[From the Transactions of the Cambridge Philosophical Society, VoL IX. Part IV.] 



IV. On the Transformation of Surfaces by Bending. 



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 thick- 



