# Why are the symbols E, F, G, L, M, and N used for the coefficients of the fundamental forms?

In differential geometry, if $$e_1$$ and $$e_2$$ are bases for a tangent space $$T_pM$$, then the coefficients of the first fundamental form is:

\begin{align}E&:=\left\\F&:=\left\\G&:=\left\end{align}

And for the second fundamental form, similar -- $$L:=\mathrm{I\!I}(e_1,e_1)$$, $$M\ldots$$, $$N\ldots$$, etc.

This is... pretty strange! Not the definitions - the nomenclature. Some more modern textbooks use E/F/G for the first coefficients and e/f/g/ for the second, but everyone seems to be in agreement that at least the letters E/F/G are to be used, and L/M/N are quite common to see.

Why were these letters used? (Or, is it known to be an arbitrary choice?)

See the paper by Gauss, Disquisitiones generales circa superficies curvas, from 1828. On page 24, we find: $$E\,dp^2 + 2F\,dp\,dq + G\,dq^2 .$$ Since then everyone continued to use the letters $$E,F,G$$. That would have been long before anyone used terms like "tangent space".
Why did Gauss choose those letters? On page 22, where letters $$E,F,G$$ are first indroduced, he had already used letters $$A,B,C,D$$. [As well as various small letters, Greek letters, letters with primes.]