1893] BAKER — SURDS AND IRRATIONALS. 203 



The same confusion is noticeable in dictionaries, encyclopedias, 

 and text books, and even in the latest authority, the Century 

 Dictionary, where surds and irrationals are treated as synonymous 

 terms ; and I know of no text book where the same errors are not 

 noticeable. 



The fallacy of this definition and the necessity for a clear 

 distinction between the terms is shown in the theorem which always 

 closely follows, viz. — The square root of a rational quantity cannot 

 be partly rational and partly a quadratic surd [irrational ). The 

 parenthesis and italics are mine. 



The proof to the contrary, objectively put, is 



1/3= 1-732 .... = I + 0-732 .... 



which shows that a quadratic surd is equal to a rational quantity plus 

 an irrational quantity. 



The truth is that the irrational part is a pure irrational which has 

 not the properties of a large class of irrationals to which some 

 distinctive name should be given, preferably that of surd. 



The correct analysis and definition should be : — 



Irrational or incommensurable numbers are divided into Surds 

 and A^oii Surds. 



Surds are the indicated or incommensurable roots of commen- 

 surable quantities, such as, ]/3, j/2, etc. 



iVon Surds are the indicated or incommensurable roots of incom- 

 mensurable numbers, /. e., all irrationals which are not surds, as 



- = y-\ -\ y 1/3—1 =0.732 . . . . , etc. 



It may be that non surds are transcendentals which can not be 

 graphically represented by the ruler and compass as can the surds, 

 1/2, j/3, etc. 



Xi'TE. Since this paper was read, Dupuis" Algebra and Van Velzer and Sclichter's Algebra 

 have appeared in which the distinction here sugg-ested has been made. So far as I know, other 

 text books conservatively adhere to the erroneous definition, though several years ago the atten- 

 tion of a number of authors had been called to the error. 



The following paper was then read by Mr. Joseph E. Putnam : 



DISCUSSION OF SOME PRACTICAL POINTS ABOUT 

 ELECTRIC MOTORS. 



The paper was illustrated by experiments. 



