In Chapter 1 we studied conics in Euclidean geometry. In the rest of the book we prove a whole range of results about figures such as lines and conics, in geometries other than Euclidean geometry. In the process of doing this, we meet two particular features of our approach to geometry which may be new to you.

The first feature is the use of transformations in geometry to simplify problems and bring out their essential character. You may have met some of these transformations previously in courses on Group Theory or on Linear Algebra.

The second feature arises from the fact that the transformations we introduce form groups. Generally, we restrict our attention to geometry in the plane, ℝ2, but even in this familiar setting there may be more than one group of transformations at our disposal. This leads to the exciting new idea that there are many different geometries!

Each geometry consists of a space, some properties possessed by figures in that space, and a group of transformations of the space that preserve these properties. For example, Euclidean plane geometry uses the space ℝ2, and is concerned with those properties of figures that depend on the notion of distance. The group associated with Euclidean geometry is the group of isometries of the plane.

This idea, that geometry can be thought of in terms of a space and a group acting on it, is called the Kleinian view of geometry, after the 19th-century German mathematician Felix Klein who proposed it first.

Geometry! For over two thousand years it was one of the criteria for recognition as an educated person to be acquainted with the subject of geometry. Euclidean geometry, of course.

In the golden era of Greek civilization around 400 BC, geometry was studied rigorously and put on a firm theoretical basis – for intellectual satisfaction, the intrinsic beauty of many geometrical results, and the utility of the subject. For example, it was written above the door of Plato's Academy ‘Let no-one ignorant of Geometry enter here!’ Indeed, Archimedes is said to have used the reflection properties of a parabola to focus sunlight on the sails of the Roman fleet besieging Syracuse and set them on flame.

For two millennia the children of those families sufficiently well-off to be educated were compelled to have their minds trained in the noble art of rigorous mathematical thinking by the careful study of translations of the work of Euclid. This involved grasping the notions of axioms and postulates, the drawing of suitable construction lines, and the careful deduction of the necessary results from the given facts and the Euclidean axioms – generally in two-dimensional or three-dimensional Euclidean space (which we shall denote by ℝ2 and ℝ3, respectively). Indeed, in the 1700s and 1800s popular publications such as The Lady's and Gentleman's Diary published geometric problems for the consideration of gentlefolk at their leisure.

My approach to the subject of causality has been self-consciously syncretic, drawing from many currents of scholarship. Yet readers may wonder if I have covered this ground in a truly comprehensive fashion. Indeed, several topics of current interest are treated schematically, or not at all, in the foregoing chapters.

In this chapter, which functions as a coda to the third part of the book, I briefly review approaches to causal inference that seem, at least on the face of things, anomalous. This includes causal-process observations, causes-of-effects, necessary/sufficient arguments, and qualitative comparative analysis (QCA). As many of these topics overlap, each of the following sections builds on the previous.