Grundlagen, Section 64: Frege's Discussion of Definitions by Abstraction in Historical Context

I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlomilch, and the tradition of textbooks in geometry for secondary schools (including a textbook Frege had used when teaching in a Privatschule in Jena in 1882-1884). In addition, I put Frege's considerations in the context of a widespread debate in Germany on 'directions' as a central notion in the theory of parallels.