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.