December 1, 1911] 



SCIENCE 



761 



A FEW MATHEMATICAL ERRORS IN THE RECENT 

 EDITION OF THE ENCYCLOP/EDIA BRITANNICA 



As a large number of students do not have 

 easy access to extensive special literature, they 

 are led to regard general vcorks, such as the 

 Encyclopedia Britannica, as the supreme au- 

 thority on many questions. It may, therefore, 

 be of interest to call attention to a few con- 

 spicuous errors in the new edition of this 

 excellent work. On page 857 of volume 19 

 (1911), we read as follows : " "What is quite 

 certain is that our present decimal system in 

 its complete form, with the zero which enables 

 us to do without the ruled columns of the 

 abacus, is of Indian origin." How far this is 

 from the truth may be inferred from the fol- 

 lowing paragraph. 



During the meetings of the second interna- 

 tional congress of mathematicians held in 

 Paris in 1900 the eminent German mathemat- 

 ical historian, Moritz Cantor, expressed the 

 opinion that the use of zero was probably dis- 

 covered by the Babylonians about 1700 B.C.' 

 In the third edition of volume I. of his classic 

 " Vorlesungen ueber Geschichte der Mathe- 

 matik," 1907, page 616, Cantor remarks that 

 according to his opinion the discovery of zero 

 is due to the Babylonians, while the deepen- 

 ing (Vertiefung) of the concept is due to 

 the Hindus. 



A more decided error is expressed on page 

 626 of volume 12, in the following sentence: 

 " The technical mathematical sense (of the 

 term group) is not older than 1870." It is 

 surprising that such a statement could ema- 

 nate from the country where Cayley worked 

 and developed the foundations of abstract 

 group theory as early as 1854. It is well 

 known that Galois (1811-32) was the first to 

 use the term group as a technical mathematical 

 term, with its present significance, and that 

 Cayley and Kirkman employed this term with 

 its technical mathematical sense in a number 

 of articles, published before 1870, in the Philo- 

 sophical Magazine and in the Memoirs and 

 Proceedings of the Literary and Philosophical 

 Society of Manchester. 



• Bulletin of the American Mathematical Society, 

 Vol. 7 (1900), p. 70. 



Closely related to the error noted in the 

 preceding paragraph is the following, which 

 appears under the word Galois: "To him 

 (Galois) is also due the notion of group of 

 substitutions." While the technical mathe- 

 matical term group is due to Galois, as we 

 observed in the preceding paragraph, the 

 notion of group is very much older. Accord- 

 ing to Frobenius and Stickelberger, the theory 

 of finite abelian groups was founded on the 

 one hand by Euler and Gauss, and on the 

 other by Lagrange and Abel; and, according 

 to Poincare, the principal foundation of 

 Euclid's demonstrations is really the existence 

 of the group and its properties.' No one 

 acquainted with the history of group theory 

 would say that the notion of group of substi- 

 tutions was due to Galois. 



In the first volume of the Encyclopaedia Brit- 

 annica under the term abscissa we find the 

 following incorrect statement : " The word 

 (abscissa) appears for the first time in a Latin 

 work written by Stefano degli Angeli (1623- 

 1697), a professor of mathematics in Eome." 

 As early as 1903 C. R. Wallner pointed out in 

 the BMiotheca Mathematica, page 37, that the 

 statement in Cantor's " Vorlesungen ueber 

 Geschichte der Mathematik," which might 

 furnish the basis of the error under considera- 

 tion, is incorrect. In a recent part of the 

 Eneyclopedie des Sciences Mathematiques, 

 tome 3, volume 3 (1911), page 1, G. Enestrom 

 points out that the origin of the word abscissa 

 goes back to the Latin translations of the 

 " Conic Sections " by Apollonius, written in 

 the third century before Christ. Enestrom 

 gives, at this place, numerous references in 

 regard to the early use of the term abscissa. 



Another incorrect statement appears in the 

 article on number theory, volume 19, page 851, 

 and reads as follows : " By totient of n, which 

 is denoted after Euler by (j>(n), we mean the 

 number of integers prime to n and not ex- 

 ceeding n." Wliile Euler studied some of the 

 properties of the totient of n he did not use 

 the symbol <t>(n). This symbol, as far as we 

 know at present, was first used by Gauss in 

 article 38 of his " Disquisitiones Arithmeticse," 



'The Monist, Vol. 9 (1898), p. 34. 



