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 Lückenausdruck, by which the matrix is expressed as a sum of a kind of products (lückenhaltig, 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 multiplication 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 Ausdehnungslehre, 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 (corresponding 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' multiplication 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 Royal 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
