We have a complete geometrical notion of a curve of a given order, viz. a curve of the order n is a curve which is met by any line whatever in n points and no more; but starting with this definition, how do we know that there exists a curve of the order n? and, further, how do we know that it depends linearly on ½n(n + 3) parameters, or, what is the same thing, that there is one and only one curve which can be drawn through ½n(n + 3) given points?

The last-mentioned property does not by itself constitute a definition of a curve of the order n; thus n = 2, we cannot define a curve of the second order as a curve which is uniquely determined by the condition of passing through 5 given points; for a cubic passing through 4 given points is a curve uniquely determined by the condition in question; but we may differentiate between these two solutions by adding the further condition that, when 3 of the 5 points are in a line, the curve of the second order shall include as part of itself this line. And we are thus led to the definition: A curve of the order n is a curve which is uniquely determined by the condition of passing through ½n(n + 3) given points; and of being moreover such that, when n + 1 of these points lie on a line, it includes as part of itself this line.