308 Notices respecting New Books, 



In the Appendix, Mr. Salter gives lists of the fossils of each for- 

 mation, with a few illustrative remarks on those of the older subdi- 

 visions, and descriptions of a large number of Lower Silurian and 

 Primordial species, most of which are very well illustrated either as 

 lithographs or engravings. 



The geological portion of the Memoir is excellently illustrated by 

 a plate of three coloured sections, one of which (No. 3) is well 

 deserving of a careful study. There are also a geologically coloured 

 index map of Wales, and a number of woodcut plans, sections, &c. ; 

 but the execution of the woodcuts is positively disgraceful, and ap- 

 pears all the worse from the excellence of the plates. 



Elements of Quaternions. By the late Sir William Rowan Ha- 

 milton, LL.D., &c. Edited by his Son, W. E. Hamilton, B.A. 

 London: Longmans and Co. 1866. 



The method of Quaternions was first announced by the late Sir 

 W. R. Hamilton in a communication made to the Royal Irish Aca- 

 demy in November 1843, in which the fundamental principles and 

 formulae of the method were laid down. Subsequently the subject 

 was fully discussed and illustrated in his work entitled " Lectures 

 on Quaternions," published in 1853. For some years previous to 

 his death he was engaged on the present work, which is an entirely 

 new exposition of the method. It is indeed (to use the author's 

 own words) " founded on the same principles as the lectures, but 

 the plan adopted is entirely new ; and the present work can in no 

 sense be considered as a second edition of that former one." 



The arrangement of the work is as follows : — The first Book treats 

 of vectors, without reference to angles or rotations. The second 

 Book treats of quaternions considered as quotients of vectors. The 

 third Book treats of quaternions considered as products or powers 

 of vectors. The object of this arrangement is to develope the method 

 from its simpler to its most general form. It would, of course, have 

 been possible to have begun with the general principles of the sub- 

 ject, and to have treated the less general parts as particular cases ; 

 and each way would have had its own advantages. In the case, 

 however, of a subject like quaternions, which might, without much 

 inaccuracy, be described as a particular kind of algebra, the rules 

 and symbols of which resemble, but are not identical in meaning 

 with, those of ordinary algebra, it is, of course, a great gain for the 

 student to see the method built up from its first principles, and to 

 observe the way in which the necessity arises for interpreting sym- 

 bols and expressions in the particular mode required by the method. 

 It is to be observed that the woik was designed by Sir W. Hamil- 

 ton for the use of students ; and, apparently for the purpose of carry- 

 ing out this design, he drew up a most elaborate table of contents to 

 the work. This Table occupies fifty-seven pages, and embraces a sort 

 of running comment on the leading portions of the work, in some cases 

 directing the reader's special attention to certain articles, in others re- 

 commending the omission of certain parts on the first perusal, in others 

 explaining the relation in which the contents of one or more articles 



