REVIEWS 331 



we doubt whether the discussions on negative quantities, on real and imaginary 

 quantities, and on quaternions are any more than skimmings from the surface. 

 In reality signless numbers, negative numbers, and neomonic numbers belong to 

 three entirely different arguments, the first to the arithmetical one, the second to 

 the linear one, the third to the rotational one. 



Thought is continuous, but words are discontinuous. One might as well try to 

 express a continuous curve by a series of isolated points as to express the whole 

 truth in any book. This is the perpetual trouble of philosophy. And this is also 

 the reason why our authors are able to find so much in the great mathematicians 

 to criticise. The great mathematicians recognised the difficulty and avoided it 

 entirely by ignoring it. That is, they trusted the mind of the reader to understand 

 more than was written. Now we are quite certain that Hamilton understood as 

 much about quaternions as our authors understand ; but he had other things to 

 do than to endeavour to plot out his curve in infinitesimal intervals. Indeed this 

 is at the bottom of the feeling that the philosophy of mathematics is not really 

 productive. We can spend a volume in proving that two and two make four, and 

 indeed we write volumes on the subject ; but somehow they appear to be un- 

 necessary. Not only the world, but, we think, many mathematicians look on with 

 some amusement as they watch philosophers trying to put infinity into boxes. 

 Nevertheless such works as these are at least interesting — though we scarcely 

 think that they are so fundamental as the authors would appear by their Preface 

 to believe. 



The second book or rather pamphlet is mostly an attack upon the tenets of 

 Mr. Bertrand Russell. When two philosophers attempt the task mentioned 

 above, their boxes are apt to be of very different sizes, so that their measurements 

 often disagree. We suspect that really there is much more agreement than might 

 be imagined, and that the trouble arises from little more than the finity of the 

 words they use. We cannot understand the authors' contention that there is more 

 than one number one. Should we say that if two billiard balls have the same 

 redness, each has a different redness ? The attributes may be the same, though 

 the things which possess them are different. Amateur. 



Euclid's Book on Divisions of Figures (ire pi biaipea-eav PfiXlov) : with a Restora- 

 tion based on Woepcke's Text and on the Practica Geometries of Leonardo 

 Pisano. By Raymond Clare Archibald, Ph.D., Assistant Professor of 

 Mathematics in Brown University, Providence, Rhode Island. [Pp. viii + 88.] 

 (Cambridge: University Press, 191 5. Price 6s. net.) 

 Of the nine works which Euclid is known to have written, approximately complete 

 texts, all carefully edited, of four are in our possession. Of four of the others we 

 only possess more or less fragmentary knowledge from mention or comment by 

 other Greek writers. The remaining work is the book On Divisions of Figures, 

 and Proclus alone among the Greeks made an explanatory reference to it : " For 

 the circle is divisible into parts unlike in definition or notion, and so is each of the 

 rectilineal figures ; this is, in fact, the business of the writer of the Eleme?its in his 

 Divisions, where he divides given figures, in one case into like figures, and in 

 another into unlike " (p. 1 of the book under review). A copy in Latin of a treatise, 

 De Divisionibus, by one " Muhammed Bagdedinnus," was given in 1563 by John 

 Dee to Commandinus, who published it in Dee's name and his own in 1570. The 

 original manuscript from which Dee's copy was made was, for all useful purposes, 

 destroyed by fire, and though the Dee manuscript is referred to by many eminent 

 historians of Euclid's works, the conclusions of Heiberg followed by Heath are : 



