Математический факультет

Text 1. P o s t u l a t e s

We have seen how from our everyday experience it is natural to imagine the existence of things called points and lines. Even though one cannot “see” a point or a line, the study of such imaginary objects turns out to be practically useful as well as interesting.

Because the objects of study in geometry are inventions of the mind, we must state very carefully the properties we wish them to possess. Then if we start with enough information about points and lines, other properties of points and lines can be proved. Many of these provable properties will be surprising, and not all obvious at the beginning. It is, of course, impossible to prove anything about points and lines unless we agree in advance about some properties that they are to have. These agreed-upon properties, or assumed properties, are called postulates, or axioms. The postulates should be simple enough to seem almost obvious and yet must be sufficient for us to prove geometric facts that seem to be true in our everyday life.

A different method of procedure would be to list all the known geometric facts and assume that they are true. Then when we had memorized all these facts, our study of geometry would be completed. There are at least three objections to such a plan. In the first place, we might not have listed all the facts needed and we would have no experience in finding new ones. In the second place, it would be a terrible strain upon the memory. To avoid memorizing everything, we need to study how the different facts are related and how the more complicated ones depend upon other simpler ones. When we prove theorems from postulates or other theorems, we are studying the way the geometric facts are related to each other; this helps us to remember them. In the third place, we would lose the great amount of pleasure that we can get out of proving from very simple and obvious assumptions that certain other things are true. It is because of this pleasure that geometry has been so attractive to men since the time of Euclid. The fact that all geometry can be established from a small list of postulates gives to geometry a kind of beauty that has appealed to men as different as Abraham Lincoln and the Greek philosopher Plato.

Q u e s t i o n s :

1) What are the objects of study in geometry?

2) What are the notions of postulates and axioms used for?

Text 2. T h o m a s E l v a E d i s o n

Edison was a thoughtful little boy. He was very inquisitive and always wanted to know how to do things. He was not very strong, and went to school when he was quite a big child. But his teacher thought him very stupid because he asked so many questions. So his mother, who was a teacher, took him away from school at the end of two months and taught him at home. With so kind a teacher, he made progress; and above all, he learned to think. His mother had some good books and among them an encyclopedia. It was probably from the encyclopedia that he first took an interest in chemistry. He liked to make experiments, so he bought some books, and made a little laboratory in the cellar of his home.

When he was twelve years old, he started to earn his living and became a newsboy on the train, which ran from Fort Huron to Detroit. There was a corner in the baggage car where he kept his stocks of newspapers, magazines and candy. To this corner he moved his little laboratory and library of chemical books, and when he was not busy, went on with his experiments. All went well for two or three years. But when he was in his sixteenth year, one day a phosphorus bottle broke on the floor. It set fire to the baggage car, and the conductor not only put the boy off the train, but soundly boxed his ear. That was the most unfortunate part of the accident, for as a result Edison gradually lost his hearing, and became almost deaf.

Once he was standing on the platform of the station in Michigan, watching a coming train, when he saw the station agent’s little boy on the track right in front of the coming engine. Another moment and the child would have been crushed; but Edison sprang to the track, seized the little one in his arms, and rolled with him to one side, just in time to escape the wheels. To show his gratitude the baby’s father offered to teach telegraphy to Edison. Working at telegraphy he at the same time spent all the spare moments in the study of chemistry and electricity. Experimenting he improved telegraph apparatus. About the same time Edison made an improvement in the transmitter of the telephone which made it easier for the waves to travel, and improved the usefulness of the telephone very much. It was just about the same time that he invented the phonograph. This is the parent idea of the gramophone, dictaphone and other instruments, but these inventions are only a small part of the work of this wonderful man.

Q u e s t i o n s :

1) How did Edison study at school?

2) Where did he work when he was 12?

3) What accident happened with Edison at the age of 16?

4) What did Edison invent?