NavList:

One example when negative numbers were frequently used in celestial navigation text books is when it comes to chronometer errors and rates. Textbooks in the Swedish language use, at least since 1861 but probably earlier as well, negative numbers for the error (or rather correction) when the chronometer is "fast" and for the rate when the chronometer is "gaining". Positive for "slow" and "losing" respectively.

Then GMT = chronometer time + correction + n · rate, where n is the number of days elapsed since the correction value was determined.

 Enclosed is an example from Pettersson's Lärobok i Navigations-Vetenskapen (4th ed 1876), which in translation reads:

"1860, December 28^th, at 2^h27^m chronometer time, the error on GMT was -20^m44.4^s and daily rate -1.93^s; what was the error on GMT on 1861 February 7^th, at 14^h50^m chronometer time?"

The answer is -22^m5.5^s, the chronometer is thus fast on GMT.

On the other hand, when it comes to latitude and declination, "same name" and "contrary name" were used for a long time. My guess is that signs instead of "names" were introduced in connection with electronic calculators, i.e. mid 1970:ies or perhaps ten years later.

Lars