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**Geschichte der Geometrie seit Hilbert. (The history of geometry since Hilbert).**
*(German)*
Zbl 0718.01003

Darmstadt: Wissenschaftliche Buchgesellschaft. x, 246 p. DM 49.00 (1988).

The authors give a historical appreciation of the development of fundamental parts of geometry. Its state until the end of the 19th century is outlined in the introduction. The startpoint of real investigations is the axiomatical explanation of geometry done by D. Hilbert, and the aim is a representation of the history of geometry in the 20th century.

The three main parts of the book under consideration concern incidence geometry (classically linear incidence geometry, special problems and modern structures, webs and nonlinear incidence geometry), axioms of order and topology (up to non-Desarguesian topological planes) and congruences (with a main emphasis on absolute geometry). Informations are given about these three chapters from the beginnings up to the newest results. A chapter “Cinematic and foundations of geometry” shall be dealt with in a 2nd volume. All notions and definitions used in the book are given, results and theorems (with references) are formulated exactly. Relations between corresponding theorems are carefully put out, as well as connections between geometries. Contributions of persons (surveyors) to the development of the geometry are indicated comprehensively, so the reader is able to follow the research process. Many statements are illustrated by figures, some including examples make curious to the following. The book is completed by 7 pages of biographical data of mentioned surveyors (often with informations about published obituaries), a bibliography of 27 p., an author index of 3 p., a subject index of 7 p., and a symbol index.

The three main parts of the book under consideration concern incidence geometry (classically linear incidence geometry, special problems and modern structures, webs and nonlinear incidence geometry), axioms of order and topology (up to non-Desarguesian topological planes) and congruences (with a main emphasis on absolute geometry). Informations are given about these three chapters from the beginnings up to the newest results. A chapter “Cinematic and foundations of geometry” shall be dealt with in a 2nd volume. All notions and definitions used in the book are given, results and theorems (with references) are formulated exactly. Relations between corresponding theorems are carefully put out, as well as connections between geometries. Contributions of persons (surveyors) to the development of the geometry are indicated comprehensively, so the reader is able to follow the research process. Many statements are illustrated by figures, some including examples make curious to the following. The book is completed by 7 pages of biographical data of mentioned surveyors (often with informations about published obituaries), a bibliography of 27 p., an author index of 3 p., a subject index of 7 p., and a symbol index.

Reviewer: W.H.Schmidt (Greifswald)

### MSC:

01A05 | General histories, source books |

01A60 | History of mathematics in the 20th century |

51-03 | History of geometry |