Читать книгу It All Adds Up - Stephen Wilson S., Mickael Launay - Страница 9

Once numbers had been invented, it did not take long for the discipline of mathematics to spread its wings. Various core branches such as arithmetic, logic and algebra gradually sprouted within it, developed to maturity and asserted themselves as disciplines in their own right.

One of these, geometry, rapidly won the popularity stakes and captivated the greatest scholars of antiquity. It was this that singled out the first celebrities of mathematics, such as Thales, Pythagoras and Archimedes, whose names still haunt the pages of our textbooks.

However, before it became a subject for great minds, geometry gained its place on the ground. Its etymology bears witness to this: it is first and foremost the science of the measurement of the Earth, and the first surveyors were hands-on mathematicians. Problems concerning the division of territory were then classics of the craft. How to divide a field into equal parts? How to determine the price of a plot of land from its area? Which of two plots is closer to the river? What route should the future canal follow to make it the shortest possible?

All these questions were paramount in ancient societies where the whole economy still revolved in a vital way around agriculture and hence around the distribution of land. In response to this, geometrical know-how was built up, enriched and transmitted from generation to generation. Anyone equipped with this know-how was certain to hold a central and indisputable place in society.

For these measurement professionals, the rope was often the primary instrument of geometry. In Egypt, ‘ropestretcher’ was a profession in its own right. When the Nile floods led to regular inundations, it was the ropestretchers who were sent for to redefine the boundaries of plots that bordered the river. Using information they recorded about the ground, they planted their stakes, stretched their long ropes across the fields, and then carried out calculations that enabled them to rediscover the boundaries erased by the floodwaters.

They were also the first port of call in constructing buildings, when they took the measurements on the ground and marked the precise location of the building based on architects’ plans. And in the case of a temple or an important monument, it was often the pharaoh in person who symbolically came to stretch the first rope.

It can be said that the rope was the all-purpose tool of geometry. Surveyors used it as a ruler, as compasses, and as a set-square.

To use it as a ruler is straightforward: if you stretch the rope between two fixed points you obtain a straight line. And if you require a graduated ruler, you just tie knots at regular intervals along your rope. For compasses, there is no magic involved either. You simply fix one of the two ends to a stake, stretch the rope and move the other end around the stake. This gives a circle. And if your rope is graduated, you can control the length of the radius exactly.

For the set-square, however, things are slightly more complicated. Let’s look at this particular problem for a few moments: what would you do to draw a right angle? With a bit of research, one can come up with several different methods. If, for example, you draw two circles that intersect each other, then the straight line that joins their centres is perpendicular to the straight line that passes through their two points of intersection. There is your right angle.

From a theoretical point of view, this construction works perfectly, but things are more complicated in practice. Imagine the surveyors, out in the fields, having to lay out two large circles precisely every time they needed to draw a right angle or, more simply, to verify that an angle which had already been constructed was actually a right angle. This was neither fast nor efficient.

The surveyors adopted a different method, which was subtler and more practical: they used their rope directly to form a triangle with a right angle (known as a right-angled triangle). The most famous one is the 3–4–5. If you take a rope divided into twelve intervals by thirteen knots, then you can form a triangle whose sides measure three, four and five intervals, respectively. And, as if by magic, the angle formed by the sides of length 3 and 4 is a perfect right angle.

Four thousand years ago, the Babylonians already had tables of numbers that could be used to construct right-angled triangles. The Plimpton 322 Tablet, which is currently in the collections of Columbia University in New York City, and dates from 1800 BC, contains a table of fifteen triples of such numbers (so-called Pythagorean triples). Apart from the 3-4-5, it has fourteen other triangles, some of which are considerably more complicated, such as the 65-72-97 or even the 1,679-2,400-2,929. Up to a few minor mistakes, such as errors in calculation or transcription, the triangles of the Plimpton Tablet are perfectly exact, and they all have a right angle.

It is difficult to know the precise period from which the Babylonian surveyors began to use their knowledge of right-angled triangles on the ground, but the use of these triangles persisted well beyond the disappearance of the Babylonian civilization. In the Middle Ages, for example, the rope with thirteen knots remained an essential tool for cathedral builders.

On our journey through the history of mathematics, it is by no means uncommon to find certain similar ideas appearing independently at removes of thousands of kilometres and in profoundly different cultural contexts. One such startling coincidence is that during the first millennium BC the Chinese civilization developed a whole mathematical know-how that corresponds remarkably to that of the Babylonian, Egyptian and Greek civilizations of the same period.

This knowledge was amassed over the centuries before being compiled under the Han dynasty, around 2,200 years ago, into one of the world’s first great mathematical works: The Nine Chapters on the Mathematical Art.

The first of these Nine Chapters is entirely devoted to the study of measurements of fields of various shapes. Rectangles, triangles, trapezia, circles, portions of circles and also rings represent geometric figures for which procedures for calculating their areas are described in minute detail. Later in the work, you discover that the ninth and final chapter deals with right-angled triangles. And guess which figure is discussed from the very first sentence of this chapter: the 3-4-5!

Good ideas are like that. They transcend cultural differences and are able to blossom spontaneously wherever human minds are ready to devise and absorb them.