Notices respecting New Books. 453 



action, interesting to some of the reading public, but not to be 

 taken as a complete or safe guide for students. They had better 

 have recourse to fuller and more complete works, which embody 

 recent researches on the mineralogy, petrology, and physics of 

 volcanoes and earthquakes. 



Principles of the Alyebra of Physics. By A. Macfarlane, M.A., 

 D.Sc, LL.D, FJIS.E. Salem," Mass., U.S.A., 1891 ; 8vo, pp. 52. 



This brochure is a reprint of a paper read before Sections A 

 and B of the American Association for the Advancement of 

 Science on the occasion of its Meeting in August 1891 at 

 Washington. 



The author, an alumnus of the University of Edinburgh, now 

 Professor of Physics in the University of Texas, published some 

 few years back a volume with the title " Physical Arithmetic," the 

 object of which was to substitute direct calculations from first 

 principles for the use of formulae, in elementary concrete examples 

 and exercises. In the present paper Dr. Macfarlane criticises 

 some of the critics of the bases of the Calculus of Quaternions as 

 laid down by Hamilton and of Grassmann's system as expounded 

 in the Ausclelinungslehre, and proposes such modifications as have 

 commended themselves to his experience. 



The objections made to the square of a vector being a negative 

 quantity, — or, more generally, to the sign of the scalar part of the 

 product of two vectors — ; the " want of harmony between the 

 notation of Quaternions and that of Determinants," and the dis- 

 cordance of sign between the square of the Hamiltonian V a °d 

 Laplace's operator are noticed in particular and commented on ; 

 as well as the limitation of the Quaternion to tridimensional space 

 in contrast to the general-dimensional character of Grassmann's 

 method, insisted on by Profs. Gibbs (Letters to ' Nature ') and 

 Hyde (' Directional Calculus '). After these preliminaries, 

 occupying some sixteen pages, Prof. Macfarlane addresses himself 

 systematically to the development of a basis for the Calculus of 

 Quaternions, beginning with * Definitions and Notation,' then pro- 

 ceeding successively to the * Addition and Subtraction of Vectors,' 

 1 Products of two Vectors ' (introducing generalized Cos. and Sin. 

 to replace Sa/5 and Va/3), of 'Three,' and of * Pour Vectors.' The 

 interpretation in the last two cases involving the consideration of 

 space of four dimensions. And here it may be remarked that the 

 author, in general full and clear in his demonstrations, in a critical 

 case at the foot of p. 88 is satisfied to tell his readers that " it can 

 be shown " that a certain expression is scalar in a space of three, 

 but directed in one of four dimensions. 



Justice could not be done to Prof. Macfarlane's views within the 

 space at command by any attempt to reproduce them here ; but 

 under the head of " Quaternions " his definition may be quoted : — 

 " By a Quaternion proper (a) is meant an arithmetical ratio (a) 



Phil. Mag. S. 5. Vol. 34. No. 210. Nov. 1892. 2 I 



