Preface
I have for a long time held the view that whereas university courses in algebra, num-
ber systems and analysis admirably consolidate the corresponding school material,
this is not the case for geometry and trigonometry. These latter topics form an im-
portant core component of mathematics, as they underpin analysis in its manifold
aspects and applications in classical applied mathematics and sundry types of sci-
ence and engineering, and motivate other types of geometry, and topology. Yet they
are not well treated as university topics, being either neglected or spread over a num-
ber of courses, so that typically a student picks up a knowledge of these incidentally
and relies mainly on the earlier intuitive treatment at school.
Clearly the treatment of geometry has seriously declined over the last fifty years,
in terms of both quantity and quality. Lecturers and authors are faced with the ques-
tion of what, if anything, should be done to try to restore it to a position of some
substance. Bemoaning its fate is not enough, and surely authors especially should
ponder what kinds of approach are likely to prove productive.
Pure or synthetic geometry was the first mathematical topic in the field and for a
very long time the best established. It was natural for authors to cover as much ground
as was feasible, and ultimately there was a large bulk of basic and further geometry.
That was understandable in its time but perhaps a different overall strategy is now
needed.
Synthetic geometry seems very difficult. In it we do not have the great benefit of
symbolic manipulations. It is very taxing to justify diagrams and to make sure of cov-
ering all cases. From the very richness of its results, it is difficult to plan a productive
approach to a new problem. In the proofs that have come down to us, extra points and
segments frequently need to be added to the configuration. It is true that, as in any
approach, there are some results which are handled very effectively and elegantly by
synthetic methods, but that is certainly not the whole story. On the other hand, what
is undeniable is that synthetic geometry really deals with geometry, and it forces at-
tention to, and clarity in, geometrical concepts. It encourages the careful layout of
sequential proof. Above all, it has a great advantage in its intuitive visualisation and
concreteness.
The plan of this book is to have a basic layer of synthetic geometry, essentially five
chapters in all, because of its advantages, and thereafter to diversify as much as pos-
sible to other techniques and approaches because of its difficulties. More than that,
we assume strong axioms (on distance and angle-measure) so as to have an efficient
approach from the start. The other approaches that we have in mind are the use of co-
ordinates, trigonometry, position-vectors and complex numbers. Our emphasis is on
clarity of concepts, proof and systematic and complete development of material. The
synthetic geometry that we need is what is sufficient to start coordinate geometry and
trigonometry, and that takes us as far as the ratio results for triangles and Pythagoras’
theorem. In all, a considerable portion of traditional ground involving straight-lines
and circles is covered. The overall approach is innovative as is the detail on trigonom-
etry in Chapter 9 and on what are termed ‘mobile coordinates’ in Chapter 11. Some