VIII. 



QUATERNIONS AND THE AUSDEHNUNGSLEHRE. 

 [Nature, vol. XLIV. pp. 79-82, May 28, 1891.] 



THE year 1844 is memorable in the annals of mathematics on account 

 of the first appearance on the printed page of Hamilton's Quaternions 

 and Grassmann's Ausdehnungslehre. The former appeared in the 

 July, October, and supplementary numbers of the Philosophical Maga- 

 zine, after a previous communication to the Royal Irish Academy, 

 November 13, 1843. This communication was indeed announced to 

 the Council of the Academy four weeks earlier, on the very day of 

 Hamilton's discovery of quaternions, as we learn from one of his 

 letters. The author of the Ausdehnungslehre, although not uncon- 

 scious of the value of his ideas, seems to have been in no haste to 

 place himself on record, and published nothing until he was able 

 to give the world the most characteristic and fundamental part of 

 his system with considerable development in a treatise of more than 

 300 pages, which appeared in August 1844. 



The doctrine of quaternions has won a conspicuous place among the 

 various branches of mathematics, but the nature and scope of the 

 Ausdehnungslehre, and its relation to quaternions, seem to be still 

 the subject of serious misapprehension in quarters where we naturally 

 look for accurate information. Historical justice, and the interests of 

 mathematical science, seem to require that the allusions to the Ausdehn- 

 ungslehre in the article on " Quaternions " in the last edition of the 

 Encyclopedia Britannica, and in the third edition of Prof. Tait's 

 Treatise on Quaternions, should not be allowed to pass without protest. 



It is principally as systems of geometrical algebra that quaternions 

 and the Ausdehnungslehre come into comparison. To appreciate the 

 relations of the two systems, I do not see how we can proceed better 

 than if we ask first what they have in common, then what either 

 system possesses which is peculiar to itself. The relative extent and 

 importance of the three fields, that which is common to the two 

 systems, and those which are peculiar to each, will determine the 

 relative rank of the geometrical algebras. Questions of priority can 

 only relate to the field common to both, and will be much simplified 

 by having the limits of that field clearly drawn. 



G. II. L 



