232 



NATURE 



[February 23, 1922 



The Pioneer of Non-Euclidean Geometry. 



Girolamo SaccherV s " Euclides Vindicatus.^' 

 Edited and translated by G. B. Halsted. 

 Pp. XXX 4- 246. (Chicago and London : The Open 

 Court Publishing Co., 1920.) 105-. net. 



THIS work is an important classic, which is 

 well worthy of inclusion in the valuable 

 series brought out by the Open Court Publishing Co. 



Sir Henry Savile, in his lectures of 1620 on 

 Euclid I., published at Oxford in 162 1, "had said 

 that in his judgment there were two blemishes 

 {naevi) or blots (labes), and no more, in the fair 

 body of geometry, the first being the parallel- 

 postulate, and the second the definition of " com- 

 pound ratio " in Book VI. (a definition now known 

 to be interpolated). Saccheri's " Euclides ab omni 

 naevo vindicatus " dealt with both naevi in parts i 

 and 2 respectively, and from the wording of his 

 title we may fairly infer that it was the Englishman 

 who gave the Italian Jesuit the motive for his 

 epoch-making treatise — that of defending Euclid 

 and proving (if he could) that Euclid's work con- 

 tained no flaw. The present edition is confined to 

 part I, on the parallel-postulate, which is alone 

 important. Saccheri must be called the pioneer of 

 non-Euclidean geometry, for, although it was his 

 object .to establish the truth of the Euclidean postu- 

 late once for all by showing that all hypotheses 

 other than that of Euclid are false, he was the 

 first to contemplate the possibility of such other 

 hypotheses and to follow them out to a number of 

 consequences. He is therefore, as Beltrami observed, 

 a true precursor of Legendre and Lobachewsky, 

 and, it might be added, of Riemann also. 



Saccheri starts with a quadrilateral formed by a 

 given straight line as base, two perpendiculars of 

 equal length erected from the extremities of the base 

 on the same side of it, and the straight line joining 

 the other extremities of the equal perpendiculars. 

 The angles made by the latter straight line with the 

 perpendiculars respectively are easily proved to be 

 equal. There are then, says Saccheri, three pos- 

 sible suppositions — the two angles may both be 

 (i) right angles (the Euclidean hypothesis), or 

 (2) obtuse angles, or (3) acute angles. Saccheri 

 calls these the hypothesis of the right angle, the 

 hypothesis of the obtuse angle, and the hypothesis 

 of the acute angle respectively, and the object of 

 his treatise is to prove the absolute falsity of the 

 last two hypotheses. His proof in the case of the 

 obtuse angle depends on the universal validity of 

 Euclid I. 16 (which excludes the Riemann hypo- 

 thesis), while his proof in the case of the acute 

 angle is even less successful. He nevertheless proves 

 certain important propositions afterwards proved by 

 Legendre, Lobachewsky, and Bolyai. 

 NO. 2730, VOL. 109] 



Saccheri lectured on philosophy and theology, as 

 well as mathematics, and he wrote an important 

 logical work, the " Logica demonstrativa," 

 brought to light by Giovanni Vailati in 1903. He 

 was otherwise an interesting personality. We are 

 told that he had a passion for truth, and in his 

 pursuit of it he would (like Archimedes) neglect his 

 person, his food, his clothes, and his comforts. As 

 a boy of five he was a calculating prodigy. Later 

 he became a great chess-player, being able to play 

 (and generally to win) three games simultaneously 

 without seeing the boards, and, as if this were no,t 

 enough, to talk to people around him and also to 

 think out some abstruse geometrical problem at the 

 same time ; afterwards he would repeat all three 

 games from memory. 



Prof. Halsted has important qualifications for 

 editing Saccheri's treatise. He is himself an 

 authority on non-Euclidean geometry, and has an 

 unbounded enthusiasm for his author, which we 

 welcome even when it leads him to write such sen- 

 tences as "So flowered the beauteous body of a 

 new geometry, mermaid-like, the latter portions 

 somewhat fishy, but, oh ! the elegant torso." It is 

 all the more unfortunate that his execution of his 

 task proves in many respects disappointing. The 

 introductory matter, mainly historical, is fairly 

 adequate, but even here there are sins of omission 

 and commission. When he says that Father 

 Manganotti, S.J., "accidentally discovered" 

 Saccheri in 1889, " re-discovered " would be a 

 better word. For Saccheri's work was thoroughly 

 examined in Kliigel's dissertation, " Conatuum 

 praecipuorum theoriam parallelarum demonstrandi 

 recensio " (1763), certain details about it are given 

 in Camerer's Euclid (vol. i, 1824), C. F. A. 

 Jacobi mentions it (1824), and it can scarcely have 

 been unknown to Gauss and Lobachewsky. 



On pp. xviii-xix Prof. Halsted pays the editor of 

 "The Thirteen Books of Euclid's Elements" the 

 compliment of quoting word for word (without in- 

 verted commas) a whole page from his description 

 of Saccheri's " Logica demonstrativa." Almost in 

 the same breath (p. i of preface) he charges the 

 same editor with supposing that Saccheri's 

 " Euclides ab omni naevo vindicatus " was a 

 " Latin edition of Euclid," a baseless charge which 

 he need not have made if he had read the other 

 passages of the raided work where the editor gives 

 not only a detailed description of the book now in 

 question, but also three long citations of proofs by 

 Saccheri of his own propositions with diagrams, 

 which certainly never appeared in any "edition" 

 of Euclid. Nor is there any excuse for Prof. 

 Halsted 's omission to mention the German trans- 

 lation of Saccheri in Engel and Stackel's " Die 



