June 9, 1899.1 



SCIENCE. 



813 



Bardwell, State Normal School, Cortland ; 

 Dr. Charles W. Dodge, University of 

 Rochester; Principal Thomas B. Lovell, 

 High School, Niagara Falls ; Professor, AV. 

 C. Peckham, Adelphi College, Brooklyn ; 

 Professor J. McKeen Cattell, Columbia 

 TJniversitj', New York ; Professor John F. 

 Woodhull, Teachers College, New York ; 

 Professor E. R. Whitney, High School, 

 Binghamton. 



Franklin "W. B arrows. 



SCIENTIFIC BOOKS. 

 Urkunden ziir Geschichte der nichteuMidischen 

 Geometrie. Von F. Engel und P. Staeckel. 

 /. Nikolai Ivanovitsch LobatschefsJci. Leipzig, 

 B. G. Teubner. 1899. 8vo. Pp. 476. 

 The name of Lobachevski is inseparably con- 

 nected with a scientific advance so fundamental 

 as actually to have changed the accepted con- 

 ception of the universe. 



Yet his first published work and his greatest 

 work have both remained for over sixty years 

 inaccessible, locked up in Russian, and are now 

 for the first time given to the world in this 

 monumental volume by Professor Engel. 



As to the precise time at which Lobachevski 

 shook himself free from Euclid's two thousand 

 years of authority there is still room for a most 

 interesting doubt. 



The first of the two treatises given in this 

 book, 'On the Elements of Geometry,' was 

 published in 1829, with this note at the foot of 

 the first page : 



"Extracted by the author himself from a 

 paper which he read February 12, 1826, in the 

 meeting of the Section for Physico-mathematic 

 Sciences, with the title : ' Exposition succiucte 

 des principes de la Geometrie, etc' " 



Again, when the four equations are reached 

 which really contain the essence of the non- 

 Euclidean geometry, Lobachevski subjoins this 

 note : " The equations (17) and all that follows 

 these the author had already appended to the 

 paper which he presented in 1826 to the Section 

 for Physico-mathematic Sciences." 



In the introduction to the second of the two 

 treatises here given, the ' New Elements of 

 Geometry,' the author says : " Everyone knows 



that in geometry the theory of parallels has 

 remained, even to the present day, incomplete. 

 " The futility of the efforts which have been 

 made since Euclid's time during the lapse of 

 two thousand years to perfect it awoke in me 

 the suspicion that the ideas employed might 

 not contain the truth sought to be demonstrated, 

 and for whose verification, as with other natural 

 laws, only experiments could serve, as, for ex- 

 ample, astronomic observations. 



" When, finally, I had convinced myself of the 

 correctness of my supposition, and believed my- 

 self to have completely solved the difficult 

 question, I wrote a paper on it in the year 

 1826, ' Exposition succincte des principes de la 

 Geometrie, aveo une dimonstration rigoureuse du 

 iheoreme des par alleles,'' read February 12, 1826, 

 in the seance of the phj'sico-mathematic Faculty 

 of the University of Kazan, but never printed." 

 No part of this French manuscript has ever,, 

 been found. The latter half of the title is 

 ominous. 



For centuries the world had been deluged 

 with rigorous demonstrations of the theorem of 

 parallels. We know that three years later 

 Lobachevski himself proved it absolutely in- 

 demonstrable. 



Yet the paper said to contain material to 

 stop forever this twenty-centuries-old striving 

 still was headed ' demonstration rigoureuse,' 

 just as Saccheri's book of 1733 containing a 

 coherent treatise on non-Euclidean geometry 

 ended by one more pitiful proof of the parallel- 

 postulate. 



If Saccheri had lived three years longer and 

 realized the pearl in his net, with the new 

 meaning, he "could have retained his old title : 

 ' Euclides ab omni naevo vindicatus,' since the 

 non-Euclidean geometry is a perfect vindica- 

 tion and explanation of Euclid. But Lobach6v- 

 ski's title is made wholly indefensible. 



A new geometry, founded on the contra- 

 dictory opposite of the theorem of parallels, and 

 so proving every demonstration of that theorem 

 fallacious, could not very well pose under 

 Lobach6vski's old title. Least said, soonest 

 mended. He never tells what he meant by it, 

 never tries to explain it. 



Yet Engel thinks that under this two thou- 

 sand years stale title, ' avec une demonstration 



