The infinite in analysis and geometry

Excerpts

• The infinite in analysis and geometry

  The point of importance is this. The infinite of analysis is a ‘limiting’ and not an ‘actual’ infinite.

• The infinite in analysis and geometry

  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.

• The infinite in analysis and geometry

  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 infinite in analysis and geometry

  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.

• The infinite in analysis and geometry

  The object of this brief note is to point out that these concepts are in no way dependent upon the analytical doctrine of limits.

• The infinite in analysis and geometry

  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.

• The infinite in analysis and geometry

  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.

• The infinite in analysis and geometry

  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’.

• The infinite in analysis and geometry

  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.

• The infinite in analysis and geometry

  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.

• The infinite in analysis and geometry

  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.

Equations

• The infinite in analysis and geometry

  ax + by + c=0

  In common Cartesian geometry a line is the set of points (x, y) whose coordinates satisfy this linear relation, with a and b not both zero.

• The infinite in analysis and geometry

  ax + by + cz = 0

  In a system of real homogeneous geometry a line is the class of points (x, y, z) satisfying this linear relation, where a, b, c are not all zero.

• The infinite in analysis and geometry

  z = 0

  The special points of real homogeneous Cartesian geometry are those with z = 0, and the single special line z = 0 is called the line at infinity.

Problems

No exercises in this chapter.