Clifford and Sylvester on the Development of Peirce’s Matrix Formulation of the Algebra of Relations, 1870–1882 Francine F. Abeles Abstract In this paper I first examine the critical factors that are essential to understanding why Charles Peirce turned to matrices in 1882, twelve years after the algebra he had developed in his 1870 paper could be realized as a quaternion algebra. I shall argue that these factors were the creative influence of Peirce’s friend and colleague, William Kingdon Clifford (1845–1879) in the years prior to 1882, and the setting in 1879–1882 when Peirce and Sylvester were together at JHU. Then I shall demonstrate that Peirce, not Sylvester, deserves the recognition for being the first of the two to show that every associative algebra can be represented by a matrix. [This paper has its roots in Anellis and Abeles, “The Historical Sources of Tree Graphs and the Tree Method in the Work of Peirce and Gentzen” in Modern Logic 1850–1950, East and West, pp. 35-97. Studies in Universal Logic. Birkhäuser, Basel, 2016] 1 Introduction Charles Sanders Peirce (1839–1914) first reported that his relational algebra can be expressed in matrix form in his 1882 papers, “On the Relative Forms of Quaternions” (Peirce 1882c), “Brief Description of the Algebra of Relatives” (Peirce 1882a), and “On a Class of Multiple Algebras” (Peirce 1882b). However, before he wrote these papers, in an 1881 addendum to his posthumous publication of his father, Benjamin Peirce’s Linear Associative Algebra (Peirce 1881), he showed that any associative algebra can be expressed in relative form that is matrix representable. There are many well-known parts to this story, including Charles’s on-going rivalry with James Joseph Sylvester (1814–1897) when they both were at Johns Hopkins University (JHU). Peirce considered Sylvester’s universal multiple algebra, a topic he included in his lectures at JHU to be similar, if not the same, as his own matrix algebra. (Sylvester published these lectures in 1884 after leaving JHU F.F. Abeles () Departments of Mathematics and Computer Science, Kean University, Union, NJ 07083, USA e-mail: fabeles@kean.edu © Springer International Publishing Switzerland 2016 M. Zack, E. Landry (eds.), Research in History and Philosophy of Mathematics, Proceedings of the Canadian Society for History and Philosophy of Mathematics/ La Société Canadienne d’Histoire et de Philosophie des Mathématiques, DOI 10.1007/978-3-319-46615-6_7 83