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’.
This is not a treatise on geometry, and there is no occasion to develop the matter in detail. The point of importance is this. The infinite of analysis is a ‘limiting’ and not an ‘actual’ infinite. The symbol ‘\(\infty\)’ has, throughout this book, been regarded as an ‘incomplete symbol’, a symbol to which no independent meaning has been attached, though one has been attached to certain phrases containing it. But the infinite of geometry is an actual and not a limiting infinite. The ‘line at infinity’ is a line in precisely the same sense in which other lines are lines.
It is possible to set up a correlation between ‘homogeneous’ and ‘common’ Cartesian geometry in which all elements of the first system, the special elements excepted, have correlates in the second. The line \(ax + by + cz = 0\), for example, corresponds to the line \(ax + by + c = 0\). Every point of the first line has a correlate on the second, except one, viz. the point for which \(z = 0\). When \((x, y, z)\) varies on the first line, in such a manner as to tend in the limit to the special point for which \(z = 0\), the corresponding point on the second line varies so that its distance from the origin tends to infinity. This correlation is historically important, for it is from it that the vocabulary of the subject has been derived, and it is often useful for purposes of illustration. It is however no more than an illustration, and no rational account of the geometrical infinite can be based upon it. The confusion about these matters so prevalent among students arises from the fact that, in the commonly used text books of analytical geometry, the illustration is taken for the reality.
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