Page： 171

The Greek scientist Eratosthenes 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.

Page： 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.

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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 inch long.

If you could pull it down like a map on a roller a distance equal to its own length 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.

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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.

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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.

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Sometimes a family's fields were not square.

some of them were like this:

or like this.

People walked across their fields; they planted them and took in the grain.

They knew how much land they had from working them before they could measure them.

They saw that a field like this

was the same size, though not the same shape, as a field like this

before they knew that they could measure how long and how wide a field was, and then get the area by taking one measure times the other.

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They saw that they could get half a field in this way...

or in this way,

before they knew how to measure rectangles or triangles.

Can you see whether these two fields have the same area?

Put in lines to prove that they are or are not equal in area.

The answer is at the bottom of Page： 178.

Page： 178

Here is a right angled triangle.

The two shorter sides are three and four units long.

How many units long is the longest side?

Can you tell without measuring?

How?

About 2500 years ago a great Greek,Pythagoras, proved that the square on the longest side of any right angled triangle is equal to the squares on the other two sides added together.

We can use his discovery to get our answer.

We multiply the length of each of the two shorter sides by itself.

We add the answers together.

Then we find a number which,multiplied by itself,gives us this number.

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Here is the answer:

When we multiply a number by itself we "square" it.

Any number is the square root of its square.

5 is the square root of 25.

Page： 180

It was not until many centuries later that people put this knowledge of geometry to wide use.

The development of science had to wait until the days of Galileo and Newton.

In the last three centuries our ways of living have been and are being deeply changed by science.

These changes can be compared only with three or four great earlier steps in the history of human development.

These are the birth of language,the use of fire and farming, and the invention of writing.

Here is a horse walking round and round the mouth of a well.

He is pulling on a strong stick of wood which is kept turning by his motion.

This moves a chain with buckets on it.

The motion of the chain carries buckets full of water up and takes empt buckets down.

The horse has a cloth over his eyes to keep him from seeing that he is walking all the time in a circle.

Would he stop if he knew he was going round in circles?