Cylindrical coordinates$ x=r\cos θ, y=r\sin θ, z=w$ seem to be a simple generalization of polar coordinates. When did they appear first? Also, who came up with the name?

  • $\begingroup$ Thanks for your comment. You're right that my answer does not exactly fit the bill and is rather about the extension of polar coordinates from the plane to three dimensional space. I have deleted my answer since it lacks the detail that you require. $\endgroup$ – Nick R May 13 at 17:27

It is interesting that Coolidge, who has an entire chapter on various coordinate systems in his History of Geometrical Methods does not even mention cylindrical coordinates. This is probably because they amount to just adding an extra variable to polar coordinates, not exactly a notable move. The earliest explicit example I recall (without the name, of course) is by Euler in De motu vibratorio tympanorum (On the motion of vibrations in drums), Novi Commentarii academiae scientiarum Petropolitanae 10, 1766. Dutka describes it as follows in On the Early History of Bessel Functions:

"Euler sets up the equations of motion for a homogeneous membrane in which the tensions in the $x$ and $y$ directions are equal with no external forces acting on it... He obtains an equation of the form $$ \frac{\partial^2z}{\partial t^2}=a^2\left(\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}\right)\hspace{1cm} (5) $$ where the right member of the equation is proportional to the sum of the forces in the $x$ and $y$ directions, and $a^2$ depends on the tension and density of the membrane.

Euler's solution for the case of a rectangular membrane is proportional to a product of sinusoidal functions of $x, y$ and $t$ respectively. He then proceeds to consider the case of a circular membrane in which the tension along the boundary is uniform. He introduces what are essentially cylindrical coordinates $(r, \theta, z)$ in place of the rectangular coordinates and substitutes into (5)."

What Euler gets is the now standard equation for the Bessel functions, also known as the cylindrical functions. This is why. He develops the, also standard, power series formula for these functions. Even before Euler (and Bessel) Daniel Bernoulli studied them already, although not nearly as thoroughly as Bessel.

As for the name, Google Scholar does not have non-spurious hits before 1870-s, Cambridge's Solutions of the Senate-House Problems and Riders (1875) is the earliest. Frost's Solid Geometry textbook (1886) already has a section named after cylindrical coordinates.

  • $\begingroup$ Great answer Conifold! Thank you so much. $\endgroup$ – J.Petrovic May 14 at 12:35

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