508 



SCIENCE. 



[N. S. Vol. II. No. 36. 



Die Theorie der Parallellinien von Euklid bis aiif 

 Gauss, eine JJrTcundensammlung zur Yorge- 

 schichte der niclitenklidischen Geometrie, in Ge- 

 meinscha/t mii Friedrich Engel, herausgegeben 

 von Paul Stackel. Leipzig, Teubner. 1895. 

 [Julyl. 



This book is a striking example of one of the 

 many beneficent characteristics of our present 

 civilization. Here all the works which show 

 the gradual but sure development of the human 

 mind toward an achievement of modern thought 

 unsurpassed for interest and importance, books 

 so rare that, so far as I know, not one is con- 

 tained in any public library on the western con- 

 tinent, are put within the reach of the poorest 

 student. Here we have, edited with the most 

 painstaking accuracy, Wallis, Saccheri, Lambert, 

 Schweikart, Taurinus, the forerunners of the 

 non-Euclidean geometry. 



The .jump made by Bolyai and Lobachevski, 

 the Magyar and the Russian, will no longer 

 seem so bewilderingly long and unanticipated. 

 How they, about the same time, 1829, came to 

 publish each a complete, a full-fledged non-Eu- 

 clidean geometry was a problem which pro- 

 voked an unfortunate pseudo-solution, a hypo- 

 thetical construction, which is still repeated, 

 and even to be found in the pages of Science. 

 [March 29, 1895, pp. 357-8.] 



After a lecture on Saccheri at the World's 

 Fair Science Congress, since published under 

 the title ' The non-Euclidean Geometry Inevit- 

 able,' in the Monist, July, 1894, pp. 483-493, 

 Professor Felix Klein, of Goettingen, who was . 

 present and said that never before had he so 

 much as heard even the name of Saccheri, was 

 asked why in his Nieht-Euklidische Geometrie, 

 1889-90, he says : " Kein Zweifel bestehen kann, 

 dass LobatschefFsky sowohl wie Bolyai die Frage- 

 stellung ihrer Untersuchungen der Gaussichen 

 Anregung verdanken " [p. 175, Zweiter Ab- 

 druck, 1893]. He answered, that he believed 

 he would be justified when Schering published 

 the ' Nachlass von Gau.ss.' Such special per- 

 sonal information from Schering perhaps is re- 

 ferred to on the preceeding page, 174, in the 

 sentence : ' ' Dies sind die saemmtlichen Notizen, 

 die man in allgemeinen Kreisen ueber die Gaus- 

 sichen Untersuchungen, betrefFend die uieh- 

 teuklidische Geometrie besitzt." 



This very question to Professor Klein, as to 

 how he could justify his ungenerous statement, 

 must have been again put to him by Engel and 

 Stackel, and he must have given essentially the 

 same answer ; for, after stating his opinion, they 

 say of it, p. 243: "Eine Entscheidung ueber 

 die Richtigkeit dieser Vermutungen wird kaum 

 moeglich sein, solange der Xachlass ^•on Gauss 

 der Forschung unzuganglich ist." 



But how little we can trust the unchecked 

 judgment of Klein in this matter is strikingly 

 sho\^'n by what he says of Gauss's letter to 

 Bolyai of 1799, on this very page 174: "Dies 

 ist das interessanteste hierher gehoerige Doku- 

 ment, da es noch ganz aus Gauss' Jugendzeit 

 stammt. In diesem letzteren Brief ist besonders 

 gesagt, dass es in der hj-perbolischen Geometrie 

 ein Maximum des Dreieckinhaltes gebe." 



This letter is given in ftill in the English 

 translation of the Science Absolute of Space 

 by Bolyai Janos, and again in the Monist, (p. 

 486), and is reproduced by Staeckel (p. 219). 

 What it really says is about as far as could 

 be well imagined from the statement of Pro- 

 fessor Klein. If Schering can do no better 

 than that, we need not wait to declare that 

 there is not the slightest particle of evidence 

 that either Bolyai or Lobachevski were even in 

 the remotest degree influenced by Gauss. 



A certain 'Gymnasiallehrer,' Richard Beez, 

 mentioned slightingly by Professor Klein on p. 

 277 of Part I. of his Nicht-Euklidische Geome- 

 trie, as incapable of grasping the subject, yet 

 presumed on p. 15 of a pamphlet published at 

 Plauen to use the expression, 'Gauss, der Lehrer 

 von Bolyai und Lobatschewsky.' This irrita- 

 ting misstatement was reproduced by Dr. Em- 

 ory McClintock in the Bulletin of the New 

 York Mathematical Society, and when written 

 to about it he asked Beez his grounds, and of 

 course found there were none. In his retrac- 

 tion, Bulletin, March, 1893, p. 146, he cites, as 

 some justification, the paragraph from Professor 

 Klein already discussed, and says: "In the 

 paper already cited I followed Beez in stating 

 too strongly the probable connection between 

 Gauss and Lobatschewsky. I am indebted for 

 my first knowledge of Beltrami's account of 

 Saccheri to a letter from Professor Beez, in 

 which he admits liis mention of Gauss as the 



