My textbook says:

We write the equation in the form:

$\displaystyle k_{(n)}=\frac{e+2f \lambda + g \lambda^2}{E+2F \lambda + G \lambda^2}$ where $\displaystyle \lambda = \frac{dv}{du}$.

From the theory of proportions we can write:

$\displaystyle k_{(n)}=\frac{(e+f \lambda) + \lambda(f+g \lambda)}{(E+F \lambda) + \lambda (F+G \lambda)}=\frac{f+g\lambda}{F+G\lambda}=\frac{e+f\ lambda}{E+F\lambda}$

Can anyone explain this use of the "theory of proportions" to me? Intuitively it makes some sense as large values of lambda give one solution and small values of lambda give the other solution. But then why not go the the extreme of saying:

$\displaystyle k_{(n)}=\frac{g}{G}=\frac{e}{E}$

You may recognise K as "Normal curvature" or the ratio of the first and second fundamental forms of a surface.