Projective geometry is most often introduced as a kind of appendix to Euclidean geometry, involving the addition of a line at infinity and other modifications so that (among other things) all pairs of lines meet in exactly one point, and all statements about lines and points are equivalent to dual statements about points and lines. But it constitutes a system in its own right, with its own axioms and logic, and one that is in many ways simpler and more elegant: as Coxeter puts it, "its primitive concepts are so simple that a self-contained account can be reasonably entertaining, whereas the foundations of Euclidean geometry are inevitably tedious".

Coxeter's approach in *Projective Geometry* is elementary, presupposing
only basic geometry and simple algebra and arithmetic, and largely
restricting itself to plane geometry, but it does assume a general
mathematical competence. He is rigorous without being too formal, with
an strong emphasis on geometric intuition — coordinates are introduced
only in the final chapter — and introduces new concepts progressively.

He begins with projectivities and perspectivities and triangles and
quadrangles ("if 4 points in a plane are joined in pairs by 6 distinct
lines, they are called the vertices of a *complete quadrangle*,
and the lines are its six sides") and quadrilaterals ("if 4 lines
in a plane meet by pairs ...") and the key Desargues Theorem ("if
two triangles are perspective from a point they are perspective from
a line"). The exposition proceeds to cover harmonic sets and nets, the
notion of duality, and the fundamental theorem of projective geometry
("a projectivity is determined when three collinear points and the
corresponding three collinear points are given") and Pappus's Theorem.

Then we are introduced to elliptic, parabolic and harmonic projectivities and involutions; collineations (mappings of the plane onto itself) projective and prospective and involutory (with period two); axes and centers and elations and homologies; and projective correlations and polarities (projective correlations of period 2). This leads to the notion of a conic, defined as the locus of self-conjugate points (or the envelope of self-conjugate lines) of a hyperbolic polarity. Coxeter gives constructions of conics touching two lines at given points, touching five lines, through five given points, and so forth.

Three final chapters explore additional topics. There are projective
geometries with only a finite number of points, and Coxeter takes a
detailed look at PG(2,5), the projective geometry in two dimensions with
six points on every line and six lines passing through each point (and
with 31 points and 31 lines in total). He explains how affine geometry
can be obtained by removing a plane from projective space, or by using
affine concepts such a *bundle* (the set of lines and planes through
a point) and an *axial pencil* (the set of planes through a line).
And in a long final chapter he introduces the use of coordinates and
looks at how they can be used to give a different perspective on the
preceding concepts and theorems.

*Projective Geometry* is a really lovely presentation, rigorous without
being pedantically formal, and prioritising motivation and pedagogy.
There's a good selection of exercises, with solutions (though in many
cases those are just substantial hints). And two pages of "Historical
Remarks" in the introduction are supplemented by quotes at the beginning
of each chapter to give a broad overview of the history of projective
geometry. I found this more than just "reasonably entertaining": both
style and subject were congenial, so I bought additional books both by
Coxeter (*Geometry Revisited*) and on projective geometry.

Aimed at higher undergraduate students, Rey Casse's *Projective Geometry:
An Introduction* takes quite a different approach. Casse deploys
coordinates from the beginning, and gives the finite geometries as much
prominence as the real projective plane. He doesn't spurn geometrical
intuition, but leaves it a secondary role, often providing small diagrams
as a marginal adjunct to a proof or derivation. The algebraic approach
offers far more generality and power, at the expense of some intuition
and motivation.

He begins with a refresher in some "assumed knowledge" of fields and linear algebra, and also assumes some group theory; only very elementary coordinate geometry is assumed. A very brief introduction to projective geometry, introducing Desargues Theorem, the Pappus configuration, the extended Euclidean plane and duality, is then followed by an abstract and quite general introduction to projective spaces and axiomatic geometry, centering on the dimension axiom.

There's a long chapter on field planes and PG(r,F), focusing on PG(2,F) and covering, among other things, homographies and collineations, the fundamental theorem of projective geometry, cross-ratios and the harmonic property, and correlations and polarities. Casse then looks at what happens if we take projective planes with various geometric structures and construct coordinates for them. This leads us to loops, nearfields, quasifields, alternative division rings, and so forth. And he constructs a variety of non-Desarguesian projective planes, using the structure of their collineation groups.

Conics are defined using Taylor's Theorem, and conics over different
kinds of fields with different characteristics are explored. And a
final chapter hints at higher dimensions but concentrates on quadrics in
PG(3, *F*), with a look at the twisted cubic.

Quite extensive exercises (and a few worked examples) follow each subsection, but there are no solutions.

Coxeter's 1963 preface hinted at the possibility of projective geometry reaching its way into secondary schools, but that seems unlikely now, when geometry of any kind barely clings to a place in syllabuses (and it wouldn't really make sense to teach it before Euclidean geometry). Projective geometry might feel like a digression for undergraduates, but it would be an entertaining one, perhaps a kind of "antidote" for those turned off by arid teaching of geometry at school, and it connects with other areas of mathematics in fascinating ways. Here I think Coxeter's approach is likely to have the most appeal, but students coming to the subject later, with more sophisticated algebra, may prefer Casse. For me, reading one after the other worked really nicely.

April 2017

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