Note by the Editor of the original
edition, Dornach, 1926
January 1, 1921
As you read
in Chapter 2, geometry began long before the Greeks became
interested in it. The “earth measurement” of the
Egyptians is an example of how geometry was used in the
earliest days of mathematics. It was used, in short, for
measuring things. The Greeks, on the other hand, liked
geometry for its own sake. They liked to draw triangles and
circles and other shapes and see what rules they could
discover for problems like finding the circumference of a
circle and the amount of space occupied by a circle, or for
working out the unknown dimensions of a triangle from known
dimensions such as the length of sides and the size of
angles, as is shown by the geometric construction on the
left.
In doing such
things the Greeks brought to geometry three new ideas that
were of great importance for the future of mathematics. Those
ideas were deduction, proof, and abstraction.
Deduction
involves using known facts, or at least facts on which we
agree, to reach conclusions that necessarily follow from
those facts. For example, let us take as the known facts, or
premises, the statements that all apples are red, and that
you are holding an apple in your hand. It necessarily follows
from the premises that the object in your hand is red. It
does not make any difference that there are also green apples
and yellow apples; the point is that for the premises that
are given, the conclusion is the correct one. Deduction, in
other words, is a reasoning process throughout which you can
build on what you know and thereby expand your knowledge.
