on 20 January 2013
Bonola's book is indeed a very good introduction to both Bolyai's "The Science of Absolute Space" and Lobachevski's "Geometrical Researches on the Theory of Parallels".
But, the two jewels of that book are the translations of the latter books, i.e. there's nothing like going to the originals !
Including those translations was a brilliant idea.
on 17 October 2011
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!