On the area of surfaces
The necessary and sufficient condition that a curve should possess a length, this length being given by the usual integral formula, is well known. The curve being defined by the equations x = x ( u ), y = y ( u ), the condition is that x ( u ) and y ( u ) should be expressible as integrals with respect to u . It may seem scarcely credible that no corresponding theorem is known with regard to the area of a surface. Such is, however, the case. And what is more surprising, no one has hitherto succeeded in giving such a definition of the area of a curved surface as permits of a determination of a sufficient condition of a general nature that the surface should possess an area, this area being given by the integral formula known to hold in the simplest cases.