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I. ON THE BENDING OF SURFACES GENERATED BY THE MOTION OF A STRAIGHT LINE IN SPACE.
II. ON THE BENDING OF SURFACES OF REVOLUTION.
III. ON THE PROPERTIES OF A SURFACE CONSIDERED AS THE LIMIT OF THE INSCRIBED POLYHEDRON.
An excerpt from the Introductory:
[Read March 13, 1854.]
Euclid has given two definitions of a surface, which may be taken as examples of the two methods of investigating their properties. That in the first book of the Elements is— "A superficies is that which has only length and breadth."
The superficies differs from a line in having breadth as well as length, and the conception of a third dimension is excluded without being explicitly introduced.
In the eleventh book, where the definition of a solid is first formally given, the definition of the superficies is made to depend on that of the solid—
"That which bounds a solid is a superficies."
Here the conception of three dimensions in space is employed in forming a definition more perfect than that belonging to plane Geometry.
In our analytical treatises on geometry a surface is defined by a function of three independent variables equated to zero. The surface is therefore the boundary between the portion of space in which the value of the function is positive, and that in which it is negative; so that we may now define a surface to be the boundary of any assigned portion of space.
Surfaces are thus considered rather with reference to the figures which they limit than as having any properties belonging to themselves.
But the conception of a surface which we most readily form is that of a portion of matter, extended in length and breadth, but of which the thickness may be neglected. By excluding the thickness altogether, we arrive at Euclid's first definition, which we may state thus—
"A surface is a lamina of which the thickness is diminished so as to become evanescent."
We are thus enabled to consider a surface by itself, without reference to the portion of space of which it is a boundary. By drawing figures on the surface, and investigating their properties, we might construct a system of theorems, which would be true independently of the position of the surface in space, and which might remain the same even when the form of the solid of which it is the boundary is changed.
When the properties of a surface with respect to space are changed, while the relations of lines and figures in the surface itself are unaltered, the surface may be said to preserve its identity, so that we may consider it, after the change has taken place, as the same surface.
When a thin material lamina is made to assume a new form it is said to be bent. In certain cases this process of bending is called development, and when one surface is bent so as to coincide with another it is said to be applied to it.
By considering the lamina as deprived of rigidity, elasticity, and other mechanical properties, and neglecting the thickness, we arrive at a mathematical definition of this kind of transformation.
"The operation of bending is a continuous change of the form of a surface, without extension or contraction of any part of it."