The Restless universe
Introduction to The restless Universe

1 The lawful Universe

2 The clockwork Universe

2.1 Mechanics and determinism 1/4

» 2.1 Mechanics and determinism 2/4

2.1 Mechanics and determinism 3/4

2.1 Mechanics and determinism 4/4

2.2 Energy and conservation 1/2

2.2 Energy and conservation 2/2

3 The irreversible Universe

4 The intangible Universe

5 The uncertain Universe

6 Closing items

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Other titles in the Physical World series

Describing motion

Predicting motion

Classical physics of matter

Static fields and potentials

Dynamic fields and waves

Quantum physics: an introduction

Quantum physics of matter

2 The clockwork Universe

2.1 Mechanics and determinism

Part 1 of 4 | Part 2 | Part 3 | Part 4

Kepler's ideas were underpinned by new discoveries in mathematics. Chief among these was the realization, by René Descartes, that problems in geometry can be recast as problems in algebra. Like most revolutionary ideas, the concept is disarmingly simple. Imagine a giant grid extending over the whole of space.
 Fig 1.4 A two-dimensional coordinate system can be used to locate the position of any point in terms of its x- and y-coordinates.Click here for larger image (6.30kb)
Figure 1.4 shows the two-dimensional case, with a grid extending over part of the page. The grid is calibrated (in centimetres) so the position of any point can be specified by giving its x- and y- coordinates on the grid. For example, the coordinates of point A are x = 3cm and y = 4cm.

This idea becomes more powerful when we consider lines and geometrical shapes. The straight line shown in Figure 1.5 is characterized by the fact that, at each point along the line, the y-coordinate is half the -coordinate. Thus, the x- and y- coordinates of each point on the line obey the equation y = 0.5x, and this is said to be the equation of the line.
 Fig 1.5 A two-dimensional coordinate system can be used to represent lines and other geometrical shapes by equations.Click here for larger image (8.96kb)
Similarly, the circle in Figure 1.5 is characterized by the equation

This is the beginning of a branch of mathematics, called coordinate geometry, which represents geometrical shapes by equations, and which establishes geometrical truths by combining and rearranging those equations. Sometimes, what is difficult to show using traditional geometry is easy to establish using algebra, so this 'mapping' of geometry into algebra gave scientists new ways of tackling geometrical problems, allowing them to go further than the greatest mathematicians of ancient Greece.
Continue on to Mechanics and determinism, part 3 of 4

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