Customer Reviews

Janos Bolyai, Non-Euclidean Geometry, and the Nature of Space

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I find Bolyai's paper quite hopeless to read; it's a strange choice for semi-popular publication. Gray's introduction is very pleasant and interesting and full of historical background, but his commentary on Bolyai's actual paper is quite short and not always clear. He does comment extensively on Bolyai's squaring of the circle but this construction is too complicated to be very enjoyable. This result, and Bolyai's entire approach, depends on hyperbolic trigonometric formulae. He saw that such formulae should exist by finding a correspondence between a hyperbolic plane and a surface in hyperbolic space whose geometry is Euclidean (F-surface, horosphere). Today we may interpret this in terms of the half-space model. As our hyperbolic plane we can take a hemisphere centred at the origin and as the horosphere we can take a plane z=c. Lines on the hemisphere are of course intersections with planes perpendicular to the x-y-plane, and lines on the horosphere are Euclidean lines. Under vertical projection of one onto the other lines go to lines and angles are preserved. So a Euclidean right-angle triangle with one vertex at the z-axis correspond to a hyperbolic right-angle triangle with one vertex at the z-axis. And by rotating about the z-axis we see that the ratio of circumferences of the circles generated by the other two vertices is the same on both surfaces. This relates side lengths and thus gives a way of transferring Euclidean trigonometry to the hyperbolic plane. But in hyperbolic geometry the circumference does not grow linearly with the radius (but rather as the hyperbolic sine of the radius, as Bolyai shows later using the angle of parallelism formula), so Euclidean trigonometry does not transfer literally.