The role of model-based reasoning in experimental and theoretical scientific thinking has been extensively studied. However, little work has been done on the role of mathematical representations in such thinking. This chapter will describe how the nature of mathematical expressions in physics can be analyzed using an extension of the metaphoric analysis of mathematics. In Where Mathematics Comes From, Lakoff and Núñez argued that embodied metaphors underlie basic mathematical ideas (e. g., the concept of number is based on the embodied operations of collecting objects), with more complex expressions developed via conceptual blends from simpler expressions (e. g., addition as combining collections). In physics, however, the need to represent physical processes and observed entities (including measurements) places different demands on the blending processes. In model-based reasoning, conceptual blends must often be based on immediately available embodiments as well as highly developed mathematical expressions that draw upon expert use of long term working memory. Thus, Faraday's representations of magnetic fields as lines of force were modeled by Maxwell as vectors. In this chapter, we compare Faraday's experimental investigation of the magnetic field within a magnet to Maxwell's mathematical treatment of the same problem. Both can be understood by unpacking the metaphoric underpinnings as physical representations. The implications for analogical and model-based reasoning accounts of scientific thinking are discussed.