The Infinite in the Finite

By Alistair Macintosh Wilson | Go to book overview

12
PROPORTION

THE GEOMETRICAL SOLUTION OF AHA PROBLEMS

Although the Greeks far surpassed their teachers, the Egyptians, in the science of geometry, they never succeeded in equalling the ancient civilizations in the treatment of aha problems. The Greeks tried to solve aha problems by using their powerful methods of geometrical reasoning. But geometrical methods are not really well suited for these problems, and they were not successful. In this chapter we shall describe what they did achieve, since this will form the basis of our later discussions of geometry.

The methods of solving aha problems described in the first part of this book passed via the Greeks to the Hindu mathematicians of India, and from them to the Islamic scholars at Baghdad, where they were finally drawn together into a new science. We shall describe how this happened in Chapters 17 and 18.

The Greek geometrical treatment of aha problems is described in the fifth and sixth books of Euclid Elements. Euclid began in a natural way by describing the theory of ratio and proportion.

THE THEORY OF PROPORTION

At the time of Euclid, the Greeks were familiar with three different kinds of proportion. The first they called arithmetical proportion. A set of line segments are in arithmetical proportion when their lengths increase by an equal amount as we go from one to another (Fig. 12.la).

The fifth book of the Elements is concerned with a different kind of proportion, which we now know as geometrical proportion. This is the kind of proportion we meet with in similar triangles. Consider Fig. 12.1b, in which lines BD and CE are parallel. We see that triangles ADB and AEC are similar, so that

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