Coolidge: Origin of Polar Coordinates
THE ORIGIN OF POLAR COORDINATES
Polar coordinates were used for special purposes and for the study of particular curves before they were appreciated as a general geometrical tool. The first writer to employ them was Bonaventura Cavalieri, who used them to find the area within an Archimedian spiral by relating it to that outside a parabola. Pascal used the same transformation to calculate the length of a parabolic are, a problem previously solved by Roberval, but his solution was not universally accepted as valid. James Gregory had a similar transformation between two individual curves, where the areas were related, while Pierre Varignon used a slightly different transformation for the study of spirals.
The first writer who looked on polar coordinates as a means of fixing any point in the plane was Newton. He, however, considered them alongside Cartesian, bipolar, and other systems, his only interest at that point being to show how the tangent could be determined when the equation of the curve was given in the one or the other system. A deeper interest was shown by Jacob Bernoulli, who went so far as to write the expression for the radius of curvature when the equation of the curve was given in polar form.
The first writer to think of polar coordinates in 3-space was Clairaut, but he merely mentions the possibility of such things. The first to develop them was Euler to whom we owe both polar and radio-angular coordinates. An interesting modification of the latter was developed by Ossian Bonnet.
Harvard University,
Cambridge, Mass., U.S.A.
JOC/EFR March 2006
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https://www-history.mcs.st-andrews.ac.uk/Extras/Coolidge_Polars.html