# Understanding Euler's paper on curvature

We've recently discussed, in the course on differential geometry which I am taking, Euler's theorem regarding curvature of sections of surfaces in $\mathbf{R}^3$. Being curious, and knowing that Euler's paper are generally fun to read, I went to read his original publication. On The Euler Archive it is listed as E333 and there is an English translation available here (the original publication in Latin is also available here).

However, I cannot seem to comprehend his use of notation. The translator attempted to include diagrams, but due to some malfunction the pictures are not showing up in the .pdf. I've attempted to recreate the figures, but Euler's notation is rather confusing, to me at least. The first diagram should be in the following place, but is obviously missing. Along with the second which should be at the top of the following page. Both of these screenshots are taken from the .pdf in the first link, above. It seems to me that Euler is describing a 2-dimensional surface in $\mathbf{R}^3$, which is parametrized as $(x,y,f(x,y))$ for some $f:\mathbf{R}^2\to\mathbf{R}$. He then considers an arbitrary point $(x,y,z)$ on the surface and a plane through it, but I am unsure as to what $A,C$ are supposed to represent. If anyone is able or willing to help me in recreating these diagrams, I would be greatly appreciative.

• instead of latin is a kind of old french – janmarqz Feb 8 '14 at 22:51

$$A = (0,0,0)$$ $$X = (x,0,0)$$ $$Y = (x,y,0)$$ $$Z = (x,y,z)$$ 