772 



SCIENCE 



[N. S. Vol. XXXVni. No. 987 



H. L. Olin, instructor in chemistry, Vassar Col- 

 lege, Poughkeepsie, N. Y. 

 E. S. Potter, research assistant, Agricultural Ex- 

 periment Station, Iowa State College, Ames, 

 Iowa. 

 E. K. Straohan, instructor in chemistry. Univer- 

 sity of Minnesota, Minneapolis, Minn. 

 G. Y. Williams, associate professor of chemistry 

 and acting head of the chemistry department 

 in the State University of Oklahoma, Norman, 

 Oklahoma. 

 P. S. Woodward, instructor, Georgia School of 

 Technology, Atlanta, Georgia. 

 The electors to the Waynflete professorsliip 

 of physiology at Oxford, vacant by the death 

 of Dr. Francis Gotch, have elected Dr. Charles 

 Scott Sherrington. Dr. Sherrington succeeded 

 Dr. Gotch as Holt professor of physiology at 

 the University of Liverpool in 1895, when Dr. 

 Gotch was called to Oxford. 



DISCUSSION AND COBBESPONDENCE 



MATHEMATICAL DEFINITIONS IN THE NEW 

 STANDARD DICTIONART 



Funk and Wagnalls's " New Standard Dic- 

 tionary of the English Language," 1913, has 

 many merits and will doubtless be used very 

 extensively. It is, therefore, of special impor- 

 tance to direct public attention to the fact that 

 this dictionary is not reliable as regards defini- 

 tions of mathematical terms. Some of these 

 definitions will doubtless interest even those who 

 remember only a little of their mathematics, 

 as they relate to elementary matters and are so 

 evidently incorrect. The following list of ex- 

 amples could easily have been extended, but 

 it is believed that it will not require many 

 examples of this type to convince the reader. 



Under the term algebra it is stated that the 

 infinitesimal calculus and the theory of func- 

 tions may be classed among " the principal 

 branches of algebra." A hundred years ago 

 such a statement might have appeared proper, 

 but it is not in accord with any of the classifi- 

 cations which have been extensively adopted 

 in recent years, such as those employed in the 

 International Catalogue of Scientific Litera- 

 ture and in the large mathematical encyclo- 

 pedias which are in the course of publication. 

 In fact, the infinitesimal calculus and the 



theory of functions are generally regarded as 

 branches of analysis. 



The explanations which follow the term 

 arithmetic include the statement that the 

 early Pythagoreans first studied arithmetic. 

 On the contrary, it is well known that the an- 

 cient Babylonians and Egyptians made con- 

 siderable use of elementary arithmetic, as may 

 be seen from the extensive mathematical 

 tables of the ancient Babylonians and the 

 large collection of examples by the Egyptian 

 scribe Ahmes. Possibly the early Pythago- 

 reans might be regarded as the first workers in 

 higher arithmetic or the theory of numbers. 



An instance of a statement which is more 

 evidently incorrect appears under the term 

 dimension. It is here stated that four-dimen- 

 sional space may be regarded as a hypothetical 

 conception to explain equations of the fourth 

 degree in analytical geometry. As a matter 

 of fact an equation of any degree in two 

 variables may be represented geometrically in 

 the plane. It is the number of the variables 

 and not the degree of an equation which corre- 

 sponds to the number of dimensions required 

 for its representation. 



Under the term equation it is stated that an 

 abelian equation is an equation "all of whose 

 roots are rational functions of one or more of 

 the roots." It is well known that the roots of 

 non-abelian equations may also be rational 

 functions of each other. In an abelian equa- 

 tion we must have the additional condition 

 that its group is commutative. 



A fractional function is defined, under the 

 term function, as one whose variable appears 

 only in its denominator; and a Galois resol- 

 vent is said to be " that resolvent of an equa- 

 tion whose roots remain the same when the 

 group of the equation is permuted in any way 

 whatever." It would be interesting to know 

 something about the new theory of permuting 

 the group of an equation. Unfortunately 

 there seems to be no clue in this dictionary as 

 regards the possible meaning of this term. 



The most original definitions seem to ap- 

 pear under the term group. A complete 

 group is defined as one in which no self -con- 

 jugate operations are possible besides the iden- 



