One of the first things taught the student in descriptive astronomy, as he endeavors to master the various systems of astronomical and geographical co-ordinates, is the uniqueness of the conditions that obtain for the astronomer or the navigator at the actual pole of rotation of the earth. The student learns that every direction is south for an observer at the North Pole, that his familiar meridian plane of lower latitudes has vanished, and with it all hour angles and azimuths in their ordinary sense, so that the time of an observation means next to nothing and the observer is in all longitudes at the same moment. North, east, and west have vanished; the stars describe circles parallel to the horizon on the celestial sphere, and so also the sun and the noon if their motions in declination are not taken into account. If the observer is as much as a mile or so from the pole, a short walk may take him from west into east longitude across the international date line of the 180th meridian and into another astronomical day.
Perhaps in part a result of such well-known items of our apparatus of instruction, there is the widespread impression that navigation or the determination of one's geographical position in the immediate vicinity of the poles of the earth is a process of mystery and high uncertainty, involving the use of complex and special methods of observation and calculation.
Nothing could well be farther from the truth. As a matter of fact, the determination of the co-ordinates of a station located within a radius of 10 miles from the pole uses essentially the same methods and is just as simple as it would be for a similar area near the equator. It is true that on the journey “up” to such a polar camp some special methods may be of occasional advantage and much assistance may be derived from such modern adjuncts as the radio time signal and the sun compass in steering on a true north course. Once arrived at the camp close to the pole, however, there is nothing new or strange; probably we shall find that we have had much more trouble at 80° N. than awaits us at 89°-55' N., for certain simplifications will be found near the pole. Time and longitude both become uncertain so that an accurate value of the time of an observation is no longer essential; our watches may be 10 to 20 minutes in error without noticeable effect. If actually at the pole, moreover, the zenith coincides with the pole of the heavens so that the simple processes of addition and subtraction take the place of the spherical trigonometry of lower latitudes. Our instrument is the familiar sextant and the mercury surface of the artificial horizon; our methods of calculation and planning of the observations will be essentially identical with those used by any navigator or any astronomer were he set down at some place in sunnier latitudes.
The success of the soviet aviators in establishing a semi-permanent camp on the ice floes in the immediate vicinity of the North Pole has aroused considerable popular interest in the question of navigational methods near the earth’s pole of rotation. In addition, there has recently been published by Professor William H. Hobbs an interesting and inspiring biography of Peary in which he touches on the methods used and on what has been fittingly termed “The Great Controversy,” that raged between the supporters of Admiral Peary and Dr. Frederick A. Cook with regard to the discovery of the North Pole (Peary, by William H. Hobbs, Macmillan, 1936).
It is no part of the present article to discuss the bitterness of the Peary-Cook controversy as manifested in the United States, which finally attained a quasi-political importance through the fact that several Congressmen (naturally authorities on navigational problems!) were strong partisans of Cook. Action was accordingly delayed on a first report on Peary’s records prepared by a strong committee of experts. Later, Peary’s data were checked by Messrs. H. C. Mitchell and C. R. Duvall, of the U. S. Coast and Geodetic Survey; their report was finally accepted, while Dr. Cook’s claims were decisively rejected by the University of Copenhagen. Even yet, however, 30 years after the event, occasional reverberations of the old controversy will be heard, and a statement will appear in the daily press to the effect that Peary never reached the pole.
Peary’s data as worked up by Messrs. Mitchell and Duvall with recomputation of the refractions, etc., appeared in a note by Thomas C. Hubbard, “To Students of Arctic Exploration,” in Atti del X Congreso Internationale di Geografia, Rome, 1913, but has been reprinted in part by Professor Hobbs as Appendix C to his biography of Peary where they are more accessible to the general reader. The full calculations were also reprinted and distributed by the National Geographic Society in 1927.
In these sources will be found the check computations for Peary’s observations of April 7, 1909 (to be referred to later as Shots 3 and 4). The position of Camp Jesup was determined by Messrs. Mitchell and Duvall in the Rome paper from Shots 3 and 4 by two independent methods of computation. Mitchell’s computation reached the result without an explicitly assumed position for Camp Jesup; Duvall, using a trial and error method, had to assume a starting latitude.
Because of the fact that the adopted zenith was less than 5' from the pole, the resulting triangles—pole-zenith-sun—were so “extreme” that 7-place logarithms were necessary in the solutions. No fault can be found with this procedure per se, but it is rather unfortunate that in the reproduction of these calculations the value of the latitude was not then rounded off to minutes or tenths of a minute of arc at most, instead of being given to hundredths of a second of arc—about 1 foot.
This point has been criticized in the review of Professor Hobb’s book published in the Geographical Journal, 89, 255 (1937). As is customary in reviews, this is signed only with the initials, A. R. H. and J. M. W. Inasmuch, however, as these initials so evidently stand for A. R. Hinks, Secretary and Editor of the R. G. S. and for J. M. Wordie, Honorary Secretary of the R. G. S., this review will be referred to, when necessary, under the names of Hinks and Wordie. In their criticism of this point, these reviewers appear to have overlooked a sentence in the original Rome paper (p. 707): “The geographic position given for Camp Jesup, namely: Lat. = 89°-55'.4 N.; Long. 137° W., is the best obtainable value for the position of that point." The writer feels that some other features of this review are "uneven" in the selection of material for criticism almost to the point of unfairness. An answer to this review was published by Mitchell and Duvall in the same Journal, 90, 164(1937), with a reply by the original reviewers, ibid., 167.
Peary's own descriptions of his methods of taking his astronomical observations near the pole, precautions necessary, the succession of observations taken and the reasons therefor, have been taken, by preference, from Peary's earliest detailed description of his polar journey given in a series of ten serial articles in Hampton's Magazine (no longer published), vols. 24and 25, between January and September,1910, under the title "The Discovery of the North Pole;" about 145 pages plus 124 half tones and 3 maps.
The uniqueness of the actual pole as a site for astronomical observations has been mentioned in the first paragraph and is so well known as to need no further emphasis. But the moment our site is moved a few miles from the pole in any direction, other factors begin to enter, though their effects are relatively small. We now have a meridian, hour angles, longitude, and time, though all of these are somewhat indeterminate in numerical value. Furthermore, queer pranks are played upon us through this proximity to the pole. We may think, as did Peary, that we are taking an observation approximately at apparent noon, while it is in reality 6:00 A.M. at our place of observation! Does this affect our results materially? While we feel sure in advance that earlier results cannot be materially changed, no one, to the best of the writer's knowledge, has ever worked over Peary's results with these points in mind, and no better way can be found to accentuate the peculiarities in the determination of a position close to the pole than to review Peary's actual observations in some detail. We shall then arrange the discussion in approximately the following order: (1) a very brief description of the navigational method effectively1 employed by Peary, the only one really usable under the conditions; (2) a discussion of the actual observations with a search for possible omissions in data or treatment; (3) a re-reduction of the observations that shall include the aforementioned peculiar features of work close to the pole. Finally, we shall ask ourselves the questions, could Peary have done better in 1909, and could we do better in1939? Our review of Peary's observations, as will be seen later, may then be described as in the nature of a second approximation, from the standpoint of the astronomer and the modern navigator.
The Saint-Hilaire method.—The Sumner method, in its original form, has been almost completely supplanted in modern navigation by the later modification known as the Saint-Hilaire method, discovered by the French Captain Marcq Saint-Hilaire in 1874 and first published in La Revue Maritime et Coloniale in 1875. The principle of this beautiful and efficient modification is shown with sufficient clearness in the two small diagrams, Figs. 1 and 2, inserted solely for the student who may be unfamiliar with the method; Fig. 1 shows the method in elevation, while Fig. 2 shows it in plan.
We have reason to think, or we assume, that we are at a certain station A on the earth’s surface, assumed to be a plane for the relatively short distances involved. Knowing the assumed co-ordinates of A and the local time, it is a simple matter to calculate what the altitude of the sun or star should be, shown by the dotted line, if we are where we think we are. An observation, however, indicates that the celestial body is really at a different altitude, shown by the full line. We must then “move” our assumed site to the location where we would have the observed altitude; in the figure, the observed altitude is the greater so that we must move toward the sun, to the point A'. Figure 2 indicates this same Saint-Hilaire shift as seen from above, and the shift must naturally be along the line of the bearing of the sun, to the point A'. If we are able to take but the single altitude, this concludes our work, and we adopt A' as the corrected position. The most probable position of the site is somewhere along the Sumner line, a short line drawn through the new position at right angles to the bearing of the sun. If we can take altitudes of two bodies, however, or of the same body at considerably different hour angles, the position will be much more accurate, for it should be at the intersection of the two resulting Sumner lines, which we term a “fix.” Any location not too far away may be taken as our assumed position; in fact in some of the modifications of the Saint-Hilaire solution subsequently made by Aquino, Dreisonstok, and others, it has been customary to take one co-ordinate of the assumed or dead-reckoning position in half or whole degrees so as to avoid troublesome interpolations in the condensed tables that are employed. Peary, in effect, used this method at the pole, and assumed the pole as his dead-reckoning site; we shall simply choose another site as a second approximation which will be closer to the actual site and which will be a small distance from the pole so as to permit the inclusion of approximate values of the actual time at the site. While, as already noted, we do not expect to secure a radically different value for the co-ordinates of his polar camp, it will serve to emphasize certain peculiar features of astronomy close to the pole.
Peary’s observations at the pole.—There cannot be the shadow of a doubt from Peary’s observations near the pole that at about 10:00 a.m. (60th meridian time) on April 6, 1909, he reached his final camp, called Camp Jesup, and that this camp was less than 5 miles from the true pole. There is no space in this note for a description of the consummate generalship of the pioneer parties sent ahead to break the trail and successively sent back when no longer needed, a plan through which the 413-mile journey was successfully traversed from Cape Columbia on the northern shore of Grant Land (longitude about 70° W.), and doubtless the only plan that could have succeeded in the days before the airplane. Peary’s “steering” and “distance” were both remarkably good; we know now that he reached a site “level” with the pole, but about 4 miles “west” of it, so far as the meridian of Cape Columbia was involved.
While near the pole, Peary took 13 altitudes of the sun. His first observation (Shot 1) was interrupted by clouds, so that it consisted of but one altitude of the sun's lower limb; it indicated that Camp Jesup (henceforth frequently abbreviated to CJ) was about 3 miles on the "south" or Cape Columbia side of the pole.2 Peary's three later observations (Shots 2, 3, and 4) each consisted of 4 altitudes, 2 measured on the upper and 2 on the lower limb of the sun.
Peary knew that to determine the position of CJ in both co-ordinates he must secure an observation preferably nearly 90° from the first one; he accordingly prepared to do this 6 hours later, but the sky was cloudy. He then journeyed about 10 miles “north” along the line from Cape Columbia and took Shot 2 at the end of this journey, a site which we shall term CJ' (see Fig. 4). This journey, and the subsequent sledge trip “poleward” between Shots 3 and 4, was carried out as follows: A double team of now well-rested dogs was hitched to a sledge whose only actual freight was a single tin of pemmican. With an Eskimo to help over the rougher spots, search for detours around hummocks or raftered ice floes, etc., quite high speed could be attained.
Returning to CJ after this sledge trip, he secured Shot 3 in a direction nearly at right angles to that of Shots 1 and 2; he thought this was a 6:00 a.m. shot, and it was such as far as the Cape Columbia meridian was involved, though we shall see later that it was actually midnight and sub polo in the apparent time at CJ. Six hours later he took another set of observations, a duplicate in its essentials of the one made at the beginning. But Shot 3 had indicated that CJ was about 4' from the pole in the “direction of Bering Strait” (168° W.). So a second sledge trip of 8 miles “poleward” was carried out between Shots 3 and 4. Four hours later he made the decision to start at once on the hazardous return journey to Cape Columbia.
The principal data with regard to the four sets of observations are assembled in Table 1, with some notes of explanation or amplification.
The conditions attending the four shots made by Peary in the vicinity of the pole are assembled in schematic form in Fig. 4. From the standpoint of the navigator or the astronomer, and bearing in mind the rather extreme field conditions, it must be admitted that Peary’s observations leave a most favorable impression, as they are adequate in plan and of very high rank as regards technical execution. No necessary datum is omitted except the index correction in Shot 2.
Though perhaps superfluous, a rough estimate of the probable accuracy of these observations may be in place. The accumulated errors in a good sextant due to eccentricity, lack of parallelism of mirrors or shade glasses, etc., will rarely exceed 12" =0.2 mile, and will generally be less. Errors due to anomalies in the refraction cannot be predicted, but are believed to be absent in the observations under consideration. An aviator may consider himself fortunate if he determines the position of his plane through astronomical methods with a precision of ±10 miles; the average rule-of-thumb navigator trusts his ship's position within 5 miles; careful sea navigation, with Sumner fixes, is generally good within 2 miles. By multiplying observations and taking numerous precautions, it is quite possible for a skilled observer to determine the position of a station on land by means of the sextant and artificial horizon well within a tenth of a mile; this accuracy, however, is not claimed for Peary's four polar shots. Shot 1, a single altitude of the sun's lower limb, would be admitted to have a possible error of 3 or 4 miles by any practical navigator. The accuracy of Shots 2, 3, and 4, each comprising 2 sights on the upper and 2 on the lower limb of the sun, is considerably higher; the writer is inclined to attribute to these an internal probable error of only approximately ± 1 mile, fora number of reasons. Peary himself seems unduly modest in his estimate of error, made in Hampton's Magazine, 25, 180 (1910): “Various authorities will doubtless give different estimates of the probable error in observations made at the pole. I am personally inclined to think that an allowance of 5 miles is an equitable one.
Re-reduction of Peary’s observations with “CJ” as origin.—As already noted, this second approximation will use as a dead-reckoning position a point in 160° W. Longitude and 89°-57’ N. Latitude with the accompanying changed co-ordinates of CJ' instead of the pole; it will include also, as a perhaps unnecessary refinement, the actual apparent times of the observations at the points of observation. There will be no other change in the data as computed by Mitchell and Duvall and recomputed by the writer, with one exception, an index correction of +2’ has been incorporated in the data of Shot 2, making the observed altitude 1’ greater.
The reduction formula employed will be the well-known haversine formula,
hav z = cos L cos d
hav t + hav (L—d),
where z = zenith distance; L = latitude; d=declination; t=hour angle. Many navigators still prefer this formula to the “short-cut” methods devised by Aquino, Dreisonstok, Ageton, and others, which divide the astronomical triangle, pole-zenith-sun, into two right-angled spherical triangles. The reason for the preference lies in the fact that a haversine is always positive, so that there is but one form of solution, and an entire absence of any ambiguity as to algebraic sign, or alternative “cases to remember.” It would be difficult to find four observations where the hour angles and other data are more “extreme” than in Peary’s four shots near the pole, yet this universally trustworthy formula handles them in straightforward fashion without the least uncertainty or brain-fag. While 4-place functions would have sufficed, 5 places have been used to be sure of tenths of a mile in the results. In connection with this second approximation, a study of the actual hour-angle conditions at the sites will be of value, as shown in Fig. 4.
Note that the shift for Shot 2, because this shot was taken about 180° from 1 and 4, really indicates a shift away from the Cape Columbia side.
It is somewhat ironical that, because of these peculiarities in hour-angle change near the pole, it turns out that Shots 1, 2, and 4, which Peary doubtless thought best determined his latitude, have less effect on that than they do on the longitude of CJ, while Shot 3, at right angles to the others, has most direct effect on the latitude. Yet each of the other three, through its Sumner line, played an important part indirectly on the latitude, and enables us to judge how close Peary came to the pole on his poleward sledge trip between Shots 3 and 4. It is indeed forunate that clouds did not prevent Peary from taking Shot 3 and then making the 8-mile sledge trip between Shots 3 and 4, for Shot 3 was all-important in locating CJ about 4 miles from the pole in the direction of Bering Strait; without it, some legislative caviller could have insisted at CJ was much farther from the pole and that Peary did not come near the pole on either of his sledge trips.
It was with very considerable hesitation that weight 4 was finally assigned to Shot 2. The uncertainty here lies not so much in the fact that a value of the index correction has been supplied as in the fact that the distance of CJ' from CJ is not known with all precision.3 The assumption that Peary sledged a distance quite close to 10 miles is, however, a highly reasonable one. Multiplying the indicated shifts by the weights in the case of Shots 1,2, and 4, we then find that CJ was about 0.8 mile toward Cape Columbia from the line of the 160° meridian, while Shot 3 places the latitude of the camp at +89°-55 ’.6.
With regard to this re-reduction of Peary’s observations near the pole, the author desires at this point to express his sincere appreciation of helpful advice and suggestions which have been generously tendered by Messrs. Mitchell and Duvall; their own reductions of Peary’s observations are, with their permission, combined with the author’s in Table 3. A least squares reduction by Mitchell and Duvall, using Curtis’ starting data, gave values very close to his, namely, +89°-55'.5 ±1'.0; 148° W. ±9°.
The writer’s thanks are also recorded here for numerous helpful suggestions made by Professor Hobbs, of special value because of his close acquaintance with Admiral Peary and his familiarity with the details of polar exploration.
We then reach the conclusion from the calculations of this paper that Peary sledged within about 0.8 mile of the pole on his second sledge trip; more than this cannot be derived from the observations. Had we assumed (as is equally probable) that the index correction to be supplied in Shot 2 was 3' instead of 2', for example, and further that the sledge journey to CJ' was about 8.5 instead of 10 miles, we would find that Peary’s second sledge trip passed within 100 feet of the pole, perhaps actually over it. While such speculations are, of course, futile, since we cannot add such to the astronomical data as recorded, this may serve to emphasize the slight and not improbable changes in the data that are needed to assume that Peary passed exactly over the pole.
The graphical Sumner fix for the four polar shots is given in the small diagram included as Fig. 5; a closely similar diagram is given by Hinks and Wordie (loc. cit., 258).4
Possible improvements, past or present.—Three limitations or conditions had to be met by Peary:
(a) He spent but 30 hours at the pole. The return journey of 413 miles to Cape Columbia lay ahead of him and this might well be more hazardous than the trip north due to the advancing season, with the opening up of new ice-leads difficult to cross; his colleague Marvin, head of one of the supporting parties, was drowned in such a lead on his return journey(unless, indeed, he was shot by his own Eskimos; cf. Hobbs, Peary, p. 347).Peary had to start back promptly or run the risk of remaining permanently.
(b) The position of CJ must be determined.
(c) Sledge trips must be made from CJ so as to pass as nearly over the pole as was practicable.
Our first question is then, given the above conditions, could Peary, in 1909, have done better? The answer must be, No. It is not difficult to think of other procedures. He might, for example, have remained continuously at CJ and taken altitude sights (given unclouded skies) every two or three hours, that the armchair explorers could subsequently solve by least squares and so locate the camp with considerable precision. But he could not then have made his sledge trips to pass more nearly over the pole. Nor is any other type of astronomical observation so efficient at the pole as the method he employed.
A second question naturally arises, given the above conditions, could we, in1939, do any better, using such modern instrumental adjuncts as the radio and the sun compass? Here again, in the opinion of the writer, the answer must be, No. Any other answer involves the negation of one or more of the given limitations. It is true that an airplane could now drop him at CJ and afford him a longer sojourn there; perhaps, also, a time of arrival could be selected when the moon at first quarter could have been used, so that the Sumner lines from both sun and moon, nearly at right angles, could be used to fix the co-ordinates of the camp in two directions. But while the radio time signal and the sun compass would have been of great service in steering the course on the way north, they could add little to the observations near the pole. Nor has any more modern navigational method been devised that would replace the Saint-Hilaire altitude shifts.
All in all, the writer, as a result of his examination of Peary's work near the pole, is far more impressed with what Peary did than with what he left out. His journey north and his dynamic activity for the 30 hours spent near the pole form a tour de force with few if any parallels in the annals of exploration. It seems impossible to plan any procedure more adequate than that actually used by Peary, and it is the measured judgment of the writer that Peary sledged within about three quarters of a mile of the earth's true pole, and perhaps even actually over that unmarked and quasi-imaginary point on the shifting ice floes of the Polar Sea.
1. Effectively, as shown by his use of the observations, though Peary does not explicitly use the words "shift" or "Saint-Hilaire" in the popular account in Hampton's Magazine. He found his "latitudes" referred to the pole as origin; the differences from 90° then motivated the two sledge trips to be mentioned later.
2. As already noted, the dead-reckoning point assumed for the determination of Saint-Hilaire shifts may be arbitrarily assumed without effect on the final results. Peary took the pole for his assumed position, first, because he felt sure it was not far distant, but Principally because this assumption relieved him of all computation other than the processes of addition or subtraction.
Accordingly the writer does not understand the statement made by Hinks and Wordie (loc. cit., 256), “He [Hobbs] does not remark that the first single observation gave not a latitude but only a position line; . . . " So far as Shot 1 was involved [and Peary had no other available datum at that time] the shift was about 3 miles toward Cape Columbia and the most probable position of CJ was on the Sumner line about at right angles to the Cape Columbia meridian; this line passed 3' "south" (on the Cape Columbia side) of the pole, his assumed dead-reckoning point. He was entirely logical in taking it as a latitude indication; if not a latitude indication in the sense described, one searches in vain for a more suitable name for it.
It seems equally difficult to understand the gratuitous and apparently unwarranted assumption made by Hinks and Wordie on p. 257, "We conclude that the times attributed to Peary are those at which he intended to take his sights; not those precisely at which he did take them." Though this assumption is softened by the statement in the next sentence that local times close to the pole have little meaning, the writer prefers the simpler assumption that Peary merely looked at his watch and jotted down the time [see the page of his notebook reproduced as Fig. 3]. Peary was a navigator of great experience and skill, and must have known enough of peculiarities of navigation near the pole to realize that this was a sufficient record. Thus the implied criticism of Hinks and Wordie higher on the same page, "The only trouble is that he does not seem to have taken the watch time of each sight" seems trivial. Peary would have done this in lower latitudes, but was a good enough navigator and astronomer to realize that such would be an unnecessary refinement where he was.
Let us assume by way of illustration, that Peary's watch had an uncorrected error of half an hour. The maximum error that could be thus introduced for a position 4' from the pole would be about half a mile!
(1) The adopted dead-reckoning position of CJ is 3' from the pole and at right angles to the line from Cape Columbia to the pole.
(2) Position CJ' assumes that Peary sledged just 10' between Shots 1 and 2, and in the continuation of the journey line from Cape Columbia.
(3) Peary states that the main difficulty in working with the sextant at these low temperatures came not so much from the cold as from the eyestrain due to the snow glare.
Some special precautions were necessary. The mercury was well warmed in advance in the igloo to prevent freezing. A special wooden trough had been made; this was filled right to the brim with mercury and so blocked up inside the artificial horizon roof that neither the roof frame nor projecting rims of the trough could interfere with the measurement of the sun at these low altitudes, all less than 7°. The observer lay prone on his stomach, with elbows and body resting upon a skin spread on the ice, thus bringing the eye sufficiently low and making it possible to hold the sextant with great steadiness.
(4) These ranges in the individual altitudes of a shot are astonishingly low, considering the difficult field conditions, and are a tribute to Peary's skill with the sextant. It would be very difficult to better these ranges in routine series of sextant observations made with an artificial horizon under much more comfortable conditions as to temperature and posture. Concomitant evidence as to Peary's proficiency with a sextant may be derived from the values of the sun's apparent diameter indicated by the differences between the altitudes of the upper and lower limbs. These are: Shot 2, 32'-38"; Shot 3, 32'-21"; Shot 4, 32'-21.The angular diameter at this date was 32'-00.
(5) Conforming to good astronomical practice, Peary determined the index correction for each set of observations [see Fig. 3, where the "On 30 Off 34" indicates that the index correction was determined through tangencies of the two solar images]. Shot 2 forms an exception, and it is not known why the index correction was omitted here. It is possible that Peary regarded it as un-necessary in finding the approximate position of the purely transient observation site at CJ'. This omitted index correction must have been either3' or 2'; the chance that it was zero or a value different from those just given is too remote for serious consideration by one familiar with sextants.
(6) Peary carried with him the pages of the refraction tables torn from Bowditch; these extend only to —10°F. The refractions were accordingly recomputed for the recorded temperatures by Mitchell and Duvall and have been again checked by the writer.
Anomalies in the refraction are of not infrequent occurrence at such low altitudes and extreme temperatures. However, because determined at Pulkova, the "low" end of the tables for refraction has been found generally trust-worthy, and the complete picture of the polar sights made by Peary indicates that no gross errors entered in from any vagaries of refraction. Any such error seems certainly less than 1 mile.
(7) It is well known that the sun assumes a somewhat oval shape near the horizon; refraction changes so rapidly at very low altitudes that the lower limb is thus raised considerably more than the upper limb. It may occur to the over-captious astronomer, as it did to the writer, that this effect should be taken into account. However, separate calculations of the refraction for the observations on the upper and lower limbs changed the values already found by quantities of the order of 1second of arc only.
(8) The odd changes between what Peary regarded as his watch times of observations and the actual apparent times at CJ and CJ', given in the last two rows of entries, may also be seen graphically in Fig. 4.
3. As a necessary adjunct to the work of exploration, Peary and his companions, as well as his Eskimos, had become very expert in estimating the distances traversed in sledge trips. It was his custom to get estimates of distance from all the members of a party, white and Eskimo alike, and then to take the average.
4. When but two Saint-Hilaire sights are available, the fix may be found either graphically or through a simple arithmetical process with the aid of Table 47 or Table 48 of Bowditch. Where more than two sights are available, practically the only method of convenient solution is the graphical one, through a diagram similar to the one mentioned above. The method is an old one and has always been used for multiple fixes in connection with Saint-Hilaire sights; Peary, since he used the pole as his dead-reckoning origin, doubtless carried it in his head, as shown by his course of action, or he may have made the roughest of sketches on a scrap of paper as a mnemonic aid.