The infinite in analysis and geometry
Some, though not all, systems of analytical geometry contain ‘infinite’ elements, the line at infinity, the circular points at infinity, and so on. The object of this brief note is to point out that these concepts are in no way dependent upon the analytical doctrine of limits.
In what may be called ‘common Cartesian geometry’ a point is a pair of real numbers (x, y). A line is the class of points which satisfy a linear relation ax + by + c = 0, in which a and b are not both zero. There are no infinite elements, and two lines may have no point in common.
In a system of real homogeneous geometry a point is a class of triads of real numbers (x, y, z), not all zero, triads being classed together when their constituents are proportional. A line is a class of points which satisfy a linear relation ax + by + cz = 0, where a, b, c are not all zero. In some systems one point or line is on exactly the same footing as another. In others certain ‘special’ points and lines are regarded as peculiarly distinguished, and it is on the relations of other elements to these special elements that emphasis is laid. Thus, in what may be called ‘real homogeneous Cartesian geometry’, those points are special for which z = 0, and there is one special line, viz. the line z = 0. This special line is called ‘the line at infinity’.