Notices respecting Neiv Books. 309 



stand to the general plan of the work, &c. We are inclined to infer, 

 from the author's evident anxiety to assist the student, that he had 

 not been altogether satisfied with the progress hitherto made by the 

 method of quaternions towards general use, and that he imputed this 

 to the difficulties to be encountered at the beginning of the subject. 

 Every part of the subject is profusely illustrated by applications, 

 showing, if any doubt existed on that head, the fertility of the me- 

 thod, and its applicability to every branch of mathematical science. 

 Without attempting to enumerate these applications, but merely to 

 give a notion of the varied contents of the work, we may mention 

 that in the first book the method of vectors is applied to geometrical 

 nets both plane and in space, to barycentres of systems of points, to 

 anharmonic equations of curves and surfaces, &c, and that, besides 

 other illustrations interspersed through the work, the last chapter, 

 comprising near upon three hundred closely printed pages, is entirely 

 taken up by applications of the method. These are, in the first 

 place, to purely geometrical questions — to tangents and to normal 

 and osculating planes of curves in space, to geodetic lines, to invo- 

 lutes and evolutes in space, &c. ; in the second place, to physical 

 questions — amongst others, to the dynamics of a rigid body, to that 

 of a system of mutually attracting points, to the undulatory theory 

 of light, &c. 



With the commencement of an article " On MacCullagh's Theorem 

 of the Polar Plane " the work abruptly terminates. The author was 

 engaged on the work at the time of his death, and the editor rightly 

 considered that his duty both to the author and the public vrould be 

 most fully discharged by simply giving to the world what had been 

 written. " I came to the conclusion," he says, " that my duty was 

 to publish the work as I found it, adding merely proof-sheets parti- 

 ally corrected by my late father, and from which I removed a few 

 tvpographical errors, and editing only in the literal sense of giving 

 forth." 



The work, though unfinished, will remain a monument both of the 

 subtlety and originality of Sir W. Hamilton's mind, and of his pro- 

 found and extensive knowledge of every branch of mathematical 

 science. Whether the method he originated will ever be largely used 

 by other mathematicians remains to be seen ; but that it is a calculus 

 of enormous power and of indefinitely extensive application cannot 

 be doubted. Indeed, in addition to the applications contained in the 

 work, several others had been contemplated by SirW. Hamilton, 

 though he well knew that he would be unable to exhaust them. We 

 learn this from the editor's statement : — " Shortly before my father's 

 death I had several conversations with him on the subject of the 

 ' Elements.' In these he spoke of anticipated applications of qua- 

 ternions to electricity, and to all questions in which the idea of po- 

 larity is involved — applications which he never in his own lifetime 

 expected to be able fully to develope." 



