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 calculated.
Claudius Ptolemy (85–165 AD) enjoyed, as his name implies, a rich mixture of Greek and Roman culture and learning. His mathematical works, principally on astronomy, were honoured with the Arabian title Almagest, meaning “The Greatest.”
Diophantus (200–284 AD) did remarkable early work on number theory, and his book Arithmetica provided a great deal of inspiration for Pierre Fermat (1601–1665). Fermat’s Last Theorem states that if we call 3 positive integers a, b, and c, then the equation an+bn=cn will only be possible if n is not greater than 2. Proving it became a leading mathematical problem for centuries, and even made its way into the Guinness Book of Records!
A brilliant Persian mathematician named Al-Khowarizmi (780–840) was also a gifted scientist and astronomer. His additional interest in astrology made him something of a numerologist — like Pythagoras — as well as a scientific mathematician. The modern word algebra was transliterated from his book Hisab al-jabr w’al-muqabala, where it
was rendered as “al-jabr.”
Francesco Pellos (1450–1500) was the inventor of the decimal point — a tremendously useful part of contemporary mathematics. The gifted Scots theologian John Napier (1550–1617) indulged in mathematics more or less as a hobby when he wanted a break from theology. He was largely responsible for creating logarithms, which were perfected by Henry Briggs. At around the same period, Sir Isaac Newton (1642–1727) published his epoch-making Principia — a masterpiece of mathematics and science. There is some justification for those who regard him as the greatest scientist who has yet lived. His work on gravitation and the 3 laws of motion are unforgettable.
Another important milestone in the history of mathematics was William Jones’s (1675–1749). He used the Greek symbol “π” to show the result of dividing the circumference of a circle by its diameter. This feat was published in his book, New Introduction to Mathematics, in 1706.
Calculus was the particular brainchild of the Italian maths genius Maria Agnesi (1718–1799). Her famous textbook on it, Istituzioni Analitiche, was an authoritative teaching aid on calculus for many years.
David Hilbert (1862–1943) was one of the most outstanding mathematical leaders in the late nineteenth and early twentieth centuries. His great contributions were to invariant theory and the axiomatization of Euclid’s geometry. One of his other theories that was essential to functional analysis was named after him as the theory of Hilbert spaces. Tragically, he and a number of other brilliant academic mathematicians at the University of Göttingen were persecuted by the Nazis.
Benoit B. Mandelbrot (1924–2010) was a superb French-American mathematician whose name is associated with the mathematical idea of fractals. The word comes from the Latin fractus, which means broken. A fractal could be described as a rough or fragmented geometric shape that can be divided repeatedly into smaller and smaller parts. These smaller parts resemble — not always exactly — the original larger whole. This characteristic is described as self-similarity. Fractals play a large