Ever since I have learnt something of the meanings of Grassmann's Ausdehnungslehre, and have at the same time learnt to regard the beauties of that system with something akin to awe, I have been persuaded that on the lines of Quaternion Algebra there is to be built a system very much like the Ausdehnungslehre, but an improvement thereon. Of course it will be matter for differing opinions whether what I call Multenions is really an improvement on the Ausdehnungslehre. I here record my own personal opinion that it is.
I do not suppose that anybody will maintain that a multitude of different kinds of multiplication within the bounds of one method can be regarded as anything but a blemish,—a blemish that may be justified by necessity and utility. The Ausdehnungslehre seems to me to have this blemish, and Multenions not to have it. Whether along with the absence of the blemish there becomes present an additional difficulty of manipulation is questionable. I have not found it so, but this may be due to the fact that I have so long been in the habit of thinking through the quaternion machinery.
I think these general remarks are all that can be usefully set down as a summary. The paper itself appears to be too condensed to admit of a true summary other than a mere table of contents.