The present volumes v, vi are an introduction to the more important of the algebraical and functional relations which are necessary for a clear and precise understanding of the principles of algebraic geometry. These relations are as the bones of the structure, to be clothed finally with a body of purely geometrical doctrine.

For the expression of these relations we make free use of coordinates, of which the justification has been examined in Vols. I and II. With their use we can define an algebraic construct (curve, surface, manifold, etc.) as the aggregate of points whose coordinates satisfy a set of algebraic equations, taken with points (limiting points, and other) which it may be necessary conventionally to add thereto. And, it is to be understood that all coordinates, and parameters, that enter in the equations employed, are capable of complex values; in particular, an aggregate will be said to be of dimension, or freedom, r, or simply to be ∞r , when it depends on the values of r parameters not restricted to real values. In the elementary geometry two figures, or two manifolds, are regarded as essentially identical when they are projectively related to one another, that is (as we have seen) transformable into one another by equations which are linear in the (homogeneous) coordinates; in general algebraic geometry, two manifolds are regarded as essentially identical when the coordinates of the points of either are expressible as rational algebraic functions of the coordinates of the points of the other, whether linear functions or not.