NavList:

Most of the NASA information on the backstaff is wrong: the backstaff (invented around the end of the 16th century) wasn't a quantitative leap because the mariner's astrolabe already measured the angle between an object and the horizontal plane and so did the cross staff (introduced into navigation around 1520-1530, but invented much earlier). The backstaff was an improvement on the cross staff as the observer did not have to look at the sun (which was dangerous to the eye) but could face away from the sun (therefore 'backstaff') and match the shadow of the shadow vane with the horizon. As the moon does not cast a strong shadow the backstaff could not be used with the moon (apart from the fact that no tables for the moon's position were available at the time, so it could only be used in navigation for predicting the tide).

By the way, one could measure angles  up to 90 degrees with a backstaff because it had 2 arcs of 30 and 60 degrees. If you want to know more, see Mörzer Bruijns, Sextants at Greenwich, Oxford 2009, 17 - 21, 94 - 102 (descriptions of the items in the NMM Greenwich). There is also a thorough discussion of the development of the backstaff by Nicolàs de Hilster in: Bulletin of the Scientific Instrument Society, Nr. 110, (2011), 14 - 22.