In Chapter 2, we described how the ancient Egyptians were forced to resurvey their fields every year by the annual flooding of the Nile. We recall that the Pharaoh's engineers carried out this process by breaking the fields into triangular plots. The study of triangles was therefore called by the Greeks 'geometria' from their words geos (Earth, land) and metres (measure).
The facts of geometry were, as described earlier, first obtained by measurement. We measure that the angles of a triangle add up roughly to 180°, that the area of a triangle is equal to half its base times its height, and so on. Measurement depends finally upon the eye. But the eye can be deceived.
Consider the following example. We have a square 8 units by 8 units (Fig. 9.1a). We divide three sides of this square into the ratio 3 to 5. Joining the division points as shown, we now take the square apart and reassemble it into a rectangle (Fig. 9.1b).
Looking at this rectangle, we see that one of its sides has length 5, the other has length 8 + 5 = 13 units. The area of our rectangle is therefore 13 × 5 = 65 units. But the area of our original square was 8 × 8 = 64 units! Where has the additional unit of area come from?
Make an 8 × 8 square, cut it up and stick it together into a rectangle. You will now see that in fact the pieces do not fit together into a rectangle. There is a hole in the middle. This hole has area one unit, and our problem is solved (Fig. 9.2a).
Next take a square 21 × 21 units, and divide its sides in the ratio 8 to 13. Reassembling the pieces, we again seem to be able to form a rectangle giving an area (21 + 13) × 13 = 442 = (21)2 + 1, one unit more than the original square (Fig. 9.2b).
Where did we pull the numbers 8 and 21 and the divisions 3 : 5 and 8 : 13 from?
Consider the sequence of numbers 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584, ..., in which each new number is obtained by adding its two predecessors.