The explorers who penetrated the Arctic regions, carried with them instruments for measuring the altitudes of celestial bodies above the horizon and also chronometers or watches intended to show, from instant to instant, the time of day at the meridian of Greenwich.
An observer who has determined the altitude of a celestial body at a given instant of time has in reality located himself at the end of a radius whose length is the zenith distance, or ninety degrees minus the altitude of the celestial body, and whose origin or center is the geographical position of the observed celestial body, or that place on the surface of the earth which is vertically under the observed body at the instant of observation. In the absence of knowledge of the precise direction of the radius, the only definite information to be obtained from the observation of the altitude of a celestial body at a given instant is that the observer is located on the circumference of a circle described by the radius.
Each separate altitude and corresponding Greenwich time of observation, whether it be of a different celestial body or of the same celestial body in a different quarter of the heavens, will result in one of these circles of position; and it is by the intersection of the circumferences of two or more of these circles that the actual geographical position of the observer is fixed.
Of course, if the observer has moved in the interval between two observations, it will be necessary, in order to find his geographical position at the instant of the second sight, to consider the center of the first circle of position to be moved in a direction represented by his true course between the stations and by an amount equal to the distance between them. Here an obstacle supervenes in finding geographical positions in the polar regions, especially on the American side of the Arctic zone, because the compass is weak and has a directive force scarcely one-fifth of its intensity in the equatorial regions of the earth, and the variation of the compass, or the pointing of the needle away from the true north, changes very rapidly from place to place and is of very large amount, as shown by the accompanying figure in which each of the variation curves passes through all places where the variation of the compass has the same value as that marked on the line. The direction of travel is, consequently, uncertain on account of the variability of the compass, and the direct distance between stations must also vary widely from accuracy on account of the capricious nature of travel over ice and the consequent uncertainty in the measurement of the distance traversed. It is, therefore, obvious that, unless the explorer observes the altitudes of two different celestial bodies at the same station or remains there long enough to observe the same body in two different quarters of the heavens, he cannot exclude from the process of fixing his geographical position the uncertainties that are visualized by considering the center of a circle of position to be moved in direction and distance by the ill-determined resultant of travel and drift between two stations.
But apart from the source of error in geographical position thus arising through finding the intersection of circles of position, it is of chief concern to examine into the particulars of a single circle of position by itself. Evidently the places on the surface of the earth which shall be passed through by the circumference of such a circle of position, i. e., the meridians of longitude and parallels of latitude which it crosses, must depend upon two factors. These are the position of the center and the length of the radius.
THE GEOGRAPHICAL POSITION OF THE CENTER OF THE CIRCLE OF POSITION.
The position of the center is always in a geographical latitude equal to the declination of the observed celestial body at the instant of observation; and in a geographical longitude equal to the hour-angle of the celestial body from the prime meridian, which, in the case of the sun, is equal to the Greenwich apparent time at the instant of observation, and, in the case of the moon or planets or stars, is equal to the difference between the Greenwich sidereal time [Greenwich mean time expressed in sidereal time + sidereal time of Greenwich mean moon (R. A. M. S.)] and the moon's or planets' or stars' right ascension. As the declinations of the celestial bodies are tabulated in the American Ephemeris and Nautical Almanac, published annually for some years in advance under the authority of the Navy Department, no element of doubt can attach to the latitude of the center of the circle of position. For the determination of its longitude, however, the position of the earth with reference to the celestial sphere must be known in respect of rotation. This requires that the Greenwich solar time or sidereal time be known, and, as there is no agency at work for determining these in Arctic explorations, excepting the chronometer, the correctness of the resulting longitude will always be dependent upon the accuracy of the chronometer time. It will not escape attention, therefore, that a circle of position—whatever its radius may be—will be displaced on the globe to the eastward or to the westward of its true position by the amount of the inaccuracy of the chronometer time at the instant of the observation of the altitude upon which the radius of the circle depends.
There are instances in which the performances of chronometers in Arctic exploration have appeared to be very satisfactory. In a letter to the makers of some of the chronometers that were used in the Grinnell Arctic Exploring Expedition, Maury stated: "The instruments have been subjected to the severest tests to which it is possible to subject instruments of such delicate construction, yet so exquisitely were they provided with adjustments and compensations for the very great extreme of temperature to which they were subjected that one of these, No. 114 Loseby, after having suffered all sorts of experience to which such instruments are liable in a polar winter, is returned with a change in its daily rate during seventeen months of only .03 sec." But it would be mischievous to spread the impression that the errors and rates of chronometers are not subject to large uncertainties in these remote regions, exposed as the instruments are to many physical influences that would naturally tend to disturb the regularity of their performance. The sources of error arise from variations of rate on account of change of barometric pressure, on account of change of temperature, and on account of change in the circumstances either of rest or motion. Under the most favorable conditions for the excellent performance of chronometers, the average change of daily rate between one week and the next exceeds 1.2 seconds; and the average liability to error in the daily rate between one week and the next is not far from 3 seconds. It would be beyond the bounds of reasonable expectation to find exhibited by chronometers in transportation a closer approach to regularity than is shown by those in use on board vessels in the trans-Atlantic passenger service. Averages obtained from a large number of observations, made under these conditions of transportation, in which all the circumstances conspire to give the most favorable results, indicate that a chronometer of average quality will, at the end of 20 days, show an error of 10 seconds; and that an error of 3.2 times this amount must be looked out for.
The quantities in the following table, compiled by Hartnup, represent the error at the end of a passage of given duration, under the average conditions of oceanic voyaging:
Mr. Hartnup made an additional investigation, obtaining the following results:
Extreme difference Variation of daily Mean of extreme
of mean daily rate rate due to change difference between any
between any 3 weeks. of temperature. 2 days of each week. S. S. S.
Mean from 297 chronometers 2.19 1.73 0.98
Mean from 40 poorest 6.28 4.55 2.80
Mean excluding 40 poorest 1.56 1.29 0.70
Under adverse conditions of transportation, a single chronometer is likely to lead to results widely divergent from the truth. For this reason it is commonly considered a necessity in navigation to carry three chronometers, and there are numerous instances in which it has been revealed that, because of abnormal vibration, all three went wrong to such an extent as to put the observer twenty, thirty, or even forty miles out of his proper position.
The liability to error of the best chronometer-watches would be at least twice as great as that which has just been indicated to be the liability to error of chronometers of average quality under the circumstances of oceanic navigation.
THE RADIUS OF THE CIRCLE OF POSITION.
The accuracy of the radius of the circle of position depends solely upon the measurement of the altitude of the observed celestial body. The sextant is the astronomical instrument which has been generally employed for the measurement of altitudes by the polar explorer, as its portability and the simplicity of its manipulation enable him to secure the data of observation with a minimum expenditure of time. Being held in the hand without fixed supports, and having small dimensions, the accuracy of fixed instruments is not to be expected from it. A comparison at observatories and other fixed stations of the true latitudes with the latitudes deduced from long series of altitudes observed by sextants in the hands of the most trained and skillful observers show that, even after the constant errors were thoroughly investigated and the resulting corrections were applied, outstanding differences of as much as 64" existed. It is not to be expected that the constant errors can be successfully eliminated from measurements made in the Arctic under the conditions of polar exploration. They include errors arising from eccentricity of the center of the radius arm and the center of the arc, from faulty graduation, and from flexure of the frame of the instrument caused by varying temperature and accidental blows. In a first-class instrument in perfect order, they may not amount to much in the resulting altitude after the index correction is applied, but, on the other hand, in an inferior instrument, and after careless treatment, it is not likely that the effect of these errors will be less than three minutes. Besides the instrumental errors, irregular or accidental errors, such as errors in the refraction in the rays of light from the observed celestial body produced by anomalous changes in the density of the atmosphere, and personal errors, arising from peculiarities of the observer and the imperfection of his senses, are always present to produce a further liability to error in the measured quantity.
On the rare occasions when the hummocky surface of an ice-covered sea does not shut out the view of the natural horizon, the use of this line as an origin of measurement of the altitude of a celestial body is attended by grave uncertainties owing to the deflecting influence of the varying atmospheric conditions through which the rays of light pass from it to the observer. The Rev. G. Fisher, astronomer of Parry's Arctic Expedition, observed a variation in the apparent place of the horizon of 18' in the Arctic regions, and there are indications that this amount may be much exceeded even amidst appearances that are devoid of every suggestion of irregularity. Sometimes these prevalent disturbing conditions of the Arctic atmosphere are made evident to the senses by the presence of mirage in consequence of which objects near the horizon are distorted in a vertical direction with the occasional appearance of a second image of the same object inverted above the first, and even a third above the others.
As general dependence upon the natural horizon is precluded, recourse must be had to the use of some form of theodolite for the measurement of altitudes of celestial bodies or else to the employment, in conjunction with the sextant, of an artificial horizon. The usual form of artificial horizon, consisting of a small rectangular shallow basin of mercury with a glass roof to protect it from the agitation of the wind, is of limited utility in the Arctic, both because mercury freezes at a temperature about—40° Fahrenheit and also because altitudes measuring less than 18° can rarely be found with it, since, when the altitude is low, the observer is obliged to increase his distance from the reflecting surface and thus incurs great difficulty in keeping sight of the image reflected in the fluid.
Oil and alcohol have been used as substitutes for mercury in the shallow basin, and, although the utility of the horizon is thus preserved against freezing, its availability for the determination of low altitudes is not thereby extended.
Instead of the liquid artificial horizon, a glass plate is sometimes used, standing upon three screws, by means of which it is levelled, a small spirit level being applied to the surface to test its horizontality. The lower surface of the plate is blackened, so that reflection of the celestial body takes place only at the upper surface. It is exceedingly difficult to level a mirror-horizon in exploratory travel, and, as the amount of its default from horizontality in the direction of the observed celestial body is transmitted undiminished into the resulting measurement of the altitude, it becomes the purpose of the observer to ascertain, by means of the spirit level, the amount of this default at the time of observation and to record this amount as a correction to be applied to the measured altitude, rather than to attempt to bring the mirror into a precisely horizontal position. The expectation would, therefore, be that observations taken with the mirror-horizon should always be accompanied by spirit-level readings of the amount of default from horizontality in the mirror, in order that the important errors thus introduced into the instrumental indications may be subsequently eliminated from the deduced altitudes.
All the usual sources of error that present themselves to mar astronomical observations under the best conditions are intensified in the frozen zones. Benumbed limbs and fatigued vision magnify the personal errors, and the instrumental errors are greatly enlarged by the effects of low temperature which causes the frost to cover the lenses and mirrors of the instruments and the intense cold to freeze the lubricating oil in the joints and to crack and granulate the silver backing of the mirrors.
The flattening of the discs of the sun and moon when near the horizon has often been noticed. This appearance indicates how strongly the rays of light near the horizon are refracted in traversing the atmosphere. The last direction of the ray, or that which it has when it reaches the eye, is that of the tangent to its curved path through the atmosphere at this point; and the difference of the direction of the ray before entering the atmosphere and this last direction is the amount by which the measured position of the celestial body is displaced in altitude from its true position. Since the amount of the refraction depends upon the density of the atmosphere, and this density varies with the pressure and temperature, which are indicated by the barometer and the thermometer, the refraction tables give the refraction for a mean state of the atmosphere; and auxiliary tables are furnished to correct the mean refraction for moderate departures of pressure and temperature from the normal values, but not for such extreme departures of temperature as are experienced on the ice-cap of the Arctic Sea. For altitudes below 5° no refraction tables can be relied upon. There occur at times anomalous deviations of the refraction from the tabular values at all altitudes; and these are most sensible at low altitudes. Unfortunately the sun and the moon are always close to the horizon in the polar regions and the reduction of measured altitudes of these bodies for the influences of refraction is subject to large uncertainties.
It has already been remarked how far the natural horizon of an ice-covered sea may appear displaced from its true position and, although this displacement may not be taken as a measure of the amount by which the rays of light from a celestial body near the horizon are deflected in their passage through the atmosphere to the eye of the observer, yet it will serve as a gauge of the amount of variability that should be expected in the tabulated refraction for small altitudes observed in low temperatures of the atmosphere. The tabulated refraction for an altitude of 5°, with the temperature at— 10° Fahrenheit, is II', but experience has shown that, with wide departures from the normal state of the atmosphere, it might on the one hand be one-half as much and on the other hand fifty per cent more.
For my own part, I have great respect for the opinion of an observer who tells me that he is unable in the Arctic to find the altitude of the sun within 5', and would conclude that he could not determine the radius of his circle of position closer than that amount.
From the other source of error, the uncertainty in the chronometer time and the consequent uncertainty in the position of the center of the circle of position, there is a measure of relief in the near approaches to the pole. For the conditions of nature intervene to make the distance between meridians less in proportion as the difficulty of transporting chronometers for time in this remote region become greater, and in latitudes near 90° the close convergence of the meridians of longitude makes it of less and less importance to ascertain the position of the earth with reference to the celestial sphere in respect of rotation. But, although an uncertainty of a few degrees of longitude will not be of moment to observers in very high latitudes, yet any error in the length of the radius of the circle of position continues to operate with full effect until the pole itself is reached.
If we express the geographical position of the explorer, as fixed by the intersection of two circles of position, in the form of a distance, Q, from some definite place on the earth's surface, then we may regard Q as some function of R and R', the radii of the circles of position, of P and P', the positions of the centers of the circles of position, and of L, the shift of the center of the first circle for the travel of the observer between the two stations at which observations were taken, thus
Q=ƒ (R, R’, P, P’, L).
The relation between the actual error of the determined geographical position, or the actual error of Q, and the errors of the quantities on the right hand side of the equation will be
d(Q)=dQ/dR eR + dQ/dR’ eR+ dQ/dP eP + dQ/dP’ eP + dQ/dL eL,
and the expression for the square of the probably error of Q will be
e2=(dQ/dR)2 e2R’ + (dQ/dR’)2 e2R’ + (dQ/dP)2 e2P + (dQ/dP’)2 e2P’ + (dQ/dL)2 e2L
Whatever the difficulties of observation may be and whatever the uncertainties in fixing the geographical position in the lower latitudes of the polar region, when the latitude of 90° is attained these difficulties vanish and there is a comparatively easy certainty of recognizing that the pole has been reached. To an observer here, the pole of the heavens coincides in position with his zenith, and the necessary and sufficient condition to be recognized is that all stars north of the celestial equator would, if the sun were not lighting the sky, remain permanently visible above the horizon, never rising nor falling at all, but moving on circles of altitude lying parallel to the horizon and coinciding with their circles of declination. At the time of year when our explorers have reached the pole, it is unending day for six months. From the 21st of March, when the sun's center has passed upward over the rational horizon, his image, lifted somewhat in the sky by the effects of refraction, has continued, as viewed from this position, to move, throughout each encircling turn, in a path varying from strict parallelism with the horizon only by the amount of the daily change in the declination. By the middle of April, the encircling courses have reached a true altitude of almost 10°, which does not alter from constancy of height above the horizon much more than 20’' throughout the twenty-four hours. It is, therefore, unnecessary for the observer to do more, for the purpose of identifying his position, than to recognize this condition. The account of time is of no farther concern to him because all meridians have converged into one point, and it is not even required of him that he should be able to measure the absolute altitude of the sun but only that he should be able to test the condition of equality of altitude within the range of 5' in the duration of 6 hours. Precision in his location cannot be expected, because, while he waits, the ice on which he stands is drifting—maybe not rapidly, but at an unascertained rate.
PROOFS OF HAVING ATTAINED THE POLE.
In the absence from the route to the North Pole of any permanent physical features that could be described or marked for identification, the only avenue of proof of having made his journey that could be open to an explorer would be through the record of the compass directions that he followed and the measured distances that he traversed along each of them, and also the original records of the astronomical observations that he made for the purpose of determining his geographical position at the various stages of his march. The instruments of observation would be of no importance in this connection excepting as a means of contributing to the collateral issue as to whether the observations were inadequate or valueless through having been made with instruments of unsuitable design or construction.
Since our moderate assumptions concerning the probable errors of the factors entering into the determination of the explorer's geographical position have shown that the uncertainty in such a determination is of the same order of magnitude as the extent of his daily march, the solution and plotting of the observations would be quite likely to make it appear that the route did not proceed by successive stages toward the destination, but that advance may have alternated with apparent retreat and have been attended with deviations now to one side and now to the other of the intended line of progress.