In an old surveying publication I saw a nice trick for adding
sexagesimal numbers on an electromechanical calculating machine. If
the minutes or seconds summed to 60 or more you added 40 to the part
that "rolled over".

Here's a simple example:

by hand         calculator
                equivalent

   3 14'            314
 + 1 48'          + 148
 -------          -----
   5 02'            462
                  +  40
                  -----
                    502

In the following example, seconds do not roll over, so 40 is added to
minutes part only:

 281 45' 32"      2814532
+ 68 59' 14"     + 685914
------------      -------
 350 44' 46"      3500446
                 +   4000
                  -------
                  3504446

Here's an example with minutes and seconds rolling over, so 40 is
added to both parts:

 316 38' 13"      3163813
+ 12 43' 54"     + 124354
------------     --------
 329 22' 07"      3288167
                 +   4040
                 --------
                  3292207

Finally, in the following example seconds roll over, and the carry
will make minutes roll over, so 40 must be added to both parts:

  82 34' 16"       823416
+ 34 25' 53"     + 342553
------------     --------
 117 00' 09"      1165969
                 +   4040
                 --------
                  1170009

The algorithm works for subtraction too; you just subtract 40 instead
of adding. Here's an example where seconds roll over. This time I'll
use a decimal point between degrees and minutes for easier reading:

 142 52' 11"      142.5211
- 21 10' 56"     - 21.1056
------------      -------
 121 41' 15"      121.4155
                 -   .0040
                  -------
                  121.4115

In general, I don't see much use for this old technique if you have a
scientific calculator. It's easier to use the sexagesimal and decimal
conversion functions and do the math in decimal. On the other hand,
the 40 method is usable on the simplest calculator. Also, it easily
handles degrees and decimal minutes, a format which has no direct
conversion on the calculators I've used.

For paper and pencil computation, I think I'll continue to do the math
in sexagesimal.


A variation on the 40 method is the 76 method for dealing with 24 hour
days. This can be combined with the 40 method:

 2 days 17 hours 27 minutes      2.1727
+1 day  20 hours 35 minutes     +1.2035
---------------------------     -------
 4 days 14 hours  2 minutes      3.3762
                                + .7640
                                -------
                                 4.1402

or for subtraction:

 2 days 17 hours 27 minutes      2.1727
-1 day  20 hours 35 minutes     -1.2035
---------------------------     -------
 0 days 20 hours 52 minutes       .9692
                                - .7640
                                -------
                                  .2052

I may try that the next time such a calculation comes along. Carrying
or borrowing 24 hours always makes me think hard, and if I'm tired
it's easy to mess up.







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