114 MULTIPLE ALGEBRA. 



and Grassmann formed their algebras, although the philosophical 

 mind of the last was not satisfied until he had produced a system 

 unfettered by any spatial relations. It is probably in connection with 

 some of these subjects that the notions of multiple algebra are most 

 widely disseminated. 



Maxwell's Treatise on Electricity and Magnetism has done so 

 much to familiarize students of physics with quaternion notations, 

 that it seems impossible that this subject should ever again be entirely 

 divorced from the methods of multiple algebra. 



I wish that I could say as much of astronomy. It is, I think, to be 

 regretted, that the oldest of the scientific applications of mathematics, 

 the most dignified, the most conservative, should keep so far aloof 

 from the youngest of mathematical methods; and standing as I do 

 to-day, by some chance, among astronomers, although not of the guild, 

 I cannot but endeavor to improve the opportunity by expressing my 

 conviction of the advantages which astronomers might gain by 

 employing some of the methods of multiple algebra. A very few of 

 the fundamental notions of a vector analysis, the addition of vectors 

 and what quaternionists would call the scalar part and the vector 

 part of the product of two vectors (which may be defined without 

 the notion of the quaternion), these three notions with some four 

 fundamental properties relating to them are sufficient to reduce 

 enormously the labor of mastering such subjects as the elementary 

 theory of orbits, the determination of an orbit from three observations, 

 the differential equations which are used in determining the best orbit 

 from an indefinite number of observations by the method of least 

 squares, or those which give the perturbations when the elements are 

 treated as variable. In all these subjects the analytical work is 

 greatly simplified, and it is far easier to find the best form for 

 numerical calculation than by the use of the ordinary analysis. 



I may here remark that in its geometrical applications multiple 

 algebra will naturally take one of two principal forms, according as 

 vectors or points are taken as elementary quantities, i.e., according as 

 something having magnitude and direction, or something having 

 magnitude and position at a point, is the fundamental conception. 

 These forms of multiple algebra may be distinguished as vector 1 

 analysis and point analysis. The former we may call a triple, the 

 latter a quadruple algebra, if we determine the degree of the algebra 

 from the degree of multiplicity of the fundamental conception. The 

 former is included in the latter, since the subtraction of points gives 

 us vectors, and in this way Grassmann's vector analysis is included in 

 his point analysis. Hamilton's system, in which the vector is the 

 fundamental idea, is nevertheless made a quadruple algebra by the 

 addition of ordinary numerical quantities. For practical purposes we 



