Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.29444

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

Stelios Negrepontis·Athanase Papadopoulos
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stelios Negrepontis, Athanase Papadopoulos. 2026-03-31. Notes on angles and solid angles, in relation with Euler's memoir De mensura angulorum solidorum. https://arxiv.org/abs/2603.29444

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Word Length Formulae, Normal Forms, Conjugation and Root-finding Algorithms in Surface Groups

In this paper, we mainly study the following symmetric presentation of the surface group $$π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle.$$ For every nontrivial element $x\in π_1(Σ_g)$ and $k\geq 2$, we obtain a uniform representative of the normal forms $\mathfrak{nf}(x^k)$ of $x^k$ under the length-lexicographical order: $$\mathfrak{nf}(x^k) = \overline{LW^{k-2}R}.$$ Building on this result, we establish a new relation among these normal forms, and then derive the following three formulae related to the word length: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. Furthermore, we extend these results to obtain a coarser analogue for every minimal geometric presentation. We then define normal forms of conjugacy classes in $π_1(Σ_g)$ and provide a criterion for determining the conjugacy of group elements. As a consequence, we provide efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications to the computation of several growth rates.

math.GT↗
arXiv2026

Plane separating continua inscribe rectangles

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

math.GT↗
arXiv2026

Every Link Has Infinitely Many Explicit Generalised T-Link Presentations

Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

math.GT↗