Popular Science Monthly/Volume 76/January 1910/How Could an Explorer Find the Pole?
| HOW COULD AN EXPLORER FIND THE POLE? |
By Professor HARRY FIELDING REID
JOHNS HOPKINS UNIVERSITY
THE claim to have reached the north pole, a point sought for several hundred years by many intrepid explorers, must, of course, be substantiated by adequate proof; and it may interest readers of this magazine to know what kind of proof is possible and necessary, and what observations the explorer must make to determine his geographical position when he is in the neighborhood of the pole.
Let us say, in the first place, that neither photographs, which only show the condition of the ice, but do not indicate whether they were taken near the pole or several hundred miles from it, nor the testimony of human beings, gives any evidence whatever that an explorer has been to the pole. Persons who were not actually with the explorer can only express their confidence in his good faith, in his knowledge of the proper astronomical observations to be made, and in his ability to make them with sufficient accuracy. Persons who accompanied him could only vouch for the fact that he did not remain in camp at a comfortable distance from the pole and manufacture observations, but that he actually traveled in the general direction of the pole, that on a certain date he claimed he was there, and that he made frequent astronomical observations on the route.
The only evidence which can at all satisfactorily show that an explorer has been near the pole is that afforded by observations on the sun or stars, capable of determining his successive positions at the times they were taken. Other evidence might prove the negative; such as inconsistencies in the narrative, inadequate time or insufficient food for the distance traveled, the description of phenomena which could not have been seen at the place where the explorer thought he was; and so on. It is impossible to foresee the many discrepancies which might show that an explorer has not been to the pole; they will not be considered here, as this article is not controversial, but merely aims to set forth, as simply as possible, what kind of observations must decide the claim of having reached the pole.
Confining our attention for the moment to observations on the sun, for the sake of simplicity of statement, we may say that the determination of one's position anywhere on the earth depends upon measuring the altitude of the sun above the horizon at two times, the second being, preferably, after the direction of the sun has changed by 90°. This becomes clear if we consider Fig. 1. Let us suppose the sun is in the direction S and is immediately over the point A of the earth's surface. Its altitude there is 90°. As we pass along the earth's surface away from A, the direction of the horizon continually changes and the altitude of the sun continually diminishes until we reach the great circle, BC, which divides the light from the dark hemisphere, and there the sun is on the horizon and its altitude is zero degrees.
As the earth is spherical, and therefore symmetrical round the line OS, if we draw a circle DE on its surface, with its plane at right angles to this line, the altitude of the sun, as seen from all parts of this circle, will be the same; at the point D this altitude will be represented by h, the angle between the direction of the sun DS'and the horizon; for the sun is so distant from the earth, that its direction is the same from the center of the earth and from any point of the surface, to the degree of accuracy required by explorers. Every part of the circle DE is at right angles to the direction of the sun. The altitude of the sun changes with, and is determined by, the distance of the circle DE from A; and, vice versa, if the altitude is known, the distance of the circle from A is determined. The point A, itself, is fixed when we know Greenwich time and the angular height d of the sun above the equator; this latter is called the declination; it is continually changing, but its value at any time can be found in the "Nautical Almanac." An explorer would always take with him a copy of this work or an abbreviation of it; and he would also be supplied with chronometers keeping Greenwich time.
If an explorer has measured the altitude of the sun, and has at the same time observed the Greenwich time by his chronometer, he has merely determined that he is somewhere on a certain circle, whose position he could plot on his map; but other considerations, such as his last determined location, and the approximate distance he had traveled from it, would make known more or less roughly in what part of the circle he was; but he could not determine his position accurately. If, some time after his first measure of the sun's altitude, he should make a second similar measure, he would determine his position on a second circle; and the intersection of those two circles would determine his position completely. The determination, of course, would be more accurate if the circles cut each. other at a high angle, and this could be insured by making the second set of observations on the sun after it had changed its direction (measured on the horizontal plane) by about 90°. Instead of making two sets of observations on the sun, we might, in the evening, observe two stars properly located with respect to each other, and we could then find two circles of position in a few minutes and completely determine our position.
If the sun is due north or south the part of the circle on which the observer is will coincide with his parallel of latitude, which is thus immediately determined; if the sun is due east or west, a part of the circle will correspond with the meridian and the longitude will be found. The old method of determining position at sea, and one still in use, was to observe the sun at noon for latitude, and to accept as local noon the poorly determined time when the sun reached its highest altitude; or to observe also in the morning or evening for time or longitude, guessing at the latitude to work out the observations. But the new method makes it possible to observe altitudes at any time and to get satisfactory results even if the sun were hidden for several hours during the middle of the day. And besides it makes clear just what information regarding our position is yielded by a single observation of the sun's altitude. This beautiful method was first used by Captain Thomas H. Sumner, of Boston, Mass., in 1837; the short parts of the circle which are drawn on the map in finding one's position are called Sumner's lines.
If an explorer were approaching the north pole, and had arrived, let us say within a degree of it, it would be necessary for him to determine his latitude in order to know his distance from the pole, and to determine the direction of the pole in order to know his course. It might be supposed that when approaching the pole he would, by means of his compass, be able to follow his meridian; but the difficulty of keeping a fixed direction when traveling over rough ice, and especially the shifting of his position by the unknown drift of the ice, would soon make a decided change in his longitude in a region where the meridians converge so rapidly.
In the neighborhood of the north pole the compass needle points approximately in the direction of the meridian 155° west of Greenwich, according to Neumayer, but the angle between the needle and the meridian changes considerably for comparatively small variations of position; especially as the distance from the pole becomes smaller.
The North Star, so closely associated in our minds with the pole, would be of no use to the explorer, for it is about a degree and a quarter from the pole, and, like the other stars, it would circle around the observer, and at times even be directly south of him. To determine its direction the explorer would have to know his own latitude and local time; moreover, it would be invisible if the sun were above the horizon.
By means of his chronometer, keeping Greenwich mean time, the explorer could determine the direction of any meridian, for the sun would be on the meridian of Greenwich at Greenwich noon, and would move 15 degrees in longitude for every hour thereafter; this knowledge would be very valuable to enable him to lay out his return course from the pole to his base of supplies, but it would not, in ignorance of his meridian, help him to find the pole; for the direction of the pole in relation to the direction of the sun, or of the compass needle, does not depend upon the general direction of the meridians, but upon the particular meridian on which he happens to be.
We have thus the apparent anomaly that the same observations would enable a person to set a satisfactory course away from the pole, but not toward it. But the anomaly is only apparent; for, suppose the base of supplies were on the 70th meridian and in latitude 83°; and suppose the explorer were near the pole and twenty miles from the 70th meridian, on one side or the other; he could lay a course parallel with the 70th meridian and this direction would only differ by about a third of a degree from the most direct line to his base of supplies; but if he kept this course accurately, he would miss his base by twenty miles. This, however, would be less important than missing the pole by the same distance.
The very simple method of determining latitude by the altitude of the sun when on the meridian would not be available to the explorer, for his meridian would not be known; and it would require a set of observations extending over several hours to learn when the sun was on his meridian. On the sixth of April the sun would circle around the horizon, at an average altitude of about 614 degrees, and would only be two degrees higher at midday than at midnight, as seen by an explorer one degree from the north pole, provided its declination were constant; this, however, is not so; but on the date mentioned we should find, superposed on the variation in altitude due to the rotation of the earth, a steady increase in altitude amounting to a little more than a third of a degree in a day. On April 21 the sun's altitude would be about ll34 degrees above the horizon, and the variations in altitude during the day would be almost the same as on the earlier date.
To determine his position, and the direction of the pole, the explorer must fall back on the method of Sumner's lines, and fortunately they can be applied with special facility in the neighborhood of the pole.
Let us suppose then that an explorer is approaching the north pole in the neighborhood of meridian 120 degrees. (See Fig. 2, where the outer circle represents a circle one degree from the pole, and the radiating lines are the meridians, 0 degree being that of Greenwich.) He determines the altitude of the sun when by his chronometer, let us say, it is in longitude 30 degrees. He now works out his latitude on the supposition that he also is in longitude 30 degrees; suppose his results give an apparent altitude of 89 degrees 50 minutes. He lays off that latitude on the 30th meridian at A, and draws a straight line
AA' at right angles to the latter; this line will practically coincide with a part of the circle at all of whose points the sun has the observed altitude at the time the observations were made; his position is therefore somewhere on this straight line, and, guessing about how far he has traveled from his last determined position, he can estimate roughly where he is; but if bad weather has prevented observations for several days, or the unknown drift of the ice has been strong, he might be many miles wrong.
If he should wait for six hours and make another similar observation of the sun's altitude when it is on the 120th meridian, he would determine a second line on which he would be; his true position would then be at the intersection of these two lines. If the second observation determined an apparent latitude of 89 degrees 40 minutes, he would lay off this latitude on the 120th meridian, draw a straight line, BB', at right angles to the latter, and his true position would be at B; this is about 22 minutes, or about 25 English miles, from the pole, and nearly on the 90th meridian. He now knows his position, and by the Greenwich time and the position of the sun he knows the direction of the 90th meridian, and therefore of the pole.
He then travels in the direction of the pole, keeping this direction by means of his compass or by the sun and his chronometer. Knowing about how fast he travels, he knows when he is in the immediate neighborhood of the pole, and he checks his position again by another pair of observations similar to the last.
Suppose, however, the drift of the ice has been quite strong; it may have carried him several miles from the line AA′ during the six hours between his observations; at the time of the second observation he would, indeed, be on the line BB′, but he would no longer be on the line AA′. If he should wait another six hours and observe the sun when on the 210th meridian, he would then find himself, let us say, on the line CC′; and, assuming a uniform drift of the ice, his position at the time of the second observation would have been on the line BB′ half way between the lines AA′ and CC′—that is, at B″; but he has drifted away from the line BB' during the six hours since he determined his position on that line, and he does not know exactly where he is on the line CC′. If he waits six hours longer, and observes the sun when on the 300th meridian, and then finds himself on the line DD', his true position at that time will be at D, and the drift of the ice during the twelve hours between his second and fourth observations will have been in the direction B″D, and it will have drifted a distance equal to the length of the line B″D on the scale of the figure.
An explorer may find his position by pure calculation, and may not use the graphic method described, but the principle in the two methods is exactly the same, and the graphic method shows more clearly what the observations mean.
An important source of error enters all these observations, namely, atmospheric refraction, or the bending down of the light rays as they pass through the atmosphere. The amount of this bending increases rapidly as the sun is nearer the horizon; it also varies with the barometric pressure, and with the temperature. On April 21, 1908, the sun was only about 1134 degrees, and on April 6, 1909, only about 614 degrees above the horizon at the north pole; on both dates the refraction was considerable, and it is hardly well enough known to prevent errors of several minutes of arc in the determinations of position. If, however, the observer should wait for twenty-four hours after his first observation and should measure a fifth altitude of the sun, he could find a fair correction for the refraction and greatly improve the determination of his position.
It would be unreasonable to expect an explorer, making a dash for the pole, to remain twenty-four or even eighteen hours at one camp for the purpose of exactly determining his position. By making daily observations on the sun, at different hours, so as not merely to fix his successive positions on a series of parallel lines, but on lines having different directions; by keeping his direction with the compass and estimating the drift of the ice and his rate of travel, he could always know where he was without too large an error. But when he was in the immediate neighborhood of the pole he should make as many observations, with the sun in different directions, as circumstances would permit.
Fatigue, severe cold, the condition of his commissariat, and the anxiety to return after having succeeded in his bold undertaking, might prevent him from making as many observations as would be desirable; but nevertheless they might be sufficient to be convincing that he had been within a few miles of the pole; it would surely be a quibble to dispute with an explorer the honor of having reached the pole if his observations showed, without reasonable doubt, that he had been within ten or fifteen miles of it.
There are two kinds of instruments used for measuring altitudes; the transit-theodolite and the sextant. The former consists of a telescope so mounted that it can turn in a vertical and in a horizontal plane; it is provided with vertical and horizontal graduated circles, to measure the angle turned through, and with leveling screws and spirit levels to adjust it in position. It is supported by a tripod, and after being properly leveled, the reading of the vertical circle gives the altitude of the object sighted through the telescope. It is by far the best instrument for an explorer on land, because it is very easy to use, and its adaptation to measure horizontal angles enables the explorer to carry on an ordinary survey.
The sextant was originally invented for use at sea, where a steady support can not be found. It consists of a telescope mounted on a frame, which is held in the hand. To measure the angle between two objects, one of them is sighted directly through the telescope, and the image of the second is reflected into the telescope by means of two mirrors, one fixed rigidly to the frame in front of the telescope, and covering half its field, and the other movable around an axis fastened to the frame. The movable mirror is turned by an arm, which moves along a graduated arc on the frame, and its reading, when the two objects appear in the telescope superposed upon each other, gives the angle beteween the objects. In determining the altitude of the sun at sea the edge of the sun is made to touch the horizon; a movement of the ship moves the sun and the horizon together and the contact is not destroyed. On land, when the sea horizon is not available, a so-called artificial horizon must be used. The ordinary mercurial artificial horizon consists of a flat dish about three inches wide and five or six long, filled to a small depth with mercury, the surface of which becomes perfectly horizontal. The image of the sun seen in the mercury will be as much below the horizontal plane as the actual sun is above it; and the angle between the sun and its image is twice the altitude of the sun. Except in very quiet air, the surface of the mercury must be protected from the wind by an accurately made glass cover.
The glass artificial horizon is a piece of perfectly flat dark glass, which will absorb the light which enters it and only reflect from its upper surface. It is provided with leveling screws and spirit levels so that it can be made perfectly horizontal. It is used in exactly the same way as a mercurial horizon.
Each form has its advantages; the glass horizon is easily transported, and can be used at temperatures below the freezing point of mercury (about 39° below zero Fahrenheit). On the other hand, it requires very careful leveling, and is liable to be broken. The mercury of a mercurial horizon is usually carried in an iron bottle; in pouring it back and forth it might be spilled and lost; and at very low temperatures it would be necessary to heat it to keep it liquid; but then it immediately takes a level surface and requires no leveling.
Lieutenant Shackleton, traveling over the Antarctic continent, determined his position by means of a small transit. Commander Peary and Dr. Cook, traveling over the floating ice of the Arctics, used sextants. The former used a mercurial and the latter a glass horizon.
It is interesting to note that if a man were taking an observation standing, with the sun about 6 degrees above the horizon and the artificial horizon on the level of his feet, it would have to be about 45 feet from him, and as he would look at it from an angle of about 6 degrees, it would only appear about half an inch long. If the altitude of the sun were 12 degrees, the artificial horizon would be 25 feet away and appear about an inch long. This can be easily imitated by putting a sheet of paper on the ground and looking at it from distances of 25 and 45 feet. Under such conditions the difficulties of making a good observation would be much increased. If, however, the artificial horizon were raised on a support, the observer would stand much closer to it, and the observation could be more easily made.
Another important instrument is the chronometer keeping mean Greenwich time; for, as has already been shown, the determination of position in general requires a knowledge of Greenwich time, though at the pole itself this is not necessary. Whenever an explorer remained as long as a week in one place he should determine, as well as he could, how much his chronometers were gaining or losing per day; and he should be most particular to determine the changes in their errors, between the times of leaving and returning to his base station.
A compass is of great value to keep one's course between observations on the sun; and an aneroid barometer and a thermometer make possible a more accurate correction for refraction. A pedometer, also, or some other form of distance meter, would be useful to estimate the distance traveled.
Although the methods of determining one's geographical position would be the same near either pole, there are slight differences in their applications; for instance, the solid land of the Antarctic continent precludes drift, and therefore this disturbance is absent. Moreover, when Lieutenant Shackleton reached his farthest south in the beginning of January, 1909, the sun was about 25 degrees above the horizon; at this altitude the refraction is not large and its value is well enough known not to introduce any great error. Near Lieutenant Shackleton's base camp, at the foot of Mount Erebus, the north pole of the compass needle pointed about 30 degrees east of south. Along the most southerly part of his route, on his dash toward the pole, the north end of the needle pointed very nearly to the south pole.
On the return trip Lieutenant Shackleton could have been guided by his compass, by the mountain range which ran very nearly parallel with his route, or by other landmarks, and, perhaps, to some extent, by his tracks; so that he found it unnecessary to make many astronomical observations. Commander Peary was guided, to a great extent, on his return by his tracks and those of his supporting parties; and Dr. Cook seems to have relied entirely on his astronomical observations.
Note.—For the sake of simplicity the sun has been generally taken, in this article, as the heavenly body on which observations are made. But the stars could serve equally well, and, for some observations, better. If the pole should be approached when the stars were visible, the altitudes of two stars lying on meridians about 90 degrees apart would determine one's position without delay; moreover, stars could be selected whose altitudes were sufficiently great to exclude errors due to refraction; or this correction could be determined by observations on a pair of stars having about the same altitude and lying on opposite sides of the zenith.
The sun's apparent motion around the earth is not uniform, and therefore a correction, known as the equation of time, must be applied to all observations on the sun; but this correction is accurately known and leads to no error.