502 



NATURE 



[January 7, 191 5 



this. On p. 159 of Mr. Maitland's translation 

 we have : — 



"Well, one day I received a visit from M. 

 Hadamard, and the conversation turned upon this 

 antinomy [of Burali-Forti] . 



"'Does not Burali-Forti 's reasoning,' I said, 

 * seem to you irreproachable? ' 



"'No,' he answered; 'and, on the contrary, 

 I have no fault to find with Cantor's. Besides, 

 Burali-Forti had no right to speak of the whole 

 of all the ordinal numbers.' 



" ' Excuse me, he had that right, since he could 

 always make the supposition that 



« = T'(No. €>). 



I should like to know who could prevent him. 

 And can we say that an object does not exist when 

 we have called it fi ? ' 



" It was quite useless ; I could not convince him 

 (besides, it would have been unfortunate if I had, 

 since he was right). Was it only because I did 

 not speak Peanian with sufficient eloquence? 

 Possibly, but, between ourselves, I do not think 

 so." 



Dr. Halsted's translation on p. 459 is as 

 follows : — 



"Now, one day M. Hadamard came to see me, 

 and the talk fell upon this antinomy. 



"' Burali'Forti's reasoning,' I said, 'does it 

 not seem to you irreproachable? ' ' No, and on 

 the contrary, I find nothing to object to in that 

 of Cantor. Besides, Burali-Forti had no right to 

 speak of the aggregate of all the ordinal numbers. ' 



" ' Pardon, he had the right, since he could 

 always put 



n = T' (No. €->). 



I should like to know who was to prevent him, 

 and can it be said a thing does not exist, when 

 we have called it ft ? ' 



" It was in vain, I could not convince him (which 

 besides would have been sad, since he was right). 

 Was it merely because I do not speak the Peanian 

 with enough eloquence? Perhaps; but between 

 ourselves I do not think so." 



Of course, Poincar^ spoke of "Ze peanien," but 

 this is scarcely a sufficient excuse for Dr. Hal- 

 sted's translation. 



On the other hand, ijt is rather evident that 

 technical terms presented a difficulty to Mr. 

 Maitland. He translates "invariant" by "in- 

 variable" on p. 35, "la theorie des congruences" 

 by "theory of congruents " on p. 41, and "zm- 

 plique" by "involves" on p. 161. Dr. Halsted 

 gives, as we should expect, the correct transla- 

 tions at the corresponding places of his book 

 (PP- 375> 379> 460). It is rather unusual to speak 

 of analytic functions as "analytical functions," as 

 Mr. Maitland does on p. 77 ; Dr. Halsted uses 

 " analytic " on p. 403. It seems to be confusing 

 to speak of a paper, published by Mr. Russell in 

 the Proceedings of the London Mathematical 

 NO. 2358, VOL. 94] 



Society, as a "treatise/' as Mr. Maitland does on 

 p. 184. Dr. Halsted has "memoir" on p. 477, 

 but has no reference on his p. 479 to the paper 

 in question, whereas Mr. Maitland has (p. 187). 

 Mr. Maitland carefully speaks of a Frenchman 

 as "M." so-and-so, of an Englishman as "Mr.," 

 and of an Italian as "Signor"; but for Germans 

 he has no regular prefix : we often come across 

 "Mr. Hilbert," and, quaintly enough, on p. 192, 

 "Signor Zermelo." Of greater importance than 

 this is that a formula of Burali-Forti, expressed 

 in Peano's symbols, is wrongly reproduced on p. 

 157 of (i), owing to the fact that Burali-Forti 's 

 symbol " Un " is translated "one," thus giving 

 some colour to Poincare's mistaken contention 

 that the idea of one is used in the very definition 

 of "i." In (2) the formula is correctly repro- 

 duced (p. 488). 



It does not seem necessary to characterise here 

 Poincare's philosophical writings : Mr. Russell, 

 in his preface to (1), has done this admirably. 

 All the four books under review will be useful 

 in spreading a knowledge of the lighter works of 

 a truly great man. As is so often the case, a 

 lively and amusing style is here, too, sometimes 

 accompanied by superficiality, and original 

 thoughts by obstinacy. Poincare's preface to (2) 

 is very welcome at the present time : it lays stress 

 on the fact that there is no reason for wishing 

 the world exclusively impressed with the charac- 

 teristics of one particular race. 



As we might expect, (3), which is the seven- 

 teenth volume of the series " Wissenschaf t und 

 Hypothese," which contains translations of the 

 first two of Poincare's works quoted above, is 

 admirably translated and provided with learned 

 and excellent notes by Prof. F. Lindemann. 



Finally, (4) is provided with a portrait of Poin- 

 care, and the translation seems well done. The 

 remark that the word " Anzahl " means the same 

 thing as " Machtigkeit " (p. 108) is wrong, and 

 a title is wrongly spelt on p. 11 1. A somewhat 

 serious mistake, which appears in the editor's 

 note on p. 124, is the failure, like Schroder's, 

 to distinguish between the notions "is contained 

 in " and "is a member of." (f>^ 



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