Lewis and Clark Astronomical Notebook.

Lewis and Clark Astronomical Notebook.

These pages were originally transcribed and posted by George Huxtable on his web site at- http://www.hux.me.uk/lewis01.htm. The current site will initially be an approximate mirror of that site and will eventually be the new repository for the transcription. Minor, editorial changes have been made in the move to this site and/or to update ageing-out web links.
This on-line transcription is intended to provide a research resource for Lewis and Clark Navigation investigators who do not have direct access to the original resource located in the Western Historical Manuscript Collection of the library of the University of Missouri, Columbia, MO.

Version 2 of the transcription was first posted on 2 March 2003.
Revision dated 8 Sept 03, making changes to the table of "examples for practice" for Problem 1st, and to the notes following those examples. Toward the end of the document, some wrong symbols representing "degrees" have been corrected.
Revision 4 Oct 03. This adds a section titled "Original Manuscript", and makes further changes to "Problem 1st", following "Form 1" and "Examples for practice"
Minor changes 31 May 2004.
Revision dated 18 August 2004, repairing some errors in "problem 5th".
Revision dated 7 December 2004, removing several errors and uncertainties throughout the text, resulting from a close study of microfilm frames by Hans Heynau.
Minor tidying to text, 18 December 2004.

This is the "Astronomical Notebook", brought back by Meriwether Lewis from the famous Lewis and Clark overland journey to the Pacific in 1804 - 1806.

It is mostly the work of the astronomer Robert M Patterson, providing astro-navigational instruction for the expedition, in the form of solutions to 5 navigational "problems".

Transcribed from manuscript in March 2003 by-
George Huxtable, 1 Sandy Lane, Southmoor. Abingdon, Oxon OX13 5HX, UK. Tel/fax +44 1865 820222.
E-mail address- george@hux.me.uk.

Return to HOME page.
Links to sections- , Start., Original Manuscript., Introduction., Patterson's Methods., Problem 1st., Problem 2nd., Problem 3rd., Problem 4th., Problem 5th., Lewis's Addition.

Original_Manuscript.

The original manuscript, from the text of which this trancript has been drawn, is held as-

Meriwether Lewis, Astronomy Notebook, [ca.1803]-1805, Western Historical Manuscript Collection-Columbia, MO.

A microfilm copy of the notebook is available upon request from that collection, at US$30 for diazo film or US$40 for the longer-lasting silver halide film. All items are shipped via the US Postal Service Global Priority Mail at an additional charge of $10-$15.

A link to the collection's website is-
http://shs.umsystem.edu/manuscripts/index.shtml

In the past, a source for detailed information about the manuscript collection was-
William T Stolz, Senior Manuscript Specialist.
address- WHMC-C, 23 Ellis Library, University of Missouri, Columbia, MO 65201.

For anyone interested, no copyrights are being broken in the present transcription. If the original authors held any copy rights to their text, those expired in the 19th century. The Manuscript Collection at Columbia may claim copyright on their photographic images that have been placed on microfilm, but none of those images is reproduced here: it is only the text that has been transcribed.

Introduction.

This is a transcript, with comments, of Meriwether Lewis's"Astronomical Notebook", taken on the Lewis and Clark expedition, up the Missouri to the Pacific, in 1804-1806. The main meat of it consists of astro-navigational instruction for the expedition , written by the astronomer Robert M Patterson, including examples from the year 1799, in the form of solutions to 5 navigational "problems".

An interesting paper by the late Richard S Preston, "The accuracy of the Astronomical Observations of Lewis and Clark", was published in the Proceedings of the American Physical Society, Vol 144, No 2, June 2000. The first page is viewable at http://www.jstor.org/pss/1515630. This has as reference 16, "Meriwether Lewis, Astronomy Notebook, Missouri state Historical Society, Columbia, C1074."

This web document is an attempt to make that Astronomical Notebook available to a wider readership.

This transcript of that notebook was made from a negative photographic image of the original manuscript, with many defects, which has presumably been subsequently photocopied more than once. Presumably, the original manuscript has suffered from age and from travel-wear over its long journey. There are numerous gaps, obliterations, and lacunae in my copy, which I have filled in or reconstructed where possible. This process was somewhat unsatisfactory, but I have done the best I can. Anyone with access to a more original document may be able to fill in or correct this transcription and I would be most grateful to receive any such feedback, or any corrections or suggestions for improvement.

Indeed, Hans Heynau has, in late 2004, kindly examined many microfim frames with a good magnifier, which has allowed me to correct several errors and remove many uncertainties.

It is over-optimistic to presume that this transcript is free from errors.

No claim is made by me for any copyright in my contribution to this document, and I give permission for it to be freely copied, quoted, and used.

I wish to thank Bruce Stark (email address- Stark4677@AOL.COM.) for providing the copy from which I have worked, and for much helpful information about astro-navigation. However, responsibility for any errors in this transcript is mine alone. I do not claim to be at all knowledgeable about Lewis and Clark's remarkable journey, my own interest being in the astro-navigation.

The tabulations of numerical data will look right only if a monospaced font is chosen for display and printout, in a size that will allow at least 91 characters to a line. This webpage is intended to specify such a font for the tables, in a size which is adequate for monitors of 640 pixels or more. If in the tables some lines appear to be wrapped, it will be necessary to select a smaller character size in your browser. To ensure that everything that was visible on the screen was contained within the width of a printed page, I have found it necessary to reduce the printer magnification (or "reduce/enlarge") slightly, from its default value of 100%.

Where a page-boundary occurs in the original manuscript, I have inserted the comment [page], in case material has been lost from the foot or the head of a page of the copy I have worked from.

A commentary, of my own, has been inserted wherever it seemed to be helpful, and is based entirely on my own opinions. It is intended for a reader who is familiar with modern astro-navigation, but not with the practices of 1800. Within the transcript, any text that is mine and not Patterson's will be enclosed within square brackets, [like this]. If everything in square brackets is deleted, the remainder will be as close as I can get it to Patterson's original text.

Patterson's spellings, abbreviations, punctuation, capitalisation, and symbols are not always consistent and I have not tried to make them so, nor attempted to preserve all such details from the original. I have felt free to alter the symbols used in the manuscript in places where I thought they were confusing, or wrong. However, I have tried to minimise any such changes, and keep to Patterson's text and table-layout as closely as possible.

Although Patterson's handwriting and figuring are in general very clear, there are many ambiguities in reading the numbers, which I have done my best to resolve by looking for consistency within the rows and columns. Where I cannot decipher illegible text or figures, this is shown by [?]

Patterson's Methods.

I have become very impressed by Patterson's competence and understanding of these complex matters: however, I doubt whether the explorers found his instructions very "user-friendly".

Patterson treats each of his 5 "problems" in a similar way-

First, the problem is defined in terms of what will be the input and what result will be obtained.

Next, in a series of "directions", he explains under what conditions the measurements should be made, what information is required from tables, and how they are to be processed.

Next there are some preliminary calculations to make small corrections to the observations and to interpolate from tables to the required time. These are all simple enough to be done by plain arithmetic without the use of logs.

Next the spherical trig algorithm is defined in terms of an example laid out as a "form". No explanation is offered and no equations are written down. Wherever Patterson mentions a trig function such as sine or cot, he will actually be referring throughout to log sine or log cot.

As is common in navigational practice, but may be unfamiliar to mathematicians, all logs have 10 added to their mathematical value, to avoid the occurrence of negative values of the log when quantities are less than 1. Nautical tables are still constructed on this basis. Therefore, after adding a number of such logs together, an appropriate amount such as 10 or 20 must be subtracted from the result.

Because these calculations are made in terms of logs, and the log of a negative quantity has no meaning, an elaborate set of rules is required to ensure that before taking logs, all the required quantities must be positive: these rules are defined in a set of "notes" which refer to each form.

After reaching the required result, Patterson, in many cases, provides further examples to be entered into the Form, so that part of the calculation may be repeated for practice.

[Lewis's"Astronomical Notebook", transcript and commentary.]

Problem 1st.

From the altitude of the Sun, together with the estimated Greenwich time, and the true hour of the day apparent time, at place of observation, to compute the Latitude of pl. obs.
[presumably, pl. obs. = "place of observation"; a common abbreviation throughout this text].

Directions.

1. At any time when the sun is not more than one or two hours from the meridian (but the nearer to the meridian the better) take its altitude and note the time per watch, making allowances for its probable error from true apparent time.

2. From the apparent altitude find the true alt., subtract this from 90° and the remainder will be the zenith distance which call north or south according as the zenith is north or south of the body observed. [page]

3 Find the true declination of the sun for the estimated Greenwich time.

4. From the time at pl. obs. find the hour angle of the Sun at the time of obs.

From the above data the latitude may be computed as in the following example.

Note. In all the following Forms, the capital letters signify the corresponding arches in the adjoining column, and the small letters, the sines, tangents, secs, &c of those arches respectively. When the small letter is omitted in the Forms the arch is found from the sine, tangent &c; but when the smaller letter is prefixed, the sine, tangent, &c, is found from the arcs.

In taking out the sine, tangent &c of an arch from the table, or in finding an arch from the sine, tangent, &c., it will be sufficient [page] to take it out to the nearest minute only.

When an ambiguous sign occurs, such as ±, or +/~ the one or the other is to be used as directed in the explanatory note to which the number in the margin refers.

[I've had to type out the sign for "sum or difference of", +/~ as 2 characters separated by a "/", but in the manuscript these are always shown one above the other as a single character.]

Example.

Suppose the app. alt. of the sun's lower limb above the southern horizon = 52° 17'. Estimated Gr. time [= Greenwich time] May 10th 1799 about 17 hours p.m. True app. time at place of obs.= 54m. p.m. Reqd. the Lat.

52° 17'   "                   17° 41' 43"   Dec. May 10 [presumably: truncated here]
        39" refr-par [a]      17° 57' 12"            11 [Dec. May 11, presumably]
----------    [subtract]      ----------   [subtract]
52° 16' 21"                       15' 29"   Diff. [in 24 hours]
    15' 53" semidiam. [a]         10' 56"   Correction [in 17 hours] [e]
----------   [add]            ----------   [add to dec at Gr. noon may 10]
52° 32' 14" True alt.         17° 52' 49"  True Dec [being May, this will be North][f]
----------  [subtract from 90°]
37° 27' 46" Zen.. dist. [a]

                                 54m [minutes of time then quartered to give degrees]
[illegible, maybe "hour angle"]  13° 30'  [illegible, maybe "degrees"] [g]

[My comments on the above example, preliminary corrections-

a. The necessary small corrections to apparent altitude for (refraction less parallax), and then for semidiameter, are shown to the left, followed by subtraction from 90° to provide zenith distance.

b. Patterson presumes, without saying so, that any necessary correction for index error of the sextant has been made beforehand.

c. Because altitudes are measured between two images of a body, one direct via the index mirror, the other reflected in a liquid surface seen through the horizon glass, and the reflection doubles the angle between them, the next step is to halve the sextant reading, after correcting index error. Patterson presumes, without saying so, that this halving has been done.

It's a pity that Patterson failed to spell out the correct procedure for correcting reflected altitudes made on-land, as shown in points b and c above. As a result of Lewis'sand Clark's subsequent misunderstanding on this matter, throughout the journey they erroneously halved the sextant reading before applying the index correction, rather than vice versa. This is made clear in-
Lawrence A Rudner and Hans A Heynau, "Revisiting Fort Mandan's Latitude", in "We Proceeded On", Vol 27 No 4, November 2001, page 27.
"We Proceeded On" is the quarterly journal of the Lewis & Clark Trail Heritage Foundation, Inc., PO box 3434; Great Falls, Montana MT59403. http://www.lewisandclark.org

d. Measurements made on land, using an artificial horizon, require no correction for dip of the horizon, as would be necessary at sea.
Patterson presumes, without saying so, that the image of the Sun via the index mirror, and the reflected image seen in the pool, are not superimposed in the sextant's view, but adjusted so that their limbs just brush, above and below. It's most important that the lower image of the Sun is the one reflected in the pool. If the images were placed the other way round, the resulting Sun altitude would be a diameter (about 30 min of arc) too small. This would be an easy error to make.

e. In the right column, declinations at two successive Greenwich noons are shown and then interpolated by a fraction 17/24 to the Greenwich time of the observation. The two columns are placed left and right simply to save paper; there is no connection between the column to the left and its neighbour to the right.

f. Although these calculations are shown at this stage worked out to a second of arc, subsequently the nearest minute will be taken for further working.

g. The time after local apparent noon of 54 minutes is converted to angle at the rate of 15° to the hour, so minutes of time divided by 4 gives degrees of local hour angle.

Some of these comments will apply to the preliminary corrections in the other problems also.]

[page]

Form I

   Hour [angle]          A  13° 30'    Sec  a  10.01217
   Decln.                B  17° 53'[N] Tan  b   9.50876    Cosec  b1  10.51275
   Zen dis,              C  37° 28'                        Cosine c1   9.89966
                                                -------
   a+b-10                D  18° 22'    Tan  -   9.52093    Sine   d1   9.49840
                                                                      -------
   b1+c1+d1 -20          E  35° 29'                        Cos    -    9.91081
                            ------
1. E +/~ D = Lat pl.obs. F  53° 51'

[Above, some new symbols b1, c1, d1 have been introduced to reduce confusion. In the original, Patterson uses a, b, etc in each column but differentiates them by employing a rather different style of script]

1. Add when zenith distance and dec. are of the same name, otherwise subtract; and the Lat. will be of the same name with the greatest number. This rule will give the Latitude in all cases where the Altitude is taken near the meridian. The rule however may be made universal thus: Add when Lat. and Dec. are of the same, and azimuth of the sun from the meridian less that 90° , also when the hour angle exceeds 90° : subtract when Lat & Decn. are of different names, also when azimuth from meridian more than 90°.

[It seems rather strange to find that zenith distance has been given a name (North or South) at all, and the adopted convention for naming it is opposite to what I would have expected. According to Raper, "Practice of Navigation", 6th ed. (1840), art. 683- "When the observer is to the N of the sun, call the zen. dist North; when he is to the S. of the sun, call it south". That's all very well for bodies when on or near the meridian, but how would the zenith distance be named if the body happened to be near to East or West?]

Note 1st. When the decn, =0, to the secant of the hour angle add the cosine of the zenith dist. and the sum (abating 10 from the index) will be the cosine of the Latitude.

Note 2nd. If an obs. be taken before and after the Sun comes to the meridian, when the alts. are both nearly equal, then mean of the lats computed from these two observations will be more accurate than that computed from one obsn., as the errors arising from an error in the watch or time, will in this case tend to correct each other.

[In the above form, the columns are related, row by row.

What is shown is an implementation of the following algorithm-

tan D = Tan dec / cos hour angle
cos E = Sin D cos zenith dist / sin dec
Take the sum or difference of D and E according to rule 1 above.

This appears to be the method described on page 169-170 of Cotter's "A History of Nautical Astronomy", as "a method of finding latitude by ex-meridian observations of the Sun, using 'direct spherics', which was given in the later editions of James Robertson's Elements of Navigation". Although the result is expressed by Cotter somewhat differently, it's equivalent to Patterson's method.

Cross-checking Patterson's result by back-tracking to find Zenith distance given lat and dec and difference in hour angle, this gives 37° 27', in good agreement.]

[page]

Examples for practice.

                 1            2            3            4            5
Hour angle    10° 40'      12° 36'       7° 17'       5° 18'      20° 24'
Decl.         18° 52'N     17° 11'N      3° 57'S      0° 00'      26° 48' S
Zen dist      27° 30'N     19° 37'N     48° 18'N     38° 15'S     20° 37' N
Lat. pl. obs. 45° 02'      31° 11'      43° 53'      37° 56'S     18° 16' S

[Here is a list of Lat. pl. obs as obtained by more precise computation-

                 1            2            3            4            5
Lat. pl. obs. 44° 53.6'N   33° 11.6'N   43° 54.2'N   37° 56.2'N   18° 19.9'S
 or else       6° 32.6'S    1° 58.2'N   51° 52.0'S   37° 56.2'S   38° 18.7'S

The expression used here to calculate precise Lat, an implementation of the procedure given for Problem 1st by Patterson, is as follows-

where L = lat., D = dec., H = hour-angle (half-interval) in degrees,
A = alt = 90 - Zenith Distance.

then L = atn (tan D/cosH) ± acs (sin(atn(tanD/cosH))*sinA/sinD)

the ± symbol indicating that the acs function has two results, equal in amount but one positive, the other negative.

This gives rise to a solution in which two alternative latitudes are usually possible, though one may sometimes be excluded as exceeding the allowable range of latitudes, ±90°. The ambiguity is not simply a mathematical one, but has a real physical basis. For the more Northerly latitude option, the body will cross the observer's meridian to his South, and vice versa. The correct alternative is usually obvious and it is easy to see which should be rejected.

The expression above appears to work in all cases, except when dec. is exactly zero, in which case an infinity, arising part way through the calculation, may cause some calculators to fail. It's possible to trap and circumvent this.

The values for precise latitude, given above, have been checked by a backwards process, and give the correct answer for altitude in all cases.

In Patterson's table of examples shown above, then,there is agreement in cases 3, 4, and 5, but errors occur in Patterson's latitude results for examples 1 and 2, which might puzzle anyone trying to follow his examples. In example 2 above, Patterson appears to have mistranscribed his result, writing clearly the erroneous 31° 11'N where it should have been 33° 11'N.]

Problem 2nd

From two alts. of the sun together with the intermediate or elapsed time between the two observations and the estimated Gr. time to compute the Lat. of the place.

[This was the "Double Altitude problem", used here for determining the latitude from two altitudes of the Sun. Presumably, this problem came into play when no noon sight could be observed, and also the local apparent time was unknown. It could become useful when the Sun's noon altitude exceeded 45 ° which when an octant was used with a reflecting pool,would otherwise require less-accurate back-observations.

No account seems to be taken of any motion of the observer between the two observations: no problem for a land traveller, but for use by a vessel under way, further corrections may be required.]

Directions.

1. For the proper time when the alts. are to be taken, consult the remarks in the book of requisite tables p. 21 & 22. [This will be a later edition of Tables Requisite than my photocopy of 1766, which does not offer such guidance. However, Maskelyne does offer guidance on this matter in his earlier work, "British Mariner's Guide", 1763, appendix page 76.]

2. From the app. alts find the true alts.

3. Find the Decn. of the sun for the middle Gr. time between the two observations. [page]

[There's no sign of direction 4, either at the foot of one page or at the head of the next. Was the missing step perhaps to take the complement of the Sun's dec. to find its polar distance?]

5. Let half the elapsed time be reduced to degrees and minutes.

From the above data the Latitude may be computed in the following example.

Example.

Suppose the app. alt. of the sun's lower limb at the time of the first obs. = 43° 28'. Time per watch. 2h 41m before noon. App. alt at second obs. 56° 21'. Time per watch 30m before noon. Estimd. Gr. time May 12th 1799 about 8 hours p.m. [Presumably, this was the Greenwich time at approximately the mid-time between the two observations.] Lat by acct. 50[?]° 19' N. Reqd. the true Lat?

43° 28' 00"   [first obs.]             18° 12' 23" Dec.,  May 12 [presumably at noon]
        54" ref-par. [subtract it]     18° 27' 16" --------   13 [24 hours later]
43° 27'  6"                                14' 53" Diff. [text shows 14°  53', in error]
    15' 52" semidiam. [add it]              4' 58"  Correction [8/24 of Diff. above]
43° 42' 58" Lesser true alt.           18° 17' 21" true Dec N
56° 21'       [second obs.]            71° 42' 39"  polar dist.
        33" [ref-par again]             2h 41m [obs 1, time before local noon.]
56° 20' 27"                                30m  [obs 2, time before local noon.]
    15' 52" [semidiam]                  2h 11m [which is then halved to give-]
56° 36' 19"  Greater true alt.          1h  5m 30sec [then min. quartered to give degrees]
           Angle of 1/2 elapsed time = 16° 22 1/2' [will be rounded to 16° 23']

[page]

Form II.

   Polar Dist.                 A  71° 43'  Tan  a  10.48097  cos   a1  9.49654
   Ang of 1/2 elap time        B  16° 23'                    Tan   b1  9.46835
1. a1+b1-10                    C  84° 44'  Cos  c   8.96305  Cot   -   8.96489
   Lesser Alt.                 D  43° 43'
   Greater alt.                E  56° 36'  Tan  e  10.18087
   1/2 (D+E)                   F  50° 10'  Tan  f  10.07875
   E~F                         G   6° 26   Cot  g  10.94786
   a+c-10                      H  15° 32'  Tan  h   9.44402
   f+g+h-20                    I  18° 42'  Cot  -  10.47063
   H~I                         K   3° 10'  Tan  k   8.74292
2. e+k-10                      L  94° 48'  Cos  -   8.92379
3. C +/~  L                    M  10°  4'  Sec  m  10.00674  Cot   m1  10.75074
   e+m-10                      N  33°   '  Cot  n  10.18761  Cosec n1  10.26389
4. A +/~ N                     O  38° 43'  Cot  o  10.09603  Sin   o1   9.79621
   m1+n1+o1-20= Hr. ang.       P   8° 47'  Cos  p   9.99488  Cot   -   10.81084
    at Greater alt.
o+p-10= Lat at last obs.       Q  50° 57'  Tan  -  10.09091

[Above, some new symbols a1, b1, m1, n1, o1, have been introduced to reduce confusion.]

8m 47sec divided by 4 gives 35 m 8 sec before noon = time at last obs. correct

[I think this should read "8° 47' multiplied by 4 gives 35min 8sec before noon = time of last obs. correct."]
[The watch, which read 30min before noon at the time of the second observation, must therefore be 5min 8sec fast on local time.] [page]

1. Set down the supplement of the angle found in the table when A is greater than 90°. Let the cosine of C be increased or decreased by as much as a1+b1-10 exceeds or wants of cot C the nearest arch in the table. [In this note, the references to cos C and cot C will, of course, actually refer to the log cos and log cot respectively.]

2. Set down the suppt. of the angle found in the table when I is greater than H

3. Subtract except A is less than the calculation by account, and when M exceeds 180° take the cot and sec of its excess above 180° and set down the remainder.

4. Subtract unless when M is greater than 90°.

Note. If both alts when corrected be equal, the Lat may be found by prob. 1st calling half the elapsed time the true hour of the day, or dist. of the body in right ascension) from the meridian or if the two alts be nearly equal you may, without any sensible error take the [mean and work ?] as above by prob. I [page]

[I am unable to relate this double-altitude calculation for latitude from the Sun to any of the methods described in Cotter for solving the same problem. It may be a variant of the rigorous solution to the double-altitude problem devised by (Facio) Duillier. Checking backwards to derive the observed altitudes from the calculated result, Patterson's result appears accurate to within a minute or so.]

Problem 3rd.

From the app. alt. of the Sun, together with the Lat pl. obs. and estimated Gr.time, to find the true hour of the day.

Directions.

1 At any time when the Sun is not less than 3 points from the meridian (but the nearer to the East or West points the better) take the alt.

2 From the app. alt find the true alt.

3 Find the dec. for the estimated Gr. time which subtract from 90° when the Lat and dec are of the same name but add to 90° when they are of different names and the result will be the polar distance.

4 From the above data the hour angle and the hour of day may be computed as in the following example. [page]

Example.

Suppose the app. alt. of sun's lower limb west of the meridian = 22° 35' Lat. pl. obs. 34° 40' N [shown in the original text as 34" 40° N, in error]. Estimated Gr. time May 24 1799 about 18 hours p.m. Reqd. the time?

22° 35'     [observed alt]       20° 49' 28" Dec May 24
    -2   6" par-ref              21° 00' 20"         25
22° 32' 54"                          10' 52" Diff [in Dec over 24 hours]
   +15' 50" semidiam                  8'   9" Correction [over 18 hours]                                                                                                     
22° 48' 44" true alt.            20° 57' 37" true dec N
                                 69°  2' 23" [polar distance]
                                 34° 31' 11" = 1/2 pol. dist.

Form III

   Lat of pl. obs           A  34° 40'  Tan   a   9.83984
   True alt                 B  22° 49'
   1/2 [A + B]              C  28° 44'  Cot   c  10.26103
   B ~ C                    D   5° 55   Tan   d   9.01550
   1/2 Polar dist.          E  34° 31'  Cot   e  10.10260
   c+d+e-20                 F  15° 22'  Tan   -   9.43913
1. E +/~ F                  G  19°  9'  Tan   g   9.54063
2. a+g-10 = hour angle      H  76°  6'  Cos   -   9.38047
   Hx4 = hour angle in time I  5h 5m 24s [degrees of arc x 4, to give minutes of time]

[This last result, for hour angle I measured in time, is slightly wrong, as multiplying 76° 6' by 4 gives 5h 4m 24s. i.e. 1 minute of time less!]

1. Subt. when A is greater than B, otherwise add

2. Take the supp to 180° when F is greater than E [page]

Note.

When the dec. = 0; then to the secant of the Latitude add the sine of the altitude, and the sum (abating 10 from the index) will be the cosine of the hour angle.

When the Lat = 0; then to the secant of the dec. add the sine of the alt. and the sum (abating 10 from the index) will be the cosine of the hour angle.

When the alt. = 0; then to the tangent of the latitude, add the tangent of the declination, and the sum (abating 10 from the index) will be the cosine of the hour angle, taking the supplement to 180° when Lat and dec are of the same name.

When both Lat and dec =0; then the complement of the alt or zen. dist. will be the hour angle.

[The above method of obtaining local time from a Sun altitude appears to differ from the methods described in Cotter, pages 243 to 254. However, back-calculating from the result above allows an altitude to be deduced that accords with the observed value to one minute of arc, so the result appears correct.] [page]

Return to HOME page.
Links to sections- ,Start., Original Manuscript., Introduction., Patterson's Methods., Problem 1st., Problem 2nd., Problem 3rd., Problem 4th., Problem 5th., Lewis's Addition.

Problem 4th.

From the Lat. of the pl. obs. together with the time at pl. obs. and the estimated Gr. time, to compute the alt. of sun, moon, or any known star.

[This is a calculation with which modern astro-navigators, using position-line navigation, will be very familiar: they would be finding an intercept by comparing an observed altitude with a computed altitude. That method was developed long after Patterson's time. Here, he is computing altitudes in the case when, for some reason, the altitude of a body cannot be observed, in order to make precise corrections to a lunar distance. From an observer's latitude, with the declination of a body and its local Hour Angle, the true altitude is computed, and the altitude that would have been observed is deduced.

Such a calculation may be familiar to modern navigators, who would be likely to use the expression-
sin alt = sin lat sin dec + cos lat cos dec cos (hour-angle)
However, Patterson goes about it, in form IV below, in a different way, introducing an angle C, where-
tan C = tan dec / cos (hour-angle).
and then deduces the altitude from-
sin alt = sin dec cos (C +/~ lat) / sin C.
which works out exactly the same in the end.]

Directions.

1. Find the dec. of the body for the estimated Gr. time.

2.Find the hour angle of the body at the given time. The altitude may then be computed by Form 4th as in the following examples.

Example 1st.

Suppose the Lat. 40° N. Gr. time May 7th 1799 about 17 hours p.m. Time at pl. obs. 3h 42m p.m. Reqd. the true and app. alt. of Sun's centre.

16° 53' 31" Dec May 7th
17°  9' 52"           8        3h 42m divided by 4 gives 55° 30' hour angle.
    16' 21"    Diff 
    11' 35"    Correction
17°  5'  6"    True Dec N

[page]

Form IV (A).

   Hour angle            A   55° 30'   sec   a  10.24687 
   Decln.                B   17°  6'   Tan   b   9.48804   sin   b1   9.46841
   a+b-10                C   28° 31'   Tan   -   9.73491   cosec c1  10.32110
   Lat                   D   40°
1. C +/~ D               E   11° 29'                    [?]cosec e1   9.99132 
   b1+c1+e1-20 true alt  F   37°  8'                       sin   -    9.78073 
2  Correctn. of alt.     G        1'
3. F±G = app. alt        H   37°  9'

[Note.
In Form IV (A) above, Patterson appears to have made rather serious errors, perhaps errors in transcription from his earlier notes. I have indicated this by the question-mark alongside his entry of "cosec e1 9.99132" in the rightmost column.
First, and most important, he has clearly written "cosec", in error, and I have copied that text. However, he should instead have written "cos". That error would make it impossible for anyone to follow these instructions and get the right answer.
Second, the number he entered was originally 9.99332, of which the first "3" has been struck-over to become a "1", to make it 9.99132, as I have written it down. However, log cos 11° 29' is actually 9.99122, which is the number Patterson should have set down. Indeed, he may well have done so in an earlier rough draft, as you can check by totting up the three logs on the right. These add up correctly to the given total of 9.78073 only if Patterson's third entry is changed from 9.99132 to 9.99122.]

1. Add when Lat. and decl. are of different names or A greater than 90°, otherwise subtract.

2. For the sun or a star take the refraction corresponding to the altitude table I. For the moon take the correction table VIII. These corrections need be taken out only to the nearest minute. [The numbered tables referred to here were presumably to be found in Tables Requisite.]

3. Add for the sun or a star subtract for the moon. [Here, the correction is made with the opposite sign to what today's navigator is accustomed to, because the apparent altitude is being calculated from the true altitude, rather than vice versa.] [page]

Example 2nd.

Suppose the Lat. 22° 40' N., Gr. time June 16th 1799 about 15 hours p.m. Time at pl.obs. 11h 32m p.m. Reqd. the true and app alt of the moon's centre.

[Comment: There has been an unstated assumption here that the longitude of the observer must be at 'about' 52° West, for the local time to be 11h 32m p.m. when the Greenwich time is 15h p.m.]

[Readers may need reminding that until 1925 the almanac used astronomers' days, which started at NOON, so that a time of 15h p.m. meant that it was actually 3 hours into the following morning at Greenwich, and still the 16th by astronomer's reckoning, though by then the civil day will have gone on to the 17th.]

[below, the angle units in the left column were shown, incorrectly, by Patterson as ' and ". I have changed them to ° and ' as they should be]

 25° 30'[S] [Moon's]Dec June 16 [a]  5h 39' 12.4" [Sun's] Rt.asc. June 16   [b]
 26° 34'[S]                  17 [a]  5h 43' 21.7"                      17   [b]
  1°  4'  Diff                           4'  9"
     16'  Correction                     2' 36"   Correction
 25° 46'  Moon's dec. S.             5h 41' 48"   [Sun's] Rt. asc.
255° 17'  Rt. asc. June 16 [a]      11h 32        time at ship [c]
  1° 58'  Correction  [d]               17h 13' 48" R.A. mid-heaven [e][then x 15]
257° 15'  Moon's Rt. asc.[d]       258° 27' ditto, in deg. & min.
                                   257° 15' Moon's R.A.
                                     1° 12' hour angle

[Comments on the above corrections. Some of these matters would be obvious to an observer looking at an 1800 almanac.

a. These values of dec and Right Ascension for the Moon are taken from an almanac which provides them at noon and midnight, so they are interpolated within the 12-hour interval that brackets the Greenwich time of the observation. Moon RA is given in degrees of arc. The Moon dec. is in fact South, but Patterson fails to say so here.

b. Sun RA is provided in the almanac for noon each day, in terms of time not angle. It is to be interpolated within a 24-hour interval, added to local time, converted to angle units.and then the angle difference from Moon RA is found.

c. Ship? This section must have been derived from a marine-navigation text.

d. This correction should have been derived by interpolating between the Moon RA at Greenwich midnight on the 16th, and at the following noon, when it is 263° 12', but Patterson fails to show this working.

e. The quaint expression "mid-heaven" appears to refer to a line through the observer's meridian.] [page]

Form IV (B).

   Hour angle[observer to Moon] A   1° 12'   Cos   a   9,99990
   Decln. [of Moon]             B  25° 46'   cot   b  10.31632   cosec b1  10.36180
   a+b-10                       C  25° 46'   cot   -  10.31632   sin   c1   9.63820
   Lat                          D  22° 40'
1. C +/~ D                      E  48° 26'                       sec   e1  10.17816
   b1+c1+e1-20 = true alt[moon] F  41° 34'                       cosec -   10.17816
2. Corr. of alt.                G      43'
3. Corr of alt of F±G=app alt   H  40° 51'

[page]

[Corr of Moon alt is combined refraction and parallax, and subtracted]

Example 3.

Suppose the lat 45° 10' N. Gr. time Jan 31 1799 about 18 hours p.m. Time at pl.obs. 12h 40m p.m. Reqd. the true and app. alt of the star Aldebaran?

16°  3'  2"                    20h 56m 44s [Sun RA, previous noon]
     2' 37" Cor.               21h 00m 49s [Sun RA, following noon]
16°  5' 39" star's dec N  [a]       4m  5s [Sun RA, change in 24h]
 4h 23m 19s                         3m  4s [Sun RA, change in 18h]
     1m  5s                    20h 59m 48s Sun's Rt. asc.
 4h 24m 24s star's Rt asc.[a]  12h 40m     time at Ship
                               33h 39m 48s
                               24h
                                9h 39m 48s  RA mid-heaven
                                4h 24m 24s
                                5h 15m 24s  divide by 4 to give hour angle
                               78° 51'  * hr. angle.

[Comment on above corrections:

a. It's rather a surprise to see corrections being made to the position of a star, which changes only very slowly. In modern almanacs, star positions are stated for each month, so no corrections are required. Presumably, the 1800 almanac did not provide up-to-date star positions, and instead these were taken from a star catalogue, which must have been dated at or near 1780. The corrections to Aldebaran are due, mostly, to the slow changes in the direction of the earth's polar axis over the 20-year interval.]

By Form 4th (A)

A   78° 51   10.71359
B   16°  6'   9.46035     9.44297
C   56° 11'  10.17394    10.08049
D   45° 10'
E   11°  1'               9.99192
F   19°  7'               9.51538
G        3'
H   19° 10'

[page]

Note When the dec = 0; then to the cosine of the Lat., add the cosine of the hour angle and the Sum (abating 10 from the index) will be the sine of the true altitude.

And when the Lat = 0, then to the cosine of the declination, add the cosine of the hour angle, and the sum (abating 10 from the index) will be the sine of the true alt.

When both Lat and dec =0, then the complement of the hour angle will be the true alt.

Examples for practice.

                    1            2             3               4
Hour angle       42° 14'       7° 18'        36° 58'         50° 18'
Decn.            10° 17'N     21° 56'N       00°             15° 27'    
Lat.             37° 49'N     46° 17N        32° 16'         00°
True alt.        43° 14'      64° 55'        42° 44'         38°

[page]

[Checking these results with a calculator, I agree, within a minute, with all but example 3, for which I find 42° 30']

Problem 5th.

To find the Longitude by Lunar observation.

Directions.

1. At any time when the sun and moon, or the moon and any of the stars from which her distance is calculated in the Nau. Alm. on the given day, are both visible, and neither of the bodies less than 5° high; your instrument being previously adjusted; or the index error ascertained, and also the error of your watch found by a previous obs. as directed prob. 3- take a set of three or more observations of the apparent angular distance of the sun and moon's nearest limbs, or of the star from the moon's nearest or furthest limb. viz. that which is fully enlightened, noting the corresponding times per watch; and at that time that you measure the distance, let two [page] assistants take the altitudes of the bodies respectively. Or the distance and alts. may all be taken by the same observer at small equal intervals of time in the following order

viz.

Distance - Time per watch
moon alt.
sun or star's alt.
Distance - Time per watch.
sun or star's alt.
moon's alt.
Distance - Time per watch.

[A more modern view would recommend a different sequence as follows-

sun or star's alt.
moon's alt.
Multiple lunar distances with times per watch.
moon's alt.
sun or star's alt.

This would bring the averaged times of all the different measurements close together]

2. Of these distances, times, and alts, take means by dividing their respective sums by the number of observations.

3.Let the mean alts be so far corrected as respects semidiameter, so as to obtain the app. alts of the centres above the true horizon.

4 To the app. dist. of sun and moon's nearest limb, add the sun's semidiameter from Alm. page III [page] of the month, and also the moon's semidiam., N.A. page VII of the month, increased by the augmentation Tab IV. When the distance of a star from the moon's nearest limb is observed, you must add the moon's semidiam. + augm; but when the star's distance from the moon's furthest limb is observed, you must subtract the semidiam. + augm.; and thus you will have the app. distance of the centres of the bodies observed.

5. If at the time of taking the above observations, one of the bodies (but especially the sun) be not less than 3 or 4 points of the compass from the meridian the time at pl. obs. may be computed from the alt. of the body as in prob. III, independently of the time per watch, and this should always be done when circumstances admit.

6. As it may frequently happen, that the alts. [page] of one or both of the bodies cannot well be taken you must then compute the app. alts. by prob 4th, and this method is generally to be prefer'd on land.

From the above data the Longitude of pl. obs. may be computed as in the following example.

Example.

Suppose the app. angular distance of the sun and the moon's nearest limbs (by taking the mean of a set of observations) to be 83° 30' 4". The app. alt. of the sun's lower limb measuring 52° 44'. And that of the moon's L.L. 24° 32'. Gr. time Sept 22 1799 about 9 hours p.m. Time at pl. obs. (allowing for error of watch from true app. solar time) 15 hours p.m.

Req'd the Long. of pl. obs. from the meridian of Greenwich?

52° 44'      sun's alt.                     83° 30'  4"
    16'      semidiam.                          16'
53°          alt of centre                      15' 22"
24° 53'[?]   moon's app. alt. [a]           84°  1' 26" ang. dist. of centres.
    15' 22"  moon semidiam. + aug.
24° 48' 22"[?] alt. of centre [a]

[page]

[a. The value stated here for moon's apparent altitude, of 24° 53', differs from the text which defines the problem, above, in which the app.alt. of the Moon's L.L. (lower limb) is stated to be 24° 32': neither of these lower-limb altitudes, when the Moon semidiameter is added of 15' 22", gives exactly the stated altitude of the Moon's centre, 24° 48' 22". These discrepancies are not understood, by me.]

Form V

   app.dist. of centres          A  84°  1' 26"
   moon app. alt.                B  24° 48'       cosec   b   10.3773
   sun or star app. alt.         C  53° 
   1/2 (B+C)                     D  38° 54'       tan     d    9.9068
   C~D                           E  14°  6'       cot     e   10.6000
   A/2                           F  42°  1'       tan     f    9.9547
   d+e+f-20                      G  70° 56'       tan     -   10.4615
1. F +/~ G                       H 112° 57'
2. F ~/+ G                       I  28° 55'       tan     i    9.7423
   moon hor. par. N.A. page VII  K      56'  3"   pr.log. k     .5067
   b+i+k-20                      L      42' 33"   pr.log. -     .6263
   Refr. of I  Tab.I             M       1' 42"
   L-M = 1st correction.         N      40' 51"
3. A±N                           O  83° 20' 35"
   Refr. (- par.) of [?] for     P          20"
    star (or sun) (2nd corr.)       
4. O±P                           Q  83° 20' 55
5. Corr. Tab. XIII (3rd corr)    R           1"
6. Q±R = true dist. of centres   S  83° 20' 56"
7. Preceding dist. in N.A.       T  83° 30' 46"
7. Following dist. in N.A.       U  82°  4'  8"
   T~S                           V       9' 50"   pr.log. v    1.2626
   T~U                           W   1° 26' 38"   pr.log. w     .3176
   v-w                           X      20' 26"   pr. log. -    .9450
8. Hour above T in N.A. + X      Y   9h 20m 26s
    = true Gr. time.
8. Time at pl. obs.              Z  15h 
9. Y~Z = Long. in time           AA  5h 39m 34s
9. AA/4 = Long in deg & minutes  BB 84° 53' 30" East

[page]

1. Add when C is greater than B, otherwise subtract.

2. Subtract when C is greater than B; otherwise add.

3. Subtract when either H or I exceeds 90°; or when H is greater than I.

4.Add when H or I exceeds 90°; or when H is less than I.

5.In Tab VIII find the correction of moon's alt., then in Table XIII under the nearest degree to Q at the top, find two numbers; one opposite to the nearest minute to moon's corr. of alt. found as above, and the other opposite the nearest min. to first correction (N) and the difference of these two numbers will be the third correction. This corrn. may, without sensible ["error" missing here?], be generally omitted.

6. Add when Q is less than 90° - otherwise subtract.

7. These are to be found in N.A. from page 8th to page 11th of the month opposite the day of the month and the sun or the star from which the moon's distance was observed; taking out the two distances, which are next greater and next less than the true distance (S) calling that the preceding distance which comes first in the order of time and the other the following distance.[page]

8. The Gr. time and time at pl. obs, must both be reckoned from the noon of the same day.

9. When the Gr. time is greater than the time at pl. obs. the Long. is west; otherwise it is east. When Long. comes out more than 12 hours or 180°, subtract it from 24 h or 360° and change its name.

[The above procedure "clears", or corrects, the observed lunar distance from the effects of semidiameters, refraction, and parallax, to give a true angle between the centres of the two bodies, for interpolation between the nearest tabulated lunar distances. This results in Greenwich time, which would be Greenwich Apparent Time in almanacs before 1834, because that time-scale was then the "argument" of the almanac. By comparison with the Local Apparent Time, the longitude results.]

["pr. log." means "proportional log", which was an invention by Maskelyne for facilitating the interpolations for time, and was tabulated in the "Tables Requisite". The proportional log is tabulated for each second, for times up to 3 hours or 10800 seconds, and can also be used for angles up to 3 degrees or 10800 arc-seconds. The proportional log is the log (to base 10) of 10800 (which is 4.0334), less the log of the number of seconds involved. For instance, the prop. log of 1 hour or 3600 seconds is 4.0334 less log (to base 10) of 3600, which is 4.0334 - 3.5563, or 0.4771. This would be tabulated as 4771]

Example 2.

Suppose the app ang. dist. of the star Regulus from the Moon's furthest limb to measure 54° 18'. app. alt. of moon's L.L. 20° 10'. app. alt. of star 58° 32'. Gr. time November 23d about 17h p.m. Time at pl.obs. allowing for error of watch 17h 20m 25s p.m. Reqd. the Longitude?

[Note. It's necessary to refer to Form V, above, to understand the entries in this example, as the labels are all omitted here. The layout is exactly the same as form V]

20° 10'
    16' 38" semidiam + aug.
20° 26' 38" moon's app. alt.

54° 18'
    16' 38" 
54°  1' 22" app.dist.

[page]

A   54°  1' 22"
B   20° 27'        10.4567
C   58° 32'
D   39° 29'         9.9159
E   19°  3'        10.4618
F   27°  1'         9.7075
G   50° 35'        10.0852
H   77° 36'
I   23° 34'         9.6397
K       60' 42"      .4721
L       48' 37"      .5685
M        2'  7"
N       46' 30"
O   53° 14' 52"
P           12"
Q   53° 14' 40"
R            6"
S   53° 14' 46"
T   52° 17' 44"
U   54°  8' 17"
V       57'  2"      .4991
W    1° 50' 33"      .2117
X    1° 32' 52"      .2874
Y   16h 32m 52s
Z   17h 20m 25s
AA      47m 33s
BB  11° 53' 15"    East

[page]

Return to HOME page.
Links to sections- ,Start., Original Manuscript., Introduction., Patterson's Methods., Problem 1st., Problem 2nd., Problem 3rd., Problem 4th., Problem 5th., Lewis's Addition.

Lewis's Addition.

[The final part of the "Astronomical Notebook" is in a different hand from the rest: presumably Lewis instead of Patterson. It consists of a ruled form with headings for "statistical tables" [?] in columns. These tables are empty; only the column headings exist. Then follows "Explanation and notes on the foregoing tables", and then a page of notes on observations. There is also a sketch-map of the headwaters of the Columbia river, with a heading " Sketch given us by [Yellept?], the principal chief of the [Wollahsoollah?] Nation.

Altogether, there are 23 columns, which spread over 5 pages. Although these are written out in the original as column headings, I have rewritten them as rows, just to save space. There's a lot of guesswork gone into interpreting these headings, as they have copied very badly.]

1. Place of observation.
2. Date of obs Astronomical
3. Hour of the day apt. Time at place obs. Astronoml P.M.
4. Hour of the day mean time P.M. astronoml
5. Hour of the day per watch P.M. as[?].
6. Error of the watch on mean time (f or s)
7. Daily gain or loss of watch on mean time (g or L)
8. Apt alt of sun and star, lower, or upper limb ( L.L., or U.L.)
9. True alt of sun or star.
10. Apt. alt. of moon's centre, lower, or upper limb (centre-symbol, L.L., U.L.)
11. True dist. of moon and star.
12. Error of the instrument employed in obsn. back or forward (B, F)
13. Star's name.
14. Apparent alt. of star.
15. App. Distance of Sun's or Moon's nearest limbs. Sun E or W.
16. App dist. of Moon & star. The star E or W.
17. True Distance of Centres.
18. Longitude of place from Greenwich.
19. Latitude of place.
20. Magnetic Azamuth of sun or pole star.
21. True azimuth of sun or pole star [?]
22. Variation of magnetic north E or W[?]
23. Miscellaneous remarks. [page]

Explanations and notes on the foregoing tables.

c. Denotes that the numbers to which it is prefixed is the result of calculations.

ob. Denotes that the numbers to which it is prefixed are the result of observation only and remain to be calculated.

f or s in column 5th. Denotes that the watch is too fast or slow. by the number of minutes or seconds &c to which it is prefixed.

g. or l. Denotes gain or loss of watch.

[a circle with a central dot to represent the Sun], ll, or ul. Denotes sun's center, his lower limb, or upper limb.

B or F Denotes the back, or fore, observation. It may be well here to remark that all the back observations are made with an octant & artificial horizon and that the apparent altitudes as set down in columns no. 8, 10,& 14 where the instrument error is designated B are always to be understood to express the number of degrees, ', &", &c shewn by the graduated limb of the octant at the time of the observation, and is the complement only of 180° or the double angle of the object observed. thus the sum given by the back observation as set down in the table are subt. from 180° and the remdr. is the double alt. of the object observed, and is the same

[page: then an unknown amount is missing, perhaps many pages, then a single page of observations follows.]

[?] ... meridian observation of the preceding observation, not made at the entrance of the Musselshell river.

Point of observation No. 21.

May 20th 1805.

At the point of land formed by the Musselshell river and the Missouri, observed meridian altitude of Sun's L.L. with octant by the back observation 59° 50' .." [This is a revealing statement. Back observations are inherently inaccurate, because they offer no way of checking the octant's index error. This is likely to become disturbed by the upsets of the journey. A sextant would have been much more accurate, for this altitude.]

Latitude deduced from this observation 47.00.24".6

Observed magnetic azimuth of Sun's centre.

Azimuth by                    Time by          Altitude by sextant.
circumferentor              chronometer        of Sun's L.L
                            h   m   s
S 85° W                     6  14  35           50°  0'  0"
S 82° W                     6  24  36           46° 37' 30"
S 80° W                     6  34  42           43° 15 30

May 21st 1805 Point of observation No 22.

On the Lard. [larboard] shore at the commencement of the 5th course of this day observed time and distances of sun's and moon's nearest limbs, the Sun East, with sextant.

Mean of a set of 12 observations. Time 9h 25m 35s AM Distance 91°45' 19"

[at this point the document becomes quite unintelligible, and soon runs out]

[The above is the sum total of the Lewis and Clark document that has reached me].

[End of transcript of Lewis's Astronomical Notebook]

George Huxtable, 2003.