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Bruce Cutting you wrote: The "subtraction" rule is from numerical analysis.  Always avoid
subtraction and division when possible.  It is best to aggregate the
numbers before doing either subtraction or division.  Te reason is that
subtraction and division produce larger errors because the precision of
the components are added for thee operations.

Bruce.  This might be true, but I’m not sure that it has much to do with the reason for + or – in the Hc formula.  The + or - is more to do with whether the star is in the same or opposite hemisphere to the observer. (You may note that even AP3270/HO249 vols 2 & 3 have separate pages for declination ‘SAME’ or ‘CONNTRARY’ to latitude).  For ‘SAME’, the side of the PZX triangle opposite the observer’s zenith Z is co-dec.  For ‘CONTRARY’, it’s 90+dec.  Now in the Hc formula, sin co-dec from the cosine formula can be written cos dec, and sin (90+dec) can also be written cos dec with no change in sign.  This is because if you look at sine and cosine curves from -360 to =360 the sine curve is 90 degrees ahead on the cosine curve.

However, when cos (90+dec) from the cosine formula changes in the Hc formula, it still magically becomes sin dec, but its value becomes negative.  E.g. Observer in Northern Hemisphere, declination of body 30S:  Therefore, cos (90+30) = cos 120 = minus 0.5= minus sin 30. DaveP