That does not seem right to me. From how I understand it, -60 seconds or -1 minute is NOT the same as -1degree 59minutes. That's like saying, in time math, that subtracting 60 seconds is the same as subtracting 1 hour 59 minutes: really, in h:m:s, -60s = -0:01:00; or, in a similar procedure in decimal math, that's like saying that subtracting 10/100ths from 0 would convert -10/100 to -1/10, and then to carry the negative sign, add 10/10 and subtract 1 unit, which would be -1.90 -- when it should really be -10/100 reduces to -1/10, and then the negative sign carries out to the unit of 0, so -0.10.

Think of it this way, in the final DMS or HMS notations, the DMS/HMS operators are a higher precedence than the unary minus operator -- so -0° 1' 0" is equivalent to -(0° 1' 0") = -(0 + 1/60 + 0/3600) = -(0 + 0/60 + 60/3600) = -60/3600 = -60 seconds

Checking a few online calculators/converters, they seem to agree with me. (I haven't yet found something more authoritative than a calculator... but it matches my reasoning above): If I go to this online DMS converter, and change -0.0167 (which is slightly more negative than -1/60 of a degree, or -1minute), it reads -0° 1' 0.12". And on another online tool, if you start with 0° 0' 0", and subtract 0° 1' 0", you get -0° 1' 0". Can you show a reference where -60seconds is displayed as -1° 59' 0"?

This is not a degree to decimal converter. This takes angles in deg,min,sec format (all integers) and adds or subtracts them.

The degree min sec format is a polynomial of the form (10^x)a + (1/60)a + (1/3600)a, if we were going to express this as decimal, where a and x are integers. This is no different from any decimal, which is in powers of 10: (10^2)a + 10b + c + (10^-1)d + (10^-2)e ...; we always simplify decimal numbers by borrowing from the next highest term if subtraction is negative, and carrying excess if addition results in a number greater than equal to the base we are working in.

The book I cited in my previous post (Trigonometry Refresher by Albert Klaf) describes the procedure on page 12. I would post the link, but I don't believe links make it through correctly. Google books has that page available.

As for online calculators -- I have found issues with them that are not consistent with what I understand to be the algebraic properties of angle addition and subtraction. Any calculator that does not reduce 60 min to 1 degree by carrying over to the next higher term, is not correct. I will consider your argument and see if I can come up with a proof of my method, or counter-example to yours, later, if the above does not persuade you.

This is no different from any decimal ...

But if I have the decimal number -345, this is
    -(3*10^2 + 4*10^1 + 5*10^0)
or
    -1 * (3*10^2 + 4*10^1 + 5*10^0)
or
    -3*10^2 + -4*10^1 + -5*10^0
or
    -3*10^2 - 4*10^1 - 5*10^0
and not
    -3*10^2 + 4*10^1 + 5*10^0
and similarly for numbers represented in sexagesimal base.

Negative numbers are only permitted in the degree column.

Unfortunately, if one has a d/m/s tuple  @dms = (3, 4, 5) one cannot negate it by writing -@dms. One has to negate each element:
    @neg_dms = map -1*$_, @dms;
or (-3, -4, -5). IIUC, the notation -3° 4' 5" is equivalent to the tuple (-3°, -4', -5") analogously to decimal notation. Klaf is describing and giving an example of base-60 subtraction with borrowing, but that is not fundamentally different from base-10 subtraction.

Update 1: Added the "Negative numbers ..." quote before the second paragraph for clarification, and various other minor wording changes and additions.

Update 2: Here's the Klaf subtraction example in terms of normalizing everything to decimal arc-seconds, doing a decimal subtraction and reducing arc-seconds to a d/m/s tuple:

c:\@Work\Perl\monks\thechartist>perl -wMstrict -le
"sub dms_2_secs {
   my ($degs, $mins, $secs) = @_;
   ;;
   return $degs * 3600 + $mins * 60 + $secs;
   }
 ;;
 sub secs_2_dms {
   use integer;
   ;;
   my ($secs) = @_;
   ;;
   my $degs = $secs / 3600;   $secs %= 3600;
   my $mins = $secs /   60;   $secs %=   60;
   ;;
   return $degs, $mins, $secs;
   }
 ;;
 my $klaf_arc_secs = dms_2_secs(83, 18, 21) - dms_2_secs(39, 41, 28);
 my @klaf_dms      = secs_2_dms($klaf_arc_secs);
 print qq{(@klaf_dms)};
"
(43 36 53)
[download]

(Update: The Klaf addition example also works correctly.) (Update: Moreover, the reversed subtraction  dms_2_secs(39, 41, 28) - dms_2_secs(83, 18, 21) yields the expected (-43 -36 -53) tuple.)

---

Give a man a fish:  <%-{-{-{-<

update: AnomalousMonk said it way better than I did. Original left for historical purposes.

---

You've misunderstood. I am claiming you are representing negative angles incorrectly. After the subtraction, if you get -60 seconds, you should have the equivalent of -1min. -1min is represented canonically as -0° 1' 0", not as -1° 59' 0". The answer you list in Re^2: How to write testable command line script? of my $test_7 = [ 0, 0, -60]; my $answer_7 = [-1, 59,  0]; is wrong. Honestly, your format doesn't allow you to use the right answer, unless you know how to get perl to store the "-0" correctly (for example, using text), because it should be my $answer_7 = [-0, 1, 0];.

regarding normal base-10 subtraction, when you subtract 0.1 from 0.0, you do NOT borrow one from the ones column, making it -1; then have 10 in the tenths column, and subtract 1, leaving 9/10ths, or -1.9. 0.0-0.1 is obviously -0.1. What you are claiming in DMS is exactly equivalent to claiming that -60sec is -1° 59' 0".

 0.0        Try borrowing: (0-1=-1).(0+10=10) => (-1).(10)
-0.1                                            -( 0).( 1)
====                                            ==========
-0.1                                             (-1).( 9)  => -1.9

That's nonsensical, so obviously not the right method for subtracting a bigger from a smaller.

To subtract a bigger number (M) from a smaller number (S) to do a difference D=M-S, you have to do -(S-M)=D
So start with the S-M:
 0.1
-0.0
====
 0.1
Then negate the answer
-0.1


The same is true in other bases, like the base 60 of DMS:
 0°00'00"
-0°00'60"
========
??°??'??"

You start by making the bigger the first,
 0°00'60"
-0°00'00"
========
????????
Then simplify the top term and subtract
 0°01'00"
-0°00'00"
========
 0°01'00"

The negate: 
-0°1'0"

However you calculate it, whether you do it in decimal degrees, or in DMS, the only difference between -60s and +60s should be the sign at the beginning. +60s is (0° 1' 0"), and -60s is -(0° 1' 0"), which gets simplified to -0° 1' 0", exactly like -(0.016) gets simplified to -0.016 -- the negative sign is placed next to the most significant digit, but implies the whole result is negative. -0.016deg converts to -(0.016deg + 0/60 + 0/3600) = -(0 + 1/60 + 0/3600) = -(0° 1' 0"). You do not interpret -0.016 = (-0) + (0/10) + (1/100) + (6/1000) -- because that math evaluates to +16/1000, which is wrong. You evaluate -0.016 as -(0.016) = -(0 + 0/10 + 1/100 + 6/1000) = -(16/1000), which is right; or, in the other direction (-16 thousandths) = -(16 thousandths) = -(1 hundredth + 6 thousandths) = -(zero ones + 0 tenths + 1 hundredths + 6 thousandths). In the same way, -60sec = -(60sec) = -(0 + 0/60 + 60/3600) = -(0 + 1/60 + 0/3600) = -(0° 1' 0") = -0° 1' 0". Negative Degress minutes seconds (-x°y'z") is not (-x°) + (y') + (z") = -x + y/60 + z/3600, it is -(x° + y' + z") -(x + y/60 + z/3600).

Have I said this in a way that makes my point more clear?

the resource https://books.google.com/books?id=YRNeBAAAQBAJ&lpg=PR1&dq=Albert%20Klaf's%20Trigonometry%20Refresher&pg=PA12#v=onepage&q&f=false did not show an example when the result is negative. I do not believe you are extending into negative results correctly.

(and linking works just fine to me: click on the LINK icon once you've navigated to the google books page, copy the URL, and paste it here between square brackets, like [https://books.google.com/books?id=YRNeBAAAQBAJ&lpg=PR1&dq=Albert%20Klaf's%20Trigonometry%20Refresher&pg=PA12#v=onepage&q&f=false].