© 2014

Euclid Vindicated from Every Blemish

Edited and Annotated by Vincenzo De Risi. Translated by G.B. Halsted and L. Allegri

  • Vincenzo De Risi
  • First complete edition of Saccheri's Euclides Vindicatus in English

  • The classical English translation by Halsted has been corrected and enriched by the important Book Two of the work (on the theory of proportions) that was missing in all the preceding editions

  • Provides a historical introduction to the birth of non-euclidean geometry


Part of the Classic Texts in the Sciences book series (CTS, volume 1)

Table of contents

  1. Front Matter
    Pages I-VII
  2. Gerolamo Saccheri
    Pages 1-58
  3. Euclid Vindicated from Every Blemish or A Geometric Endeavor in which are Established the Foundation Principles of Universal Geometry

    1. Front Matter
      Pages 60-69
    2. Gerolamo Saccheri
      Pages 70-191
    3. Gerolamo Saccheri
      Pages 192-245
  4. Notes to the text

    1. Front Matter
      Pages 247-247
    2. Gerolamo Saccheri
      Pages 249-347
  5. Back Matter
    Pages 349-381

About this book


This first complete English language edition of Euclides vindicatus presents a corrected and revised edition of the classical English translation of Saccheri's text by G.B. Halsted. It is complemented with a historical introduction on the geometrical environment of the time and a detailed commentary that helps to understand the aims and subtleties of the work.

Euclides vindicatus, written by the Jesuit mathematician Gerolamo Saccheri, was published in Milan in 1733. In it, Saccheri attempted to reform elementary geometry in two important directions: a demonstration of the famous Parallel Postulate, and the theory of proportions. Both topics were of pivotal importance in the mathematics of the time. In particular, the Parallel Postulate had escaped demonstration since the first attempts at it in the Classical Age, and several books on the topic were published in the Early Modern Age. At the same time, the theory of proportion was the most important mathematical tool of the Galilean School in its pursuit of the mathematization of nature. Saccheri's attempt to prove the Parallel Postulate is today considered the most important breakthrough in geometry in the 18th century, as he was able to develop for hundreds of pages and dozens of theorems a system in geometry that denied the truth of the postulate (in the attempt to find a contradiction). This can be regarded as the first system of non-Euclidean geometry. Its later developments by Lambert, Bolyai, Lobachevsky and Gauss eventually opened the way to contemporary geometry.

Occupying a unique position in the literature of mathematical history, Euclid Vindicated from Every Blemish will be of high interest to historians of mathematics as well as historians of philosophy interested in the development of non-Euclidean geometries.


history non-euclidean geometry parallel postulate theory of proportions

Authors and affiliations

  1. 1.BaselSwitzerland

Editors and affiliations

  • Vincenzo De Risi
    • 1
  1. 1.History of ScienceMax Planck InstituteBerlinGermany

Bibliographic information

  • Book Title Euclid Vindicated from Every Blemish
  • Book Subtitle Edited and Annotated by Vincenzo De Risi. Translated by G.B. Halsted and L. Allegri
  • Authors Gerolamo Saccheri
  • Editors Vincenzo De Risi
  • Series Title Classic Texts in the Sciences
  • Series Abbreviated Title Classic Texts in the Sciences
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2014
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-05965-5
  • Softcover ISBN 978-3-319-37791-9
  • eBook ISBN 978-3-319-05966-2
  • Edition Number 1
  • Number of Pages VII, 381
  • Number of Illustrations 83 b/w illustrations, 6 illustrations in colour
  • Topics History of Mathematical Sciences
    History of Science
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking


From the book reviews:

“This is the first complete English edition of Saccheri’s Euclides vindicatus of 1733, and, with 55 pages of introduction, 99 pages of notes, 11 pages of bibliography, the definitive one. … The material in the introduction and the notes is of great interest to both historians of mathematics and to mathematicians … .” (Victor V. Pambuccian, zbMATH, Vol. 1303, 2015)