14.  The last four problems are a study of the consequences of the following transformation:  A point O is fixed in the plane.  Then to any point P is made to correspond the line p at right angles to OP and bisecting it.  In this correspondence, what happens to p when P moves along a straight line?  What corresponds to the theorem that two lines have only one point in common?  What to the theorem that the angle sum of a triangle is two right angles?  Etc.

CHAPTER X — ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY

161.  Ancient results. The theory of synthetic projective geometry as we have built it up in this course is less than a century old.  This is not to say that many of the theorems and principles involved were not discovered much earlier, but isolated theorems do not make a theory, any more than a pile of bricks makes a building.  The materials for our building have been contributed by many different workmen from the days of Euclid down to the present time.  Thus, the notion of four harmonic points was familiar to the ancients, who considered it from the metrical point of view as the division of a line internally and externally in the same ratio(1) the involution of six points cut out by any transversal which intersects the sides of a complete quadrilateral as studied by Pappus(2); but these notions were not made the foundation for any general theory.  Taken by themselves, they are of small consequence; it is their relation to other theorems and sets of theorems that gives them their importance.  The ancients were doubtless familiar with the theorem, Two lines determine a point, and two points determine a line, but they had no glimpse of the wonderful law of duality, of which this theorem is a simple example.  The principle of projection, by which many properties of the conic sections may be inferred from corresponding properties of the circle which forms the base of the cone from which they are cut—­a principle so natural to modern mathematicians—­seems not to have occurred to the Greeks.  The ellipse, the hyperbola, and the parabola were to them entirely different curves, to be treated separately with methods appropriate to each.  Thus the focus of the ellipse was discovered some five hundred years before the focus of the parabola!  It was not till 1522 that Verner(3) of Nuernberg undertook to demonstrate the properties of the conic sections by means of the circle.

162.  Unifying principles. In the early years of the seventeenth century—­that wonderful epoch in the history of the world which produced a Galileo, a Kepler, a Tycho Brahe, a Descartes, a Desargues, a Pascal, a Cavalieri, a Wallis, a Fermat, a Huygens, a Bacon, a Napier, and a goodly array of lesser lights, to say nothing of a Rembrandt or of a Shakespeare—­there began to appear certain unifying principles connecting the great mass of material dug out by the ancients.  Thus, in 1604 the great astronomer Kepler(4) introduced the notion that parallel lines should be considered as meeting at an infinite distance, and that a parabola is at once the limiting case of an ellipse and of a hyperbola.  He also attributes to the parabola a “blind focus” (caecus focus) at infinity on the axis.