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XLII. Notices respecting New Books. 



Elements of Quaternions. By the late Sir William Kowan 

 Hamilton. Second Edition, edited by C. J. Joly. (Longmans, 

 Green & Co. Vols. I & II. : 1900, 1901.) 

 T)B,OEESSOR JOLT is to be congratulated on having accom- 

 -*- plished his great task of issuing a new edition of Hamilton's 

 classic work. By virtue of its larger type and broader page, the 

 new edition, with its two handsome volumes beautifully printed at 

 the Dublin University Press, is an improvement upon the original 

 book, dear though that was in its very compactness to students of 

 quaternions. The distinction which Hamilton himself made 

 between the large and small type portions is sacrificed ; but 

 in other respects, even to the characteristic use of italics and 

 capitals, the new edition seems to be a faithful reproduction of 

 the earlier. 



The position to be assigned to quaternions in the mathematical 

 developments of the last century is still matter of dispute. The 

 pure mathematician looks askance at it, and will not take the pains 

 to master its transformations. Cayley, who himself in the pages 

 of this Journal nearly sixty years ago gave the subject some 

 attention, seems never to have considered the geometrical significance 

 of the Quaternion. After Tait had shown how to extract the 

 square root of the linear vector function, that is, of a particular 

 matrix, Cayley investigated the matter by his analytical methods 

 and expressed surprise that his result agreed with Tait's ! More 

 recently he has contrasted coordinates and quaternions as 

 geometrical methods and has given his verdict in favour of the 

 former. Nevertheless, is there any mathematical treatise that can 

 be compared to Hamilton's Elements for its wealth of geometrical 

 applications, or is there any analytical method that is so incisive 

 as the quaternionic ? As with a search-light Hamilton directs his 

 symbols into every region of tridimensional geometry, illuminating 

 wherever he touches. There is, we believe, no other single treatise 

 from which a student, familiar with ordinary mathematical 

 processes in their elementary applications, could gain so wide or 

 so deep a knowledge of geometry as from Hamilton's great work. 

 Even familiarity with ordinary mathematical processes might in 

 special cases be dispensed with. Tait, the great example of a man 

 who got, as Maxwell expressed it, " the Quaternion mind directly 

 from Hamilton," used to speculate on the possibility of training a 

 mathematically gifted mind wholly along quaternionic lines. To 

 such a mind all artificial systems of coordinates w r ould appear in 

 their true light, as valuable aids for mathematical counting but not 

 as fundamental necessities for mathematical thinking — these being 

 according to Maxwell the two kinds of work of mathematicians. 

 There is no doubt as to the necessity of thinking on the part of 

 the man who would use quaternions. But surely the thinking is 

 well bestowed when two or three lines of appropriate trans- 

 formations are the equivalent of a page of Cartesian eliminations. 



