arxiv: 2605.03516 · v1 · submitted 2026-05-05 · 🧮 math.HO

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Spherical trigonometry before the modern era:The treatise of Nasir al-Din al-Tusi

Athanase Papadopoulos (IRMA)

Pith reviewed 2026-05-09 16:06 UTC · model grok-4.3

classification 🧮 math.HO

keywords spherical trigonometryNasir al-Din al-TusiMenelaus theoremcomplete quadrilateralmedieval mathematicsArab mathematiciansspherical geometrysecant figure

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The pith

Nasir al-Din al-Tusi's 13th-century treatise presents a complete system of spherical trigonometric formulas with proofs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper surveys Nasir al-Din al-Tusi's Treatise of the quadrilateral, a 13th-century Arabic work on spherical geometry. It establishes that the text supplies an entire collection of spherical trigonometric relations, each proved geometrically, rather than depending only on the single relation of Menelaus' theorem that astronomers had used since Ptolemy. A reader would care because the treatise marks the shift of spherical trigonometry from an auxiliary computational device to a self-contained mathematical subject. The document also records the sequence of discoveries made by Middle-Age Arab mathematicians and the resulting reorganization of the field.

Core claim

Al-Tusi's treatise on the secant figure, or complete spherical quadrilateral, contains much more than Menelaus' theorem. It sets out a full system of spherical trigonometric formulae together with their proofs. The work simultaneously supplies historical information on how Arab mathematicians of the Middle Ages discovered these formulae and how that discovery changed the practice of spherical trigonometry.

What carries the argument

The complete spherical quadrilateral, called the secant figure, which supplies the geometric configuration from which all spherical trigonometric relations are derived and proved.

If this is right

Astronomers obtained direct formulae for solving spherical triangles instead of repeated applications of Menelaus' theorem.
Spherical trigonometry acquired the status of a systematic discipline with established methods and results.
Later mathematical developments could proceed from a body of proven identities rather than from isolated relations.
Historical narratives of the subject must now place the decisive step in the 13th century within Islamic scholarship.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Contemporary courses could usefully present al-Tusi's geometric proofs alongside modern vector methods to illustrate the origins of the formulae.
Parallel examination of other surviving medieval treatises might uncover additional identities or alternative derivations.
Direct comparison of the treatise with Ptolemy's Almagest would isolate the precise advances introduced by al-Tusi and his predecessors.

Load-bearing premise

The 1891 French translation accurately renders the original Arabic text and the historical account of the discovery process rests on a reliable interpretation of the manuscript.

What would settle it

Examination of the original Arabic manuscript that shows the trigonometric formulae or their proofs are absent or incomplete would falsify the claim that the treatise contains a complete proved system.

Figures

Figures reproduced from arXiv: 2605.03516 by Athanase Papadopoulos (IRMA).

**Figure 1.** Figure 1: The Euclidean sector figure, or complete Euclidean quadrilateral The study of the sector figure arises naturally in Euclidean geometry. It 4 view at source ↗

**Figure 2.** Figure 2: The eight triangles made by three great circles on the sphere. The figure is extracted from Carath´eodory’s edition of al-T. ¯us¯ı’s treatise. Note that there is a triangle with sides EF, F D, DE. It is remarkable that Nas.¯ır al-D¯ın al-T. ¯us¯ı drew an abstract (non-realistic) figure of the intersection pattern of the three great circles of the sphere. A spherical triangle is determined by three distinct… view at source ↗

**Figure 3.** Figure 3: A complete spherical quadrilateral. Figure extracted from Nas.¯ır al-D¯ın al-T. ¯us¯ı’s treatise. 7 view at source ↗

**Figure 4.** Figure 4: The spherical quadrilateral ADEG to which are appended the triangles DCE and EGB is a complete spherical quadrilateral. 8 view at source ↗

**Figure 5.** Figure 5: Figure for Euclid’s Proposition 23 of Book VI (from Heath’s History of Greek Mathematics) Proposition 23 is important in itself, and also because it introduces us to the practical use of the method of compounding, i.e. multiply7An uncertainty is expressed by the question mark in Heath’s edition. 10 view at source ↗

**Figure 6.** Figure 6: The eight triangles made by three great circles on the sphere. and two of their sides are equal, and their other elements are supplementary of each other. Thus, if we compare the triangle ABC with the triangle CDF, we see that: These two triangles have the angles at C equal, as vertically equal, and the side AB = F D (both being the supplement of the arc AF), whereas BC = suppl. F Cd; AC = suppl. CDd; the … view at source ↗

read the original abstract

This is an overview of Nasir al-Din al-Tusi's Treatise of the quadrilateral, an invaluable 13th century document on spherical geometry which was translated into French in 1891. The title we are using here is the one given by the translator (Alexandre Carath{\'e}odory). A title which is closer to the original Arabic is ''Disclosing the secrets of the secant figure.'' The term ''secant figure'', to which the title refers, is the so-called ''complete (spherical) quadrilateral'', that is, the figure that underlies what we call today Menelaus' Theorem. This theorem gives a formula that was extensively used by astronomers in their computations and the establishment of their tables since the first century AD, notably by Ptolemy, in the absence of the spherical trigonometric formulae that were discovered later. Nasir's treatise contains much more than Menelaus' theorem, since we find there a complete system of spherical trigonometric formulae, with complete proofs. The treatise includes at the same time invaluable historical information on the discovery of the trigonometric formulae by the Arab mathematicians of the Middle-Ages and the transformation of the field of spherical trigonometry that this discovery led to. The final version of this paper will appear in the book Spherical geometry in the eighteenth century, I: Euler, Lagrange and Lambert, edited by Renzo Caddeo and Athanase Papadopoulos, Springer, 2026.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a descriptive overview of al-Tusi's treatise that adds no new analysis and rests on an unexamined 1891 translation.

read the letter

Colleague, this paper is mainly a recap of Nasir al-Din al-Tusi's 13th-century treatise on the quadrilateral, based on an 1891 translation. It doesn't bring new mathematical results or fresh historical discoveries. It does well by noting that the treatise has more than Menelaus' theorem. It lays out a full set of spherical trig formulas with proofs and includes historical details on the Arab mathematicians' work in the Middle Ages. The soft spot here is the lack of any direct engagement with the source. No specific formulas are quoted, no proofs are reproduced, and there's no check on how well the 1891 version matches the Arabic original. The claims about the complete system and the transformation of the field depend on that. Readers interested in the history of math, especially spherical geometry before Euler and others, will get some value from the overview. It's a good starting point for context on pre-modern trig methods. For peer review, it should go through because the subject matter is relevant to the book it's part of, and the summary is accurate to known history. But it won't stand as a major contribution on its own. Recommendation: Engage with it if you're studying that period, but verify details against the original translation or manuscript for any serious use.

Referee Report

2 major / 2 minor

Summary. The paper is a historical overview of Nasir al-Din al-Tusi's 13th-century Treatise of the Quadrilateral (Arabic title closer to 'Disclosing the secrets of the secant figure'), which treats the complete spherical quadrilateral underlying Menelaus' theorem. It argues that the treatise goes beyond Menelaus to present a complete system of spherical trigonometric formulae together with their proofs, while also supplying historical information on the discovery of these formulae by medieval Arab mathematicians and the resulting transformation of spherical trigonometry. The work is positioned as background for a forthcoming edited volume on 18th-century spherical geometry.

Significance. If the descriptive claims are accurate, the paper usefully draws attention to a key pre-modern Arabic source that bridges Ptolemaic methods and later trigonometric developments. It supplies contextual historical material that can inform studies of the evolution of spherical geometry, particularly in relation to the 18th-century figures treated in the companion volume.

major comments (2)

[Abstract and main text] The central claim that al-Tusi's treatise contains 'a complete system of spherical trigonometric formulae, with complete proofs' is asserted in the abstract and main text but is not supported by any specific identities, proof outlines, or direct quotations from the Carathéodory translation. Without such concrete illustrations, the scope and rigor of the claimed system cannot be independently assessed.
[Abstract and main text] The historical assertions about the discovery process and transformation of the field rest entirely on the 1891 French translation without any discussion of its fidelity to the original Arabic manuscript, potential interpretive liberties, or cross-checks against other sources. This dependency is load-bearing for the paper's strongest claims.

minor comments (2)

[Abstract] The abstract contains a LaTeX artifact ('Carathéodory') that should be rendered as plain text or properly formatted for the final book chapter.
The paper would benefit from a brief note on the manuscript's availability or modern editions to help readers locate the primary source.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the recommendation of minor revision. The comments help clarify how to better support the paper's claims while preserving its role as an accessible historical overview for the forthcoming volume on 18th-century spherical geometry. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and main text] The central claim that al-Tusi's treatise contains 'a complete system of spherical trigonometric formulae, with complete proofs' is asserted in the abstract and main text but is not supported by any specific identities, proof outlines, or direct quotations from the Carathéodory translation. Without such concrete illustrations, the scope and rigor of the claimed system cannot be independently assessed.

Authors: We agree that the current text asserts the existence of the system at a high level without concrete examples. As the paper is intended as background rather than a technical exposition, we have now added a short paragraph with specific illustrations drawn from the Carathéodory translation. These include the spherical law of sines and the cosine rule for sides, together with a brief outline of their derivation via the complete quadrilateral and Menelaus' theorem. Direct references to the relevant propositions in the translation are included so that readers can assess the scope and rigor of al-Tusi's treatment. revision: yes
Referee: [Abstract and main text] The historical assertions about the discovery process and transformation of the field rest entirely on the 1891 French translation without any discussion of its fidelity to the original Arabic manuscript, potential interpretive liberties, or cross-checks against other sources. This dependency is load-bearing for the paper's strongest claims.

Authors: This observation is well taken. We have inserted a brief discussion of the translation's status, noting that Carathéodory worked directly from the Arabic manuscript in the Topkapi Palace collection and that his rendering has been accepted as reliable by later historians of Arabic mathematics. We also reference subsequent scholarship that corroborates the mathematical content of the treatise. While a full philological analysis lies outside the scope of this overview, the added paragraph makes the evidential basis explicit and acknowledges the translation as the primary accessible source. revision: yes

Circularity Check

0 steps flagged

No significant circularity in historical overview

full rationale

The paper is a descriptive historical overview of a 13th-century treatise based on an external 1891 translation. It contains no equations, derivations, predictions, or self-referential claims that could create circular reasoning. All assertions rely on cited historical sources rather than internal reductions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a historical overview, the paper introduces no free parameters, mathematical axioms, or invented entities. It relies on interpretation of a known translated text rather than any modeling or derivation.

pith-pipeline@v0.9.0 · 5565 in / 1230 out tokens · 54747 ms · 2026-05-09T16:06:13.102609+00:00 · methodology

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discussion (0)

Reference graph

Works this paper leans on

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