2 Preliminaries Ch. 1
between them were worked out.
Among prominent countries,The Elementslasted longest in its original style in
the U.K., until about 1890. They had started chipping away at it in France in the
16th century, beginning with Petrus Ramus 1515-1572. There is a very readable ac-
count of the changes which were made in France in Cajori [3, pages 275 – 289].
These changes mainly involved disimprovements logically; authors brought in con-
cepts which are visually obvious, but they did not provide an account of the properties
of these concepts. Authors in France, and subsequently elsewhere, started using our
algebra to handle the quantities and this was a major source of advance. One very
prominent textbook of this type wasElements of Geometryby Legendre, (first edi-
tion 1794), which was very influential on the continent of Europe and in the U.S.A.
All in all, these developments in France shook things up considerably, and that was
probably necessary before a big change could be made.
AlthoughThe Elementswas admired widely and for a long time for its logic,
there were in fact logical gaps in it. This was known to the leading mathematicians
for quite a while, but it was not until the period 1880-1900 that this geometry was put
on what is now accepted as an adequate logical foundation. Another famous book
Foundations of Geometryby Hilbert (1899) was the most prominent in doing this.
The logical completion made the material very long and difficult, and this type of
treatment has not filtered down to school-level at all, or even to university undergrad-
uate level except for advanced specialised options.
Another sea-change was started in 1932 by G.D. Birkhoff; instead of building
up the real number system via geometrical quantities, he assumed a knowledge of
numbers and used that from the start in geometry; this appeared in his ‘ruler postulate’
and ‘protractor postulate’. His approach allowed for a much shorter, easier and more
efficient treatment of geometry.
In the 1960’s there was the world-wide shake-up of the ‘New Mathematics’, and
since then there are several quite different approaches to geometry available. In this
Chapter 1 we do our best to provide a helpful introduction and context, and suggest
a re-familiarisation with the geometrical knowledge already acquired.
Logically organised geometry dates from c. 600-300 B.C. in Greece;
by c. 350B.C. there was already a history of geometry by Eudemus. From the same
period and earlier, date positive integers and the treatment of positive fractions via
ratios. The major mathematical topics date from different periods: geometry as just
indicated; full algebra from c. 1600 A.D.; full coordinate geometry from c. 1630A.D.;
full numbers (negative, rational, decimals) from c. 1600 A.D.; complex numbers from
c. 1800 A.D.; calculus from c. 1675 A.D.; trigonometry from c. 200 B.C., although
circles of fixed length of radius were used until c. 1700 A.D. when ratios were intro-
duced.
There is an account of the history of geometry of moderate length by H. Eves in
[2, pages 165-192]
It should be clear from this history that the Greek contribution to geometry greatly
influenced all later mathematics. It was transmitted to us via the Latin language,
and we have included a Glossary on pp. xix-xx showing the Greek or Latin roots of