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.

早在有数字系统或制造或使用指南针之前，人们就已经在地球上行走，翻山越岭，顺流而下，横渡海洋。 没有人知道是谁发明了指南针。 中国人、阿拉伯人、希腊人和意大利人，以及其他国家的人，都说他们做到了。 当人们能够计算出线和空间之间的关系，并且能够测量距离和角度时，几何科学——测量地球——开始了。 那时，人们不再测量田地和土地，而是测量地球本身的大小。

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.

希腊科学家埃拉托色尼（公元前276-194）是第一个计算出地球大小的人。 他听说有一口深井，一年中有一天，太阳的光一直照到井底。 他从距离井500英里的另一个地方取同一时刻太阳的角度，用几何学计算出地球的周长约为29,000英里。 科学家们现在告诉我们，地球的大小大约是2.5万英里。

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.

172几何从线和空间的概念开始。 这里有两个圆形和两个正方形。 左边的圆圈在一个正方形里面。 这就是圆和正方形的关系。 右边的正方形在一个圆里面。 这是它与圆的关系。 这些是关于这一页上的圆圈和正方形的事实。 符合事实的陈述是真实的。 与事实不符的陈述是不正确的。 右边的正方形在圆的外面，这是不正确的。 说它是会做一个错误的陈述。

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.

什么是圆？看到它是什么很容易，但说出它是什么并不容易。 这是一条半英寸长的直线。 如果你能把这条线像表针一样转过来，它就能绕一个圈了。 当线的另一端转弯时，线的一端必须保持在同一个地方。 这是另一条同样长度的线；它有半英寸长。 如果你能像把地图放在滚轴上一样把它往下拉，拉的距离等于它本身的长度（1 / 2英寸），那么它就会变成一个边长为半英寸的正方形。 这不是正方形，虽然它的边是相等的。 为什么不呢?因为它的角不是直角。 这不是一个正方形，虽然它的两个角都是直角。 为什么不呢?因为它的两边并不都相等。

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.

17.4六千年前，在埃及，人们通过他们对正方形和三角形的知识，知道了如何测量他们的土地。 这个正方形有多大？它的大小是多少？ 因为正方形是在正方形纸上，所以很容易看出它的大小。 我们数一数大正方形中小正方形的数目。 这个数字是正方形的面积。 如果小正方形是一英寸的正方形，那么大正方形的面积就是十六平方英寸。 如果它们是一平方英尺，那么大广场的面积将是16平方英尺。 如果它们是1码见方，那么大广场的面积将是16平方码。 无论用什么计量单位，边长与面积的关系都是一样的。

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.

175 .人们从他们的身体中取出了第一个单位的长度。 男人的拇指末端大约有一英寸长。 一个高个子男人的脚大约有12英寸或1英尺长。 一个长台阶大约是三英尺或一码长。 测量短距离最简单的方法是步进。 这些长度量单位对我们帮助很大。 它们使我们能够测量和比较长度、面积和体积。 测量可以让我们建造一个我们想要的大小和形状的房间，例如，20英尺长，16英尺宽，12英尺高。