CHAPTER XII - PLANE CONTENT AND AREA

Summary

The Theory of Content, as developed in Chap. V, is again unaltered in all its main features when we come to deal with two or more dimensions; it receives, however, a vast extension from the fact that we have to distinguish between linear content, plane content, three-dimensional content, and so forth.

The theory of plane content in the plane. This theory is practically the same as that of linear content in the straight line, and generally as w-dimensional content in space of n dimensions. The definitions and properties given in Ch. V require little more than the substitution of the word regions for intervals, to be valid as they stand. As however there are one or two alterations of a more than verbal character which slightly complicate the issues, the discussion is here shortly given for the plane, and can then without difficulty be modified to suit higher space.

The principles which govern this theory are the same as in the straight line:—

1. (1) The content of a plane set of points, where it exists, is to be a non-negative quantity, and depends only on the relative position of the points of the set, not on its position as a whole in the fundamental region.

2. (2) The content of the sum of two sets having no common points is to be the sum of their contents.

As in the straight line we started with the content of intervals and sets of intervals as the basis of the whole discussion, so now we start with regions, and, as simplest form of regions, with triangles.