“God ever geometrizes,” the Greek scholar Plato said. Plato was probably thinking about the shapes of fruit, of leaves, of the moon, of many things in nature that have interesting and beautiful forms. In any case, his remark tells something about the interest the Greeks had in geometry. It was mainly through geometry that the Greeks changed the face of mathematics in a period of a few hundred years before the birth of Christ. Greek mathematics did include some other things-arithmetic, by which they meant the study of numbers; logistics which was what we would now call arithmetic, or calculation for practical purposes; and a kind of algebra, with which they described number relationships in words. Today we use letters and other symbols. But above all, it included geometry, which was where their real interest lay. (excerpt fromrom first lecture given 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.