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On 2018-11-18 21:45, Steve E. Bryant wrote:
>
> When performing calculations involving degrees and minutes, I much prefer 
converting the minutes to degrees, proceeding with the additions and 
subtractions, and then converting the decimal degrees back to minutes.

A calculator can do sexagesimal addition and subtraction directly with a
trick from the days of mechanical calculating machines. For example, to
calculate 1°05′ + 2°10′, enter the operands as 1.05 + 2.10 = 3.15.

Of course it's not always that simple. A carry may occur, for example in
1°55′ + 2°30′. The calculator says 3.85. Clearly 85 minutes is illegal.
To force a carry add 40′ (.4 on the calculator) to obtain 4.25. (Back in
the day, some computers called this trick the "rule of 40.")

When the minutes field in the sum is 60 or greater it obviously needs
adjustment. I call that a "type 1" carry. On the other hand, 50′ + 50′
comes out as 1.00 on the calculator. That's a "type 2" carry — if you
look only at the result you don't see anything wrong. However, the type
2 can be anticipated because it occurs only when the minutes sum is 100
or greater. That requires a large number of minutes in both operands —
one in the 50s and the other in the 40s or 50s.

Degrees (or hours) minutes seconds are no problem. For example, add 1h
28m 50.4s to 2h 31m 15.8s. Sum on the calculator is 3.59662. Since the
carry from seconds will force a carry from minutes, you can save time by
adding 40 to both minutes and seconds with a single operand (.404).
Result is 4h 00m 06.2s. Note that a carry also occurred from the
fractional part of seconds, but that's a decimal number so the carry
took care of itself.

Subtraction is similar to addition, except that you subtract 40 to force
a borrow. For instance, work 1°25.2′ - 48.8′ as 1.252 - .488 = .764.
Subtract 40 from minutes to obtain 36.4′.

A type 2 borrow occurs when the minutes differ by more than 40. For
example, 1°01′ - 42′ appears on the calculator as .59. From inspection
of both operands it's obvious a borrow must occur, so subtract 40′ to
get the correct 19′.

The rule of 40 can deal with negative numbers too. By experimentation
with simple calculations you can work out the rules. With a stack type
calculator (Hewlett-Packard) it's easy to swap subtraction operands to
dodge these issues.

I use the decimal point to separate degrees from minutes, but a
sexagesimal can be entered as an integer. That's what they did with
mechanical calculators. A disadvantage is that when an operand is a
whole number of degrees (as when computing 90 minus something) you have
to pad the right side with zeros.

Unfortunately I know no easy way to multiply and divide sexagesimal
values on a calculator.


> There may be some statistical reason for not doing it that way; or, is the 
practice perfectly acceptable.
> There are those professionals who teach navigation that insist upon working 
only with degrees and minutes, unless it's necessary to use decimal degrees 
for calculator entry in which case they carry the decimal out to the fifth 
place.

I think four places are enough. One minute is about .02°, so a tenth is
.002°. Thus a conversion to degrees with four decimal places properly
rounded has far less error than tables rounded to tenth minute precision.

   

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