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Trudeau, with obvious pleasure, demonstrates how each proposal turns on some assertion logically equivalent to Postulate 5. Finally the other lapses into exasperated silence.

The conclusion sanctioned by Trudeau is that intuition should be given up in favour of formalism. But this is complete bogus-history. Which is why Trudeau is reduced to fighting, "with obvious pleasure," a fictitious opponent. While the result was not intuitive, the nature of the formal system suggested that it could be proved. Therefore, if any lessons are to be learned from the history of non-Euclidean geometry it is the exact opposite of that sanctioned by Trudeau. Among the many other things which Trudeau gets backwards because of his doctrinal blindness are the relation of logic to mathematics p.

But I do not have the patience to detail these things. As indicated in my other reviews, my views of the subject differ from accepted ones, and I will try to explain them further in relation to this book. The book assumes a somewhat condescending attitude, with imaginary dialogues between the author and presumably a student, possibly from the author's experience. The student asks supposedly "common sense" questions, and the author answers with lengthy explanations, sounding to me like excuses that make the teacher come out the loser.

Thus in a section about "points" pp. The concept of points as the constituents of lines is indeed recent and questionable.

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The author like others overlooks Euclid's definition 3, "The extremities of a line are points", and points are in fact used to delimit lines, as do "breadthless" lines delimit areas, either usage not adding to dimension. Another dialogue discusses "line" p. Today the word is used for "straight line", while contrariwise the term is also applied to curves like great circles on a sphere.

But the sticking point to me is the way it is justified to leave "primitive terms" undefined and then "interpret" them as desired e.


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The justification is roughly as follows. Basic logical principles are so general that one needn't specify what they are about, and then can apply them to particular cases. This is indeed true if a principle holds for anything whatsoever. But, for instance, Euclid's 5th postulate applies specifically to straight lines in a plane, which is why to reinterpret those terms as curvatures, and say the postulate then does not apply and is hence unprovable, commits the fallacy of equivocation.

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What is disappointing is that undoubtedly good heads so carelessly perpetuate illogicalities while laying claim to increased rigor. See all 3 reviews. Amazon Giveaway allows you to run promotional giveaways in order to create buzz, reward your audience, and attract new followers and customers. Learn more about Amazon Giveaway.

Set up a giveaway. There's a problem loading this menu right now. Learn more about Amazon Prime. The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. Richard Trudeau begins The Non-Euclidean Revolution with an admission: that he first studied non-Euclidean geometry only when he was asked to develop a course on the subject. He then went on to write a fascinating look at this area of mathematics.

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In The Non-Euclidean Revolution , we have a mathematically rigorous explanation of this sea change in mathematics which is at the same time suitable for any educated reader. Yes, the mathematics is present and at the center of the exposition, but great mathematical proficiency is no prerequisite for benefitting from this book. My honors students come from throughout my college; few if any are mathematics majors or even all that interested in mathematics. Indeed, one major advantage of representing mathematics with non-Euclidean geometry is that the students have that opportunity to live through a revolution in their own understanding of mathematics.

Gauss shared his thoughts on this topic and asked them not to disclose this information but Gauss never published them. It has been proposed by historians that Gauss was concerned that these concepts were too radical for acceptance by mathematicians at that time. And if this was the case, it probably was correct since the two founders of non-Euclidean geometry, Bolyai and Lobechevsky, received very little acceptance until after their deaths.

In his early days at the university, he did try to find a proof of the parallel postulate, but later changed direction. As early as , he made use of the hypothesis of the acute angle already developed by Saccheri and Lambert in his lecture noting that two parallels to a given point can be drawn from a point where the sum of the angles of the triangle is less than two right angles. He later completed his work in one French and two German publications.


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  • Lobachevsky developed his Pan-Geometry on the 28 propositions of Euclidean geometry and the negation of the parallel postulate. He developed the concepts for non-Euclidean geometry by introducing two new figures—the horocycle and the horoscope. Using these two concepts and some transformation formulas, he developed his new geometry. Although Lobachevsky continued throughout his career improving the development of non-Euclidean geometry, Johann Bolyai, the other mathematician given credit for its development apparently only spent slightly over a decade in his mathematical considerations.

    As indicated previously, Johann's father suggested that he not waste his time working on the complex problems of the parallel postulate. However, Johann and his friend Carl Szasz worked on the theory of parallels while students at the Royal College for Engineers at Vienna from In Bolyai discovered the formula for the transformation which connected the angle of parallelism to the corresponding line.

    He continued with his development and sent his manuscript to his father who published it in Prior to its publication, Johann's father had sent the paper to Gauss for his consideration. It is reported that the paper originally sent in to Gauss was lost. Three months after the publication, the article was sent again to Gauss and in his father received his reply. Gauss indicated that he was impressed by the work but noted that he had been working on the same problem with similar results and was pleasantly surprised to have the development completed by his friend's son.

    Johann was deeply suspicious of this reply and apparently suspected Gauss of trying to take credit for his work. However, in this instance there was no problem, since Gauss had no publications on the topic and could not claim priority but Johann continued to be suspicious. After the publication of his work, Johann did very little significant mathematical research.

    And even though he was interested in having his work published before Lobachevsky when he heard of Lobachevsky's contributions, he never completed the necessary research to report to the mathematical journals. The most important conclusions of Bolyai's research in non-Euclidean geometry were the following: 1 The definition of parallels and their properties independent of the Euclidian postulate.

    The geometry of the sphere of infinite radius is identical with ordinary plane geometry. Direct demonstration of the formula. Applications to the calculation of areas and volumes. Squaring the circle, on the hypothesis that the fifth postulate is false. A later development following that of Bolyai's and Lobachevsky's hyperbolic non-Euclidean geometry was that of elliptic non-Euclidian geometry.

    How new geometries reshaped our world

    Based on the foundations that Riemann had introduced, Klein was able to further develop elliptic non-Euclidean geometry and was actually the mathematician who defined this new field as Elliptic non-Euclidian geometry. Klein's Erlanger Program made a significant contribution in providing major distinguishing features among parabolic Euclidean geometry , hyperbolic, and elliptic geometries. Bonola, R. Non-Euclidean Geometry. New York : Dover Publications, Greenberg, M.