Descartes' conjugate equation, magnification

Principal points H and H' are taken to have for origin :

And

So that

And

Replacing in (21), the conjugate equation becomes :

Following figure 38, the ray (green) issued from B and going through H forms an angle with the axis. As previsously shown, its image going from H' forms an angleθ' such that nθ= n'θ'. Since θ= y/z and θ' = y'/z', we deduct from it the transversal magnification gy :

With (22) and (23) :

To one sole value of z (or of z') corresponds only one value of gy and conversely. These formula are identical to that of diopters'. For one diopter, the principal planes are confounded and situated on the dioopter surface.

The same kind of reasonning leads to an angular magnification :

And to an axial magnification gz :