Non-Euclidean Geometry: A Critical and Historical Study of its Development [Paperback]

Roberto Bonola (Author), Nicholas Lobachevski (Author), John Boylai (Author), H. S. Carslaw (Translator), Federigo Enriques (Introduction)

Book Description

Publication Date: November 18, 2010

Examines various attempts to prove Euclid's parallel postulate — by the Greeks, Arabs and Renaissance mathematicians. Ranging through the 17th, 18th, and 19th centuries, it considers forerunners and founders such as Saccheri, Lambert, Legendre, W. Bolyai, Gauss, Schweikart, Taurinus, J. Bolyai and Lobachewsky. Includes 181 diagrams.

Product Details

• Paperback: 431 pages
• Publisher: Dover Publications (November 18, 2010)
• Language: English
• ISBN-10: 0486600270
• ISBN-13: 978-0486600277
• Product Dimensions: 7.9 x 5.3 x 1 inches
• Shipping Weight: 1 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,276,472 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

5.0 out of 5 stars The most enjoyable book on mathematics I have read, October 19, 2011

By 

Yngvar Hartvigsen, 6872 Luster, Norway - See all my reviews

This review is from: Non-Euclidean Geometry: A Critical and Historical Study of its Development (Paperback)

The starting point is of course Euclid's Parallell Postulate, from which can be deduced:

i. Through a given point outside a given line, one and only one line can be drawn which is parallell to the given line.

Bonola shows us the many attempts that through history have been made to prove the postulate, that is: to show that it is a consequence of the other postulates. This is done by presenting many short and easy to follow proofs, all shown to be resting on different assumptions. The conclusion beeing that the Parallell Postulate can not be deduced from Euclid's other postulates, but may be replaced by other assumptions which will yield the same results.

To follow the proofs the reader only needs some basic knowledge of geometry, just to know some basic nomenclature and properties, and some familiarity with how geometric properties are proved.
Thus by small and easy proofs you discover a lot of interesting connections, and understand why Euclids Parallell Postulate is equivalent to statements like these:

"The sum of the angles in a triangle equals two right angles".
"The distance between two parallell (that is nonintersecting) lines is constant".
"From every point inside an angle a line can be drawn that cuts both arms of the angle".
"A circle can always be drawn through three points not on a straight line".
"It is possible to make two figures with the same shape, but of different size."
"It is possible to make a rectilinear triangle with area greater than any given area."

Since Euclid's Postulate could not be proved, the next step was to assume it wrong, and try to deduce a contradiction. Thus in principle are explored the consequenses of replacing Euclid's assumption with:

ii. Through a given point outside a given line, an infinity of lines can be drawn, all parallell to the given line.

iii. Through a given point outside a given line, no line can be drawn which is parallell to the given line.

Suprisingly, the assumptions ii and iii does not lead to any contradictions, and thus produces two entirely new geometries.

As the story moves forward the mathematics becomes gradually more difficult, demanding more mathematical knowledge from the reader. But the presentation is always clear, and most of it should not be to complicated to follow, and the conclusions are easy to grasp.

In the last part of the book are included two original works by Bolyai (1832) and Lobachevski (1840). Having worked through the first part of the book, these original papers can be read with pure joy.

This is simply the most enjoyable book on mathematics that I have read!

7 of 18 people found the following review helpful:

3.0 out of 5 stars A very old classic, July 28, 2001

By 

R. Bagula "Roger L. Bagula" (Lakeside, Ca United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Non-Euclidean Geometry: A Critical and Historical Study of its Development (Paperback)

In Einstein's day this might have been a very good read! It is very well written. It is like reading a Spanish concurrent to Russell. A little reading finds it is a translation of a 1912 text. With general Relativity being a product of the understanding of the velocity based non Euclidean geometry of Lorentz who based his work on Poincare who based his work on Klein who based his work on... you see that history is important in an axiomatic development like this has been! But for a modern student of geometry, this book is much like buying a copy of Euclid's book on geometry: a reference that might help with understanding, but is so far out of date that it can be very little help in current problem!