History of Western Philosophy - Bertrand Russell [124]
The square root of 2, which was the first irrational to be discovered, was known to the early Pythagoreans, and ingenious methods of approximating to its value were discovered. The best was as follows: Form two columns of numbers, which we will call the a's and the b's; each starts with 1. The next a, at each stage, is formed by adding the last a and b already obtained; the next b is formed by adding twice the previous a to the previous b. The first 6 pairs so obtained are (1, 1), (2, 3), (5, 7), (12, 17), (29, 41), (70, 99). In each pair, 2a2 – b2 is 1 or –1. Thus b/a is nearly the square root of two, and at each fresh step it gets nearer. For instance, the reader may satisfy himself that the square of 99/70 is very nearly equal to 2.
Pythagoras—always a rather misty figure—is described by Proclus as the first who made geometry a liberal education. Many authorities, including Sir Thomas Heath,1 believe that he probably discovered the theorem that bears his name, to the effect that, in a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides. In any case, this theorem was known to the Pythagoreans at a very early date. They knew also that the sum of the angles of a triangle is two right angles.
Irrationals other than the square root of two were studied, in particular cases by Theodorus, a contemporary of Socrates, and in a more general way by Theaetetus, who was roughly contemporary with Plato, but somewhat older. Democritus wrote a treatise on irrationals, but very little is known as to its contents. Plato was profoundly interested in the subject; he mentions the work of Theodorus and Theaetetus in the dialogue called after the latter. In the Laws (819–820), he says that the general ignorance on this subject is disgraceful, and implies that he himself began to know about it rather late in life. It had of course an important bearing on the Pythagorean philosophy.
One of the most important consequences of the discovery of irrationals was the invention of the geometrical theory of proportion by Eudoxus (ca. 408–ca. 355 B.C.). Before him, there was only the arithmetical theory of proportion. According to this theory, the ratio of a to b is equal to the ratio of c to d if a times d is equal to b times c. This definition, in the absence of an arithmetical theory of irrationals, is only applicable to rationals. Eudoxus, however, gave a new definition not subject to this restriction, framed in a
manner which suggests the methods of modern analysis. The theory is developed in Euclid, and has great logical beauty.
Eudoxus also either invented or perfected the 'method of exhaustion', which was subsequently used with great success by Archimedes. This method is an anticipation of the integral calculus. Take, for example, the question of the area of a circle. You can inscribe in a circle a regular hexagon, or a regular dodecagon, or a regular polygon of a thousand or a million sides. The area of such a polygon, however many sides it has, is proportional to the square on the diameter of the circle. The more sides the polygon has, the more nearly it becomes equal to the circle. You can prove that, if you give the polygon enough sides, its area can be got to differ from that of the circle by less than any previously assigned area, however small. For this purpose, the 'axiom of Archimedes' is used. This states (when somewhat simplified) that if the greater of two quantities is halved, and then the half is halved, and so on, a quantity will be reached, at last, which is less than the smaller of the original two quantities. In other words, if a is greater than b, there is some whole number n such that