How David Thompson Navigated
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Message to the Reader

I apologize for the considerable space in this volume of Northwest Journal which is devoted to early 19th century navigation, as many readers may have only a passing interest in this topic. However, please recognize that the vast majority of the information presented in these articles has never before appeared in print. Such a complete assessment of David Thompson's techniques and skill has never been done before. I hope that the information in these articles will spur new interest in assessing the skills and contributions made by all North American geographers of that time. —J. Gottfred

Contents

See also Period Navigation

Art I. Understanding How Thompson Navigated— Introduction to the Case Study By J. Gottfred.

Art II. Assessing David Thompson's Surveying Skill— Guidelines for Historians. by J. Gottfred.

Art III. The Location of David Thompson's 'Goods Shed' on the Athabasca River, by J. Gottfred.

Art IV. Using David Thompson's Sextant Index Error to Show his Diligence and High Accuracy. By J. Gottfred.

Art V. Recomputing Thompson's Data— Latitude by Double Meridian Altitude. By J. Gottfred.

Art VI. Recomputing Thompson's Data— Latitude by Double Altitudes. By J. Gottfred.

Art VII. Recomputing Thompson's Data— Greenwich Apparent Time by Lunar Distance. By J. Gottfred.

Art VIII. Recomputing Thompson's Data— Longitude from GT, Local Time, Magnetic Variation. By J. Gottfred.

Art IX. A New Latitude for the Goods Shed Computed from Thompson's Data. By J. Gottfred.

Art X. Glossary and References. By J. Gottfred.

 

 

The Author lecturing on Thompson's techniques at Old Fort William. (Photo by A. Gottfred)

 

Art I. Understanding How Thompson Navigated— Introduction to the Case Study By J. Gottfred.

Introduction

David Thompson is famous for his early exploration and mapping of western Canada and the northwestern United States. From 1790 to 1812, he traveled the Northwest using a sextant and compass to record valuable navigational information. He used this information to make some of the earliest detailed maps of the northwestern U.S. and western Canada. Paradoxically, although his navigational skills gave Thompson his claim to fame, they are poorly understood by both historians and geographers. How did he calculate his latitude and longitude, and how accurate was he? In this issue, I will use examples from Thompson's notes to illustrate and explain the navigational methods that he used.

There are a number of challenges to studying Thompson's navigational methods. The techniques of celestial navigation have changed significantly in the last two hundred years, but there are still a number of common denominators. Marine navigators still rely on sextants today, although there is increasing reliance on other navigational instruments such as GPS receivers. (Sextants are no longer used for land navigation and mapping.) Modern navigators use the Marcq St. Hilaire method, which requires two sextant observations and a highly accurate knowledge of Greenwich Mean Time, to find their latitude and longitude simultaneously. A nautical almanac (ephemeris) provides the positions of the celestial bodies for the current year, in a format designed to be easily used with modern navigational methods. Electronic calculators or tables of pre-calculated solutions are used to simplify the calculations and ensure accuracy. In Thompson's day, finding latitude and longitude were two separate problems, requiring a number of different sextant observations made at different times. A nautical almanac provided the positions of the celestial bodies for the current year, including data on lunar distances for use in determining longitude. A large array of tables such as those in Neville Maskelyne's Tables Requisite (e.g. proportional logarithms, log-trig tables, double altitude tables, etc.) were used to simplify calculations and ensure accuracy.

In this volume of Northwest Journal, I present a case study of David Thompson's navigational methods, skill, and accuracy. The case study examines Thompson's journey from Boggy Hall to the Whirlpool River, from October 19, 1810 until January 7, 1811. This is an important period because Thompson was near the end of his fur trade career (he retired in 1812) and had twenty years of navigational experience behind him. He was about to make two of his most important journeys : crossing the Athabasca Pass and descending the Columbia River to its mouth. So it is important to have a baseline appraisal of Thompson's skill, by examining his first few steps along the trail and evaluating his ability to accurately plot his position.

The case study provides an example of each type of sextant observation made by Thompson during this period. I explain the purpose of the observation and show how he used it to help determine his latitude or longitude. In addition, I demonstrate that his 'goods shed' (possibly the first 'Henry House' on the Athabasca) was not located at the south end of Brûlé Lake, contrary to the currently accepted position.

To my knowledge, most of the information presented in these articles has never before been published. Smyth (1981) and Sebert (1981) both discuss methods used by Thompson, but do not seem to be heavily based on Thompson's data ; instead, both authors turn to an examination of methods used by marine navigators of Thompson's time. Stewart (1936) and Smith (1961) compared the accuracy of Thompson's maps to modern maps of the same areas. In his journals, Thompson recorded all the information about his astronomical observations that he would need to re-do his calculations and double-check them at a later date. He also recorded key steps (intermediate solutions) in his calculations. Using Thompson's own data and recomputing his answers step by step, I will show :

—How he periodically recomputed the index error of his sextant, and how this information can be used to show that the accuracy of his instrument and his observational skill were excellent. (Article IV, p. 15-16).

—How he found his latitude by observing meridian altitudes of the sun, and that the observation made at the 'goods hut' was made using his reflecting artificial horizon and was accurately computed. (Article V, p. 16-17).

—How he found his latitude by double altitudes— observing two altitudes of the sun or other star, separated by a time interval measured on his watch, and how this data can be used to show that Thompson was capable of making highly accurate runs of sight while away from the conveniences of fort life. (Article VI, p. 17-20).

—How he computed Greenwich Apparent Time by observing lunar distances and how lunar distance longitudes can be made by only one person who observes only the lunar distance and the altitude of some other body. I also show that this was the technique actually used by Thompson (Articles VII, p. 21-28, & Article VIII, p. 28-31).

—How, with a knowledge of Greenwich Apparent Time, he computed longitude by observing the sun or another body, how he kept his watch set correctly to local time, and how he computed magnetic declination ('compass variation'). In addition I provide an estimate of the accuracy of his watch and how he used his watch in general. (Article VIII, p. 28-31).

Along with these demonstrations, I also :

— Describe, in general, Thompson's navigational routine and discuss my assessment of his diligence (Article II, p. 4-7).

—Provide guidelines for assessing the accuracy of any position computed by Thompson (Article II).

—Re-examine the location of the 'goods shed' constructed by Thompson and his men December 5-29, 1810, in the light of the case study (Article III, p. 7-14).

—Compute a completely new latitude for the goods shed which, in conjunction with Thompson's latitude, demonstrates that the accepted location is incorrect (Article IX, p. 31-34).

—Provide a complete explanation of each kind of notation (including marginal notations) in Thompson's astronomical notes for the period of the case study (passim).

—Provide a glossary of navigational terms (Article X, p. 35-37).

It should also be noted that Articles V through IX assume a basic knowledge of modern celestial navigation as used on small boats today. For background information on navigational instruments and the general principles of celestial navigation during Thompson's time, see Gottfred, 'Period Navigation'. More details on Thompson's navigational instruments are in Gottfred, 'Life', p. 3-6, and in Smyth.

Resources

By far the most accessible way for club members to see Thompson's data is through Columbia Journals, edited by Barbara Belyea. Columbia Journals is an excellent transcription of selections from David Thompson's daily notes from October 1800 to September 1811. Belyea has done an outstanding job of deciphering Thompson's often faint and nearly illegible handwriting, and presenting this information in a clear form true to the original. With this volume, Belyea has made a large portion of Thompson's textual descriptions of where he is and what he is doing available to the reader.

Columbia Journals is not intended to be a complete transcription of all of Thompson's material for this period. Belyea has selected portions of the journals which focus on Thompson's efforts to cross the Rocky Mountains and establish fur posts and trade routes between the Rockies and the Pacific. She omits most of Thompson's navigational information. Consequently, for the period of the case study, the manuscripts journals contain much additional useful information about all the celestial observations.

For the data presented in this case study, I have used Thompson's course and distance information as it appears in Columbia Journals, and his celestial navigation information as it appears in the original manuscripts. A complete list of references and resources is provided at the end of Article X.

 

Art II. Assessing David Thompson's Surveying Skill— Guidelines for Historians. by J. Gottfred.

Sloppy...

I have heard comments to the effect that Thompson was a sloppy surveyor. Is this so? In my opinion, for the period of the case study, the answer is yes.

During the period of the case study, Thompson observed eleven lunar distances. He computed nine of these correctly, and in two he either made significant errors in computing the true distance, or he copied down the wrong data.

He incorrectly writes 'N' instead of 'S' for solar declinations four times during this period, but he does not use them incorrectly in his calculations (Nov. 2, 3, 13, 21).

He mis-identifies stars on two occasions and confuses himself to the point that he throws out good data (see Art. III).

At one point he writes down 27º 49' 45" as 27º 29' 45", and ruins a day's work before he notices the error. However, he does go back and correct (almost) everything (Art. III).

In his lunar distance calculations he sometimes records right ascensions in hours (which they should be), and sometimes he records the value after he has converted hours to degrees. Confusingly, he usually fails to note which of these units he is using (p. 23, 24).

Almost every page of computations has values scratched out and changed or written over. In this regard it is not a 'neat' copy.

Thompson does record everything required to go back and recompute his answers at a later date. Every observation has the temperature recorded, as well as each time and altitude pair taken during the run of sights.

It must be remembered that Thompson likely suffered from a shortage of paper, and this is reflected by the fact that many intermediate steps in his calculations are missing from the pages, and the data is crammed into as small a space as possible.

Overall, the notes strike one as being made by an individual for his own use. Certain items which were obvious to the note-keeper have not been recorded for posterity. Anyone viewing this material with an eye to information content for posterity would be justified in calling the work 'sloppy'.

Yet Reliable

Does this mean that Thompson's work is unreliable? In my opinion, no. The information is recorded in a fashion which made sense to the owner. Over the last ten years I have used my own sextant under many different conditions to replicate all of the techniques that Thompson used. I can honestly say that his notes make sense to me, and that his observational skill was probably better than mine. How he records information in his notebooks says nothing about his observational skill and I have found much evidence that his skill was considerable.

Broad-based comparisons made by looking at Thompson's positions compared to modern positions may fail to account for the varying reliability of different observational techniques, and the possibility of errors in calculation. On the other hand, if one takes the time to carefully examine Thompson's observations, to recalculate them to check for errors, to assess the accuracy of the method used for each observation, and the conditions under which they were made, then error bars may be placed around each observation. In fact, it is even possible to find additional useful data. For example, in the case study I found two additional latitudes which are perfectly sound, and which were helpful in reconstructing his movements (Article IX). The error-bounded observations should then form the control points through which his course and distance positions must pass.

General assessments of the accuracies of Thompson's observations should be avoided; instead, each observation should be independently assessed. However, if Thompson's calculations are accurate, if a mercury reflecting artificial horizon was used for all observations (except measuring lunar distance), and if the apparent altitudes of the bodies weren't too close to the horizon, then I feel that it should be generally safe to assume that any latitude by meridian transit observation would be correct within 1½ nautical miles, and any latitude from a double altitude observation should be correct to within 2 nm. This is based on my own experience using Thompson's methods in the field.

Synopsis of Thompson's Navigational Routine

Thompson's general routine for determining his position using the sun and the moon (for example) was as follows :

Upon arrival at a new camp, Thompson would try to obtain an accurate latitude. If possible, he would observe a meridian transit of the sun ('noon sight') (Art. V). If this was not possible, he would make two observations of the sun one hour apart, which he would then use to compute a latitude with the double altitude method (Art. VI).

If the moon was in a convenient location, Thompson would observe the distance between the moon and the sun (Art. VII). Then, within half an hour or so, he would observe the altitude of the sun— a 'time shot' (Art. VIII). When making these lunar distance observations, he checked the index error of his sextant to make sure that it had not changed (Art. IV).

Using the observed altitude of the sun, the latitude computed earlier, and the declination of the sun as determined from the nautical almanac for the approximate Greenwich time as based on his ded reckoning longitude of the observation, Thompson computed the local apparent time. He then reset his watch to the correct local apparent time. This helped to ensure that he did not miss the next day's noon sight due to the inaccuracy of his watch.

Sometimes, Thompson would note the compass bearing to the sun at the instant of the time shot. From his knowledge of his latitude, the sun's declination, and the observed altitude, he could compute the sun's true bearing (azimuth). The difference between the true bearing and the magnetic compass bearing was the magnetic variation (declination) at his position (Art. VIII).

From his knowledge of his latitude, the local apparent time, and the declination of the sun, he then computed a close approximation of the sun's altitude at the instant of the lunar distance observation. Then, from his knowledge of the local apparent time, his latitude, the declination of the moon based on his approximation of the time in Greenwich, and the difference in the right ascensions of the sun and the moon at the approximate Greenwich time of his observation, he computed a close approximation of the true altitude of the moon at the instant of the lunar distance observation (Art. VII).

From the close approximations of the moon's and sun's altitudes, combined with a highly accurate observation of the lunar distance, he then 'cleared the distance' of the effects of refraction and lunar parallax to determine an accurate true lunar distance between the sun and the moon for the local apparent time of the observation. He then used the nautical almanac to determine the apparent time in Greenwich at which the moon would be at the distance that he observed. The difference between his local apparent time and the apparent time in Greenwich, converted to degrees, resulted in his longitude (see Art. VII & VIII).

Thompson also used the stars to compute lunar distances and double altitudes. The techniques are generally the same, but with a slight complication for the computation of local apparent time.

 

Art III. The Location of David Thompson's 'Goods Shed' on the Athabasca River, by J. Gottfred.

The Location of David Thompson's 'Goods Shed'

In December 1810, while preparing for his epic journey over the Athabasca Pass, David Thompson built a 'goods shed' near the Athabasca River in the vicinity of Brûlé Lake (near today's Jasper National Park). Thompson explains why he built the shed in his memoirs :

'...our Guide told me it was of no use at this late season to think of going any further with Horses...but from this place prepare ourselves with Snow Shoes and Sleds to cross the Mountains : Accordingly the next day we began to make Log Huts to secure the Goods, and Provisions, and shelter ourselves from the cold and bad weather...' (Glover, 318)

Thompson and his men built at least two buildings here : a storehouse and a 'meat shed'. They spent December 5-29 at this spot. On December 30, Thompson left the goods shed, leaving behind North West Company clerk William Henry and half of his twenty-four men.

It is generally accepted that this shed was located at the south end of Brûlé Lake (Glover, 318n; Belyea, 253), but the precise location has not been found. Using my knowledge of his navigational methods I have come to the conclusion that the goods shed was actually located approximately three miles up Solomon Creek at the north end of Brûlé Lake.

David Thompson's movements from late October to early December, 1810 took him from Boggy Hall to this 'goods shed'. While traveling, he used his compass to note courses and recorded distance measurements. He also made frequent observations using his sextant. Before trying to follow his trail it is necessary to try to get a feel for the accuracy of these observations.

The course observations can be tricky to interpret. Thompson generally uses a compass to determine his course. His compass readings are magnetic headings, so correcting them for plotting on a modern map requires some knowledge of the magnetic declination of the area as it was in 1810. (Magnetic declination changes slowly over time as the north magnetic pole wanders.) In this regard I must caution the reader, for
Columbia Journals
contains many declination values which seem to be magnetic declinations. In fact, these values are the declinations of celestial bodies (usually the sun) and therefore cannot be used to correct magnetic compass headings. This is a perfectly understandable error for anyone unfamiliar with nautical astronomy.

Thompson confuses matters by occasionally noting solar declinations as north ; these declinations are south during the fall and winter months. He does not use them in his calculations as north declinations, and in other places (most notably in the lunar distance calculations) he indicates that they are south declinations. To add to the confusion on this point, the text in Columbia Journals has an 'N' after the declination values listed under November 26 and December 6 which do not appear in the manuscript, and an 'N' after the declination for November 28 which is actually an 'S' in the manuscript. (All other declinations in Columbia Journals for the period of the case study are correct.)

Incidentally, in Thompson's time, magnetic declination was called 'variation' and that term is used by marine navigators today (Belyea, 183; Bowditch, 85). (For an example of how Thompson differentiates between magnetic and celestial declinations see Belyea, 78.) During the period of the case study, Thompson only records one variation : '22º or 23º East', measured while at the 'Goods Shed'.

Usually what Thompson is recording is a dead reckoning course : an estimated position allowing for deviations around obstructions, minor bends in the river, &c. What he is really saying is 'we went down this river through a bunch of twists and turns and around a mountain over a distance of maybe ten miles, but I figure that our new position is six miles south west of our last recorded position.' (Modern navigators call this kind of course ded ('deduced') reckoning.) Not only are these bearings his estimate as a navigator, but sometimes he estimates his bearings by the sun rather than using his compass. (Thompson's compass may not be entirely reliable (Bowditch, 9-10)). I generally assume that his courses are correct to within plus or minus twenty degrees.

Thompson estimates his distance traveled, and I usually assume that his distances are accurate to plus or minus ten percent. In the few journeys that I have plotted, I have been impressed by his ability to judge distances. I feel that his distances estimates are more reliable than his course bearings.

The sextant observations are more interesting. The general accuracy of celestial navigation 200 years ago was basically identical to that of today. The tables of astronomical data and physical phenomena required for navigation appear to have been just as accurate as today's. A commander in the Royal Navy in 1899 stated that 'the maximum error of lunar table[s] may now be considered to be 10 seconds' (Sebert, 412). My modern nautical almanac states that the error in the position of the moon alone may reach 18 seconds! (United States Naval Observatory (USNO), 261)

The sextants were also just as accurate as modern ones. My modern Astra IIIB sextant reads to 12" of arc and has an accuracy of plus or minus 20". David Thompson's sextant could read to 15" of arc ; I don't know its accuracy. However, my instrument in conjunction with modern tables is barely accurate enough to compute longitudes by lunar distance, and my results are generally not as good as Thompson's. Therefore Thompson's instrument and tables must have been at least as good as mine, and probably better. (I have found good evidence to support this by looking at Thompson's index error calculations. See Art. IV) Thompson's most inaccurate instrument was his timepiece. Accurate chronometers were very expensive, and Thompson did not have one. Instead, he used two or more 'common watches' (pocket watches) with second hands.

Today, navigators can compute both latitude and longitude from a pair of observations. In Thompson's time, due to the lack of accurate watches, latitude and longitude observations were done separately.

The accuracy of any given observation depends upon what type it was. Thompson uses three different techniques for observing latitudes. The first is called a double meridian altitude observation, usually of the sun. This means that he is observing the height of the sun when it is at high noon (crossing the meridian), and he is doing it using his parallel glasses, a reflecting artificial horizon made by placing a glass cover over a bowl of mercury. ('Double' refers to the fact that the altitude measured with an artificial horizon is twice what would have been measured using a sea horizon.) Using the same techniques today, I generally compute latitudes to an accuracy better than half of a nautical mile. (1 nm = 1852 meters). On good days my accuracy is within 300 meters. Based on my comparisons with David Thompson's data from Rocky Mountain House, as well as the data in the case study, I feel that for any of Thompson's double meridian altitude observations it is reasonable to assume that his observation would place him within 1½ nautical miles of his true latitude.

Thompson sometimes mentions a second type of latitude observation which he calls a meridian altitude. This is an ambiguous term. In most cases, it means that he made the observation using his artificial horizon, but in some cases it means that he has used a local body of water as an artificial horizon. This technique involves using the far shore of a lake or long river as a level, and estimating the distance to the far shore. The navigator then applies a correction to the observation based on this estimate. This technique is more properly known as the dip short method. It is far less reliable than using an artificial horizon because of the difficulty of accurately estimating the distance to the far shore. Thompson did not make any dip short observations during the case study period.

The third method of observing for latitude is called a double altitude observation. This means that the navigator has made two observations of the same celestial body from the same position, separated by a time interval measured on a common watch. The idea is that the common watch, although not accurate enough to keep time over several days, is accurate enough to keep the time for about an hour. This means that two observations of the same star, separated by a known time interval, allow the navigator to compute his latitude directly using spherical trigonometry. The accuracy of this method depends upon a few variables. First, if both observations were made using an artificial horizon then the results can be very accurate. If one or both observations used the dip short method, then the result will be fairly suspect. For observations made using an artificial horizon and from one to two hours apart, it should be safe to say that the accuracy of the observation would be plus or minus two nautical miles.

Finding longitude involves making highly accurate observations of lunar distances. Thompson makes eleven observations for longitude at three locations during the period of the case study. The method which he used is fully explained in articles VI and VII. Unfortunately, the accuracy of lunar distance longitudes is poor at best. In general, any single observation will be no better than ±20' of longitude. If more observations from the same spot are averaged together, the result will be more reliable. More than a dozen observations must be averaged together to obtain accuracies within four or five minutes of longitude. During the period of the case study Thompson does not stay in one spot long enough to observe enough longitudes to pin his location down to better than ±20' of longitude.

Thompson's Trail

I began by tracing Thompson's route from a position near the Athabasca River on November 26, 1810 (Belyea, 125). On that day, Thompson and his party camp at 12:15 p.m., and Thompson soon makes a double altitude observation. He computes a latitude of N53º 30' 39". The next two day's travels take him northwest around a 'long lake' and along a brook to near the Athabasca River. On November 28, he observes a double meridian altitude of the sun and computes a latitude of N53º 37' 54". There is no reason to suspect that either latitude would be farther than 2 nm from his actual position. In his November 27 journal entry, he draws a picture of the 'long lake'. It seems clear that this lake is Summit Lake, and the creek that they followed to the Athabasca River is Obed Creek.

Summit Lake— David Thompson's drawing of the lake they passed on November 27, 1810. It seems clear that this is Summit Lake. Obed Creek is shown emerging from the north west corner of the lake.

Thompson then begins to ascend the Athabasca, and on November 29 they camp near a large island in the river (¾ mile long.) River islands are usually poor landmarks, since they erode rapidly, but this island would probably be big enough to persist. Such an island appears on my topographic map within one statute mile of where Thompson's course and distance information would place him on November 29. This camp would be on the Athabasca River at about N53º 31'.

The next day, Thompson continues up the river. A position for his camp based on his course and distance information on November 30 would be one mile north of the river at the junction of the Athabasca River and Maskuta Creek. This position tallies with his description : 'the river run around a large Point & is dist from our road thro' the Willow Plain & from the camp at abt 1M' (Belyea, 127). The Athabasca runs fairly straight up to this point, where it begins a series of curves into the mountains.

On the evening of November 30 he makes observations for latitude and longitude. He chooses the star Procyon for his longitude calculation. Unfortunately this star was still below the horizon when he made his observations! I suspect that he confused the star Pollux with Procyon. He notes in his journal that he has observed the 'wrong star.' On the evening of December 1, while at the same camp, he tries once more but again throws out the data noting 'wrong star'. (There is another possible explanation— perhaps he didn't observe the wrong star, perhaps he used the wrong star. Thompson's nautical almanac may not have included lunar distance tables for another star he observed that night, Algenib, making the observation useless.) Two of the December 1 observations are of Aldebaran. I find it hard to imagine how Thompson could mis-identify this very bright (magnitude one) red star. The altitude that Thompson measures for the star is very close to what it should be for Aldebaran on that date and time. For these reasons, I think Thompson correctly identified Aldebaran. I see nothing else wrong with this Aldebaran observation. When I recompute it, I obtain the result N53º 25' 10". (Thompson calculated N53º 23' before throwing out all of his data for that night.) This latitude lies right on the spot that I obtain by plotting Thompson's course and distance information. I see no reason to think that it is not very close to the actual position.

On December 2 they set out again. It is at this point that I feel Thompson's course diverges from the accepted one. Belyea states that he follows Maskuta Creek to the south end of Brulé Lake (253). However, Thompson says that they traveled south 1 mile to the bank, then southwest through plains and over brooks, and at the end of the day they had gone about 6 miles 'going to the right in curves' and the river was about one-third of a mile away. I feel that this indicates that they followed the course of the Athabasca River, not Maskuta Creek.

They then set off towards the southwest, and met up with the banks of the river, which they followed. They continued southwest until they reached 'the entrance to the Flats which appear like a Lake' (Belyea, 129). I believe that this is the north end of Brulé Lake. Here they met some hunters, who took them to a Native hut on the lake. To get to this hut, they traveled southwest. I believe that they were traveling along the northwest shore of Brulé Lake. My estimate of their position would place them at modern-day Swan Landing. It may be significant that this spot is a hamlet today, as places where people meet tend to persist.

Thompson says that the hunters' hut was small and dirty, and there was no grass for the horses, so they moved the next day. He says that they went north-northwest about 5 miles through aspen forest and camped 'near a small Fountain of Water amongst Pines & Aspins' (Belyea, 129). This is where they decided to build the 'goods shed'.

If they were at Swan Landing on the northwest shore of Brûlé Lake the previous day, then Thompson's course and distance information would suggest a position for the hut somewhere up Solomon Creek at about N53º 23', Lo. 117º 53' W. (Solomon Creek flows southeast before emptying into the northwest end of Brûlé Lake.) I should note here that, although Thompson travels along Brûlé Lake, he doesn't say he is on or near a lake. This is likely because Brûlé Lake is very shallow, and would be at low water and frozen in December. Later, he does not mention the larger Jasper Lake. Both of these lakes are really just widenings in the river.

On December 6, Thompson records a double meridian altitude of the sun, giving a latitude of N53º 23' 27", which is less than one nautical mile from the course and distance position. By examining Thompson's observations from the previous evening, I realized that I could use his observations of the stars Capella and Vega to compute a new latitude using the double meridian altitude method (see Art. IX for details). The position that I obtain is N 53º 21' 22", or two nautical miles from Thompson's December 6 position. Both of these two latitudes are about eight nautical miles from the south end of Brulé Lake, and effectively rule out that position as the location of the goods shed.

This brings us to a possible error. Thompson compiled a table of observations which is reproduced in Belyea (314). Under December 1-6 he made the note 'Athabasca River, at the Shed Depot of Goods (Longitude of 4 observations)' and gives the position N53º 33' 33" Lo. 117º 36' 34". This latitude seems quite wrong. Where did it come from?

On December 6, Thompson incorrectly copies the value 27° 49' 45" as 27° 29' 45". This causes him to compute a latitude for the goods shed of N53º 33' 33". He then seems to catch the error, and changes the copy to reflect the correct latitude. The incorrect value must have been copied to another journal or log and not corrected.

The reader might be tempted to dismiss all of these calculations due to the seemingly large number of errors that I have described. To do so would be a mistake. I have recalculated Thompson's observations and, except as noted, they are correct. Also, an error of this sort is relatively easy for the navigator to catch, as Thompson did, because it gives a result that is markedly inconsistent with the other observations. I see nothing in the journals to indicate that the other values are not reliable. With the corroborating evidence of the newly computed latitude, I feel confident Thompson's value of N53º 23' 27" is within 1½ nm of his actual position.

If we assume that the goods shed is on Solomon Creek, then is this consistent with an analysis of his journey to the Whirlpool River? I believe that it is. They departed the goods shed on December 30 and traveled southeast for five miles. Yet the Athabasca flows southwest from the south end of Brulé Lake. It would seem that they were actually traveling back to the north end of Brulé Lake. As mentioned earlier, Thompson does not describe this place as a lake, but as 'full of small Flats & Isles', in other words, a braided stream. His courses from here to the Whirlpool are not very accurate. This may be explained by the fact that on December 31 he gives an approximate course as judged by the sun. Again, I think taking his course directions too literally would be a mistake.

However, there is a good way to judge whether or not a hut position on Solomon Creek is plausible, and that is to look at the distance traveled. Between December 30 and January 7 (arrival at the mouth of the Whirlpool), Thompson notes distances traveled, generally over good ground and in straight lines. They total 54½ miles. The distance from the proposed position of the hut on Solomon Creek to the mouth of the Whirlpool River is about 50 miles. This is about a 10% error on Thompson's part, which is consistent with his ability to judge distances over good ground.

If the goods shed was located at the south end of Brulé Lake, then the actual distance from the hut to the mouth of Whirlpool River would be about 39 miles. Thompson's journal says he traveled 54½ miles. This is about a 30% error, which strikes me as being an unreasonably large error given the skill of the navigator and the type of terrain that he is covering. It is also inconsistent with the accuracy of the distances stated by Thompson on the first part of this journey.

In summary, the latitude listed in Thompson's table for the 'goods shed' is erroneous, and should be discarded. In addition, I suggest that a position for the 'goods shed' approximately three miles up Solomon Creek is consistent with the navigational information supplied by Thompson.

 

Art IV. Using David Thompson's Sextant Index Error to Show his Diligence and High Accuracy. By J. Gottfred.

The index error of a sextant is a correctable instrument error. Index error is caused by the sextant mirrors being not quite parallel, a normal condition for sextants even today. The navigator can easily determine this index error and correct for it in calculations by observing a distant star and noting what the instrument reads when the star images are properly aligned. In a perfectly tuned instrument, the angle should be zero, but most sextants will show a small positive or negative angle which must then be applied to every observation made with that instrument. The index error should also be monitored with every observation and periodically recomputed to ensure that the mirrors have not been bumped in transit &c. Failure to note the correct index error or to notice a change in the value over time is the most likely way for a systematic observational error to result in positions which are many nautical miles in error. On November 3, 21, and 26, David Thompson makes marginal notations in his calculations which clearly show that he is checking his index error on a regular basis, and modifying its value over time as the instrument reacts to the rigors of travel and changing climactic conditions.

David Thompson's journal entry for November 3, 1810 has an excellent example of an index error calculation which also demonstrates just how accurate his eye and his instrument are. In the corner of the page he makes the notes shown in the box at the right.

Thompson is using the sun to compute the instrument's index error. He does not do this by superimposing the two sun images as one does with a star. This is because the sun is not a point source of light, and judging when the two sun images are precisely overlapped is difficult. To obtain the maximum accuracy, Thompson first aligns the bottom of the sun image with the top and records a measurement of 35' 52". He then reverses the sun images and records 28' 45". Note that the second measurement is actually a negative measurement. He sums the two numbers and then takes their difference, which is 7' 7". He divides the difference by two in order to compute the index error for the center of the body, with the result of 3' 34". Noting that this is a positive value (on the arc), and therefore he must subtract this value from any observation that he makes, he records his instrument error correction as –3' 34", the value he actually uses.

By summing the two values he obtains 64' 37". Dividing by two will yield the sun's diameter. Dividing by two once again yields the sun's semi-diameter, 16' 9.4" (or 16.16'). Semi-diameters for the sun for each day are listed in the nautical almanac, because they change gradually throughout the year, as the earth revolves around the sun.

The following table lists sun semi-diameters computed from Thompson's index error notes for the period of the case study (rounded to the nearest 0.1') compared to the actual values of the sun's semi-diameter for those dates, as listed in a 1996 nautical almanac.

Date

DT's S.D.

Actual S.D.

Nov. 3, 1810 16.2' 16.2'
Nov. 21, 1810 16.2' 16.2'
Nov. 26, 1810 16.3' 16.2'

Note that the almanac only gives values to the nearest 0.1' as this is the limit of the resolution of the eye. (This limit corresponds to a maximum theoretical accuracy of 185 meters on the ground.)

This demonstrates that Thompson and his instrument can measure the semi-diameter of the sun to an accuracy of 0.1 of a minute. Because this process is visually identical with actually making an altitude observation using a reflecting artificial horizon, this also demonstrates that both Thompson and his instrument were capable of measuring the height of the sun to an accuracy of 0.1'.

 

Art V. Recomputing Thompson's Data— Latitude by Double Meridian Altitude. By J. Gottfred.

On December 6, 1810, while at the 'goods shed', Thompson records the following information :

Thompson is saying that he has observed a meridian altitude of the sun's lower limb to determine his latitude, and that the sun's declination was 22º 30' 50" when he made the observation at 7 hours 48 minutes Greenwich Time by his watch. (The astronomical day started at noon in Thompson's time (Belyea, 274)). On December 6, local apparent noon at Brûlé Lake would be about 19:48 Greenwich Time.

A meridian altitude of the sun ('noon shot') is still used by navigators at sea to determine latitude. When an artificial horizon is used, this type of observation may be called a double meridian altitude, because the angle measured by the sextant is twice what it would have been if a sea horizon had been used.

We can use the data given by Thompson to recompute a latitude for his position (Thompson's data in italics) :

27º 49' 45"     Height of sun times 2 (artificial horizon),

–3'                 less index correction (see Art. IV),

27º 46' 45"     gives corrected height of the sun.

13º 53' 22"     Divide by 2 to get height of the sun,

–3' 48"           less refraction of the air correction (ignore f),

13º 49' 34"     gives height of sun corrected for refraction.

+16' 18"        Adding semi-diameter of sun for Dec. 6

14º 5' 52"      gives height of the sun as observed.

90º 0' 0"       Angle between the zenith and horizon,

14º 5' 52"–    less the height of sun,

75º 54' 8"      gives zenith distance.

22º 30' 50"–   Subtract south declination of sun for

53º 23' 18"     final latitude.

(Please note that 'height of the sun' is height of the sun above the horizon.)

This latitude differs from Thompson's by only 9 seconds, or 278 meters on the ground. Since I don't know what Thompson used as a refraction coefficient (f) to correct for local density altitude, the answers are essentially the same.

This example clearly shows that Thompson used a reflecting artificial horizon for this observation and that he did not make any mathematical errors in calculating this latitude.

 

Art VI. Recomputing Thompson's Data— Latitude by Double Altitudes. By J. Gottfred.

On November 3, 1810, David Thompson records some observations which were used to compute latitudes by double altitude. In this article, I present an example calculation for latitude using Thompson's data. Note that I will show a mathematical technique for computing the latitude which illustrates how it is done. Thompson would have used tables to simplify his calculations.

Thompson's first observation is an upper limb observation of the sun (Thompson's data in italics) :

0h 33m 53s 43º 44' 0"  
0h 34m 41s 43º 42' 0"  
0h 35m 27s 43º 40' 0"  
0h 34m 40s 43º 42' 0" Average of values,
  –3' 34" less index correction,
  43º 38' 26" gives Hs x 2 corrected.
  21º 49' 13" Divide by 2 to find Hs.
  –2' 30" Less refraction correction.
  21º 46' 43" Corrected for refraction.
  –16' 12" Less semi-diameter of sun
  21º 30' 31" Height observed #1 (Ho1)

The reason for averaging the time and sextant readings is that the average values will fall on a line which is the best fit through all three observations. Thompson provides a second run of sights roughly an hour later :

1h 30m 54s 39º 25' 45"  
1h 31m 37s 39º 21' 30"  
1h 32m 21s 39º 16' 0"  
1h 31m 37s 39º 21' 5" Average of values,
  –3' 34" less index correction,
  39º 17' 31" gives Hs x 2 corrected.
  19º 38' 46" Divide by 2 & find Hs.
  –2' 48" Less refraction correction
  19º 35' 58" Corrected for refraction.
  –16' 12" Less semi-diameter of sun.
  19º 19' 46" Height observed #2 (Ho2)

Thompson notes the declination of the sun at the time of the first observation is 15º 4' 20" south. I computed a declination at the time of the second observation based on the rate of change in the sun's declination for this time of the year as follows :

The difference in time between the first and second observations is 56m 57s. The rate of change in the declination of the sun is 44" per hour on this day of the year, so the declination for the second observation would be 15º 5' 2" S.

We now convert the time difference between the two observations (56m 57s) to arc in order to find the meridian hour angle t (See figure 1). By dividing the time interval by 4 minutes per degree of longitude, we can express angle t in degrees, and so we find that t is 14º 14' 15".

Find d

Now find d. For all of the following computations we will use the law of cosines for spherical triangles. It states that for any spherical triangle XYZ, consisting of sides of length x, y, z :

Therefore, substituting for the triangle Pn-Sun1-Sun2

Figure 1 — The line segments connecting the points are all great circles on the surface of the earth. Pn— The north pole. Z— The observer's zenith, Sun1— the geographical position (GP) of the sun at the time of observation #1. Sun2— the geographical position of the sun at the time of observation #2. t—the meridian hour angle between Sun1 and Sun2. PD1— The polar distance of the sun at the time of observation #1. PD2— The polar distance of the sun at the time of observation #2. d— The distance between the GP's of Sun1 and Sun2. co-L— the observer's co-latitude (90º– latitude). co-Ho1— The co-height of the sun measured by observation #1. co-Ho2— The co-height of the sun measured by observation #2. A1— Angle Pn-Sun2-Sun1. A2— Angle Z-Sun2-Sun1 A3— Angle Pn-Sun2-Z.

We note that the polar distances PD1 and PD2 are the sum of 90º plus the declinations of sun declinations 1 and 2 respectively. Solving, we find d = 13º 44' 44.6"

Find A1

The next step is to find the angle Pn-Sun2-Sun1. Again, from the law of cosines for spherical triangles we can write:

Solving, we find A1 = 91º 48' 45.0"

 

 

Find A2

Next, we find the angle Z-Sun2-Sun1 in spherical triangle Z-Sun2-Sun1. Once again we use the law of cosines for spherical triangles to write:

 

Because sin x = cos (90º–x) and cos x = sin (90º–x) :

Solving, we find A2 = 78º 23' 26.2"

 

Find A3

Now we find the angle Pn-Sun2-Z in triangle Pn-Sun2-Sun1 by simply subtracting A2 from A1:

Find L

Finally, we find the latitude, L. Once again from the law of cosines for spherical triangles we can write:

Using the same trigonometric rules earlier we can write:

Rearranging, we get:

Solving, we find L = 53º 8' 22".

This value for L differs from Thompson's computation (53º 7' 57") by only 25". This deviation is easily explained by the fact that we did not use exactly the same declination as Thompson, nor did we apply the refraction correction coefficient f, nor did we account for Thompson's watch error rate. Even so, this nearly identical result clearly shows that Thompson was using his reflecting artificial horizon for these observations, and that his calculations are correct.

Upon recalculating three possible latitude pairs for observations on that day which are separated by up to 2h 28m I found the mean value to be 53º 8' 36" ± 26". This is only ± 0.4 nm, which is excellent shooting for double altitude observations, and demonstrates that Thompson was able to accurately measure the sun's changing altitude over the course of three hours while away from the conveniences of fort life.

 

Art VII. Recomputing Thompson's Data— Greenwich Apparent Time by Lunar Distance. By J. Gottfred.

On November 21, 1810, David Thompson recorded three lunar distance observations in order to determine Greenwich Apparent Time (GAT). He needed to know GAT time, in conjunction with local time, to compute his longitude (see article VIII for details).

Observing the motion of the moon to compute the time in Greenwich, England is based on the fact that the moon's proper motion relative to the stellar background is about 30' of arc per hour. This means that in about 12 seconds of time the moon will move far enough, relative to another object on the ecliptic, for the distance it moved to be measured. Since the moon's motion is predicted with high accuracy in the nautical almanac, this means that in theory an observer could determine at what GAT time the moon would be seen at the observed distance to an accuracy within 12 seconds of time. This would allow the observer (in conjunction with another observation) to compute a longitude to a theoretical accuracy of about 3'. In practice, accuracies of ± 20' are all that can be achieved. However, several observations taken from the same place and averaged together can yield significantly higher accuracy.

Although the basic idea is simple, it is complicated by the distorting effects of the refraction of the earth's atmosphere. The refraction and parallax corrections for an observation taken perpendicular to the earth's surface are easily obtained from tables. However, lunar distance observations cut across the sky at oblique angles, and computing the effects of refraction and lunar parallax are more challenging. Correcting for these effects is called 'clearing the distance', and an example of Thompson's method is given below.

Contemporary books describing lunar distance observations for mariners recommended that four observers and three sextants be used to obtain the maximum accuracy. One observer measured the distance between the moon and another object on the ecliptic, another observed the altitude of the moon above the horizon, a third observed the height of the second body above the horizon, and the fourth called out the time so that all these observations could be made simultaneously.

Thompson worked alone (he only had one sextant), so how he could have made these observations has been a puzzle. In his article on Thompson's method for longitude, Sebert notes that :

'On smaller ships, and in Thompson's case, it was the custom for one observer to read all the angles, assisted only by a locally trained timekeeper. The readings, in the order taken, were the air temperature (for calculating refraction), the moon's elevation and time, the star's elevation and time, the lunar distance and time, the star's elevation and time, and the moon's elevation and time.' (Sebert, 408. cf. Garnett, 31-32.)

Smyth also mentions this method (Smyth, 14-15).

All of these observations should be made as quickly as possible. The idea is to determine the rate of change in the lunar and stellar altitudes, and compute what their altitudes would have been at the instant of the lunar distance observation.

I have used this technique myself and it works admirably. However, David Thompson, although diligently recording all other observations, makes no such 'bracketing observations' of his lunar distances. In fact, for the period of the case study he never observes the moon's elevation at all. In short, Thompson's journals for the period of the case study do not show that he made his longitude observations in the manner Sebert & Smyth both suggest.

The solution to this puzzle is found in Thompson's marginal notes for each lunar distance that he observes. For each distance observation, he notes the right ascension of the sun and moon, as well as their declinations. This information is not required for clearing a distance if the altitudes have been observed. If, however, the altitudes of the moon and second celestial body have not been observed, then they can be calculated from knowledge of the observer's known latitude and estimated longitude, using the right ascensions and declinations given in the nautical almanac for the approximate GAT of the observation.

At first glance this may seem like a paradox— if Thompson knows where he is then why is he trying to figure out where he is? It is important to understand that Thompson knows where he is within some ever-widening circle of uncertainty. The object of all of these observations is to refine this estimate of his position and shrink the circle of uncertainty. After all, the art of navigation is the art of staying found.

The mathematics of clearing lunar distances seem to be quite tolerant of errors in the computed altitudes of the moon and second body. What is critical is that the distance between the two be measured with high accuracy. Some preliminary calculations that I have done suggest that as long as his dead reckoning longitude was correct to within 30', it makes little or no difference to the cleared distance. However, more study is required before this figure can be accepted.

Sebert notes that this method was used in Thompson's time :

'...in the more advanced navigational texts there is a method given for the case where the horizon cannot be seen and only the lunar distance is read. The solution involves first calculating the true altitude of the two bodies, and then applying the corrections for refraction and parallax in reverse. It is a very long problem.' (Sebert, 412)

This was exactly what Thompson was doing for every lunar distance I have examined, yet Sebert does not seem aware of Thompson's use of this method. (Readers of Sebert's paper should know that he makes a couple of minor errors in his calculations. On p. 409, he renders 51º 28' 35" as 51.4676º. The correct value is 51.4764º. The moon's refraction on p. 410 is 4' 20" which is 4.33', not 4.2'. These early errors throw off Sebert's calculations for both examples as well as his conclusion. The answer obtained using Borda's method should give the same result as using Young's formula given at the end of this article.)

On November 21, 1810 Thompson observes a lunar distance between the sun and the moon's near limbs. In the margin he notes the following information :

H ' "

¤ 's AR [sun's right ascension]— 15 .. 46 .. 16

Dec [sun's declination] — 19 .. 54½ S

's AR [moon's right ascension]— 176 .. 15 .. 15

Dec [moon's declination] — 57 1/6' N

S.D.[moon's semi-diameter]— 15' 9½"

HP [moon's horizontal parallax]— 55 .. 37

¤ 's TA [sun's true altitude]— 11 .. 44 .. 9

¤ 's AA [sun's apparent altitude]— 11 .. 48 .. 50

's TA [moon's true altitude]— 32 .. 20 .. 53

's AA [moon's apparent altitude]— 31 .. 35 .. 10

TD [true lunar distance]— 62 .. 32 .. 36

+ 2-5 + 2-20 + 9"

117º 13' W

Thompson lists these values using little abbreviated symbols, the meaning of which should be clear if you know what the abbreviations are. All of the values are actually in degrees, minutes, and seconds of arc except for the sun's AR which is in hours, minutes and seconds of time. The moon's HP is in minutes and seconds of arc. Of these values, the AR, dec, and HP values come from the nautical almanac. The moon's S.D. may have come from the almanac, or may have been computed using the standard formula S.D. = 0.2724º * HP. The values for the true altitudes of the sun and moon would have been computed by Thompson in the following manner :

From his observation of the altitude of the sun taken fifteen minutes after the lunar distance observation, Thompson would have computed the local apparent time as described in Article VIII. The only information he needs to compute the local apparent time is his ded reckoning longitude and his latitude, which he determines using one of the latitude methods discussed previously. From his dead reckoning longitude he determines the approximate time in Greenwich. For example, if he assumed that he was at Lo. 116º 30' W, then at 15º per hour, he would be 7 hours 46 minutes behind Greenwich. Knowing the approximate time at Greenwich allows him to compute the declination of the sun from the information in his nautical almanac. Using the declination of the sun (d) and his latitude (L = 53º 24' 52"), as well as the height of the sun above the horizon (Ho) that he observed, he computed a local apparent time of 22h 9m 31s for his 'time shot' (see Art. VIII). Using the difference in his watch time between the lunar distance observation and the time observation, he now knows the local apparent time of his lunar distance observation, in this case 21h 53m 15s. Remember, Thompson is using astronomical time, so this corresponds to 9h 53m 15s a.m., or 2h 6m 45s before noon, local time. At 15º per hour, this means that the sun's meridian angle (the angular distance from noon at Thompson's position) is 31.6875º. Thompson now knows his latitude (L), the polar distance of the sun (PD = 90º + declination), and the meridian hour angle (t). This allows him to compute the apparent altitude of the sun at his location at the instant of the lunar distance observation using the formula:

In this case, I compute a true altitude for the sun of 11º 44' 17". The 8" difference from Thompson's value is probably explained by his use of tables to compute the result.

He then computes the true altitude of the moon. Again, he knows his latitude (L), and he can find the declination of the moon from the nautical almanac for his estimate of the time at Greenwich. To find the meridian hour angle (t) of the moon, he needs the right ascensions of both the sun and the moon. Again, he gets these from the nautical almanac for the approximate GAT time of his observation. The difference in the right ascensions of the sun and the moon tell him the meridian hour angle between the sun and the moon, in this case 4h 1m 15s. At 15º per hour, this is 60.3125º. Incidentally, the moon's AR is really 11h 45' 1", but Thompson writes 176 .. 15 .. 15. This is because he has already converted it to degrees.

The sun was not yet at Thompson's meridian, but the moon has already gone by. This means that the moon's meridian hour angle will be 60.3125º minus the sun's meridian hour angle of 31.6875º. The answer I get is 28.625º. Plugging these values into the formula above yields a true lunar altitude of 32º 26' 33". This differs from Thompson's answer by 5' 39". (For some reason, all of my recomputed lunar true altitudes for this day differ from Thompson's by about 5' 40".) However, this is sufficiently close to Thompson's answer that it seems clear that this is the method that he was using.

Calculating Apparent Altitudes

Now that Thompson has calculated his true altitudes from his dead reckoning position, he calculates the apparent altitudes from the true altitudes. First he calculates the apparent altitude of the sun.

11º 44' 9" S (computed by Thompson),
+4' 31" plus refraction correction,
11º 48' 40" gives s— apparent altitude of sun.

This differs from Thompson's value by 10". Next, we want to find the moon's apparent altitude. However, before we proceed, we must compute the moon's parallax in altitude (PA), which is the cosine of the apparent altitude, times the horizontal parallax (HP). The value I obtain is 46' 59".

32º 20' 53" M (computed by Thompson),
–46' 59" less PA,
+1' 30" plus refraction correction,
31º 35' 24" gives m— apparent altitude of moon.

This differs from Thompson's value by 14". The small difference notwithstanding, these examples demonstrate how apparent altitudes can be computed from the true altitudes. For the rest of the example, I will use Thompson's apparent altitudes so that the final answer can be compared to his.

Finding Apparent Distance—d

The first step in clearing the distance is to find the apparent distance, d, between the moon and another celestial body (sun, star, or planet). (See fig. 1). Thompson has measured the distance between the nearest limbs (edges) of the sun and moon to be 62º 0' 5", so this value must be corrected for the semi-diameters of both bodies to find the distance from center of body to center of body. This is computed as follows (Thompson's data in italics):

62º 0' 5" Average observed in sight run,
–3' 34" less index error correction, gives
61º 56' 31" actual angle measured to near limbs.
+15' 9" Add moon's semi-diameter (S.D.) and
+16' 12" add sun's S.D. for date, for
62º 27' 52" d— the apparent distance.

Thompson does not record this value in his notes. Nor does he note any d values for any of his lunar distances.

Figure 1 — Clearing the distance. Z— observer's zenith. HO— The observer's horizon. M— the true altitude of the moon's center. m— The apparent altitude of the moon's center. S— The true altitude of the sun's center. s— The apparent altitude of the sun's center. D— Line SM, the true distance between the sun and moon's centers. d— Line sm, the apparent distance between the sun and moon's centers.

The Approximation Method

Thompson uses an approximation method to determine a solution to the distance d. The answers should not materially differ from computations using a more rigorous method, and in Thompson's day they were much easier to perform.

Referring again to figure 1, perpendiculars (xM and yS) are drawn from the line connecting ms to the true positions M and S. The idea is that the distance xy is a close approximation to the distance MS. Angles y and x are 90º, and their sides are so small that they can be treated as plane triangles for the purposes of finding the lengths of xm and ys. We can now solve the spherical triangle sZm and compute the angles at m and s. (note that Zsm and YsS are congruent).

Thompson provides the following data :

Finding Angles m and s

The first step is to find the value of angle m. For all of the following computations we will use the law of cosines for spherical triangles (see Art. VI). Using this formula for triangle sZm we can write :

Solving, we find angle m = 92.8388º. Similarly we can write:

Solving, we find
s = 62.4644º

 

Finding Segments xm and ys

The line segments xm and ys can be found from plane trigonometry.

xm = 2' 16"
ys = 2' 10"

 

From figure 1 it can be seen that if angle m is less than 90º (acute), then the length of line segment xm should be subtracted from d. If it is greater that 90º (obtuse), then it should be added to d. If angle s is less than 90° (acute), then the length of segment ys should be added to d, and if s is obtuse then the ys should be subtracted from d.

In this case, m is obtuse, so we must add xm to d, and s is acute, so we must also add ys to d. This yields a true distance of 62º 32' 18".

Thompson's value for D is 62º 32' 36", only an 18" difference. Interestingly enough, we could write our corrections as '+ 2-16 + 2-10' This is clearly what Thompson is writing at the bottom of the page where he notes '+ 2-5 + 2-20 + 9"' Thompson's values differ slightly from our results, but it must be kept in mind that there were several variations of this approximation method. When I clear this distance employing a rigorous mathematical method by Young (Cotter, 214) using the formula :

I obtain the value 62º 32' 40" which is within 4" of Thompson's value.

Finding GMT

Thompson now has a true distance between the center of the moon at the center of the sun as measured when his watch said 21h 53m 15s (11:53:15 am). He would now turn to his nautical almanac, where he would find true lunar distances for every three hours GAT for various bodies close to the ecliptic. Thompson would then use linear interpolation, assisted by proportional log tables, to compute the GAT time for the distance which he observed. Thompson now knows the time GAT that corresponds to his watch time of 21h 53m 15s. How he uses this time to compute longitude is the subject of the following article.

 

Art VIII. Recomputing Thompson's Data— Longitude from GT, Local Time, Magnetic Variation. By J. Gottfred.

In addition to the various observations discussed in the previous articles, Thompson also computed longitudes from his knowledge of Greenwich and Local Apparent Times, set his watches to local apparent time by observing the sun or other stars, and computed the magnetic variation at his locale.

To demonstrate how these values were determined, I will use a hypothetical case (since Thompson leaves us no calculations) using the data from November 3, 1810.

On this day he observed two lunar distances, and made four observations of the sun's altitude. From figure 1, and using the law of cosines for spherical triangles, it can be seen that

This can be simplified to :

Figure 1 — Pn— The north pole. Z— The observer's zenith. Sun— The geographical position of the sun. PD— The polar distance of the sun. co-L— Observer's co-latitude (90º – Latitude). co-Ho— The co-height of the sun (90º – Ho). t— The meridian hour angle. z— Azimuth angle.

This observation can be used to compute the observer's local apparent time, even if the observer does not know what time it is in Greenwich. The meridian hour angle t can be converted into hours with the formula:

 

 

 

For an afternoon observation of the sun this converts directly into local apparent time p.m. If Thompson has done a lunar distance observation within a hour of this 'time shot' (to reduce the effects of an inaccurate watch), then he now knows the difference between the local apparent time in Greenwich, and the local apparent time at his position. This time difference is simply converted into degrees at 15º per hour to find his longitude west of Greenwich.

Even if Thompson does not have a lunar distance to go along with his time shot, he will still make such time observations in order to keep his watch set to local apparent time. This allows him to know exactly when noon will occur (subject to watch error and how far he has moved in longitude since he last set his watch). This allows him to plan his day's events so that he does not miss a double meridian altitude observation, the most important daily observation for any navigator to make.

Watch Rate Computed

There is clear evidence that Thompson's watches were next to useless as navigational tools. First, he never computes a watch rate, nor uses one in his calculations. Secondly, he always keeps time-critical observation pairs as close together in time as possible to ensure the maximum accuracy. Thirdly, his notes show that at nearly every opportunity he computed the local apparent time and reset his watch. An excellent example of this is provided in his observational notes for November 26, 1810. In these notes he lists the following values (Thompson's data in italics) :

H ' "

º '

Line #

55 .. 9

28 .. 21½

1

55 .. 42

18¼

2

56 .. 13

15¼

3

1[6]7 .. 4

55

4

55 .. 41

28 .. 18 .. 20

5

+18 .. 58

–3

6

1 .. 14 .. 39

28 .. 15 .. 20

7

Lines 1 to 3 are Thompson's observation pairs where he is recording his watch time (i.e. 0 hours (noon), 55 minutes, 9 seconds) and the height of the body as measured by the sextant (i.e. 28º 21.5'). On line 4 he writes down the sum of the values, and in line 5 the average values. This gives him a point on a line which is the best fit through all three values— a standard technique of the day for improving sight accuracy (Garnett, 31).

The following step is not clear from Thompson's notes, for the values on line six are not filled in at the same time. What Thompson does next is to write down his sextant index error correction under the right-hand column on line six. He then finds the final sextant altitude which he records on line 7. At this point he then computes the local apparent time of the observation as discussed above. He then writes the corresponding local apparent time next to the sextant altitude to which it applies. This is the value in the left hand column on line 7. Thompson then computes the difference between what his watch said and the actual local apparent time at the instant of the observation. In this case, his watch is 18 minutes 58 seconds slow. He notes this value in the space on line 6 in order to conserve paper and to keep things neat. Thompson would no doubt immediately set his watch forwards by 19 minutes so that on the next day, he would know to within a few minutes when local apparent noon is going to occur so that he can get set up to make a meridian altitude observation of the sun— the navigator's most important daily observation. Indeed, in the notes for this day he records that 'Examined watch moved 20' forward'. Normally he does not bother to record the fact that he has reset his watch.

When you first look at these columns of numbers it appears at first glance that the time value on line 6 is some sort of correction which applies to get the time on line 7, but this is not so. This example in his notes is a good one to examine, as it seems that the values in the left-hand column on lines 6 and 7 were written in later, as he is using a pen with a different width. This same pattern can be seen in many other entries where the time correction value is squeezed into too small a space or does not line up correctly with the other numbers in the column.

Using two observations from November 3, I calculate that Thompson's watch was gaining about 3¼ seconds per hour on that day. When I look at his watch corrections overall for the period of the case study, correct them for changes in his longitude, and assume that he reset his watch each time he made a time observation, I find that his average watch rate was 4 seconds per hour fast, ±9 seconds. This is quite poor. In 1806, Garnett remarks that :

'...Dr. Maskelyne observes, that a watch that can be depended upon within a minutes for 6 hours is absolutely necessary ; but I would recommend to have at least one pocket chronometer or time piece, for connecting the observation for finding the time, with that for the distance. They are made by Mr. Arnold in London as low as 25 guineas ; also by Mr. Earnshaw and Mr. Broeckbank ; and would be extremely useful for a variety of purposes both for the longitude and latitude, and in discovering currents.' (Garnett, 30)

Thompson's watches from Joseph Jolly were worth only 12 guineas each in 1794 (Smyth, 8). These would be better-than-average watches for the time, but apparently not in the 'pocket chronometer' league. It would also appear that his watches were never upgraded, and that 16 years later he still had not acquired a pocket chronometer. Thompson himself noted on August 2, 1811 that :

'All the Obsns made going to the Sea was with a com[mon] Watch that went very badly, losing time— on my return also with a com[mon] Watch that went tolerable well'. (Belyea, 163)

It would seem clear that his watches were only useful for tracking the general time of day, the Greenwich time to the nearest half-hour, and the time separations between double altitude or lunar distance longitude pairs for intervals of less than an hour.

Compass Variation

If the observer records the bearing of the body along with its altitude, then it is possible to compute the magnetic declination (variation). From figure 1, it can be seen that:

This becomes :

Angle Z is the true bearing of the sun at the time of the observation (called the azimuth), in this case in degrees west of north. In this example this can be converted to a standard compass bearing by subtracting z from 360º. The difference between the calculated bearing and the bearing measured by the compass is the variation.

 

Art IX. A New Latitude for the Goods Shed Computed from Thompson's Data. By J. Gottfred.

On December 5, 1810, while at the location of his 'goods shed' on the Athabasca, David Thompson observed a series of lunar distances and two time shots to be used for the longitude component of the two lunar distances. On December 6, he observed a meridian altitude of the sun and computed a latitude of N 53º 23' 27" for the location of the goods shed. He then used this latitude to compute the longitudes observed on the previous evening. This latitude is quite different from the latitude of the accepted location of Thompson's goods shed.

In Article V, I examined Thompson's latitude observation from December 6 and showed how it was made with his reflecting artificial horizon, and that Thompson did not make any computational errors in performing the calculation. This suggests that the observation should be a very good one, and it should be close to the truth.

Unfortunately, this single observation says nothing about how accurately he took the measurement. Any number of arguments may be made to suggest that this single value is not to be relied upon— perhaps he was having an 'off day' and was careless with the observation. Perhaps he misread the index vernier. Maybe his parallel glasses were not quite parallel and produced a significant distortion. Perhaps there was a strong temperature inversion caused by a nearby storm which strongly affected the refraction of the earth's atmosphere. It was partly cloudy on that day, so perhaps the cloud interfered with his ability to accurately see the edge of the sun's limb. All of these arguments can legitimately cast some doubt on the accuracy of any single observation. However, if another latitude for the site can be computed using different celestial objects observed on a different day under different conditions and at different azimuths, then, should the results agree to a reasonable extent, all of the above arguments can be shown to be unfounded.

Thompson's time shots from December 5 allow a new latitude to be computed. This latitude was never computed by Thompson ; either he never realized that it could be done, or he didn't bother. If he did not realize that it could be done, then this suggests that he was well grounded in all of the standard techniques as outlined in the navigational texts, but that he really did not have a firm grasp of how he actually arrived at his answers— he just followed the instructions for 'standard sights'. I favor this interpretation because he computed all the other permutations of his sights on this journey, and since he did not have a latitude to use that evening he was forced to wait until noon the next day before he could do the calculations.

Thompson's observations of the stars Vega ('Lyræ') and Capella allow me to compute a latitude using the double altitude method as described in article VI. Table XXX of Garnett's Tables Requisite... from 1806 lists the right ascensions and declinations of the principle navigational stars, as well as how much these values change per year. This information is summarized as follows :

Star

RA

Var./yr

Dec.

Var./yr

Vega

18h30m20s

2.03s

38º 36' 27" N

+2.6"

Capella

5h 2m 18s

4.41s

45º 47' 16" N

+5.0"

From this information, the right ascensions and declinations of these two stars can be estimated for the time of Thompson's observations in 1810 :

Star

RA

Dec.

Vega 18h 30m 28s 38º 36' 37" N
Capella 5h 2m 36s 45º 47' 36" N

We also have Thompson's observations and watch times which I provide as they appear in his journal (corrected for index error).

Star

WT

Hs x2 corr.

Vega

7h 14m 53s

77º 18' 30"

Capella

7h 56m 39s

88º 24' 56"

From the values of right ascension (RA) we can compute the difference in RA between the two stars in 1810. Note that Capella was visible in the east, and Vega was visible in the west, therefore the time separation is 24h - 18h 30m 28s + 5h 2m 36s = 10h 32m 8s. Multiplying by 15º per hour results in an angle between the two stars of 158º 2' 0".

Thompson observed Vega first, which you can visualize as 'pinning' its location to the heavens at the instant of the observation. He observed Capella 41m 46s later. During that time, as the heavens appear to rotate overhead, Capella moves closer to the 'pinned' position of Vega. Again, converting time to arc and subtracting this from the angle separating the two stars we get the meridian hour angle (t) which is 147º 35' 30".

Thompson noted that the air temperature was –2ºF, and from a topographic map of his approximate position I feel that his elevation was just under 1100 meters. Assuming average atmospheric conditions this results in a refraction factor f of 0.98. True refraction = mean refraction * f. The altitudes of these two stars above the horizon can therefore be computed using modern refraction tables in the following manner :

 

Vega

Capella

Hs x 2 corr. 77º 18' 30" 88º 24' 56"
Divide by 2 38º 39' 15" 44º 12' 28"
R –1' 10" – 59"
Height observed Hv = 38º 38' 5" Hc = 44º 11' 29"

 

From figure 1, and using the data above as well as the formulae presented in the article on computing latitude from double altitudes, the following values can be computed :
d = 90.72550º , A1 = 21. 94522º, A2 = 25.50479º. Note that in this case Z is south of the great circle connecting Vega and Capella, so A3 = A1 + A2. The final latitude computed is N 53º 21' 27"

Figure 1 Pn— The north pole. Z— Thompson's Zenith. co-Dv— 90º– declination of Vega. co-Dc— 90º– declination of Capella. d— distance between Vega and Capella along a great circle on the celestial sphere. t— The meridian hour angle between Vega and Capella. co-Hv—90º– observed altitude of Vega. co-Hc— 90º– the observed height of Capella. co-L— 90º – the latitude of the observer.

This latitude is 2' (2 nm) south of the latitude that Thompson computes on December 6. This observation is probably not quite as accurate as the double meridian altitude observation of December 6, but if we assume for the sake of argument that they are equally valid, then the position of the 'goods shed' must lie at latitude N53º 22' 27" ± 1'. The close correlation between the latitudes observed on December 5 and 6 effectively removes the possibility of significant systematic or random error in the execution of the observations. As the south end of Brûlé Lake is roughly 7.8' south of this position, this rules it out as a possible location for Thompson's goods shed.

Postscript

For those readers familiar with celestial navigation who are eager to try Thompson's methods, you may write the author care of Northwest Journal for additional information, assistance, etc., as well as to obtain copies of the author's Tables Useful for Celestial Navigation. This booklet contains : mean refraction tables circa 1781; refraction factor (f) tables circa 1781; modern mean refraction tables; modern refraction factor tables; conversion between mb, inches Hg and mm Hg; barometer corrections for altitude above sea level; a table of barometric pressure by boiling point of water (insert) which when used with the barometer correction table yields the observer's altitude; conversion of arc to time; temperature conversion; and miscellaneous data and formulae. A full explanation of each table is included along with information on how they were computed and the data source used.

Modern plastic practice sextants are available which are accurate enough for computing latitudes. (The Davis Mk 15, $107 US, looks like a good bet.) The Nautical Almanac sells for $16.95 US. All the stuff you require (except for the land navigation tables) can be ordered through Celestaire at 1-800-727-9785.

Art X. Glossary and References. By J. Gottfred.

Apparent altitude (Ha) - The height of the body above the horizon as it appears to the observer once mechanical measuring errors have been eliminated. See also Observed Altitude (Ho), and Sextant Altitude (Hs).

Azimuth angle (Z) - The angle between the sides co-latitude and co-calculated altitude of the spherical triangle connecting the north pole, the observer's zenith, and the geographical position of the observed body.

Azimuth (Zn) - The angle between true north and the body, measured clockwise from true north. Azimuth is always positive, and between 0° and 360° . Zn is computed from azimuth angle (Z).

Cleared lunar distance (D) - The angular distance between the moon's center and another body as measured from the center of the earth. Obtained from the lunar distance (d).

Co-declination (co-d) - One side of a spherical triangle equal to 90º minus the declination of the body.

Co-latitude (co-L) - One side of spherical triangle equal to 90º minus the latitude of the observer.

Declination (dec) - The position of a celestial body on the celestial sphere measured in degrees north or south of the celestial equator. It is exactly equivalent to latitude and is measured the same way. For example, if at some instant the declination of a body is S 15º 32' 4", then at that instant the geographical position of the body is at latitude S15º 32' 4". Magnetic declination is the difference between magnetic north (the direction the compass needle points) and true north (roughly Polaris). Also called variation or compass variation.

f - Correction factor applied to mean refraction to correct for non-standard atmospheric pressure and temperature.

Geographical position (GP) - The intersection of a line connecting the center of the Earth to the center of a celestial body and the surface of the Earth. For any instant of time this spot is the position on the Earth's surface at which the body is at the zenith. The GP may be expressed in terms of declination and right ascension/hour angle.

Great circle - The shortest distance between two points on the surface of a sphere. A great circle is described by the intersection of a plane cutting through the center of a sphere and the surface of the sphere.

Greenwich Apparent Time (GAT) - This is the local apparent time at the Greenwich meridian, which is defined as being at zero degrees of longitude. GAT differs from Greenwich Mean Time (GMT) by the difference of the equation of time for that day. Thompson did not use GMT.

Ha - See apparent altitude

Horizontal parallax (HP) - The parallax of the moon when it is observed at the horizon.

Ho - See observed altitude.

Hs - See sextant altitude.

Index correction (IC) - The correction to applied to sextant altitude (Hs) to correct for registration error of the instrument. IC is opposite in sign to index error (IE).

Index error (IE) - The registration error of the sextant caused by the horizon and index mirrors being non-parallel. IE is positive if the error is on the arc, and negative if the error is off the arc. See also index correction (IC).

Latitude (L) - Imaginary parallel lines on the earth's surface at right angles to the earth's axis of rotation. The equator is 0° latitude, the north pole is 90° north latitude, and the south pole is 90° south latitude. See also declination.

Longitude (Lo) - Imaginary lines on the earth's surface which are described by great circles passing through the north and south poles. The prime meridian is 0° longitude and is located in Greenwich, England. Longitude is measured east and west of Greenwich to 180º.

Lunar distance (d) - The angular distance between the moon's limb and another celestial body, usually on or near the ecliptic as measured with a sextant. Also, a longitude calculated using this measurement. See also cleared distance (D).

Mean refraction () - The refraction of the atmosphere at a standard temperature of +7° C and a pressure of 1010 mb.

Meridian - The meridian is the line of longitude which passes through the zenith. When the sun is on the observer's meridian, it is local apparent noon.

Meridian angle (t) - The smallest angular distance between the meridian at the observer's position (Z) and the meridian of the geographical position (GP) of a celestial body. Also the smallest angle between the meridian of any two positions or bodies. Meridian angle is measured east or west and is always positive.

Nautical mile (nm) 1nm = 1' of latitude = 1852 meters.

Observed altitude (Ho) - The altitude of the body above the horizon as measured by the observer, once all corrections have been applied to the observation.

Parallax in altitude (PA) - The component of the moon's horizontal parallax which applies for altitudes greater than 0° . PA is computed as :

PA = HP × cos (Ha)

Refraction correction (R) - A correction to the apparent altitude of a body which accounts for the bending of light from the body as it travels through the Earth's atmosphere. The refraction correction is computed as:

Semi-diameter correction (SD) - A correction to the apparent altitude of a body which adjusts for observations on the limb of a body. For stars and planets, no SD correction is required. For the sun and moon, SD corrections are listed for each day in the nautical almanac. For the moon, SD can also be computed as:

The value of SD is positive for a lower limb observation, and negative for an upper limb observation.

Sextant - A hand-held instrument for measuring the angle between two distant observed objects.

Sextant altitude (Hs) - The height (altitude) of a body as measured by the sextant and prior to applying any instrument or artificial horizon corrections. See also apparent altitude.

Spherical triangle - A triangle drawn on the surface of a sphere consisting of sides which are segments of great circles. The length of any side of a spherical triangle is the angle of arc described by that side as measured from the center of the earth. The angle between two sides of a spherical triangle is the angle as measured on the surface of the sphere.

Z - See azimuth angle.

Zenith - The point on the celestial sphere which is directly overhead.

References & Bibliography

Alberta Forestry, Lands & Wildlife. Edson 83F. [Topographic Map.] Provincial Mapping Section, Land Information Services Division. 1988.

Bowditch, Nathaniel, LL.D., The American Practical Navigator : An Epitome of Navigation, Volumes I, II. Defense Mapping Agency Hydrographic/Topographic Center Pub. No. 9, 1984.

Cotter, Charles H. A History of Nautical Astronomy American Elsevier Publishing Company : New York, 1968.

Garnett, John. Tables Requisite to be Used with the Nautical Almanac for the Finding of Latitude and Longitude at Sea. John Garnett : New Brunswick, New Jersey, 1806.

Gottfred, J. 'Period Navigation', in Northwest Journal, Vol. III, April to July 1995. pp. 11-18.

Gottfred, J. Tables Useful for Celestial Navigation over Land, 2d ed. J. Gottfred : Calgary, 1995.

Gottfred, A. & J. 'The Life of David Thompson', in Northwest Journal, Vol. V, November 1995 to January 1996, pp. 1-19.

Her Majesty's Nautical Almanac Office. The Star Almanac for Land Surveyors. London, HMSO.

Henry, Alexander (the Younger). New Light on the Early History of the Northwest : The Manuscript Journals of Alexander Henry... Elliot Coues (ed.) Reprint-Ross & Haines : Minneapolis, 1965. Originally published 1897.

Marriott, C. A. Skymap V2.2.4. Planetarium Software for Windows. 1992, 1994.

Sebert, L. M. 'David Thompson's Determination of Longitude in Western Canada', in Canadian Surveyor. Vol. 35, March 1981, no. 4 : 405-414

Smith, Allan H. 'An Ethnohistorical Analysis of David Thompson's 1809-1811 journeys in the Lower Pend Oreille Valley, Northwestern Washington', in Ethnohistory, Vol. 8, No. 4, Fall 1961.

Smyth, David. 'David Thompson's Surveying Instruments and Methods in the Northwest 1790-1812.' Cartographica 18, no. 4 (1981), pp. 1-17.

Stewart, W. M. 'David Thompson's Surveys in the North-West', in Canadian Historical Review, 1936.

Thompson, David. Columbia Journals. Barbara Belyea (ed.) McGill-Queen's : Montreal, 1994.

Thompson, David. Original manuscript journals, Archives of Ontario volume 25. Unpublished. Archives of Ontario.

United States Naval Observatory (USNO). Nautical Almanac 1996. Paradise Cay Publications : Middletown, California, 1996.

 

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