Lewis’s Astronomical Instrument Calibration

Methods and Assessment


by Hans A. Heynau


Descriptions of the astronomical instruments carried by Lewis and Clark are readily available[1].  The results of the explorer’s periodic calibration measurements to determine the zero offset of the sextant and octant instruments were recorded in their journals.  The details of the specific calibration process were not, however included.  The “standing error” values for the sextant and octant were used in calculations of latitude and longitude.  The accuracy of these values, as well as how they are applied in the calculations, directly affects the accuracy of the results.  Most modern navigation texts describe the calibration process in terms of techniques that are primarily for use at sea or on the edge of a large body of water[2].  This article describes the land based calibration processes available to the explorers, the most likely ones used by them based on the data and results in the journals, and the implications of possible errors in the measurement process to the apparent accuracy of the journal results[3]

Fig. 1 Sextant with attached magnifier to help read nonius (vernier) scale (as with Lewis' sextant)

Sextant Calibration Methods

The primary alignment of the optical components of a sextant is typically carried out by the instrument’s manufacturer.  In the case of a previously owned sextant, primary alignment may be preformed by an instrument technician, perhaps where the instrument is purchased; or by a knowledgeable owner.  On a more routine basis, the instrument is calibrated to determine the index, or mirror zero offset, error (IE). This error represents the systematic error value that is either added or subtracted to all measurements, depending on the sign of the error, to correct the measured values.  Lewis recorded the results of his sextant calibration measurements in the journals on several occasions and added the notation that the values were to be considered the standing error.  That is, the same recorded value was to be used over periods of many days or months since he did not feel that the value varied over that period.  In modern practice, the index error is typically checked before each usage even though it may not vary over long periods of time.  Lewis did not record whether this was also his practice or whether he simply trusted the instrument to be constant so long as it was not subjected to any abuse or accidents.


In a maritime situation, the sextant’s IE is easily determined by sighting on the far sea horizon and determining the instrument’s reading when it’s arc has been adjusted so that the horizon appears to be at the same level in both halves of the horizon mirror[4]


Since a sharp, uniform, distant (several miles) horizon or similar feature was not available to the explorers, other techniques would have to be employed to calibrate the sextant.  One of the reference books carried by the expedition[5] included some suggested alternatives for the calibration procedure.  In the reference’s discussion, it was indicated that using the sun as the sharp, distant object “is incomparably the best object for this purpose” and either centering the two suns (one in either half of the horizon mirror due to the direct and index mirror measurement paths) or “a still better method” by measuring the sun’s diameter twice (once with the direct sun above and once with the direct sun below), with half the difference in the reading representing the measured error.  The reason that this method (sun diameters) is preferred over the method of superimposing one sun image at the same level as the other is that the sun’s edge is a more sharply defined feature and hence the measurement may be made more accurately, just as an upper or lower limb (top or bottom edge of sun) horizon measurement is a more accurate measurement of the sun’s altitude than a measurement made by trying to estimate when the center of the sun is at horizon.  Figures 2a and 2b illustrate how the horizon mirror would appear for the two sun diameter measurements.  An example from the expedition’s reference book is a sun diameter measurement of 30’ on the arc (a reading greater than 0º  0’) and one of 33’ off the arc (reading less than 0º  0’).  The result is that the sextant’s correction would be 1.5’ (1/2 of the 3’ difference), which should be added to observations made for this calibration result.


It would appear likely that the latter calibration method was employed by Lewis as evidenced by his description recorded for July 22, 1804 that the calibration error resulted “ from a series of observations ”.  Lewis’s other procedures typically involve averaging several observations to improve the accuracy due to random errors.  It thus seems likely that he would have made multiple sun diameter observations, perhaps alternating on the arc diameter measurements and off the arc ones, to use in average error computation.  This multi-measurement process was also recommended by the previously cited reference book.


(a)                               (b)

Fig. 2.Horizon mirror views for a land based sextant calibration technique.  (a) Sun’s diameter for an upper limb appearing measurement. (b) Sun’s diameter for a lower limb appearing measurement[6].


Sextant Calibration Assessment

One way to assess Lewis’s sextant calibration measurements is to compare the results of computations utilizing the calibration values with modern estimates for the same parameters.  To the extent that systematic errors appear to be present, the original measurements may have been biased.  Table 1 presents the recorded sextant error value results during the expedition and summarizes the typical, or average, error in deduced latitude as Lewis worked his sight data, as well as similar latitude results when the original sight data are worked with an appropriately revised effective error value[7].  The corrected results would have been readily obtained by anyone working Lewis’s raw sight data who correctly applied the sextant’s index error for the artificial horizon sight method.  The modern latitude values used in the comparisons were based on the maps of Plamondon[8].  As evidenced by results in the corrected latitude error column, calculations incorporating the corrected sextant error calibration values result in small average error values with no significant bias (+ or -).  This leads to the conclusion that the calibration measurements were carefully made and contained no significant bias errors.  It may also be noted that the sextant error value was constant for most of the journey which is a tribute to the care taken with the instruments under the many times adverse field conditions.


Table 1.  Sextant error values from expedition calibration measurements and their impacts on estimates of latitude




Recorded error value[9]

Lewis Latitude avg. error[10]

Corrected error value for Latitude use

Latitude avg. error

July 22, 1804[11]

(-) 8’ 45”


(-) 4’ 22.5”


April 12, 1805[12]

(-) 8’ 45”


(-) 4’ 22.5”


Feb. 4, 1806[13]

(-) 5’ 45”


(-) 2’ 52.5”


June 9, 1806[14]

(-) 6’ 15”


(-) 3’ 07.5”



Fig. 3 Octant (with back sight capability)


Octant Fore Observation Mode Calibration Methods

A basic octant is similar in construction and operation to a sextant, except that the octant is only capable of measuring angles between 0 and 90º, whereas a sextant has a measurement range of 0 to 120º.  The elements and optical path for the octant employed in the fore observation mode are illustrated in figure 5a.  A basic octant, or the fore observation mode of a dual capability octant, can be calibrated for index mirror offset error using the techniques described above for a sextant.  In addition, on February 16, 1806 Lewis carried out additional calibration measurements aimed at determining if the octant error was constant over the entire measurement range.  As indicated in the sextant discussion, this process would typically be carried out by the manufacturer or an instrument technician.  Of particular interest is the fact that Lewis apparently performed this calibration because he was concerned about the octant having a “fracture” in the scale around the 55º 20’ location on the scale.  By way of background, wooden octants typically had an ivory inlay in which the scale markings were engraved.  Lewis did not mention such a scale imperfection in his original equipment description so it is not clear whether it developed during the trip or was there all along and Lewis simply decided to determine if it was significant to his celestial measurements.  As noted in his journal entry, his method for calibrating the octant error at various angles was “by a comparison with my sextant the error of which had been previously ascertained.”  The likely approach for making these cross-calibration measurements would be to measure the horizontal angle between various sharply defined distant objects (the sextant and octant would be oriented in a horizontal plane to make these measurements), perhaps between sharply defined coastal features such as some of Oregon’s pinnacle islands off Cannon Beach, just south of the Salt Maker’s Camp.  Figure 4 depicts the geometry of this method.  The sun at this time of year would not provide a large enough direct elevation angle for this purpose. Also, the calibration results do not exhibit the characteristics that would be apparent if the measurements had been made utilizing an artificial horizon.



Figure 4.  Fore observation mid scale calibration utilizing horizontally spaced terrestrial objects (one step of a cross-calibration process).


Octant Fore Observation Mode Calibration Assessment

In the earlier sextant assessment discussion, latitude computation results using the recorded error values were used to assess the quality of the expedition’s calibration measurements.  In the case of the octant fore observation mode, latitude determination sights were not made during the main part of the expedition.  Clark made a few octant fore observation sights prior to leaving St. Louis[16], but these and others of he and Lewis’s pre-trip observations are problematical in that many errors and inconsistencies appear in them and it is best to consider them as pre-expedition practice sessions where their observation routines could be worked out in preparation for the official expedition.

What can be noted from the summary of fore error values in Table 2 is that the fore error values near the 0 scale reading remained unchanged throughout the entire trip.  Again, this is a tribute to the care taken with the instruments under the many times adverse field conditions.  Also, although not all of the optics are the same in the fore and back observation modes, the fact that the fore mode error remained constant indicates that no accident occurred with the octant to change its fore error and therefore there is some confidence that the much more difficult to measure back observation error would likely be fairly constant.  Although the explorers do not record checking the octant or sextant error on a regular basis, it is possible that they may have performed a rough check on the sun and, finding essentially the same value as their multi-reading averaged computations, felt confident in continuing to use their standing error value.


The slightly different error value recorded on February 16, 1806 for angles > 55º 20’ could have implications for back observation measurements since different portions of the same, cracked scale were used for both modes.  The small difference is, however, not large compared to random observation and other potential errors[17].  Since the potential effect would be of less than 1’ in deduced latitudes and the flaw was not noted during the 1804-05 portions of the trip, it is not included in the back observation assessment of the following section.


Table 2.  Octant fore observation error readings from expedition calibration measurements


Recorded error value


July 22, 1804

 (+) 2º   

Likely measured spring 1804

April 12, 1805

 (+) 2º --’” –“

May have been measured before this date

Feb. 16, 1806

 (+) 2º   
 (+) 2º 01’”45“

For angles from 0º to 55º 20’
For angles > 55º 20’


Octant Back Observation Mode Calibration Methods

The octant carried by the expedition was described by Lewis on July 22, 1804 as “… prepared for both fore and back observation…”.  The measurement range for the octant in the back mode is from 90 to 180º, thus extending the octant’s measurement capability between the fore and back modes to cover the entire sky from horizon to horizon.  Octants constructed with a back observation capability were essentially two instruments in one with the index mirror, index arm, and calibrated arc elements being common to both observational modes.


The elements and optical path for the octant employed in the back observation mode are illustrated in figure 5b.  Among other things, this capability would allow a maritime observer to measure the angular distance of a celestial body to the opposite horizon when the typical, near horizon was obscured by a fog bank.  As indicated in the figure, the position of the celestial body being observed is actually behind the observer, hence the term back observation.  When operating in this mode, the 0 to 90º inscriptions on the instrument’s arc actually correspond to the supplement[18] of the observed angle (180 degrees minus the value read from the arc).  The capability to measure large angles (particularly greater than the 120º capability of the sextant) was extensively utilized during the months of May thru August by the expedition to measure the meridian, or local noon, altitude of the sun.  The meridian altitude measurements were used to calculate the latitude of the point of observation.  During the summer months the angles to be measured using the reflective artificial horizon device (true elevation angles are doubled) were greater than the 120º capability of the sextant.  For example, the meridian sun angle to be measured utilizing the artificial horizon would have been approximately 140º on May 18, 1804.


Fig. 5a Standard (fore observation) octant operation (0 - 90º)


Fig. 5b Back observation octant operation (90 – 180º)


At the start of the trip, the error for the back observation mode was stated as 2º 11’ 40.3”+. The (+) sign again indicates that the value is to be added to all measured values taken in the back observation mode.  Since the back observation mode angular range of 90-180º does not include the zero value, the measurement and/or calibration of its index error is quite challenging.  The known reference books carried by the expedition[19] did not include a description of how to measure the index error for the back observation mode.  It is not clear as to whether Lewis was instructed on this topic by Andrew Ellicott during his stay in Lancaster, PA, or if Lewis created the procedures he used on his own.  In a contemporary reference, Bowditch[20] describes back observation error measurement and adjustment methods, but only for conditions where sea horizons are available both in front of and behind the observer.  In one method the front and back horizon images are matched against each other, after allowing for twice the dip angle (angle of the horizon below horizontal, based on the observer’s height of eye above the water).  A second method involved observing the sun’s lower limb meridian altitude, when the altitude is at its maximum and seems to stay at the same altitude for a period of time, first by the fore observation and then by the back observation method on the opposite horizon.  In this method, twice the dip angle is again taken into account.  The discussion of this method notes that the measurements “…must be done quickly, before the sun sensibly alters in altitude…”.  Lewis, had neither a fore nor a back sea horizon available for use in his calibration process. He needed to utilize other techniques to determine the value of his octant back observation mode error.


There are a number of back observation calibration techniques that Lewis could have used, however several may be ruled out based on his recorded values and how he used them to reduce his meridian sun data throughout the journals.  For example, he could have measured the horizontal angle between two distant objects with first the sextant and then the octant (for angles between 90 and 120º), as was discussed relative to the fore observation calibration technique for February 16, 1806.  Had this method been used, it should have resulted in a calibration value of over 4º, rather than the stated value of 2º 11’ 40.3” [21].  Since Lewis’s latitude calculations with back observation sights during the first year of the trip give fairly accurate results, it may be deduced that his calibration technique must have involved artificial horizon aided angle measurements.  One or more of the following calibration procedures, all involving the use an artificial horizon, may have been used.  By understanding these techniques, examining the journal entries, and examining some apparent problems with Lewis’s latitude computations using his various calibration values determined during the expedition, a fair idea of the likely techniques that were used for particular calibration values may be obtained.


Back Observation Calibration Candidate #1 – Terrestrial object

In his journal entry for June 9, 1806, Lewis records “Error of Octant by back observation on the distant fragment of the broken limb” and the value “2º 30’ 4.5” + a additive”[22].  Figure 6 depicts measuring the vertical angle of a broken limb extending from a tree using the sextant and artificial horizon.  After the angle was measured and recoded for the sextant, the measurement of the same angle would be repeated with the octant in the back observation mode.  An elevation angle, relative to horizontal, of between 45 and 60º would result in a measurement angle between 90 and 120º, due to the doubling effect with the use of the artificial horizon.  Since this measurement range is within the capabilities of both the sextant and the back observation octant, the two instruments should measure the same angle, provided that the broken limb is sufficiently far away and the errors of the two instruments are taken into account.  The basic procedure would be to correct the sextant reading for that device’s index error, measured separately, and then, to the extent that the supplement of the octant’s scale reading varies from the corrected sextant value[23]  the result would be the error for the octant in the back observation mode. 


The critical qualifier with this procedure is the requirement that the object used in the measurement process be located sufficiently far away.  Anyone who has taken a picture with a camera that has a separate viewfinder and found that the final picture is missing parts that looked to be in the field of view will understand the problem of optical parallax.  In the case of the sextant and octant, the optical path from the index mirror (located at the top of the instrument) to a nearby object is not parallel to the optical path from the horizon mirror (located part way between the top and the bottom of the instrument.  Notice in figure 6 how the path from the top of the sextant and the one from the artificial horizon are not parallel to each other.  For a celestial object, such as the sun, the two optical paths would be parallel.  Lewis did not record how far way his broken limb was located or the dimensions of the mirror locations for the sextant, octant, or the geometry for his use of the artificial horizon.  For this reason, a computation of the parallax error contained in his octant error calibration value may only be conjectured.  What is found is that Lewis’s latitude estimates based on this value for June 9, July 19, and July 23, 1806 are approximately 20 to 27 stature miles south of modern estimates of his location on these dates.  This error is consistent with an uncorrected difference in parallax error between the measurements by the two instruments used in the cross-calibration of approximately 23’.  In fact, Lewis would have obtained better results if he had used either his summer 1804 error value (either corrected or uncorrected for his application of the sextant error as will be discussed later) or his summer 1805 value, if it were corrected as discussed relative to Calibration Candidate #2 in the following sub-section. 


Fig. 6.  Back observation calibration utilizing a terrestrial object (one step of a cross-calibration process).


The conditions where back observation candidate #1 might be the one of choice would be when a suitable sighting object is available and conditions are less than favorable for the other candidate considered in this section.  For example, on June 9, 1806, the meridian altitude of the sun would have been approximately 66º.  This angle would require an artificial horizon aided measurement capability of 132º.  While this is within the capability of the octant’s back sight optics, it is beyond the capability of the sextant.  Therefore candidate #2 below would not be able to be utilized for this particular case.


Back Observation Calibration Candidate #2 –Meridian Sun

This procedure utilizes the altitude of the meridian sun as an angle whose sextant and back observation octant values may be compared in order to calibrate the octant’s back observation error.  This could be thought of as a variation of the maritime procedure described by Bowditch discussed earlier.  The difference is that the sextant and octant measurements are cross correlated, rather than using the octant to make fore and back mode measurements to opposite horizons.  As illustrated in Fig. 7, using the meridian (local noon) time for the calibration measurements utilizes the sun’s altitude at its maximum when seems to stay at the same value for a period of time[24].  For example, as the Corps of Discovery was departing Ft. Mandan on April 12, 1805, the meridian sun would appear to stay at its maximum altitude (within 0.5’) for approximately 8 minutes.  This would allow time to switch back and forth between the sextant and the octant, while utilizing the same artificial horizon to make both measurements.


As in candidate #1, the data reduction procedure would be to correct the sextant reading for that device’s index error, measured separately, and then the error for the octant in the back observation mode would be the difference relative to the octant’s reduced value for the same angle.  This direct comparison of data would apply for the case where the same relative point of the sun, upper limb (UL) or lower limb (LL), was measured by both instruments.  If opposite limbs were measured, then the measured values would need to be corrected to a common basis, such as the sun’s center, before comparison.


As an example, in the remarks of his April 12, 1805 journal entry for Point of Observation No. 1[25], Lewis recorded the error of the octant in back observation mode as additive 2º 40’ --.  On that date, the sun’s observed meridian altitude, using the artificial horizon, would have been approximately 103º.  This angle is well within the capabilities of both the sextant (0-120º) and the back observation mode of the octant (90-180º)[26]


Fig. 7.  Back observation calibration utilizing meridian sun observations (one step of cross-calibration process).



(a)                                        (b)

Fig 8.  Horizon glass images for meridian sun measurements with different instruments, limbs, and/or sight optics.  (a) sextant UL with inverting telescope or octant LL (typical winter/summer observations, respectively) (b) octant LL or sextant LL with no telescope or a non-inverting telescope.


Figure 8 shows the sextant and octant images that the explorers would have seen with the two instruments and different measurement options.  In figure 8(a) the lower, half moon shape on the bottom is the sun image that is transmitted thru the horizon mirror and the upper circle shape is the image of the sun coming from the upper, or index, mirror.  The grayed region in the upper circle is meant to represent the dimmed portion of the image where it is reflecting from the clear glass portion of the horizon mirror, in contrast to the right hand portion that is the portion of the index mirror image that is being reflected from the full mirror portion of same mirror on the instruments.  As alluded to in the figure (a) caption, Lewis preferred using an inverting telescope with the sextant so that an Upper Limb (UL) sextant image will have the same appearance as a Lower Limb (LL) octant image.  With few exceptions, all of the winter month sextant meridian transit measurements were sun UL and all of the summer (sun at high altitude) back observation octant meridian measurements were sun LL.  Thus the images they used for their meridian transit measurements would tend to look the same, even though they represented different sun limb configurations.  Figure 8 (b) illustrates the expected instrument image for the exceptional cases of either an octant LL or a non-inverting telescope sextant LL observation.


To calibrate back observation octant error using octant and sextant measurements at sun meridian transit time, either both instruments must be set to measure the same limb or opposite limb measurements must be corrected to a common reference, such as sun center.  Also, the effect of the previously measured sextant error must be properly accounted for, depending on what point in a sight reduction (mathematical manipulation and/or correction of the raw data to a net angle, such as for latitude computations) process the two instrument readings are compared in order to calibrate the error of the octant.


Octant Back Observation Mode Calibration Assessment

As in the case of the earlier sextant calibration assessment, Lewis’s octant calibration measurements were compared to the results of computations utilizing the calibration values with modern estimates and examined for any apparent systematic errors that might have led to biased results.  Table 3 presents the recorded octant back observation error value results during the expedition and summarizes the typical, or average, error in deduced latitude as Lewis did, or would have, worked his sight data, as well as similar latitude results when the data is worked with effective error values based on the correction of certain apparent systematic bias errors.  Unlike the sextant systematic errors, the allowance for the source of the errors is based on what likely occurred, based on the journal data and apparent calibration procedures, rather than correcting for a simple misapplication of a standard artificial horizon calculation procedure as in the case of the sextant assessment results.  This uncertainty as to the source of the errors would have made it extremely difficult for Lewis and Clark’s contemporary data reviewers[27] to discover and address the problem.  The large errors associated with the summer of 1805 and 1806  (approximately ½ degree) might have come to light if the latitude results from the octant data were compared with similar results from the somewhat less accurate, but independently arrived at, latitudes based on  the sun equal altitude (chronometer calibration) data approach favored by Andrew Ellicott[28].  Lewis’s own tabular data from late in the expedition seems to indicate that he had also become suspicious of some of the back octant calibration values[29] even before the end of the trip.  Since there appears to be a different situation associated with the error values for each of the four calibration entries in table 3, each of the four will be discussed separately.


There is evidence that, although the first recorded entry for octant back observation error was not until July 22, 1804, the calibration was made prior to May 29, 1804, perhaps even before the start of the trip[30].  In the May 1804 time frame, the altitude of the meridian sun would have been suitable for measurement by both the sextant and back observation octant so that the meridian sun cross calibration technique (candidate calibration method #2) would have been entirely suitable and able to provide an accurate calibration value.  In this method, however, the sextant error value must be applied, and in a manner appropriate to the artificial horizon measurement technique used with both the sextant and the octant.  The bias error in Lewis’s latitude results during the summer of 1804 is consistent with an oversight in accounting for the sextant error in the cross calibration.  In table 3, the corrected effective error value, as Lewis would have applied it, reflects correcting for this oversight.  As seen in the corrected error column of Table 3, there appears to still be some bias error.


The octant calibration data for the April 12, 1805 journal entry was likely taken in the April 1 to April 11, 1805 time period[31].  During that time at, or near, Fort Mandan the sun’s elevation would have been suitable for sextant-octant cross calibration measurements during the meridian transit of the sun (calibration candidate #2).  Lewis’s extremely large latitude errors indicated in Table 3 are in sharp contrast to his relatively accurate measurements with the sextant and even the somewhat less accurate results with the octant for the summer of 1804.  There are a number of possible explanations for this large bias error.  One possibility is that the apparent transcription error in the journal entry for June 9, 1805 in Table 3 may have been a better indication of the value recorded in his working papers for the back observation calibration.  Although the use of the postulated June 9 yields better results, Lewis’s latitude computation results on the summer 1805 journal dates indicate he consistently used the octant error value he recorded on April 12. 


Another, and based on the revised computational results, more likely error scenario is that there was an undetected confusion during the cross calibration process as to which limb of the sun was being measured by both instruments.  The calibration process would be expected to utilize measurements of the same limb of the sun made by both the sextant and the octant for comparison.  While alternate limbs could be used and simply accounted for in the cross calibration process, the most straight forward approach would be to measure the angle for the same limb with both instruments, which would mean the reduction mathematics would also proceed in the same manner for both sight reductions.  Unfortunately, the sextant was typically used with an image inverting telescope and the octant without a telescope.  Therefore, as illustrated in Figure 8(a), the orientation of the sun images for a sextant UL measurement looks exactly the same as the image orientation for an octant LL measurement.  In fact, the vast majority of the explorer’s meridian sights were taken in just that manner.  That is the sextant meridian sights were almost always UL and the octant sights were LL.  Although rare, there are examples in the journal data where, based on the data and modern latitude estimates, the explorers recorded the opposite limb designation compared to what they must have actually measured[32].  The result in the cross calibration computations would be a bias error equal to the diameter of the sun on the date the measurements were taken.  In early April 1805 the sun’s diameter value was approximately 32’[33].  As with the other artificial horizon calibration measurements, Lewis would have corrected for the full amount of sextant index error whereas effectively only ½ the value would apply at the point he used it in his calculations.  If an offset for that error is combined with the 30’ value for the mix-up in sun limb identification, a net adjustment to Lewis’s recorded value (2º 40’) of 28’ 37.5” yields an effective index error value, as Lewis would have applied it, of 2º 12’ 22.5”.  The corrected index error value, when applied to Lewis’s raw data recorded in the journals, results in errors, relative to modern latitude estimates, that are nearer to what he achieved with the sextant.  It should be noted that no account has taken of a possible contribution to the error calibration due to the crack in the octant scale.


The parallax error difficulties associated with calibration candidate #1, which the journals indicate was the method used for the June 9, 1806 entry, have already been discussed.  Although parallax errors will produce the type of bias offset seen in the 1806 calibration value, the exact dimensions and distances can only be conjectured.  For this reason the corrected effective error value selected to evaluate the three latitude observations made during the summer of 1806 was simply set equal to the corrected value used for the summer of 1805.  The resulting error values of (+)0.4’, (-) 1.8’ and (-) 5.9’ were averaged to produce the 2.4’ result listed in Table 3.  The small corrected average error values seem to confirm that the octant index error remained relatively constant over the course of the trip and that the expedition took the proper care in storage and transport of their celestial instruments.


Table 3.  Octant back observation error readings from expedition calibration measurements and impacts on estimates of latitude





Recorded error value

Lewis Latitude avg. error

Corrected error effective value for Latitude use

Latitude avg. error (corrected)

July 22, 1804

(+)2º 11’ 40.3”


(+)2º 07’ 17.8”


April 12, 1805

(+)2º 40’ ---”


(+)2º 12’ 22.5”


June 9, 1805

(+)2º 4º [sic]
{(+)2º 4 ?}


{(+)2º 8’ 22.5”?}


June 9, 1806

(+)2º 30’  4.5”


(+)2º 12’ 22.5”




Summary and Conclusions

Throughout the expedition Lewis used many different techniques to determine the fixed offset errors of his celestial observation instruments.  Many of these needed to be different than the maritime techniques described in his reference books.  One even was aimed at determining the amount of error associated with a crack in the measurement scale of the octant.  Although sometimes large bias errors are evident in Lewis’s estimates of his latitude, when the calibration techniques are understood and likely field misunderstandings are allowed for the corrected values are found to be quite near modern estimates for the latitudes of the respective points of observation.  The only unfortunate point is that these errors could not have been determined in the original review and assessments of the results so that they could be incorporated and earlier generations of readers could appreciate their instrument and observation skill.  The tight grouping of the error results confirms the consistent and accurate observation techniques of the observers.  Also, the effective error correction values are found to remain nearly constant throughout the trip, which is a tribute to the care taken with the instruments under the many times adverse field conditions.  The detailed journal records of the expedition allowed these interesting results to be obtained even at this long separated time in the future.  The captain’s are again to be commended for their diligence.


© 2005-2008 Hans A. Heynau  All rights reserved.

[1] Silvio A. Bedini, “The Scientific Instruments of the Lewis and Clark Expedition” Great Plains Quarterly, ”, Vol. 4, No. 1 (Winter 1984) pp. 54-69.  Gary E. Moulton, ed, The Journals of the Lewis and Clark Expedition, Vol. 2 (Lincoln: University of Nebraska Press, 1987), p. 411. Entry for July 22, 1804.  All quotations or references to journal entries in the ensuing text are from Moulton, by date, unless otherwise indicated.

[2] G.D. Dunlap and H.H. Shufeldt, Dutton’s Navigation and Piloting Twelfth Edition, (Annapolis, Md., Naval Institute Press) pp. 393-4

[3] Moulton, for example, in the summary volume The Lewis and Clark Journals: An American Epic of Discovery, periodically records the explorer’s estimate of latitude and also records his value for the modern estimate of the same location.  Although no comment is made about the apparent wide difference for some cases, the sources of the differences may provide readers with additional insight into the explorer’s results.

[4] Lawrence A. Rudner and Hans A. Heynau, “Revisiting Fort Mandan’s Latitude”, We Proceeded On, November 2001, pp. 27-30 An Illustration of maritime calibration horizon views appears in the sidebar on p. 28 this article.

[5] Nevil Maskelyn, Tables Requisite to be used with the Nautical Ephemeris, for finding the Latitude and Longitude at sea, published by the Commissioners of Longitude, 3rd ed., 1802, General Introduction (following Table XXX), pp. 1-2. Lewis and Clark carried some edition of this work.  This edition was the most recently published prior to their departure.

[6] If Lewis used his preferred image inverting sextant telescope, the labels for the two illustrations would be reversed, but the procedure would remain the same.

[7] The previously cited Rudner and Heynau. article points out that, for the way Lewis applied his sextant correction value, the result would be in error by ½ the index error value.  Table 1 addresses the correction of this flaw by providing an “effective” index error for latitude calculations, as Lewis would have worked them, that is half the value recorded in the journal.

[8] Martin Plamondon II, Lewis and Clark Trail Maps (Pullman, Washington: Washington State University Press, 2000-2004), Volumes I, II, and III.  The accuracy of placing and/or reading these values from the Plamondon’s cartographic reconstructions is estimated to be +/- 0.1’.  These maps were also used in the octant assessments.

[9]  The (-) sign indicates that the value is to be subtracted from all measured values with the sextant.  Plus (+) values, such as in the two octant observation modes are to be added to the values measured by the instrument.

[10] (+) error values indicate the calculated value is north (greater latitude value), and (-) error values are south, of the modern estimate of observation latitude. A few latitude values that exhibited significant errors, due to transcription or other in determinant problems, were excluded from the latitude error averaging process in this and subsequent tables.

[11] Although the entry was marked July 22, Lewis’s Latitude for their 1803-04 winter camp indicates that he had determined this value much earlier, probably at the winter camp.

[12]  No indication as to whether the calibration was carried out on that date or earlier in 1805.

[13] The wording seems to imply that the calibration was made that day.

[14] No indication as to whether the calibration was carried out on that date or earlier in 1806.

[15] No subsequent sextant meridian transit data available for use in latitude comparisons.

[16] Examples appear in Clark’s entries for December 18 and 19, 1803.

[17] The fact that the difference in error values would correspond to an overlap of the scale segments, rather than a small gap that a crack would imply, also points toward a conclusion that the difference may have been statistical, rather than real.

[18] Lewis incorrectly describes this feature on July 22, 1804 as the “complyment” (90 degrees minus scale value), however in his calculations with the observed data he appears to correctly use the observed arc value as the angle’s supplement.  He used the term “compliment” again in his April 12, 1805 entry, but made his intent clear by going on to make it read “is the compliment of 180º of the double altitude of the object observed”.

[19] Donald D. Jackson, “Some Books Carried by Lewis and Clark,” Missouri Historical society Bulletin, Vol. XVI, No. 1. October 1959, p. 24. The primary reference of the books carried for celestial observations would have been the Tables Requisite, which is silent on back observation. Another reference item was the Meriwether Lewis, Astronomy Notebook, Missouri State Historical Society, Columbia, C1074. This reference contains a set of formulae prepared by Robert Patterson of Philadelphia for Lewis to use in his astronomical calculations.  It contains no information relative to instrument calibration other than to commend the Tables Requisite volume.

[20] Nathaniel Bowditch, The New American Practical Navigator; … (Newburyport, MA, for Brown & Stansbury, New York, 1st edition, 1802) p 142.

[21] The misapplication of the index error in artificial horizon sight reductions was introduced in the previously cited Rudner-Heynau WPO article.  If the true index error in back observation mode had been measured, Lewis’s latitude calculations would have led to differences relative to modern latitude estimates on the order of 120 nautical miles, which is not the case.

[22] Moulton records this data for the stated date with the footnote that it was taken from the front flyleaf of Codex L.

[23] Hans A. Heynau, “Finding Latitude” (Letters), We Proceeded On, February 2004, pp. 4-5.  This letter presents an initial extension of the sextant error effect to back observation calibration values. Note that the sextant index error must be applied before dividing the sextant reading by two or the error must be divided by two before applying to ½ sextant reading for measurements made with the aid of an artificial horizon.  The results presented here supercede the preliminary assessment presented in the letter.

[24]  If the sun is not at meridian, its elevation changes by as much as 1’ in altitude every 4 seconds so that accurate, simultaneous instrument measurements are almost impossible, particularly with only one liquid artificial horizon.

[25] Lewis started again numbering from No. 1 for his observation points after departing Fort Mandan in the spring of 1805.

[26] These conditions would have been satisfied any time from approximately March 26 on.

[27] Ferdinand Rudolph Hassler to [Robert Patterson], August 12, 1810, Donald Jackson, Letters of the Lewis and Clark Expedition with Related Documents, 1783-1854 (Urbana: University of Illinois Press, 1962) p 556-559. In this letter Hassler, who was trying to analyze the expedition’s celestial data, expresses some of his frustrations in trying to reconcile the data provided to him with the expedition preliminary chart and other maps of the time.

[28] Andrew Ellicott to Jefferson, March 6, 1803, Donald Jackson Letters pp. 23-25. In this letter Ellicott, who was one of Lewis’s two primary instructors in celestial navigation, notes his use and high regard for the equal altitude method for determining latitude. Patterson provided sight reduction formulae for this method that were included in Lewis’s Astronomy Notebook citer earlier.

[29] Excerpts circa March 22, 1806 in Jls, Part 3: Miscellany.  The Lewis latitude table for Fort Mandan to Fort Clatsop contains notations that some were calculated with error 2º 11’ 40” (nearer to the corrected value presented in this article) and some with 2º 40’—“, when all would require 2º 40’ based on the dates of observation and the standing error recorded by Lewis as applicable to these dates.

[30] Although Lewis transcribed his latitude data and results in codex O starting with a July 22, 1804 entry, Clark, in his daily log entry for May 29 mistakenly records Lewis’s octant latitude results as the altitude of the measurement.  In order for this to happen, Lewis would have had to have worked up this result, using the July 22 recorded calibration value, at the time that Clark was making his daily entries.

[31] Even though Lewis did not record data in his journal entries for this period (March 30, 1805 was the date of the last celestial data in Codex O), Clark recorded celestial data for April 3, 1805, while they were packing and preparing to proceed on.

[32] The Nov. 25, 26, 27 and Dec. 3, 1803 meridian entries are labeled lower limb, but must have been upper limb (George Huxtable, private communication). The journal editor’s markings for the Nov. 20 entry of “Upper <lower> Limb” indicate that Lewis crossed out his designation of lower. Perhaps he corrected it to upper after computing the latitude per the observation and, after re-checking, determined that it must have been upper.  The later journal entries did not include a deduced latitude so he would not have noticed the disparity compared to a dead reckoning latitude estimate based on the Nov. 20 value.

[33] John Garnet, The Nautical Almanac and Astronomical Ephemeris, for the year 1805, published in London by the Commissioners of Longitude April 27, 1801, third American impression August 20, 1803.  The sun’s diameter for April 1 was tabulated as 16’ 01.2”; April 7 15’ 59.5”; and April 13 15’ 57.9”.  Since the exact date the calibration was made was not documented, the round value of 16’ (32’ diameter) was selected for use in the analysis.