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    Re: Proportional Logarithms
    From: Ed Popko
    Date: 2015 Feb 28, 12:13 -0800

    Although I'm one of Frank's recent Lunars workshop students, I have not posted to the NavList till now even though he prompted me to do so some time back.

    I would like to a couple of footnotes to this chain with the hopes of making proportional logs better understood. Proportional logs are quite useful because they use only addition and subtraction to compute a proportion which we would, today, do with multiplication and division. It's quite clever when you see how it works. Nevil Maskelyne was no dummy when he proposed proportional logs. Although he gets a somewhat bad rap for obstructing Harrison's chronometer solution to longitude, you have to recognize it took between 50 and 100 years for chronometers to be accurate, mass produced and aboard every ship. So Lunars was a very important contribution to navigation until time-based navigation (time-sights, Sumner Lines, and Intercepts) took over.

    - there are two commonly used 'bases' for proportional logs in old lunars literature. Both relate to the fact that almanac lunar distances were tabulated every three hours.
    The first 'base' is 3 because the tabulations were three hours apart. The second is 10800, this is the number of seconds in three hours. Either 'base' can be used but must be used consistently.

    - the posted response to what the proportional log is and how to compute it for the differences in lunar distances is good.

    Here, I mention how to convert a proportional log time back to  a time expressed in hours:min:seconds:

    Using the formula in the post, let's see what the pLog of 1hr 30min is and then do a round robin example to convert the pLog back to this time.

    1hr 30min is 5400 seconds

    pLog(5400) = .30103        

    Now, lets convert the pLog value back to seconds and then to HMS:
     
    time in seconds = ALOG(LOG(10800)-pLog(x))
                           = ALOG(LOG(10800)-.30103)
                           = ALOG(4.03343476 - .30103)
                           = ALOG(3.773239376)
                           = 5400     number of seconds what we started with

    One additional division makes this number useful, divide by 3600 (number of seconds in an hour) gives us decimal hours, then convert it to HMS which is 1:30:00.

     

       
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