NavList:

Darden Rives, you wrote:
"I am just wondering if anyone would recognize the text or method he was using this rule for"

First, a public "welcome aboard".

Those equations are the standard basic equations of celestial navigation. The first, for example, is the standard "law of cosines" formula for calculating the angular altitude, h, of the Sun or any celestial body given the observer's latitude, L, the Sun's declination, d, and the local hour angle of the Sun, t. The next equation gives the azimuth, z, from d, t, and the previously calculated h.  There's some repetition here in the later formulae. This suggests that these may have been written down at an early stage in the learning process.

It's worth noting that these equations cannot be used with any genuine accuracy using a common slide rule, so your father-in-law was probably experimenting with the equations rather than using them in practice. Common slide rules can be used for dead reckoning calculations and some other low accuracy computations. They were not regularly used for celestial navigation. Cylindrical slide rules, which you've seen discussed in some recent posts, were specifically developed for the higher level of accuracy demanded by celestial navigation computations, and the Bygrave and other cylindrical slide rules were moderately popular for a few decades in the early twentieth century.

Frank Reed
Conanicut Island USA