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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:

$$\begin{align}E&:=\left<e_1,e_1\right>\\F&:=\left<e_1,e_2\right>\\G&:=\left<e_2,e_2\right>\end{align}$$

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?)

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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.]

Thanks to Mark Yasuda for the reference.

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