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    Re: Lunars: how to use the prop.log tables?
    From: David Pike
    Date: 2016 Sep 1, 15:48 -0700
    Readers will already have noticed that this writer is not exactly the brightest star in the sky, so the discovery that there is something called proportional logarithms came a source of wonder to one who forgot how to use ordinary logarithms a long time ago.  There’s not much about them on Google, so they must be rare indeed.  Definitions like Logistic logarithms, or Proportional logarithms, certain
          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 18th and 19th century celestial navigation, as I see it, 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 this becomes: 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 so, but in the 18th and 19th century I’m sure some would have found the tables very useful.  DaveP

       
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