Josiah Willard Gibbs. 195 



units, each of which is an indeterminate product of two 

 factors. On the other hand, C. S. Peirce, who was the first to 

 recognize (1870) the quadrate linear associative algebras identi- 

 cal wdth matrices, uses for the units a letter pair^ but does not 

 regard this combination as a product. In addition. Professor 

 Gibbs, following the spirit of Grassmann's system, does not 

 confine himself to one kind of multiplication of dyadics, as do 

 Hamilton and Peirce, but considers two sorts, both originating 

 with Grassmann. Thus it may be said that quadrate, or 

 matricular algebras, are brought entirely within the wonderful 

 system expounded by Grassmann in 1814. 



As already remarked, the exposition of the theory of 

 dyadics given in the vector analysis is not in accord with 

 Grassmann's system. In a footnote to the address referred 

 to above. Professor Gibbs shows the slight modification neces- 

 sary for this purpose, while the subject has been treated in detail 

 and in all generality in his lectures on multiple algebra deliv- 

 ered for some years past at Yale University. 



Professor Gibbs was much interested in the applications of 

 vector analysis to some of the problems of astronomy, and 

 more than once he called attention to the great saving of labor 

 which the use of this method would cause in such subjects as 

 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. 



Between the years 1882 and 1889, five papers appeared in 

 this Journal upon certain points in the electromagnetic theory 

 of light and its relations to the various elastic theories. These 

 are remarkable for the entire absence of special hypotheses as 

 to the connection between ether and matter, the only supposi- 

 tion made as to the constitution of matter being that it is fine- 

 grained with reference to the wave-length of light, but not infin- 

 itely tine-grained, and that it does disturb in some manner the 

 electrical fluxes in the ether. By methods whose simplicity and 

 directness recall his thermodynamic investigations, the author 

 shows in the first of these articles that, in the case of perfectly 

 transparent media, the theory not only accounts for the disper- 

 sion of colors (including the '^ dispersion of the optic axes " in 

 doubly refracting media), but also leads to FresneFs laws of 

 double refraction for any particular wave-length without 

 neglect of the small quantities which determine the dispersion 

 of colors. He proceeds in the second paper to show that circu- 

 lar and elliptical polarization are explained by taking into 

 account quantities of a still higher order, and that these in turn 



