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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: David Thompson's Navigational Technique
From: Frank Reed CT
Date: 2004 Jun 1, 19:43 EDT
From: Frank Reed CT
Date: 2004 Jun 1, 19:43 EDT
I wrote earlier:
> (1/2)*dh_moon^2*cot(d) - (1/2)*(dh_moon*A)^2*cot(d)
> so by entering table 10 twice, first with the Moon's altitude
> correction and then with the correction multiplied by A (which we've
> already calculated!), you get the complete quadratic correction.
Ken M quoting Maskelyne:
>>>Call the principal effect of parallax, found by the preceding rule, the
parallax in distance; and find the parallax answering to the Moon's
altitude. Then to the constant logarithm 0.941 add the logarithm of
the sum of the parallax in altitude and the parallax in distance, the
logarithm of the difference of the same parallaxes, and the cotangent
of the observed distance of the Moon from the star (corrected for
refraction, and the principal effect of parallax), the sum, abating 13
from the index, is the logarithm of the number of seconds required,
being the second correction of parallax; and is always to be added
to the distance of the Moon from the star, first corrected for refraction,
and the principal effect of parallax found above, in order to obtain the
true distance; unless the distance exceeds 90 degrees, in which case
it is to be subtracted.<<<
And added:
"If I'm reading this right, he has
8.73 * (dh_moon + A) * (dh_moon - A) * cot(d) * 1/2"
Two things:
1) if we're sticking with my notation for "A" (not because it's perfect or anything, but just to stick with one notation), then where you've written A, it should be dh_moon*A. This product is the straight linear term in the series expansion that's calculated at an earlier point in the clearing process and it's what Maskelyne means by the "principal effect of parallax" or "parallax in distance". It's the linear term in the series expansion. This product is the portion of the Moon's altitude correction that acts along the arc of the lunar (A is a pure ratio).
2) I can't see where you got a factor of 1/2 out of Maskelyne's instructions. There *is* a factor of 1/2 but it's folded into the "8.73" factor. That factor (besides the 1/2) arises from working in minutes and seconds of arc instead of pure angles (a.k.a. radians). You'll find that 8.73 is pi/360 (with a factor of 1000) which is why Maskelyne says to "abate 13" from the index instead of the more common 10 or 20.
In that 'Easy Lunars' post, I mentioned that you have to divide by 3438 (the number of minutes of arc in a unit "one radian" angle). That's different from Maskelyne's 0.00873 only because it was prefered back then to work in minutes and seconds instead of the minutes and tenths of minutes which are now standard in navigation.
And:
"and he's left out the dh_moon from the 2nd term in square brackets."
It's in there. He's talking about the actual correction along the arc rather than the intermediate quantity "A".
After accounting for those "two things" above, unless I've missed something, I think you'll find that Maskelyne's instructions agree with what I described regarding Moore's Table 10.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
> (1/2)*dh_moon^2*cot(d) - (1/2)*(dh_moon*A)^2*cot(d)
> so by entering table 10 twice, first with the Moon's altitude
> correction and then with the correction multiplied by A (which we've
> already calculated!), you get the complete quadratic correction.
Ken M quoting Maskelyne:
>>>Call the principal effect of parallax, found by the preceding rule, the
parallax in distance; and find the parallax answering to the Moon's
altitude. Then to the constant logarithm 0.941 add the logarithm of
the sum of the parallax in altitude and the parallax in distance, the
logarithm of the difference of the same parallaxes, and the cotangent
of the observed distance of the Moon from the star (corrected for
refraction, and the principal effect of parallax), the sum, abating 13
from the index, is the logarithm of the number of seconds required,
being the second correction of parallax; and is always to be added
to the distance of the Moon from the star, first corrected for refraction,
and the principal effect of parallax found above, in order to obtain the
true distance; unless the distance exceeds 90 degrees, in which case
it is to be subtracted.<<<
And added:
"If I'm reading this right, he has
8.73 * (dh_moon + A) * (dh_moon - A) * cot(d) * 1/2"
Two things:
1) if we're sticking with my notation for "A" (not because it's perfect or anything, but just to stick with one notation), then where you've written A, it should be dh_moon*A. This product is the straight linear term in the series expansion that's calculated at an earlier point in the clearing process and it's what Maskelyne means by the "principal effect of parallax" or "parallax in distance". It's the linear term in the series expansion. This product is the portion of the Moon's altitude correction that acts along the arc of the lunar (A is a pure ratio).
2) I can't see where you got a factor of 1/2 out of Maskelyne's instructions. There *is* a factor of 1/2 but it's folded into the "8.73" factor. That factor (besides the 1/2) arises from working in minutes and seconds of arc instead of pure angles (a.k.a. radians). You'll find that 8.73 is pi/360 (with a factor of 1000) which is why Maskelyne says to "abate 13" from the index instead of the more common 10 or 20.
In that 'Easy Lunars' post, I mentioned that you have to divide by 3438 (the number of minutes of arc in a unit "one radian" angle). That's different from Maskelyne's 0.00873 only because it was prefered back then to work in minutes and seconds instead of the minutes and tenths of minutes which are now standard in navigation.
And:
"and he's left out the dh_moon from the 2nd term in square brackets."
It's in there. He's talking about the actual correction along the arc rather than the intermediate quantity "A".
After accounting for those "two things" above, unless I've missed something, I think you'll find that Maskelyne's instructions agree with what I described regarding Moore's Table 10.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
