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This is a response to Ken Meldrew, who said (inter alia)-.

>I'm interested in learning about the navigational techniques used by
>land geographers in the late 18th and early 19th centuries;
>particularly those used by the fur traders exploring the Canadian
>West at that time (such as David Thompson, Peter Fidler, Philip
>Turnor, etc.).

and later-

>Sorry for following up my own post but it seems I was mistaken in
>thinking that Merrifield's approximate method was used by these
>explorers as it wasn't developed until about 50 years later (Cotter, A
>History of Nautical Astronomy, 1968). Jeff Godfred apparently uses
>William Hall's approximate method in his Northwest Journal article on
>David Thompson, although this method was only published in 1903.
>It doesn't much matter if one is using a calculator, but in order to get
>a feeling for the difficulty and time required to do the calculation, the
>actual method needs to be used.
>
>Cotter says that two approximate methods (the first due to Lyons
>and the second due to Dunthorne) were published in the original
>Nautical Almanac. A brief description of Dunthorne's method is
>given in Cotter and I am also slogging through a paper by Mendoza
>from the Philosophical Transaction of the Royal Society published in
>1797 where 40 exact methods are given and about 10 pages,
>generously sprinkled with algebra, are devoted to approximate
>methods. If anyone knows of other sources for the approximate
>methods used ca. 1800 (especially those discussed in Robertson's
>text, Elements of Navigation, as Thompson was known to have
>possessed a copy) I would be very grateful for suggestions.

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

Reply from George Huxtable.

Don't believe everything you read in Cotter. Jan Kalivoda and I, with other
help, have put together a long list of known (or at least thought-to-be)
errors in Cotter's History of Nautical Astronomy. The book contains a lot
of careless errors, and some others which I take to be failures of
understanding, though its many virtues remain. I will copy a
recently-updated error list below.

Dunthorne's method isn't an approximate method, as Cotter states, but a
rigorous one. That's one of the Cotter errors.

Are you aware of Arthur Pearson's website    www.ld-DEADLINK-com
which contains many useful references?

You refer to Jeff Gottfred's papers on Thompson.. This information is
valuable and useful, but I have made a number of comments and queries, the
first being this-

******Gottfred promises, at the very end of Art. 7, to show "how he uses
this time to compute longitude" and again in the heading of Art.8 promises
to do this using the data from Nov 3, but the vital last step of going from
lunar distance and LAT to longitude seems to have got missed.*********

This matter is relevant to your statement "where a longitude calculation is
fully worked". Trouble is, although that's promised, it doesn't happen.

Do you know of the interesting paper by the late Richard.S.Preston,"the
accuracy of the Astronomical Observations of Lewis and Clark", downloadable
from www.aps-pub.com/proceedings/jun00/Preston.pdf

In that paper, Preston refers to an "astronomical notebook" manuscript
compiled by the astronomer Robert.M.Patterson, which Lewis and Clark took
with them for advice. To make that notebook more readily available I have
transcribed it with a commentary to make it more intelligible to modern
readers. It's on www.huxtable.u-net.com/lewis01.htm


>Through
>the excellent articles in the archives, as well as a remarkable paper
>by Jeff Gottfred in the Northwest Journal
>(http://www.northwestjournal.ca/dtnav.html) where a longitude
>calculation is fully worked, I have come to have an understanding of
>the technique.
>
>I would really like to gain an appreciation of the mechanical aspects
>of the calculation, as it was done ca. 1800. I know that they used
>Merrifield's approximate method for clearing the lunar distance,
>using calculated altitudes of the moon and star (with a double
>altitude of the sun to establish latitude).

Not sure I agree with all the above

>But I don't know what their
>method of manual calculation was (although I have used log tables to
>perform mechanical calculations...long ago). I would like to know
>what tables were used by these navigators. Does anyone here know
>what tables were included in Maskelyne's Tables Requisite as it
>would have appeared in the late 1700's and what precision the
>numbers were given to?

I have in photocopy Maskelyne's Tables Requisite of 1767 and also his
British Mariner's guide of 1763.. Neither contains the log trig tables that
I think you are asking for. Those are required to be obtained separately.
In the Tables requisite there are "proportional logs", a special trick for
easing interpolation within the three hours between times for tabulated
lunar distances, but those are not what you are asking about, I think.

In the British Mariner's Guide Maskelyne says on pages 47 "In making these
calculations seven places of figures must be used, besides the index, and
proportion must be made for seconds, except Gardiner's logarithms be used,
out of which the logarithms may be taken at sight, to the nearest ten
seconds", In the following table he shows log sin 25deg 53' 24" as being
9.6401282., and so on.

(In my own opinion 6 figures are probably adequate in most nautical
applications.)

The quote above shows that 7-figure tables were available at that time. I
have never seen Gardiner's logarithms, or handled any such ancient tables
in a library. It's surprising to me that such accurate tables were
available at that date. Presumably, they were intended for astronomers. Who
else would require such accuracy at that early date?

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

Here is a list of known or suspected errors in Cotter's "History of
Nautical Astronomy", compiled by Jan Kalivoda and George Huxtable, with
additions from Herbert Prinz. I plan to edit this list shortly into full
page-order and repost it to Nav-L.

==============================
This first part is from Jan.

The second number marks the line of the page - excuse me, if I had missed
the numbering a bit.


--------------------------
83,10 from above

Alae sui -> Alae sive

--------------------------
84,15 from below

the longest side -> the lower shorter side ???

---------------------------
101,1 from below

the true zenith distance theta_1 -> the apparent zenith distance theta_1 at
the point O_1 (i.e. "O with the index 1")

----------------------------
107,5 from below

To the formula at this line a remark would be useful: "(mi - 1) = U".
Otherwise the deduction from the formulas at the page 105 is not clear.

----------------------------
108,3 from above

Here I am not certain. But the deduction of this formula from the formulas
III and IV on the page 105 seems to be wrong. Cotter proceeds, as "2 PZ" on
page 105 would equal "lambda" (geographical latitude !) in the picture 2,
page 104. But this not the case!

Can anybody help with this derivation? Maybe it's my fault.

-----------------------------
110,11 from above

the first edition (of Maskelyne's "Requisite Tables") -> the second edition

-----------------------------
113,6 from above

One would add to this line: "And h much less than R" (otherwise R/(R+H)
would not equal to (R-h)/R)

-----------------------------
121,11 from below

1/4 (approx.) -> 4/1 (approx.)

-----------------------------
135,9

15th century -> 16th century

-----------------------------
151, 5-7 from above

The words from "Moreover, ..." to the end of paragraph seem to be wrong to
me. When reducing the measured altitude of the Sun to the time of the first
observation by the run and azimuth of the Sun, the observer should not make
any other reductions acording to the run between observations?

------------------------------
158,16 from above

ZY is the great circle -> XY is the great circle

------------------------------
163,12 from below

after the meridian altitude -> before the meridian altitude ???

-------------------------------
217,16-18 from below

This was true only if the navigator used the table of "logarithmic
differences", giving the value of the equation X from the page 216 by
inspection. Such tables were in use, but introduced an error of 3-6
arc-seconds into result. This was tolerable in the times when lunar
positions itself (hence the true lunar distances, too) were tabulated with
the error of 15-30 arc-seconds in alamanacs, owing to deficiences of the
used theories of lunar motion.

-------------------------------
225 below, 231 passim

Cotter explains the fundamentals of the lunar distances clearly and
sufficiently. But his historical sketch of their evolution is very
unsatisfactory. With exception of Borda's method and his direct successors,
he describes only the methods of ending 19th century and neglects the most
esteemed methods from the first half of 19th century when the importance of
lunars was the greatest. Elford, Bowditch, Turner, Thomson, Cambridge
Tables - all are missing. Maybe, I will write a modest supplement to this
subject.

In particular, the Dunthorne's method mentioned on these pages was not a
indirect method, but the first and the most succesful direct method,
preceding the Borda's in time and a simpler one. It was the most common
method of lunars in the continental Europe, apart from France.

--------------------------------
239,6 from below

prop. log SMALL delta being given in the almanac -> prop.log. GREAT delta
being given in the almanac  !!!

--------------------------------
241,8 from above

the value 50' of error when neglecting the second differences is grossly
exaggerated - George Huxtable emphasized it rightly some time ago.

--------------------------------
248,5 from above

vers (PX ? PX) -> vers (PZ ? PX)             (in the numerator on the right
side of the equation)

--------------------------------
265,7 from below

delta d ->  delta h

--------------------------------
265,3 from below

tan l cos h ->  tan l csc h

--------------------------------
266,2 from above

cos h -> csc h

--------------------------------
266,9-13 from below

This text seems completely confused - either by Cotter or by the author of
original. One could try to correct it, but it would better to verify it in
the Nautical Magazine for 1848, which I haven't at my disposal.

--------------------------------
272,19 from below

if the two hour angles -> if the difference of two hour angles

--------------------------------
308,3 from below

Now the least important -> Not the least important

--------------------------------
310

The manual of Pedro Nunez (mentioned elsewhere in the book) "De arte et
ratione navigandi" from 1522 is missing - it was very important and very
early publication, the example for many other authors

--------------------------------
311

The manual of Martin Cortes appeared in 1551

--------------------------------
366

Nos 238 and 240 of bibliography seem uncomplete to me. The first (1767?)
and fourth (1811?) editions of Maskelyne's "Requisite Tables" are not
mentioned.

--------------------------------

Jan Kalivoda

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

EARLIER CORRECTIONS FROM GEORGE HUXTABLE
=======================================

Here's an updated list of some things I suspect are wrong in Charles H
Cotter's otherwise-excellent book "A History of Nautical Astronomy".

Here goes-

===========================
page 49. The third paragraph starts- "The civil day at sea commenced at
midnight", which is correct. In the next paragraph Cotter states "The civil
day commenced when the Mean Sun culminated at noon." which is
contradictory, and wrong

==========================
page 118, foot of. Cotter states
"Augmentation = Moon's semidiameter x sine apparent altitude". This is wrong.
It would be roughly true to state instead-
Augmentation (in minutes) = Moon's semidiam. (in degrees) x sine apparent
altitude
but more accurate to say-
Augmentation (in minutes) = Moon''s semidiam. (in minutes) x sine apparent
altitude / 55.

==========================
page 120, 2nd line, Cotter says- "...  body Y, which has the same APPARENT
place as body X". but fig 5 shows body Y at the same TRUE place as body X.

==========================
page 210-212, Borda's method. Here I think Cotter has got into a real mess
with his trig. The equation that precedes equation (Y) is given as -
(sin D/2)^2 = sin{(M+S)/2 + theta} sin {(M+S)/2 - theta}

Here, he has got the last term the wrong way round and it should be-
(sin D/2)^2 = sin{(M+S)/2 + theta} sin {theta- (M+S)/2}
so in consequence, in equation (Y), the second sine term in the product of
two sines is also reversed.

Similarly in the last equation on page 210, for log sin D/2, the last term
in the sum should end up as log sin (theta- (M+S)/2), not log sin ((M+S)/2
- theta), as Cotter gives it. If you slavishly follow Cotter's steps, you
will end up taking the log of a negative quantity, which is an
impossibility.

I think Cotter has realised there's something wrong, without being sure
what it is, because on page 211 he states the rules in words for clearing
the distance, and in rule 5 he says- "Find the sum of and difference
between theta and phi". Because he hasn't defined here which way round to
take that difference, the navigator will presume that he should subtract in
such a direction as to give a positive answer. So that bit of "fudging" has
got Cotter out of his problem. In fact the subtraction should ALWAYS be
theta - phi, and NEVER as stated at the foot of 210, phi - theta.

On line 3 of page 212, that's what he has written down in the calculation,
theta - phi, just as it should be.

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

There's an additional error in Rule 5, page 211, in that the last sentence
should not read-
"The result is the sine of half the true lunar distance, that is D/2.",
but instead-
"The result is the LOG sine of half the true lunar distance, that is D/2."

========================
page 226, Dunthorne's method. Dunthorne's is in fact a rigorous method of
clearing the lunar distance, but Cotter has included it in his list of
approximate methods

========================
page 237. For a navigator, it may be useful to know that alpha Aquilae is
more familiar as Altair, alpha Arietis as Hamal, and alpha Pegasi as
Markab.

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

page 250.  The four equations shown on this page all use the quantity s,
but I cannot find any definition of s. I presume that s is half the
perimeter of the PZX triangle, so-
s = 1/2 (ZX + PZ + PX)

In the third expression, for cos P/2, a quantity s with a subscript 2
appears. That little 2 appears to be a misprint should be erased.

========================
page 264. Cotter says, about finding the moment of noon by equal Sun altitudes-

"By taking the equal-altitude sights shortly before and after noon the
necessity for applying a correction for the change in the Sun's declination
in the interval is obviated, since any such change will be trifling."

I disagree with Cotter's analysis here. It seems to me that the correction
necessary for a changing declination does not reduce as the interval chosen
gets closer to noon.

===========================
page 265. Re Hall's rule. Cotter introduces delta-d as the correction in
seconds of time..
This ought to be delta-h
and in the equation near the foot of the page, cos h should be cosec h.

page 266. Similarly, in the equation at the top of this page cos h should
be cosec h.

==========================
page 354. Napier's rule. I suspect that the second expression, shown as-

cos x = cos y cos z  is wrong, and should be-
cos y = cos x cos z

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

Below are some additions to the error-list, which have recently come to light.

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

page 213, log computation, bottom half of page. The logs column is
mis-ordered. The line to the right, Ar comp cos, which is placed alongside
the value for d, relates to s, and should be placed alongside that
quantity.

page 247, near bottom of page. Equation for hav P should have its
square-root sign removed.

page 250. Mid-page. Cotter says "It can be demonstrated that the first is
suitable for cases in which P is near 90 degrees." It seems to me that the
first formula is completely UNsuitable in those circumstances, giving an
inaccurate and ambiguous result.

page 275, paragraph quoting from Sumner. Herbert Prinz has pointed out that
"defect" should become "difficulty", and "observation" should become
"projection".

=============================
end of list of errors in Cotter.

George Huxtable

================================================================
contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
================================================================

   

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