Foundations of Geometry

For the best translation into English (not Townshend's translation) see Hilbert's "Foundations of Geometry" as extended by Paul Bernays, Open Court Publishing Co, second edition in English, 1971. (This should be a translation of the 10th and final edition in German by Bernays, which dates from 1968.) Bernays was Hilbert's assistant at Göttingen beginning in 1917 and his co-author of "Grundlagen der Mathematik" (1934-39). Hardcover ISBN: 0875481639 Paperback ISBN: 0875481647

This is the first book ever to present the axiomatic foundations of euclidean geometry. The first edition appeared in the nineties of the nineteenth century. Most of the book can be read and appreciated by someone who is mature in elementary euclidean geometry (in fact the material was originally conceived to be used in a summer school for mathematics teachers in Germany). If you expect to find a treatment that will fill up all the gaps in the elementary books you will be disappointed, it does not. If you are looking for a text that does fill all the gaps try to get a copy Forders' book The foundations of Euclidean geometry,. This edition is not based on the last German edition that is available and does not contain the appendices by Hilbert and the supplements by Paul Bernays, so as a text on the foundations of euclidean geometry it is not useless but it is surely crippled. I do not dare to give a book with Hilberts name on it less than five stars.

Hilbert gives his new system of axioms and studies their consistency, independence and necessity. Consider for example the theorem that the angle sum in any triangle cannot be greater than two right angles. We can prove it as follows. Consider a triangle ABC with the angles labelled so that ABC<=ACB. Let D be the midpoint of BC. Draw AD and extend it to E so that AD=DE. By SAS, ACD=BDE, so that angle CAD=angle BAE and angle DBE=angle ACB. Thus ABC has the same angle sum as ABE. ABC<=ACB means that AC=BE<=AB, so angle BAE<=angle AEB, so angle BAE<=angle BAC/2. In other words: for any angle A in any triangle we can construct a new triangle with equal angle sum that has as one of its angles A/2. By repeating this process we can make the angle A as small as we like. Thus, if the angle sum of some triangle was greater than two right angles, and we applied this procedure, we would get a new triangle where two of the angles are greater than two right angles, which is impossible. The "as small as we like" part gives away the fact that we are relying on Archimedes' axiom, which is necessary. "The investigation of this matter which [Max] Dehn has undertaken at my urging led to a complete clarification of this problem. ... If Archimedes' axiom is dropped then from the assumption of infinitely many parallels through a point it does not follow that the sum of the angles in a triangle is less than two right angles. Moreover, there exists a geometry (the non-Legendrian geometry) in which it is possible to draw through a point infinitely many parallels to a line and in which nevertheless the theorems of Riemannian (elliptic) geometry hold. On the other hand there exists a geometry (the semi-Euclidean geometry) in which there exists infinitely many parallels to line through a point and in which the theorems of Euclidean geometry still hold. From the assumption that there exist no parallels it always follows that the sum of the angles in a triangle is greater than two right angles." Another interesting topic is the connection between laws of algebra and the theorems of Pappus (which Hilbert calls Pascal's) and Desargues. Geometrically, we can multiply two numbers a and b using only the axioms of projective geometry as follows. We choose a line to be the "x-axis" and call one of its points the origin O and another of its points the unit 1. Mark Oa and Ob on this line. Draw another line, the "y-axis", through O. Pick some point i on the y-axis. Connect 1 and i, and draw the parallel to this line through b, meeting the y-axis at b' (as usual, "parallel to l" means: meets l at an arbitrarily designated line called the line at infinity). Connect a and 1 and draw the parallel to this line through b'. In Euclidean geometry this line cuts the x-axis at ab. In general, then, we may define multiplication in this way. The algebraic identity ab=ba now becomes a geometric theorem. This is the beautiful part: ab=ba is not just any old geometric theorem, it is in fact equivalen

This historic book is available for free from Project Gutenberg http://www.gutenberg.org. Search for Geometry. This book is one of a few books available. This is the complete Open Court text. It is available both as a pdf file and a TeX file.

Unlike other books of geometry , the author of this book constructed geometry in a axiomatic method . This is the feature which differ from other books of geometry and the way I like . Let's see how the author constructed axiomization geometry . Intuition and deduction are two powerful ways to knowledge . The axioms are the intuitive principles which are needless to be proved . The theorems are the demonstrated propositions which are deduced from axioms . Although axioms are intuitive , they may have the demonstrated propositions called theorems which contradict . If they do , the system of the axiomization geometry would break down . Because it has some false propositions if you think the contradictory ones as truth , and vice versa . There are all the discussions of the problems above in chapter 2 called consistency which is very important in an axiomatic system .