# 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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