TRANSLATOR'S NOTE. 



Hj ^HE essays and recreations constituting this volume are by one of the foremost 

 * mathematicians and text-book writers of Germany. The monistic construc- 

 tion of arithmetic, the systematic and organic development of all its consequences 

 from a few thoroughly established principles, is quite foreign to the general run of 

 American and English elementary text-books, and the first three essays of Professor 

 Schubert will, therefore, from a logical and esthetic side, be full of suggestions for 

 elementary mathematical teachers and students, as well as for non-mathematical 

 readers. For the actual detailed development of the system of ari f bmetic here 

 sketched, we may refer the reader to Professor Schubert's volume Arithmctik 

 und Algebra, recently published in the Goschen-Sammlung (GSschen, Leipsic), 

 an extraordinarily cheap series containing many other unique and valuable text- 

 books in mathematics and the sciences. 



The remaining essays on "Magic Squares," "The Fourth Dimension," and 

 "The History of the Squaring of the Circle," will be found to be the most com- 

 plete generally accessible accounts in English, and to have, one and aty, a distinct 

 educational and ethical lesson. 



In all these essays, which are of a simple and popular character, and designed 

 for the general public, Professor Schubert has incorporated much of his original 

 research. 



THOMAS J. McCoRMAOC. 



LA SALLE, 111., December, zi 



