Beyond Geometry: Classic Papers from Riemann to Einstein (Dover Books on Mathematics) [Paperback]

Mathematics (Author)

Book Description

ISBN-10: 0486453502 | ISBN-13: 978-0486453507 | Publication Date: December 15, 2006

Eight essays trace seminal ideas about the foundations of geometry that led to the development of Einstein's general theory of relativity. This is the only English-language collection of these important papers, some of which are extremely hard to find. Contributors include Helmholtz, Klein, Clifford, Poincaré, and Cartan.

Product Details

• Paperback: 224 pages
• Publisher: Dover Publications (December 15, 2006)
• Language: English
• ISBN-10: 0486453502
• ISBN-13: 978-0486453507
• Product Dimensions: 9.2 x 6.2 x 0.5 inches
• Shipping Weight: 9.6 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #786,121 in Books (See Top 100 in Books)

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21 of 23 people found the following review helpful:

5.0 out of 5 stars Very interesting, May 6, 2007

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Viktor Blasjo - See all my reviews
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This review is from: Beyond Geometry: Classic Papers from Riemann to Einstein (Dover Books on Mathematics) (Paperback)

There are many interesting articles in this collection. I shall give some highlights.

Helmholtz, "The Origin and Meaning of Geometrical Axioms". "We can ... infer how the objects in a pseudospherical world, were it possible to enter one, would appear to an observer whose eye-measure and experiences of space had been gained like ours in Euclid's space. Such an observer would continue to look upon rays of light or the lines of vision as straight lines, such as are met with in flat space and as they really are in the spherical representation of pseudospherical geometry [i.e., the three-dimensional version of the projective disc model]. The visual image of the objects in pseudospherical space would thus make the same impression upon him as if he were at the center of Beltrami's sphere. He would think he saw the most remote objects round about him at a finite distance, let us suppose a hundred feet off. But as he approached these distant objects, they would dilate before him ... while behind him they would contract. He would know that his eye judged wrongly. If he saw two straight lines that in his estimate ran parallel to his world's end, he would find on following them that the farther he advanced the more they diverged" (p. 65). "There would be an illusion of the opposite description, if, with eyes practised to measure in Euclid's space, we entered a spherical space of three dimensions. We should suppose the more distant objects to be more remote and larger than they are, and should find on approaching them that we reached them more quickly than we expected from their appearance. ... The strangest sight, however, in the spherical world would be the back of our own head" (p. 66). For the two-dimensional case we should imagine mapping the sphere from its center onto a tangent plane ("gnomonic projection"), which of course sends great circles to lines. The inverse of this map is approximated by the image on a spherical mirror, so it is easy to imagine what a Euclidean world would look like to a spherical creature: he would think distant objects were pretty close (about a quarter of the circumference of the sphere) and very small. His illusion is thus the opposite of ours, as it would be in the hyperbolic case as well (a hyperbolic creature would impose the projective disc metric on our Euclidean plane, so things would seem distant and big).

Clifford, "The Postulates of the Science of Space", interprets Euclidean axiomatics using modern concepts, e.g. "all right angles are equal" is a postulate to exclude singular points such as the vertex of a cone: "I can make two lines cross at the vertex of a cone so that the four adjacent angles shall be equal, and yet not one of them equal to a right angle" (p. 81).

Poincaré, "On the Foundations of Geometry": "Our sensations cannot give us the notion of space. That notion is built up by the mind from elements which pre-exist in it ... What could a man see who possessed but a single immovable eye? ... Suppose that two points A and B are very near to each other, and that the distance AC is very great. Would our hypothetical man be cognisant of the difference? We perceive it, who can move our eyes, because a very slight movement is sufficient to cause an image to pass from A to B. But for him the question whether the distance AB was very small as compared with the distance AC would not only be insoluble, but would be devoid of meaning" (pp. 117-118), just as our sense of taste does not enable us to say whether water is further from milk than wine is from beer. Geometry is thus a creation of the mind, and it is not "imposed by experience. It is simply guided by experience. ... To ask whether the geometry of Euclid is true or that of Lobachevsky is false, is as absurd as to ask whether the metric system is true and that of the yard, foot, and inch, is false." (p. 145). This is one of the many articles that have been translated before and OCRed for this printing, creating a few glitches such as "superf1ous" for "superfluous" (p. 133).

Pesic adds many notes, usually full of references. Many are useful, some are strange (I disagree with note 12, p. 70, and find note 1, p. 33, presumptuous). Pesic's reference hysteria is especially marked in his disorganised 16-page introduction, which has 49 footnotes. I think many readers would have benefited more from some pictures; this is a geometry book after all---why not some pictures of models of hyperbolic geometry, for instance?