136 Prof. A. Macfarlane on 4 Outlines of Quaternions.' 



does not belong to it. That which in Hamilton is an 

 "outburst of genius" is in Colonel Hime "contrary to the 

 whole genius of common sense." 



The manner in which the author founds the calculus is not 

 satisfactory to your reviewer. This is not surprising ; on 

 examination I find that it is based on a collection of defini- 

 tions. The fundamental rules are explained as follows : — In 

 i j the former symbol is a quadrantal versor and the latter a 

 vector, the effect of the former on the latter is to change it 

 into k. Logically, we expect that in ii the former symbol is 

 a quadrantal versor and the latter a vector, giving i as the 

 result ; but it is not so : both symbols are now to be con- 

 sidered quadrantal versors, giving r=— 1. These two 

 inconsistent theories are simultaneously applied to find the 

 product of two vectors, 



mii + m 2 j + >»3& and n^i + n 2 j -f njc. 



The product consists of two kinds of terms ; m x ni ii is a type of 

 the one, and mi n 2 ij of the other. We are asked to believe that 

 in 7n 1 7iiii both unit-vectors become quadrantal versors, while 

 in miTt^ij it is only the former of the two. The operator and 

 operand theory is applied to one set of terms, and a double 

 operator theory to the other set of terms ; while in the 

 vectors themselves there is no operator at all. 



In my papers on Space-analysis I have attempted to lay 

 down the fundamental principles in a logical manner. When 

 that is done, the meaning of every expression and equation 

 becomes clear, and there is no need for a special chapter such 

 as we find in the i Outlines,' on the " Interpretation of Qua- 

 ternion Expressions. " The best proof of the correctness of 

 my principles is that they have enabled me to carry the 

 analysis beyond the point where Hamilton left it. 



In conclusion I wish to say that I yield to no one in 

 admiration of the works of Hamilton and Tait. The attitude 

 of mind of your reviewer appears to be that of " wondering 

 awe." In that I cannot follow him ; I stand in awe of the 

 truth only. 



Alexander Macfarlane. 



Ithaca, N. Y, 



