460 THE POPULAR SCIENCE MONTHLY 



the following statement : " The matrix has also significance apart from 

 its development into a determinant." In view of the fact that a matrix, 

 as commonly used by mathematicians, has no possible development, it is 

 clear that the given sentence does not convey any information. In fact, 

 the other remarks under this term are almost equally objectionable, and 

 they raise the question whether a philosopher should be selected to 

 define the mathematical terms of a standard work. 



It is not implied that such an extensive work as a large dictionary 

 could be expected to be free from defects, but there is always a limit 

 to the number and the type of those which appear excusable. When 

 one reads in such a dictionary that " algebra is that branch of mathe- 

 matics which treats the relations and properties of quantity by means 

 of letters and symbols," and then turns to page 22 of volume 1 of the 

 large French mathematical encyclopedia and reads that " it is con- 

 venient, in arithmetic, to represent any number by a letter, it being 

 understood that this letter denotes a single and the same number when- 

 ever one remains in the same subject," it becomes evident that the given 

 definition of algebra is not supported by some of the highest authorities. 



In fact, such terms as arithmetic, algebra and geometry are used 

 with such a wide range of meanings by eminent authorities that it 

 seems impossible to give satisfactory definitions of them, and our dic- 

 tionaries would convey more reliable information about mathematical 

 terms by stating this fact, together with some indication of what broad 

 subjects are generally classed under these terms, than by giving cate- 

 gorical definitions which can be accepted only by those who have a 

 meager knowledge of mathematics. 



Without implying that Webster's New International Dictionary is 

 any less reliable with respect to mathematical matters than most others, 

 we shall refer to one more instance of misleading statements in this 

 work. On page 2547 we read as follows : " The cipher was originally 

 a dot, used as a mere arbitrary sign to mark place or local value." Such 

 a definite statement seems strange in view of the fact that the origin of 

 zero is one of the unsettled questions of the history of mathematics. 

 It is of interest to note in this connection that Cantor changed his view 

 with respect to the origin of this concept and this symbol, in the third 

 edition of Volume I. of his classic " Vorlesungen ueber Geschichte der 

 Mathematik," where he states that the symbol for zero and the posi- 

 tional arithmetic are probably due to the Babylonians instead of to the 

 Hindus, as he had stated in the earlier editions of this work, and as is 

 stated in a large number of other works. 



Our encyclopedias also frequently exhibit careless editing along the 

 line of mathematical terms, and the choice of editors for such work 

 often seems to indicate that the general editor regarded the choice of 

 the mathematical editors as a matter of little consequence. Possibly 



