It is generally recognised that the proofs given in the school text-books on Geometry are incomplete and that many theorems are assumed because they are intuitively ‘obvious,’ and one of my aims this afternoon is to show wherein this incompleteness lies and to state the ‘obvious’ assumptions.

It has been suggested that a sketch, limited to the real plane, of the theory of homogeneous coordinates and points at infinity which is developed for complex space in my Prolegomena to Analytical Geometry, would be of interest to teachers who without wanting to go deeply into the matter are humanly discontented with the crude treatment that makes of infinity “the place where things happen that don’t”.

Every student of Mechanics or Applied Mathematics has at some time or other considered the motion of a particle projected into space with a known velocity. He is generally content to ignore air resistance and like contingencies, and to take the subsequent motion as that resulting from the combination of the constant velocity in the given direction of projection with the acceleration due to gravitational attraction upon the particle. Mathematical manipulation of the statements giving the position in space of the particle after an interval of time shows:

(a) That if the particle be projected into space with a given velocity the range on a given plane is a maximum if the direction of projection bisects the angle between the plane and the vertical. Further, the direction of motion of the particle when it strikes the plane at this point of maximum range is at right angles to the direction of projection;

(b) Also, if two directions of projection be obtained such that the range on the given plane for a certain velocity of projection is the same for each, then it can be derived that these directions make equal angles with the bisector of the angle between the plane and the vertical.