Navigational Arithmetic



Doing navigational arithmetic

Arithmetic is arithmetic, right?  Well, yes it is.  However, celestial navigation works with two kinds of arithmetic that are not common to people and which are not easily accomplished using ordinary calculators:

adding and subtracting time, and

adding and subtracting degrees, minutes and tenths of minutes.

Adding and subtracting time

As you know, there are 60 seconds in a minute and 60 minutes in an hour.  That means that when you "carry" or "borrow" in ordinary addition and subtraction you have to be careful when you carry and you don't borrow 10.  Let's look at a couple of samples.


When adding 32 seconds and 48 seconds, you arrive at 80 seconds.  You would report that as 1 minutes 20 seconds.




Notice that when you reached 60 seconds (not 100 seconds) you carried a minute. Similarly when adding minutes, you will carry an hour once you've reached or exceeded 60 minutes.  Try this one:


12h 43m 37s

+4h 32m 52s

If you got 17h 16m 29s you did well.  If you got 16h 75m 89s you forgot to carry when you exceeded 60 seconds and 60 minutes.  Try this one:


22h 28m 12s

+6h 37m 22s

Hmm, what do you do about 29 hours since there are only 24 hours in a day?  Well, you've moved into the next calendar day, haven't you?  Therefore, a good answer for this addition problem would be

5h 05m 34s, next day


As with carrying, the trigger value must be 60 when dealing with both seconds and minutes and 24 when dealing with hours.  Try this one:

14h 39m 17s

- 8h 52m 43s

Until you get to the point where you can do this in your head, you might consider rewriting the problem like this

13h 98m 77s

-8h 52m 43s


'See what's happened?  We've borrowed one hour (60 minutes) from the hours column and added it to the minutes column.  We've also borrowed one minute (60 seconds) from the minutes column and added it to the second column.  Now the subtraction is straight forward.  If you got 5h 46m 34s, you did a great job.

Degrees and minutes

When you add and subtract degrees and minutes, you'll do the same things.  When borrowing to subtract minutes convert one degree to 60 minutes.  When carrying add a degree when you reach or exceed 60 minutes.

The funny thing about angular measure units in celestial navigation is that minutes of arc are subdivided into tenths of minutes, not the seconds of arc you might expect.  The reason this is done has to do with reading the micrometer drum of a sextant.  Light is often poor when sights are taken and seconds of arc are tiny subdivisions to try to read while preserving your night vision, so the micrometer drum on sextants is marked off in minutes of arc and tenths of minutes.  That's easy enough except that calculators don't deal with mixed units like that very well, so you'll be best off just doing the arithmetic long hand.

Let's try one:

229o 37.8'

+  4o 52.6'


When adding this one, the 8 and 6 give you 14.  You carry a 1 since you're working in decimal units in this column. 7 + 2 + 1 = 10.  Again, you put down a 0 and carry a 1, still working with decimal units in this column.  3 + 5 + 1 = 9.  Now, you're working with minutes, so you put down a 3 , and carry 1 degree forward to the degrees column.  If you got 234o 30.4', you did well.