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**The legacy of Mario Pieri in geometry and arithmetic.**
*(English)*
Zbl 1126.01021

Basel: Birkhäuser (ISBN 978-0-8176-3210-6/hbk). xix, 494 p. (2007).

This book is the first in a projected set of three volumes devoted to the Italian mathematician Mario Pieri (1860–1913). The authors begin with biographical information on Pieri, summarize his research output, and provide sixty pages of biographical sketches of nearly one hundred other mathematicians whose lives and/or works relate to Pieri’s story. There follows a more detailed survey of his contributions to axiomatic developments of various geometries, such as projective, inversive, and Euclidean. The book’s centerpiece is a 111-page English translation of Pieri’s 1908 paper in which he reduced Euclidean geometry to two primitive notions: point and a ternary equidistance relation. The authors also treat Pieri’s work on axiomatizing the natural numbers. Presentation of material in this book is thorough, if somewhat repetitious, reflecting the authors’ conviction (seconded by Ivor Grattan-Guinness in the foreword) that Pieri’s contributions have been unduly neglected.

Reviewer: Leon Harkleroad (Wilton)

### MSC:

01A70 | Biographies, obituaries, personalia, bibliographies |

01A55 | History of mathematics in the 19th century |

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

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

51-03 | History of geometry |

03-03 | History of mathematical logic and foundations |