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Stuick Summaries Gauss on Surfaces

Struik writes, ". . .(Gauss) saw in a surface not so much the boundary of a solid body, as a fleece or film, a two-dimensional entity not necessarily attached to a three-dimensnioal body. A piece of such surface can be bent and we can ask for the properties of the fleece which do not change under bending. A two-dimensional being, living on the surface, and unaware of any outside space — like the beings of Abbott's Flatland, which live in the plane unaware of any space of which the plane may be a part, would not be able to find out what asymptotic lines or lines of curvature are. But he would be able to find the road of shortest distance between two points measured along the surface, or the angle of two directions on the surface, that is, the intrinsic properties of the surface. Thus, with his characteristic understanding of theory and practice alike, Gauss drew from his work as a surveyor the inspiration for his profound reappraisal of the general theory of surfaces."
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