1
Preliminaries
1.1 Historicalnote.............................
This is one in a long line of textbooks on geometry. While all civilisations seem to
have had some mathematical concepts, the most significant very old ones historically
were the linked ones of Sumer, Akkad and Babylon, largely in the same region in
what is now southern Iraq, and the separate one of Egypt. These are the ones which
have left substantial traces of their mathematics, which was largely arithmetic, and
geometrical shapes and measurement.
The outstanding contribution to mathematics was in Greece about 600 B.C.–
200 B.C. The earlier mathematics conveyed techniques by means of examples, but
the Greeks stated general properties of the mathematics they were doing, and or-
ganised proof of later properties from ones taken as basic. There was astonishing
progress in three centuries and the fruit of that was written up in Euclid’sThe Ele-
ments, c. 300B.C. He worked in Alexandria in Egypt, which country had come into
the Greek sphere of influence in the previous century.
Euclid’sThe Elementsis one of the most famous books in the world, certainly
the most famous on mathematics. But it was influential widely outside mathematics
too, as it was greatly admired for its logical development. It is the oldest writing on
geometry of which we have copies by descent, and it lasted as a textbook until after
1890, although it must be admitted that in lots of places and for long periods not very
many people were studying mathematics. It should probably be in the Guinness Book
of Records as the longest lasting textbook in history.
The Elementsshaped the treatment of geometry for 2,000 years. Its style would
be unfamiliar to us today, as apart from using letters to identify points and hence
line-segments, angles and other figures in diagrams, it consisted totally of words.
Thus it did not use symbols as we do. It had algebra different from ours in that it
said things in words written out in full. Full symbolic algebra as we know it was not
perfected until about 1600 A.D. in France, by Vieta and later Descartes. Another very
significant feature ofThe Elementswas that it did not have numbers ready-made, and
used distance or length, angle-measure and area as separate quantities, although links
Geometry with Trigonometry
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