Testing Patterson's Formula

Page 6 of 6
Finding the Longitude Using Robert Patterson's Method1
Computing the True Distance
A = Apparent angular distance center of the Moon to center of the Sun =
Observed mean distance - Index Error + sun's semidiameter +
moon's semidiameter + moon's augmentation =
102°29'15" - 6'15" + 15'48" + 16'01" + 8" =
102°54'58.5"
B = Moon's "observed" altitude =
 32°54'04"
C = Sun's "observed" altitude =
 31°39'00"
D = 1⁄2 (B+C) = 1⁄2 (32°54'04"-31°39'00") = 1⁄2 (64°33'04") =
 32°16'32"
E = C+ Å`D = 31°39'00" - 32°16'32" =
  0°37'32"
F = A ÷ 2 = 102°54'58" ÷ 2 =
 51°27'29.25"
G = tan-1 tangent D x cotangent E x tangent F =
tan-1 tan 32°16'32" x cot 0°37'32" x tan 51°27'29" =
tan-1 0.63157678 x 91.5881929 x 1.255284945 =
tan-1 72.61192752 =
 89°12'39.5"
H = F ± G =F - G = 51°27'29.25" - 89°12'39.5" =
 37°45'10.2"
I = F ± G =F + G = 51°27'29.25"+ 89°12'39.5" =
140°40'08.8"
K = proportional log (180 ÷ moon's horizontal parallax = 180 ÷ 58'47.8" =
 3.060891915
L = proportional log (180 ÷ cosecant B x tangent I x K =
180 ÷ cosecant 32°54'04"x tangent 140°40'08.8" x 3.060891915 =
180 ÷ 1.840973972 x 0.819391192 x 3.060891915 =
180 ÷ 4.61728768 = 38.98392573 =
    38'59.0"
M = refraction of I = ref of 180°-140°40'08.8 = ref of 39°19'51.2" =
     1'06.7"
N = L - M = 38'59.0" - 1'06.7" =
    37'52.3"
O = A - N = 102°54'58.5" - 37'52.3" =
102°17'06.2"
P = refraction + parallax of H =ref + par of 37°45'10.2" =
     1'03.6"
Q = O + P = 102°17'06.2" + 1'03.6" =
102°18'09.8"
R = Tables Requisite (can be ignored)
102°18'10"
S = true distance =
102°18'10"

1. Thomas Jefferson asked Robert Patterson (1743-1824), professor of mathematics at the University of Pennsylvania, to help Lewis develop his skills in making celestial observations. Jefferson wrote to Patterson on May 2, 1803: "He [Lewis] has been for some time qualifying himself for taking observations of longitude & latitude to fix the geographical points of the line he will pass over, but little means are possessed here [in Washington City] of doing that; and it is the particular part in which you could give him valuable instruction, & he will receive it thankfully & employ it usefully." For the use of his pupil Patterson wrote an instruction manual now known as the Astronomy Notebook. Donald Jackson, Letters of the Lewis and Clark Expedition with Related Documents, 1783-1854 (2 vols., Urbana: University of Illinois Press: 1978), 1:21, 28-31.

Funded in part by the Idaho Governor's Lewis and Clark Trail Committee.