Box 1, Folder 9: Notebook

Amplitudes

their difference in the variation; but if one be north and the other be South, their sum is the variation. -

In the preceding example, the Mag. Amp. is 90' - 71' = 19' N from the E. True Amp. as found E 37' 50' N difference 18' 50' true variation, westerly, because the true Amp, is to the left of the magnetic amp. -

The true Amp. may also be found by this. Rule. To the Sec. of the latitude, rejecting the index, Add the sine of the Suns declination; Their Sum will be the Sine of the true amplitude. -

Example. Sept 24th 1832, in Lat 26' 32 N and Low 78' W the Suns center was observed to set, W 6' 15' 5. abot 6 P.M - Required the variation of the Compass.

Suns dec. S . . . . . 0'. 29' Cor. for low. 78' W . . . . . + .05 Cor. for time 6h P.M. . . . . . + .06 Reduced dec. . . . . . 0. 40' Sine 8.06578 Latitude . . . . . 26: 32' Sec. .04833 Sine of true Amp. . . . . . 0. 45' S 8.11411 Mg. Amp. . . . . . 6: 15' S. Variation 5' 30' 6. because the true amplitude is to the right of the magnetic.-

The same example by the 1st Rule.- Suns dec. S. . . . . . 0: 29' Cor. for Low. . . . . . + .05 Cor. for time . . . . . + .06 Reduced dec. . . . . = 0. 40

As Co. S. Lat. . . . . . 26' 32 . . . 9.95167 : S. sums dec. . . . . . 0' 40' S. 8.07650 :: Ra. . . . . . 90'00 10.00000 : S. of true Amp . . . . . 0' 45' S. 8.12483 Mg Amp . . . . . 6' 15' S. Variation 5' 30' E. -

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No 3 The semidiameter of the Sun

January March May July September November 16:19 16.10 15.54 15.47 15.55 16.11 16:19 16.09 15.53 15.47 15.56 16.13 Suns 16:19 16.07 15.52 15.47 15.58 16.14 semidi 16:18 16.06 15.51 15.48 15.59 16.15 16:17 16.04 15.50 15.48 16.00 16.16

February April June August October December 16.16 16.02 15.49 15.49 16.03 16.17 16.15 16.01 15.48 15.50 16.04 16.18 Suns 16.14 15.59 15.47 15.51 16.06 16.18 semidi 16.13 15.57 15.47 15.52 16.08 16.19 16.12 15.56 15.47 15.53 16.09 16.19

No 4" The suns No 5 Parallax in altitude Dip of the Horizon Alti- Paral Height Dip Height Dip tude lax feet ' " feet ' " o " 1 0.58 19 4.11 0 9 2 1.21 20 4.17 10 9 3 1.40 21 4.23 20 8 4 1.56 22 4.30 30 8 5 2.09 23 4.36 40 7 6 2.21 24 4.42 50 6 7 2.33 26 4.52 55 5 8 2.44 28 5.05 60 4 9 2.53 30 5.15 65 4 10 3.02 35 5.39 70 3 11 3.10 40 6.04 75 2 12 3.19 45 6.27 80 2 13 3.27 50 6.46 85 1 14 3.36 60 7.25 90 0 15 3.42 70 8.01 16 3.50 80 8.34 17 3.57 90 9.00 18 4.04 100 9.35

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No 6 Dip of the Hor. at different differences from the observer Height of the [cyl?] in feet. Miles 5 10 15 20 25 30 40 50 60 70 80 90 100 1/12 11 23 34 45 57 1:08 1/2 6 12 17 23 28 0.34 3/4 4 8 12 15 19 .23 1 3 6 9 12 15 .17 1 1/4 3 5 7 10 12 .14 1 1/2 2 4 6 8 10 .12 2 2 4 5 7 8 .09 2 1/2 " 3 4 6 7 .08 3 " " 3 1/2 5 6 .7 4 " " " 4 6 .7 5 " " " " 5 .6 6 " " " " " .6 7 " " " " " .5 8 " " " " " " 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

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No 7 Mean Refraction of Celestial Bodies. Appt Appt Appt Appt Appt Alt Refr Alt Refr Alt Refr Alt Refr Alt Refr 0.00 33.00 2.30 16.24 5.00 9.54 7.30 6.53 10.00 5.15 .05 32.10 2.35 16.04 5.05 9.46 7.35 6.49 10.10 5.10 .10 31.22 2.40 15.45 5.10 9.38 7.40 6.45 10.20 5.05 .20 30.35 2.45 15.27 5.15 9.30 7.45 6.41 10.30 5.00 .15 29.50 2.50 15.09 5.20 9.23 7.50 6.37 10.40 4.56 .20 29.06 2.55 14.52 5.25 9.15 7.55 6.33 10.50 4.51 .30 28.23 3.00 14.36 5.30 9.08 8.00 6.29 11.00 4.47 .35 27.41 3.05 14.20 5.35 9.01 8.05 6.25 11.10 4.43 .40 27.00 3.10 14.04 5.40 8.54 8.10 6.22 11.20 4.39 .45 26.20 3.15 13.49 5.45 8.47 8.15 6.18 11.30 4.34 .50 25.42 3.20 13.34 5.50 8.41 8.20 6.15 11.40 4.31 .55 25.05 3.25 13.20 5.55 8.34 8.25 6.11 11.50 4.27 1.00 24.29 3.30 13.06 6.00 8.28 8.30 6.08 12.00 4.23 1.05 23.54 3.35 12.53 6.05 8.21 8.35 6.05 12.10 4.20 1.10 23.20 3.40 12.40 6.10 8.15 8.40 6.01 12.20 4.16 1.15 22.47 3.45 12.27 6.15 8.09 8.45 5.58 12.30 4.13 1.20 22.15 3.50 12.15 6.20 8.03 8.50 5.55 12.40 4.09 1.25 21.44 3.55 12.03 6.25 7.57 8.55 5.52 12.50 4.06 1.30 21.15 4.00 11.51 6.30 7.51 9.00 5.48 13.00 4.03 1.35 20.46 4.05 11.40 6.35 7.45 9.05 5.45 13.10 4.00 1.40 20.18 4.10 11.29 6.40 7.40 9.10 5.42 13.20 3.57 1.45 19.51 4.15 11.18 6.45 7.35 9.15 5.39 13.30 3.54 1.50 19.25 4.20 11.08 6.50 7.30 9.20 5.36 13.40 3.51 1.55 19.00 4.25 10.58 6.55 7.25 9.25 5.34 13.50 3.48 2.00 18.35 4.30 10.48 7.00 7.20 9.30 5.31 14.00 3.45 2.05 18.11 4.35 10.39 7.05 7.15 9.35 5.28 14.10 3.43 2.10 17.48 4.40 10.29 7.10 7.11 9.40 5.25 14.20 3.40 2.15 17.26 4.45 10.20 7.15 7.06 9.45 5.23 14.30 3.38 2.20 17.04 4.50 10.11 7.20 7.02 9.50 5.20 14.40 3.35 2.25 16.44 4.55 10.02 7.25 6.57 9.55 5.18 14.50 3.33 2.30 16.24 5.00 9.54 7.30 6.53 10.00 5.15 15.00 3.30

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No 7 Mean Refraction of Celestial Bodies. Appt Appt Appt Appt Appt Alt Refr Alt Refr Alt Refr Alt Refr Alt 15.00 3.30 20.00 2.35 25.00 2.02 34.00 1.24 58.00 15.10 3.28 20.10 2.34 25.10 2.01 34.30 1.23 59. 15.20 3.26 20.20 2.32 25.20 2.00 35.00 1.21 60. 15.30 3.24 20.30 2.31 25.30 1.59 35.30 1.20 61. 15.40 3.21 20.40 2.29 25.44 1.58 36.00 1.18 62. 15.50 3.19 20.50 2.28 25.50 1.57 36.30 1.17 63 . 16.00 3.17 21.00 2.27 26.00 1.56 37.00 1.16 64. 16.10 3.15 21.20 2.26 26.10 1.55 37.30 1.14 65. 16.20 3.12 21.20 2.25 26.20 1.55 38.00 1.13 66. 16.30 3.10 21.30 2.24 26.30 1.54 38.30 1.11 67. 16.40 3.08 21.40 2.23 26.40 1.53 39.00 1.10 68. 16.50 3.06 21.50 2.21 26.50 1.52 39.30 1.09 69. 17.00 3.04 22.00 2.20 27.00 1.51 40.00 1.08 70. 17.10 3.03 22.10 2.19 27.15 1.50 41.00 1.05 71. 17.20 3.01 22.20 2.18 27.30 1.49 42.00 1.03 72. 17.30 2.59 22.30 2.17 27.45 1.48 43.00 1.01 73. 17.40 2.59 22.40 2.16 28.00 1.47 44.00 0.59 74. 17.50 2.55 22.50 2.15 25.15 1.46 45.00 0.57 75. 18.00 2.54 23.00 2.14 28.30 1.45 46.00 0.55 76. 18.10 2.52 23.10 2.13 28.45 1.44 47.00 0.53 77. 18.20 2.51 23.20 2.12 29.00 1.42 48.00 0.51 78. 18.30 2.49 23.30 2.11 29.30 1.40 49.00 0.49 79. 18.40 2.47 23.40 2.10 30.00 1.38 50.00 0.48 80. 18.50 2.46 23.50 2.09 30.30 1.37 51.00 0.46 81. 19.00 2.44 24.00 2.08 31.00 1.35 52.00 0.44 82. 19.10 2.43 24.10 2.07 31.30 1.33 53.00 0.43 83. 19.20 2.41 24.20 2.00 32.00 1.31 54.00 0.41 84. 19.30 2.40 24.30 2.05 32.30 1.30 55.00 0.40 85. 19.40 2.38 24.40 2.04 33.00 1.28 56.00 0.38 86. 19.50 2.37 24.50 2.03 33.30 1.26 57.00 0.37 88. 20.00 2.35 25.00 2.02 34.00 1.24 58.00 0.35 90.00

52 Asimuth

The altitude of the sun, observed by a quadrant or sextant, requires four corrections in order to obtain the true altitude; from which to ascertain the Latitude or a true asimuth. These are the semidiameter, dip, refraction and parallax; and are to be attained by means of tables No 3. 4. 5. 6 & 7, and the accompanying explanations

Having observed with a quadrant or sextant the altitude of the suns lower limb above the visable horizon - or the line of seperation of the land from the water, when that horizon is obstructed by land - add thereto the semidiameter taken from table No 3. at the given day of the month, or the one nearest to it, and from this sum subtract the dip taken from table No 5 or 5 corresponding to the height of the observers eye above the surface of the water; and this result will be the apparant altitude of the Suns center. Then take the refraction from table No 7. corresponding to this altitude, and the parallax from table No 4, of the same altitude; the difference of these quantities, called the correction, being subtracted from the apparant altitude, the remainder will be the suns true altitude.-

When the observation has been made by bringing the suns image in the quadrant or sextant, to a just coincidence with its image in an artifical horizon, half the angle shown on the instrument is the Suns apparant alt. which must be corrected by the corresponding refraction and parallax only, in order to obtain the true altitude. -

To find the variation of the Compass by an Asimuth. - Reduce the Suns declination to the time and place of observation, and compute the true altitude of the Suns center.

Subtract the Suns declination from 90' when the latitude and declination are of the same name, or add it to 90' when they are of contrary names; and the sum or remainder will be the Suns Polar distance.

Thus having the latitude of the place, the Suns declination, and its altitude given, the true asimuth is ob-

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Azimuth

obtained by the following astronomical proportions.- As the Tan. of half the complement of the Lat. Is to the Tan of half the suns polar dist. and Comp. of the Alt. So is the Tan, of the dif. of the Suns polar dist. and Comp. of the alt, To the Tan. of a fourth arch. -

Add this fourth arch and half the complement of the latitude together, their sum will give a fifth arch; from which, if the complement of the latitude be taken, will give a sixth arch.- Then

As Radius Is to the Tan of the Alt, So is the Tan. of the 6th Arch To the Co. Sine of the Suns true Asimuth: Which is to be counted from the South in north latitude, and from the north in South latitude: towards the east in the morning, and towards the west in the afternoon.-

If both asimuths are east or west, their difference is the variation; but if one be east, and the other west, their sum is the variation. And to know if the variation be east or west; just as with the amplitudes, let the observers face be turned to the Sun; then if the true asimuth be to the right hand of the magnetic, the var. is eastly, but if to the left, westerly.-

Example. In Lat. 53' 20' N. the suns dec. being 19' 03 N. I find by observation, the suns alt. to be 37' 30 and its mag. azimuth to be S 51' E. Required the variation.

90' - 53. 20' = 36. 40', The comp. of the Lat. 18. 20', 1/2 the comp of Lat. 90' - 37. 30' = 52. 30', The comp of the Alt. 90' - 19. 03' = 70. 57', The suns Po. list. 123. 27', Sum of suns Po. dist. and Comp of Alt. 61. 43', 1/2 sum " " 18. 27' , dif of sums Po. dist, and Comp. of Alt 9. 13' , 1/2 difference " " Then as

[left page] 54 Azimuth

as Tan 1/2 Comp Lat. . . . . . 18. 20' -- 9.52031 : Tan 1/2 Sum of S. Po, dis & Comp. Alt. . . . . . 61. 43' - 10.26916 :: Tan 1/2 their dif. . . . . . 9. 13' -- 9.21022 = 19.47938 : Tan of a 4th arc. . . . . . 42. 18' 9.95907 1/2 Comp Lat. . . . . . + 18. 20' The 5th arc. . . . . . 60. 38 Comp Lat . . . . . -- 36.40' The 6th arch . . . . . 23. 58 Then as Rad . . . . . 90 - 10.00000 : Tan. alt. . . . . . 37. 30 - 9.88498 :: Tan 6th arch. . . . . . 23. 58 - 9.64790 : To. S. Sums true azimuth . . . . . S 70. 04 E. 9.53288 Mag. az. . . . . . S 51. 00' E Variation 19' 04' W. -

The true azimuth being to the left of the magnetic one, the variation is westerly. -

To find true azimuth by another Rule. Having the latitude of the place, the Suns dec and its altitude given, as in the preciding can: the true azimuth is found as follows.

Add together the Suns polar distance, the lat. of the place, and the altitude of the Sun; take the difference between half their Sum, and the polar distance, and note the remainder. Then add together The secant of the altitude ) The secant of the latitude ) rejecting the secant indices The Co. Sine of the half sum And Co. Sine of the remainder.

Half the sum of these four logarithms will be the sine of an arch, which doubled, will be the Suns true azimuth.-

Example Nov 2 1832, in Lat 25' 32 N and low 75' W

[right page] Azimuth 55 the altitude of the Suns lower limb was observed to be 15' 36' about 4 h 10m P.M. his magnetic azimuth at that time being S 58' 32' W. and the height of the eye 18pt.

Required the variation of the Compass Suns dec. Tab 1 . . . . . 14. 48' S Alt of Suns lower limb 15. 36' Cor. for Low. 75' W . . . . . + .04 Semidi 16' Cor. for time 4h 10m P.M. . . . . . + .03 dip - -- 4 = + .12' Reduced dec. - 14. 55' S 15. 48 + 90' refraction 3. 21' Polar dist. . . . . . 104. 55' parallex.- 08 = + -- 0. 3' 13" True Alt ...................... 15.45' 15.45 Latitude ...................... 25.32' Sum ..............= 146.12 1/2 sum ............ 73.06' Po. dist .......... -- 104.55' Remainder ......= 31.49' then add together the Lec. alt. ............................................ 15.45' ........ 0.01662 Lec. lat.............................................. 25.32' ........ 0.04463 Co. 8. 1/2 sum ................................... 73.00' ........ 9.46345 Co. S. remainder ............................... 31.49' ........ 9.92929 sum of logs. 19.45399 Sim of arc. 32.14' .. 9.72699 1/2 sum log x 2 True as. .... S. 64.28' W. Mag. as. .... S. 58.32' W. Varation... 5.56' E. because the true azimuth is to the right of the magnetic one.-

The same - Example - by the 1st. rule. Let the equation of the sums declination and altitude be made as before: We have then given The Lat .................... 25. 32' The Suns Alt ............. 15. 45' The Suns dec ............ 14. 55' S The Mag. az. ........... S 58. 32' W Then,