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      logarithmic numbers used to shorten the calculation of the

      fourth term of a proportion of which one of the terms is a

      given constant quantity, commonly one hour, while the

      other terms are expressed in minutes and seconds; -- not

      now used.

      [1913 Webster]

didn’t help much.   Frank put me on the right track with ‘The math problem they help us solve is no more complicated than this: T = 40.1 x 180/97.’ 

Edward Riddle in ‘Treatise on Navigation and Nautical Astronomy 1824 https://books.google.co.uk/books?id=fPlKAAAAYAAJ  also helped with an example on P251 and the tables on P198 onwards.

So for the benefit of those as backwards as me in 18^th and 19^th century celestial navigation, proportional logarithms can be used to solve problems such as: If a body moves though angle A1 in time T1, what angle will it move though in time T2?  The ratio is T1/A1 = T2/A2, so A2 = (A1xT2)/A1 , using logs: logA2 = (logA1 + logT2) – logA2.

The main value of prop.log tables as far as I can see is that they save you having to convert a time which might include hours, minutes, and seconds into a common unit e.g. seconds before you can work with it.  The same is true for angles.  Some scientific calculators do this automatically of course if you can remember how to make them do it, but in the 18^th and 19^th century I’m sure some would have found the tables very useful.  DaveP