DEVELOPMENT OF MATHEMATICAL THOUGHT. 655 



calculus was shown to be of special use in expressing 

 the relations of spherical trigonometry. Two terms 

 expressing definite notions special to geometry, by which 

 science has been enriched and practical application greatly 

 simplified, are an outcome of this line of research. These 

 are the terms " vector," to express the notion of directed 

 magnitude i.e., of direction and magnitude combined as 

 distinguished from magnitude and position alone; and 

 the notion of an " operator " which changes direction and 

 magnitude as an ordinary multiplier changes magnitude 

 only. 1 It was shown by Argand and others that the 



1 These two notions, which have 

 their origin in the writings of 

 Hamilton on the one side and the 

 Calculus of Operations on the 

 other, belong to this country and to 

 a period during which mathematical 

 researches were carried on in a frag- 

 mentary manner, and much out of 

 contact with the contemporary 

 mathematics of the Continent. 

 Both the Calculus of Quaternions 

 of Hamilton and the Calculus of 

 Operations were looked upon for a 

 long time as curiosities (as was also 

 the Barycentric Calculus of Mbbius 

 in Germany). Gradually, however, 

 the valuable ideas which were con- 

 tained in them became recognised 

 as much from the practical as from 

 the theoretical point of view. In 

 the former interest the application 

 of Vector Analysis or the Algebra 

 of Directed Quantities received a 

 great impetus when the need was 

 felt of having an algebra of " phy- 

 sical quantities." This found ex- 

 pression in the writings of Clerk- 

 Maxwell. (See his 'Treatise on 

 Electricity and Magnetism,' vol. i. 

 p. 8, 2nd ((].. as also his paper on 

 " The Mathematical Classification of 

 Physical Quantities," 1871. 'Coll. 

 Papers,' vol. ii. p. 257.) In the prac- 

 tical application of electrical theories 



these notions have since become in- 

 dispensable, and the subject has re- 

 ceived increasing attention, notably 

 in America, which holds a foremost 

 place in the development of electrical 

 science and its application. Mathe- 

 maticians of the first order, such 

 as J. Willard Gibbs, have pub- 

 lished text-books on the subject, 

 whilst other electricians of emin- 

 ence, such as Mr Oliver Heaviside, 

 have elaborated special forms of the 

 Directional Calculus to serve their 

 purposes. In Dynamics the Dublin 

 School, represented after the death 

 of Hamilton by Sir Robert S. Ball 

 (in his 'Theory of Screws,' 1876), 

 has had an important influence in 

 the introduction of novel and more 

 appropriate methods which have 

 gradually permeated the general 

 treatment of the subject. Whilst 

 there is no doubt that for a long 

 time the Calculus of Quaternions 

 was the only methodical elaboration 

 of these novel and useful ideas, it 

 was overlooked that simultaneously 

 and quite independently H. Grass- 

 mann of Stettin (see above, vol. i. 

 p. 243) had worked out a much more 

 comprehensive and fundamental 

 calculus, of which the method of 

 quaternions and all the different 

 forms of Vector Analysis can be 



