How did Scott and Amundsen detect the direction to the South Pole during their expedition? How did they determine the exact South Pole on reaching there?
Comparison of the Amundsen and Scott expeditions doesn't give this information.
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I'm certain that Scott and Amundsen didn't detect the exact South Pole but more less reached a position that explorers would think was close enough to count.
One technique they probably used was to measure their latitude and find that were, for example, about 1,000 feet from the pole. Then walk forward for about 2,000 feet and measure their latitude again. If it was close to the previous measurement they should have walked closer to the pole than either of those two points.
Then they could turn at a right angle, walk another 2,000 feet, measure latitude, walk another 2,000 feet measure latitude, and son on until they had walked approximately 8,000 feet along the sides of a rough square.
If each of the corner points had a latitude north of the south pole and those latitudes were close to each other, they could deduce that the south pole was within the square and then walk as close to the center of the square as they could measure and try measuring their latitude again, and maybe walk in a much smaller square around the south pole.
How would polar explorers measure latitude? With the same sorts of instruments that navigators at see and surveyors on land measured latitude. Measuring latitude is necessary for ships to reach their proper destinations without wasting time and not run into rocks, and for surveyors to put the borders of parcels of land at the right positions.
I think they would probably use a sextant to measure the height of the Sun above the horizon.
Professional sextants use a click-stop degree measure and a worm adjustment that reads to a minute, 1/60 of a degree. Most sextants also include a vernier on the worm dial that reads to 0.1 minute. Since 1 minute of error is about a nautical mile, the best possible accuracy of celestial navigation is about 0.1 nautical miles (190 m). At sea, results within several nautical miles, well within visual range, are acceptable. A highly skilled and experienced navigator can determine position to an accuracy of about 0.25-nautical-mile (460 m).4
I think they would probably use a portable chronometer to measure what time it was at the longitude which the chronometer was set for - when they got close the north or south pole it wouldn't matter much which longitude the chronometer was set for.
Show they would look up the apparent height of the Sun above the horizon at that time of day at various latitudes and find the latitude which corresponded to the height of the Sun at that time, which would be the latitude they were in.
And of course there was a limit to the accuracy of their chronometers and their sexants and their techniques of using them.
The published first person accounts of Peary and those who later reached the North pole, and Amundsen and those who reached the south pole after him, and Scott's diary entry for the day they reached the south pole should record some details of how they measured their latitude.
Here is a link to a photo of the ceremonial south pole erected at the Amundsen-Scott South Pole Station.
However, the ceremonial pole is not at the exact geographical south pole.
It would be interesting to plot the approximate paths of Amundsen and Scott on a map of the south pole base.
And of course historians of polar exploration should be able to give much better answers.
I want to add to this answer some details about "measuring latitude". The instruments used are sextant (for measuring altitude of a star (or Sun or Moon) over the horizon, and chronometer for measuring time. Under normal conditions, a student of a naval academy passing the exam was expected to find his coordinates at sea with these instruments with 1-2 (nautical) mile accuracy.
But the conditions near the S Pole cannot be called normal. First a clearly visible horizon may not be available, and one may expect various irregularities of refraction near the horizon. On the other hand, observations are performed on land, so they almost certainly used an artificial horizon. This increases accuracy.
The simplest method would be to try to determine the maximal and minimal altitude of the Sun during the day, and make sure that they are equal. But this is not practical since this altitude changes too slowly during the day (if you are exactly at the pole it does not change at all).
Surely they were familiar with more advanced methods of "positions lines", when any two stars are used instead of the Sun, and in a good weather, with reliable chronometer, and enough time for repeated observations, they could probably determine their positions to a fraction of a mile.
Myself, with a sextant, chronometer and artificial horizon, I was able to do this on land with accuracy 0.2 mile (less than 300 m). The crucial condition is weather: the stars must be visible! Another important condition is the ability to transport the chronometer during the travel so that it keeps the time well.
More details can be found here
For more technical analysis, including the assessment of their accuracy see
The Observations of Amundsen and Scott at the South pole by Arthur R. Hinks, The Geographical Journal , Apr., 1944, Vol. 103, No. 4 (Apr., 1944), pp. 160-180