In textbooks on differential geometry, the first fundamental form looks like $E^2+2FG+H^2$, and its length is calculated through the help of the dot product. However, the inner product did not exist. The analytic formula was known that $x_1x_2+y_1y_2+z_1z_2$ . Nor did the vector exist. So how did he prove it? Same story with the unit normal: in textbooks they use the cross product $r(u) \times r(v)$ But this did not exist either. Again, how did he manage to calculate it?