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 the center C of the big circle, but about a point called the equant E – on the other side of the Earth from C.
This system was very complicated. It was also reasonably accurate, however, and lasted as a guide for navigation, astronomy, calendar setting, and so on, for well over 1,000 years. Its accuracy was not challenged until the insistence on circular motion was abandoned and other curves were considered.
A model for planetary motion.
See: The Sun-Centered Universe, page 60; The Sun-Centered Universe Again, pages 64–65
628 AD India
Zero
Brahmagupta (598–665 AD)
The recognition that zero is a proper number.
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Nowadays zero is an ordinary part of how we write numbers and calculate with them. It took a long time for it to be accepted as a proper number.
For most of recorded history the number system started with one. There was no number zero (0) when the system for dating years was devised in 525 AD. The system goes straight from 1 BC to 1 AD – there is no 0 AD.
There is no need for zero when using Roman numerals. The number 105 is CV (100 + 5) and we do not have to show there are no tens. By contrast, in a place value system, some indication is required.
Before modern times, arithmetic was done using a counting board, with one column for units, one column for 10s, one column for 100s, and so on. A symbol for zero isn’t required – just a space in the relevant column for the missing digit.
When writing numbers, it is useful to have a special symbol to show that a number is missing – such a symbol was used in Central America by indigenous peoples centuries ago. The Babylonians used two small wedges in their sexagesimal system.
Using a symbol for a missing digit in a place value system is not the same as recognizing zero as a number in its own right. That was invented by Indian mathematicians. In 628 AD Brahmagupta wrote out a set of rules for zero including:
• The sum of a zero and a positive number is positive.
• Zero divided by zero is zero.
Now we agree with the first but not the second. Division by zero is forbidden. If a car travels