November 28, 1919] 



SCIENCE 



491 



brevity and elegance resulting from the use 

 of determinants. 



Fortunately tlie desire to excel is common 

 to the yoimg and old. I have often wondered 

 to vfhat extent the deep interest which women 

 exhibit towards children, especially towards 

 babies, is enhanced by the fact that in them 

 they find human beings who do not pretend 

 to know as much or more than they them- 

 selves do. At any rate the interest of yoimg 

 students can often be most easily aroused by 

 guiding them so that they can experience 

 that in at least some particulars they can 

 make improvements on the work of others. 

 The occasional discussion of possible improve- 

 ments on the text-book or on articles in stan- 

 dard works of reference may serve a useful 

 purpose if it is conducted in the right spirit. 

 That some of the best works afford opportuni- 

 ties for such discussions may be illustrated by 

 the article on " mathematical signs and sym- 

 bols " in the new edition of the Encyclopedia 

 Americana which is now almost completed 

 and is noteworthy on account of its valuable 

 mathematical articles. 



The history of science is also useful because 

 it instils optimism. That the history of sci- 

 ence is in the main a history of progress 

 needs scarcely to be emphasized in these days 

 of rapid economic charges due to new scien- 

 tific discoveries. The progress of science is 

 in part reflected in the many new inventions 

 and improvements contributing to our com- 

 fort in sickness and in health. The fact that 

 these inventions and improvements are finally 

 based on the work of such a large number of 

 scientific investigators directs attention to the 

 vast opportunities of rendering useful service 

 in the field of science, and one of the most 

 striking elements of the history of science is 

 the fact that our rich scientific heritage is 

 due to the work of millions for world better- 

 ment. 



In the study of the history of science as 

 well as in the study of science itself many 

 students meet with the dilemma that what 

 they would most like to know they can not 

 know and what they can know they care little 

 about. In both cases real progress is usually 

 coupled with a willingness to work where 



progress seems possible. One of the most 

 striking instances of this fact is furnished by 

 the history of our common nimierals. For 

 centuries mathematical historians have been 

 interested in the origin of these numerals and 

 for a long time there was almost complete 

 agreement that they were of Hindu origin 

 and were transmitted to Europe by the 

 Arabians. Hence the common name Hindo- 

 Arabic numerals. 



During the last dozen years various mathe- 

 matical historians have re-investigated this 

 question and have reached the conclusion that 

 these numerals originated in Europe and not 

 in Asia. One of the most active supporters 

 of this new theory is G. E. Kaye, an English- 

 man residing in India, who wrote a book on 

 "Indian Mathematics" (1915), and is in- 

 clined to give little credit to the Hindus for 

 originality in mathematics. Instead of call- 

 ing our common numerals " Hindu- Arabic " 

 or " Babylonic-Hindu " it would be more in 

 accord with our present state of knowledge to 

 admit that they are of unknown origin, and 

 if a student of the history of mathematics 

 insists on knowing the origin of zero before 

 taking up other historical questions it is 

 likely that his knowledge of this history will 

 remain zero. 



As Cajori's history, to which we referred 

 above, will probably be used widely as a text- 

 book it seems desirable to refer here to a 

 peculiarity which might otherwise cause per- 

 plexity. The author of this history speaks 

 at various places about the origin of our com- 

 mon numerals and at all of these places save 

 one he supports the theory that they are of 

 Hindu origin. This single exception appears 

 in a note on page 98 where he acknowledges 

 our ignorance in regard to the origin of these 

 numerals without, however, acknowledging ex- 

 plicitly his recent conversion to this view. 

 It therefore happens that both those who sup- 

 port the theory of the Hindu origin of our 

 common numerals and those who acknowledge 

 agnosticism as regards their origin can find 

 support of their views in different parts of 

 the same book. 



This singular fact seems to deserve public 

 notice also because Cajori's work is the 



