https://sguzman.github.io/marginalia/tags/euler/Recent content in Euler on MarginaliaHugoen-usThu, 12 Feb 2026 00:00:00 +0000https://sguzman.github.io/marginalia/posts/complex-plane-culture/Thu, 12 Feb 2026 00:00:00 +0000https://sguzman.github.io/marginalia/posts/complex-plane-culture/A cultural-intellectual history of the complex plane from the mid-18th century to the mid-2020s, tracing how “imaginary” numbers moved from disputed algebraic fictions to a stable geometric picture and then into the practical core of physics, engineering, computing, and visual culture. The essay follows key conceptual shifts (symbol → plane → toolkit → canvas), highlights major historical actors and applications, and argues that the complex plane became a durable bridge between abstraction and reality.https://sguzman.github.io/marginalia/posts/number-theory-from-euler-to-today/Thu, 12 Feb 2026 00:00:00 +0000https://sguzman.github.io/marginalia/posts/number-theory-from-euler-to-today/A historical and thematic survey of number theory from Euler’s late-18th-century breakthroughs through modern developments, emphasizing the field’s expansion across analytic, algebraic, geometric, probabilistic, and computational methods, with a focus on Europe and the United States and on major landmark results and milestones.https://sguzman.github.io/marginalia/posts/analysis-semantics/Thu, 12 Feb 2026 00:00:00 +0000https://sguzman.github.io/marginalia/posts/analysis-semantics/A historical and conceptual survey of what mathematicians have meant by “analysis” from the Newton–Leibniz era to the present. The essay traces how analysis shifted from a general method of discovery contrasted with synthesis into a distinct discipline centered on limits, continuity, infinite processes, and the continuum. It follows key semantic pivots driven by calculus, the rise of the function concept, the 19th-century program of rigor (Cauchy, Weierstrass), and later expansions into complex, Fourier, functional, and modern applied/abstract analysis, highlighting how boundaries with algebra, geometry, topology, probability, and computation repeatedly blurred and re-formed.https://sguzman.github.io/marginalia/posts/geometry-semantics/Thu, 12 Feb 2026 00:00:00 +0000https://sguzman.github.io/marginalia/posts/geometry-semantics/This report traces the evolving meanings of “geometry” in European and American mathematics from the era of Leonhard Euler (18th century) to the present day. Over nearly three centuries, “geometry” has expanded from denoting the classical study of shapes and Euclidean space to a sprawling family of subfields and methodologies. We examine what mathematicians of each era understood “geometry” to mean, how new theories and external pressures reshaped those meanings, and how “geometry” functioned both as a subject area and a style of reasoning. Key transitions include the rise of analytic and coordinate methods, the introduction of projective and non-Euclidean geometries, the 19th-century unification of geometry via transformation groups (Klein’s Erlangen Program), the axiomatization of geometry by Hilbert, the 20th-century branching into differential, topological, algebraic, and computational geometries, and the influence of physics and computing. We also explore enduring tensions—synthetic vs. analytic methods, intuition vs. rigor, local vs. global perspectives, continuous vs. discrete structures, algebraic vs. geometric mindsets—and how the term “geometry” at times unified mathematicians and at other times fragmented into specialized “geometries.” Through historical narrative, case studies, and quotations from major figures, we show how “geometry” evolved from the study of tangible figures in Euclidean space to a unifying language of mathematical structure, and why it remains a plurality of approaches today.