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I think Brad has got a few aspects of this matter wrong.

I was writing about just Cook's North Atlantic crossings in the years 1764
to 67, at a time when he had no way of assessing longitude except (as Brad
says) incrementally; working out how far his longitude had shifted, from
one noon to the next, by dead reckoning, and integrating it up. In that
respect, it was quite unlike latitude, which was measured anew each time
the Sun could be seen at noon. Such Westing was subject to all sorts of
errors, with systematic and random compnents: leeway, current drift, poor
helming, errors in assumed magnetic variation, errors in log and hourglass
calibration, any drag of the log with respect to the water. As well as the
numerical errors that Brad refers to, arising from digitising the hourly
log readings to tenths of a knot (but maybe sometimes eighths; it wasn't
always clear) of a knot, and course bearings to a half point.

The end result of the traverse table day's-work would be given, and noted,
to 0.01 miles,as change in  lat and long, in, say, 90 miles of daily
westing., a far greater precision than the initial data warranted. So I
think we can exclude eny affect of round-up and digitising error in the
overall result.

====================

Where Brad is quite wrong is in writing- "I remain convinced that Cook
would not have changed his navigational method for any voyage, short or
long, navigational methods being so conservative." Not a bit of it!

After, in his earlier years, being limited to latitude sailing, and
longitude by dead-reckoning, Cook learned the craft of navigation by lunars
at the start of his first circumnavigation in 1768.  Then, add the start of
his second, in 1772, he had to adapt to using a chronometer, which was
principally checked against his lunars. It was that combination which
became invaluable to Cook. So, contrary to Brad, no mariner has ever had to
be more adaptable to new ideas than was Cook.

Brad continued-
"George pointed out that he reckoned his longitude from EACH departure, not
necessarily just the Lizard.  I interpreted that to mean that when he left
(took his departure) from Australia, he reckoned his longitude from there.
That would indeed result in an accumulation of error."

No, on two counts. Leaving for an ocean voyage, such as from the English
Channel across the North Atlantic, a navigator would take his "deparure"
from the last headland seen, such as Start Point or the Lizard, and start
streaming the log each hour and summing each "day's work". But how did he
know that that would be the last headland seen? He didn't, but it's simple.
If, later, he  happened to see another recognisable waypoint, such as St
Agnes' light on Scilly, then the previous work was discarded, and a new
departure taken from that known point. North Atlantic crossings were such
that there would be no further waypoints after St. Agnes, until
Newfoundland was reached.

Long before Cook visited Australia, he was using lunars, and because those
lunar distances in the Almanac were based on the longitude of Greenwich,
all Cook's longitudes were then worked from Greenwich.

Brad referred to my joint work on Cook, with Ian Jackson, as "recently
presented in the UK at their equivalent Navigation Symposium". What has
happened, so far, is that this has just been published as a paper in the
current issue of The Journal of Navigation (not to be confused, as it often
is, with the US journal, "Navigation"). The details are, for anyone
interested, "Journey to Work: James Cook's Transatlantic Voyages in the
"Grenville", 1764-1767, by George Huxtable and Ian Jackson. It's under
copyright, but I will be happy to send it as an attachment to anyone who
asks.

As for the Greenwich Symposium, that isn't until 6-7 May, and I've been
asked to present a short talk there which will be loosely based on that
work. Nor will that meeting be any sort of equivalent to the gathering at
Mystic, concentrating as it will on historical matters: it seems to be
attracting historians fom various disciplines, as well as navigators.

George.
.
contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
----- Original Message -----
From: "Brad Morris" 
To: 
Sent: Sunday, March 21, 2010 5:03 PM
Subject: [NavList] Re: Cook's Incremental Reckoning


Hi Frank

Perhaps you mis-understood what I was trying to convey.

I would assume that he used incremental reckoning on his several year
voyages. George pointed out the cross atlantic voyages as a function of his
familiarity with them and his detailed research of them, recently presented
in the UK at their equivalent Navigation Symposium.  I remain convinced
that Cook would not have changed his navigational method for any voyage,
short or long, navigational methods being so conservative.

George pointed out that he reckoned his longitude from EACH departure, not
necessarily just the Lizard.  I interpreted that to mean that when he left
(took his departure) from Australia, he reckoned his longitude from there.
That would indeed result in an accumulation of error.

You wrote:
Round-off error only becomes an issue when the navigation is MORE accurate
than the precision of the calculations

Here we dis-agree.  Whatever the navigation calculations show, there is
almost always a truncation or roundoff error at the last digit.  Your
absolute, true position will not be properly represented by the digits you
show.  For example, suppose that your position is represented as W40d 29m
32s.  Is that a 32 seconds with 100 trailing zeros after the decimal point?
Or, as I contend, just your current representation of position, to the
nearest second.


Then you wrote:
Round-off adds a small "random walk" to the data. But if the step size in
that random walk is well below the normal uncertainties in the process of
DR navigation itself, then the round-off is irrelevant.

Ah! So we do agree that there is some roundoff!  Yes, it will be irrelevant
when we consider journeys of only a few days or weeks. Of course, this is
also a function of the step size of record.  Does Cook record to the
nearest second, minute or 1/4 degree? Roundoff can and will pose a greater
challenge when the period extends to years.

An analoguous situation occurs in robotic equipment.  It can be placed into
absolute positioning mode (relative to reference location, like the
Greenwich meridian) or into incremental mode (relative to where you are
right now).  Placing the equipment into incremental mode, we command it to
go plus 1.234 inches and then back 1.234 inches.  With only a few passes,
external calibration equipment does not show any accumulation of error.
However, over long term, those round off errors DO accumulate, and the
calibration equipment will show a significant error.  It is a well known
problem with incremental positioning and robotics, requiring that the robot
find the reference point periodically or face defects in manufacture.

As Cook journeyed through the Pacific, using his incremental reckoning,
there could (note: I don't say absolutely must have been) be an
accumulation of error. He journeys from England, around the Pacific and
returns to England after a long period of time.  IF there was an
accumulation of error, then he could have a large undershoot (too far out
into the Atlantic when he reaches the appropriate latitude) or large
overshoot (crashes into Africa).  Even with emergent navigational methods
like the one of the first chronometers and Maskelynes Tables, finding his
absolute longitude would have been a challenge.

So my inquiry remains, do his long term journeys show this accumulation?
With today's knowledge of longitude, do his longitudes of Pacific locations
agree, or do they show a relative drift with time.

I do hope this clarified what I was trying to convey!

Best Regards
Brad

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