May 28, 1891] 



NATURE 



79 



a system. Why should they ? Does the desire to beat com- 

 petitors stimulate a desire for knowledge? Does it stimulate 

 originality? I (or one would willingly see them non-existent. 



Up to a certain point, the acquisition of knowledge of facts 

 should be, as at present, tested by examination ; but I am 

 convinced that the system is at present pushed to an extreme, 

 and that much better results would be gained by giving a degree 

 for training, and that can be done only by the trainer — the 

 teacher. He will, as a rule, be glad to share his responsibility 

 with, and to benefit by the advice of, an outsider ; but with him 

 should ultimately rest the decision as to the merit or demerit of 

 a candidate, as he is the only person able to judge. Under 

 such a system, there would be little plucking ; for the student 

 would be advised not to present himself, unless he had suffi- 

 ciently qualified. 



It may also be said that undue advantage would be taken 

 by the teacher in recommending unfit students for graduation. 

 Teachers in such positions are, I believe, generally honourable 

 men ; they are chosen after the most careful inquiry into their 

 past career. It is not held fitting in commercial circles to 

 appoint a clerk or an accountant on good recommendations, 

 and after sufficient apprenticeship, and then to surround him 

 with safeguards, in case he turn out incapable or dishonest. 



The objection may possibly be raised, that under such a 

 system the standard of degrees would be very uneven ; but 

 what of that ? As at present, anyone applying for a post of any 

 kind would furnish a reference to his teachers ; and a private 

 letter from one well acquainted with the candidate turns the 

 scale, for or against, in spite of every degree in the United 

 Kingdom. 



In plain English, degrees, as at present given, are not valued 

 by that portion of the public qualified to judge ; and we must 

 face this fact, and endeavour to render a degree a real mark of 

 merit. 



I believe, with Mr. Dickins, that the examinations of the 

 University of London have done much in disseminating know- 

 ledge, and they have therefore proved of great service, but 

 -except in the case of the higher degrees before mentioned, and of 

 the degrees in the Faculty of Medicine where evidence of training 

 is a sine qtid non, I greatly doubt whether they have contributed 

 towards the creation of knowledge, or training in originality. 

 And from the very nature of the constitution of the University 

 of London, it is impossible that it should be otherwise. This 

 very morning, I happened to ask a student attending my lectures 

 on organic chemistry why he, a B.Sc. in chemistry, was attend- 

 ing my lectures. His reply was characteristic. " I scamped up 

 enough of the subject privately, sir, to squeeze through ; but 

 now I wish to know it." In any right system, such a proceeding 

 should be impossible. 



It is therefore with the hope that the creation of a teaching 

 University for London might tend to remedy such evils, that I, 

 for one, would welcome it. I would urge that the distinguished 

 names mentioned by Mr. Thiselton Dyer are surely guarantees 

 that the London Colleges recently possessed men capable of 

 imparting the highest standard of knowledge, and of stimulating 

 true originality ; yet I believe that it is by no means "cutting 

 cheese with a razor " to employ just such men in watching over 

 the development even of junior students ; and it is not without 

 advantage to the most able men of science and of letters to be 

 obliged periodically to devote consideration to "elements" and 

 to pass in review first principles. It counteracts the tendency 

 towards specialization, which, however valuable, always limits 

 the mental horizon. I will undertake to say that the quality of 

 the most advanced teaching in biology and physiology in 

 University College when the chairs were occupied by Burdon 

 Sanderson, by Michael Foster, and by Lankester knew no 

 limit ; and I greatly doubt the wisdom of appointing teachers 

 whose attention is to be devoted exclusively to research. As my 

 predecessor, Prof. Williamson, often remarked, it is more 

 difficult to teach junior than to teach senior students ; and 

 while the superintendence of exercise and laboratory work may 

 well be shared by assistants, in order that the professor may 

 have time to devote to research, and to superintendence of 

 advanced students, it would be a serious calamity were the 

 influence of such minds to be withdrawn wholly from the 

 juniors. 



It is precisely by such a federation of Colleges such as 

 University and King's, and of other sufficiently qualified 

 institutions which have the will and the power to join, that 

 specialization may ultimately be effected. The future occupants 



of the chairs may be chosen so as to represent every side of a 

 subject ; and anyone wishing to pursue research in any special 

 branch would have no difficulty in selecting that particular 

 college where his specially was also the specially of the teacher. 



William Ramsay. 



NO. I 126, VOL. 44] 



No well-wisher of the University can feel otherwise than 

 grateful to you for affording a portion of your valuable space for 

 the letters of Mr. Thiselton Dyer and Mr. Dickins on this sub- 

 ject. No two men could be found to speak with greater authority 

 from first-hand knowledge of the facts. The arguments on the 

 subject have been too much of an ex parte character hitherto, not 

 seldom based on insufficient information or erroneous impressions. 

 Nothing, for example, could be further from the truth than the 

 statement in the "rimes of May 13, by the writer of what was 

 upon the whole a fair and comprehensive leading article, that 

 " there is no reason why the highest honours of the University 

 of London should not be obtained by a person who never set 

 foot in London or even in England." Many, who like myself 

 voted for the projected scheme of the Senate, must have felt, as I 

 did, as a result of a wide and" varied educational experience, that 

 it was potential with great good in the future, and could be ac- 

 cepted as the working basis of the future development of the 

 University, although we felt that the one serious blot in it was 

 the abandonment of uniformity in the examinations for the pass 

 degrees. I verily believe that this was the one thing fatal to its 

 success in Convocation ; that it was so far in excess of the re- 

 commendations of the Koyal Commission as to be unwarrantable ; 

 and that it put a lever into the hands of the opposition, of which 

 — as the event proved — a practised disputant like Mr. Bompas 

 did not fail to make most effective and disastrous use. 



Wellington College, Berks, May 25. A. Irving. 



Quaternions and the " Ausdehnungslehre." 



The year 1844 is memorable in the annals of mathematics on 

 account of the first appearance on the printed p^e 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 com- 

 munication to the Royal Irish Academy, November 13, 1843. 

 This communication was indeed announced to the Council of the 

 Academy four weeks earlier, on thfe very day of Hamilton's 

 discovery of quaternions, as we learn from one of his letters. 

 The author of the "Ausdehnungslehre," although not un- 

 conscious 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 funda- 

 mental 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 qua- 

 ternions, 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 "Ausdehnungslehre" 

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

 "Encyclopaedia Britannica," and in the third edition of Prof. 

 Tail's " Treatise on Quaternions," should not be allowed to pass 

 without protest. 



It is principally as systems of geometrical algebra that qua- 

 ternions 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 geo- 

 metrical 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. 



Geometrical addition in three dimensions is common to the 

 two systems, and seems to have been discovered independently 

 both by Hamilton and Grassmann, as well as by several other 

 persons about the same time. It is not probable that any 

 especial claim for priority with respect to this principle will be 

 urged for either of the two with which we are now concerned. 



