A Course in Old and New Geometry II: Basic Euclidean Geometry is a detailed exploration of a renowned mathematical approach to constructing arithmetic of segments, or international standard mathematical formulas and its implications for the relationship between geometry, trigonometry, and algebra. The method of creating a field from a geometric model, thereby establishing a bridge between geometry, trigonometry, and algebra.
The main focus is on the arithmetic of segments, which is developed in the context of affine planes and Pythagorean planes. The discussion is framed around key theorems, definitions, and proofs, including theorems of Pappus and Desargues, which play a central role in determining the algebraic properties of the constructed fields. Hilbert constructs a field from congruence classes of segments, leveraging geometric principles such as the similarity of triangles and parallelism. The construction avoids reliance on continuity axioms and instead uses weaker geometric assumptions, such as the theorem of Pappus or Desargues.
Further, it provides rigorous definitions of key concepts, such as Pythagorean planes, Pythagorean fields, geometry, algebra, and fields in general. Detailed proofs are presented for the commutative, associative, and distributive laws of segment arithmetic, often relying on geometric constructions and configurations like the Pappus hexagon.
Then, it provides a thorough and rigorous treatment of the topic, with detailed proofs and logical explanations. It effectively connects geometric principles, and trigonometry, to algebraic structures, showcasing the depth of Hilbert’s approach.
Overall, the use of geometric constructions and configurations helps to ground abstract concepts in concrete examples. By presenting multiple approaches to segment arithmetic, formulas, and mathematical creation, the book highlights the flexibility and generality of Hilbert and other renowned mathematical experts’ methods.
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