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Old 20-09-2013, 06:06   #1
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Capn Cooks' Chronometer

Recently someone asked me to calibrate a chart of Easter Island created during Captain Cooks second voyage in 1772 to see how accurate it was compared to Google earth. The chart can be found here:

File:Map and sketches of Easter Island, in 1772-5.jpg - Wikimedia Commons

Attached is a screenshot of Google earth showing where Capn Cooks' chart thought Easter Island was and where Easter Island actually is. (Google earth is accurate because the airport is at the right coordinates).

Turns out he was off by about 20 nm in longitude and 3 nm in latitude.

He was using the new K1 Chronometer on this voyage. I would have thought it would be more accurate.

If you look at the easterly tip of the island (C2) on Capn Cooks chart has a longitude of 109 40' while its actually 109 14' on Google Earth an error of 26'.

Question: How many seconds was the chronometer off?
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Old 20-09-2013, 06:15   #2
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Haven't had my second cup of coffee yet so mentally not up to the task, but two good related books below.

http://www.amazon.com/Blue-Latitudes.../dp/0312422601

http://www.amazon.com/Longitude-Geni.../dp/080271529X
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Old 20-09-2013, 06:28   #3
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1 min 44 sec of error in terms of time.

Is this a trick question or a simple troll?
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Old 20-09-2013, 06:47   #4
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Re: Capn Cooks' Chronometer

Unless there is some trick I don't see, basic celestial navigation 4 seconds of time equates to 1 nm of longitude
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Old 20-09-2013, 07:00   #5
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Re: Capn Cooks' Chronometer

Quote:
Originally Posted by skipmac View Post
Unless there is some trick I don't see, basic celestial navigation 4 seconds of time equates to 1 nm of longitude
There’s 360 degrees in a circle around the Earth, and we have 24 hours (1440 min) in a day (which is a full rotation of these 360 degrees).
Hence (24Hrs x60 min) ÷ 360 deg = 4 min,
or 1 degree of longitude is equal to about 4 minutes of time.
Or, am I confused?
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Old 20-09-2013, 08:08   #6
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Re: Capn Cooks' Chronometer

Quote:
Originally Posted by GordMay View Post
There’s 360 degrees in a circle around the Earth, and we have 24 hours (1440 min) in a day (which is a full rotation of these 360 degrees).
Hence (24Hrs x60 min) ÷ 360 deg = 4 min,
or 1 degree of longitude is equal to about 4 minutes of time.
Or, am I confused?
No I was confused. First should have said minutes of longitude, not nm of longitude which is a meaningless term. Then I reversed the numbers, should have been 4 mins long per one second of time instead of 1 min long per four seconds.

If you do the math by your answer 1 degree of longitude = 60 mins of long. 4 mins of time = 240 seconds of time. 60 mins long / 240 seconds time = 1 min long per 0.25 secs time or 4 mins long per 1 sec time.

I should have a second cup of coffee before doing complex math, like adding and subtracting.
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Old 20-09-2013, 10:06   #7
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Re: Capn Cooks' Chronometer

That would only apply at the equator, the further south or north you get the further off you will be.
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Old 20-09-2013, 10:51   #8
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Re: Capn Cooks' Chronometer

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Originally Posted by captain58sailin View Post
That would only apply at the equator, the further south or north you get the further off you will be.
Not sure what you're saying exactly. Off in which regard?

Since a degree of longitude is smaller as you get further north or south of the equator then a nautical mile would equal more degrees of longitude at 40 N vs at the equator.

However since a point on the earth rotates the same number of degrees around the earth's axis no matter how far north or south then a certain amount of time will result in the same number of degrees of longitude to pass the GP of a body but a shorter distance.

So, at first thought, if your ship's time is off by X number of seconds when you calculate your position wouldn't you be more nautical miles in error at the equator than at higher latitudes since time relates to degrees of longitude or rotation?
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Old 20-09-2013, 11:53   #9
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Re: Capn Cooks' Chronometer

At the equator 1 minute of Longitude and 1 minute of Latitude are roughly the same, the further north or south you go the more difference there is, depending on how far away from the equator you are the clock error will have more or less impact on your position error. Albeit fairly small with in 20 degrees latitude of the equator.
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Old 20-09-2013, 12:37   #10
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Re: Capn Cooks' Chronometer

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Originally Posted by captain58sailin View Post
At the equator 1 minute of Longitude and 1 minute of Latitude are roughly the same, the further north or south you go the more difference there is, depending on how far away from the equator you are the clock error will have more or less impact on your position error. Albeit fairly small with in 20 degrees latitude of the equator.

Yes I think that's what I meant to say. Longitude varies with latitude, the two about equal at the equator but at higher latitudes a degree of longitude is much smaller than a degree of latitude.

I was trying to relate that fact with errors in navigation based on a error in time. So wouldn't a time error when plotting a fix near the equator make for a larger error than the same time error when plotting a fix at high latitudes where a degree of longitude is smaller?
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Old 20-09-2013, 12:58   #11
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Re: Capn Cooks' Chronometer

Maybe I am looking at this from the wrong side of the equation. In my mind if you have a time error in the higher latitudes, then minutes of Longitude because they are closer together would generate a greater error. You would move farther Longitudinally in the higher latitudes with the time error.
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Old 20-09-2013, 13:25   #12
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Re: Capn Cooks' Chronometer

Quote:
Originally Posted by captain58sailin View Post
Maybe I am looking at this from the wrong side of the equation. In my mind if you have a time error in the higher latitudes, then minutes of Longitude because they are closer together would generate a greater error. You would move farther Longitudinally in the higher latitudes with the time error.
Gee, you're in Alaska and I'm in FL where the equator is a lot closer. I was hoping you would be certain about this.

Here's my logic. In four minutes your position anywhere on the earth relative to the stars will change 1 degree of longitude. At the equator, assuming long approximately equals lat that will be about 60 nm. If you move north until 1 degree of long is about half 1 degree of lat then your position relative to the stars will only move 30 nm. So a time error on board would only give half the distance error in your plot compared to the equator.

Does that make sense?

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Old 20-09-2013, 13:57   #13
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Re: Capn Cooks' Chronometer

Have you checked Cook's original log from his second voyage to see if his Easter Island position was based on sights taken at the location or dead reckoning from a position obtained by sights some time earlier? Could account for a small error, although he had been at sea for around 18 months when he went to Easter Island so only a small chronometer error would have been mounting up by then if he had not be able to account and correct for it. According to his journal he did spend several days at Easter Island and landed several times so he could have taken sights ashore but this is not noted in the journal - you would need the log for that.
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Old 20-09-2013, 14:22   #14
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Re: Capn Cooks' Chronometer

Cook was almost certainly correcting his chronometer by lunars during his voyage. His log might record these corrections. It would require research to determine when exactly he could have shot lunars during his voyage and of course he would have needed a clear sky when those times occurred. So you can't assume his watch time was incorrect.
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Old 21-09-2013, 07:10   #15
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Re: Capn Cooks' Chronometer

No its not a trick question.

You can see William Wales, the astronomer on board, log here:

Papers of the Board of Longitude : Log book of HMS 'Resolution'

He took a number of longitude readings see attachment. The first seems to be dead reckoning, the second seems to be the one for the K1 chronometer, not sure what the other two are but one may be by lunars.

I doubt that he would be correcting the K1 by lunar calculations, probably just comparing them as the point was to see how accurate the chronometer was. This was nearly two years after they left England so I guess the K1 being off by less than two minutes is understandable.
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