Euclid vindicated from every blemish. Edited and annotated by Vincenzo De Risi. Translated from the Italian by G. B. Halsted and L. Allegri (Q2449335)

This is the first complete English edition of Saccheri's \textit{Euclides vindicatus} of 1733, and, with 55 pages of introduction, 99 pages of notes, 11 pages of bibliography, the definitive one. The text itself is in its original Latin side by side with an English translation. The translation of the first part is that of \textit{G. B. Halsted} [Euclides vindicatus; translated with introduction and notes by G. B. Halsted in a Latin-English edition. Chicago: Open Court (1920; JFM 47.0035.03)], modified and corrected by the editor. The English translation of the second part is that of \textit{Linda Allegri}, from her unpublished 1960 Columbia University doctoral dissertation, with a large number of corrections and revisions by the editor. The introductory essay -- which, together with the notes was translated from the Italian edition [\textit{G. Saccheri}, Euclide vendicato da ogni neo (Italian, Latin). Pisa: Edizioni della Normale (2011; Zbl 1227.01004)] -- treats in sufficient depth the philosophical intentions of Saccheri and their effect on the structure of the \textit{Euclides vindicatus}, the theory of parallels and of proportions during the 17th century, as well as in the works of Muslim mathematicians such as al-Tusi and Khayyam, Saccheri's use of \textit{consequentia mirabilis}, a discussion of the results and (minor) errors the book contains, its reception in its time and in later centuries, and a brief survey of the previous editions and translations of \textit{Euclides vindicatus}. The information contained in the introductory essay and in the notes is supplemented by a complete bibliography of works related to Saccheri or to mathematicians whose work dealt with the two themes Saccheri deals with: the Euclidean Fifth Postulate and the Eudoxian theory of proportions, in particular the tacit assumption of the existence of the fourth proportional, i.e. that, given three magnitudes \(A\), \(B\), and \(C\), there always exists a magnitude \(D\) with \(A:B::C:D\). The material in the introduction and the notes is of great interest to both historians of mathematics and to mathematicians, given that, in the case of the first, geometrical, part of the book, the reader is being told a significant part of the story of the parallels up to the present day. This includes the major breakthrough that was \textit{W. Pejas}'s [Math. Ann. 143, 212--235 (1961; Zbl 0109.39001)] -- which enabled the purely algebraic study of questions related to the Euclidean parallel postulate that interested Saccheri -- as well as several later results that use Pejas's classification.