# Who introduced cylindrical coordinates?

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?

• 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. – Nick R May 13 at 17:27

"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)."