The Theory of the Imaginary in Geometry, together with the Trigonometry of the Imaginary

Abstract

WHEN we interpret ϕ(x, y)=o, ψ (x, y)-o as the point-equations of two loci, we are bound to consider any values (x_i, y_i) which satisfy both equations as the co-ordinates of a point common to both curves. The simplest case is when ϕ,ψ are polynomials with ordinary integral coefficients; here the values (x_i, y₁) are determinate, and can be calculated, either exactly or to any desired degree of approximation. Abstractly, (x₁, y₁) are a perfectly definite set of couples of algebraic numbers. A couple (x₁, y₁) may be real, and then corresponds to a real point; but it may be, and often is, complex. What is the most appropriate and fruitful way, from a geometrical point of view, of interpreting these complex solutions of the given pair of equations? This is one of the fundamental problems of analytical geometry, and there are two ways in which it may be attacked. Suppose that the coefficients of ϕ, ψ are real, complex intersections (x₁, y₁) fall into conjugate pairs. The usual analytical formula gives a real line as the join of two conjugate points, and we may call this a common chord of the two loci. The visible result of combining ϕ = o, ψ= o may be said to be a certain number of real intersections and a certain number of real lines which, from an algebraical point of view, are to be regarded as common chords. The most familiar case is that of two circles and their radical axis; and here we have a geometrical definition of the radical axis which applies whether it meets the two circles or not. We can construct a definition of a common chord of two conics by analogy, whether it meets them in two real or two conjugate complex points; but the procedure is artificial, and there is no obvious way of extending it to higher curves.

The Theory of the Imaginary in Geometry, together with the Trigonometry of the Imaginary.

By Prof. J. L. S. Hatton. Pp. vii + 215. (Cambridge: At the University Press, 1920.) Price 18s. net.