Sec. 1.4 Our approach 9
1.4 Ourapproach
1.4.1 Typeofcourse.............................
Very scholarly courses in geometry assume as little as possible, and as a result are
long and difficult. Shorter and easier courses have more or stronger assumptions, and
correspondingly less to prove. What is difficult in a thorough course of geometry is
not the detail of proof usually included, but rather is, first of all, locational viz. to
prove that points are where diagrams suggest they are, that is to verify the diagrams,
and secondly to be sure of covering all cases.
In particular, the type of approach which assumes that distance, angle-measure
and area are different ‘quantities’ leads to a very long and difficult treatment of ge-
ometry. To make things much easier and shorter, we shall suppose that we know what
real numbers are, and deal with distance/length and angle-measure as basic concepts
given in terms of real numbers, and develop their properties. Moreover, we shallde-
finearea in terms of lengths.
What we provide, in fact, is a combination of Euclid’s original course and a modi-
fication of an alternative treatment due to the American mathematician G.D. Birkhoff
in 1932.
1.4.2 Needforpreparation..........................
What this course aims to do is to revise and extend the geometry and trigonometry
that has been done at school. It gives a careful, thorough and logical account of famil-
iar geometry and trigonometry. At school, a complete, logically adequate treatment
of geometry is out of the question. It would be too difficult and too long, unattractive
and not conducive to learning geometry; it would tend rather to put pupils off.
Thus this is not a first course in geometry. It is aimed at third level students, who
should have encountered the basic concepts at secondary, or even primary, school.
It starts geometry and trigonometry from scratch, and thus is self-contained to that
extent.
But it is demanding because of a sustained commitment to deductive reasoning.
In preparation the reader is strongly urged to start by revising the geometry and
trigonometry which was done at school, at least browsing through the material. It
would also be a good idea to read in some other books some descriptive material on
geometry, such as the small amount in Ledermann and Vajda [10, pages 1 – 26], or the
large amount in Wheeler and Wheeler [13, Chapters 11 – 15]. Similarly trigonometry
and vectors can be revised from McGregor, Nimmo and Stothers [11, pages 99 – 123,
279 – 331].
It would moreover be helpful to practise geometry by computer, e.g. by using soft-
ware systems such asThe Geometer’s SketchpadorCabri-Géomètre. Material which
can be found in elementary books should be gone over, and also a look forwards
could be had to the results in this book.