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 Magazine, 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 unconscious 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 Ausdehnungselehre, 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 Ausdehnungselehre in the article on "Quaternions" in the last edition of the Encyclopædia 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 Ausdehnungselehre 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.
