170 QUATERNIONS AND THE ALGEBRA OF VECTORS. 



only a secondary object with Hamilton to express the geometrical 

 relations of vectors, secondary in time, and also secondary in this, 

 that it was never allowed to give shape to his work. 



But this relates to the past. In regard to the present status, I beg 

 leave to quote what Mr. McAulay has said on another occasion (see 

 Phil. Mag., June 1892) : " Quaternions differ in an important respect 

 from other branches of mathematics that are studied by mathe- 

 maticians after they have in the course of years of hard labour laid 

 the foundation of all their future work. In nearly all cases these 

 branches are very properly so called. They each grow out of a 

 definite spot of the main tree of mathematics, and derive their 

 sustenance from the sap of the trunk as a whole. But not so with 

 quaternions. To let these grow in the brain of a mathematician, 

 he must start from the seed as with the rest of his mathematics 

 regarded as a whole. He cannot graft them on his already flourishing 

 tree, for they will die there. They are independent plants that 

 require separate sowing and the consequent careful tending." 



Can we wonder that mathematicians, physicists, astronomers, and 

 geometers feel some doubt as to the value or necessity of something 

 so separate from all other branches of learning? Can that be a 

 natural treatment of the subject which has no relations to any other 

 method, and, as one might suppose from reading some treatises, has 

 only occurred to a single man ? Or, at best, is it not discouraging 

 to be told that in order to use the quaternionic method, one must 

 give up the progress which he has already made in the pursuit of 

 his favourite science, and go back to the beginning and start anew 

 on a parallel course ? 



1 believe, however, that if what I have quoted is true of vector 

 methods, it is because there is something fundamentally wrong in 

 the presentation of the subject. Of course, in some sense and to some 

 extent it is and must be true. Whatever is special, accidental, and 

 individual, will die, as it should; but that which is universal and 

 essential should remain as an organic part of the whole intellectual 

 acquisition. If that which is essential dies with the accidental, it 

 must be because the accidental has been given the prominence which 

 belongs to the essential. For myself, I should preach no such doctrine 

 to those whom I wish to convert to the true faith. 



In Italy, they say, all roads lead to Rome. In mechanics, kine- 

 matics, astronomy, physics, all study leads to the consideration of 

 certain relations and operations. These are the capital notions ; these 

 should have the leading parts in any analysis suited to the subject. 



If I wished to attract the student of any of these sciences to an 

 algebra for vectors, I should tell him that the fundamental notions 

 of this algebra were exactly those with which he was daily con- 



