134 



SCIENCE. 



[N. S. Vol. III. No. 56. 



Skating ponds illuminated by natural gas are 

 among the possibilities of the future. 



Ira Remsen. 

 Baltimore, January 14, 1896. 



'professors' gabnee and gates. 

 The daily papers state that Mr. Richard L. 

 Oarner, whose alleged investigation of tlie 

 speech of monkeys has been so prominently 

 advertised, is again expected in America. Ac- 

 counts of the alleged investigations of Mr. 

 Elmer Gates on the development of the brain 

 are also being extensively reported. It is per- 

 haps the duty of a scientific journal to state 

 that neither of these gentlemen has as yet pub- 

 lished scientific work deserving serious consid- 

 eration. J. McK. C. 



SCIENTIFIC LITERATURE. 

 The Psychology of Number and Its Applications to 



Methods of Teaching Arithmetic : By James A. 



McLellan, A.m., LL.D., and John Dewey, 



Ph.D. International Educational Series. D. 



Appleton & Co. , New York. 



This book makes a false analysis of the num- 

 ber concept, but advocates methods in teach- 

 ing arithmetic which are in the main good. 

 The conviction of its authors that the difiicul- 

 ties which children have with arithmetic are 

 due to the neglect of teachers to lay sufficient 

 stress on the metrical function of number has 

 carried them to the extreme of maintaining that 

 number is essentially metrical in its nature and 

 origin. The conviction is well founded, inas- 

 much as the first serious difficulties of children 

 are with fractions whose primitive function was 

 unquestionably metrical and to which men in 

 general attach no other than a metrical mean- 

 ing; but there is no reason for drawing the con- 

 clusion that because the fraction, which is but 

 a secondary concept of arithmetic, is metrical, 

 its primary concept, the integer, is metrical also, 

 or even that because a child can hardlj^ be made 

 to understand fractions without associating them 

 with measurement, he requires the same help 

 with integers. Nevertheless, the authors of 

 this book maintain, in the most unqualified 

 manner, that the integer is essentially metrical 

 and should be taught accordingly. Thus they 

 account as follows for the origin of number : 

 Man found himself in a world in which the 



supply of almost everything that he needed was 

 limited. To obtain what he required, there- 

 fore, an economy of effi3rt, a careful adjustment 

 of means to an end, was necessary. But the 

 process of adjusting means to an end is valuable 

 in the degree in which it establishes an exact 

 balance between them. "In the effort to attain 

 such a balance, the vague quantitative ideas of 

 smaller and greater j(. * ^j. were transformed 

 into the definite quantitative ideas of just so 

 distant, so long .^ j<. ^ . This demands the in- 

 troduction of the idea of number. Number is 

 the definite measurement, the definite valuation 

 of a quantity falling within a given limit." 



They define counting, the fundamental num- 

 erical operation as but measuring with an unde- 

 fined unit. ' ' We are accustomed to distinguish 

 counting from measuring. Nevertheless, all 

 counting is measuring and all measuring count- 

 ing. The diflcrence is that in what is ordinarily 

 termed counting, as distinct from measuring, 

 we woi'k with an undefined unit ; it is vague 

 measurement because our unit is unmeasured. 



* s ^(. If I count off four books, 'book,' 

 the unit which serves as unit of measurement, is 

 only a qualitative, not a quantitative unit." 



And they formally define number as ' the 

 repetition of a certain magnitude used as the 

 unit of measurement to equal or express the 

 comparative value of a magnitude of the same 

 kind,' a definition which, so far as it goes, 

 agrees, it is true, with that given by Newton in 

 his Arithmeiica Universalis, viz, ' the abstract 

 ratio of any quantity to another quantity of the 

 same kind taken as unit,' though Newton's pur- 

 pose having been to formulate a working defini- 

 tion comprehensive enough to include the irra- 

 tional number, it is anything but evident that 

 this statement represents his analysis of the 

 notion of number in the primary sense. 



The immediate objection to all this is that it 

 is much too artificial to be sound. And in fact it 

 requires but a little reflection to be convinced 

 that pure number is not metrical and that count- 

 ing is not measuring, but something so much 

 simpler that men must have counted long before 

 they knew how to measure in any proper sense. 



It is not enough to say that counting is the 

 simplest mathematical operation; it is one of the 

 simplest of intellectual acts. For to count a 



