George Lakoff and I first presented the Invariance Hypothesis implicitly in 'More than Cool Reason' (Lakoff and Turner 1989). In "An Image-Schematic Constraint on Metaphor", I developed an explicit version of the Invariance Hypothesis. George Lakoff developed what appeared to be a different version of the Invariance Hypothesis, in the paper he delivered at the 1989 International Symposium on Cognitive Linguistics at Duisburg, "The Invariance Hypothesis: Do Metaphors Preserve Cognitive Topology?" (available from the Linguistic Agency at the University of Duisburg). Lakoff and I have discussed these two variant formulations and have now agreed that they differ only in the extent to which my phrasing makes explicit parts of the hypothesis Lakoff assumed. By the time this comment is printed, Lakoff will have presented a new phrasing of our Invariance Hypothesis in the first issue of Cognitive Linguistics. In an attempt to clarify some of the issues raised by the Invariance Hypothesis, I will here summarize the main points of my earlier article. In a conceptual metaphor, we map a source domain onto a target domain. But that mapping is not arbitrary. What constrains a metaphorical mapping? (Mark Turner)