Lancelot Thomas Hogben

Explains mathematics from counting to calculus in the light of man's changing social achievements

Mathematics for the Million

Stefano Zorzi

Form a later date people stopped using nothing but pictures to record how numbers behave. They began to use letters, and compiled dictionaries in which you can find the meaning of the words used. Such diction- aries are called tables. There is nothing more mysterious about understanding the meaning of sin 15 than the meaning of the French word “écoutille." Sin 15° is found by looking up the number given for 15° in the tables of natural sines. This is an essential part of a ship's equipment. In the tables it is given as 0-2588, just as the meaning of écoutille" is given as companion-hatch" in the French dictionary. True, this does not tell you how to use sin l5°. Neither does knowing that écoutille" means “companion-hatch tell you how to use the word écoutille," unless you happen to know what part of a ship a companion-hatch is. In both cases, knowing how to use the word nrt of knowing all about a ship.

If you want to get the best out of this book, do not make the mistake of teachers who multiply our difficulties by extolling the powers of the human mind. Make the fullest use of two helps, one of which has been illus- trated in this chapter. If you are looking for a numerical rule or are trying to apply one to solve a practical-problem, test the rule or its- application by making up a simple example or analogous problem to see whether you get the right answer. This will help you to see whether you understand the rule, which is the same thing as being able to use it, or whether you understand the problem, which is equivalent to knowing what sort of rule to apply. Most tests given at the end of the chapters in this book can be checked. So no answers are given. 

By then Greek geometry had already become the plaything of a pros- perous class with slaves to do its work. Brainwork and handwork reflected the stratification of social classes. At the very point when it had created a new instrunent for man's conquest of the world in which he lives, geometry degenerated into a mere toy. Only when Greek civilization was destroyed were the fruits of Greek geometry harvested.

 The belief that the earth is spherical is of great antiquity among these seafaring peoples. It was a cardinal point in the teaching of the Phoenician Pythagoras. The calendar makers knew it, because they were acCustomed to look at the circular rim of the earth's shadow in a lunar eclipse. The seafaring people did not take long to see that it giyes a simple explanation of something whicb every traveller sees he strains for sight of land, or watches the shore recede.

The Greek geometry which Euclid brought to Alexandria about 300 B.C. had already reached its climax. It embodied all the necessary principles from which the Alexandrians and Arabs derived simple rules of calcutation, and a more economical method of surveying, astronomical and geographical as well as architectural and domestic. The failure of the Greeks to make any noteworthy discoveries by following up the daring and brilliant surmise of Anaxagoras with actual measurements ike those of Aristarchus and Eratosthenes was due to the fact that geometry had become a hobby of intellectuals who had lost interest in the social achievements of the craftsman and the mariner.

When you come to think about it, what could be a more fitting text for such a lesson than the Bridge of Asses? In the fifth proposition of the first book of Euclid the most dramatic dichotomy in human history lies buried. The priestly superstitions which had grown up with the calendar demanded that temple sites should greet the vision of their august and celestial patrons. To make hotels comfortable and luxurious enough for star gods to visit, slaves toiled on burning sand. This brought into being a craft which possessed the secret of making houses fit for men to live in. Greek literature made their knowledge the articulate possession of mankind. The whole after-history of human thought may be viewed as a struggle between these two ways of dealing with the world. One is the way of the practical man who uses knowledge to change the world. The other is the culture of a caste with leisure to contemplate the world.

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