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SWEDENBORG ON LONGITUDE
On Jul 7, 5:42 pm, (Steve Willner) wrote:
[Irrelevant newsgroups snipped.] In article , George Hammond writes: http://www.thenewphilosophyonline.or...hp?page=1042&i.... The article is hard to follow because of the obsolete terminology, and I haven't made much effort. As far as I can tell, though, the basic idea is to use the Moon as a clock. The Moon moves its own diameter with respect to the background stars every hour or so, and thus measuring an accurate lunar position gives the time. Combining time with altitudes of stars gives longitude. There are second-order corrections for "horizontal parallax," but the basic method should work. The difficulty I see is that calculating accurate lunar positions in advance is very difficult because the Moon's orbit is perturbed by the Sun and by the non-sphericity of the Earth. This would make lunar positions unsuited to navigation until at least the 20th century, by which time better methods existed. The "lunar clock" could have been used for figuring out the longitudes of fixed places on Earth (by, say, after-the-fact comparison with lunar positions measured at Greenwich), but in practice using Jupiter's satellites turned out to be much easier both because the calculations were easier and because advance ephemerides could easily be computed. What worries me about my understanding of the article is on the second page: "there is no need to know the moon's apparent position, but only the position of the point (d) in the zodiac, which is obtained from the known latitude and longitude of the stars." I don't understand this, but if Swedenborg thought he could determine longitude only by measuring positions of fixed stars (without a chronometer), he was wrong. As I say, it's entirely possible that I've misunderstood the whole article, but I'd expect the general comments on using lunar positions for longitude to be more or less right. -- Steve Willner As discussed here Tobias Mayer or his widow received part of the famous Longitude prize for a similar method: Precomputed lunar distance tables. http://www.math.uu.nl/people/wepster/ldtab.html I became interested in the story of the longitude after seeing a Nova episode based on Dava Sobel's book "Longitude". One extraordinary episode recounted in the book told of an entire English fleet lost in heavy fog while just a few miles off the coast of Britain because they did not have an accurate knowledge of their longitude. Sobel described that a common sailor came forward to give the admirals and captains of the ships who assembled together to decide their location his opinion of their position and for this he was promptly hanged on the spot. The naval laws were strict about sailors even attempting to do their own calculations because this was so critical it was felt mutiny would arise if the ships crew did not trust the captains judgment on the matter. It turned out that that sailors calculations were correct and had the admirals and captains listened the fleet would have been saved. Sobels dramatic telling led me to speculate about some methods mariners of the time might have been able to accurately determine their longitude. I copied two emails below that I wrote to the author of the "Precomputed lunar distance tables" web site about a suggested method. It also concerned the position of the Moon. But I believe it would have been easier to implement than requiring accurate determination of the angular distance between the Moon and stars. Bob Clark ============================================== Date : Wed, Nov 29, 2006 02:50 AM EST From : "Robert Clark" ****@**** To : Subject : The apparent size of the Moon to solve the "problem of the longitude"? Hello. I saw your web page discussing "lunars" after a web search. I like many other scientists was fascinated by the story described by Dava Sobel about the quest for an accurate determination of the longitude, finally solved in the 18th century. I wondered if it would have been possible to get it from the change in the apparent size of the Sun according to distance. However, the change in this distance is relatively small. So I thought instead about the Moon. The idea is that a pinhole camera projects an image whose size is proportional to the size of the source and inversely proportional to its distance: Finding the Size of the Sun and Moon. http://cse.ssl.berkeley.edu/AtHomeAs...tivity_03.html Note that this can still work to measure the apparent size of the Moon even when it's not a full Moon as long as part of the limb is visible. As you described on your web page on the calculation of the "lunars", Tobias Mayer calculated very accurate positions for the Moon. I presume this means he also would have known very accurate distances from the Earth to the Moon. Note that the distances are not always the same even for the perigee or for the apogee; so these predictions would have to be made for each particular day in each particular year. Here's a modern calculator for this using the most up to date data for the Moon's orbit: Lunar Perigee and Apogee Calculator. http://www.fourmilab.ch/earthview/pacalc.html And this web page shows the distance to the Moon changes by as much as 14% during its orbit, resulting in a dramatic change in its apparent size: Lunar Image Gallery - Scenic Phenomenon http://www.perseus.gr/Astro-Lunar-Sc...po-Perigee.htm So instead of tables of angular distances between the Moon and certain stars, there would be tables of the size of the Moon at at a certain fixed longitude, and you could deduce how far away you were from that location by the size you measured at your own location. There would be "measuring boxes" made of a uniform size so the image projected would be the same size for the same apparent size Moon, with gradations marked on the inside to easily read off the size. As discussed on the pinhole camera page, the size of the projected image is dependent on the distance between the pinhole and the projection surface. So this length would have to be greatly and accurately standardized. You would also need the boxes to be made of material that expanded very little with temperature variations. This is a problem John Harrison encountered for the production of his accurate watches. I believe this is a much easier problem with a static box than with a complicated moving mechanism like a watch. For instance glass is used for telescope lenses and mirrors because of its stability against size variations on temperature change. It's much easier to make a static glass box than a watch made of glass. The change in distance and therefore apparent size is 1 part in 7 over half an orbit, 14 days. So it's 1 part in 7*14 = 98 per day, assuming the distance changing uniformly. (We could include in our calculations the deviation from the uniformity assumption by taking into account the elliptical shape of the orbit.) Then it's 1 part in 98*24= 2352 per hour. If we wanted to reach the accuracy required for the Longitude Prize by *direct measurement* we would have to multiply this number by 15 to get it to 1 part in 35,280. However, we could instead use interpolation as used for the "lunars". The Moon's diameter is about 3500 km and the distance at perigee is about 350,000. So the ratio of actual size to distance is about 1 to a 100. This means the ratio of size of projected image to length of the box would also be 1 to 100. If the measuring box was 10 meters long, the size of the image would be about 10 cm. So at the 1 part in 2352 accuracy level we would need to measure to within an accuracy of 42 microns across the 10 cm image. This is about the width of a human hair, which should be within the measuring accuracy available for the 18th century. For visually observing this small distance magnifying lenses would be sufficient. How badly would diffraction of the atmosphere effect the accuracy of this method? ============================================= Date : Tue, Dec 05, 2006 07:00 PM EST From : "Robert Clark" ****@**** To : Subject : Re[2]: The apparent size of the Moon to solve the "problem of the longitude"? Thanks for the response. I didn't think of the fact that the distance would change little at the max and min distance. However, I thought of a way to make the measurements easier in general. What you could do would be to use a telescope to make the image larger. The telescope would be used like a film projector to make a larger image on a screen. This page shows pretty decent scopes were made in the 18th century: 18th-century telescopes. http://www.antiquetelescopes.org/18thc.html The use of the telescope for astronomy dates back to Galileo of course in the 17th century: 17th-century telescopes. http://www.antiquetelescopes.org/17thc.html I don't know the relationship between the size of the image and the size of the objective but I presume it would also depend on the focal length of the scope and the distance to the screen. The presumption is you could make a larger image say 1 meter size at a shorter distance to the screen, so that you wouldn't need a 10 meter distance like I first suggested, by using a larger lens or mirror and the appropriate focal length. Tables would be used as before to indicate the expected size of the image at the reference location according to the time of day at the reference location. For finding local time required in the calculation, my reading of Sobel's book suggests portable clocks of the time would be accurate to within a few minutes within a single day, which would be all that is required for determination of longitude. (Harrison's accomplishment was to create a clock that would be accurate to within a few seconds per day so over a sea voyage it only be off by a few minutes.) So you would just set your clock at the local noon say and you would only need it to be accurate to within a few minutes at the night time observations. Another possibility occurs to me for finding the expected size according to the time you were observing. Wouldn't the position of the Moon from North, determined by the Pole star or compass, change as the night progressed? It seems to me you could have the tables for the reference location give the distance in degrees from North at a particular time and also give the expected size at that position. Then for the mariners making their observations they would find the angular distance of the Moon from north, check the table for the expected size at this angular distance, then compute their longitude from the deviation of their measured size from the size given in the table. As for the required calculations, I was startled by this discovery of the capabilities of this calculating machine for determining positions of the Moon and known planets dating from 100 to 200 B.C.: Ancient calculator demystified at last Greeks’ 2,100-year-old Antikythera Mechanism was used in astronomy. http://www.msnbc.msn.com/id/15953550/ The device worked by a complicated combination of interconnected gears. This was certainly within the capabilities of the 18th century. Admittedly it's construction details were lost until revealed recently. But there were human-like "automatons" made of gears made in the 18th century and I believe calculating devices could also have been made at this time if someone had thought of it. Bob Clark ========================================== |
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SWEDENBORG ON LONGITUDE
"Robert Clark" wrote in message ... On Jul 7, 5:42 pm, (Steve Willner) wrote: [Irrelevant newsgroups snipped.] In article , George Hammond writes: http://www.thenewphilosophyonline.or...hp?page=1042&i... The article is hard to follow because of the obsolete terminology, and I haven't made much effort. As far as I can tell, though, the basic idea is to use the Moon as a clock. The Moon moves its own diameter with respect to the background stars every hour or so, and thus measuring an accurate lunar position gives the time. Combining time with altitudes of stars gives longitude. There are second-order corrections for "horizontal parallax," but the basic method should work. The difficulty I see is that calculating accurate lunar positions in advance is very difficult because the Moon's orbit is perturbed by the Sun and by the non-sphericity of the Earth. This would make lunar positions unsuited to navigation until at least the 20th century, by which time better methods existed. The "lunar clock" could have been used for figuring out the longitudes of fixed places on Earth (by, say, after-the-fact comparison with lunar positions measured at Greenwich), but in practice using Jupiter's satellites turned out to be much easier both because the calculations were easier and because advance ephemerides could easily be computed. What worries me about my understanding of the article is on the second page: "there is no need to know the moon's apparent position, but only the position of the point (d) in the zodiac, which is obtained from the known latitude and longitude of the stars." I don't understand this, but if Swedenborg thought he could determine longitude only by measuring positions of fixed stars (without a chronometer), he was wrong. As I say, it's entirely possible that I've misunderstood the whole article, but I'd expect the general comments on using lunar positions for longitude to be more or less right. -- Steve Willner As discussed here Tobias Mayer or his widow received part of the famous Longitude prize for a similar method: Precomputed lunar distance tables. http://www.math.uu.nl/people/wepster/ldtab.html I became interested in the story of the longitude after seeing a Nova episode based on Dava Sobel's book "Longitude". One extraordinary episode recounted in the book told of an entire English fleet lost in heavy fog while just a few miles off the coast of Britain because they did not have an accurate knowledge of their longitude. Sobel described that a common sailor came forward to give the admirals and captains of the ships who assembled together to decide their location his opinion of their position and for this he was promptly hanged on the spot. The naval laws were strict about sailors even attempting to do their own calculations because this was so critical it was felt mutiny would arise if the ships crew did not trust the captains judgment on the matter. It turned out that that sailors calculations were correct and had the admirals and captains listened the fleet would have been saved. Sobels dramatic telling led me to speculate about some methods mariners of the time might have been able to accurately determine their longitude. I copied two emails below that I wrote to the author of the "Precomputed lunar distance tables" web site about a suggested method. It also concerned the position of the Moon. But I believe it would have been easier to implement than requiring accurate determination of the angular distance between the Moon and stars. Bob Clark ============================================== Date : Wed, Nov 29, 2006 02:50 AM EST From : "Robert Clark" ****@**** To : Subject : The apparent size of the Moon to solve the "problem of the longitude"? Hello. I saw your web page discussing "lunars" after a web search. I like many other scientists was fascinated by the story described by Dava Sobel about the quest for an accurate determination of the longitude, finally solved in the 18th century. I wondered if it would have been possible to get it from the change in the apparent size of the Sun according to distance. However, the change in this distance is relatively small. So I thought instead about the Moon. The idea is that a pinhole camera projects an image whose size is proportional to the size of the source and inversely proportional to its distance: Finding the Size of the Sun and Moon. http://cse.ssl.berkeley.edu/AtHomeAs...tivity_03.html Note that this can still work to measure the apparent size of the Moon even when it's not a full Moon as long as part of the limb is visible. As you described on your web page on the calculation of the "lunars", Tobias Mayer calculated very accurate positions for the Moon. I presume this means he also would have known very accurate distances from the Earth to the Moon. Note that the distances are not always the same even for the perigee or for the apogee; so these predictions would have to be made for each particular day in each particular year. Here's a modern calculator for this using the most up to date data for the Moon's orbit: Lunar Perigee and Apogee Calculator. http://www.fourmilab.ch/earthview/pacalc.html And this web page shows the distance to the Moon changes by as much as 14% during its orbit, resulting in a dramatic change in its apparent size: Lunar Image Gallery - Scenic Phenomenon http://www.perseus.gr/Astro-Lunar-Sc...po-Perigee.htm So instead of tables of angular distances between the Moon and certain stars, there would be tables of the size of the Moon at at a certain fixed longitude, and you could deduce how far away you were from that location by the size you measured at your own location. There would be "measuring boxes" made of a uniform size so the image projected would be the same size for the same apparent size Moon, with gradations marked on the inside to easily read off the size. As discussed on the pinhole camera page, the size of the projected image is dependent on the distance between the pinhole and the projection surface. So this length would have to be greatly and accurately standardized. You would also need the boxes to be made of material that expanded very little with temperature variations. This is a problem John Harrison encountered for the production of his accurate watches. I believe this is a much easier problem with a static box than with a complicated moving mechanism like a watch. For instance glass is used for telescope lenses and mirrors because of its stability against size variations on temperature change. It's much easier to make a static glass box than a watch made of glass. The change in distance and therefore apparent size is 1 part in 7 over half an orbit, 14 days. So it's 1 part in 7*14 = 98 per day, assuming the distance changing uniformly. (We could include in our calculations the deviation from the uniformity assumption by taking into account the elliptical shape of the orbit.) Then it's 1 part in 98*24= 2352 per hour. If we wanted to reach the accuracy required for the Longitude Prize by *direct measurement* we would have to multiply this number by 15 to get it to 1 part in 35,280. However, we could instead use interpolation as used for the "lunars". The Moon's diameter is about 3500 km and the distance at perigee is about 350,000. So the ratio of actual size to distance is about 1 to a 100. This means the ratio of size of projected image to length of the box would also be 1 to 100. If the measuring box was 10 meters long, the size of the image would be about 10 cm. So at the 1 part in 2352 accuracy level we would need to measure to within an accuracy of 42 microns across the 10 cm image. This is about the width of a human hair, which should be within the measuring accuracy available for the 18th century. For visually observing this small distance magnifying lenses would be sufficient. How badly would diffraction of the atmosphere effect the accuracy of this method? ============================================= Date : Tue, Dec 05, 2006 07:00 PM EST From : "Robert Clark" ****@**** To : Subject : Re[2]: The apparent size of the Moon to solve the "problem of the longitude"? Thanks for the response. I didn't think of the fact that the distance would change little at the max and min distance. However, I thought of a way to make the measurements easier in general. What you could do would be to use a telescope to make the image larger. The telescope would be used like a film projector to make a larger image on a screen. This page shows pretty decent scopes were made in the 18th century: 18th-century telescopes. http://www.antiquetelescopes.org/18thc.html The use of the telescope for astronomy dates back to Galileo of course in the 17th century: 17th-century telescopes. http://www.antiquetelescopes.org/17thc.html I don't know the relationship between the size of the image and the size of the objective but I presume it would also depend on the focal length of the scope and the distance to the screen. The presumption is you could make a larger image say 1 meter size at a shorter distance to the screen, so that you wouldn't need a 10 meter distance like I first suggested, by using a larger lens or mirror and the appropriate focal length. Tables would be used as before to indicate the expected size of the image at the reference location according to the time of day at the reference location. For finding local time required in the calculation, my reading of Sobel's book suggests portable clocks of the time would be accurate to within a few minutes within a single day, which would be all that is required for determination of longitude. (Harrison's accomplishment was to create a clock that would be accurate to within a few seconds per day so over a sea voyage it only be off by a few minutes.) So you would just set your clock at the local noon say and you would only need it to be accurate to within a few minutes at the night time observations. Another possibility occurs to me for finding the expected size according to the time you were observing. Wouldn't the position of the Moon from North, determined by the Pole star or compass, change as the night progressed? It seems to me you could have the tables for the reference location give the distance in degrees from North at a particular time and also give the expected size at that position. Then for the mariners making their observations they would find the angular distance of the Moon from north, check the table for the expected size at this angular distance, then compute their longitude from the deviation of their measured size from the size given in the table. As for the required calculations, I was startled by this discovery of the capabilities of this calculating machine for determining positions of the Moon and known planets dating from 100 to 200 B.C.: Ancient calculator demystified at last Greeks’ 2,100-year-old Antikythera Mechanism was used in astronomy. http://www.msnbc.msn.com/id/15953550/ The device worked by a complicated combination of interconnected gears. This was certainly within the capabilities of the 18th century. Admittedly it's construction details were lost until revealed recently. But there were human-like "automatons" made of gears made in the 18th century and I believe calculating devices could also have been made at this time if someone had thought of it. Bob Clark ========================================== If Earth, like Mars, had two smaller moons instead one large one how much easier would it be to locate latitude and longitude with an astrolabe? Or better yet, enough moons so that three or four could be seen at all times instead of only when one is above the horizon? Suppose we put a radio transmitter on every moon so that we were not dependent on a visual sighting, but could "see" right through cloud whenever we wanted to? Suppose each moon could tell us where it was on that radio signal so that we didn't need an astrolabe? An even better improvement would be to have each moon carry an accurate clock and tell us the exact time it was there when it sent the radio transmission. This would be sci-fi, of course... unless it was called GPS, finally solved in the 20th century. How badly would diffraction (oops - refraction) of the atmosphere effect the accuracy of this method? Quite a lot really, you could easily be up to 100 feet out of position vertically and 30 feet off horizontally. How terrible. |
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