NATURE 



[November 3, 1892 



and the quaternion notion of a vector being also a quad- 

 rantal versor is not entertained at all. 



The author of this pamphlet devotes a portion of it to 

 the consideration of quaternions, which he holds 

 should form a distinct algebra by themselves, and he 

 suggests a special notation for them. He restricts a 

 quaternion proper to a pure number (a stretching factor) 

 combined with a certain amount of turning. A vector, on 

 the contrary, may be a quantity of any dimensions, 

 possessing direction, with no suggestion of turning 

 attached to it. 



He clearly shows that the objectionable minus 

 which occurs in scalar products in quaternions arises 

 from the attempt to use the same symbol both for a 

 quadrantal versor and for a vector, so that the laws 

 established for dealing with one set of quantities may 

 hold also for the other set, or for a combination of the 

 two. 



It may be worth while to notice that this minus sign of 

 the quaternionists would disappear as an expHcit 

 symbol if they considered the second vector as being 

 drawn from the end of the first, as AB, BC, and then 

 took the angle ABC as being the angle between the 

 vectors — that is to say, if, in a polygon of vectors, they 

 were to define the angles between the successive vectors 

 to be the internal angles of the polygon. Indeed, by 

 many the internal angles of a polygon (or triangle) are 

 considered as being the angles between the sides, though 

 there is loss of real naturalness and of symmetry caused 

 by so considering them : for instance, the connection 

 between A, B, C and a, d, c in a. spherical triangle would 

 be greatly simplified if A, B, C were to denote the external 

 angles. However, if we consider these internal angles 

 to be the angles considered by the quaternionists, the 

 reason for the square of a vector being negative appears 

 at once ; for if a be the quantitative part (freed from the 

 notion of direction) of a vector A, we have A A = <2^ cos 

 180°, A and A being consecutive sides of the polygon 

 which have straightened out till the internal angle 

 between them is 1 80°. 



It may therefore be contended that the quaternionists' 

 minus is not quite irrational in vector algebra (though it 

 cannot be said not to be inconvenient there), and that 

 the advantage of being able to treat a vector as a quad- 

 rantal versor without having to establish a new set of 

 formulse far more than compensates for the loss of sym- 

 metry. On the other hand, the advocates of vector 

 algebra without the minus would probably reply that they 

 have to deal with vectors which are not in any sense the 

 same as quadrantal or any other kind of versors, and that 

 the imaginary completeness gained does not in any degree 

 whatever compensate for the loss of naturalness and loss 

 of symmetry involved in the mitius. 



The author differs from Prof. Gibbs and Mr. Heaviside 

 in the mode in which he defines the product of two 

 vectors, as he considers the complete product formed on 

 the understanding that the multiplication shall obey the 

 distributive law. He then finds that this complete pro- 

 duct consists of a non-directed part, and of a directed or 

 vector part, the former consisting of the product of the 

 two quantities into the cosine of the angle between them, 

 and the latter of the product of the two quantities into 

 the sine of the same angle, having as axis the normal to 

 NO. 1 201, VOL. 47] 



the plane containing the two vectors. The angle is the 

 angle through which the first vector (occurring on the 

 left-hand side of the product) would have to turn to make 

 its direction coincide with that of the second. 



Prof. Gibbs and Mr. Heaviside, on the contrary, define 

 the scalar product and the vector product as if they were 

 entirely distinct and independent quantities. Finally the 

 same result is attained, but Prof. Macfarlane's mode 

 of introducing these partial products as arising naturally 

 from applying the distributive law of multiplication would 

 seem to have an advantage from the point of view of a 

 student. 



Prof. Macfarlane dwells emphatically on the importance 

 of considering dimensions of vectors, as well as their 

 direction, and to emphasize this he separates his vector,, 

 not into tensor and unit-vector, but into quantity and 

 direction. Thus in the equation X =\xi, x is the quan- 

 tity, and /denotes the axis. Hence the equation 7,^ = i 

 is not a violation of dimensions, but is merely a conven- 

 tion as to the interpretation of a composite direction, a 

 convention, moreover, which could only be adopted in 

 space of three dimensions, and is the statement that the 

 plane in which / and k lie has its orientation sufficiently 

 indicated by the normal direction /, with the further 

 convention that the angle from/ to k shall be considered 

 positive. 



The author's notation is novel, and forms a very im- 

 portant feature in his treatment of the subject. The 

 scalar product of AB, which is ab cos (a<^),he calls cos (AB) 

 and the vector product he calls Sin AB, its magnitude, 

 irrespective of direction, being denoted by sin AB. 

 Possibly an improvement in this latter would be to denote 

 it by sin ab, and then the capital letter in the complete 

 vector would become unnecessary. 



The particular symbol used to denote a scalar or a 

 vector product is a matter of secondary importance, but 

 is a matter which must sooner or later be settled if vector- 

 algebra is to come into general use. Lord Kelvin is of 

 opinion that a function-symbol should be written with not 

 less than three letters, and Prof. Macfarlane's notation 

 obeys that law, and is moreover easy to work with, but 

 is incomplete, being applicable to products of two- 

 vectors only. Mr. Heaviside uses no prefix at all to a 

 scalar product, but considers that AB means the scalar 

 product. He uses the quaternionic expression t^'AB for 

 the vector product. Prof. Gibbs uses no prefix for either, 

 but denotes the scalar product by A . B, and the vector 

 product by AxB. The three-lettered prefix seems the 

 clearest in both cases to denote the special product 

 intended, and the symbols cos .and sin are more or less 

 suggestive. 



In forming a product of three vectors, Prof. Macfarlane 

 makes the convention that ABC shall mean (AB)C, the 

 combination commencing on the left. In his notation 

 this product expands into 



(cos AB + Sin AB)C 

 = cos(cosAB.C + SinAB.C) + Sin(cosAB. C + SinAB-d). 

 = cos (Sin AB . C) + Sin (cos AB • C) + Sin (Sin AB . C) 

 = volABC + C.cos AB + Sin(SinAB.C) 



which finally becomes 



=vo1ABC + CcosAB-hBcosAC-AcosBC; 



