this principle, replacing 10 by 108. He called all numbers up to 108 numbers of the first order. Using 108 as a starting point, he took successive multiples of this new unit. He called all numbers between 108 and 108 x 108 = 1016 numbers of the second order. This is continued to numbers of the third order and so on, ending with numbers of the myriad myriadth order, which starts at Now Archimedes had his number system and he concluded that the number of grains required is 10,000,000 units of the eighth order, which is 1063.
This anticipates much of modern ways of writing large numbers. It also includes a tantalizing reference to the claim of another Greek scientist, Aristarchus, that the Earth travels round the Sun.
2nd century BC Greece
Trigonometry
Hipparchus (190–120 BC)
Trigonometry is concerned with calculating sides of triangles from angles and early development was mainly used in astronomy.
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Trigonometry is taught throughout high schools. It is based on three functions: sine, cosine, and tangent.
The original trigonometric function was the chord function. Start with an isosceles triangle rather than a right-angled one. Let the equal sides each have length one unit. The chord function, crd(Ø), gives the third side of the triangle.
A definition of the chord function.
It is easy to convert between the chord and sine functions, with these formulae, which involve only doubling and halving: crd(Ø) = 2 sin(1/2 Ø) sin(Ø) = 1/2 crd(2Ø)
No trigonometrical work of Hipparchus survives but he is known to have compiled the first trigonometric table – a table of values of the crd function.
The oldest surviving table is in the Almagest of Ptolemy, a work of astronomy. The table is a feat of numerical complexity: starting with results like crd(60°) = 1 and crd(90°) = √2, and using formulae for crd(A + B) and crd(A – B), the chords of angles are found for every 1/2°, to an accuracy of up to 6 decimal places.
The familiar functions of sine, cosine, and tangent, introduced by Indian and Arab mathematicians, are now used but the methods remain the same.
2nd century BC China
Negative Numbers
The extension to numbers less than zero.
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Negative numbers make sense in some contexts, but not in others. Where they are relevant they save a lot of time but mathematicians did not accept them as proper numbers until comparatively recently.
It makes no sense to say “There are minus five people in the room.” However, in many other contexts it is useful to have numbers which are less than zero. One familiar example is temperature: at 0°C water freezes and we require numbers to describe temperatures lower than that figure. In commerce, too, it is useful to have negative numbers to describe a debt.
Chinese mathematicians were the first to accept negative numbers. They did their arithmetic on a chequerboard using short rods for the numbers. Red rods were used for positive numbers and black for negative, whereas the modern way of describing whether a bank-balance is positive or negative is the opposite way round.
Greek mathematicians did not recognize negative solutions of equations. In the 3rd century, Diophantus rejected x + 10 = 5, saying it was not a proper equation. Indian mathematicians came closer to accepting negative numbers, finding negative roots of quadratic equations. However, Bhaskara II (1114–1185) the leading mathematician in the 12th century, rejected these solutions, stating that people did not approve of them.
Nowadays, negative numbers are an essential part of mathematics. An example of their usefulness is in solving quadratic equations.
The equation ax2 + bx + c = 0 is solved by the formula:
Here a, b, and c can be positive or negative and the same formula fits all cases.
If we do not allow negative numbers, a, b, and c must be positive. There are several separate cases to consider:
ax2 + bx = c
ax2 + c = bx
ax2 = bx + c
Each of these separate cases will have a different formula. That makes three formulae to remember instead of just one!
The product of two negative numbers is positive. This fact is a part of school mathematics that is famously difficult to justify. Teachers have to rely on the following:
Minus times minus equals a plus.
The reason for this we shall not discuss.
See: Quadratic Equations, pages 11–12
150 AD Greece
The Earth-Centered Universe
Claudius Ptolemy (83–161 AD)
This is a system of the universe.
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In Ptolemy’s system the Earth is at the center and all other bodies revolve around it.
The earliest model for the motion of heavenly bodies about the Earth involved their moving in circles with the Earth at the center. More accurate observations showed that this was incorrect and modifications had to be made. These suggested that:
1. The Earth is not the center of the circle.
2. The bodies move in smaller circles (called epicycles) which were themselves moving in circles around the Earth.
3. The speed of the body was not constant as it moved around the circle.
In Ptolemy’s Mathematike Syntaxis, the mathematical compilation, better known by its Arabic name Almagest, all these modifications are used.
The diagram on the opposite page illustrates this model. A planet moves in a small circle. The center of this small circle moves in a big circle around a point C – this is not the Earth – and the angular speed is constant, not about the Earth and not about
