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Re: Errors in Cotter's book
From: Jan Kalivoda
Date: 2003 Jan 2, 22:39 +0100

```The remarks are inserted below.

----- Original Message -----
From: "George Huxtable"
To:
Sent: Sunday, December 29, 2002 11:02 PM
Subject: Errors in Cotter's book

> 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."
>
> This appears to be quite contradictory, and my guess is that Cotter had
> intended to say "The ASTRONOMICAL day commenced when the Mean Sun
> culminated at noon."

I rather suppose that "The civil day commenced when the Mean Sun culminated at
MIDNIGHT" is meant as a repetition of the same thing twice. The next (last)
paragraph on the page 49 would then start quite logically.

>
> ==========================
> 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}
>
> I think 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.
>
>

Yes, you are right in my opinion. Cotter had wrongly neglected minus in the
formula for the difference of two cosines of different angles, which can be
disposed of only by multiplication of the 9th line from bottom of the page
210 by "-1". This renders both sides of the 7th line from bottom positive
after an arrangement, but with the terms of sum and difference reversed. The
5th rule on the page 211 confirms it: "Find ... difference between theta and
phi.", as you say further.

>
> 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, as it should be.
>
> =========================
>
> There's an additional error in Rule 5, page 211, in that the last sentence
>
> "The result is the sine of half the true lunar distance, that is D/2.",
>
>
> "The result is the LOG sine of half the true lunar distance, that is D/2."

Just so.

>
> ========================
>
> 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.

It is very funny that Bowditch from 1888 (thanks to Dan Allen!) doesn't
mention the names "Altair", "Hamal" and "Markab", it calls just these three
stars as "alfas" only! This is perhaps a wonderfull and long text tradition.

>
> =========================
>
> 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)  Can anyone confirm that?
>
> In the third expression, for cos P/2, a quantity s with a subscript 2
> appears. I have no idea where that subscript appears from and presume it's
> just a misprint. That little 2 should I think be erased.

Yes, you are right in both cases. "s" is very commonly used as the half
circumference of a spherical triangle in all textbooks of spherical
trigonometry and Cotter had forgotten to mention it.

The index "2" is to be deleted, it's a pure mistake.

>
> ========================
>
> 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 change in declination will be exactly the same whether the
> equal altitudes are measured over a short interval or a long one. Anyone
> disagree with that?

Here I agree with Cotter - the change of declination during a short interval
to noon and to the second altitude is to be smaller than during a longer
interval and its effect on the deduced time of culmination smaller, too,
isn't it? (This is the real question, not a tag.)

But near the meridian the change of altitude of the Sun is small and the time
deduced from it inaccurate, unless the zenith distance is very small.
Therefore Lecky says that this method is appropriate only for a sailing ship
becalmed (without any own motion to noon) near the equator.

>
> ===========================
>
> 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
>

Just so, the spherical cosine formula leads to this expression for a right-angled spherical triangle.

>I think there may be a further error in Cotter, on page 233, 5 lines down,
>in which he says "corrections to apparent distance are -mX and -sY."
>Shouldn't this be "-mX and +sY"?

Yes, I think so.

Jan Kalivoda

```
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