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The idea of this system was developed in 1637 in two writings by Descartes:
The modern Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is commonly defined by two axes, at right angles to each other, forming a plane (an xyplane). The horizontal axis is labeled x, and the vertical axis is labeled y. In a three dimensional coordinate system, another axis, normally labeled z, is added, providing a sense of a third dimension of space measurement. The axes are commonly defined as mutually orthogonal to each other (each at a right angle to the other). (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles.) All the points in a Cartesian coordinate system taken together form a socalled Cartesian plane.
The point of intersection, where the axes meet, is called the origin normally labeled O. With the origin labeled O, we can name the x axis Ox and the y axis Oy. The x and y axes define a plane that can be referred to as the xy plane. Given each axis, choose a unit length, and mark off each unit along the axis, forming a grid. To specify a particular point on a two dimensional coordinate system, you indicate the x unit first (abscissa), followed by the y unit (ordinate) in the form (x,y), an ordered pair. In three dimensions, a third z unit is added, (x,y,z).
The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.
An example of a point P on the system is indicated in the picture below using the coordinate (5,2).
The arrows on the axes indicate that they extend forever in the same direction (i.e. infinitely). The intersection of the two xy axes creates four quadrants indicated by the roman numerals I, II, III, and IV. Conventionally, the quadrants are labeled counterclockwise starting from the northeast quadrant. In Quadrant I the values are (x,y), and II:(x,y), III:(x,y) and IV:(x,y). (see table below.)
Quadrant  x values  y values 

I  > 0  > 0 
II  < 0  > 0 
III  < 0  < 0 
IV  > 0  < 0 
Three dimensional coordinate system Sometime in the early 19th century the third dimension of measurement was added, using the z axis.
A three dimensional coordinate system is usually depicted using what is called the righthand rule, and the system is called a righthanded coordinate system (see handedness). By holding up the middle finger, index finger, and thumb of the right hand, you will see the orientation of the X, Y, and Z axes, respectively (the thumb being the Z axis). The fingers each point toward the positive direction of their representative axes. In the picture above, we see a righthanded coordinate system. Less common, but still in use (normally outside of the physical sciences) is the lefthanded coordinate system.
When the z axis is depicted as pointing upward, this is sometimes called a world coordinates orientation. However, the important thing is which direction the axes point in the positive direction with respect to each other. If we drew an image in the righthanded system and then plotted the image, point for point in a lefthanded system, you would have a mirror image.
The coordinates in a three dimensional system are of the form (x,y,z). An example of two points plotted in this system are in this picture, point P(5,0,2) and Q(5,5,10):
The three dimensional coordinate system is popular because it provides the physical dimensions of space, of height, width, and length, and this is often referred to as "the three dimensions". It is important to note that a dimension is simply a measure of something, and that, for each class of features to be measured, another dimension can be added. Attachment to visualizing the dimensions precludes understanding the many different dimensions that can be measured (time, mass, color, cost, etc.). It is the powerful insight of Descartes that allows us to manipulate multidimensional object algebraically, avoiding compass and protractor for analyzing in more than three dimensions.
It may be interesting to note that some have indicated that the master artists of the Renaissance used a grid, in the form of a wire mesh, as a tool for breaking up the component parts of their subjects they painteda trade secret. That this may have influenced Descartes is merely speculative.
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