Januaky 17, 1896.] 



SCIENCE. 



99 



for scholarships. The residue of the estate 

 (some $20,000) is left to Clark University. 



Dr. C. a. Strong, associate professor of 

 psychology in the University of Chicago, has 

 been elected lecturer on psychology in Colum- 

 bia College. ' 



Prof. L. S. Luther, of Trinity College, 

 Hartford, has been elected president of Kenyon 

 College, Gambler, Ohio. Professor Theodore 

 Stirling, the professor of natural science, has 

 been during the last four years acting president. 



Prof. Theodore von der Goltz has been 

 appointed professor of agriculture in the Uni- 

 versity at Bonn in the place of Prof. Dunkel- 

 berg, who has retired. 



DISCUSSION AND COBBESPONDENCE. 

 QUATERNIONS. 



Editor of Science : The circular letter of 

 Dr. Molenbroek and Mr. Kimura published in 

 the issue of your journal for October 18th ap- 

 pears to me to be a distinct improvement upon 

 their preceding letter published in Nature for 

 October 3d. In the former letter they assume 

 that Hamilton's Quaternions is a much more 

 perfect method than it really is, and they affirm 

 that the newer forms of vector theory invented 

 by physicists are founded on definitions which 

 are established by Quaternions, and are systems 

 of notation rather than logical developments of 

 a mathematical idea. They also advise the 

 "many who are prejudiced against the calculus 

 of quaternions and maintain the opinion that it 

 is hard to understand and that it contains a 

 great deal which is useless in addition to things 

 immediately applicable " to " approach the cal- 

 culus with proper care and meekness in the as- 

 surance that they will ere long rejoice in hav- 

 ing at their disposal an instrument of research 

 mightier far than they had the slightest notion 

 of so long as they were in the domain of carte- 

 sian mathematics." 



In recent years I have published a series of 

 papers on Space Analysis, the express object of 

 which is to unify and harmonize the several 

 vector methods with one another and with the 

 ordinary analysis. I exclude neither the idea 

 of a vector nor the idea of a quaternion, and I 

 do not attempt to make Nature simpler than 



she really is by identifying ideas that are difler- 

 ent though complementary to one another. I 

 look upon vector-analysis not as an independent 

 and rival plant, but as a development of the old 

 tree of mathematical analysis. 



The greatest impediments to the progress of 

 the method of Quaternions are not prejudice 

 and false opinion in those to whom it is pre- 

 sented, but rather imperfections, mistakes and 

 errors in the method itself. Hamilton ought to 

 be reverenced for what he did accomplish, but 

 that ought not to blind us to what he did not 

 accomplish. It is an error to identify, as Hamil- 

 ton does, vectors with quadrantal quaternions. 

 It is an error to confound, as Hamilton does, 

 successive with simultaneous addition ; for 

 thereby he failed to discover the generalization 

 for space of the Exponential Theorem and of 

 Taylor' s Theorem. It is a mistake to introduce, 

 as Hamilton does, a new notation which has no 

 relation to the established notation of trigonom- 

 etry, or to adopt conventions which do not har- 

 monize with the established conventions of 

 analysis. 



To the amended proposal for an ' Inter- 

 national Association for promoting the study of 

 Quaternions and allied systems of Mathe- 

 matics ' there is no room for objection ; for it 

 does not assume the perfection and finality of 

 Hamilton's work, but rather invites to the de- 

 velopment and study of vector-analysis in its 

 broadest sense. It will, I hope, receive a favor- 

 able response from all who are interested in the 

 development or the teaching of space analysis. 

 It is inevitable that there should be diversity of 

 notation and warm discussion of principles 

 among the pioneers in this region, but inasmuch 

 as all are zealous for the truth, the proposed 

 association would accelerate the progress to 

 definite decisions, and thereby smooth the way 

 for the spread of this, the highest development 

 of the art of algebra. 



Messrs. Molenbroek and Kimura refer to the 

 remarkable advance in Electrical theory. That 

 advance has been due in large measure to the 

 practical manner in which electricians have dis- 

 cussed the principles and definitions of their 

 science, finally settling all definitions by an 

 authorized Congress. Doubtless the proposed 

 association would eventually accomplish an 



