TITLE: Why are affine functions called "affine" functions?
QUESTION [5 upvotes]: I am learning about affine functions and I do not understand why a certain type of function (functions that are in the form of $f(x)=ax+b$) are called affine functions. I read about the word "affine" and I know that it means related, but I do not understand how is it related to this type of function and what was the reason behind this naming. I believe that understanding the reason behind namings can help one better understand a concept.

REPLY [6 votes]: According to Jeff Miller, the adjective affinis (in Latin: "bordering", "adjacent", "linked by marriage") was introduced in geometry by Euler, in analogy to similis, to denote a weaker equivalence relation  than the latter: "Because curves originated this way do keep a certain Affinity between them, we will name these curves  affine". 

AFFINE. Affinis and affinitas were first used by Leonhard Euler in
  Introductio in analysin infinitorum (1748) Chapter XVIII: De
  similitudine et affinitate linearum curvarum. He also wrote (II.
  xviii. 239): "Quia Curvae hoc modo ortae inter se quandam Affinitatem
  tenent, has Curvas affines vocabimus."