the show was totally convinced that everything was ending. He had given up a good professional job and his London home so that he could go back to the village where he had been born in order to await the end there. That is the kind of serious damage that misguided eschatological obsessions can induce. The authors — with over 50 years’ experience of investigating the paranormal and the anomalous — are totally confident that nothing bad will happen to our Earth in December 2012. Mayan mathematics was exceptionally advanced for its time, but well-designed as it was, their Long Count Calendar certainly does not herald the end of all things.
Thales of Miletus lived from 624–546 BC. He was a pre-Socratic philosopher and a master of mathematics and astronomy, who also had a keen interest in ethics and metaphysics. Experts regard him as one of the traditional 7 Sages of Greece. He set out to explain natural phenomena through causes and effects, as opposed to the mythological explanations so prevalent at the time. It was characteristic of popular Greek thought during this period to depend upon exegetical myths — those that attempted to explain the origins of everything. For example, the changing of seasons was thought to be a result of the Greek earth goddess, Demeter, searching for her missing daughter, Persephone. Numerology features here in the myth of Persephone and the 6 pomegranate seeds that she ate out of temptation, leading to her curse of having to spend 6 out of every 12 months with Hades in the Underworld, which corresponds to the number of winter months we experience each year. Another example is provided by Aeolus, or Aiolos, who was appointed by Zeus to take charge of the storm winds. These were released when the gods wanted to cause damage and disaster. Instead of relying on these myths to explain the natural phenomena happening around him, Thales looked instead for rational explanations that he did his best to examine open-mindedly and objectively. This gave him the title of the Father of Science, although supporters of Democritus felt that he deserved the title instead.
One of Thales’s excellent ideas was to calculate the height of a tall building, such as a pyramid, by standing in a position where he could measure his own shadow and the shadow of the target building. When his shadow coincided exactly with his own height, he measured the pyramid’s shadow and argued that if his shadow gave his height, then the pyramid’s shadow would reveal its height. He was also responsible for a number of important theorems: that any diameter bisects a circle; that the angle from the diameter to the circumference in a semi-circle is a right-angle; that the base angles of an isosceles triangle are equal; that the opposite angles formed by 2 intersecting straight lines are equal; and that triangles are congruent if 1 side and 2 angles are equal.
Pythagoras lived from approximately 570–495 BC and was the leader of a group known as the Pythagoreans. He and his followers in Croton (what is now Crotone in southern Italy) lived like a monastic brotherhood and were all vegetarians. All their joint mathematical discoveries were attributed to Pythagoras, so it is impossible to tell how much of the work was his alone. Because of this cult aspect of his life, it may be more appropriate to think of him as a numerologist than as a scientific mathematician. The safest conclusion is that he was both. The theorem associated with him is that in any right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other 2 sides. The proof of this theory is a very interesting one. Draw a right-angled triangle with sides a, b, and c as shown below. The longest side, the hypotenuse opposite the right-angle, is side c. What has to be proven is that a²+b²=c². Now draw the square of side c as shown in the diagram, and draw the original 90-degree triangle, “a, b, c,” in the 4 corners as illustrated here.
The big square with the c² inside it has the area (a+b)², or we can say:
A (the area of the big square)=(a+b)(a+b).
The area of the tilted internal square is c².
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Diagram of Pythagoras' Theorem
Each of the right-angled triangles has the area 1/2ab. There are 4 of these, so their total area is: 4(1/2ab). We can re-write this as 2ab. The entire area of the tilted square and the 4 right-angled triangles is A=c²+2ab. This then becomes (a+b)(a+b)=c²+2ab. The term (a+b)(a+b) can be multiplied out to produce a²+2ab+b², further resulting in the equation a²+2ab+b²=c²+2ab. The next step is to subtract the 2ab from each side. This leaves a²+b²=c², which is the proof that we were looking for!
One of Pythagoras’s most able disciples was Parmenides, who applied his mathematical skills to cosmology and came up with the idea of a spherical Earth inside the spherical universe. One of his disciples, Zeno of Elea, devoted his mathematical and philosophical skills to creating paradoxes, and some of these shaped the development of mathematics for centuries after his death. One of Zeno’s best-known paradoxes is called Achilles and the Tortoise. Zeno argued that a fast runner like Achilles could never overtake the tortoise. If it started 16 metres ahead of Achilles, but could run at only half his speed, by the time he covered those 16 metres, the tortoise would have gone 8. By the time he had covered those 8 metres, the tortoise would have gone another 4. By the time he had covered those 4, the tortoise would have gone another 2 … and so on indefinitely.
Another important early Greek mathematician was Hippocrates of Chios (470–410 BC) who, like many others since his day, attempted to square the circle: that is, to construct a square with the same area as a given circle. It is impossible to do this with perfect accuracy because π (3.14159…) is not a rational number.
The curve, known as the “quadratrix,” or “trisectrix,” was the work of Hippias of Elias around about 430 BC. It can be used to trisect an angle, or to divide an angle into a given number of equal parts. He, too, was one of the early mathematicians involved in unsuccessful attempts to square the circle.
Eudoxus of Cnidas (410–350 BC) developed a system of geometric proofs, based on what became known as the exhaustion method. When he attempted to show that 2 areas, a and b, were equal, he would begin by trying to prove that a was greater than b.
When that proof failed (was exhausted) he would attempt to prove that b was greater than a. When that proof also failed (was also exhausted) Eudoxus argued that as neither area was bigger or smaller than the other, they must be equal.
Eudemus of Rhodes (350–290 BC) was not so much a great early mathematician in his own right as an historian of mathematics. His 3 very informative books in this genre were histories of arithmetic, geometry, and astronomy. He also wrote another volume dealing with angles.
Euclid of Alexandria (325–265 BC) has gone down in mathematical history as the greatest mathematical teacher of all time. His ideas are still quoted authoritatively today. In addition to all his well-known work on geometry, his theory of the infinite number of prime numbers has stood the test of time.
Aristarchus of Samos (310–230 BC) was a remarkably able mathematician and astronomer, who came up with a well-argued heliocentric theory of the universe. Nearly 2,000 years later, Copernicus delved into Aristarchus’s work and agreed with his conclusions.
Apollonius of Perga (262–190 BC) focussed his mathematical skills on cones and the curves that are derived from slicing them. His book on conics introduced the terms “parabola,” “ellipse,” and “hyperbola.”
Hipparchus of Rhodes (190–120 BC) worked mainly as the pioneer of trigonometry. Every angle has a sine, a cosine, and a tangent, and these can be used to find angles or sides in a 90-degree triangle. The 3 sides are referred to as the hypotenuse, the adjacent, and the opposite. The basic trigonometrical formulae are:
sine=opposite÷hypotenuse
cosine=adjacent÷hypotenuse
tangent=opposite÷adjacent
These formulae make it clear that whenever any 2 of the measurements are known, the third can readily be
