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    Re: Natural haversines: the interpolation table?
    From: Frank Reed
    Date: 2017 Sep 28, 11:31 -0700

    Tony Oz, you asked how to use the interpolation table in the haversine table that Greg uploaded. Honestly, it couldn't be simpler! There's something about the expression "haversine" that creates its own confusion. It's a mathematically trivial concept with a long and mysterious-sounding name ...from "ye olde times" :). And somehow the illusion of mystery makes it harder to see the obvious.

    Specific instructions for an example case:
    Suppose you want the haversine of 78° 42.6'. Now the easy, modern way to do this on a calculator or smartphone or in a spreadsheet is to convert the angle to a pure fraction (a.k.a. "radians"): 
       x=78+42.6/60 and y=x/57.29578, 
    and then calculate 
       hav=(1-cos(y))/2, which for this angle yields 0.4021125. 

    To calculate the same thing with the table, you enter with degrees along the header of each page and minutes of arc down the left side (if you found the degrees in the footer of the page, then you would use the minutes of arc on the right side). You go to 78°42' and note the values for both 42' and 43'. They are 0.40203 and 0.40217. Since our angle end in 42.6', we need to interpolate 60% of the way from 0.40203 to 0.40217. You could easily do this "by inspection" or "by eyeball" which just means working out 0.6·14 in your head --where 14 is the difference between the last two digits of the tabulated values. By that inspection interpolation, you would probably conclude that the correct value is 40203+8 or 0.40211. But if that's too much work or you worry about a fine difference, looking at the interpolation table, it should be immediately obvious that you enter with the tenths of the minute on the left side and with the difference between the final digits of tabulated values along the top. In this case you have 0.6 minutes and a difference of 14 so the tabulated interpolated value is 8, which you add to the starting values final digits just as in the "eyeball" interpolation. The final result is 0.40211 just as in the calculator computation (which is, of course, the gold standard) and in the interpolation by inspection. 

    Some minor flaws with this table as posted: 

    • In the interpolation table, why range from 0.1 to 1.0? Either 0.1 to 0.9 or 0.0 to 1.0 would be consistent.
    • Also, why have the minutes in the interpolation range downward from 1.0 down to 0.1?
    • In the main table, the left margin is labeled '\° while the right margin is labeled °/". Sure, it's just a typo, but someone is bound to get confused and think they can look up arguments in seconds of arc.

    Finally, haversines are over-blown, over-sold, and of minor significance historically. Unfortunately, they have become a fetish property in celestial navigation, like owning a first edition of a comic book. With a few rare exceptions, they are a waste of time. 

    Frank Reed
    Conanicut Island USA

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