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Cartesian Co-ordinate System

  According to legend, René Descartes (1596 - 1650) saw the shadow of the lattice of a window falling on a table, and this inspired him to design the first co-ordinate system, which gives numerical values to points in space. His system is still used today, and is known as Cartesian in his honour.

In one dimension (in other words, on a line), Cartesian co-ordinates are defined by choosing a point known as the origin (with co-ordinate 0), and specifying a distance measure (how far away the point with co-ordinate 1 is from the origin). The points on the line now all have uniquely defined real number values; for example, if the point 1 is 1cm away from the origin, the point -243 is 243cm away from the origin in the opposite direction.

The two-dimensional system is a generalization from the one-dimensional system. Here, the aim is to define co-ordinates for the plane. As before, the origin is chosen, but this time also a line (known as the abscissa) through the origin. One-dimensional co-ordinates are then defined on the abcissa. A second line, the ordinate, is drawn through the origin at right-angles to the first line, and its co-ordinate system is defined in terms of that on the abscissa. If someone\'s right hand has thumb and forefinger extended, with the forefinger pointing along the abscissa from 0 to 1, then the point 1 on the ordinate is that at the same distance from the origin in the direction the thumb is pointing in. (This may sound confusing, but when drawn on a piece of paper it becomes easy to see what is going on.) Each point on the plane now has a unique co-ordinate, (x,y), where x is the length of the shortest straight line joining the point to the ordinate and y is the length of the shortest straight line joining the point to the abscissa.

The same process is repeated to give a third dimension, a co-ordinate system for space. It can also be generalized to any number of dimensions, something which originally seemed pointless but which now has many applications in today\'s theories of cosmology. It is worth noting that in order to co-ordinatize the motion of a body in space, such as the Earth, three dimensions are needed to give the position and three to give the velocity—six dimensions in all. In the same way cylindrical, spherical and other polar co-ordinates are developed using ideas of angular (as opposed to linear) measurements.

Algebraic geometry as a subject has advanced considerably since Descartes, but its importance to the development of mathematical thought stems mainly from the idea of the co-ordinate system. Cartesian co-ordinates are not the only ones which can be developed. SMcL



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