Fermat's Last Theorem Read Online Free PDF

bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist, and so on. However, there is a natural process which will curtail the chaos: increasing loopiness will result in rivers doubling back on themselves and effectively short-circuiting. The river will become straighter and the loop will be left to one side forming an ox-bow lake. The balance between these two opposing factors leads to an average ratio of π between the actual length and the direct distance between source and mouth. The ratio of π is most commonly found for rivers flowing across very gently sloping plains, such as those found in Brazil or the Siberian tundra.
    Pythagoras realised that numbers were hidden in everything, from the harmonies of music to the orbits of planets, and this led him to proclaim that ‘Everything is Number’. By exploring the meaning of mathematics, Pythagoras was developing the language which would enable him and others to describe the nature of the universe. Henceforth each breakthrough in mathematics would give scientists the vocabulary they needed to better explain the phenomena around them. In fact developments in mathematics would inspire revolutions in science.
    As well as discovering the law of gravity, Isaac Newton was a powerful mathematician. His greatest contribution to mathematics was his development of calculus, and in later years physicists would use the language of calculus to better describe the laws of gravity and to solve gravitational problems. Newton’s classical theory of gravity survived intact for centuries until it was superseded by Albert Einstein’s general theory of relativity, which developed a more detailed and alternative explanation of gravity. Einstein’s own ideas were only possible because of new mathematical concepts which provided him with a more sophisticatedlanguage for his more complex scientific ideas. Today the interpretation of gravity is once again being influenced by breakthroughs in mathematics. The very latest quantum theories of gravity are tied to the development of mathematical strings, a theory in which the geometrical and topological properties of tubes seem to best explain the forces of nature.
    Of all the links between numbers and nature studied by the Brotherhood, the most important was the relationship which bears their founder’s name. Pythagoras’ theorem provides us with an equation which is true of all right-angled triangles and which therefore also defines the right angle itself. In turn, the right angle defines the perpendicular, i.e. the relation of the vertical to the horizontal, and ultimately the relation between the three dimensions of our familiar universe. Mathematics, via the right angle, defines the very structure of the space in which we live.

    Figure 1. All right-angled triangles obey Pythagoras’ theorem.
    It is a profound realisation and yet the mathematics required to grasp Pythagoras’s theorem is relatively simple. To understand it, simply begin by measuring the length of the two short sides of a right-angled triangle (
x
and
y
), and then square each one (
x
2 ,
y
2 ). Then add the two squared numbers (
x
2 +
y
2 ) to give you a final number. If you work out this number for the triangle shown in Figure 1 , then the answer is 25.
    You can now measure the longest side
z
, the so-called hypotenuse, and square this length. The remarkable result is that this number
z
2 is identical to the one you just calculated, i.e. 5 2 = 25. That is to say,
    In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.
    Or in other words (or rather symbols):

    This is clearly true for the triangle in Figure 1 , but what is remarkable is that Pythagoras’ theorem is true for every right-angled triangle you can possibly imagine. It is a universal law of mathematics, and you can rely on it whenever you come across any triangle with a right