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 user-friendly Euclidean geometry, basically in line with Hilbert. applicable for liberal arts scholars, potential highschool lecturers, math. majors, or even vibrant highschool scholars. the 1st 8 chapters are usually available to any trained reader; the final chapters and the 2 appendices include extra complex fabric, comparable to the category of motions, hyperbolic trigonometry, hyperbolic buildings, class of Hilbert planes and an creation to Riemannian geometry.
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Extra resources for Euclidean and Non-Euclidean Geometries: Development and History (4th Edition)
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. 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.
22 EUCLID'S GEOMETRY congruent. ) The parallel postulate is different in that we cannot verify empiri cally whether two drawn lines meet since we can draw only segments, not complete lines. We can extend the segments further and further to see if the lines containing them meet, but we cannot go on extending them forever. Our only recourse is to verify parallelism indirectly by using criteria other than the definition. What is another criterion to test whether l is parallel to m? , a line t that intersects both l and m in distinct points) and considering the interior angles a one side of t.
Books I-IV and VI are about plane geometry. • Books XI-XIII are about solid geometry. • Book V gives Eudoxus' theory of proportions. • Books VII-IX treat the theory of whole numbers. The last propo sition of Book IX (Proposition 36) provides a method of construct ing a perfect number-a number that is equal to the sum of its proper divisors, such as 6, 28, or 496. To this day no other method has been found. • Book X presents Theaetetus' classification of certain types of ir rationals; curiously, Euclid did not include a proof that the diago nal of a square is incommensurable with its side, though the Ital ian translation by Commandino in 1575 does add a proof of that.