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William Lambert to Thomas Jefferson, 8 December 1809

From William Lambert

City of Washington, Decemr 8th 1809.

Sir,

I have the honor to transmit herewith, two astronomical tables; one for computing the Moon’s longitude, latitude, Etc. for every hour, and the other to find the Moon’s hourly velocity at any intermediate time between 0 and 12 hours, by which the motion for a given number of minutes and seconds between the hours, may be accurately obtained. The table which I formerly presented to you, was constructed to find the relative motion in 12 hours, and that being ascertained, the Moon’s longitude, &c. at any interval of time between noon and midnight, or midnight and noon, may be had by simple proportion. But the hourly velocity at 1 hour, is very nearly equal to the difference of the Moon’s longitude, latitude, Etc. computed at 1. h. 30. m., and 0. h. 30. m respectively: the hourly velocity at 4h. 30m. to the difference found at 5h. and 4h. . . hence, half the interval in minutes and Seconds from the preceding hour, or the numbers corresponding thereto, must be taken out of Table II, according to the principles and Series on which it has been formed. This is an essential element to ascertain the true interval of time between the beginning or end of a Solar eclipse, (or the immersion and emersion of a fixed Star), and the ecliptical conjunction, for the purpose of determining the longitude of a place with due precision; and ought, therefore, in all such cases, to be computed with great accuracy. When the Moon’s place at the commencement of a given hour is found by Table I, the motion for the minutes and seconds between that and the succeeding hour, may be had by Table II, as will appear from the following examples:—

Required the Moon’s longitude, December 21st 1809, at 4 hours, 48 minutes, P.M. at Greenwich?

1809. s. ° ′ ″ 
Decem: 20th   Noon,  2.12.53.58. A. ° ′ ″ 
+ 5.56.21. a1. ′ ″ 
〃 Midn: 2.18.50.19. B. – 0.11. a2 ′ ″
+ 5.56.10. b1. + 0.9 a3. ″  
21. Noon, 2.24.46.29. C. – 0. 2. b2 } + 0. a4.
+ 5.56. 8. c1. + 0.9 b3.
〃 Midn: 3. 0.42.37. D. + 0. 7. c2
+ 5.56.15. d1.
22. Noon, 3. 6.38.52. E.

In this example, the third differences being equal to each other, the fourth difference vanishes or becomes nothing.

The differences c1, c2, b3, and a4, are always to be used, according to the principles on which both Tables have been constructed.

s. ° ′ ″ dec. 
Moon’s longitude, Dec.21st, at noon, 2.24.46.29.000 
+ c1. × ⁴⁄₁₂. +  1.58.42.667.
– ,1111111. (Table I) × c2. –  0. 0. 0.778.
+ ,0617284. × b3. +  0. 0. 0.555.
Moon’s longitude at 4 hours, (Table I.) 2.26.45.11.444.
 ′ ″ dec. 
¹⁄₁₂c1. 29.40.6667.
– ,0111111. × c2. – 0.0778
+ ,0028240  × b3. +    0.0254.
 ☽’s hourly velocity, 29.40.6143.
 proportion for 48 minutes,  23.44. 491.

which added to 2. s. 26.° 45.′ 11.″ 444 dec., gives 2. s. 27.° 8.′ 55.″ 935 dec., the Moon’s longitude at 4 hours 48 minutes.

☞  The numbers corresponding to 4h. 24m. (half the interval from the preceding hour) have been taken from Table II.
The truth of the foregoing result may be tested by the following process—
s. ° ′ ″ dec. 
 Moon’s longitude at noon, 2.24.46.29.000 
c1 × ⅖ + 2.22.27.200 
– ³⁄₂₅ × c2 – 0. 0. 0.840.
+ ⁸⁄₁₂₅ × b2 + 0. 0. 0.576 
 Moon’s longitude at 4 hours, 48 min:  2.27. 8.55.936.

The last result is strictly correct, supposing the positions at noon and midnight to be true, and differs only ¹⁄₁₀₀₀ part of a second from the former.

By Mr Garnett’s second Theorem, (Naut. Alm. Am. impressn)

° ′ ″ dec. 
 b1. 5.56.10.000 
 +½b2  – 0. 1.000.
 –a3 + b3.⁄12 – 0. 1.500 
(M) 5.56. 7.500 
 M. × x⁄y, 2.22.27.000.
+½b2. × xx⁄yy, – 0. 0.160.
+a3+b3⁄12. × xxx⁄yyy, + 0. 0.096.
 Moon’s motion in 4h. 48 min. 2.22.26.936.
 〃 Longitude at noon, + 2.24.46.29. – 
 Moon’s longitude, as above, 2.27.  8.55.936.

I have been favored by Mr Garnett, with another method, not published in the American edition of the Nautical Almanac, as follows:—

  ° ′ ″ dec 
b1 + c1./2. = N, 5.56.  9.000 
N, × x⁄y, 2.22.27.600.
+ xx⁄2yy ×  – 0. 0.160.
– x + xxx⁄36yyy. × a3 + b3⁄2, – 0. 0.504 
2.24.46.29.000 
 Moon’s longitude, as before, 2.27. 8.55.936.

We shall now prove the accuracy of the numbers in Table II, by Mr Garnett’s method, applied to the foregoing example.

° ′ ″ dec. 
 (M) 5.56. 7.5000 
+ x⁄y = ¹¹⁄₃₀ × b2 – 0. 0.7333.
+ xx⁄yy, ¹²¹⁄₉₀₀ × a3 + b3⁄4 + 0. 0.6050.
Moon’s velocity in 12 hours.  12)5.56.  7.3717.
 Moon’s hourly velocity at 4h. 24 min:      29.40.6143.

In the same Example, the Moon’s hourly velocity at 9. h. 20. m. is required?

′ ″ dec. 
¹⁄₁₂c1, 29.40.6667 
+ ,0231482. × c2 + 0.1620.
– ,0118313. × b3 – 0.1065.
 Moon’s hourly velocity at 9. h. 20 m.  29.40.7222.

By Mr Garnett’s method.

′ ″ dec. 
(M.) = 5.° 56.′ 7½″ × ¹⁄₁₂ 29.40.6250.
+ x⁄12y, = ⁷⁄₁₀₈ × b2 – 0.1297.
+ xx⁄12yy, = ⁴⁹⁄₉₇₂ × a3 + b3⁄4 + 0.2268
Moon’s hourly velocity at 9. h. 20. m.  29.40.7221.

differing only ¹⁄₁₀₀₀₀ part of a second.

 

Let the Moon’s latitude, December 27th 1809, be required, at 7 hours, 12 minutes, by the meridian of Greenwich?

1809. th ° ′ ″ 
Decem. 26.  Noon  4.10.41. A. ′ ″ 
– 19.18. a1 ′ ″ 
〃 Midn: 3.51.23. B. – 2.51. a2 ″ 
– 22. 9. b1 + 11 a3. ″
27. Noon, 3.29.14. C. – 2.40. b2. – 1 a4.
> – 24.49. c1 + 10. b3.
〃 Midn: 3. 4.25. D. – 2.30. c2
– 27.19. d1
28. Noon, 2.37. 6. E.
° ′ ″ dec. 
 Moon’s latitude at noon, 3.29.14.0000.  South.
– c1. = 24.′ 49.″ × ⁷⁄₁₂ – 14.28.5833.
– ,1215278  × c2 = 150″ + 0.18.2291.
+ ,0573881. × b3 + 0. 0.5739.
+ ,0227161. × a4 – 0. 0.0227.
 Moon’s latitude at 7 hours, by Table I.  3.15. 4.1970.  South

By Table II.

′ ″ dec. 
¹⁄₁₂.c1 – 2. 4.0833 
x. + ,0076388 (at 7. h. 6. m.) × c2 – 1.1458 
y, – ,0069415. × b3,  – 0.0694.
z, – ,0015807 × a4 + 0.0016 
Moon’s hourly velocity  – 2. 5.2969.
proportion for 12 minutes,  – 25.0594.

which subtracted from 3.° 15.′ 4.″ 1970 dec, leaves 3.° 14.′ 39.″ 1376. dec. South, the Moon’s latitude at 7h. 12m. the time required.

The given time 7. h. 12. m is = ⅗ of 12 hours;—hence

° ′ ″ dec 
 Moon’s latitude at noon, 3.29.14.0000.  S.
c1. × ⅗ – 14.53.4000.
– ³⁄₂₅ × c2 + 0.18.0000.
+ ⁷⁄₁₂₅ × b3 + 0. 0.5600.
+ ¹⁴⁄₆₂₅ × a4 – 0. 0.0224.
 Moon’s latitude at 7. h. 12 m.  3.14.39.1376.  South.

By Mr Garnett’s Second Theorem.

′ ″ dec. 
  b1 – 22. 9.0000 
+ ½b2 –  1.20.0000.
– a3 + b3⁄12 –  1.7500.
(M)  – 23.30.7500.
 M × x⁄y, – 14. 6.4500.
+ xx⁄yy × + ½b2 – ¹⁄₂₄ a4 –  0.28.7850.
+ xxx⁄yyy, × a3 + b3⁄12 +  0. 0.3780.
+ xxxx⁄yyyy × ¹⁄₂₄ a4 –  0. 0.0054 
Motion in latitude in 7. h. 12 m  – 14.34.8624 
 Moon’s latitude at noon  3.29.14.0000.
do  at 7h. 12m. as above,  3.14.39.1376.  South.

The utility of the tables may be seen by the foregoing examples;—and as they are of a permanent nature, the more accurate the Moon’s positions at noon and midnight are given in the Nautical Almanac, the greater dependence may be had in the precision with which the Moon’s true place can be obtained at any intermediate time.

I have the honor to be, with perfect respect and esteem, Sir, Your most obedt servant,

William Lambert.1

  Table I.
for computing the Moon’s longitude, latitude, Etc. for every hour between 0 and 12 hours.

 h. m.  x. y. z.  
0. 0.   –,0000000.   +,0000000.   +,0000000. 
1. 0.  –,0381944.  +,0244020.  +,0066089. 
2. 0.  –,0694444.  +,0424383.  +,0123778. 
3. 0.  –,0937500.  +,0546875.  +,0170898. 
4. 0.  –,1111111.  +,0617284.  +,0205761. 
5. 0.  –,1215278.  +,0641397.  +,0227161. 
6. 0.  –,1250000.  +,0625000.  +,0234375. 
7. 0.  –,1215278.  +,0573881.  +,0227161. 
8. 0.  –,1111111.  +,0493827.  +,0205761. 
9. 0.  –,0937500.  +,0390625.  +,0170898. 
10. 0.  –,0694444.  +,0270062  +,0123778. 
11. 0.  –,0381944.  +,0137924.  +,0066089. 
12. 0.  –,0000000.  +,0000000.  +,0000000. 

Table II.

for computing the Moon’s hourly velocity in longitude, latitude, right ascension and declination, at every ten minutes of intermediate time between 0 and 12 hours, extended to differences of the fourth order, and to seven places of figures in the decimal fractions, as auxiliary to Table I.

☞ Five positions of the Moon at noon and midnight, three preceding, and two following the time required, are always to be taken.
  Equation of
2d differences. 		Third diff. Fourth diff.  
 h. m.  x. y. z.
0. 0.   –,0416667.   +,0277778.   +,0069444. 
10.  –,0405093.  +,0266284.  +,0068439. 
20.  –,0393519.  +,0254951.  +,0067357. 
30.  –,0381944.  +,0243779.  +,0066199. 
40.  –,0370370.  +,0232768.  +,0064967. 
50.  –,0358796.  +,0221917.  +,0063664. 
1. 0.  –,0347222.  +,0211227.  +,0062291.   
10.  –,0335648.  +,0200698.  +,0060851. 
20.  –,0324074.  +,0190329.  +,0059347. 
30.  –,0312500.  +,0180122.  +,0057780. 
40.  –,0300926.  +,0170075.  +,0056153. 
50.  –,0289352.  +,0160189  +,0054468. 
2. 0.  –,0277778.  +,0150463.  +,0052726.   
10.  –,0266204.  +,0140898.  +,0050931. 
20.  –,0254630.  +,0131494.  +,0049085. 
30.  –,0243056.  +,0122251.  +,0047190. 
40.  –,0231482  +,0113169.  +,0045248. 
50.  –,0219907.  +,0104247.  +,0043261. 
3. 0.  –,0208333.  +,0095486.  +,0041232.   
10.  –,0196759.  +,0086886.  +,0039163. 
20.  –,0185185.  +,0078446.  +,0037056. 
30.  –,0173611.  +,0070167.  +,0034913. 
40.  –,0162037.  +,0062050.  +,0032737. 
50.  –,0150463.  +,0054093.  +,0030529. 
4. 0.  –,0138889.  +,0046296.  +,0028292.   
10.  –,0127315.  +,0038660.  +,0026028. 
20.  –,0115741.  +,0031185.  +,0023740. 
30.  –,0104167.  +,0023871.  +,0021430. 
40.  –,0092592.  +,0016718.  +,0019099. 
50.  –,0081018  +,0009725.  +,0016751. 
5. 0.  –,0069444.  +,0002893  } +,0014387.   
10.  –,0057870.  –,0003778.  +,0012010. 
20.  –,0046296.  –,0010288.  +,0009621.   
30.  –,0034722.  –,0016638  +,0007223. 
 
40.  –,0023148.  –,0022827.  +,0004819. 
50.  –,0011574.  –,0028855.  +,0002411 
6. 0.  ∓,0000000.  –,0034722.  ±,0000000.   
10.  +,0011574.  –,0040429.  –,0002411. 
20.  +,0023148.  –,0045975.  –,0004819. 
30.  +,0034722.  –,0051360.  –,0007223. 
40.  +,0046296.  –,0056584  –,0009621. 
50.  +,0057870.  –,0061648.  –,0012010. 
7. 0.  +,0069444.  –,0066551.  –,0014387.   
10.  +,0081018.  –,0071293.  –,0016751. 
20.  +,0092592  –,0075874.  –,0019099. 
30.  +,0104167.  –,0080295.  –,0021430. 
40  +,0115741.  –,0084555.  –,0023740. 
50.  +,0127315.  –,0088654.  –,0026028. 
8. 0.  +,0138889.  –,0092592.  –,0028292.   
10  +,0150463.  –,0096370.  –,0030529. 
20  +,0162037.  –,0099987.  –,0032737. 
30.  +,0173611.  –,0103443.  –,0034913. 
40  +,0185185.  –,0106738.  –,0037056. 
50  +,0196759.  –,0109873.  –,0039163. 
9. 0.  +,0208333.  –,0112847.  –,0041232.   
10.  +,0219907.  –,0115660.  –,0043261. 
20.  +,0231482.  –,0118313.  –,0045248. 
30.  +,0243056.  –,0120804.  –,0047190. 
40.  +,0254630.  –,0123135.  –,0049085. 
50.  +,0266204.  –,0125305.  –,0050931. 
10. 0.  +,0277778.  –,0127315.  –,0052726.   
10.  +,0289352.  –,0129163.  –,0054468. 
20.  +,0300926.  –,0130851.  –,0056153. 
30.  +,0312500.  –,0132378.  –,0057780. 
40.  +,0324074.  –,0133745.  –,0059347. 
50.  +,0335648.  –,0134951.  –,0060851. 
11. 0.  +,0347222.  –,0135995.  –,0062291.   
10.  +,0358796.  –,0136879.  –,0063664. 
20.  +,0370370.  –,0137603.  –,0064967. 
30.  +,0381944.  –,0138166.  –,0066199. 
40.  +,0393519.  –,0138567.  –,0067357. 
50.  +,0405093.  –,0138808.  –,0068439. 
12. 0.  +,0416667.  –,0138889.  –,0069444. 

To interpolate the numbers or decimal fractions in Table II, for every minute of intermediate time, if required.

The first differences of x, being uniform, or very nearly so, throughout the table, simple proportion only is necessary.—For y and z;—

Take two numbers next before, and two immediately following the time given; find their first and second differences, to which prefix their proper signs.—then multiply a mean of the second differences—.

min.
For  1 by  –,045. } The product being added to, or subtracted from
the proportional part of the first difference, according
as the signs direct, will give the correction, to be
applied to the number or decimal fraction next
preceding the intermediate time.
2 –,08.
3, –,105.
4, –,12.
5 –,125.
6 –,12.
7 –,105.
8, –,08.
9 –,045.

Examples.

Required the numbers x, y, and z, at 4. h. 24. m.

The proportional part of 11574, the first difference ⁴⁄₁₀, is = 4630, nearly, which subtracted from ,0115741, leaves ,0111111, the value of x, at 4. h. 24m.

h. m.
4.10   +,0038660    +,0026028.
–7475. –2288.
20 ,0031185  +161. ,0023740. –22  }
–7314. –2310. Mean 21.5.
30 ,0023871. +161. ,0021430. –21.
–7153  –2331.
40 ,0016718. ,0019099.
h. m.   h. m.
4.20 +,0031185.  4.20 +,0023740.
–7314 × ⁴⁄₁₀, – 2925.6 –2310 × ⁴⁄₁₀ – 924.
+161. × –,12,  – 19.3 –21.5. × –,12.  + 2.
(y.) +,0028240.  (z) +,0022818.

Required x, y, and z, at 9h. 18m.

x, is found to be +,0229167.

h. m. y. z.
9. 0   –,0112847. –,0041232.
–2813. –2029.
10 –,0115660. +160 } –,0043261. +42 }
–2653. +161  –1987. +43.5
20 –,0118313. +162. –,0045248. +45
–2491  –1942 
30 –,0120804. –,0047190.
h. m.   h. m.
9.10 –,0115660 9.10 –,0043261.
–2653 × ⁸⁄₁₀, –, 2122.4 –1987 × ⁸⁄₁₀, – 1589.6
+101 × –,08  – [12.6] +43.5 × –,08  – 3.4
(y.)  –,0117795 (z.)  –,0044854 –
Required x, y, and z, at 7. h. 6. m.
By a similar process,  x, will be  +,0076388.
y, –,0069415.
z, –,0015807.

The foregoing examples and operations are sufficient to explain the method by which any number or decimal fraction not contained in the Table, may be found; from which the equations arising from the second, third and fourth differences, respectively, may be obtained with great precision, at any intermediate time between 0 and 12 hours.

William Lambert.

City of Washington, }
December 8th 1809.

RC (PPAmP: Robert M. Patterson Papers); on four folio sheet; torn at crease; ellipsi in original; repeated column headings and hour numbers at head of pp. 6–7 omitted; adjacent to first signature: “Thomas Jefferson, late President of the U.S. and President of the American Philosophical Society”; endorsed by TJ as received 14 Dec. 1809 and so recorded in SJL. Enclosed in a brief letter of transmittal from TJ to Robert Patterson, Monticello, 19 Dec. 1809 (RC in PPAmP: APS description begins American Philosophical Society description ends Archives, Manuscript Communications to the APS description begins American Philosophical Society description ends , with notation on verso: “Jefferson Thos Dec. 19. 1809 read Janry 19. 1810 Accompanyg Tables for computg Moon’s Longitude”; not recorded in SJL).

A letter from Lambert to TJ of 27 Jan. 1810, not found, is recorded in SJL as received from Washington on 31 Jan. 1810.

1. Verso of sheet two ends here.

Index Entries

  • American Philosophical Society; and W. Lambert search
  • astronomy; and lunar calculations search
  • Garnett, John; Nautical Almanac and Astronomical Ephemeris search
  • Greenwich Observatory, England; and prime meridian search
  • Lambert, William; letters from search
  • Lambert, William; letters from accounted for search
  • Lambert, William; lunar calculations search
  • moon; calculations of motion, position, and distance of search
  • Patterson, Robert; letters to accounted for search
  • sun; and astronomical calculations search
  • The Nautical Almanac and Astronomical Ephemeris (J. Garnett); and W. Lambert search

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Jefferson Papers
Title
William Lambert to Thomas Jefferson, 8 December 1809
Author
Lambert, William
Recipient
Jefferson, Thomas
Date
8 December 1809

Reference

Cite as
“William Lambert to Thomas Jefferson, 8 December 1809,” Founders Online, National Archives, https://founders.archives.gov/documents/Jefferson/03-02-02-0042. [Original source: The Papers of Thomas Jefferson, Retirement Series, vol. 2, 16 November 1809 to 11 August 1810, ed. J. Jefferson Looney. Princeton: Princeton University Press, 2005, pp. 60–68.]

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