Affine geometry is a generalization of Euclidean geometry in which distance can be scaled along parallel directions, though relative distances in different directions may be incommensurable. This article presents a new procedure for testing the intrinsic affine structure of a psychological space by having subjects perform bisection judgments over multiple directions. If those judgments are internally consistent with one another, they must satisfy a theorem first proved by Pierre Varignon around 300 years ago. In the experiment reported here, this procedure was employed to measure the perceived structure of a visual ground surface. The results revealed that observers' judgments were systematically distorted relative to the physical environment, but that the judged bisections in different directions had an internally consistent affine structure. Implications of these findings for other possible response tasks are considered.