April 18, 1913] 



SCIENCE 



609 



The Natural History of Clay. By Alfred B. 



Searle. 

 The Origin of Earthquakes. By Charles 



Davison. 

 Bochs and their Origins. By Geenville A. 



J. Cole. 

 The Modern Locomotive. By C. Edgar 



Allen. 



Considering that high general level of excel- 

 lence, together with the very moderate prices, 

 it would seem that almost any public library 

 or large high school would do well to obtain 

 both series. The treatment, usually difierent 

 from that of the conventional text-book, is 

 likely to interest many readers, some in one 

 subject, some in another. There is not as 

 much duplication in the two series as some of 

 the titles might suggest ; thus " Anthropol- 

 ogy " and " Man " in the one do not at all 

 take the place of " Prehistoric Man " in the 

 other. T. D. A. Cockerell 



XJNIVEESITY OF COLOKADO 



On the Foundation and Technic of Arithmetic. 



By George Bruce Halsted. Chicago, The 



Open Court Publishing Company. 1912. 



Pp. 133. 



The main purpose of this book is to place 

 the number concept of modern mathematics 

 within easy reach of the teacher in the grades. 

 That there is reason for the existence of such 

 a text is apparent from the fact that people in 

 general and to some extent even teachers of 

 arithmetic still look upon mathematics as 

 " the science of quantity." The primitive 

 number concept of modern mathematics has 

 nothing to do with quantity. Mathematical 

 research on this subject has been slow in com- 

 manding the attention of non-mathematicians. 

 It is not very many years ago that a promi- 

 nent American psychologist published a book 

 in which the simple act of " counting " was 

 declared to be an act of " measuring." It is 

 not very long since, that a series of arith- 

 metics was published in which the primitive 

 idea of number was presented as being that of 

 '' ratio." Dr. Halsted brings out clearly and 

 strongly the fact that primitive number, 

 whether considered from the standpoint of its 



modern logical exposition, or from its historic 

 development, is wholly divorced from meas- 

 urement, and that number viewed as a ratio 

 presupposes counting and is a more involved 

 concept. The book under review contains an 

 able presentation of fundamental concepts. 

 This every one familiar with Dr. Halsted's 

 earlier works had reason to expect. 



The leading topics discussed in the book are 

 as follows : The genesis of number, counting, 

 genesis of our number notation, addition, 

 multiplication, subtraction, division, deci- 

 mals, fractions, measurement, mensuration, 

 order, ordered sets, ordinal number, the psy- 

 chology of reading a number, arithmetic as a 

 formal calculus, suggestions on the teaching of 

 arithmetic. 



Halsted makes the interesting observation 

 that, besides the " ordinal number " and 

 " cardinal number," modern civilization has 

 introduced " nominal number " used as a 

 proper noun, as in the telephone service. 

 " Since the size of the number and its place in 

 the number series are here alike irrelevant, the 

 whole stress falls upon its recognition as a 

 unique name." 



The text contains numerous historical state- 

 ments, some of which are open to criticism as 

 not embodying the latest researches. More- 

 over, there is a frequent lack of bibliograph- 

 ical reference to authorities. Thus Halsted 

 gives 7r = 3.14+ and ir= 3.1416 — and then 

 adds: 



This is historically the first meaning of the 

 signs + and — , which arose from the marks 

 chalked on chests of goods in German warehouses, 

 to denote excess or defect from some standard 

 weight. 



In view of the fact that historians have been 

 in doubt as to the exact origin of -|- and • — , 

 the authority for Halsted's categorical state- 

 ment would be interesting. Cantor' and 

 Tropfke° both express themselves with great 

 reserve on the validity of the explanation en- 

 dorsed by Halsted. Enestrom in a later re- 



^ Cantor, "Gesehichte der Mathematik, " Vol. 

 II. (2), 1900, pp. 230, 231, 320. 



- J. Tropf ke, ' ' Gesehichte der Elementar-Mathe- 

 matik," Vol. I., 1902, p. 134. 



