# Euclidean and Non-Euclidean Geometries: Development and by Marvin J. Greenberg

By Marvin J. Greenberg

This can be the definitive presentation of the heritage, improvement and philosophical importance of non-Euclidean geometry in addition to of the rigorous foundations for it and for hassle-free Euclidean geometry, basically based on Hilbert. applicable for liberal arts scholars, potential highschool lecturers, math. majors, or even vivid highschool scholars. the 1st 8 chapters are ordinarily available to any proficient reader; the final chapters and the 2 appendices comprise extra complex fabric, comparable to the category of motions, hyperbolic trigonometry, hyperbolic buildings, category of Hilbert planes and an advent to Riemannian geometry.

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Additional info for Euclidean and Non-Euclidean Geometries: Development and History (4th Edition)

Example text

It follows that AB AC-in cases 2 and 3 by addition and in case 4 by subtraction. The triangle is therefore isosceles. " J. L. Lagrange, the great master of dynamics after Newton, prided himself that his Analytic Mechanics (published in 1788) contained not a single diagram. " But Hilbert did include diagrams in his Grundlagen der Geometrie. � � � The Power of Diagrams Geometry, for human beings, is a visual subject, and many people think visually more than symbolically. Correct diagrams can be extremely helpful in understanding proofs and in discovering new results.

DEFINITION. An "angle with vertex A" is a point A together with two � � distinct non-opposite rays AB and AC (called the sides of the angle) emanating from A (see Figure 1. 7). 6 AB and AC. However, by bringing in set theory, as Hilbert did, we are sullying Euclid. To avoid that, many of the terms we define as sets would have to be left undefined and new axioms would have to be added to characterize them. The Greeks believed that lines and circles were rwt made up of points. " We eliminated those expressions be­ cause most of the assertions we will make about angles do not apply to them.

The Power of Diagrams Geometry, for human beings, is a visual subject, and many people think visually more than symbolically. Correct diagrams can be extremely helpful in understanding proofs and in discovering new results. For ex­ ample, the great physicist Richard Feynman invented a new type of di­ agram (now named after him) to understand and do research in quan­ tum electrodynamics. 14, which reveals immediately the validity of the Pythagorean theorem in Euclidean geometry. 15 is a simpler diagram suggesting a proof by dissection.