Green lines and keys indicate only addition and subtraction are used in those steps which can be done with pencil and paper using an appropriate form.
Red lines and keys indicate more complex formulas but, as we will see, in most cases simple tables can be used or a handheld calculator.
Hs is the height read off the sextant.
UTC is Universal Time, read from a chronometer.
lat: assumed latitude
Refr.: Refraction can be found using lookup tables or the formula provided
Ho: height observed
All the previous steps are common to all the different variants of the St. Hillaire method. We now come to the resolution of the spherical triangle which can be done by many different ways. The formulas which permit the calculation of Hc and Z are the two lower ones below the diagram. They can be easily calculated using a handheld calculator with trig functions or lookup tables such as HO229 can be used. Lookup tables were the only method before the advent of calculators. Today, a cheap, programmable calculator will do the calculation instantly by just entering the three arguments: dec., LHA (Local Hour Angle) and lat.
With those three arguments (dec., LHA & lat.) we obtain:
The difference between Ho and Hc is the "Intercept" or error and Zn is Z reduced to the correct quadrant.
This process can be done quite mechanically using an appropriate form where one just fills the boxes sequentially
With Hc and Zn we can plot a LOP (Line of Position). Take several LOPs and advance them all to a common time and you have a fix.