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    Re: Practical Haversine Use
    From: Frank Reed
    Date: 2021 Nov 14, 10:11 -0800

    By the way, these "10+" logarithms were simply "standard logarithms" for many practical users from the late 18th century and into the 1960s. They were not exotic or suprising or worthy of note in any way. This practice of dropping the leading digits was workaday math for almost all navigators for nearly two centuries. It came as naturally as "dotting an i" while composing a sentence, and many navigators did it without even a notation in their work. I have taught this "step" in the process for over a decade in my "Celestial Navigation in the Age of Sail" classes. It was known back then as "rejecting the tens". If you open almost any online edition of a navigation manual (a Norie or a Bowditch or any of dozens) from the nineteenth century and search on the word "reject", you'll find examples.

    Tibor Miseta already gave an excellent description of the logic for this process. Here's my short version:

    • navigators needed to multiply various trig functions with up to four or five (maybe more) significant digits
    • historically multplication was performed by adding logarithms
    • some key trig functions, like sine and cosine, have values (absolute) that are less than one.
    • logs of numbers less than one are by definition negative so adding those logarithms is subtraction
    • subtraction is time-consuming and error-prone compared to addition - avoid if possible!
    • fix: take all the numbers we need to multiply together and multiply them by 10 biillion (1e10)
    • all numbers to be multiplied are now greater than one and their logs are never negative
    • proceed as usual adding up logs, no subtraction necessary
    • at the end, come back out of logs and divide out as many factors of 10 billion as there are in the product
    • or (and this was obviously the practice employed) drop (or "reject") any extra tens in the final sum of logs

    Like I say, this was simply common practice for nearly two centuries! No one had to think about it. No one had to know about the logical steps I've listed above. A practical author like Rantzen could comfortably tell his readers 'don't worry about why... trust the mathematicians who invented this system long before any of us were born'. It was a property of log tables. 

    Frank Reed

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