The Snake-Columbia Confluence Observations

On the morning of 27 September 1805, wood chips flew as expedition axes bit into five large pine trees near Clearwater Canoe Camp. When the trees finally crashed to the ground, workers cut the trunks into lengths suitable for canoes then they began to hollow the tree trunks with axes, hatchets—and, not least, with fire—a method they learned from the Nez Perce. By the afternoon of 6 October 1805, the work parties had completed five dugout canoes. During this same period of time, Lewis and Clark took at least seven sets of celestial observations and would take two more sets several hours after sunset on the 6th. (see Clearwater Canoe Camp celestial observations). Loading the canoes began early the morning of 7 October 1805 and by afternoon the expedition started down the Clearwater River. Rapids, riffles and rocks were plentiful, and it required both careful watching and quick maneuvering to avoid these dangers. Nevertheless, incidents and accidents occurred, and repairs to the canoes had to be made. On 10 October 1805, after traveling about 40 river miles from its canoe camp, the expedition reached the junction of the Clearwater and Snake Rivers. By the afternoon of 16 October 1805, the Snake’s swift current and the men’s well-plied paddles had brought the expedition another 140 miles downstream to the mouth of the Snake River. Before them flowed the largest river they had seen since wintering at Camp Dubois seventeen months earlier. There was no doubt in the captains’ minds that this was the Great River of the West, the Columbia (or “Oregon,” as it was known before Captain Gray renamed it)—and that they needed to take celestial observations here.

At the Snake’s Mouth

AM Equal Altitudes, 17 October 1805

On 17 October, at the junction of the Snake and Columbia River, sunrise comes about 6:20 a.m. Local Mean Time. In 1805, on that date and at that time the sky was clear—perfect for taking celestial observations. The captains waited, however, until the sun had burned off some of the morning chill and had risen higher into the sky. Then, about 7:30 a.m., Lewis set out the artificial horizon and got the sextant out of its case. Meanwhile one of the men^[1]Throughout the Expedition, the only celestial observations taken without an assistant were the Meridian observations of the sun. For observations where the chronometer‘s time was required, … Continue reading stood by with the chronometer, ready to read and record the time of Lewis’s measurements of the sun’s altitude. The first measurement was at 7:40:13 a.m.^[2]Unless noted otherwise, all observation times are those shown by the chronometer, which was 9 minutes 15 seconds fast on Local Mean Time at noon on the 17th, based on the Equal Altitudes observation … Continue reading, and the double altitude was 22°25’15” (this becomes 11°03’35” after halving the angle and correcting for sextant’s index error and for refraction).^[3]Refraction makes a light rays from an object in the sky bend upwards. The effect of refraction is especially strong near the horizon where the atmosphere is the densest; cold temperatures add to this … Continue reading Three and one-half minutes later Lewis had completed the last of the AM measurements for the Equal Altitudes observation. After taking this last altitude measurement, however, the captains must have reevaluated their observation plans. It was standard practice to leave the sextant’s index arm locked at the AM setting until completing the PM set of observations. Those PM observations, however, could not be taken until about 5:30 p.m., during which time the sextant would be unavailable for other observations. If the sky clouded up later in the day or if it was cloudy the next day, the captains might miss an opportunity to take Lunar Distance observations for longitude and other observations of the sun for Magnetic Declination. The captains opted to reject the AM Equal Altitudes observation they had just completed and take the other observations.

Lunar Distance from the Sun, 17 October 1805

When Lewis began to take his first measurement of the angular distance between the moon and the sun, the sun was about 14° above the horizon and bore S60°E true. The moon (about two days past its last quarter) stood more than 54½° above the horizon and was nearly due south. Lewis completed the first angular-distance measurement for longitude at 7:51:43 a.m. The distance (uncorrected) between the near limbs of the two bodies, by his sextant, was 60°47’15”. Sixteen minutes later (at 8:07:52) he obtained the twelfth angular-distance measurement. By that time the near limbs of the two bodies had moved 5’45” closer together. This rapid motion of the moon with respect to other celestial bodies is why Lunar Distance observations came to be the first practical observational method to determine longitude.

Magnetic Declination (Variation of the Compass Needle), 17 October 1805

The captains took a short break before beginning the observations to determine the magnetic declination. An assistant readied the chronometer while Clark set up and leveled the six-inch-diameter surveying compass. Meanwhile Lewis selected the proper sunshades on the sextant, then, looking at the artificial horizon through the horizon glass, he found the image of the sun on the water’s surface. Next he moved the index arm with its mirror until his eye caught the image of the sun reflected from that mirror to the mirror on the horizon glass. When he had brought the upper limbs of the two images into contact he signaled to Clark and to the assistant at the chronometer, who recorded the time—8:15:45 a.m. Clark then read the bearing of the sun’s center (S75°E), and this, too, was duly recorded together with the uncorrected double altitude of the sun’s upper limb (33°04’30”). The captains took a second set of altitudes and bearings four minutes later, and the observations for magnetic declination at this junction were completed. The time was 8:19:43 a.m.

AM and PM Equal Altitudes Observation, 17 October 1805

Less than four minutes after completing the second observation for magnetic declination, Lewis began (at 8:23:00 a.m.) taking another set of Equal Altitude observations to replace the set taken earlier that morning. The final observation that morning was taken at 8:26:49 a.m. The sextant’s index arm was locked at the altitude observed: 35°09’30” (true altitude of sun’s center = 17°27’30.5″ after corrections). The afternoon observations for Equal Altitudes were completed at 3:25:42 p.m.

Meridian Observation of the Sun with Octant, 17 October 1805

Neither Lewis nor Clark recorded taking a Meridian Observation of the sun on 17 October, but is highly likely that they would have taken this observation using the octant in case the sky was cloudy the next day. Whitehouse, in the redraft of his journals for 18 October, seems to confirm this: “Our officers delayed until after 12 o’Clock A.M. to compleat & prove the observation that they had taken Yesterday.” This strongly suggests that Lewis took a Meridian Observation of the sun with the octant on 17 October, but took the opportunity to check on that observation with the more precise sextant the next day.

Double Altitudes of the Sun, 18 October 1805

On the morning of 18 October, the captains took two altitudes of the sun and noted the time (8:01:24 a.m. and 10:03:59 a.m.). Between the time of these two observations the sun had risen more than 15° higher in the sky. These observations could be used either to check the chronometer error or determine the latitude.

Lunar Distance from the Sun, 18 October 1805

In the interval between the two observations of the sun’s altitude mentioned above, the captains took a Lunar Distance observation of the sun. This observation consisted of twelve sets of measurements (from 9:37:46 a.m. to 9:53:46 a.m.) of the angular-distance between the moon and sun’s nearest limbs. At the mid-time of the observation (9:45:44 a.m.) the sun was at 27°27′ above the horizon and bore about S35½°E true; the moon was 46°56′ above the horizon and bore about S21°W; the uncorrected angular distance between the near limbs of the two bodies, was 47°12’33”.

Meridian Observation of the Sun, 18 October 1805

The captains needed one last observation before the expedition set out down the Columbia. This was the Meridian Observation of the sun for the latitude of this junction. At noon, Lewis took the double altitude of the sun’s upper limb with the sextant and artificial horizon. The sun’s uncorrected double altitude, as he measured it, was 68°57’30”.

Calculations Made from the Expedition’s Observations

The junction point between the Snake and Columbia rivers in 1805 was at or near 119°02½’ W. On the Lewis and Clark’s map of 1806 and that of 1814 (Moulton, Journals, vol. 1, Map 123 and 126), however, this junction is shown at about 119°40′ W. If the captain’s celestial observations had been calculated before the topographers of the Northern Railroad Surveys made their observations (1854), calculations from Lewis and Clark’s observations likely would have been made by assuming a longitude of 119°40′ or, more probably, 119°45′.^[4]The calculations for latitude, longitude and magnetic declination in this article were made using 119°45′. The time at 119°45′ W is 7 hours 59 minutes earlier than at Greenwich. Put another way, the Apparent (solar) Time at Greenwich for an observation made at Local Noon at the 1805 mouth of Snake River would have been 7:59:00 p.m.

From Meridian Sun

At noon on 18 October, Lewis used his sextant and artificial horizon to obtain the meridian altitude of the sun’s upper limb. This observation produced a double altitude of 68°57’30” from which the Lewis calculated a latitude of 46°15’13.9″.^[5]The latitude of the mouth of Snake River is shown at about 46°15 N on the Lewis and Clark map of 1806, Clark’s map of 1810, and the Lewis and Clark map of 1814. David Thompson, on 8 July … Continue reading

Most of the 3’18”-difference just noted comes from a recurring mistake the captains made in their calculations when using the sextant and artificial horizon. Their procedure was to divide the observed altitude by two, then subtract the sextant’s full index error. They either should have (a) subtracted the full index error from the observed altitude before dividing the altitude by two or (b) subtracted half the index error after dividing the altitude by two. This mistake, by itself, results in a latitude that is 4’22½” (5 statute miles) too far north.

But, as seen above, the captains’ latitude for the observation of 18 October is only 3’18” too far north—not 4’22½”. Somewhere in the process of calculating the latitude they also must have made what is called a “compensating error.” Unfortunately, as they did not save their calculations it is not possible to find this “error.” Most likely they either 1) made a simple mistake in adding, subtracting or dividing or 2) incorrectly determined refraction, parallax, sun’s semidiameter or the sun’s declination.

The 2 arc seconds difference between the recalculated latitude and that derived from map and aerial photo interpretation is equivalent to about 200 feet. Considering that the smallest angle that Lewis’s sextant was capable of displaying was 7½”, one might conclude that this day’s Meridian Altitude observation either was first-rate and the sextant’s index error continued to be +8’45” as it had been since the fall of 1803 or there were some unusual compensating errors in this observation.

Double Altitudes of the Sun

At about 8 a.m. and 10 a.m. on 18 October, Lewis took observations of the sun’s altitude. Those two observations, together, are generally called Double Altitudes of the sun^[6]Not to be confused with the double altitude of the sun which results from the use of the artificial horizon. and commonly are used to determine latitude. These observation pairs, however, can be used to determine the chronometer’s error on Local Time provided the latitude is known.

Because the captains took a Meridian Altitude observation of the sun less than two hours after the second observation of Double Altitudes, it is clear that they took this set of Double Altitude observations to find the error of the chronometer and its rate of loss since noon on 17 October; see Lewis: 1805, July 20.^[7]“Having lost my post-meridian observations for equal altitudes in consequence of a cloud which obscured the sun for several minutes about that time, I had recourse to two altitudes of the sun … Continue reading

When the sun’s declination is changing rapidly (a month or so on either side of the Equinoxes) and the time between observations is more than about 3 hours, the declination of the sun should be determined for each observation. At other seasons, a simple average generally is adequate to obtain a latitude to within plus or minus a few arc minutes. Nevertheless, a latitude derived from a Double Altitude observation, even when made with great care, tends to be less reliable than a latitude derived from a Meridian observation.

Equal Altitudes

The captains completed only one Equal Altitudes observation at the mouth of Snake River. Calculations made from this observation reveal that the chronometer, at solar noon, would have showed a time of 11:54:40 a.m (that is, it would have been 5m20s slow on Local Apparent Time but 9m15s fast on Local Mean Time). If Lewis’s chronometer either kept perfect time or gained or lost time a consistent rate, this one observation would have provided all the data necessary to determine the true Local Apparent Times for the observations for Lunar Distance and Magnetic Declination taken at the mouth of Snake River. Lewis’s chronometer, however, manifested a time-variable nature despite being housed in a special box and suspended on gimbals within it. Dust, extreme heat and cold, improper winding, severe jolts while being transported all took their toll on a device that was meant to travel in a navigator’s cabin at sea. During 1805, the chronometer’s daily rate of loss, as calculated from two or more Equal Altitude observations made at the same location, ranged was from 15 to 65 seconds per day. These rates, however, reflect the both the accuracy of the Equal Altitudes observation and the actual daily loss. The two cannot be distinguished easily.

Although the daily (or hourly) rate-of-loss of Lewis’s chronometer cannot be determined from a single AM-PM Equal Altitudes observation, there are other means to approximate its rate-of-loss. The most common method is to calculate the Local Apparent Time of two observations taken several hours to several days apart at the same point of observation.^[8]See, for example, Robert Patterson; 1803, Astronomy Notebook for Meriwether Lewis, Problem III. In order to make this calculation it is necessary to know (or be able to calculate) the following:

1. the latitude correct to within several arc minutes,
2. the sextant’s index error^[9]Since the mouth of the Ohio in November 1803, Lewis has stated that his sextant error is 8’45” (it reads too high by that amount). The latitude derived from the sextant observation of the … Continue reading (to obtain the true altitude of the sun’s center for the observation),
3. the refraction correction based on the altitude determined in item 3 above,
4. a good approximation of the observer’s longitude.

If all the above values are reasonably well known, the calculated Local Apparent Time of the observation usually will be within five seconds of the actual time. The time so derived is then subtracted from the time the chronometer showed at the observation. The result is the chronometer’s error on Local Apparent Time for that observation. Next it is necessary to find the chronometer’s error for a second observation. Then, from the time elapsed between the two observations, the chronometer’s rate of loss can be calculated and the Local Apparent Time of a time-dependant observation can be determined.^[10]For best results, the observation (Lunar distance, Magnetic Declination) for which the correct time is needed should lie between the two observations for which the Local Apparent Time has been … Continue reading

Example

Given: Average chronometer time for Lunar Distance observation, 17 October 1805: 7:59:15.7.
Wanted: “True” Local Apparent Time for that Lunar Distance observation.

In the 17m17.7s since the 1st AM Equal Altitudes the chronometer, thus, has lost another 1.3 seconds. Therefore it was 5m12.3s slow (5m11s + 0m1.3s) on Local Apparent Time at the average time of the Lunar Distance observation, and the “true” Local Apparent Time should have been 7:59:15.7 + 0:05:12.3 = 8:04:28.

Local Apparent Time

The example above merely demonstrates the procedure. In actuality, the sun’s altitude during the 1st AM Equal Altitudes observation on 17 October, was still too low and still was being strongly influenced by refraction. This means that the sun’s altitude, as calculated from this observation, might be subject to a significant error and might not provide a reliable Local Apparent Time. This observation probably would not have been used only if more reliable observations were available. Fortunately, several more reliable observations were available, and the calculations for longitude and magnetic declination in this article were made from an evaluation of the Local Apparent Times they provided. Nevertheless, after calculating the Local Apparent Time for all the observations for which the captains gave the sun’s declination and double altitude and applying the results to the Lunar Distance observation of 17 October, the projected true Local Apparent Time was 8:04:29.0—a difference of 1 second, and too small to make an appreciable difference in the longitude derived from that observation.

Time-of-observation calculations were made for the two AM and one PM Equal Altitudes observations and the averaged times for Magnetic Declination. These times, together with the calculated time for noon on 17 October from the Equal Altitudes observation, suggest that the chronometer’s rate of loss for 17 October was 33 seconds per day on Local Mean Time. The time-of-observation calculations made from the two double altitudes observations, however, show a different rate of loss: only 10.6 seconds in 24 hours. This difference in rate may not be real inasmuch as only 2 hours had elapsed between the two observations whereas 7 hours and 42 minutes had elapsed between the first and last observations on the 17th. Nevertheless:

A difference of one second of time represents a difference of 15″ (arc seconds) of longitude. If the chronometer’s average rate of loss of 30 seconds per day was used for all observations, the error in longitude would be plus or minus 7.5′ (arc minutes) solely from the time element of the observation. Considering all other possible errors in making observations and calculations for longitude, most navigators of the day would have been pleased with a longitude error of “only” 15 arc minutes (about 12 statute miles at 45° latitude).

Longitude

Shortly before 8 a.m. on 17 October, the captains, with sextant and chronometer, began their observations of the Lunar Distance from the sun for longitude. They also took a second set of observations of the Lunar Distance from the sun the next morning. As usual, they did not calculate the longitude from their observations. The longitudes given below were calculated using the data that Clark’s recorded.

Sequence of Calculations

1. A plot (graph) was made from the data pairs (angular distances and chronometer times) for each observation to evaluate how consistent the data were.
2. The data pairs that produced the longest, most-nearly straight line were selected and averaged.
3. he chronometer time for each observation average then was adjusted to Local Apparent Time based on calculations made from the Equal Altitudes observation and other observations for which the captains obtained the chronometer time of the sun’s altitude.
4. The Greenwich Apparent Time of each observation average then was obtained assuming a longitude of 119°45′ for the mouth of Snake River as is shown on Lewis and Clark’s 1806 and 1814 maps. This makes a time difference from Greenwich of 7 hours 59 minutes.
5. The sun-moon data from the Nautical Almanac for October 1805 then were adjusted to what they would have been at the assumed Greenwich Apparent Time of the observation.
6. Calculations then were made to determine the sun and moon’s true and apparent altitude.
7. The average angular separation of the sun and moon for each observation was corrected for the sextant’s index error, the sun and moon’s semidiameters and the moon’s augmentation.
8. The calculation for true angular sun-moon distance was then made using Chevalier Jean Borda’s method. The distance thus obtained was compared to the distances and times given in the Nautical Almanac for every three hours of Greenwich Time and the true Greenwich Apparent Time for the true separation was obtained.
9. The true Local Apparent Time of the observation was subtracted from the true Greenwich Apparent Time and the result, multiplied by 15 gives the longitude.

As can be seen below, the longitudes calculated by Steps 1-9 above are too far west by an average of 52 arc minutes.

As a check, longitude calculations were made using the true longitude of the 1805 mouth of Snake River of 119°02’29”. These calculations moved the longitudes a mere 5 arc minutes farther to the east.

The captains’ Lunar Distance observations with the sun usually produce longitudes within 25 arc minutes of the true longitude. In addition, their Lunar Distance observations made at mouth of Kansas, Fort Mandan, mouth of Marias, the Three Forks of the Missouri and Clearwater Canoe Camp have produced longitude averages that were too far east.^[11]One of the five observations at Fort Mandan was too far west by 27 arc minutes and one of the two observations at Fortunate Camp was too far west by 13 arc minutes. Why do both of the observations made at the mouth of Snake River result in a longitude that is too far west by about 52 arc minutes? The sun and moon on both days were aligned so as to make the observation an easy one to take, so that should not have been the problem. Furthermore is not likely that the captains misread the angles on the sextant’s graduated arc by nearly the same amount on both days, nor is it likely that they misread the vernier in the same manner on both days. A possibility, however, is that the sextant’s index error changed and no longer read high by 8’45”. An arc minute or so in the angular distance read during the observation would make a sizeable difference in the calculated longitude. Using the average time and angular distance from the two Lunar Distance observations taken at the mouth of Snake River (see Step 2, above), what sextant index error would have given: a) the correct longitude and b) a longitude too far east by 30 arc minutes?:

1. about 6’55” too high and,
2. about 5’50” too high.

From the mouth of the Ohio River all the way to the Pacific, the captains always gave the sextant’s index error as 8’45”. Did they merely assume that this error remained unchanged or did they know it actually remained the same based on test observations?^[12]A common and simple test for index error is to sight on a bright star through the sextant’s horizon glass then find that same star in index mirror and bring the two images to match. The angle … Continue reading Lewis gives the first notice of a change in the sextant’s index error on February 4, 1806, twenty-six months after leaving the mouth of the Ohio River: “By the mean of several observations found the error of the Sextant to be Subtractive  —5’45”.” That is, the sextant read high by 5’45” and this amount must be subtracted from all angles measured. This index error, 5’45”, looks suspiciously like a mis-copy for the sextant’s standing error of 8’45”, but that index error, in the original text, clearly is 5’45”. Four months later, on June 9, 1806, Lewis gave yet another index error for his sextant: 6’15” too high. The large too-far-west error at the mouth of Snake River, however, may not be related to a change in the sextant’s index error. The latitude re-calculated from the sextant observation Lewis took at the mouth of Snake River comes as close to the actual latitude as any the captains made. Inasmuch as it differs from the actual latitude by only 2 arc seconds, not 2 or 3 arc minutes, it suggests the index error, indeed, was still 8’45”. Furthermore, it will be shown in the article on celestial observations taken at Station Camp (in present-day Washington nearly opposite Astoria, Oregon) that the longitudes calculated from the captains’ Lunar Distance observations taken there by sextant are too far east again. Additionally, the latitude from their sextant observation at that location differs from the actual latitude by only 1 arc minute. The journal entries do not specify whether it was Lewis or Clark who took the Lunar Distance observations. If it was Clark, was he out of practice, bringing the images of the sun and moon’s near limbs together incorrectly (too much overlap, making the angular distance less)? It appears that these observations are anomalous, but the reason for it is not clear at this time.

Magnetic Declination

The captains took two observations of the sun’s altitude with the sextant and artificial horizon while simultaneously observing the sun’s magnetic bearing with the circumferentor. Meanwhile another member of the expedition recorded the times shown by the chronometer for those observations. The captains did not calculate the magnetic declination of the compass from these observations. To determine the magnetic declination listed below, the two observations were averaged and a single calculation was made to find the sun’s true azimuth at the averaged time.

  123°21'57"

– 115°30'

  017°51'57" E =   

18°

22°

21°

16½°

In the morning of 18 October 1805, while Lewis and an assistant were taking celestial observations, Clark and several others were taking measurements to determine the width of the Columbia and Snake rivers near their junction. Clark used triangulation, and his method was relatively straightforward.

Measured the width of the Columbia River, from the point across to a point of view is S22°W; from the point up the Columbia to a point of view is N84°W, 148 poles, thence across to the 1st point of view is S28½°E. Measured the width of the Ki moo e nim River, from the point across to an object on the opposite side is N41½°E; from the point up the river is N8°E, 82 poles, thence across to the point of view is N79°E. Distance across the Columbia is 960-3/4 yards water Distance across the Ki moo e nim 575 yards water.
–William Clark, 18 October 1805

Survey Instruments—Two instruments were needed to make the surveys for river width: a circumferentor (six-inch-diameter surveying compass) and a two-pole chain (a measuring device 33-foot long, composed of 50 chain links, each 7.92 inches long).

The Survey—Clark, having already chosen the junction point of the rivers as his starting point (call it “A”) from which to begin his survey, examined both rivers upstream from the junction. He needed to make sure the shorelines were straight enough and long enough to provide good baselines^[16]For best results the baseline should not be significantly shorter than the estimated width or distance to be measured. for the triangulation. He then looked for a conspicuous landmark such as a large rock or distinctive tree upstream on the opposite side of the Columbia (“B”) and another upstream along the Snake (“D”). While the men, with the two-pole chain, measured the distances between “A” and “B” (148 poles) and “A” and “D” (82 poles), Clark set up and carefully leveled his surveying compass on the point (“A”). From “A” Clark shot the magnetic bearing (South 22° West) to a prominent object (“C”) on or close to the far shore of the Columbia River, then shot a bearing upstream along the base line (North 84° West) to “B”. After recoding the bearing he had observed, he moved his surveying compass to “B” and shot the bearing (South 28½° East) across the river to “C” and recorded that bearing and that of the distance of the base line. That done, the surveying part of the operation for the Columbia River was complete.

Plotting instruments – 1) a ruler to draw straight lines and measure line lengths and 2) a protractor to mark off the angles that he had measured.

This is how Clark might have proceeded to determine the rivers’ widths:

Starting with his survey notes:

From “A”: S22ºW to a point across the Columbia (“C”)
From “A”: N84ºW, 148 poles to “B”
From “B”: S 28½º to a point across the Columbia (“C”)

Clark used his ruler and plotted the base line for the Columbia River at a convenient scale.^[17]For example: Suppose Clark had only a ruler marked off in sixteenths of an inch. To obtain a measuring precision of no worse than plus or minus 2 poles, he could have plotted his survey at a scale as … Continue reading Then, setting the center mark of his protractor over “A” and aligning the protractor so that the mark for N84°W fell on the extension of the line through “B”, he marked off the magnetic bearing he had shot from “A” to “C” (S22°W) and drew a line from “A” through that mark. He then placed the center mark of the protractor over “B” and aligned the protractor so the base line going back through “A” fell along the mark for S84°E. He then marked off a bearing of S28½°E and drew a line from “B” through that mark. The intersection of the lines from “A” and “B” marked the location of “C” on the opposite shore. With his ruler, Clark then measured the shortest distance between the base line and “C”, then converted that measurement to actual distance. In doing so he obtained the river’s width without getting his feet wet.

Accuracy

Actually, the surveying operations at the mouth of Snake River were a little more involved than outlined above. Clark also had designated a point (“D”) on the shore upstream along the Snake River to be the end of a second base line, and the men also measured the distance from “A” to “D” with the two-pole chain. Clark, before moving to “B” also shot a bearing from “A” to “D” and to “E” (a point on the far shore of the Snake River). After shooting the bearing to “C” from “B”, Clark moved his surveying compass to “D” and shot a bearing to “E” on the far shore of the Snake River. He then plotted his survey of the Snake River’s width in the same manner as he did for the Columbia River.

In triangulation where precision is not critical, using a ruler and protractor is a convenient means of determining the approximate distance to an object not easily reached. For precise work, such as in Geodetic Triangulation, trigonometry is used. By trigonometry, the width of the rivers (from Clark’s survey data) are:

Columbia: 151.9 poles = 836 yards
Snake: 74.3 poles = 409 yards^[18]The bearing that Clark shot from D across the Snake River to E does not make a right angle with the river’s width (shortest distance). By measurement, the width (using Clark’s data) is … Continue reading

Depending upon the scale of the plot Clark made, the widths that he derived by scale and protractor easily should have been within plus or minus 2 poles (33 feet) of the actual distance. Yet, he recorded a width of 960¾ yards for the Columbia and 575 yards for the Snake. It is interesting to note that the distances that Ordway, Gass and Whitehouse recorded in their journals from Clark’s data, 860 yards and 475 yards, are more nearly correct when compared to the widths derived by trigonometry. Did Clark miscopy his own work?