NavList:

While most of this exchange occurred I happened to be spending my free time after work refreshing my memory of linear algebra, doing some exercises from an old textbook which had me manipulating matrices with pencil and paper. 

My biggest problem behind transcription errors was missed sign changes, and even the transcription errors were themselves usually a missed negative sign.

Regardless of education requirements in prior centuries, dealing with negative numbers itself is a source of error, so there's motivation to either avoid them entirely or at least delegate them to a dedicated, "cleaning up" step (e.g. "Which quadrant should this angle be in?").

I imagine there's also the problem of different authorities assigning negative numbers to different directions.  SHA increases to the west while right ascension (e.g. 2102-D star finder) increases to the east.  Avoiding negative numbers can help avoid headaches when using references from different publishers.

David C:  Regarding you asking about why there were different testing requirements for multiplying numbers greater and smaller than 12, I imagine sailors who were paid in pre-deicmal shillings had a special place in their hearts for factors and multiples of 12, and it might have involved techniques too specalized to work well with arbitrary multipliers.