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.

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.

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.

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'.

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.

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.

'...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)

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.

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

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 :

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".

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

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

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

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 :

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.

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.

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".

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.

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):

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.

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 :

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

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.

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.

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 :

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.

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 :

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

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

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 :

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"