part 18

170 People made their way about on the earth, over mountains, down rivers and across seas long before they had a number system or could make or use a compass. Nobody knows who invented the compass. The Chinese, Arabs, Greeks and Italians, among others, say they did. When people became able to work out the relations of lines and spaces to one another, and could measure distances and angles, the science of geometry, earth measuring, began. People went on then from measuring fields and bits of land, to measure the size of the earth itself.

171 The Greek scientist Eratosthenes (276–194 B.C.) was the first man to work out the size of the earth. He heard that there was a deep well into which on one day of the year the sun's light went all the way down to the bottom. He took the angle of the sun at the same hour from another place 500 miles from the well and worked out by geometry that the earth was about 29,000 miles round. The size of the earth, scientists now tell us, is about 25,000 miles round.

172 Geometry starts with ideas about lines and spaces. Here are two circles and two squares. The circle on the left is inside a square. That is the relation of that circle to that square. The square on the right is inside a circle. That is its relation to the circle. These are facts about the circles and squares on this page. Statements which tally with facts are true. Statements which don't tally with facts are not true. It is untrue that the square on the right is outside the circle. To say it is would be to make a false statement.

173 What is a circle? It is easy to see what it is, but not equally easy to say what it is. Here is a straight line half an inch long. If you could turn the line right round like the hand of a watch, it would have covered a circle. One end of the line would have to keep in the same place while the rest of the line was turning. Here is another line the same length; it is half an inch long. If you could pull it down like a map on a roller a distance equal to its own length (1⁄2 inch) then it would make a square with sides half an inch long. This is not a square though its sides are equal. Why not? Because its angles are not right angles. This is not a square though its angles are right angles. Why not? Because its sides are not all equal.

174 Six thousand years ago in Egypt there were people who saw how to measure their land through their knowledge about squares and triangles. How large is this square? What is its size? Because the square is on squared paper, it is easy to see what its size is. We count the number of small squares in the large square. This number is the area of the square. If the small squares were an inch square, the area of the large square would be sixteen square inches. If they were one foot square, the area of the large square would be sixteen square feet. If they were one yard square, the area of the large square would be sixteen square yards. Whatever the unit of measure used the relation of side to area is the same.

175 People took the first units of long measure from their bodies. The end of a man's thumb is about one inch long. A tall man's foot is about twelve inches or one foot long. A long step is about three feet or one yard long. The simplest way of measuring a short distance is to step it. These units of long measure have been a great help to us. They have made it possible for us to measure and compare lengths and areas and volumes. Measuring lets us build a room the size and shape we want it, for example, twenty feet long, sixteen feet wide and twelve feet high.