GEORGE BRUCE HALSTED — NEW ELEMENTS OF GEOMETRY. 
17 
Rigorously speaking, plane curvilinear areas, as also solids 
bounded by curved surfaces, can not be measured so long as they 
are compared those to a square, these to a cube. However, if one 
only proposes to find a limit toward which tend the results of a 
direct operation, it will suffice, for the attainment of this end, to 
demonstrate that such a limit exists necessarily in each particular 
case, and then to explain in what way we ought to understand the 
operation of measuring, and how to attain the amount of exacti- 
tude desired. 
To satisfy these conditions recourse must be had to certain aux- 
iliary propositions, admitted under the name of axioms; namely: 
1. Two plane figures are equal when they are decomposable 
into parts identical, though differently arranged. 
2. Of two plane figures, that is the smaller which is entirely 
contained in the interior of the contour of the other, without, how- 
ever, filling this completely. 
3. The magnitude of a triangle decreases indefinitely when one 
of its sides diminishes indefinitely. 
This latter proposition is necessary for the evaluation of plane 
areas themselves. 
It is necessary also to have recourse to analogous axioms when 
one wishes to measure solids. 
