NavList:

At the beginning of this discussion, Rommel asked:
"Question I have would be if Bligh had an octant or a sextant and if he had either did he do the math in his head?"

Yes and yes. The navigation in the boat after the mutiny would have been essentially identical to the navigation of Joshua Slocum a century later. Bligh used dead reckoning for longitude, and Noon Sun sights for latitude.

Dead reckoning in higher latitudes involves the mathematical complexity of dealing with the differing sizes of latitude and longitude degrees, but in the tropics where Bligh and his compatriots were sailing, it's simple: longitude and latitude are just like x and y on a basic graph. You work out how many miles you've travelled west and how many miles north and you add (or subtract as appropriate) directly from the previous day's latitude and longitude. Dead reckoning usually requires a simple timepiece to keep track of the number of hours spent on each heading, if there is any change of heading. Bligh, like Slocum, had a simple watch. I've also seen speculation that Bligh could have kept track of elapsed time using time sights for local time. I consider that an improbable fantasy.

Latitude by Noon Sun is simple enough. They had declination tables for the Sun in their copy of Moore's New Practical Navigator. As long as they didn't lose track of the date, and as long as they had a very rough idea of longitude (to estimate approximate Greenwich time), they could pull out the Sun's declination accurate to within a few miles. Then it's just Lat = ZD + Dec (where ZD is the corrected zenith distance, which is 90° - the corrected altitude). Measuring the altitude would have been no problem either with the old octant, which Bligh talked about openly in his report, or with the fancier sextant, which Bligh did not describe in his report. The sextant may well have provided something that Mark Coady alluded to recently: a morale boost. Sure, the octant works great, but wouldn't one's confidence be enhanced by the finer quality of the metal sextant? But why did Bligh not mention the sextant? One possibility here is that he knew that those reading the report would know just enough about navigation to reach the wrong conclusions. They might have seen luxury and even ease in the word sextant. Perhaps they would have assumed that longitude was then possible, since sextants in this era were almost exclusively reserved for lunars and therefore longitude. Omitting the sextant from his report was no lie even though it may appear at least deceptive. The sextant provided no advantage. They could have used it as a knife and a hammer to prepare fish for dinner and never used it for navigation at all with the same result in the end. The octant was sufficient for latitude at noon.

Note that Joshua Slocum a century later did shoot a lunar once (one single lunar on the circum-navigation). In that open boat, Bligh probably would have loved to shoot and work up a lunar, but the mutineers refused to give him the almanac, and without that data lunars were impossible. 

I'll reiterate here that Bligh did do the math "in his head" -- it was trivial math that anyone could do. Bligh's computational skill was not required in this survival story, yet it surely gave him the confidence to complete the journey without self-doubt.

Frank Reed