264 POPULAR SCIENCE MONTHLY. 



any one of them be assumed the others are readily proved. The 

 theorem that the sum of the three angles of a triangle is equal to two 

 right angles belong to this set. Ptolemy (Claudius Ptolemaeus, sec- 

 ond century A. D.) seems to have been the first to publish an attempted 

 proof of this postulate of Euclid. Almost all mathematicians down to 

 the beginning of the nineteenth century have given more or less atten- 

 tion to this question, and the account of their efforts to prove the postu- 

 late forms one of the most interesting chapters in the history of mathe- 

 matics. Cajori, in his 'History of Elementary Mathematics/ says, 

 page 270: "They all fail, either because an equivalent assumption is 

 implicitly or explicitly made, or because the reasoning is otherwise 

 fallacious. On this slippery ground good and bad mathematicians alike 

 have fallen. We are told that the great Lagrange, noticing that the 

 formulas of spherical trigonometry are not dependent upon the paral- 

 lel-postulate, hoped to frame a proof on this fact. Toward the close 

 of his life he wrote a paper on parallel lines and began to read it before 

 the Academy, but suddenly stopped and said: 'II faut que j'y songe 

 encore' (I must think it over again); he put the paper in his pocket and 

 never afterwards publicly recurred to it." 



About the time to which I have referred, the end of the eighteenth 

 aud beginning of the nineteenth century, the idea began to force itself 

 upon mathematicians that perhaps there was more in the question than 

 appeared on the surface. It was one of the many instances which have 

 occurred in all branches of human knowledge where some truth of 

 fundamental importance has begun to force itself simultaneously on a 

 number of minds. We leave the significance of this aspect of the ques- 

 tion to the psychologists. Another curious fact to be noted in connec- 

 tion with the writings which have finally shown us the true meaning, 

 of the parallel-postulate is that either they attracted little or no gen- 

 eral attention when they first appeared, or else they remained unpub- 

 lished. The names of Lobatchewsky and the Bolyais have been made 

 immortal by their writings on this subject, but it was not until long 

 after they were published that their vast importance was recognized. 

 The inimitable Gauss wrote on the same subject, but left his work un- 

 published, and Cajori (ibid., p. 274) mentions two writers of much 

 earlier date who anticipated in part the theories of Lobatchewsky and 

 the Bolyais. These are Geronimo Saccheri (1667-1733), a Jesuit father 

 of Milan, and Johann Heinrich Lambert (1728-1777), of Muhlhausen, 

 Alsace. 



Lobatchewsky (Nicholaus Ivanovitch Lobatchewsky, 1793-1856) 

 conceived the brilliant idea of cutting loose from the parallel-postulate 

 altogether and succeeded in building up a system of geometry without 

 its aid. The result is startling to one who has been taught to look upon 

 Ihe facts of geometry (that is, of the Euclidean geometry) as incon- 



