358 



SCIENCE. 



[N. S. Vol. I. No. 13. 



and protecting friend of Lobachevsky, con- 

 firms the supposition that the first impulse 

 to these studies came to him, at least indi- 

 rectly, from Gauss. To the same source of 

 inspiration must be traced the almost simul- 

 taneous, but independent, researches of 

 the Hungarian "Wolfgang Bolyai and his 

 son Johann. Gauss himself never pu^blished 

 anything on the subject of non-Euchdean 

 geometry ; but we know from his letters to 

 Schumacher that he had spent much thought 

 on these questions, which had occupied him 

 from his earliest youth, and had arrived at 

 practically the same results as Lobachevsky 

 and the Bolyais. 



In the later development of non-Euclid- 

 ean geometry and the closely related the- 

 ory of M-dimensional spaces or manifold- 

 nesses we find among others the names of 

 Grassmann, Riemann, Hehnholtz, Cayley, 

 Klein, Lie ; and in these the uninitiated 

 may find a sufiicient guarantee for the value 

 of the subject. 



In conclusion, a few words must be said 

 of the present English translation. The 

 original has been followed so faithfully that 

 anybody possessed of an adequate knowl- 

 edge of the Russian language will under- 

 stand the translation very readily. The 

 reading of such unidiomatic English is, 

 however, exceedingly painful. Were it not 

 for the direct statement on the title-page, 

 we should never have ascribed this transla- 

 tion to Professor Halsted, whose vigorous 

 command of the English language is well 

 known. It seems almost incredible that a 

 person whose native language is English 

 should have written, or even passed in the 

 proof, such sentences as these : (p. 3) " So 

 in celebrating this day to Lobachevskj'^, we 

 must remember with gratitude his teach- 

 ers." (ib.) "His destiny was to be the 

 teacher and protector not only of Loba- 

 chevsky, but of the scientist of our century 

 most influential on the development of 

 mathematics, Gauss, (ib.) "The mathe- 



matical ability of the boy-genius awakened 

 the attention of the science-hungry Bar- 

 tels." (p-4.) "... he received the grade 

 of 'Magister' July 10, 1811, for extraordi- 

 nary advance in mathematics and physics." 

 (ib.) "... the question of the lowering 

 of the gi-ade of a two-termed equation ..." 



The transliteration of Russian names is 

 faulty and inconsistent ; thus we find Pouch- 

 kin for Pushkin, Demidef for Demidov, Ka- 

 ramzen for Karamzin, Simenov for Simonov, 

 etc. It is inconceivable why the name of the 

 well-known astronomer Littrow should be 

 persistently misspelled Lettrov. On p. 1, 

 for ' November 9, 1807 ' read ' January 9, 

 1807.' The statement in the preface, p. 

 vii., that " in 1500 Copernicus was enjoying 

 the friendship of Regiomontanus and fulfill- 

 ing with distinction the duties of a chair of 

 mathematics " is singularlj' incorrect. Re- 

 giomontanus died in 1476, when Copernicus 

 was three years of age ; and, although Rhae- 

 ticus, in speaking of the residence at Rome 

 in 1500, refers to Copernicus as ' professor 

 mathematum,' it is now, in the absence of 

 any direct evidence, generally accepted that 

 the author of the De revolutionibus was never 

 connected as teacher with any scientific in- 

 stitution. Alexander Ziwet. 



UNnrEESiTY OF Michigan. 



Laboratory Exercises in Botany, designed for 

 the use of colleges and other schools in 

 which Botany is taught by laboratory 

 methods, by Edson S. Bastin, Am. Pro- 

 fessor of Materia Medica and Botany and 

 Director of the Microscopical Laboratoiy 

 in the Philadelphia CoUege of Pharmacy. 

 Philadelphia. 1895. $2.50. 

 In a review of this volume it should be 

 considered for whom it was written and 

 from that standpoint an estimate should be 

 made whether the purpose has been really 

 accomplished. Being designed for students 

 who are beginners, it leads them from the 

 simple to the complex, and does it, we think. 



