Sight reduction process for celestial navigation by the Intercept method.
Also inexactly called Marcq St. Hilaire method.

This is a schematic diagram of the basic process of sight reduction by the Marc St. Hillaire method.
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.
IC is the sextant's Index Correction.
He is the height of eye above the water from which Dip correction can be obtained using lookup tables or the simple formula mentioned in the diagram.
Ha is obtained by adding algebraically the three previous.

UTC is Universal Time, read from a chronometer.
Body is the name of the celestial body observed.
With the body name and UTC as arguments we extract from the almanac the following:
GHA: Greenwich Hour Angle
dec.: declination (GHA & dec. are the coordinates of the body)
SD: SemiDiameter (only applicable to moon and sun).
HP: Horizontal Parallax (mostly moon only).
This step is the most complex in the sense that it cannot be done by any simple calculations locally. One needs to extract the information from the almanac or use a computer.

lat: assumed latitude
lon: Assumed longitude

Refr.: Refraction can be found using lookup tables or the formula provided
PA: Parallax in Altitude = HP * Cos(Ha)
(Augmentation is a very minor correction for the Moon only)

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:
Hc: height calculated
Z: Azimuth angle

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.