Notes on angles and solid angles, in relation with Euler's memoir De mensura angulorum solidorum

Abstract: We provide some historical context to the study of solid angles carried out by Euler in his memoir \emph{De mensura angulorum solidorum} (On the measure of solid angles). We extend our study to the general notion of angle (not only solid). While doing so, we explore some works by Ancient Greek mathematicians and others by Arabs mathematicians of the Middle-Ages as well as some later Western authors from the Renaissance. In particular, we review the Pythagorean anthyphairetical perspective on angles which establishes the basis of the important relation between the mathematical notion of angle and the philosophical concept of finitization of the Infinite. In doing so, we shall show that questions addressed by Euler lead us to questions raised about 2500 years ago. At the same time, we highlight the fact that mathematics in those times is also today's mathematics. The reader can also see in this study the intermingling between mathematics and philosophy. This paper will appear in the book \emph{Spherical geometry in the Eighteenth Century, I: Euler, Lagrange and Lambert}, ed. R. Caddeo and A. Papadopoulos, Springer, 2026.

• The paper reveals the deep historical and philosophical roots of angles, emphasizing the use of anthyphairesis and recursive approximations to distinguish various angle types.
• It details Euler's methodical innovations in measuring solid angles through spherical geometry and explicit formula derivations analogous to Heron’s formula.
• It addresses ongoing challenges in defining angles as magnitudes, highlighting unresolved issues in comparability and measurement in higher-dimensional contexts.

Historical and Philosophical Analysis of Angles and Solid Angles

Overview

This comprehensive study investigates the mathematical and philosophical underpinnings of angles and solid angles, anchored by Euler's De mensura angulorum solidorum. The exploration traverses a historical arc from ancient Greek geometry through medieval Arabic scholarship to post-Renaissance and modern mathematical developments. A central theme is the entanglement between the formalization of the angle concept and the interplay with philosophical constructs, notably the finitization of the infinite—an idea pervasive in Pythagorean mathematics and its descendants.

Mathematical and Philosophical Origins

The text establishes the multifaceted origins of the angle concept, emphasizing its cross-cultural emergence and subsequent formalization in Greek mathematics. While planar and solid angles were intuited early in human civilization, rigorous axiomatic and philosophical analysis arose prominently in the works of the Pythagoreans, where mathematics, ontology, and epistemology intermingled. The authors elucidate the Pythagorean use of anthyphairesis (reciprocal subtraction, akin to continued fractions) both to demonstrate incommensurability and to relate acute, right, and obtuse angles to the principles of the infinite (apeiron) and finite (peras). This relation is formalized through convergent and divergent sequences of rational approximations to $\sqrt{2}$, which underpin the philosophical demarcation between the types of angles.

Plato and Aristotle further embed the angle concept in their philosophical discourse. Plato’s discussions on the receptacle in the Timaeus exhibit a geometric metaphysics wherein regular polyhedra, their construction, and properties (including solid angles) assume a fundamental ontological status. Aristotle’s extensive treatment focuses on the categorial placement of angles (quantity, quality, relation) and the admissibility of their operations.

Treatment in Greek Mathematics

In Euclid’s Elements, angles are axiomatized both as geometric objects (via definition and postulates) and as magnitudes subject to ordering and comparison. However, solid angles receive a more ambiguous definition, situated between surface-based and plane-based constructs. The treatise carefully reconstructs the development and limitations of these formalizations, highlighting issues such as the cornicular (horn-like) angle, which resists incorporation within the standard Archimedean framework.

The study attributes to the Pythagoreans a recursive algebraic mechanism to distinguish acute and obtuse angles based on anthyphairesis. This mechanism is demonstrated with explicit Pell-type recursion yielding alternately acute and obtuse rational approximations to right angles. The authors argue that this recursive hierarchy influenced both Greek and later philosophical-mathematical treatments, as evidenced in Proclus and Plato.

Euler’s Contributions and the Calculation of Solid Angles

A major focus is placed on Euler’s operationalization of the notion of the solid angle. Euler extends the analytic paradigm by defining the measure of a solid angle at a point as the area subtended on the unit sphere by the intersecting planes (spherical triangle). The memoir investigates Euler’s derivation of formulae analogous to Heron’s formula for spherical triangles, and the calculation of explicit values for the solid angles of the regular polyhedra. It highlights the historical nuance that, while the classification of regular polyhedra is achieved in Elements, the precise measure of their solid angles is carried out only by Euler.

The discussion also critically assesses the history of the “angle excess” formula, customarily attributed to Girard, but with antecedents in Harriot and Cavalieri, indicating a complex lineage for key results in spherical geometry.

Problems of Numerical Magnitude and Comparability

A recurring issue throughout the development of angle concepts is their problematic status as magnitudes. The work meticulously traces debates—commencing with Aristotle and culminating in the Renaissance and Arabic treatises—on whether angles (and, by extension, solid angles) are homogeneous magnitudes, and on the legitimacy of their comparison, addition, and multiplication. Particular attention is given to mixed and “contact” angles (e.g., the angle between a tangent and a circle), which challenge standard commensurability and ratio theory.

The algebraic and axiomatically subtle concept of magnitude (as found, e.g., in Eudoxian theory) underpins many of these debates, with angles often inhabiting a liminal space in classification schemes.

Angles in Spherical and Non-Euclidean Geometry

A distinct section addresses the extension of the angle concept to spherical and non-Euclidean contexts. Here, the work of Menelaus is foregrounded in establishing spherical geometry as an intrinsic discipline, with novel definitions and properties of spherical angles and triangles. Later, Euler, Lagrange, and others provided both synthetic and analytic results for angles and their sums in such geometries, and engaged deeply with related problems such as the parallel postulate.

Arabic Mathematical Tradition

The Arabic mathematical tradition receives extensive treatment as a crucial vector for the preservation, elaboration, and transmission of geometric ideas. The study draws on large corpora of previously underexplored manuscripts, highlighting original treatments of both plane and solid angles, as well as philosophical disquisitions on magnitude, comparability, and infinite divisibility. Contributions by al-Sijzī, Ibn al-Haytham, al-Samaw'al, Nasīr al-Dīn al-Tūsī, and others are expounded, many of whom engaged creatively with subtleties in the Elements and advanced the theory of solid angles—particularly in the context of infinitesimal calculus and isoperimetric problems.

Developments in the Renaissance and Modernity

The Renaissance marked renewed analytical and combinatorial interest in solid angles, as seen in the work of Kepler, Descartes, de Beaune, de Gua, and others. Of note is the transition, culminating in Euler, from purely geometric investigations to the explicit calculation of measures for solid angles. The text notes ongoing technical refinements in the definitions and computational techniques for solid angles and their applications in optics, physics, and higher-dimensional geometry.

In the modern period, the formal treatment of angles within synthetic, projective, and metric geometry (e.g., the work of Hilbert, Klein, and Alexandrov) clarified and systematized the axiomatic and operational character of the angle, with angles assuming roles as undefined primitive objects, as in Birkhoff’s geometry, or being associated with group actions and symmetries, as in Klein’s Erlangen Program and Thurston’s work on orbifolds and cone points.

Implications and Future Developments

The essay demonstrates the enduring complexity and subtlety underlying the concept of angle in mathematics, a concept whose definition, measure, and status as a magnitude are deeply tied to the historical development of mathematical rigor, abstraction, and the philosophical pursuit to finitize the infinite. The interplay of philosophical and mathematical mechanisms, especially the use of anthyphairesis and recursive approximations, is indicative of the deep methodological connections between ancient and modern mathematics.

From a practical standpoint, the historical problems associated with defining and measuring angles and solid angles foreshadow many open questions in higher-dimensional geometry, measure theory, and mathematical physics. Furthermore, the continued investigation of “problematic” angles, such as mixed or contact angles, remains relevant for modern topics in analysis, differential geometry, and singularity theory.

Continued research into the manuscript traditions, particularly those of the Arabic mathematicians, is likely to yield further insights into both the technical and philosophical developments of the angle concept, as well as new interpretations of classical works.

Conclusion

This study presents an authoritative and detailed analysis of the concept of angle and its higher-dimensional generalization, the solid angle, from both a historical and a philosophical perspective. It traces their development across diverse intellectual traditions and mathematical frameworks, elucidating the philosophical commitments that informed technical definitions and theorems. By grounding the most advanced analytical and synthetic results—exemplified by Euler’s explicit measurements—in a context that includes both philosophical reflection and deep analytic methods, the work illuminates the foundational importance of the notion of angle, and its continued influence on the structural evolution of mathematics.