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# Secondary Magazine - Issue 43: Focus on

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Created on 14 September 2009 by ncetm_administrator
Updated on 29 September 2009 by ncetm_administrator

# Focus on...coordinates

The commonly-used Cartesian coordinate system was first used by René Descartes in 1637 (it was also independently developed by Fermat, although Fermat’s version, which he didn’t publish, used three dimensions) in the appendix of his Discourse on the Method. Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides. He further expanded this idea in La Géométrie.

Descartes’ contributions to philosophy, mathematics and science are well known but it is perhaps less well known that he was the first to use superscripts as notation for powers (such as the 7 in x7).

The Discourse on the Method (the full name of which is Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences) in which Descartes explores Cartesian coordinates, is also the source of perhaps his most famous quote, I think therefore I am.

The invention of Cartesian Coordinates is one of the foundations which allowed Newton and Leibniz to develop calculus.

In Cartesian coordinates in two dimensions, the coordinate axis split the plane into four quadrants. The quadrant in which both x and y are positive is the first quadrant and the second, third and fourth quadrants are labelled anticlockwise from here. In three dimensions, the first octant is the one where x, y and z are all positive but there is no convention for naming the other seven octants.

The polar coordinate system uses a distance from a fixed point and an angle from a fixed direction as the two variables. They first appeared in their recognisable form in the mid-17th century, with the Belgian mathematician Grégoire de Saint-Vincent and the Italian mathematician Bonaventura Francesco Cavalieri independently using them. The name polar coordinates has been attributed to the Italian mathematician Gregorio Fontana (1735 – 1803). You can see the relationship between polar and rectangular coordinates in this demonstration from Wolfram Mathworld.

Cylindrical coordinates are a three dimensional extension of polar coordinates in which the distance from a fixed point and an angle from a fixed direction, are combined with a height above a fixed plane to allow three variables to be represented. Although there is no single convention for writing the three coordinates, (ρ, φ, z) are commonly used.

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