Читать книгу It All Adds Up: The Story of People and Mathematics - Stephen Wilson S., Mickael Launay - Страница 10

Questions about fields, about architecture, or more generally about land planning led the scholars of antiquity to set themselves a great diversity of geometric problems of which the following are examples.

The following statement, taken from the Babylonian Tablet BM 85200, shows that the Babylonians were not content with plane geometry, but also thought in the space dimension.

An excavation. So much as the length, that is the depth. 1 the earth I have torn out. My ground and the earth I have accumulated, 1´10. Length and width: '50. Length and width, what?¹

As you will have gathered, the style of the mathematicians of Babylon was telegraphic in nature. If we expand it further, this same statement might look as follows:

The depth of an excavation is twelve times greater than its length.² If i dig further so that my excavation has one more unit of depth, then its volume will be equal to 7/6. If i add the length to the width i obtain 5/6.³ What are the dimensions of the excavation?

This problem was accompanied by the detailed method for solving it, ending with the solution: the length is ½, the width ⅓ and the depth 6.

Let’s now take a short trip down the Nile. As a matter of course, in the case of the Egyptians, we find problems about pyramids. The following statement is an extract from a famous papyrus, the Rhind papyrus, copied by the scribe Ahmes, dating from the first half of the sixteenth century BC.

A pyramid has a base side of 140 cubits and an inclination ⁴ of 5 palms and 1 digit, what is its altitude?’

The cubit, the palm and the digit were units of measurement of 52.5 centimetres, 7.5 centimetres and 1.88 centimetres, respectively. Ahmes also gave the solution: 93 + ⅓ cubits. In this same papyrus, the scribe also took on the geometry of the circle.

Example of the calculation of a round field with a diameter of 9 khet. What is the value of its area?

The khet is also a unit of measurement representing approximately 52.5 metres. To solve this problem, Ahmes stated that the area of this circular field is equal to that of a square field with a side of length 8 khet. The comparison is extremely useful, for it is much easier to calculate the area of a square than that of a circle. His solution was 8 × 8 = 64. However, the mathematicians who succeeded Ahmes came to discover that his result was not quite exact. The areas of the circle and the square do not quite agree. Since then, many people have attempted to answer the question: how do you construct a square with an area equal to that of a given circle? Many have worn themselves out in vain in this pursuit, but for a reason. Without knowing it, Ahmes was one of the first to tackle what would become one of history’s classic mathematical conundrums: the squaring of the circle.

In China too, people sought to calculate the area of circular fields. The following problem is taken from the first of the Nine Chapters.

Suppose one has a circular field with circumference 30 bu and diameter 10 bu. The question is how big is the field?⁵

Here, a bu is equal to about 1.4 metres. And, as in Egypt, the mathematicians kept tripping over the rug with this figure. The original statement was already known to be false, since a circle of diameter 10 has a circumference slightly greater than 30. However, that did not prevent Chinese scholars from putting an approximate value on the area (75 bu), or from complicating the task further for themselves by continuing on to questions about circular rings!

Suppose one has a field in the shape of a ring with internal circumference 92 bu, external circumference 122 bu, and transverse diameter 5 bu. The question is: how big is the field?

It seems likely that there were never any ring-shaped fields in ancient China – these latter problems suggest that the scholars of the Central Kingdom were into geometry and raised these questions as purely theoretical challenges. Research into ever more improbable and weird-looking geometric figures in order to study and understand them remains a favourite pastime of our mathematicians to this day.

Among the ranks of professional geometers, one must also include the Bematists. While it was the job of surveyors and other ropestretchers to measure fields and buildings, the Bematists had a much grander view of things. In Greece, it was the job of these men to measure long distances by counting their steps.

And sometimes their work could take them a long way from home. For example, in the fourth century BC, Alexander the Great took several Bematists with him on his campaign in Asia, which led him as far as the boundaries of what is now India. These walking geometers thus had to measure routes of several thousand kilometres in length.

Step back a little and imagine for a moment the strange spectacle of these men walking in quick time, traversing the immense landscapes of the Middle East. See them cross the plateaux of Upper Mesopotamia; walk through the arid yellow settings of the Sinai Peninsula to reach the fertile banks of the Nile Valley; then turn back, setting off to brave the mountainous massifs of the Persian Empire and the deserts of what is now Afghanistan. Can you see them, calmly walking on and on, with a dry and monotonous rhythm, and passing through the foothills of the gigantic mountains of the Hindu Kush to return via the shores of the Indian Ocean? And all the while they counted their paces tirelessly.

The image is captivating, and the outrageous challenge of their undertaking seems insane. And yet, their results were remarkably accurate: less than 5 per cent difference on average between their measurements and the true distances we know today. Alexander’s Bematists thus made it possible to describe the geography of his kingdom in a way that had never before been achieved for so vast a region.

Two centuries later, in Egypt, a scholar of Greek origin named Eratosthenes conceived an even larger project – that of measuring nothing less than the circumference of the Earth. Of course, there was no question of sending poor Bematists on a tour of the planet. However, based on skilful observations of the difference in the inclination of the sun’s rays between the towns of Syene (the present-day Aswan) and Alexandria, Eratosthenes calculated that the distance between the two towns must represent one fiftieth of the total circumference of the Earth.

Naturally, he then called upon the Bematists to undertake the measurement. Unlike their Greek counterparts, the Egyptian Bematists did not count their own steps, but those of a camel they took with them. The animal is renowned for the regularity of its walking. After long days of travelling along the Nile, the verdict was issued: the two towns were 5,000 stadia apart and the girth of our planet was therefore 250,000 stadia, or 39,375 kilometres. Once again, the result was astonishingly accurate, as we know today that this circumference measures exactly 40,008 kilometres. The error was less than 2 per cent.

Perhaps more than any other ancient people, the Greeks gave geometry pride of place in the heart of their culture. It was recognized for its rigour and its ability to form the mind. For Plato, it was an obligatory rite of passage for any budding philosopher, and legend has it that the slogan ‘Let no one ignorant of geometry enter’ was engraved at the entrance to his academy.

Geometry was so much in vogue that it eventually spilled over into other disciplines. The arithmetical properties of numbers were thus interpreted in geometric language. See, for example this definition by Euclid extracted from the seventh book of his Elements of Mathematics dating from the third century BC:

And when two numbers multiplying one another make some other number then the number so created is called plane, and its sides are the numbers which multiply one another.

If I take the product 5 × 3, the numbers 5 and 3 are thus called, according to Euclid, the ‘sides’ of the multiplication. Why is this? It is simply because a multiplication can be represented as the area of a rectangle. If the latter has a width equal to 3 and a length of 5, its area is 5 × 3. The numbers 3 and 5 are precisely the sides of the rectangle. The result of the multiplication, 15, is for its part called the ‘plane’, since it corresponds geometrically to a surface.

Similar constructions relate to other geometric figures. For example, a number is said to be triangular if it can be represented in the form of a triangle. The first triangular numbers are 1, 3, 6 and 10.

The last of the above triangles with ten points is nothing other than the famous tetractys that Pythagoras and his disciples employed as the symbol of the harmony of the cosmos. On the same principle, we also find the square numbers whose first representatives are 1, 4, 9 and 16.

And we could of course continue for a long time like that with all kinds of figures. Thus, the geometric representation of numbers makes it possible to render visual and evident properties that are otherwise seemingly incomprehensible.

By way of example, have you ever tried adding the odd numbers successively, one after another: 1 + 3 + 5 + 7 + 9 + 11 + …? No? Well, something completely surprising happens. Look:

Can you see something special about the numbers that emerge? In the order in which they occur: 1, 4, 9, 16 … These are the square numbers!

And you can continue as long as you wish; this rule will never be broken. Add the first ten odd numbers from 1 to 19, if you are brave enough, and you will reach 100, which is the tenth square:

Isn’t that surprising? But why? What kind of miracle makes this property always hold? Of course, it would be possible to give a numerical proof, but there is a much simpler way. Using the geometrical representation, you just have to slice up the square numbers and the explanation becomes plain to see.

Each slice adds an odd number of balls, and increases the side of the square by one unit. This completes the simple and lucid proof.

In the kingdom of mathematics, geometry was the queen, and no assertion could be verified without coming under its scrutiny. Its reign lasted far beyond antiquity and the Greeks. It was almost two thousand years before the scholars of the Renaissance launched a vast movement to modernize mathematics which would dethrone geometry in favour of a completely new language: that of algebra.