Bridging Disciplines: An In-Depth Exploration of Segment Arithmetic in A Course in Old and New Geometry II

A Course in Old and New Geometry II: Basic Euclidean Geometry offers an in-depth examination of a well-established mathematical framework for constructing the arithmetic of segments. This framework also encompasses internationally recognized mathematical formulas and explores their implications for the interplay between geometry, trigonometry, and algebra. The course emphasizes the creation of a field derived from a geometric model, effectively bridging these three mathematical disciplines.

The primary emphasis is on the arithmetic of segments, which is developed within the contexts of affine and Pythagorean planes. The discourse revolves around essential theorems, definitions, and proofs, including the pivotal theorems of Pappus and Desargues, which are crucial for understanding the algebraic characteristics of the constructed fields. Hilbert's approach involves constructing a field from congruence classes of segments, utilizing geometric principles such as triangle similarity and parallelism. Notably, this construction does not depend on continuity axioms but instead relies on less stringent geometric assumptions, including theorems from Pappus and Desargues.

The course provides precise definitions of fundamental concepts, such as Pythagorean planes, Pythagorean fields, geometry, algebra, and fields in general. It includes comprehensive proofs of the commutative, associative, and distributive laws of segment arithmetic, often drawing on geometric constructions and configurations like the Pappus hexagon.

The book delivers a thorough and methodical exploration of the subject, complete with detailed proofs and logical reasoning. It adeptly connects geometric principles and trigonometry to algebraic structures, showcasing the depth of Hilbert’s methodology.

The course grounds abstract concepts in tangible examples by employing geometric constructions and configurations. Through the presentation of various approaches to segment arithmetic, formulas, and mathematical creation, the book underscores the versatility and generality of Hilbert's methods and those of other esteemed mathematicians.