JOSIAH WILLAKD GIBBS. xix 



printed in 1881 and 1884 a very concise account of the vector analysis 

 which he had developed, and this pamphlet was to some extent circu- 

 lated among those especially interested in the subject. In the develop- 

 ment of this system the author had been led to study deeply the 

 Ausdehnungslehre of Grassmann, and the subject of multiple algebra 

 in general ; these investigations interested him greatly up to the time 

 of his death, and he has often remarked that he had more pleasure in 

 the study of multiple algebra than in any other of his intellectual 

 activities. His rejection of quaternions, and his championship of 

 Grassmann's claim to be considered the founder of modern algebra, 

 led to some papers of a somewhat controversial character, most of 

 which appeared in the columns of Natwre. When the utility of 

 his system as an instrument for physical research had been proved 

 by twenty years' experience of himself and of his pupils, Professor 

 Gibbs consented, though somewhat reluctantly, to its formal publi- 

 cation in much more extended form than in the original pamphlet. 

 As he was at that time wholly occupied with another work, the task 

 of preparing this treatise for publication was entrusted to one of his 

 students, Dr. E. B. Wilson, whose very successful accomplishment of 

 the work entitles him to the gratitude of all who are interested in 

 the subject. 



The reluctance of Professor Gibbs to publish his system of vector 

 analysis certainly did not arise from any doubt in his own mind as 

 to its utility, or the desirability of its being more widely employed ; 

 it seemed rather to be due to the feeling that it was not an original 

 contribution to mathematics, but was rather an adaptation, for special 

 purposes, of the work of others. Of many portions of the work this 

 is of course necessarily true, and it is rather by the selection of 

 methods and by systematization of the presentation that the author 

 has served the cause of vector analysis. But in the treatment of the 

 linear vector function and the theory of dyadics to which this leads, 

 a distinct advance was made which was of consequence not only in 

 the more restricted field of vector analysis, but also in the broader 

 theory of multiple algebra in general. 



The theory of dyadics* as developed in the vector analysis of 1884 

 must be regarded as the most important published contribution of 

 Professor Gibbs to pure mathematics. For the vector analysis as an 

 algebra does not fulfil the definition of the linear associative algebras 

 of Benjamin Peirce, since the scalar product of vectors lies outside the 

 vector domain; nor is it a geometrical analysis in the sense of 



* The three succeeding paragraphs are by Professor Percey F. Smith ; they form part 

 of a sketch of Professor Gibbs's work in pure mathematics, which Professor Smith con- 

 tributed to the Bulletin of the, American Mathematical Society, vol. x, p. 34 (October, 

 1903). 



