

QUATERNIONS AND THE AUSDEHNUNGSLEHRE. 167 



may be treated as extensive quantities, capable of addition as well as 

 of multiplication. This idea, however, is older than the memoir of 

 1858. The Luckenausdruck, by which the matrix is expressed as a 

 sum of a kind of products (luckenhaltig, or open), is described in a note 

 at the end of the first Ausdehnungslehre. There we have the matrix 

 given not only as a sum, but as a sum of products, introducing a 

 multiplicative relation entirely different from the ordinary multipli- 

 cation of matrices, and hardly less fruitful, but not lying nearly so 

 near the surface as the relations to which Prof. Sylvester refers. The 

 key to the theory of matrices is certainly given in the first Ausdehn- 

 ungslehre, and if we call the birth of matricular analysis the second 

 birth of algebra, we can give no later date to this event than the 

 memorable year of 1844. 



The immediate occasion of this communication is the following 

 passage in the preface to the third edition of Prof. Tait's Quaternions : 



" Hamilton not only published his theory complete, the year before 

 the first (and extremely imperfect) sketch of the Ausdehnungslehre 

 appeared ; but had given ten years before, in his protracted study of 

 Sets, the very processes of external and internal multiplication (corre- 

 sponding to the Vector and Scalar parts of a product of two vectors) 

 which have been put forward as specially the property of Grassmann." 



For additional information we are referred to art. "Quaternions," 

 Encyc. Brit., where we read respecting the first Ausdehnungslehre : 



" In particular two species of multiplication (' inner ' and ' outer ') of 

 directed lines in one plane were given. The results of these two 

 kinds of multiplication correspond respectively to the numerical and 

 the directed parts of Hamilton's quaternion product. But Grassmann 

 distinctly states in his preface that he had not had leisure to extend 

 his method to angles in space. . . . But his claims, however great 

 they may be, can in no way conflict with those of Hamilton, whose 

 mode of multiplying couples (in which the ' inner ' and ' outer ' multi- 

 plication are essentially involved) was produced in 1833, and whose 

 quaternion system was completed and published before Grassmann had 

 elaborated for press even the rudimentary portions of his own system, 

 in which the veritable difficulty of the whole subject, the application 

 to angles in space, had not even been attacked." 



I shall leave the reader to judge of the accuracy of the general 

 terms used in these passages in comparing the first Ausdehnungslehre 

 with Hamilton's system as published in 1843 or 1844. The specific 

 statements respecting Hamilton and Grassmann require an answer. 



It must be Hamilton's Theory of Conjugate Functions or Algebraic 

 Couples (read to the Eoyal Irish Academy, 1833 and 1835, and 

 published in vol. xvii of the Transactions) to which reference is made 

 in the statements concerning his "protracted study of Sets" and 



