An introduction to the geometry of numbers.
Repr. of the 1971 ed.

*(German)*Zbl 0866.11041
Classics in Mathematics. Berlin: Springer. x, 344 p. (1997).

The first edition of J. W. S. Cassels’ book appeared in 1959 (see the review in Zbl 0086.26203), and it was the first advanced textbook in the geometry of numbers since H. Minkowski’s basic work in 1896 (JFM 27.0127.09). The second edition in 1971 (Zbl 0209.34401) contained some minor changes and corrections, and now, 38 years after the first edition, the 1971 edition is reissued, which indicates the timeless quality of this book.

Of course, there has been much progress in various parts of the geometry of numbers during the last three or four decades. So the reprint does not contain the results on dense sphere packings, based on coding theory, no lattice point inequalities and nothing about lattice polytopes including Ehrhart’s polynomial and his reciprocity law. Further, recent progress in reduction theory and its application in optimization is missing.

But the book still contains the main stream of the geometry of numbers including the British school, which is characterized by the names of Mordell, Davenport, Rankin, Rogers and Swinnerton-Dyer. And there is no doubt about this book remaining a standard reference, and it still can serve as an additional advanced textbook for all types of courses in the geometry of numbers.

Of course, there has been much progress in various parts of the geometry of numbers during the last three or four decades. So the reprint does not contain the results on dense sphere packings, based on coding theory, no lattice point inequalities and nothing about lattice polytopes including Ehrhart’s polynomial and his reciprocity law. Further, recent progress in reduction theory and its application in optimization is missing.

But the book still contains the main stream of the geometry of numbers including the British school, which is characterized by the names of Mordell, Davenport, Rankin, Rogers and Swinnerton-Dyer. And there is no doubt about this book remaining a standard reference, and it still can serve as an additional advanced textbook for all types of courses in the geometry of numbers.

Reviewer: J.M.Wills (Siegen)

##### MSC:

11Hxx | Geometry of numbers |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

01A75 | Collected or selected works; reprintings or translations of classics |

52C05 | Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry) |

52C07 | Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) |

52C15 | Packing and covering in \(2\) dimensions (aspects of discrete geometry) |

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |

11E25 | Sums of squares and representations by other particular quadratic forms |

11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |

11H06 | Lattices and convex bodies (number-theoretic aspects) |

11H31 | Lattice packing and covering (number-theoretic aspects) |

11J20 | Inhomogeneous linear forms |

11H46 | Products of linear forms |

11H56 | Automorphism groups of lattices |

11H60 | Mean value and transfer theorems |

11H50 | Minima of forms |