching. This research is bounded by the witnesses named here; survival in a book and a modern edition date are not a continuous history of practical adoption. ^{19 3}

The exposition uses a sphere, tangent and right triangle, with knowledge of how length ratios and angles correspond. It starts with a small surveyable quantity, not an already known Earth radius. It also situates mountain-height measurement among methods for determining distances before using it for Earth's size: carrying an instrument to a summit is insufficient without the height entering the proportion. ⁶

Sanad's named predecessor appears in the browser chain; earlier arc-surveying appears as another context, without crediting all trigonometry to one person. Plate and alidade implement observations, while geometry explains their connection. Separating them lets a reader question each stage instead of confusing a valid proof with the accuracy of every reading. ¹

The sources read here establish no direct dependency chain ending in satellites or GPS. The dependent-invention field therefore contains no fabricated genealogy. What can be described is the method's life as a historical reconstruction problem: Maslikov's abstract discusses reconstructing method and instruments while warning against inflated accuracy and uncertain historical length conversion. We have not read the full Russian article. ³

The modern illustration helps explain the relation, but is neither a document from al-Biruni's workshop nor independent proof of his numbers. Teaching can reuse a proof without establishing every later measurement technology's descent from it. We distinguish explanatory revival from historical dependency rather than turning a useful diagram into evidence of medieval diffusion. ^{20 6}

The book places Earth measurement within correcting distances between cities, connecting local position to larger calculations of locations and distances. The passage's explicit advantage is avoiding long desert traverses to obtain circumference, in exchange for a suitable mountain, visible surface, height survey and dip observation. That is a change in working requirements, not a promise of easy measurement for every traveller. ²¹

The pages read supply no figures for money or travel time saved and do not establish ordinary people's daily reliance on the method. A narrower effect can be shown: writing down steps that let a reader follow observation into calculation. Distinguishing what was measured from what was inferred makes the account discussable rather than simply praise of genius. ²²

The witness brings calculation, instrument, place and an earlier account into one exposition. The author gives more than a final number: triangle relations, a height procedure, then angle, length unit and result. Historians can examine every link, treating an old computation as something to understand rather than an isolated numerical contest. ¹

MacTutor offers a favourable6339.6km assessment and a comparison with the West. We identify that as the biography authors' assessment alongside Maslikov's caution about accuracy and units. We do not choose a number and announce a spectacularly tiny error. Establishing what a cubit means and how ratios were approximated is part of the historical question, not detail to discard after seeing the answer. ^{23 3}

The translator's reading is explicit: Ali calls deriving circumference from horizon dip the author's invention. The main text is also explicit in reporting Sanad ibn Ali applying the same principle under al-Ma'mun. We do not erase either statement for a quicker answer; the record is marked “contested attribution” while retaining al-Biruni's specific contribution. ²⁴

Priority of an idea differs from leaving a detailed exposition and application. The inspected witness supports describing the latter, not claiming to have witnessed humanity's first experiment or established the origin of every geometrical ratio involved. Fame does not confer invention of a spherical Earth: its circumference is already a subject of measurement before this story within the text itself. ¹

The method can teach the relation between a mathematical model, measurements and the limits of historical evidence. The angle is small, as is the difference between total sine and the other side; rounded numbers are not automatic proof of exceptional accuracy. We display the historical sexagesimals separately from our modern algebra and the kilometre comparison. ^{9 3}

The accompanying illustration is Nevit Dilmen's2010 work, made with GeoGebra and Inkscape and licensed CC BY-SA3.0. It shows tangent sightline, angle and height with conspicuously exaggerated scale. It is not a photograph of Nandana's mountain or a manuscript copy. We use its geometry and licence, not historical precision claims in the file description. ²⁵

The witness actually read is an illustrated English translation by Jamil Ali, published by the American University of Beirut, with a February1967 preface. Its local filename does not make it an Arabic edition. Small numbers refer to Cairo-edition pages alongside printed English pagination; our183–189 citations are to the English pages, with figures55–56 illustrating the relevant exposition. ²⁶

The preface itself warns about computational and figure-letter problems, sometimes difficult to assign to author or copyist. Tümer describes the manuscript witness and1962 edition; we have not inspected the original manuscript, read the whole book or treated the2019 abstract as the complete paper. Those are limits of the reading performed, not manufactured doubt about the text's existence. ^{18 2 3}

A simple headline is tempting: a scholar climbs a mountain and finds the world's size. The text is richer: an assumed spherical Earth, named predecessor, height-survey instrument, sexagesimal angle and local unit. Preserving these does not diminish the achievement; it restores the traceable work instead of making it a context-free numerical miracle. ¹

The distance between celebration and examination appears in the university biography foregrounding6339.6km and the research abstract questioning accuracy and units. The museum leaves both named voices available without pretending to have read everything they cite or issuing a numerical verdict unavailable from the evidence actually inspected. ^{23 3}

Imagine looking from high up toward the distant line where ground and sky seem to meet. Al-Biruni asks: if we know our height and how far that line dips, what can triangle geometry tell us about Earth? The diagram enlarges mountain and curvature to make the idea visible; it is not Earth drawn to that scale. You need not climb a mountain or look at the sun to learn the problem. ^{6 20}

The lesson is not that one scholar knew everything: he names someone earlier. A large number alone is not accurate to every digit; ask what was measured, what was calculated and which unit was used. Those questions make the story understandable while keeping al-Biruni's contribution visible without deleting predecessors. ²⁷

• Tahdid, Jamil Ali translation, prefaceix–xii,183–191, especially188–189 and figures55–56; selected passages, not the entire book.
• Günay Tümer, Biruni,6(1992),206–215; selected biography, affiliation and works passages.
• Britannica, al-Biruni, explicit Muslim classification.
• O'Connor and Robertson, MacTutor, geodesy paragraph, November1999.
• Maslikov, Biruni's Method of Measuring the Earth,80(7),2019,57–64; only abstract and bibliographic record read.
• Nevit Dilmen illustration metadata,2010, CC BY-SA3.0; modern non-scale schematic.