Projective geometry, by Oswald Veblen and John Wesley Young.

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DEGREE OF A GEOMETRIC PROBLEM

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device for drawing lines anrd planes in space. But a picture (which is the
section by a plane of a pr-ojectionl froml a point) of the lines and points of
intersection of linearly constructed planes may be constructed with a straightedge (cf. tlie definition of a plane).
As examples of linear problems we mention: (a) the determination
of the point homologous with a given point in a projectivityv on a
line of which three pairs of homologous points are given; (1b) the
determination of the sixth point of a quadrangular set of which five
points are given; (c) the determination of the second double point
of a projectivity on a line of which one double point and two pairs of
homologous points are given (this is equivalent to (b)); (d) the determinatioll of the second point of intersection of a line with a conic, one
point of intersection and four other points of the conic being given, etc.
The analytic relations existing between geometric elements offer
a convenient means of classifying geometric problems.* Confining
ourselves, for the sake of brevity, to problems in a plane, a geometric
problem consists in constructing certain points, lines, etc., which bear
given relations to a certain set of points, lines, etc., which are supposed given in advance. In fact, we may suppose that the elements
sought are points only; for if a line is to be determined, it is sufficient
to determine two points of this line; or if a conic is sought, it is sufficient to determine five points of this conic, etc. Similar considerations may also be applied to the given elements of the problem,
to the effect that we may assume these given elements all to be
points. This merely involves replacing any given elements that are
not points by certain sets of points having the property of uniquely
determining these elements. Confining our discussion to problems in
which this is possible, any geometric problem may be reduced to
one or more problems of the following form: Given in a plane a
certain finite number of points, to construct a point which shall bear
to the given points certain given relations.
In the analytic formulation of such a problem the given points
are supposed to be determined by their coordinates (homogeneous or
nonhomogeneous), referred to a certain frame of reference. The vertices of this frame of reference are either points contained among the
given points, or some or all of them are additional points which we
* The remainder of this section follows closely tle discussion given in Castelnuovo, Lezioni di geometria, Rome-Milan, Vol. I (1904), pp. 467 ff.