That's a tough one - and an interesting challenge! Almost everything I could think of either didn't come with a risk of disaster (or at least, no risk of major disaster), or has had major disasters.
The best I came up with was determining longitude based on the lunar distance. It could be disaster for a single ship that got it wrong, but I don't think there were major disasters during the 80 years (2-3 generations) it was in service.
I can think that a fleet might have ended up in the wrong place (like the Scilly naval disaster of 1707), so there could be a major disaster, but a fleet also has enough independent people doing their own reckonings that it would be self-correcting. I looked through the UK and French lists of ship disasters, but found no multi-ship disasters based on location errors during 1767 to ~1850.
(If you want a physical technology, rather than mathematical technique, then Hadley's octant, integrated with Vernier's scale, which made the octant precise enough that sailors could use to judge longitude usefully. In turn, it was effectively replaced in the 1800s by the sextant.)
But I don't know enough about nautical history to know if there were major disasters caused by octant/sextant use which were comparable to pre-lunar distance methods.
The best I came up with was determining longitude based on the lunar distance. It could be disaster for a single ship that got it wrong, but I don't think there were major disasters during the 80 years (2-3 generations) it was in service.
I can think that a fleet might have ended up in the wrong place (like the Scilly naval disaster of 1707), so there could be a major disaster, but a fleet also has enough independent people doing their own reckonings that it would be self-correcting. I looked through the UK and French lists of ship disasters, but found no multi-ship disasters based on location errors during 1767 to ~1850.
(If you want a physical technology, rather than mathematical technique, then Hadley's octant, integrated with Vernier's scale, which made the octant precise enough that sailors could use to judge longitude usefully. In turn, it was effectively replaced in the 1800s by the sextant.)
But I don't know enough about nautical history to know if there were major disasters caused by octant/sextant use which were comparable to pre-lunar distance methods.